MOPAC Manual (Seventh Edition)

Dr James J. P. Stewart

PUBLIC DOMAIN COPY (NOT SUITABLE FOR

PRODUCTION WORK) January 1993

This document is intended for use by developers of semiempirical programs and software. It is not intended for use as a guide to MOPAC.

All the new functionalities which have been donated to the MOPAC project during the period 1989-1993 are included in the program. Only minimal checking has been done to ensure conformance with the donors’ wishes. As a result, this program should not be used to judge the quality of programming of the donors. This version of MOPAC-7 is not supported, and no attempt has been made to ensure reliable performance.

This program and documentation have been placed entirely in the public domain, and can be used by anyone for any purpose. To help developers, the donated code is packaged into files, each file representing one donation.

In addition, some notes have been added to the Manual. These may be useful in understanding the donations.

If you want to use MOPAC-7 for production work, you should get the copyrighted copy from the Quantum Chemistry Program Exchange. That copy has been carefully written, and allows the donors’ contributions to be used in a full, production-quality program.

Contents

1 Description of MOPAC 1 1.1 Summary of MOPAC capabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Copyright status of MOPAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Porting MOPAC to other machines . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 Relationship of AMPAC and MOPAC . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.5 Programs recommended for use with MOPAC . . . . . . . . . . . . . . . . . . . . . 3 1.6 The data-file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.6.1 Example of data for ethylene . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.6.2 Example of data for polytetrahydrofuran . . . . . . . . . . . . . . . . . . . 6

2 Keywords 9 2.1 Specification of keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Full list of keywords used in MOPAC . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Definitions of keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Keywords that go together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 Geometry specification 41

3.1 Internal coordinate definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.1 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Gaussian Z-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3 Cartesian coordinate definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.4 Conversion between various formats . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.5 Definition of elements and isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.6 Examples of coordinate definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4 Examples 49 4.1 MNRSD1 test data file for formaldehyde . . . . . . . . . . . . . . . . . . . . . . . . 49 4.2 MOPAC output for test-data file MNRSD1 . . . . . . . . . . . . . . . . . . . . . . 50

5 Testdata 55 5.1 Data file for a force calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.2 Results file for the force calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.3 Example of reaction path with symmetry . . . . . . . . . . . . . . . . . . . . . . . 63

6 Background 65 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.2 AIDER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.3 Correction to the peptide linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6.4 Level of precision within MOPAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.5 Convergence tests in subroutine ITER . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.6 Convergence in SCF calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.7 Causes of failure to achieve an SCF . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 6.8 Torsion or dihedral angle coherency . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.9 Vibrational analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.10 A note on thermochemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.10.1 Basic Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.10.2 Thermochemistry from ab initio MO methods . . . . . . . . . . . . . . . . . 72

6.11 Reaction coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.12 Sparkles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.13 Mechanism of the frame in FORCE calculation . . . . . . . . . . . . . . . . . . . . 86 6.14 Configuration interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.15 Reduced masses in a force calculation . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.16 Use of SADDLE calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.17 How to escape from a hilltop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.17.1 EigenFollowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.17.2 Franck-Condon considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.18 Outer Valence Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.18.1 Example of OVGF calculation . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.19 COSMO (Conductor-like Screening Model) . . . . . . . . . . . . . . . . . . . . . . 99 6.20 Solid state capability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7 Program 101 7.1 Main geometric sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.2 Main electronic flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 7.3 Control within MOPAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.3.1 Subroutine GMETRY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

8 Error messages produced by MOPAC 105

9 Criteria 113 9.1 SCF criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 9.2 Geometric optimization criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

10 Debugging 117 10.1 Debugging keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

11 Installing MOPAC 121 11.1 ESP calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

A Names of FORTRAN-77 files 127

B Subroutine calls in MOPAC 129

C Description of subroutines 137

D Heats of formation 147

E References 149 • New Functionalities:

– Michael B. Coolidge, The Frank J. Seiler Research Laboratory, U.S. Air Force Academy, CO 80840, and James J. P. Stewart, Stewart Computational Chemistry, 15210 Paddington Circle, Colorado Springs, CO 80921-2512. (The Air Force code was obtained under the Freedom of Information Act) Symmetry is used to speed up FORCE calculations, and to facilitate the analysis of molecular vibrations.

– David Danovich, The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel. Ionization potentials are corrected using Green’s Function techniques. The resulting I.P.s are generally more accurate than the conventional I.P.s. The point–group of the system is identified, and molecular orbitals are characterized by irreducible representation.

– Andreas Klamt Bayer AG, Q18, D-5090 Leverkusen-Beyerwerk, Germany. A new approach to dielectric screening in solvents with explicit expressions for the screening energy and its gradient has been added.

• Existing Functionalities:

– Victor I. Danilov, Department of Quantum Biophysics, Academy of Sciences of the Ukraine, Kiev 143, Ukraine. Edited the MOPAC 7 Manual, and provided the basis for Section 6.17.2, on excited states.

– Henry Kurtz and Prakashan Korambath, Department of Chemistry, Memphis State University, Memphis TN 38152. The Hyperpolarizability calculation, originally written by Prof Kurtz, has been improved so that frequency dependent non-linear optical calculations can be performed. (Prakashan Korambath, dissertation research)

– Frank Jensen, Department of Chemistry, Odense Universitet, Campusvej 55, DK– 5230 Odense M, Denmark. The efficiency of Baker’s EF routine has been improved.

– John M. Simmie, Chemistry Department, University College, Galway, Ireland. The MOPAC Manual has been completely re-formatted in the LaTeX document preparation system. Equations are now much easier to read and to understand.

– Jorge A. Medrano, 5428 Falcon Ln., West Chester, OH 45069, and Roberto Bochicchio (Universidad de Buenos Aires). The BONDS function has been extended to allow free valence and other quantities to be calculated.

– George Purvis III, CAChe Scientific, P.O. Box 500, Delivery Station 13-400, Beaverton, OR 97077. The STO-6G Gaussian expansion of the Slater orbitals has been expanded to Principal Quantum Number 6. These expansions are used in analytical derivative calculations.

• Bug-reports/bug-fixes:

– Victor I. Danilov, Department of Quantum Biophysics, Academy of Sciences of the Ukraine, Kiev 143, Ukraine. Several faults in the multi-electron configuration interaction were identified, and recommendations made regarding their correction.

CONTENTS

Chapter 1

Description of MOPAC

MOPAC is a general-purpose semi-empirical molecular orbital package for the study of chemical structures and reactions. The semi-empirical Hamiltonians MNDO, MINDO/3, AM1, and PM3 are used in the electronic part of the calculation to obtain molecular orbitals, the heat of formation and its derivative with respect to molecular geometry. Using these results MOPAC calculates the vibrational spectra, thermodynamic quantities, isotopic substitution effects and force constants for molecules, radicals, ions, and polymers. For studying chemical reactions, a transition state location routine and two transition state optimizing routines are available. For users to get the most out of the program, they must understand how the program works, how to enter data, how to interpret the results, and what to do when things go wrong.

While MOPAC calls upon many concepts in quantum theory and thermodynamics and uses some fairly advanced mathematics, the user need not be familiar with these specialized topics. MOPAC is written with the non-theoretician in mind. The input data are kept as simple as possible so users can give their attention to the chemistry involved and not concern themselves with quantum and thermodynamic exotica.

The simplest description of how MOPAC works is that the user creates a data-file which describes a molecular system and specifies what kind of calculations and output are desired. The user then commands MOPAC to carry out the calculation using that data-file. Finally the user extracts the desired output on the system from the output files created by MOPAC.

1. This is the “sixth edition”. MOPAC has undergone a steady expansion since its first release, and users of the earlier editions are recommended to familiarize themselves with the changes which are described in this manual. If any errors are found, or if MOPAC does not perform as described, please contact Dr. James J. P. Stewart, Frank J. Seiler Research Laboratory, U.S. Air Force Academy, Colorado Springs, CO 80840–6528.

2. MOPAC runs successfully on normal CDC, Data General, Gould, and DEC computers, and also on the CDC 205 and CRAY–XMP “supercomputers”. The CRAY version has been partly optimized to take advantage of the CRAY architecture. Several versions exist for microcomputers such as the IBM PC-AT and XT, Zenith, etc.

1.1 Summary of MOPAC capabilities

1. MNDO, MINDO/3, AM1, and PM3 Hamiltonians.

2. Restricted Hartree-Fock (RHF) and Unrestricted Hartree-Fock (UHF) methods.

3. Extensive Configuration Interaction

(a) 100 configurations

Description of MOPAC

(b) Singlets, Doublets, Triplets, Quartets, Quintets, and Sextets (c) Excited states

(d) Geometry optimizations, etc., on specified states

4. Single SCF calculation

5. Geometry optimization

6. Gradient minimization

7. Transition state location

8. Reaction path coordinate calculation

9. Force constant calculation

10. Normal coordinate analysis

11. Transition dipole calculation

12. Thermodynamic properties calculation

13. Localized orbitals

14. Covalent bond orders

15. Bond analysis into sigma and pi contributions

16. One dimensional polymer calculation

17. Dynamic Reaction Coordinate calculation

18. Intrinsic Reaction Coordinate calculation

1.2 Copyright status of MOPAC

At the request of the Air Force Academy Law Department the following notice has been placed in MOPAC.

Notice of Public Domain nature of MOPAC.

“This computer program is a work of the United States Government and as such is not subject to protection by copyright (17 U.S.C. # 105.) Any person who fraudulently places a copyright notice or does any other act contrary to the provisions of 17 U.S. Code 506(c) shall be subject to the penalties provided therein. This notice shall not be altered or removed from this software and is to be on all reproductions.”

I recommend that a user obtain a copy by either copying it from an existing site or ordering an ‘official’ copy from the Quantum Chemistry Program Exchange, (QCPE), Department of Chemistry, Indiana University, Bloomington, Indiana, 47405. The cost covers handling only. Contact the Editor, Richard Counts, at (812) 855–4784 for further details.

1.3 Porting MOPAC to other machines

MOPAC is written for the DIGITAL VAX computer. However, the program has been written with the idea that it will be ported to other machines. After such a port has been done, the new program should be given the version number 6.10, or, if two or more versions are generated, 6.20, 6.30, etc. To validate the new copy, QCPE has a test-suite of calculations. If all tests are passed, within the tolerances given in the tests, then the new program can be called a valid version of MOPAC 6. Insofar as is practical, the mode of submission of a MOPAC job should be preserved, e.g., 1.4 Relationship of AMPAC and MOPAC

  (prompt) MOPAC <data-set> [<queue-options>...]

Any changes which do not violate the FORTRAN–77 conventions, and which users believe would be generally desirable, can be sent to the author.

1.4 Relationship of AMPAC and MOPAC

In 1985 MOPAC 3.0 and AMPAC 1.0 were submitted to QCPE for distribution. At that time, AMPAC differed from MOPAC in that it had the AM1 algorithm. Additionally, changes in some MNDO parameters in AMPAC made AMPAC results incompatable with MOPAC Versions 1-3. Subsequent versions of MOPAC, in addition to being more highly debugged than Version 3.0, also had the AM1 method. Such versions were compatible with AMPAC and with versions 1–3 of MOPAC.

In order to avoid confusion, all versions of MOPAC after 3.0 include journal references so that the user knows unambiguously which parameter sets were used in any given job.

Since 1985 AMPAC and MOPAC have evolved along different lines. In MOPAC I have endeavoured to provide a highly robust program, one with only a few new features, but which is easily portable and which can be relied upon to give precise, if not very exciting, answers. At Austin, the functionality of AMPAC has been enhanced by the research work of Prof. Dewar’s group. The new AMPAC 2.1 thus has functionalities not present in MOPAC. In publications, users should cite not only the program name but also the version number.

Commercial concerns have optimized both MOPAC and AMPAC for use on supercomputers. The quality of optimization and the degree to which the parent algorithm has been preserved differs between MOPAC and AMPAC and also between some machine specific versions. Different users may prefer one program to the other, based on considerations such as speed. Some modifications of AMPAC run faster than some modifications of MOPAC, and vice versa, but if these are modified versions of MOPAC 3.0 or AMPAC 1.0, they represent the programming prowess of the companies doing the conversion, and not any intrinsic difference between the two programs.

Testing of these large algorithms is difficult, and several times users have reported bugs in MOPAC or AMPAC which were introduced after they were supplied by QCPE.

Cooperative Development of MOPAC

MOPAC has developed, and hopefully will continue to develop, by the addition of contributed code. As a policy, any supplied code which is incorporated into MOPAC will be described in the next release of the Manual, and the author or supplier acknowledged. In the following release only journal references will be retained. The objective is to produce a good program. This is obviously not a one-person undertaking; if it was, then the product would be poor indeed. Instead, as we are in a time of rapid change in computational chemistry, a time characterized by a very free exchange of ideas and code, MOPAC has been evolving by accretion. The unstinting and generous donation of intellectual effort speaks highly of the donors. However, with the rapid commercialization of computational chemistry software in the past few years, it is unfortunate but it seems unlikely that this idyllic state will continue.

1.5 Programs recommended for use with MOPAC

MOPAC is the core program of a series of programs for the theoretical study of chemical phenomena. This version is the sixth in an on-going development, and efforts are being made to continue its further evolution. In order to make using MOPAC easier, five other programs have also been written. Users of MOPAC are recommended to use all four programs. Efforts will be made to continue the development of these programs.

HELP

HELP is a stand-alone program which mimics the VAX HELP function. It is intended for users on UNIX computers. HELP comes with the basic MOPAC 6.00, and is recommended for general use.

DRAW

DRAW, written by Maj. Donn Storch, USAF, and available through QCPE, is a powerful editing program specifically written to interface with MOPAC. Among the various facilities it offers are:

1. The on-line editing and analysis of a data file, starting from scratch or from an existing data file, an archive file, or from a results file.

2. The option of continuous graphical representation of the system being studied. Several types of terminals are supported, including DIGITAL, TEKTRONIX, and TERAK terminals.

3. The drawing of electron density contour maps generated by DENSITY on graphical devices.

4. The drawing of solid-state band structures generated by MOSOL.

5. The sketching of molecular vibrations, generated by a normal coordinate analysis.

DENSITY

DENSITY, written by Dr. James J. P. Stewart, and available through QCPE, is an electrondensity plotting program. It accepts data-files directly from MOPAC, and is intended to be used for the graphical representation of electron density distribution, individual M.O.’s, and difference maps.

MOHELP

MOHELP, also available through QCPE, is an on-line help facility, written by Maj. Donn Storch and Dr. James J. P. Stewart, to allow non-VAX users access to the VAX HELP libraries for MOPAC, DRAW, and DENSITY.

MOSOL

MOSOL (Distributed by QCPE) is a full solid-state MNDO program written by Dr. James J. P. Stewart. In comparison with MOPAC, MOSOL is extremely slow. As a result, while geometry optimization, force constants, and other functions can be carried out by MOSOL, these slow calculations are best done using the solid-state facility within MOPAC. MOSOL should be used for two or three dimensional solids only, a task that MOPAC cannot perform.

1.6 The data-file

This section is aimed at the complete novice — someone who knows nothing at all about the structure of a MOPAC data-file.

First of all, there are at most four possible types of data-files for MOPAC, but the simplest data-file is the most commonly used. Rather than define it, two examples are shown below. An explanation of the geometry definitions shown in the examples is given in the chapter “GEOMETRY SPECIFICATION”.

1.6 The data-file

1.6.1 Example of data for ethylene

     Line   1 :     UHF PULAY MINDO3 VECTORS DENSITY LOCAL T=300
     Line   2 :      EXAMPLE OF DATA FOR MOPAC
     Line   3 :        MINDO/3 UHF CLOSED-SHELL D2D ETHYLENE
     Line   4a:    C
     Line   4b:    C    1.400118  1
     Line   4c:    H    1.098326  1  123.572063  1
     Line   4d:    H    1.098326  1  123.572063  1  180.000000  0   2  1  3
     Line   4e:    H    1.098326  1  123.572063  1   90.000000  0   1  2  3
     Line   4f:    H    1.098326  1  123.572063  1  270.000000  0   1  2  3
     Line   5 :

As can be seen, the first three lines are textual. The first line consists of keywords (here seven keywords are shown). These control the calculation. The next two lines are comments or titles. The user might want to put the name of the molecule and why it is being run on these two lines.

These three lines are obligatory. If no name or comment is wanted, leave blank lines. If no keywords are specified, leave a blank line. A common error is to have a blank line before the keyword line: this error is quite tricky to find, so be careful not to have four lines before the start of the geometric data (lines 4a-4f in the example). Whatever is decided, the three lines, blank or otherwise, are obligatory.

In the example given, one line of keywords and two of documentation are shown. By use of keywords, these defaults can be changed. Modifying keywords are +, &, and SETUP. These are defined in the KEYWORDS chapter. The following table illustrates the allowed combinations:

  Line 1      Line 2     Line 3  Line 4   Line 5  Setup used
  Keys        Text       Text   Z-matrix Z-matrix  not used
  Keys +      Keys       Text    Text    Z-matrix  not used
  Keys +      Keys +     Keys    Text    Text      not used
  Keys &      Keys       Text   Z-matrix Z-matrix  not used
  Keys &      Keys &     Keys   Z-matrix Z-matrix  not used
  Keys SETUP  Text       Text   Z-matrix Z-matrix 1 or 2 lines used
  Keys +      Keys SETUP Text    Text    Z-matrix 1 line used
  Keys &      Keys SETUP Text   Z-matrix Z-matrix 1 line used

No other combinations are allowed.

The proposed use of the SETUP option is to allow a frequently used set of keywords to be defined by a single keyword. For example, if the default criteria are not suitable, SETUP might contain:

       " SCFCRT=1.D-8  SHIFT=30 ITRY=600 GNORM=0.02 ANALYT "
       "                                                   "

The order of usage of a keyword is:

Line 1 > Line 2 > Line 3.
Line 1 > SETUP.
Line 2 > SETUP.
SETUP > built in default values.

The next set of lines defines the geometry. In the example, the numbers are all neatly lined up; this is not necessary, but does make it easier when looking for errors in the data. The geometry is defined in lines 4a to 4f; line 5 terminates both the geometry and the data-file. Any additional data, for example symmetry data, would follow line 5.

Summarizing, then, the structure for a MOPAC data-file is:

Line 1 Keywords. (See chapter 2 on definitions of keywords)

Line 2 Title of the calculation, e.g. the name of the molecule or ion.

Line 3 Other information describing the calculation.

Lines 4 Internal or cartesian coordinates (See chapter on specification of geometry)

Line 5 Blank line to terminate the geometry definition.

Other layouts for data-files involve additions to the simple layout. These additions occur at the end of the data-file, after line 5. The three most common additions are:

• Symmetry data: This follows the geometric data, and is ended by a blank line.

• Reaction path: After all geometry and symmetry data (if any) are read in, points on the reaction coordinate are defined.

• Saddle data: A complete second geometry is input. The second geometry follows the first geometry and symmetry data (if any).

1.6.2 Example of data for polytetrahydrofuran

The following example illustrates the data file for a four hour polytetrahydrofuran calculation. As you can see the layout of the data is almost the same as that for a molecule, the main difference is in the presence of the translation vector atom “Tv”.

Line 1 :T=4H
Line 2 :      POLY-TETRAHYDROFURAN (C4 H8 O)2
Line 3 :
Line 4a:   C    0.000000  0    0.000000  0    0.000000  0   0  0  0
Line 4b:   C    1.551261  1    0.000000  0    0.000000  0   1  0  0
Line 4c:   O    1.401861  1  108.919034  1    0.000000  0   2  1  0
Line 4d:   C    1.401958  1  119.302489  1 -179.392581  1   3  2  1
Line 4e:   C    1.551074  1  108.956238  1  179.014664  1   4  3  2
Line 4f:   C    1.541928  1  113.074843  1  179.724877  1   5  4  3
Line 4g:   C    1.551502  1  113.039652  1  179.525806  1   6  5  4
Line 4h:   O    1.402677  1  108.663575  1  179.855864  1   7  6  5
Line 4i:   C    1.402671  1  119.250433  1 -179.637345  1   8  7  6
Line 4j:   C    1.552020  1  108.665746  1 -179.161900  1   9  8  7
Line 4k:  XX    1.552507  1  112.659354  1 -178.914985  1  10  9  8
Line 4l:  XX    1.547723  1  113.375266  1 -179.924995  1  11 10  9
Line 4m:   H    1.114250  1   89.824605  1  126.911018  1   1  3  2
Line 4n:   H    1.114708  1   89.909148  1 -126.650667  1   1  3  2
Line 4o:   H    1.123297  1   93.602831  1  127.182594  1   2  4  3
Line 4p:   H    1.123640  1   93.853406  1 -126.320187  1   2  4  3
Line 4q:   H    1.123549  1   90.682924  1  126.763659  1   4  6  5
Line 4r:   H    1.123417  1   90.679889  1 -127.033695  1   4  6  5
Line 4s:   H    1.114352  1   90.239157  1  126.447043  1   5  7  6
Line 4t:   H    1.114462  1   89.842852  1 -127.140168  1   5  7  6
Line 4u:   H    1.114340  1   89.831790  1  126.653999  1   6  8  7
Line 4v:   H    1.114433  1   89.753913  1 -126.926618  1   6  8  7
Line 4w:   H    1.123126  1   93.644744  1  127.030541  1   7  9  8
Line 4x:   H    1.123225  1   93.880969  1 -126.380511  1   7  9  8
Line 4y:   H    1.123328  1   90.261019  1  127.815464  1   9 11 10
Line 4z:   H    1.123227  1   91.051403  1 -125.914234  1   9 11 10
Line 4A:   H    1.113970  1   90.374545  1  126.799259  1  10 12 11

  Line 4B:   H    1.114347  1   90.255788  1 -126.709810  1  10 12 11
  Line 4C:  Tv   12.299490  1    0.000000  0    0.000000  0   1 11 10
  Line 5 :   0    0.000000  0    0.000000  0    0.000000  0   0  0  0

Polytetrahydrofuran has a repeat unit of (C_{4}H_{8}O)_{2}; i.e., twice the monomer unit. This is necessary in order to allow the lattice to repeat after a translation through 12.3 Å. See the section on Solid State Capability for further details.

Note the two dummy atoms on lines 4k and 4l. These are useful, but not essential, for defining the geometry. The atoms on lines 4y to 4B use these dummy atoms, as does the translation vector on line 4C. The translation vector has only the length marked for optimization. The reason for this is also explained in the Background chapter.

Description of MOPAC

Keywords

2.1 Specification of keywords

All control data are entered in the form of keywords, which form the first line of a data-file. A description of what each keyword does is given in Section 2.3. The order in which keywords appear is not important although they must be separated by a space. Some keywords can be abbreviated, allowed abbreviations are noted in Section 2.3 (for example 1ELECTRON can be entered as 1ELECT). However the full keyword is preferred in order to more clearly document the calculation and to obviate the possibility that an abbreviated keyword might not be recognized. If there is insufficient space in the first line for all the keywords needed, then consider abbreviating the longer words. One type of keyword, those with an equal sign, such as, BAR=0.05, may not be abbreviated, and the full word needs to be supplied.

Most keywords which involve an equal sign, such as SCFCRT=1.D-12 can, at the user’s discretion, be written with spaces before and after the equal sign. Thus all permutations of SCFCRT=1.D-12,

such as SCFCRT =1.D-12, SCFCRT = 1.D-12, SCFCRT= 1.D-12, SCFCRT  =  1.D-12, etc. are al-

lowed. Exceptions to this are T=, T-PRIORITY=, H-PRIORITY=, X-PRIORITY=, IRC=, DRC= and TRANS=. ‘ T=’ cannot be abbreviated to ‘ T ’ as many keywords start or end with a ‘T’; for the other keywords the associated abbreviated keywords have specific meanings.

If two keywords which are incompatible, like UHF and C.I., are supplied, or a keyword which is incompatible with the species supplied, for instance TRIPLET and a methyl radical, then error trapping will normally occur, and an error message will be printed. This usually takes an insignificant time, so data are quickly checked for obvious errors.

2.2 Full list of keywords used in MOPAC

 &        - TURN NEXT LINE INTO KEYWORDS
 +        - ADD ANOTHER LINE OF KEYWORDS
 0SCF     - READ IN DATA, THEN STOP
 1ELECTRON- PRINT FINAL ONE-ELECTRON MATRIX
 1SCF     - DO ONE SCF AND THEN STOP
 AIDER    - READ IN AB INITIO DERIVATIVES
 AIGIN    - GEOMETRY MUST BE IN GAUSSIAN FORMAT
 AIGOUT   - IN ARC FILE, INCLUDE AB-INITIO GEOMETRY
 ANALYT   - USE ANALYTICAL DERIVATIVES OF ENERGY WRT GEOMETRY
 AM1      - USE THE AM1 HAMILTONIAN
 BAR=n.n  - REDUCE BAR LENGTH BY A MAXIMUM OF n.n
 BIRADICAL- SYSTEM HAS TWO UNPAIRED ELECTRONS
 BONDS    - PRINT FINAL BOND-ORDER MATRIX
 C.I.     - A MULTI-ELECTRON CONFIGURATION INTERACTION SPECIFIED

\underline{Keywords}

CHARGE=n - CHARGE ON SYSTEM = n (e.g. NH4 => CHARGE=1)
COMPFG   - PRINT HEAT OF FORMATION CALCULATED IN COMPFG
CONNOLLY - USE CONNOLLY SURFACE
DEBUG    - DEBUG OPTION TURNED ON
DENOUT   - DENSITY MATRIX OUTPUT (CHANNEL 10)
DENSITY  - PRINT FINAL DENSITY MATRIX
DEP      - GENERATE FORTRAN CODE FOR PARAMETERS FOR NEW ELEMENTS
DEPVAR=n - TRANSLATION VECTOR IS A MULTIPLE OF BOND-LENGTH
DERIV    - PRINT PART OF WORKING IN DERIV
DFORCE   - FORCE CALCULATION SPECIFIED, ALSO PRINT FORCE MATRIX.
DFP      - USE DAVIDON-FLETCHER-POWELL METHOD TO OPTIMIZE GEOMETRIES
DIPOLE   - FIT THE ESP TO THE CALCULATED DIPOLE
DIPX     - X COMPONENT OF DIPOLE TO BE FITTED
DIPY     - Y COMPONENT OF DIPOLE TO BE FITTED
DIPZ     - Z COMPONENT OF DIPOLE TO BE FITTED
DMAX     - MAXIMUM STEPSIZE IN EIGENVECTOR FOLLOWING
DOUBLET  - DOUBLET STATE REQUIRED
DRC      - DYNAMIC REACTION COORDINATE CALCULATION
DUMP=n   - WRITE RESTART FILES EVERY n SECONDS
ECHO     - DATA ARE ECHOED BACK BEFORE CALCULATION STARTS
EF       - USE EF ROUTINE FOR MINIMUM SEARCH
EIGINV   -
EIGS     - PRINT ALL EIGENVALUES IN ITER
ENPART   - PARTITION ENERGY INTO COMPONENTS
ESP      - ELECTROSTATIC POTENTIAL CALCULATION
ESPRST   - RESTART OF ELECTROSTATIC POTENTIAL
ESR      - CALCULATE RHF UNPAIRED SPIN DENSITY
EXCITED  - OPTIMIZE FIRST EXCITED SINGLET STATE
EXTERNAL - READ PARAMETERS OFF DISK
FILL=n   - IN RHF OPEN AND CLOSED SHELL, FORCE M.O. n
           TO BE FILLED
FLEPO    - PRINT DETAILS OF GEOMETRY OPTIMIZATION
FMAT     - PRINT DETAILS OF WORKING IN FMAT
FOCK     - PRINT LAST FOCK MATRIX
FORCE    - FORCE CALCULATION SPECIFIED
GEO-OK   - OVERRIDE INTERATOMIC DISTANCE CHECK
GNORM=n.n- EXIT WHEN GRADIENT NORM DROPS BELOW n.n
GRADIENTS- PRINT ALL GRADIENTS
GRAPH    - GENERATE FILE FOR GRAPHICS
HCORE    - PRINT DETAILS OF WORKING IN HCORE
HESS=N   - OPTIONS FOR CALCULATING HESSIAN MATRICES IN EF
H-PRIO   - HEAT OF FORMATION TAKES PRIORITY IN DRC
HYPERFINE- HYPERFINE COUPLING CONSTANTS TO BE CALCULATED
IRC      - INTRINSIC REACTION COORDINATE CALCULATION
ISOTOPE  - FORCE MATRIX WRITTEN TO DISK (CHANNEL 9 )
ITER     - PRINT DETAILS OF WORKING IN ITER
ITRY=N   - SET LIMIT OF NUMBER OF SCF ITERATIONS TO N.
IUPD     - MODE OF HESSIAN UPDATE IN EIGENVECTOR FOLLOWING
K=(N,N)  - BRILLOUIN ZONE STRUCTURE TO BE CALCULATED
KINETIC  - EXCESS KINETIC ENERGY ADDED TO DRC CALCULATION
LINMIN   - PRINT DETAILS OF LINE MINIMIZATION
LARGE    - PRINT EXPANDED OUTPUT
LET      - OVERRIDE CERTAIN SAFETY CHECKS
LOCALIZE - PRINT LOCALIZED ORBITALS

2.2 Full list of keywords used in MOPAC

MAX      - PRINTS MAXIMUM GRID SIZE (23*23)
MECI     - PRINT DETAILS OF MECI CALCULATION
MICROS   - USE SPECIFIC MICROSTATES IN THE C.I.
MINDO/3  - USE THE MINDO/3 HAMILTONIAN
MMOK     - USE MOLECULAR MECHANICS CORRECTION TO CONH BONDS
MODE=N   - IN EF, FOLLOW HESSIAN MODE NO. N
MOLDAT   - PRINT DETAILS OF WORKING IN MOLDAT
MS=N     - IN MECI, MAGNETIC COMPONENT OF SPIN
MULLIK   - PRINT THE MULLIKEN POPULATION ANALYSIS
NLLSQ    - MINIMIZE GRADIENTS USING NLLSQ
NOANCI   - DO NOT USE ANALYTICAL C.I. DERIVATIVES
NODIIS   - DO NOT USE DIIS GEOMETRY OPTIMIZER
NOINTER  - DO NOT PRINT INTERATOMIC DISTANCES
NOLOG    - SUPPRESS LOG FILE TRAIL, WHERE POSSIBLE
NOMM     - DO NOT USE MOLECULAR MECHANICS CORRECTION TO CONH BONDS
NONR     -
NOTHIEL  - DO NOT USE THIEL'S FSTMIN TECHNIQUE
NSURF=N  - NUMBER OF SURFACES IN AN ESP CALCULATION
NOXYZ    - DO NOT PRINT CARTESIAN COORDINATES
NSURF    - NUMBER OF LAYERS USED IN ELECTROSTATIC POTENTIAL
OLDENS   - READ INITIAL DENSITY MATRIX OFF DISK
OLDGEO   - PREVIOUS GEOMETRY TO BE USED
OPEN     - OPEN-SHELL RHF CALCULATION REQUESTED
ORIDE    -
PARASOK  - IN AM1 CALCULATIONS SOME MNDO PARAMETERS ARE TO BE USED
PI       - RESOLVE DENSITY MATRIX INTO SIGMA AND PI BONDS
PL       - MONITOR CONVERGENCE OF DENSITY MATRIX IN ITER
PM3      - USE THE MNDO-PM3 HAMILTONIAN
POINT=N  - NUMBER OF POINTS IN REACTION PATH
POINT1=N - NUMBER OF POINTS IN FIRST DIRECTION IN GRID CALCULATION
POINT2=N - NUMBER OF POINTS IN SECOND DIRECTION IN GRID CALCULATION
POLAR    - CALCULATE FIRST, SECOND AND THIRD ORDER POLARIZABILITIES
POTWRT   - IN ESP, WRITE OUT ELECTROSTATIC POTENTIAL TO UNIT 21
POWSQ    - PRINT DETAILS OF WORKING IN POWSQ
PRECISE  - CRITERIA TO BE INCREASED BY 100 TIMES
PULAY    - USE PULAY'S CONVERGER TO OBTAIN A SCF
QUARTET  - QUARTET STATE REQUIRED
QUINTET  - QUINTET STATE REQUIRED
RECALC=N - IN EF, RECALCULATE HESSIAN EVERY N STEPS
RESTART  - CALCULATION RESTARTED
ROOT=n   - ROOT n TO BE OPTIMIZED IN A C.I. CALCULATION
ROT=n    - THE SYMMETRY NUMBER OF THE SYSTEM IS n.
SADDLE   - OPTIMIZE TRANSITION STATE
SCALE    - SCALING FACTOR FOR VAN DER WAALS DISTANCE IN ESP
SCFCRT=n - DEFAULT SCF CRITERION REPLACED BY THE VALUE SUPPLIED
SCINCR   - INCREMENT BETWEEN LAYERS IN ESP
SETUP    - EXTRA KEYWORDS TO BE READ OF SETUP FILE
SEXTET   - SEXTET STATE REQUIRED
SHIFT=n  - A DAMPING FACTOR OF n DEFINED TO START SCF
SIGMA    - MINIMIZE GRADIENTS USING SIGMA
SINGLET  - SINGLET STATE REQUIRED
SLOPE    - MULTIPLIER USED TO SCALE MNDO CHARGES
SPIN     - PRINT FINAL UHF SPIN MATRIX
STEP     - STEP SIZE IN PATH

STEP1=n  - STEP SIZE n FOR FIRST COORDINATE IN GRID CALCULATION
STEP2=n  - STEP SIZE n FOR SECOND COORDINATE IN GRID CALCULATION
STO-3G   - DEORTHOGONALIZE ORBITALS IN STO-3G BASIS
SYMAVG   - AVERAGE SYMMETRY EQUIVALENT ESP CHARGES
SYMMETRY - IMPOSE SYMMETRY CONDITIONS
T=n      - A TIME OF n SECONDS REQUESTED
THERMO   - PERFORM A THERMODYNAMICS CALCULATION
TIMES    - PRINT TIMES OF VARIOUS STAGES
T-PRIO   - TIME TAKES PRIORITY IN DRC
TRANS    - THE SYSTEM IS A TRANSITION STATE
           (USED IN THERMODYNAMICS CALCULATION)
TRIPLET  - TRIPLET STATE REQUIRED
TS       - USING EF ROUTINE FOR TS SEARCH
UHF      - UNRESTRICTED HARTREE-FOCK CALCULATION
VECTORS  - PRINT FINAL EIGENVECTORS
VELOCITY - SUPPLY THE INITIAL VELOCITY VECTOR IN A DRC CALCULATION
WILLIAMS - USE WILLIAMS SURFACE
X-PRIO   - GEOMETRY CHANGES TAKE PRIORITY IN DRC
XYZ      - DO ALL GEOMETRIC OPERATIONS IN CARTESIAN COORDINATES.

2.3 Definitions of keywords

The definitions below are given with some technical expressions which are not further defined. Interested users are referred to Appendix E of this manual to locate appropriate references which will provide further clarification.

There are three classes of keywords:

1. those which CONTROL substantial aspects of the calculation, i.e., those which affect the final heat of formation,

2. those which determine which OUTPUT will be calculated and printed, and

3. those which dictate the WORKING of the calculation, but which do not affect the heat of formation. The assignment to one of these classes is designated by a (C), (O) or (W), respectively, following each keyword in the list below.

& (C)

An ‘␣&’ means ‘turn the next line into keywords’. Note the space before the ‘&’ sign. Since ‘&’ is a keyword, it must be preceeded by a space. A ‘␣&’ on line 1 would mean that a second line of keywords should be read in. If that second line contained a ‘␣&’, then a third line of keywords would be read in. If the first line has a ‘␣&’ then the first description line is omitted, if the second line has a ‘␣&’, then both description lines are omitted.

Examples: Use of one ‘&’

 VECTORS DENSITY RESTART & NLLSQ T=1H SCFCRT=1.D-8 DUMP=30M ITRY=300
 PM3 FOCK OPEN(2,2) ROOT=3 SINGLET SHIFT=30

Test on a totally weird system: Use of two ‘&’s

 LARGE=-10 & DRC=4.0 T=1H SCFCRT=1.D-8 DUMP=30M ITRY=300 SHIFT=30
 PM3 OPEN(2,2) ROOT=3 SINGLET NOANCI ANALYT  T-PRIORITY=0.5 &
 LET GEO-OK VELOCITY KINETIC=5.0

2.3 Definitions of keywords

+ (C)

A ‘␣+’ sign means ‘read another line of keywords’. Note the space before the ‘+’ sign. Since ‘+’ is a keyword, it must be preceeded by a space. A ‘␣+’ on line 1 would mean that a second line of keywords should be read in. If that second line contains a ‘␣+’, then a third line of keywords will be read in. Regardless of whether a second or a third line of keywords is read in, the next two lines would be description lines.

Example of ‘␣+’ option

    RESTART T=4D FORCE OPEN(2,2) SHIFT=20 PM3 +
    SCFCRT=1.D-8 DEBUG + ISOTOPE FMAT ECHO singlet ROOT=3
    THERMO(300,400,1) ROT=3

Example of data set with three lines of keywords. Note: There are two lines of description, this and the previous line.

0SCF (O)

The data can be read in and output, but no actual calculation is performed when this keyword is used. This is useful as a check on the input data. All obvious errors are trapped, and warning messages printed.

A second use is to convert from one format to another. The input geometry is printed in various formats at the end of a 0SCF calculation. If NOINTER is absent, cartesian coordinates are printed. Unconditionally, MOPAC Z-matrix internal coordinates are printed, and if AIGOUT is present, Gaussian Z-matrix internal coordinates are printed. 0SCF should now be used in place of DDUM.

1ELECTRON (O)

The final one-electron matrix is printed out. This matrix is composed of atomic orbitals; the array element between orbitals i and j on different atoms is given by:

H(i, j) = 0.5 × (β_{i} + β_{j}) × overlap(i, j)

The matrix elements between orbitals i and j on the same atom are calculated from the electronnuclear attraction energy, and also from the U(i) value if i = j.

The one-electron matrix is unaffected by (a) the charge and (b) the electron density. It is only a function of the geometry. Abbreviation: 1ELEC.

1SCF (C)

When users want to examine the results of a single SCF calculation of a geometry, 1SCF should be used. 1SCF can be used in conjunction with RESTART, in which case a single SCF calculation will be done, and the results printed.

When 1SCF is used on its own (that is, RESTART is not also used) then derivatives will only be calculated if GRAD is also specified.

1SCF is helpful in a learning situation. MOPAC normally performs many SCF calculations, and in order to minimize output when following the working of the SCF calculation, 1SCF is very useful.

AIDER (C)

AIDER allows MOPAC to optimize an ab-initio geometry. To use it, calculate the ab-initio gradients using, e.g., Gaussian. Supply MOPAC with these gradients, after converting them into kcal/mol. The geometry resulting from a MOPAC run will be nearer to the optimized ab-initio geometry than if the geometry optimizer in Gaussian had been used.

AIGIN (C)

If the geometry (Z-matrix) is specified using the Gaussian-8X, then normally this will be read in without difficulty. In the event that it is mistaken for a normal MOPAC-type Z-matrix, the keyword AIGIN is provided. AIGIN will force the data-set to be read in assuming Gaussian format. This is necessary if more than one system is being studied in one run.

AIGOUT (O)

The ARCHIVE file contains a data-set suitable for submission to MOPAC. If, in addition to this data-set, the Z-matrix for Gaussian input is wanted, then AIGOUT (ab initio geometry output), should be used.

The Z-matrix is in full Gaussian form. Symmetry, where present, will be correctly defined. Names of symbolics will be those used if the original geometry was in Gaussian format, otherwise ‘logical’ names will be used. Logical names are of form <t><a><b>[<c>][<d>] where <t> is ‘r’ for bond length, ‘a’ for angle, or ‘d’ for dihedral, <a> is the atom number, <b> is the atom to which <a> is related, <c>, if present, is the atom number to which <a> makes an angle, and <d>, if present, is the atom number to which <a> makes a dihedral.

ANALYT (W)

By default, finite difference derivatives of energy with respect to geometry are used. If ANALYT is specified, then analytical derivatives are used instead. Since the analytical derivatives are over Gaussian functions—a STO-6G basis set is used—the overlaps are also over Gaussian functions. This will result in a very small (less than 0.1 kcal/mole) change in heat of formation. Use analytical derivatives (a) when the mantissa used is less than about 51–53 bits, or (b) when comparison with finite difference is desired. Finite difference derivatives are still used when non-variationally optimized wavefunctions are present.

AM1 (C)

The AM1 method is to be used. By default MNDO is run.

BAR=n.nn (W)

In the SADDLE calculation the distance between the two geometries is steadily reduced until the transition state is located. Sometimes, however, the user may want to alter the maximum rate at which the distance between the two geometries reduces. BAR is a ratio, normally 0.15, or 15 percent. This represents a maximum rate of reduction of the bar of 15 percent per step. Alternative values that might be considered are BAR=0.05 or BAR=0.10, although other values may be used. See also SADDLE.

If CPU time is not a major consideration, use BAR=0.03.

BIRADICAL (C)

Note: BIRADICAL is a redundant keyword, and represents a particular configuration interaction calculation. Experienced users of MECI (q.v.) can duplicate the effect of the keyword BIRADICAL by using the MECI keywords OPEN(2,2) and SINGLET.

For molecules which are believed to have biradicaloid character the option exists to optimize the lowest singlet energy state which results from the mixing of three states. These states are, in order, (1) the (micro)state arising from a one electron excitation from the HOMO to the LUMO, which is combined with the microstate resulting from the time-reversal operator acting on the parent microstate, the result being a full singlet state; (2) the state resulting from de-excitation from the formal LUMO to the HOMO; and (3) the state resulting from the single electron in the formal HOMO being excited into the LUMO.

                       Microstate 1          Microstate 2      Microstate 3

                  Alpha Beta   Alpha Beta    Alpha  Beta       Alpha  Beta

      LUMO         *                 *                           *    *
                  ---  ---     ---  ---       ---  ---          ---  ---

                             +

      HOMO              *       *              *    *
                  ---  ---     ---  ---       ---  ---          ---  ---

A configuration interaction calculation is involved here. A biradical calculation done without C.I. at the RHF level would be meaningless. Either rotational invariance would be lost, as in the D2d form of ethylene, or very artificial barriers to rotations would be found, such as in a methane molecule “orbiting” a D2d ethylene. In both cases the inclusion of limited configuration interaction corrects the error. BIRADICAL should not be used if either the HOMO or LUMO is degenerate; in this case, the full manifold of HOMO × LUMO should be included in the C.I., using MECI options. The user should be aware of this situation. When the biradical calculation is performed correctly, the result is normally a net stabilization. However, if the first singlet excited state is much higher in energy than the closed-shell ground state, BIRADICAL can lead to a destabilization. Abbreviation: BIRAD. See also MECI, C.I., OPEN, SINGLET.

BONDS (O)

The rotationally invariant bond order between all pairs of atoms is printed. In this context a bond is defined as the sum of the squares of the density matrix elements connecting any two atoms. For ethane, ethylene, and acetylene the carbon-carbon bond orders are roughly 1.00, 2.00, and 3.00 respectively. The diagonal terms are the valencies calculated from the atomic terms only and are defined as the sum of the bonds the atom makes with other atoms. In UHF and nonvariationally optimized wavefunctions the calculated valency will be incorrect, the degree of error being proportional to the non-duodempotency of the density matrix. For an RHF wavefunction the square of the density matrix is equal to twice the density matrix.

The bonding contributions of all M.O.’s in the system are printed immediately before the bonds matrix. The idea of molecular orbital valency was developed by Gopinathan, Siddarth, and Ravimohan. Just as an atomic orbital has a ‘valency’, so has a molecular orbital. This leads to the following relations: The sum of the bonding contributions of all occupied M.O.’s is the same as the sum of all valencies which, in turn is equal to two times the sum of all bonds. The sum of the bonding contributions of all M.O.’s is zero.

C.I.=n (C)

Normally configuration interaction is invoked if any of the keywords which imply a C.I. calculation are used, such as BIRADICAL, TRIPLET or QUARTET. Note that ROOT= does not imply a C.I. calculation: ROOT= is only used when a C.I. calculation is done. However, as these implied C.I.’s involve the minimum number of configurations practical, the user may want to define a larger than minimum C.I., in which case the keyword C.I.=n can be used. When C.I.=n is specified, the n M.O.’s which ‘bracket’ the occupied- virtual energy levels will be used. Thus, C.I.=2 will include both the HOMO and the LUMO, while C.I.=1 (implied for odd-electron systems) will only include the HOMO (This will do nothing for a closed-shell system, and leads to Dewar’s half-electron correction for odd-electron systems). Users should be aware of the rapid increase in the size of the C.I. with increasing numbers of M.O.’s being used. Numbers of microstates implied by the use of the keyword C.I.=n on its own are as follows:

    Keyword        Even-electron systems           Odd-electron systems
                No. of electrons, configs       No. of electrons, configs
                Alpha   Beta                    Alpha Beta
     C.I.=1       1      1          1            1     0             1
     C.I.=2       1      1          4            1     0             2
     C.I.=3       2      2          9            2     1             9
     C.I.=4       2      2         36            2     1            24
     C.I.=5       3      3        100            3     2           100
     C.I.=6       3      3        400            3     2           300
     C.I.=7       4      4       1225            4     3          1225
     C.I.=8   (Do not use unless other keywords also used, see below)

If a change of spin is defined, then larger numbers of M.O.’s can be used up to a maximum of 10. The C.I. matrix is of size 100 x 100. For calculations involving up to 100 configurations, the spinstates are exact eigenstates of the spin operators. For systems with more than 100 configurations, the 100 configurations of lowest energy are used. See also MICROS and the keywords defining spin-states.

Note that for any system, use of C.I.=5 or higher normally implies the diagonalization of a 100 by 100 matrix. As a geometry optimization using a C.I. requires the derivatives to be calculated using derivatives of the C.I. matrix, geometry optimization with large C.I.’s will require more time than smaller C.I.’s.

Associated keywords: MECI, ROOT=, MICROS, SINGLET, DOUBLET, etc.

C.I.=(n,m)

In addition to specifying the number of M.O.’s in the active space, the number of electrons can also be defined. In C.I.=(n,m), n is the number of M.O.s in the active space, and m is the number of doubly filled levels to be used. Examples:

  Keywords           Number of M.O.s  No. Electrons
  C.I.=2                   2             2 (1)
  C.I.=(2,1)               2             2 (3)
  C.I.=(3,1)               3             2 (3)
  C.I.=(3,2)               3             4 (5)
  C.I.=(3,0) OPEN(2,3)     3             2 (N/A)
  C.I.=(3,1) OPEN(2,2)     3             4 (N/A)
  C.I.=(3,1) OPEN(1,2)     3           N/A (3)

Odd electron systems given in parentheses.

CHARGE=n (C)

When the system being studied is an ion, the charge, n, on the ion must be supplied by CHARGE=n. For cations n can be 1, 2, 3, etc, for anions −1 or −2 or −3, etc. Examples:

      ION               KEYWORD              ION          KEYWORD
      NH4(+)           CHARGE=1             CH3COO(-)      CHARGE=-1
      C2H5(+)          CHARGE=1             (COO)(=)       CHARGE=-2
      SO4(=)           CHARGE=-2            PO4(3-)        CHARGE=-3
      HSO4(-)          CHARGE=-1            H2PO4(-)       CHARGE=-1

DCART (O)

The cartesian derivatives which are calculated in DCART for variationally optimized systems are printed if the keyword DCART is present. The derivatives are in units of kcals/Angstrom, and the coordinates are displacements in x, y, and z.

DEBUG (O)

Certain keywords have specific output control meanings, such as FOCK, VECTORS and DEN-SITY. If they are used, only the final arrays of the relevant type are printed. If DEBUG is supplied, then all arrays are printed. This is useful in debugging ITER. DEBUG can also increase the amount of output produced when certain output keywords are used, e.g. COMPFG.

DENOUT (O)

The density matrix at the end of the calculation is to be output in a form suitable for input in another job. If an automatic dump due to the time being exceeded occurs during the current run then DENOUT is invoked automatically. (see RESTART)

DENSITY (O)

At the end of a job, when the results are being printed, the density matrix is also printed. For RHF the normal density matrix is printed. For UHF the sum of the alpha and beta density matrices is printed.

If density is not requested, then the diagonal of the density matrix, i.e., the electron density on the atomic orbitals, will be printed.

DEP (O)

For use only with EXTERNAL=. When new parameters are published, they can be entered at run-time by using EXTERNAL=, but as this is somewhat clumsy, a permanent change can be made by use of DEP.

If DEP is invoked, a complete block of FORTRAN code will be generated, and this can be inserted directly into the BLOCK DATA file.

Note that the output is designed for use with PM3. By modifying the names, the output can be used with MNDO or AM1.

DEPVAR=n.nn (C)

In polymers the translation vector is frequently a multiple of some internal distance. For example, in polythene it is the C1–C3 distance. If a cluster unit cell of C6H12 is used, then symmetry can be used to tie together all the carbon atom coordinates and the translation vector distance. In this example DEPVAR=3.0 would be suitable.

DFP (W)

By default the Broyden–Fletcher–Goldfarb–Shanno method will be used to optimize geometries. The older Davidon–Fletcher–Powell method can be invoked by specifying DFP. This is intended to be used for comparison of the two methods.

DIPOLE (C)

Used in the ESP calculation, DIPOLE will constrain the calculated charges to reproduce the cartesian dipole moment components calculated from the density matrix and nuclear charges.

DIPX (C)

Similar to DIPOLE, except the fit will be for the X-component only.

DIPY (C)

Similar to DIPOLE, except the fit will be for the Y-component only.

DIPZ (C)

Similar to DIPOLE, except the fit will be for the Z-component only.

DMAX=n.nn (W)

In the EF routine, the maximum step-size is 0.2 (Angstroms or radians), by default. This can be changed by specifying DMAX=n.nn. Increasing DMAX can lead to faster convergence but can also make the optimization go bad very fast. Furthermore, the Hessian updating may deteriorate when using large stepsizes. Reducing the stepsize to 0.10 or 0.05 is recommended when encountering convergence problems.

DOUBLET (C)

When a configuration interaction calculation is done, all spin states are calculated simultaneously, either for component of spin=0 or 1/2. When only doublet states are of interest, then DOUBLET can be specified, and all other spin states, while calculated, are ignored in the choice of root to be used.

Note that while almost every odd-electron system will have a doublet ground state, DOUBLET should still be specified if the desired state must be a doublet.

DOUBLET has no meaning in a UHF calculation.

DRC (C)

A Dynamic Reaction Coordinate calculation is to be run. By default, total energy is conserved, so that as the ‘reaction’ proceeds in time, energy is transferred between kinetic and potential forms.

DRC=n.nnn (C)

In a DRC calculation, the ‘half-life’ for loss of kinetic energy is defined as n.nnn femtoseconds. If n.nnn is set to zero, infinite damping simulating a very condensed phase is obtained.

This keyword cannot be written with spaces around the ‘=’ sign.

DUMP (W)

Restart files are written automatically at one hour cpu time intervals to allow a long job to be restarted if the job is terminated catastrophically. To change the frequency of dump, set DUMP=nn to request a dump every nn seconds. Alternative forms, DUMP=nnM, DUMP=nnH, DUMP=nnD for a dump every nn minutes, hours, or days, respectively. DUMP only works with geometry optimization, gradient minimization, path, and FORCE calculations. It does not (yet) work with a SADDLE calculation.

ECHO (O)

Data are echoed back if ECHO is specified. Only useful if data are suspected to be corrupt.

EF (C)

The Eigenvector Following routine is an alternative to the BFGS, and appears to be much faster. To invoke the Eigenvector Following routine, specify EF. EF is particularly good in the end-game, when the gradient is small. See also HESS, DMAX, EIGINV.

EIGINV (W)

Not recommended for normal use. Used with the EF routine. See source code for more details.

ENPART (O)

This is a very useful tool for analyzing the energy terms within a system. The total energy, in eV, obtained by the addition of the electronic and nuclear terms, is partitioned into mono- and bi-centric contributions, and these contributions in turn are divided into nuclear and one- and two-electron terms.

ESP (C)

This is the ElectroStatic Potential calculation of K. M. Merz and B. H. Besler. ESP calculates the expectation values of the electrostatic potential of a molecule on a uniform distribution of points. The resultant ESP surface is then fitted to atom centered charges that best reproduce the distribution, in a least squares sense.

ESPRST (W)

ESPRST restarts a stopped ESP calculation. Do not use with RESTART.

ESR (O)

The unpaired spin density arising from an odd-electron system can be calculated both RHF and UHF. In a UHF calculation the alpha and beta M.O.’s have different spatial forms, so unpaired spin density can naturally be present on in-plane hydrogen atoms such as in the phenoxy radical.

In the RHF formalism a MECI calculation is performed. If the keywords OPEN and C.I.= are both absent then only a single state is calculated. The unpaired spin density is then calculated from the state function. In order to have unpaired spin density on the hydrogens in, for example, the phenoxy radical, several states should be mixed.

EXCITED (C)

The state to be calculated is the first excited open-shell singlet state. If the ground state is a singlet, then the state calculated will be S(1); if the ground state is a triplet, then S(2). This state would normally be the state resulting from a one-electron excitation from the HOMO to the LUMO. Exceptions would be if the lowest singlet state were a biradical, in which case the EXCITED state could be a closed shell.

The EXCITED state will be calculated from a BIRADICAL calculation in which the second root of the C.I. matrix is selected. Note that the eigenvector of the C.I. matrix is not used in the current formalism. Abbreviation: EXCI.

Note: EXCITED is a redundant keyword, and represents a particular configuration interaction calculation. Experienced users of MECI can duplicate the effect of the keyword EXCITED by using the MECI keywords OPEN(2,2), SINGLET, and ROOT=2.

EXTERNAL=name (C)

Normally, PM3, AM1 and MNDO parameters are taken from the BLOCK DATA files within MOPAC. When the supplied parameters are not suitable, as in an element recently parameterized, and the parameters have not yet installed in the user’s copy of MOPAC, then the new parameters can be inserted at run time by use of EXTERNAL=<filename>, where <filename> is the name of the file which contains the new parameters.

<filename> consists of a series of parameter definitions in the format:

<Parameter> <Element> <Value of parameter>

where the possible parameters are USS, UPP, UDD, ZS, ZP, ZD, BETAS, BETAP, BETAD, GSS, GSP, GPP, GP2, HSP, ALP, FNnm, n=1,2, or 3, and m=1 to 10, and the elements are defined by their chemical symbols, such as Si or SI.

When new parameters for elements are published, they can be typed in as shown. This file is ended by a blank line, the word END or nothing, i.e., no end-of-file delimiter. An example of a parameter data file would be (put at least 2 spaces before and after parameter name):

  Line  1:     USS      Si      -34.08201495
  Line  2:     UPP      Si      -28.03211675
  Line  3:     BETAS    Si       -5.01104521
  Line  4:     BETAP    Si       -2.23153969
  Line  5:     ZS       Si        1.28184511
  Line  6:     ZP       Si        1.84073175
  Line  7:     ALP      Si        2.18688712
  Line  8:     GSS      Si        9.82
  Line  9:     GPP      Si        7.31
  Line 10:     GSP      Si        8.36
  Line 11:     GP2      Si        6.54
  Line 12:     HSP      Si        1.32

Derived parameters do no need to be entered; they will be calculated from the optimized parameters. All ”constants” such as the experimental heat of atomization are already inserted for all elements.

NOTE: EXTERNAL can only be used to input parameters for MNDO, AM1, or PM3. It is unlikely, however, that any more MINDO/3 parameters will be published.

See also DEP to make a permanent change.

FILL=n (C)

The n’th M.O. in an RHF calculation is constrained to be filled. It has no effect on a UHF calculation. After the first iteration (NOTE: not after the first SCF calculation, but after the first iteration within the first SCF calculation) the n’th M.O. is stored, and, if occupied, no further action is taken at that time. If unoccupied, then the HOMO and the n’th M.O.’s are swapped around, so that the n’th M.O. is now filled. On all subsequent iterations the M.O. nearest in character to the stored M.O. is forced to be filled, and the stored M.O. replaced by that M.O. This is necessitated by the fact that in a reaction a particular M.O. may change its character considerably. A useful procedure is to run 1SCF and DENOUT first, in order to identify the M.O.’s; the complete job is then run with OLDENS and FILL=nn, so that the eigenvectors at the first iteration are fully known. As FILL is known to give difficulty at times, consider also using C.I.=n and ROOT=m.

FLEPO (O)

The predicted and actual changes in the geometry, the derivatives, and search direction for each geometry optimization cycle are printed. This is useful if there is any question regarding the efficiency of the geometry optimizer.

FMAT

Details of the construction of the Hessian matrix for the force calculation are to be printed.

FORCE (C)

A force-calculation is to be run. The Hessian, that is the matrix (in millidynes per Angstrom) of second derivatives of the energy with respect to displacements of all pairs of atoms in x, y, and z directions, is calculated. On diagonalization this gives the force constants for the molecule. The force matrix, weighted for isotopic masses, is then used for calculating the vibrational frequencies. The system can be characterized as a ground state or a transition state by the presence of five (for a linear system) or six eigenvalues which are very small (less than about 30 reciprocal centimeters). A transition state is further characterized by one, and exactly one, negative force constant.

A FORCE calculation is a prerequisite for a THERMO calculation.

Before a FORCE calculation is started, a check is made to ensure that a stationary point is being used. This check involves calculating the gradient norm (GNORM) and if it is significant, the GNORM will be reduced using BFGS. All internal coordinates are optimized, and any symmetry constraints are ignored at this point. An implication of this is that if the specification of the geometry relies on any angles being exactly 180 or zero degrees, the calculation may fail.

The geometric definition supplied to FORCE should not rely on angles or dihedrals assuming exact values. (The test of exact linearity is sufficiently slack that most molecules that are linear, such as acetylene and but-2-yne, should not be stopped.) See also THERMO, LET, TRANS, ISOTOPE.

In a FORCE calculation, PRECISE will eliminate quartic contamination (part of the anharmonicity). This is normally not important, therefore PRECISE should not routinely be used. In a FORCE calculation, the SCF criterion is automatically made more stringent; this is the main cause of the SCF failing in a FORCE calculation.

GEO-OK (W)

Normally the program will stop with a warning message if two atoms are within 0.8 Angstroms of each other, or, more rarely, the BFGS routine has difficulty optimizing the geometry. GEO-OK will over-ride the job termination sequence, and allow the calculation to proceed. In practice, most jobs that terminate due to these checks contain errors in data, so caution should be exercised if GEO-OK is used. An important exception to this warning is when the system contains, or may give rise to, a Hydrogen molecule. GEO-OK will override other geometric safety checks such as the unstable gradient in a geometry optimization preventing reliable optimization.

   See also the message "GRADIENTS OF OLD GEOMETRY, GNORM= nn.nnnn".

GNORM=n.nn (W)

The geometry optimization termination criteria in both gradient minimization and energy minimization can be over-ridden by specifying a gradient norm requirement. For example, GNORM=20 would allow the geometry optimization to exit as soon as the gradient norm dropped below 20.0, the default being 1.0.

For high-precision work, GNORM=0.0 is recommended. Unless LET is also used, the GNORM will be set to the larger of 0.01 and the specified GNORM. Results from GNORM=0.01 are easily good enough for all high-precision work.

GRADIENTS (O)

In a 1SCF calculation gradients are not calculated by default: in non-variationally optimized systems this would take an excessive time. GRADIENTS allows the gradients to be calculated.

Normally, gradients will not be printed if the gradient norm is less than 2.0. However, if GRA-DIENTS is present, then the gradient norm and the gradients will unconditionally be printed. Abbreviation: GRAD.

GRAPH (O)

Information needed to generate electron density contour maps can be written to a file by calling GRAPH. GRAPH first calls MULLIK in order to generate the inverse-square-root of the overlap matrix, which is required for the re-normalization of the eigenvectors. All data essential for the graphics package DENSITY are then output.

HESS=n (W)

When the Eigenvector Following routine is used for geometry optimization, it frequently works faster if the Hessian is constructed first. If HESS=1 is specified, the Hessian matrix will be constructed before the geometry is optimized. There are other, less common, options, e.g. HESS=2. See comments in subroutine EF for details.

H-PRIORITY (O)

In a DRC calculation, results will be printed whenever the calculated heat of formation changes by 0.1 kcal/mole. Abbreviation: H-PRIO.

H-PRIORITY=n.nn (O)

In a DRC calculation, results will be printed whenever the calculated heat of formation changes by n.nn kcal/mole.

IRC (C)

An Intrinsic Reaction Coordinate calculation is to be run. All kinetic energy is shed at every point in the calculation. See Background.

IRC=n (C)

An Intrinsic Reaction Coordinate calculation to be run; an initial perturbation in the direction of normal coordinate n to be applied. If n is negative, then perturbation is reversed, i.e., initial motion is in the opposite direction to the normal coordinate. This keyword cannot be written with spaces around the ‘=’ sign.

ISOTOPE (O)

Generation of the FORCE matrix is very time-consuming, and in isotopic substitution studies several vibrational calculations may be needed. To allow the frequencies to be calculated from the (constant) force matrix, ISOTOPE is used. When a FORCE calculation is completed, ISOTOPE will cause the force matrix to be stored, regardless of whether or not any intervening restarts have been made. To re-calculate the frequencies, etc. starting at the end of the force matrix calculation, specify RESTART.

The two keywords RESTART and ISOTOPE can be used together. For example, if a normal FORCE calculation runs for a long time, the user may want to divide it up into stages and save the final force matrix. Once ISOTOPE has been used, it does not need to be used on subsequent RESTART runs.

ISOTOPE can also be used with FORCE to set up a RESTART file for an IRC=n calculation.

ITRY=NN (W)

The default maximum number of SCF iterations is 200. When this limit presents difficulty, ITRY=nn can be used to re-define it. For example, if ITRY=400 is used, the maximum number of iterations will be set to 400. ITRY should normally not be changed until all other means of obtaining a SCF have been exhausted, e.g. PULAY CAMP-KING etc.

IUPD=n (W)

IUPD is used only in the EF routine. IUPD should very rarely be touched. IUPD=1 can be used in minimum searches if the message

   "HEREDITARY POSITIVE DEFINITENESS ENDANGERED.  UPDATE SKIPPED THIS CYCLE"

occurs every cycle for 10–20 iterations. Never use IUPD=2 for a TS search! For more information, read the comments in subroutine EF.

K=(n.nn,n) (C)

Used in band-structure calculations, K=(n.nn,n) specifies the step-size in the Brillouin zone, and the number of atoms in the monomeric unit. Two band-structure calculations are supported: electronic and phonon. Both require a polymer to be used. If FORCE is used, a phonon spectrum is assumed, otherwise an electronic band structure is assumed. For both calculations, a density of states is also done. The band structure calculation is very fast, so a small step-size will not use much time.

The output is designed to be fed into a graphics package, and is not ‘elegant’. For polyethylene, a suitable keyword would be K=(0.01,6).

KINETIC=n.nnn (C)

In a DRC calculation n.nnn kcals/mole of excess kinetic energy is added to the system as soon as the kinetic energy builds up to 0.2 kcal/mole. The excess energy is added to the velocity vector, without change of direction.

LARGE (O)

Most of the time the output invoked by keywords is sufficient. LARGE will cause less-commonly wanted, but still useful, output to be printed.

1. To save space, DRC and IRC outputs will, by default, only print the line with the percent sign. Other output can be obtained by use of the keyword LARGE, according to the following rules:

LARGE Print all internal and cartesian coordinates and cartesian velocities.

LARGE=1 Print all internal coordinates.

LARGE=-1 Print all internal and cartesian coordinates and cartesian velocities.

LARGE=n Print every n’th set of internal coordinates.

LARGE=-n Print every n’th set of internal and cartesian coordinates and cartesian velocities.

If LARGE=1 is used, the output will be the same as that of Version 5.0, when LARGE was not used. If LARGE is used, the output will be the same as that of Version 5.0, when LARGE was used. To save disk space, do not use LARGE.

LINMIN (O)

There are two line-minimization routines in MOPAC, an energy minimization and a gradient norm minimization. LINMIN will output details of the line minimization used in a given job.

LET (W)

As MOPAC evolves, the meaning of LET is changing.

Now LET means essentially “I know what I’m doing, override safety checks”.

Currently, LET has the following meanings:

1. In a FORCE calculation, it means that the supplied geometry is to be used, even if the gradients are large.

2. In a geometry optimization, the specified GNORM is to be used, even if it is less than 0.01.

3. In a POLAR calculation, the molecule is to be orientated along its principal moments of inertia before the calculation starts. LET will prevent this step being done.

LOCALIZE (O)

The occupied eigenvectors are transformed into a localized set of M.O.’s by a series of 2 by 2 rotations which maximize ⟨ψ^{4}⟩. The value of 1/⟨ψ^{4}⟩ is a direct measure of the number of centers involved in the MO. Thus the value of 1/⟨ψ^{4}⟩ is 2.0 for H2, 3.0 for a three-center bond and 1.0 for a lone pair. Higher degeneracies than allowed by point group theory are readily obtained. For example, benzene would give rise to a 6-fold degenerate C–H bond, a 6-fold degenerate C–C sigma bond and a three-fold degenerate C–C pi bond. In principle, there is no single step method to unambiguously obtain the most localized set of M.O.’s in systems where several canonical structures are possible, just as no simple method exists for finding the most stable conformer of some large compound. However, the localized bonds generated will normally be quite acceptable for routine applications. Abbreviation: LOCAL.

MAX

In a grid calculation, the maximum number of points (23) in each direction is to be used. The default is 11. The number of points in each direction can be set with POINTS1 and POINTS2.

MECI (O)

At the end of the calculation details of the Multi Electron Configuration Interaction calculation are printed if MECI is specified. The state vectors can be printed by specifying VECTORS. The MECI calculation is either invoked automatically, or explicitly invoked by the use of the C.I.=n keyword.

MICROS=n (C)

The microstates used by MECI are normally generated by use of a permutation operator. When individually defined microstates are desired, then MICROS=n can be used, where n defines the number of microstates to be read in.

Format for Microstates

After the geometry data plus any symmetry data are read in, data defining each microstate is read in, using format 20I1, one microstate per line. The microstate data is preceded by the word “MICROS” on a line by itself. There is at present no mechanism for using MICROS with a reaction path.

For a system with n M.O.’s in the C.I. (use OPEN=(n1,n) or C.I.=n to do this), the populations of the n alpha M.O.’s are defined, followed by the n beta M.O.’s. Allowed occupancies are zero and one. For n=6 the closed-shell ground state would be defined as 111000111000, meaning one electron in each of the first three alpha M.O.’s, and one electron in each of the first three beta M.O.’s.

Users are warned that they are responsible for completing any spin manifolds. Thus while the state 111100110000 is a triplet state with component of spin = 1, the state 111000110100, while having a component of spin = 0 is neither a singlet nor a triplet. In order to complete the spin manifold the microstate 110100111000 must also be included.

If a manifold of spin states is not complete, then the eigenstates of the spin operator will not be quantized. When and only when 100 or fewer microstates are supplied, can spin quantization be conserved.

There are two other limitations on possible microstates. First, the number of electrons in every microstate should be the same. If they differ, a warning message will be printed, and the calculation continued (but the results will almost certainly be nonsense). Second, the component of spin for every microstate must be the same, except for teaching purposes. Two microstates of different components of spin will have a zero matrix element connecting them. No warning will be given as this is a reasonable operation in a teaching situation. For example, if all states arising from two electrons in two levels are to be calculated, say for teaching Russel-Saunders coupling, then the following microstates would be used:

         Microstate       No. of alpha, beta electrons  Ms  State
           1100                    2     0              1   Triplet
           1010                    1     1              0   Singlet
           1001                    1     1              0   Mixed
           0110                    1     1              0   Mixed
           0101                    1     1              0   Singlet
           0011                    0     2             -1   Triplet

Constraints on the space manifold are just as rigorous, but much easier to satisfy. If the energy levels are degenerate, then all components of a manifold of degenerate M.O.’s should be either included or excluded. If only some, but not all, components are used, the required degeneracy of the states will be missing.

As an example, for the tetrahedral methane cation, if the user supplies the microstates corresponding to a component of spin = 3/2, neglecting Jahn-Teller distortion, the minimum number of states that can be supplied is 90 = (6!/(1!5!))(6!/(4!2!)).

While the total number of electrons should be the same for all microstates, this number does not need to be the same as the number of electrons supplied to the C.I.; thus in the example above, a cationic state could be 110000111000.

The format is defined as 20I1 so that spaces can be used for empty M.O.’s.

MINDO/3 (C)

The default Hamiltonian within MOPAC is MNDO, with the alternatives of AM1 and MINDO/3. To use the MINDO/3 Hamiltonian the keyword MINDO/3 should be used. Acceptable alternatives to the keyword MINDO/3 are MINDO and MINDO3.

MMOK (C)

If the system contains a peptide linkage, then MMOK will allow a molecular mechanics correction to be applied so that the barrier to rotation is increased (to 14.00 kcal/mole in N-methyl acetamide).

MODE (C)

MODE is used in the EF routine. Normally the default MODE=1 is used to locate a transition state, but if this is incorrect, explicitly define the vector to be followed by using MODE=n. (MODE is not a recommended keyword). If you use the FORCE option when deciding which mode to follow, set all isotopic masses to 1.0. The normal modes from FORCE are normally mass-weighted; this can mislead. Alternatively, use LARGE with FORCE: this gives the force constants and vectors in addition to the mass-weighted normal modes. Only the mass-weighted modes can be drawn with DRAW.

MS=n

Useful for checking the MECI calculation and for teaching. MS=n overrides the normal choice of magnetic component of spin. Normally, if a triplet is requested, an MS of 1 will be used; this excludes all singlets. If MS=0 is also given, then singlets will also be calculated. The use of MS should not affect the values of the results at all.

MULLIK (O)

A full Mulliken Population analysis is to be done on the final RHF wavefunction. This involves the following steps:

1. The eigenvector matrix is divided by the square root of the overlap matrix, S.

2. The Coulson-type density matrix, P, is formed.

3. The overlap population is formed from P (i, j)S(i, j).

4. Half the off-diagonals are added onto the diagonals.

NLLSQ (C)

The gradient norm is to be minimized by Bartel’s method. This is a Non-Linear Least Squares gradient minimization routine. Gradient minimization will locate one of three possible points:

(a) A minimum in the energy surface. The gradient norm will go to zero, and the lowest five or six eigenvalues resulting from a FORCE calculation will be approximately zero.

(b) A transition state. The gradient norm will vanish, as in (a), but in this case the system is characterized by one, and only one, negative force constant.

(c) A local minimum in the gradient norm space. In this (normally unwanted) case the gradient norm is minimized, but does not go to zero. A FORCE calculation will not give the five or six zero eigenvalues characteristic of a stationary point. While normally undesirable, this is sometimes the only way to obtain a geometry. For instance, if a system is formed which cannot be characterized as an intermediate, and at the same time is not a transition state, but nonetheless has some chemical significance, then that state can be refined using NLLSQ.

NOANCI (W)

RHF open-shell derivatives are normally calculated using Liotard’s analytical C.I. method. If this method is NOT to be used, specify NOANCI (NO ANalytical Configuration Interaction derivatives).

NODIIS (W)

In the event that the G-DIIS option is not wanted, NODIIS can be used. The G-DIIS normally accelerates the geometry optimization, but there is no guarantee that it will do so. If the heat of formation rises unexpectedly (i.e., rises during a geometry optimization while the GNORM is larger than about 0.3), then try NODIIS.

NOINTER (O)

The interatomic distances are printed by default. If you do not want them to be printed, specify NOINTER. For big jobs this reduces the output file considerably.

NOLOG (O)

Normally a copy of the archive file will be directed to the LOG file, along with a synopsis of the job. If this is not wanted, it can be suppressed completely by NOLOG.

NOMM (C)

All four semi-empirical methods underestimate the barrier to rotation of a peptide bond. A Molecular Mechanics correction has been added which increases the barrier in N-methyl acetamide to 14 kcal/mole. If you do not want this correction, specify NOMM (NO Molecular Mechanics).

NONR (W)

Not recommended for normal use. Used with the EF routine. See source code for more details.

NOTHIEL (W)

In a normal geometry optimization using the BFGS routine, Thiel’s FSTMIN technique is used. If normal line-searches are wanted, specify NOTHIEL.

NOXYZ (O)

The cartesian coordinates are printed by default. If you do not want them to be printed, specify NOXYZ. For big jobs this reduces the output file considerably.

NSURF (C)

In an ESP calculation, NSURF=n specifies the number of surface layers for the Connolly surface.

OLDENS (W)

A density matrix produced by an earlier run of MOPAC is to be used to start the current calculation. This can be used in attempts to obtain an SCF when a previous calculation ended successfully but a subsequent run failed to go SCF.

OLDGEO (C)

If multiple geometries are to be run, and the final geometry from one calculation is to be used to start the next calculation, OLDGEO should be specified. Example: If a MNDO, AM1, and PM3 calculation were to be done on one system, for which only a rough geometry was available, then after the MNDO calculation, the AM1 calculation could be done using the optimized MNDO geometry as the starting geometry, by specifying OLDGEO.

OPEN(n1,n2) (C)

The M.O. occupancy during the SCF calculation can be defined in terms of doubly occupied, empty, and fractionally occupied M.O.’s. The fractionally occupied M.O.’s are defined by OPEN(n1,n2), where n1 = number of electrons in the open-shell manifold, and n2 = number of open-shell M.O.’s; n1/n2 must be in the range 0 to 2. OPEN(1,1) will be assumed for odd-electron systems unless an OPEN keyword is used. Errors introduced by use of fractional occupancy are automatically corrected in a MECI calculation when OPEN(n1,n2) is used.

ORIDE (W)

Do not use this keyword until you have read Simons’ article. ORIDE is part of the EF routine, and means “Use whatever Λ’s are produced even if they would normally be ‘unacceptable’.” J. Simons, P. Jorgensen, H. Taylor, J. Ozment, J. Phys. Chem. 87:2745 (1983).

PARASOK (W)

Use this keyword with extreme caution! The AM1 method has been parameterized for only a few elements, less than the number available to MNDO or PM3. If any elements which are not parameterized at the AM1 level are specified, the MNDO parameters, if available, will be used. The resulting mixture of methods, AM1 with MNDO, has not been studied to see how good the results are, and users are strictly on their own as far as accuracy and compatibility with other methods is concerned. In particular, while all parameter sets are referenced in the output, other programs may not cite the parameter sets used and thus compatibility with other MNDO programs is not guaranteed.

PI (O)

The normal density matrix is composed of atomic orbitals, that is s, px, py and pz. PI allows the user to see how each atom-atom interaction is split into σ and π bonds. The resulting “density matrix” is composed of the following basis-functions:- s-σ, p-σ, p-π, d-σ, d-π, d-δ. The on-diagonal terms give the hybridization state, so that an sp^{2} hybridized system would be represented as s-σ: 1.0, p-σ: 2.0, p-π: 1.0.

PM3 (C)

The PM3 method is to be used.

POINT=n (C)

The number of points to be calculated on a reaction path is specified by POINT=n. Used only with STEP in a path calculation.

POINT1=n (C)

In a grid calculation, the number of points to be calculated in the first direction is given by POINT1=n. ‘n’ should be less than 24; default: 11.

POINT2=n (C)

In a grid calculation, the number of points to be calculated in the second direction is given by POINT2=n. ‘n’ should be less than 24, default: 11;

POTWRT (W)

In an ESP calculation, write out surface points and electrostatic potential values to UNIT 21.

POLAR (C)

The polarizability and first and second hyperpolarizabilities are to be calculated. At present this calculation does not work for polymers, but should work for all other systems. Two different options are implemented: the older finite field method and a new time-dependent Hartree-Fock method.

Time-Dependent Hartree-Fock

This procedure is based on the detailed description given by M. Dupuis and S. Karna (J. Comp. Chem. 12, 487 (1991)). The program is capable of calculating:

Frequency Dependent Polarizability  alpha(-w;w)
Second Harmonic Generation  beta(-2w;w,w)
Electrooptic Pockels Effect  beta(-w;0,w)
Optical Rectification  beta(0;-w,w)
Third Harmonic Generation  gamma(-3w;w,w,w)
DC-EFISH  gamma(-2w;0,w,w)
Optical Kerr Effect  gamma(-w;0,0,w)
Intensity Dependent Index of Refraction gamma(-w;w,-w,w)

The input is given at the end of the MOPAC deck and consists of two lines of free-field input followed by a list energies. The variables on the first line are:

Nfreq = How many energies will be used to calculate

the desired quantities.

Iwflb = Type of beta calculation to be performed.

This valiable is only important if iterative beta calculations are chosen. 0 - static 1 - SHG 2 - EOPE 3 - OR

Ibet = Type of beta calculation:

0 - beta(0;0) static 1 - iterative calculation with type of beta chosen by Iwflb. 1 - Noniterative calculation of SHG -2 - Noniterative calculation of EOPE -3 - Noniterative calculation of OR

Igam = Type of gamma calculation:

0 - No gamma calculation 1 - THG 2 - DC-EFISH 3 - IDRI 4 - OKE

The vaiables on the second line are:

Atol =  Cutoff tolerance for alpha calculations

(1.0e-4 seems reasonable)

Maxitu = Maximum number of iteractions for beta

calculations

Maxita = Maximum number of iterations for alpha

calculations

Btol =  Cutoff tolerance for beta calculations

Nfreq lines follow, each with an energy value in eV’s at which the hyperpolarizabilites are to be calculated.

POWSQ (C)

Details of the working of POWSQ are printed out. This is only useful in debugging.

PRECISE (W)

The criteria for terminating all optimizations, electronic and geometric, are to be increased by a factor, normally, 100. This can be used where more precise results are wanted. If the results are going to be used in a FORCE calculation, where the geometry needs to be known quite precisely, then PRECISE is recommended; for small systems the extra cost in CPU time is minimal. PRECISE is not recommended for experienced users, instead GNORM=n.nn and SCFCRT=n.nn are suggested. PRECISE should only very rarely be necessary in a FORCE calculation: all it does is remove quartic contamination, which only affects the trivial modes significantly, and is very expensive in CPU time.

PULAY (W)

The default converger in the SCF calculation is to be replaced by Pulay’s procedure as soon as the density matrix is sufficiently stable. A considerable improvement in speed can be achieved by the use of PULAY. If a large number of SCF calculations are envisaged, a sample calculation using 1SCF and PULAY should be compared with using 1SCF on its own, and if a saving in time results, then PULAY should be used in the full calculation. PULAY should be used with care in that its use will prevent the combined package of convergers (SHIFT, PULAY and the CAMP-KING convergers) from automatically being used in the event that the system fails to go SCF in (ITRY-10) iterations.

The combined set of convergers very seldom fails.

QUARTET (C)

RHF interpretation: The desired spin-state is a quartet, i.e., the state with component of spin = 1/2 and spin = 3/2. When a configuration interaction calculation is done, all spin states of spin equal to, or greater than 1/2 are calculated simultaneously, for component of spin = 1/2. From these states the quartet states are selected when QUARTET is specified, and all other spin states, while calculated, are ignored in the choice of root to be used. If QUARTET is used on its own, then a single state, corresponding to an alpha electron in each of three M.O.’s is calculated.

UHF interpretation: The system will have three more alpha electrons than beta electrons.

QUINTET (C)

RHF interpretation: The desired spin-state is a quintet, that is, the state with component of spin = 0 and spin = 2. When a configuration interaction calculation is done, all spin states of spin equal to, or greater than 0 are calculated simultaneously, for component of spin = 0. From these states the quintet states are selected when QUINTET is specified, and the septet states, while calculated, will be ignored in the choice of root to be used. If QUINTET is used on its own, then a single state, corresponding to an alpha electron in each of four M.O.’s is calculated.

UHF interpretation: The system will have three more alpha electrons than beta electrons.

RECALC=n

RECALC=n calculates the Hessian every n steps in the EF optimization. For small n this is costly but is also very effective in terms of convergence. RECALC=10 and DMAX=0.10 can be useful for difficult cases. In extreme cases RECALC=1 and DMAX=0.05 will always find a stationary point, if it exists.

RESTART (W)

When a job has been stopped, for whatever reason, and intermediate results have been stored, then the calculation can be restarted at the point where it stopped by specifying RESTART. The most common cause of a job stopping before completion is its exceeding the time allocated. A saddle-point calculation has no restart, but the output file contains information which can easily be used to start the calculation from a point near to where it stopped.

It is not necessary to change the geometric data to reflect the new geometry. As a result, the geometry printed at the start of a restarted job will be that of the original data, not that of the restarted file. A convenient way to monitor a long run is to specify 1SCF and RESTART; this will give a normal output file at very little cost.

Note 1: In the FORCE calculation two restarts are possible. These are (a) a restart in FLEPO if the geometry was not optimized fully before FORCE was called, and (b) the normal restart in the construction of the force matrix. If the restart is in FLEPO within FORCE then the keyword FORCE should be deleted, and the keyword RESTART used on its own. Forgetting this point is a frequent cause of failed jobs.

Note 2: Two restarts also exist in the IRC calculation. If an IRC calculation stops while in the FORCE calculation, then a normal restart can be done. If the job stops while doing the IRC calculation itself then the keyword IRC=n should be changed to IRC, or it can be omitted if DRC is also specified. The absence of the string ”IRC=” is used to indicate that the FORCE calculation was completed before the restart files were written.

ROOT=n (C)

The n’th root of a C.I. calculation is to be used in the calculation. If a keyword specifying the spin-state is also present, e.g. SINGLET or TRIPLET, then the n’th root of that state will be selected. Thus ROOT=3 and SINGLET will select the third singlet root. If ROOT=3 is used on its own, then the third root will be used, which may be a triplet, the third singlet, or the second singlet (the second root might be a triplet). In normal use, this keyword would not be used. It is retained for educational and research purposes. Unusual care should be exercised when ROOT= is specified.

ROT=n (C)

In the calculation of the rotational contributions to the thermodynamic quantities the symmetry number of the molecule must be supplied. The symmetry number of a point group is the number of equivalent positions attainable by pure rotations. No reflections or improper rotations are allowed. This number cannot be assumed by default, and may be affected by subtle modifications to the molecule, such as isotopic substitution. A list of the most important symmetry numbers follows:

            ----    TABLE OF SYMMETRY NUMBERS    ----

       C1 CI CS     1      D2 D2D D2H  4       C(INF)V   1
       C2 C2V C2H   2      D3 D3D D3H  6       D(INF)H   2
       C3 C3V C3H   3      D4 D4D D4H  8       T TD     12
       C4 C4V C4H   4      D6 D6D D6H  12      OH       24
       C6 C6V C6H   6      S6          3

SADDLE (C)

The transition state in a simple chemical reaction is to be optimized. Extra data are required. After the first geometry, specifying the reactants, and any symmetry functions have been defined, the second geometry, specifying the products, is defined, using the same format as that of the first geometry.

SADDLE often fails to work successfully. Frequently this is due to equivalent dihedral angles in the reactant and product differing by about 360 degrees rather than zero degrees. As the choice of dihedral can be difficult, users should consider running this calculation with the keyword XYZ. There is normally no ambiguity in the definition of cartesian coordinates. See also BAR=.

Many of the bugs in SADDLE have been removed in this version. Use of the XYZ option is strongly recommended.

SCALE (C)

SCALE=n.n specifies the scaling factor for Van der Waals’ radii for the initial layer of the Connolly surface in the ESP calculation.

SCFCRT=n.nn (W)

The default SCF criterion is to be replaced by that defined by SCFCRT=.

The SCF criterion is the change in energy in kcal/mol on two successive iterations. Other minor criteria may make the requirements for an SCF slightly more stringent. The SCF criterion can be varied from about 0.001 to 1.D-25, although numbers in the range 0.0001 to 1.D-9 will suffice for most applications.

An overly tight criterion can lead to failure to achieve a SCF, and consequent failure of the run.

SCINCR=n.nn

In an ESP calculation, SCINCR=n.nn specifies the increment between layers of the surface in the Connolly surface. (default: 0.20)

SETUP (C)

If, on the keyword line, the word ‘SETUP’ is specified, then one or two lines of keywords will be read from a file with the logical name SETUP. The logical file SETUP must exist, and must contain at least one line. If the second line is defined by the first line as a keyword line, and the second line contains the word SETUP, then one line of keywords will be read from a file with the logical name SETUP.

SETUP=name (C)

Same as SETUP, only the logical or actual name of the SETUP file is ‘name’.

SEXTET (C)

RHF interpretation: The desired spin-state is a sextet: the state with component of spin = 1/2 and spin = 5/2.

The sextet states are the highest spin states normally calculable using MOPAC in its unmodified form. If SEXTET is used on its own, then a single state, corresponding to one alpha electron in each of five M.O.’s, is calculated. If several sextets are to be calculated, say the second or third, then OPEN(n1,n2) should be used.

UHF interpretation: The system will have five more alpha electrons than beta electrons.

SHIFT=n.nn (W)

In an attempt to obtain an SCF by damping oscillations which slow down the convergence or prevent an SCF being achieved, the virtual M.O. energy levels are shifted up or down in energy by a shift technique. The principle is that if the virtual M.O.’s are changed in energy relative to the occupied set, then the polarizability of the occupied M.O.’s will change pro rata. Normally, oscillations are due to autoregenerative charge fluctuations.

The SHIFT method has been re-written so that the value of SHIFT changes automatically to give a critically-damped system. This can result in a positive or negative shift of the virtual M.O. energy levels. If a non-zero SHIFT is specified, it will be used to start the SHIFT technique, rather than the default 15eV. If SHIFT=0 is specified, the SHIFT technique will not be used unless normal convergence techniques fail and the automatic “ALL CONVERGERS … ” message is produced.

SIGMA (C)

The McIver-Komornicki gradient norm minimization routines, POWSQ and SEARCH are to be used. These are very rapid routines, but do not work for all species. If the gradient norm is low, i.e., less than about 5 units, then SIGMA will probably work; in most cases, NLLSQ is recommended. SIGMA first calculates a quite accurate Hessian matrix, a slow step, then works out the direction of fastest decent, and searches along that direction until the gradient norm is minimized. The Hessian is then partially updated in light of the new gradients, and a fresh search direction found. Clearly, if the Hessian changes markedly as a result of the line-search, the update done will be inaccurate, and the new search direction will be faulty.

SIGMA should be avoided if at all possible when non-variationally optimized calculations are being done.

If the Hessian is suspected to be corrupt within SIGMA it will be automatically recalculated. This frequently speeds up the rate at which the transition state is located. If you do not want the Hessian to be reinitialized — it is costly in CPU time — specify LET on the keyword line.

SINGLET (C)

When a configuration interaction calculation is done, all spin states are calculated simultaneously, either for component of spin = 0 or 1/2. When only singlet states are of interest, then SINGLET can be specified, and all other spin states, while calculated, are ignored in the choice of root to be used.

Note that while almost every even-electron system will have a singlet ground state, SINGLET should still be specified if the desired state must be a singlet.

SINGLET has no meaning in a UHF calculation, but see also TRIPLET.

SLOPE (C)

In an ESP calculation, SLOPE=n.nn specifies the scale factor for MNDO charges. (default=1.422)

SPIN (O)

The spin matrix, defined as the difference between the alpha and beta density matrices, is to be printed. If the system has a closed-shell ground state, e.g. methane run UHF, the spin matrix will be null.

If SPIN is not requested in a UHF calculation, then the diagonal of the spin matrix, that is the spin density on the atomic orbitals, will be printed.

STEP (C)

In a reaction path, if the path step is constant, STEP can be used instead of explicitly specifying each point. The number of steps is given by POINT. If the reaction coordinate is an interatomic distance, only positive STEPs are allowed.

STEP1=n.nnn (C)

In a grid calculation the step size in degrees or Angstroms for the first of the two parameters is given by n.nnn. By default, an 11 by 11 grid is generated. See POINT1 and POINT2 on how to adjust this number. The first point calculated is the supplied geometry, and is in the upper left hand corner. This is a change from Version 5.00, where the supplied geometry was the central point.

STEP2=n.nnn (C)

In a grid calculation the step size in degrees or Angstroms for the second of the two parameters is given by n.nnn.

STO3G (W)

In an ESP calculation STO3G means “Use the STO-3G basis set to de-orthogonalize the semiempirical orbitals”.

SYMAVG (W)

Used by the ESP, SYMAVG will average charges which should have the same value by symmetry.

SYMMETRY (C)

Symmetry data defining related bond lengths, angles and dihedrals can be included by supplying additional data after the geometry has been entered. If there are any other data, such as values for the reaction coordinates, or a second geometry, as required by SADDLE, then it would follow the symmetry data. Symmetry data are terminated by one blank line. For non-variationally optimized systems symmetry constraints can save a lot of time because many derivatives do not need to be calculated. At the same time, there is a risk that the geometry may be wrongly specified, e.g. if methane radical cation is defined as being tetrahedral, no indication that this is faulty will be given until a FORCE calculation is run. (This system undergoes spontaneous Jahn-Teller distortion.)

Usually a lower heat of formation can be obtained when SYMMETRY is specified. To see why, consider the geometry of benzene. If no assumptions are made regarding the geometry, then all the C–C bond lengths will be very slightly different, and the angles will be almost, but not quite 120 degrees. Fixing all angles at 120 degrees, dihedrals at 180 or 0 degrees, and only optimizing one C–C and one C–H bond-length will result in a 2–D optimization, and exact D_{6h} symmetry. Any deformation from this symmetry must involve error, so by imposing symmetry some error is removed.

The layout of the symmetry data is:

   <defining atom> <symmetry relation> <defined atom> <defined atom>,...

where the numerical code for <symmetry relation> is given in the table of symmetry functions below.

For example, ethane, with three independent variables, can be defined as:

     SYMMETRY
     ETHANE, D3D                                   NA NB NC
      C
      C    1.528853 1                               1
      H    1.105161 1  110.240079 1                 2  1
      H    1.105161 0  110.240079 0  120.000000 0   2  1  3
      H    1.105161 0  110.240079 0  240.000000 0   2  1  3
      H    1.105161 0  110.240079 0   60.000000 0   1  2  3
      H    1.105161 0  110.240079 0  180.000000 0   1  2  3
      H    1.105161 0  110.240079 0  300.000000 0   1  2  3
      0    0.000000 0    0.000000 0    0.000000 0   0  0  0
       3,    1,    4,    5,     6,     7,     8,
       3,    2,    4,    5,     6,     7,     8,

Here atom 3, a hydrogen, is used to define the bond lengths (symmetry relation 1) of atoms 4,5,6,7 and 8 with the atoms they are specified to bond with in the NA column of the data file; similarly, its angle (symmetry relation 2) is used to define the bond-angle of atoms 4,5,6,7 and 8 with the two atoms specified in the NA and NB columns of the data file. The other angles are point-group symmetry defined as a multiple of 60 degrees.

Spaces, tabs or commas can be used to separate data. Note that only three parameters are marked to be optimized. The symmetry data can be the last line of the data file unless more data follows, in which case a blank line must be inserted after the symmetry data.

The full list of available symmetry relations is as follows:

<Symmetry                  SYMMETRY FUNCTIONS
 relation>
   1     BOND LENGTH    IS SET EQUAL TO THE REFERENCE BOND LENGTH
   2     BOND ANGLE     IS SET EQUAL TO THE REFERENCE BOND ANGLE
   3     DIHEDRAL ANGLE IS SET EQUAL TO THE REFERENCE DIHEDRAL ANGLE
   4     DIHEDRAL ANGLE VARIES AS  90 DEGREES - REFERENCE DIHEDRAL
   5     DIHEDRAL ANGLE VARIES AS  90 DEGREES + REFERENCE DIHEDRAL
   6     DIHEDRAL ANGLE VARIES AS 120 DEGREES - REFERENCE DIHEDRAL
   7     DIHEDRAL ANGLE VARIES AS 120 DEGREES + REFERENCE DIHEDRAL
   8     DIHEDRAL ANGLE VARIES AS 180 DEGREES - REFERENCE DIHEDRAL
   9     DIHEDRAL ANGLE VARIES AS 180 DEGREES + REFERENCE DIHEDRAL
  10     DIHEDRAL ANGLE VARIES AS 240 DEGREES - REFERENCE DIHEDRAL
  11     DIHEDRAL ANGLE VARIES AS 240 DEGREES + REFERENCE DIHEDRAL
  12     DIHEDRAL ANGLE VARIES AS 270 DEGREES - REFERENCE DIHEDRAL
  13     DIHEDRAL ANGLE VARIES AS 270 DEGREES + REFERENCE DIHEDRAL
  14     DIHEDRAL ANGLE VARIES AS THE NEGATIVE OF THE REFERENCE
         DIHEDRAL
  15     BOND LENGTH VARIES AS HALF THE REFERENCE BOND LENGTH
  16     BOND ANGLE VARIES AS HALF THE REFERENCE BOND ANGLE
  17     BOND ANGLE VARIES AS 180 DEGREES - REFERENCE BOND ANGLE
  18     BOND LENGTH IS A MULTIPLE OF REFERENCE BOND-LENGTH

Function 18 is intended for use in polymers, in which the translation vector may be a multiple of some bond-length. 1,2,3 and 14 are most commonly used. Abbreviation: SYM.

SYMMETRY is not available for use with cartesian coordinates.

T= (W)

This is a facility to allow the program to shut down in an orderly manner on computers with execution time cpu limits.

The total cpu time allowed for the current job is limited to nn.nn seconds; by default this is one hour, i.e., 3600 seconds. If the next cycle of the calculation cannot be completed without running a risk of exceeding the assigned time the calculation will write a restart file and then stop. The safety margin is 100 percent; that is, to do another cycle, enough time to do at least two full cycles must remain.

Alternative specifications of the time are T=nn.nnM, this defines the time in minutes, T=nn.nnH, in hours, and T=nn.nnD, in days, for very long jobs. This keyword cannot be written with spaces around the ‘=’ sign.

THERMO (O)

The thermodynamic quantities, internal energy, heat capacity, partition function, and entropy can be calculated for translation, rotation and vibrational degrees of freedom for a single temperature, or a range of temperatures. Special situations such as linear systems and transition states are accommodated. The approximations used in the THERMO calculation are invalid below 100K, and checking of the lower bound of the temperature range is done to prevent temperatures of less than 100K being used.

Another limitation, for which no checking is done, is that there should be no internal rotations. If any exist, they will not be recognized as such, and the calculated quantities will be too low as a result.

In order to use THERMO the keyword FORCE must also be specified, as well as the value for the symmetry number; this is given by ROT=n.

If THERMO is specified on its own, then the default values of the temperature range are assumed. This starts at 200K and increases in steps of 10 degrees to 400K. Three options exist for overriding the default temperature range. These are:

THERMO(nnn) (O)

The thermodynamic quantities for a 200 degree range of temperatures, starting at nnnK and with an interval of 10 degrees are to be calculated.

THERMO(nnn,mmm) (O)

The thermodynamic quantities for the temperature range limited by a lower bound of nnn Kelvin and an upper bound of mmm Kelvin, the step size being calculated in order to give approximately 20 points, and a reasonable value for the step. The size of the step in Kelvin degrees will be 1, 2, or 5, or a power of 10 times these numbers.

THERMO(nnn,mmm,lll) (O)

Same as for THERMO(nnn,mmm), only now the user can explicitly define the step size. The step size cannot be less than 1K.

T-PRIORITY (O)

In a DRC calculation, results will be printed whenever the calculated time changes by 0.1 femtoseconds. Abbreviation, T-PRIO.

T-PRIORITY=n.nn (O)

In a DRC calculation, results will be printed whenever the calculated time changes by n.nn femtoseconds.

TRANS (C)

The imaginary frequency due to the reaction vector in a transition state calculation must not be included in the thermochemical calculation. The number of genuine vibrations considered can be: 3N − 5 for a linear ground state system, 3N − 6 for a non-linear ground state system, or 3N − 6 for a linear transition-state complex, 3N − 7 for a non-linear transition-state complex.

This keyword must be used in conjunction with THERMO if a transition state is being calculated.

TRANS=n (C)

The facility exists to allow the THERMO calculation to handle systems with internal rotations. TRANS=n will remove the n lowest vibrations. Note that TRANS=1 is equivalent to TRANS on its own. For xylene, for example, TRANS=2 would be suitable.

This keyword cannot be written with spaces around the ‘=’ sign.

TRIPLET (C)

The triplet state is defined. If the system has an odd number of electrons, an error message will be printed.

UHF interpretation

The number of alpha electrons exceeds that of the beta electrons by 2. If TRIPLET is not specified, then the numbers of alpha and beta electrons are set equal. This does not necessarily correspond to a singlet.

RHF interpretation

An RHF MECI calculation is performed to calculate the triplet state. If no other C.I. keywords are used, then only one state is calculated by default. The occupancy of the M.O.’s in the SCF calculation is defined as (… 2,1,1,0,…), that is, one electron is put in each of the two highest occupied M.O.’s.

See keywords C.I.=n and OPEN(n1,n2).

TS (C)

Within the Eigenvector Following routine, the option exists to optimize a transition state. To do this, use TS. Preliminary indications are that the TS method is much faster and more reliable than either SIGMA or NLLSQ.

TS appears to work well with cartesian coordinates.

In the event that TS does not converge on a stationary point, try adding RECALC=5 to the keyword line.

UHF (C)

The unrestricted Hartree-Fock Hamiltonian is to be used.

VECTORS (O)

The eigenvectors are to be printed. In UHF calculations both alpha and beta eigenvectors are printed; in all cases the full set, occupied and virtual, are output. The eigenvectors are normalized to unity, that is the sum of the squares of the coefficients is exactly one. If DEBUG is specified, then ALL eigenvectors on every iteration of every SCF calculation will be printed. This is useful in a learning context, but would normally be very undesirable.

VELOCITY (C)

The user can supply the initial velocity vector to start a DRC calculation. Limitations have to be imposed on the geometry in order for this keyword to work. These are (a) the input geometry must be in cartesian coordinates, (b) the first three atoms must not be coaxial, (c) triatomic systems are not allowed (See geometry specification - triatomic systems are in internal coordinates, by definition.)

Put the velocity vector after the geometry as three data per line, representing the x, y, and z components of velocity for each atom. The units of velocity are centimeters per second.

The velocity vector will be rotated so as to suit the final cartesian coordinate orientation of the molecule.

If KINETIC=n.n is also specified, the velocity vector will be scaled to equal the velocity corresponding to n.n kcal/mole. This allows the user to define the direction of the velocity vector; the magnitude is given by KINETIC=n.n.

WILLIAMS (C)

Within the ESP calculation, the Connolly surface is used as the default. If the surface generation procedure of Donald Williams is wanted, the keyword WILLIAMS should be used.

X-PRIORITY (O)

In a DRC calculation, results will be printed whenever the calculated geometry changes by 0.05 Å. The geometry change is defined as the linear sum of the translation vectors of motion for all atoms in the system. Abbreviation, X-PRIO.

X-PRIORITY=n.nn (O)

In a DRC calculation, results will be printed whenever the calculated geometry changes by n.nn Å.

XYZ (W)

The SADDLE calculation quite often fails due to faulty definition of the second geometry because the dihedrals give a lot of difficulty. To make this option easier to use, XYZ was developed. A calculation using XYZ runs entirely in cartesian coordinates, thus eliminating the problems associated with dihedrals. The connectivity of the two systems can be different, but the numbering must be the same. Dummy atoms can be used; these will be removed at the start of the run. A new numbering system will be generated by the program, when necessary.

XYZ is also useful for removing dummy atoms from an internal coordinate file; use XYZ and 0SCF.

If a large ring system is being optimized, sometimes the closure is difficult, in which case XYZ will normally work.

Except for SADDLE, do not use XYZ by default: use it only when something goes wrong! In order for XYZ to be used, the supplied geometry must either be in cartesian coordinates

or, if internal coordinates are used, symmetry must not be used, and all coordinates must be flagged for optimization. If dummy atoms are present, only 3N-6 coordinates need to be flagged for optimization.

If at all possible, the first 3 atoms should be real. Except in SADDLE, XYZ will still work if one or more dummy atoms occur before the fourth real atom, in which case more than 3N-6 coordinates will be flagged for optimization. This could cause difficulties with the EF method, which is why dummy atoms at the start of the geometry specification should be avoided. The coordinates to be optimized depend on the internal coordinate definition of real atoms 1, 2, and 3. If the position of any of these atoms depends on dummy atoms, then the optimization flags will be different from the case where the first three atoms defined are all real. The geometry is first converted to cartesian coordinates and dummy atoms excluded. The cartesian coordinates to be optimized are:

    Atoms  R R R  R R X  R X R  X R R  R X X  X R X  X X R  X X X
           X Y Z  X Y Z  X Y Z  X Y Z  X Y Z  X Y Z  X Y Z  X Y Z
   Atom 1
        2  +      +      + +    + +    + + +  + +    + + +  + + +
        3  + +    + + +  + + +  + + +  + + +  + + +  + + +  + + +
     4 on  + + +  + + +  + + +  + + +  + + +  + + +  + + +  + + +

Where R and X apply to real and dummy atoms in the internal coordinate Z-matrix, and atoms 1, 2, 3, and 4 are the real atoms in cartesian coordinates. A ‘+’ means that the relevant coordinate is flagged for optimization. Note that the number of flagged coordinates varies from 3N − 6 to 3N − 3, atom 1 is never optimized.

2.4 Keywords that go together

2.4 Keywords that go together

Normally only a subset of keywords are used in any given piece of research. Keywords which are related to each other in this way are:

1. In getting an SCF: SHIFT, PULAY, ITRY, CAMP, SCFCRT, 1SCF, PL.

2. In C.I. work: SINGLET, DOUBLET, etc., OPEN(n,m), C.I.=(n,m), LARGE, MECI, MS=n, VECTORS, ESR, ROOT=n, MICROS.

3. In excited states: UHF with (TRIPLET, QUARTET, etc.), C.I.=n, C.I.=(n,m).

4. In geometry optimization:

(a) Using BFGS: GNORM=n.n, XYZ, PRECISE.

(b) Using EF: GNORM=n.n, XYZ, PRECISE

(c) Using NLLSQ: GNORM=n.n, XYZ, PRECISE

(d) Using SIGMA: GNORM=n.n, XYZ, PRECISE

5. In Gaussian work: AIGIN, AIGOUT, AIDER.

6. In SADDLE: XYZ, BAR=n.n Keywords

Geometry specification

FORMAT: The geometry is read in using essentially “Free-Format” of FORTRAN-77. In fact, a character input is used in order to accommodate the chemical symbols, but the numeric data can be regarded as “free-format”.indexdata!free-format This means that integers and real numbers can be interspersed, numbers can be separated by one or more spaces, a tab and/or by one comma. If a number is not specified, its value is set to zero.

The geometry can be defined in terms of either internal or cartesian coordinates.

3.1 Internal coordinate definition

For any one atom (i) this consists of an interatomic distance in Angstroms from an already-defined atom (j), an interatomic angle in degrees between atoms i and j and an already defined k, (k and j must be different atoms), and finally a torsional angle in degrees between atoms i, j, k, and an already defined atom l (l cannot be the same as k or j). See also dihedral angle coherency.

Exceptions:

1. Atom 1 has no coordinates at all: this is the origin.

2. Atom 2 must be connected to atom 1 by an interatomic distance only.

3. Atom 3 can be connected to atom 1 or 2, and must make an angle with atom 2 or 1 (thus 3–2–1 or 3–1–2); no dihedral is possible for atom 3. By default, atom 3 is connected to atom 2.

3.1.1 Constraints

1. Interatomic distances must be greater than zero. Zero Angstroms is acceptable only if the parameter is symmetry-related to another atom, and is the dependent function.

2. Angles must be in the range 0.0 to 180.0, inclusive. This constraint is for the benefit of the user only; negative angles are the result of errors in the construction of the geometry, and angles greater than 180 degrees are fruitful sources of errors in the dihedrals.

3. Dihedrals angles must be definable. If atom i makes a dihedral with atoms j, k, and l, and the three atoms j, k, and l are in a straight line, then the dihedral has no definable angle. During the calculation this constraint is checked continuously, and if atoms j, k, and l lie within 0.02 Angstroms of a straight line, the calculation will output an error message and then stop. Two exceptions to this constraint are:

(a) if the angle is zero or 180 degrees, in which case the dihedral is not used.

(b) if atoms j, k, and l lie in an exactly straight line (usually the result of a symmetry constraint), as in acetylene, acetonitrile, but-2-yne, etc.

Geometry specification

If the exceptions are used, care must be taken to ensure that the program does not violate these constraints during any optimizations or during any calculations of derivatives - see also FORCE.

Conversion to Cartesian Coordinates

By definition, atom 1 is at the origin of cartesian coordinate space—be careful, however, if atom 1 is a dummy atom. Atom 2 is defined as lying on the positive X axis — for atom 2, Y=0 and Z=0. Atom 3 is in the X-Y plane unless the angle 3–2–1 is exactly 0 or 180 degrees. Atom 4, 5, 6, etc. can lie anywhere in 3-D space.

3.2 Gaussian Z-matrices

With certain limitations, geometries can now be specified within MOPAC using the Gaussian Z-matrix format.

Exceptions to the full Gaussian standard

1. The option of defining an atom’s position by one distance and two angles is not allowed. In other words, the N4 variable described in the Gaussian manual must either be zero or not specified. MOPAC requires the geometry of atoms to be defined in terms of, at most, one distance, one angle, and one dihedral.

2. Gaussian cartesian coordinates are not supported.

3. Chemical symbols must not be followed by an integer identifying the atom. Numbers after a symbol are used by MOPAC to indicate isotopic mass. If labels are desired, they should be enclosed in parentheses, thus "Cl(on C5)34.96885".

4. The connectivity (N1, N2, N3) must be integers. Labels are not allowed.

Specification of Gaussian Z-matrices

The information contained in the Gaussian Z-matrix is identical to that in a MOPAC Z-matrix. The order of presentation is different. Atom N, (real or dummy) is specified in the format:

 Element   N1   Length  N2  Alpha   N3  Beta

where Element is the same as for the MOPAC Z-matrix. N1, N2, and N3 are the connectivity, the same as the MOPAC Z-matrix NA, NB, and NC: bond lengths are between N and N1, angles are between N, N1 and N2, and dihedrals are between N, N1, N2, and N3. The same rules apply to N1, N2, and N3 as to NA, NB, and NC.

Length, Alpha, and Beta are the bond lengths, the angle, and dihedral. They can be ‘real’, e.g. 1.45, 109.4, 180.0, or ‘symbolic’. A symbolic is an alphanumeric string of up to 8 characters, e.g. R51, A512, D5213, CH, CHO, CHOC, etc. Two or more symbolics can be the same. Dihedral symbolics can optionally be preceeded by a minus sign, in which case the value of the dihedral is the negative of the value of the symbolic. This is the equivalent of the normal MOPAC SYMMETRY operations 1, 2, 3, and 14.

If an internal coordinate is real, it will not be optimized. This is the equivalent of the MOPAC optimization flag “0”. If an internal coordinate is symbolic, it can be optimized.

The Z-matrix is terminated by a blank line, after which comes the starting values of the symbolics, one per line. If there is a blank line in this set, then all symbolics after the blank line are considered fixed; that is, they will not be optimized. The set before the blank line will be optimized.

Example of Gaussian Z-matrix geometry specification 3.3 Cartesian coordinate definition

 Line 1    AM1
 Line 2  Ethane
 Line 3
 Line 4    C
 Line 5    C     1     r21
 Line 6    H     2     r32       1     a321
 Line 7    H     2     r32       1     a321      3  d4213
 Line 8    H     2     r32       1     a321      3 -d4213
 Line 9    H     1     r32       2     a321      3   60.
 Line 10   H     1     r32       2     a321      3  180.
 Line 11   H     1     r32       2     a321      3  d300
 Line 12
 Line 13      r21        1.5
 Line 14      r32        1.1
 Line 15      a321     109.5
 Line 16      d4313    120.0
 Line 17
 Line 18      d300     300.0
 Line 19

3.3 Cartesian coordinate definition

A definition of geometry in cartesian coordinates consists of the chemical symbol or atomic number, followed by the cartesian coordinates and optimization flags but no connectivity.

MOPAC uses the lack of connectivity to indicate that cartesian coordinates are to be used. A unique case is the triatomics for which only internal coordinates are allowed. This is to avoid conflict of definitions: the user does not need to define the connectivity of atom 2, and can elect to use the default connectivity for atom 3. As a result, a triatomic may have no explicit connectivity defined, the user thus taking advantage of the default connectivity. Since internal coordinates are more commonly used than cartesian, the above choice was made.

If the keyword XYZ is absent every coordinate must be marked for optimization. If any coordinates are not to be optimized, the keyword XYZ must be present. The coordinates of all atoms, including atoms 1, 2 and 3 can be optimized. Dummy atoms should not be used, for obvious reasons.

3.4 Conversion between various formats

MOPAC can accept any of the following formats: cartesian, MOPAC internal coordinates, and Gaussian internal coordinates. Both MOPAC and Gaussian Z-matrices can also contain dummy atoms. Internally, MOPAC works with either a cartesian coordinate set (if XYZ is specified) or internal coordinates (the default). If the 0SCF option is requested, the geometry defined on input will be printed in MOPAC Z-matrix format, along with other optional formats.

The type(s) of geometry printed at the end of a 0SCF calculation depend only on the keywords XYZ, AIGOUT, and NOXYZ. The geometry printed is independent of the type of input geometry, and therefore makes a convenient conversion mechanism.

If XYZ is present, all dummy atoms are removed and the internal coordinate definition remade. All symmetry relations are lost if XYZ is used.

If NOXYZ is present, cartesian coordinates will not be printed.

If AIGOUT is present, a data set using Gaussian Z-matrix format is printed.

Note: (1) Only if the keyword XYZ is absent and the keyword SYMMETRY present in a MOPAC internal coordinate geometry, or two or more internal coordinates in a Gaussian Zmatrix have the same symbolic will symmetry be present in the MOPAC or Gaussian geometries output. (2) This expanded use of 0SCF replaces the program DDUM, supplied with earlier copies of MOPAC.

3.5 Definition of elements and isotopes

Elements are defined in terms of their atomic numbers or their chemical symbols, case insensitive. Thus, chlorine could be specified as 17, or Cl. In Version 6, only main-group elements and transition metals for which the ‘d’ shell is full are available.

Acceptable symbols for MNDO are:

         Elements                      Dummy atom, sparkles and
                                         Translation Vector
   H
  Li  *          B  C  N  O  F
  Na' *         Al Si  P  S Cl           +                       o
   K' * ...  Zn  * Ge  *  * Br         XX  Cb  ++   +  --   -  Tv
  Rb' * ...   *  * Sn  *  *  I         99 102 103 104 105 106 107
  *   * ...  Hg  * Pb  *
'  These symbols refer to elements which lack a basis set.
+  This is the dummy atom for assisting with geometry specification.
*  Element not parameterized.
o  This is the translation vector for use with polymers.

Old parameters for some elements are available. These are provided to allow compatibility with earlier copies of MOPAC. To use these older parameters, use a keyword composed of the chemical symbol followed by the year of publication of the parameters. Keywords currently available: Si1978, S1978.

For AM1, acceptable symbols are:

       Elements                      Dummy atom, sparkles and
                                       Translation Vector
 H
 *  *          B  C  N  O  F
Na' *         Al Si  P  S Cl           +                       o
 K' * ...  Zn  * Ge  *  * Br         XX  Cb  ++   +  --   -  Tv
Rb' * ...   *  * Sn  *  *  I         99 102 103 104 105 106 107
*   * ...  Hg  *  *  *

If users need to use other elements, such as beryllium or lead, they can be specified, in which case MNDO-type atoms will be used. As the behavior of such systems is not well investigated, users are cautioned to exercise unusual care. To alert users to this situation, the keyword PARASOK is defined.

For PM3, acceptable symbols are:

        Elements                      Dummy atom, sparkles and
                                        Translation Vector
  H
  *  Be          *  C  N  O  F
 Na' Mg         Al Si  P  S Cl           +                       o
  K' * ...  Zn  Ga Ge As Se Br         XX  Cb  ++   +  --   -  Tv
 Rb' * ...  Cd  In Sn Sb Te  I         99 102 103 104 105 106 107
 *   * ...  Hg  Tl Pb Bi

3.5 Definition of elements and isotopes

      Diatomics Parameterized within the MINDO/3 Formalism
     H   B   C   N   O   F  Si   P   S  Cl     A star (*) indicates
   -----------------------------------------   that the atom-pair is
  H  *   *   *   *   *   *   *   *   *   *     parameterized within
  B  *   *   *   *   *   *                     MINDO/3.
  C  *   *   *   *   *   *   *   *   *   *
  N  *   *   *   *   *   *           *   *
  O  *   *   *   *   *   *       *   *
  F  *   *   *   *   *   *       *
 Si  *       *               *
  P  *       *       *   *       *       *
  S  *       *   *   *               *   *
 Cl  *       *   *               *   *   *

Note: MINDO/3 should now be regarded as being of historical interest only. MOPAC contains the original parameters. These do not reproduce the original reported results in the case of P, Si, or S. The original work was faulty, see G. Frenking, H. Goetz, and F. Marschner, J. Am. Chem. Soc., 100:5295 (1978). Re-optimized parameters for P–C and P–Cl were derived later which gave better results. These are:

• Alpha(P–C): 0.8700 G. Frenking, H. Goetz, F. Marschner,

• Beta(P–C): 0.5000 J. Am. Chem. Soc., 100:5295-5296 (1978).

• Alpha(P–Cl): 1.5400 G. Frenking, F. Marschner, H. Goetz,

• Beta(P–Cl): 0.2800 Phosphorus and Sulfur, 8:337-342 (1980).

Although better than the original parameters, these have not been adopted within MOPAC because to do so at this time would prevent earlier calculations from being duplicated. Parameters for P–O and P–F have been added: these were abstracted from Frenking’s 1980 paper. No inconsistency is involved as MINDO/3 historically did not have P–O or P–F parameters.

Extra entities available to MNDO, MINDO/3, AM1 and PM3:

      +     A 100% ionic alkali metal.
     ++     A 100% ionic alkaline earth metal.
      -     A 100% ionic halogen-like atom
     --     A 100% ionic group VI-like atom.
     Cb     A special type of monovalent atom

Elements 103, 104, 105, and 106 are the sparkles; elements 11 and 19 are sparkles tailored to look like the alkaline metal ions; Tv is the translation vector for polymer calculations. See “Full description of sparkles” in Chapter 6.

Element 102, symbol Cb, is designed to satisfy valency requirements of atoms for which some bonds are not completed. Thus in “solid” diamond the usual way to complete the normal valency in a cluster model is to use hydrogen atoms. This approach has the defect that the electronegativity of hydrogen is different from that of carbon. The “capped bond” atom, Cb, is designed to satisfy these valency requirements without acquiring a net charge.

Cb behaves like a monovalent atom, with the exception that it can alter its electronegativity to achieve an exactly zero charge in whatever environment it finds itself. It is thus all things to all atoms. On bonding to hydrogen it behaves similar to a hydrogen atom. On bonding to fluorine it behaves like a very electronegative atom. If several capped bond atoms are used, each will behave independently. Thus if the two hydrogen atoms in formic acid were replaced by Cb’s then each Cb would independently become electroneutral.

Capped bonds internal coordinates should not be optimized. A fixed bond-length of 1.7 Å is recommended, if two Cb are on one atom, a contained angle of 109.471221 degrees is suggested, and if three Cb are on one atom, a contained dihedral of −120 degrees (note sign) should be used.

Element 99, X, or XX is known as a dummy atom, and is used in the definition of the geometry; it is deleted automatically from any cartesian coordinate geometry files. Dummy atoms are pure mathematic points, and are useful in defining geometries; for example, in ammonia the definition of C3v symmetry is facilitated by using one dummy atom and symmetry relating the three hydrogens to it.

Output normally only gives chemical symbols.

Isotopes are used in conjunction with chemical symbols. If no isotope is specified, the average isotopic mass is used, thus chlorine is 35.453. This is different from some earlier versions of MOPAC, in which the most abundant isotope was used by default. This change was justified by the removal of any ambiguity in the choice of isotope. Also, the experimental vibrational spectra involve a mixture of isotopes. If a user wishes to specify any specific isotope it should immediately follow the chemical symbol (no space), e.g., H2, H2.0140, C(meta)13, or C13.00335.

The sparkles ++, +, –, and − have no mass; if they are to be used in a force calculation, then appropriate masses should be used.

Each internal coordinate is followed by an integer, to indicate the action to be taken.

      Integer                        Action
        1                Optimize the internal coordinate.
        0                Do not optimize the internal coordinate.
       -1                Reaction coordinate, or grid index.

Remarks: Only one reaction coordinate is allowed, but this can be made more versatile by the use of SYMMETRY. If a reaction coordinate is used, the values of the reaction coordinate should follow immediately after the geometry and any symmetry data. No terminator is required, and free-format-type input is acceptable.

If two “reaction coordinates” are used, then MOPAC assumes that the two-dimensional space in the region of the supplied geometry is to be mapped. The two dimensions to be mapped are in the plane defined by the “−1” labels. Step sizes in the two directions must be supplied using STEP1 and STEP2 on the keyword line.

Using internal coordinates, the first atom has three unoptimizable coordinates, the second atom two, (the bond-length can be optimized) and the third atom has one unoptimizable coordinate. None of these six unoptimizable coordinates at the start of the geometry should be marked for optimization. If any are so marked, a warning is given, but the calculation will continue.

In cartesian coordinates all parameters can be optimized.

3.6 Examples of coordinate definitions

Two examples will be given. The first is formic acid, HCOOH, and is presented in the normal style with internal coordinates. This is followed by formaldehyde, presented in such a manner as to demonstrate as many different features of the geometry definition as possible.

 MINDO/3
 Formic acid
 Example of normal geometry definition
   O                                        Atom 1 needs no coordinates.
   C    1.20 1                              Atom 2 bonds to atom 1.
   O    1.32 1  116.8 1            2  1     Atom 3 bonds to atom 2 and
                                            makes an angle with atom 1.
   H    0.98 1  123.9 1    0.0 0   3  2  1  Atom 4 has a dihedral of 0.0
                                            with atoms 3, 2 and 1.
   H    1.11 1  127.3 1  180.0 0   2  1  3
   0    0.00 0    0.0 0    0.0 0   0  0  0

Atom 2, a carbon, is bonded to oxygen by a bond-length of 1.20 Angstroms, and to atom 3, an oxygen, by a bond-length of 1.32 Angstroms. The O-C-O angle is 116.8 degrees. The first 3.6 Examples of coordinate definitions

hydrogen is bonded to the hydroxyl oxygen and the second hydrogen is bonded to the carbon atom. The H-C-O-O dihedral angle is 180 degrees.

MOPAC can generate data-files, both in the Archive files, and at the end of the normal output file, when a job ends prematurely due to time restrictions. Note that the data are all neatly lined up. This is, of course, characteristic of machine-generated data, but is useful when checking for errors.

Format of internal coordinates in ARCHIVE file

O    0.000000 0    0.000000 0    0.000000 0    0   0   0
C    1.209615 1    0.000000 0    0.000000 0    1   0   0
O    1.313679 1  116.886168 1    0.000000 0    2   1   0
H    0.964468 1  115.553316 1    0.000000 0    3   2   1
H    1.108040 1  128.726078 1  180.000000 0    2   1   3
0    0.000000 0    0.000000 0    0.000000 0    0   0   0

Polymers are defined by the presence of a translation vector. In the following example, polyethylene, the translation vector spans three monomeric units, and is 7.7 Angstroms long. Note in this example the presence of two dummy atoms. These not only make the geometry definition easier but also allow the translation vector to be specified in terms of distance only, rather than both distance and angles.

Example of polymer coordinates from ARCHIVE file:

T=20000
  POLYETHYLENE, CLUSTER UNIT :  C6H12
C    0.000000  0    0.000000  0    0.000000  0    0   0   0
C    1.540714  1    0.000000  0    0.000000  0    1   0   0
C    1.542585  1  113.532306  1    0.000000  0    2   1   0
C    1.542988  1  113.373490  1  179.823613  1    3   2   1
C    1.545151  1  113.447508  1  179.811764  1    4   3   2
C    1.541777  1  113.859804  1 -179.862648  1    5   4   3
XX    1.542344  1  108.897076  1 -179.732346  1    6   5   4
XX    1.540749  1  108.360151  1 -178.950271  1    7   6   5
H    1.114786  1   90.070026  1  126.747447  1    1   3   2
H    1.114512  1   90.053136  1 -127.134856  1    1   3   2
H    1.114687  1   90.032722  1  126.717889  1    2   4   3
H    1.114748  1   89.975504  1 -127.034513  1    2   4   3
H    1.114474  1   90.063308  1  126.681098  1    3   5   4
H    1.114433  1   89.915262  1 -126.931090  1    3   5   4
H    1.114308  1   90.028131  1  127.007845  1    4   6   5
H    1.114434  1   90.189506  1 -126.759550  1    4   6   5
H    1.114534  1   88.522263  1  127.041363  1    5   7   6
H    1.114557  1   88.707407  1 -126.716355  1    5   7   6
H    1.114734  1   90.638631  1  127.793055  1    6   8   7
H    1.115150  1   91.747016  1 -126.187496  1    6   8   7
Tv    7.746928  1    0.000000  0    0.000000  0    1   7   8
0    0.000000  0    0.000000  0    0.000000  0    0   0   0

Geometry specification

Examples

In this chapter various examples of data-files are described. With MOPAC comes two sets of data for running calculations. One of these is called MNRSD1.DAT, and this will now be described.

4.1 MNRSD1 test data file for formaldehyde

The following file is suitable for generating the results described in the next section, and would be suitable for debugging data.

Line  1:         SYMMETRY
Line  2:  Formaldehyde, for Demonstration Purposes
Line  3:
Line  4:   O
Line  5:   C 1.2 1
Line  6:   H 1.1 1 120 1
Line  7:   H 1.1 0 120 0 180 0 2 1 3
Line  8:
Line  9:   3 1 4
Line 10:   3 2 4
Line 11:

These data could be more neatly written as:

Line  1:         SYMMETRY
Line  2:  Formaldehyde, for Demonstration Purposes
Line  3:
Line  4:   O
Line  5:   C    1.20  1                         1
Line  6:   H    1.10  1  120.00  1              2  1
Line  7:   H    1.10  0  120.00  0  180.00  0   2  1  3
Line  8:
Line  9:   3,   1,   4,
Line 10:   3,   2,   4,
Line 11:

These two data-files will produce identical results files.

In all geometric specifications, care must be taken in defining the internal coordinates to ensure that no three atoms being used to define a fourth atom’s dihedral angle ever fall into a straight line. This can happen in the course of a geometry optimization, in a SADDLE calculation or in following a reaction coordinate. If such a condition should develop, then the position of the dependent atom would become ill-defined.

\underline{Examples}

4.2 MOPAC output for test-data file MNRSD1

****************************************************************************
** FRANK J. SEILER RES. LAB., U.S. AIR FORCE ACADEMY, COLO. SPGS., CO. 80840
****************************************************************************
                               MNDO CALCULATION RESULTS              Note 1
****************************************************************************
*          MOPAC:  VERSION  6.00               CALC'D.  4-OCT-90     Note 2
*  SYMMETRY - SYMMETRY CONDITIONS TO BE IMPOSED
*   T=      - A TIME OF  3600.0 SECONDS REQUESTED
*  DUMP=N   - RESTART FILE WRITTEN EVERY  3600.0 SECONDS
********************************************************************043BY043
    PARAMETER DEPENDENCE DATA
       REFERENCE ATOM      FUNCTION NO.    DEPENDENT ATOM(S)
           3                  1             4
           3                  2             4
            DESCRIPTIONS OF THE FUNCTIONS USED
  1      BOND LENGTH    IS SET EQUAL TO THE REFERENCE BOND LENGTH
  2      BOND ANGLE     IS SET EQUAL TO THE REFERENCE BOND ANGLE
        SYMMETRY                                                     Note 3
 Formaldehyde, for Demonstration Purposes
   ATOM   CHEMICAL  BOND LENGTH    BOND ANGLE     TWIST ANGLE
  NUMBER  SYMBOL    (ANGSTROMS)     (DEGREES)      (DEGREES)
   (I)                  NA:I          NB:NA:I      NC:NB:NA:I    NA NB NC
     1      O                                                        Note 4
     2      C         1.20000 *                                  1
     3      H         1.10000 *      120.00000 *                 2   1
     4      H         1.10000        120.00000     180.00000     2   1   3
         CARTESIAN COORDINATES
   NO.       ATOM         X         Y         Z
    1         O        0.0000    0.0000    0.0000
    2         C        1.2000    0.0000    0.0000                    Note 5
    3         H        1.7500    0.9526    0.0000
    4         H        1.7500   -0.9526    0.0000
 H: (MNDO):  M.J.S. DEWAR, W. THIEL, J. AM. CHEM. SOC., 99, 4899, (1977)
 C: (MNDO):  M.J.S. DEWAR, W. THIEL, J. AM. CHEM. SOC., 99, 4899, (1977)
 O: (MNDO):  M.J.S. DEWAR, W. THIEL, J. AM. CHEM. SOC., 99, 4899, (1977)
         RHF CALCULATION, NO. OF DOUBLY OCCUPIED LEVELS =  6
           INTERATOMIC DISTANCES
                 O  1       C  2       H  3       H  4
------------------------------------------------------
    O    1   0.000000
    C    2   1.200000   0.000000
    H    3   1.992486   1.100000   0.000000                          Note 6
    H    4   1.992486   1.100000   1.905256   0.000000
CYCLE:   1 TIME:   0.75 TIME LEFT:   3598.2 GRAD.:     6.349 HEAT:-32.840147
CYCLE:   2 TIME:   0.37 TIME LEFT:   3597.8 GRAD.:     2.541 HEAT:-32.880103
HEAT OF FORMATION TEST SATISFIED                                     Note 7
PETERS TEST SATISFIED                                                Note 8
---------------------------------------------------------------------------
        SYMMETRY                                                     Note 9
 Formaldehyde, for Demonstration Purposes                            Note 10

4.2 MOPAC output for test-data file MNRSD1

PETERS TEST WAS SATISFIED IN BFGS            OPTIMIZATION        Note 11
SCF FIELD WAS ACHIEVED                                           Note 12
                     MNDO    CALCULATION                    Note 13
                                             VERSION  6.00
                                              4-OCT-90
FINAL HEAT OF FORMATION =        -32.88176 KCAL             Note 14
TOTAL ENERGY            =       -478.11917 EV
ELECTRONIC ENERGY       =       -870.69649 EV
CORE-CORE REPULSION     =        392.57733 EV
IONIZATION POTENTIAL    =         11.04198
NO. OF FILLED LEVELS    =          6
MOLECULAR WEIGHT        =     30.026
SCF CALCULATIONS  =               15
COMPUTATION TIME =   2.740 SECONDS                          Note 15
ATOM   CHEMICAL  BOND LENGTH    BOND ANGLE     TWIST ANGLE
NUMBER  SYMBOL    (ANGSTROMS)     (DEGREES)      (DEGREES)
(I)                  NA:I          NB:NA:I      NC:NB:NA:I     NA   NB   NC
1      O
2      C         1.21678 *                                   1  Note 16
3      H         1.10590 *      123.50259 *                  2    1
4      H         1.10590        123.50259     180.00000      2    1    3
  INTERATOMIC DISTANCES
        O  1       C  2       H  3       H  4
------------------------------------------------------
O    1   0.000000
C    2   1.216777   0.000000
H    3   2.046722   1.105900   0.000000
H    4   2.046722   1.105900   1.844333   0.000000
        EIGENVALUES
-42.98352 -25.12201 -16.95327 -16.29819 -14.17549 -11.04198  0.85804  3.6768
3.84990   7.12408                                                  Note 17
    NET ATOMIC CHARGES AND DIPOLE CONTRIBUTIONS
ATOM NO.   TYPE          CHARGE        ATOM  ELECTRON DENSITY
 1          O          -0.2903          6.2903
 2          C           0.2921          3.7079              Note 18
 3          H          -0.0009          1.0009
 4          H          -0.0009          1.0009
DIPOLE           X         Y         Z       TOTAL
POINT-CHG.     1.692     0.000     0.000     1.692
HYBRID         0.475     0.000     0.000     0.475                   Note 19
SUM            2.166     0.000     0.000     2.166
CARTESIAN COORDINATES
NO.       ATOM               X         Y         Z
1         O                  0.0000    0.0000    0.0000
2         C                  1.2168    0.0000    0.0000
3         H                  1.8272    0.9222    0.0000
4         H                  1.8272   -0.9222    0.0000
ATOMIC ORBITAL ELECTRON POPULATIONS
1.88270   1.21586   1.89126   1.30050   1.25532   0.86217   0.89095  0.69950
1.00087   1.00087                                                    Note 20
TOTAL CPU TIME:             3.11 SECONDS
== MOPAC DONE ==

Note 1: The banner indicates whether the calculation uses a MNDO, MINDO/3, AM1 or PM3 Hamiltonian; here, the default MNDO Hamiltonian is used.

Note 2: The Version number is a constant for any release of MOPAC, and refers to the program, not to the Hamiltonians used. The version number should be cited in any correspondence regarding MOPAC. Users’ own in-house modified versions of MOPAC will have a final digit different from zero, e.g. 6.01.

All the keywords used, along with a brief explanation, should be printed at this time. If a keyword is not printed, it has not been recognized by the program. Keywords can be in upper or lower case letters, or any mixture. The cryptic message at the right end of the lower line of asterisks indicates the number of heavy and light atoms this version of MOPAC is configured for.

Note 3: Symmetry information is output to allow the user to verify that the requested symmetry functions have in fact been recognized and used.

Note 4: The data for this example used a mixture of atomic numbers and chemical symbols, but the internal coordinate output is consistently in chemical symbols.

The atoms in the system are, in order:

• Atom 1, an oxygen atom; this is defined as being at the origin.

• Atom 2, the carbon atom. Defined as being 1.2 Angstroms from the oxygen atom, it is located in the +x direction. This distance is marked for optimization.

• Atom 3, a hydrogen atom. It is defined as being 1.1 Angstroms from the carbon atom, and making an angle of 120 degrees with the oxygen atom. The asterisks indicate that the bond length and angle are both to be optimized.

• Atom 4, a hydrogen atom. The bond length supplied has been overwritten with the symmetry-defined C-H bond length. Atom 4 is defined as being 1.1 Å from atom 2, making a bond-angle of 120 degrees with atom 1, and a dihedral angle of 180 degrees with atom 3. None of the coordinates of atom 4 are marked for optimization. The bond-length and angle are symmetry-defined by atom 3, and the dihedral is group-theory symmetrydefined as being 180 degrees. (The molecule is flat.)

Note 5: The cartesian coordinates are calculated as follows:

Stage 1: The coordinate of the first atom is defined as being at the origin of cartesian space, while the coordinate of the second atom is defined as being displaced by its defined bond length along the positive x-axis. The coordinate of the third atom is defined as being displaced by its bond length in the x-y plane, from either atom 1 or 2 as defined in the data, or from atom 2 if no numbering is given. The angle it makes with atoms 1 and 2 is that given by its bond angle.

The dihedral, which first appears in the fourth atom, is defined according to the IUPAC convention. Note: This is different from previous versions of MNDO and MINDO/3, where the dihedral had the opposite chirality to that defined by the IUPAC convention.

Stage 2: Any dummy atoms are removed. As this particular system contains no dummy atoms, nothing is done.

Note 6: The interatomic distances are output for the user’s advice, and a simple check made to insure that the smallest interatomic distance is greater than 0.8 Å.

Note 7: The geometry is optimized in a series of cycles, each cycle consisting of a line search and calculation of the gradients. The time given is the cpu time for the cycle; time left is the total time requested (here 100 seconds) less the cpu time since the start of the calculation (which is earlier than the start of the first cycle!). These times can vary slightly from cycle to cycle due to different options being used, for example whether or not two or more SCF calculations need to be done to ensure that the heat of formation is lowered. The gradient is the scalar length in kcal/mole/Angstrom of the gradient vector.

Note 8: At the end of the BFGS geometry optimization a message is given which indicates how the optimization ended. All “normal” termination messages contain the word “satisfied”; other terminations may give acceptable results, but more care should be taken, particularly regarding the gradient vector.

Notes 9, 10: The keywords used, titles and comments are reproduced here to remind the user of the name of the calculation.

Notes 11, 12: Two messages are given here. The first is a reminder of how the geometry was obtained, whether from the Broyden–Fletcher–Goldfarb–Shanno, Eigenvector Following, Bartel’s or the McIver–Komornicki methods. For any further results to be printed the second message must be as shown; when no SCF is obtained no results will be printed.

Note 13: Again, the results are headed with either MNDO or MINDO/3 banners, and the version number. The date has been moved to below the version number for convenience.

Note 14: The total energy of the system is the addition of the electronic and nuclear terms. The heat of formation is relative to the elements in their standard state. The I.P. is the negative of the energy level of the highest occupied, or highest partially occupied molecular orbital (in accordance with Koopmans’ theorem).

Note 15: Advice on time required for the calculation. This is obviously useful in estimating the times required for other systems.

Note 16: The fully optimized geometry is printed here. If a parameter is not marked for optimization, it will not be changed unless it is a symmetry-related parameter.

Note 17: The roots are the eigenvalues or energy levels in electron volts of the molecular orbitals. There are six filled levels, therefore the HOMO has an energy of -11.041eV; analysis of the corresponding eigenvector (not given here) shows that it is mainly lone-pair on oxygen. The eigenvectors form an orthonormal set.

Note 18: The charge on an atom is the sum of the positive core charge; for hydrogen, carbon, and oxygen these numbers are 1.0, 4.0, and 6.0, respectively, and the negative of the number of valence electrons, or atom electron density on the atom, here 1.0010, 3.7079, and 6.2902 respectively.

Note 19: The dipole is the scalar of the dipole vector in cartesian coordinates. The components of the vector coefficients are the point-charge dipole and the hybridization dipole. In formaldehyde there is no z-dipole since the molecule is flat.

Note 20: MNDO AM1, PM3, and MINDO/3 all use the Coulson density matrix. Only the diagonal elements of the matrix, representing the valence orbital electron populations, will be printed, unless the keyword DENSITY is specified.

Extra lines are added as a result of user requests:

1. The total CPU time for the job, excluding loading of the executable, is printed.

2. In order to know that MOPAC has ended, the message == MOPAC DONE == is printed.

Examples

Testdata

TESTDATA.DAT, supplied with MOPAC 6.00, is a single large job consisting of several small systems, which are run one after the other. In order, the calculations run are:

1. A FORCE calculation on formaldehyde. The extra keywords at the start are to be used later when TESTDATA.DAT acts as a SETUP file. This unusual usage of a data set was made necessary by the need to ensure that a SETUP file existed. If the first two lines are removed, the data set used in the example given below is generated.

2. The vibrational frequencies of a highly excited dication of methane are calculated. A nondegenerate state was selected in order to preserve tetrahedral symmetry (to avoid the Jahn-Teller effects).

3. Illustration of the use of the & in the keyword line, and of the new optional definition of atoms 2 and 3

4. Illustration of Gaussian Z-matrix input.

5. An example of Eigenvector Following, to locate a transition state.

6. Use of SETUP. Normally, SETUP would point to a special file which would contain keywords only. Here, the only file we can guarantee exists, is the file being run, so that is the one used.

7. Example of labelling atoms.

8. This part of the test writes the density matrix to disk, for later use.

9. A simple calculation on water.

10. The previous, optimized, geometry is to be used to start this calculation.

11. The density matrix written out earlier is now used as input to start an SCF.

This example is taken from the first data-file in TESTDATA.DAT, and illustrates the working of a FORCE calculation.

5.1 Data file for a force calculation

Line  1   nointer  noxyz + mndo dump=8
Line  2    t=2000 + thermo(298,298) force isotope
Line  3  ROT=2
Line  4    DEMONSTRATION OF MOPAC - FORCE AND THERMODYNAMICS CALCULATION
Line  5    FORMALDEHYDE, MNDO ENERGY = -32.8819  See Manual.
Line  6    O

\underline{Testdata}

Line  7    C    1.216487  1                                 1  0  0
Line  8    H    1.106109  1  123.513310  1                  2  1  0
Line  9    H    1.106109  1  123.513310  1  180.000000  1   2  1  3
Line 10    0    0.000000  0    0.000000  0    0.000000  0   0  0  0

5.2 Results file for the force calculation

****************************************************************************
** FRANK J. SEILER RES. LAB., U.S. AIR FORCE ACADEMY, COLO. SPGS., CO. 80840
****************************************************************************
                               MNDO CALCULATION RESULTS

***************************************************************************
*          MOPAC:  VERSION  6.00               CALC'D. 12-OCT-90
*   T=      - A TIME OF  2000.0 SECONDS REQUESTED
*  DUMP=N   - RESTART FILE WRITTEN EVERY     8.0 SECONDS
*  FORCE    - FORCE CALCULATION SPECIFIED
*  PRECISE  - CRITERIA TO BE INCREASED BY 100 TIMES
*  NOINTER  - INTERATOMIC DISTANCES NOT TO BE PRINTED              Note 1
*  ISOTOPE  - FORCE MATRIX WRITTEN TO DISK (CHAN. 9 )
*  NOXYZ    - CARTESIAN COORDINATES NOT TO BE PRINTED
*  THERMO   - THERMODYNAMIC QUANTITIES TO BE CALCULATED
*  ROT      - SYMMETRY NUMBER OF  2 SPECIFIED
*******************************************************************040BY040
 NOINTER  NOXYZ + MNDO DUMP=8
  T=2000 + THERMO(298,298) FORCE ISOTOPE
ROT=2  PRECISE
  DEMONSTRATION OF MOPAC - FORCE AND THERMODYNAMICS CALCULATION
  FORMALDEHYDE, MNDO ENERGY = -32.8819  See Manual.
 ATOM   CHEMICAL  BOND LENGTH    BOND ANGLE     TWIST ANGLE
NUMBER  SYMBOL    (ANGSTROMS)     (DEGREES)      (DEGREES)
 (I)                  NA:I          NB:NA:I      NC:NB:NA:I    NA  NB  NC
 ATOM   CHEMICAL  BOND LENGTH    BOND ANGLE     TWIST ANGLE
   1      O
   2      C         1.21649  *                                 1
   3      H         1.10611  *     123.51331  *                2   1
   4      H         1.10611  *     123.51331  *  180.00000  *  2   1   3
 H: (MNDO):  M.J.S. DEWAR, W. THIEL, J. AM. CHEM. SOC., 99, 4899, (1977)
 C: (MNDO):  M.J.S. DEWAR, W. THIEL, J. AM. CHEM. SOC., 99, 4899, (1977)
 O: (MNDO):  M.J.S. DEWAR, W. THIEL, J. AM. CHEM. SOC., 99, 4899, (1977)

         RHF CALCULATION, NO. OF DOUBLY OCCUPIED LEVELS =  6

         HEAT OF FORMATION =  -32.881900 KCALS/MOLE

5.2 Results file for the force calculation

         INTERNAL COORDINATE DERIVATIVES
  ATOM  AT. NO.  BOND           ANGLE          DIHEDRAL
    1     O
    2     C     0.000604
    3     H     0.000110    -0.000054
    4     H     0.000110    -0.000054     0.000000

         GRADIENT NORM =   0.00063                             Note 2

         TIME FOR SCF CALCULATION =    0.45

         TIME FOR DERIVATIVES     =    0.32                    Note 3
         MOLECULAR WEIGHT =   30.03
          PRINCIPAL MOMENTS OF INERTIA IN CM(-1)
         A =    9.832732   B =    1.261998   C =    1.118449

          PRINCIPAL MOMENTS OF INERTIA IN UNITS OF 10**(-40)*GRAM-CM**2
         A =    2.846883   B =   22.181200   C =   25.028083

        ORIENTATION OF MOLECULE IN FORCE CALCULATION
   NO.       ATOM         X         Y         Z
    1         8       -0.6093    0.0000    0.0000
    2         6        0.6072    0.0000    0.0000
    3         1        1.2179    0.9222    0.0000
    4         1        1.2179   -0.9222    0.0000

   FIRST DERIVATIVES WILL BE USED IN THE CALCULATION OF SECOND DERIVATIVES
         ESTIMATED TIME TO COMPLETE CALCULATION =    36.96 SECONDS
STEP:   1 TIME =     2.15 SECS, INTEGRAL =      2.15 TIME LEFT:   1997.08
STEP:   2 TIME =     2.49 SECS, INTEGRAL =      4.64 TIME LEFT:   1994.59
STEP:   3 TIME =     2.53 SECS, INTEGRAL =      7.17 TIME LEFT:   1992.06
STEP:   4 TIME =     2.31 SECS, INTEGRAL =      9.48 TIME LEFT:   1989.75
STEP:   5 RESTART FILE WRITTEN, INTEGRAL =     11.97 TIME LEFT:   1987.26
STEP:   6 TIME =     2.43 SECS, INTEGRAL =     14.40 TIME LEFT:   1984.83
STEP:   7 TIME =     2.32 SECS, INTEGRAL =     16.72 TIME LEFT:   1982.51
STEP:   8 TIME =     2.30 SECS, INTEGRAL =     19.02 TIME LEFT:   1980.21
STEP:   9 RESTART FILE WRITTEN, INTEGRAL =     22.17 TIME LEFT:   1977.06
STEP:  10 TIME =     2.52 SECS, INTEGRAL =     24.69 TIME LEFT:   1974.54

STEP:  11 TIME =     2.25 SECS, INTEGRAL =     26.94 TIME LEFT:   1972.29
STEP:  12 TIME =     3.15 SECS, INTEGRAL =     30.09 TIME LEFT:   1969.14

      FORCE MATRIX IN MILLIDYNES/ANGSTROM
0
             O  1       C  2       H  3       H  4
------------------------------------------------------
O    1   9.557495
C    2   8.682982  11.426823
H    3   0.598857   2.553336   3.034881
H    4   0.598862   2.553344   0.304463   3.034886

     HEAT OF FORMATION =  -32.881900 KCALS/MOLE

      ZERO POINT ENERGY      18.002 KILOCALORIES PER MOLE      Note 4
THE LAST 6 VIBRATIONS ARE THE TRANSLATION AND ROTATION MODES
THE FIRST THREE OF THESE BEING TRANSLATIONS IN X, Y, AND Z, RESPECTIVELY

      NORMAL COORDINATE ANALYSIS

                                                               Note 5
ROOT NO.    1           2           3           4           5           6
 1209.90331  1214.67040  1490.52685  2114.53841  3255.93651  3302.12319
1   0.00000     0.00000    -0.04158    -0.25182     0.00000     0.00067
2   0.06810     0.00001     0.00000     0.00000     0.00409     0.00000
3   0.00000    -0.03807     0.00000     0.00000     0.00000     0.00000
4   0.00000     0.00000    -0.03819     0.32052     0.00000    -0.06298
5  -0.13631    -0.00002     0.00000     0.00000     0.08457     0.00000
6  -0.00002     0.15172     0.00000     0.00000     0.00000     0.00000
7  -0.53308    -0.00005     0.55756     0.08893    -0.39806     0.36994
8   0.27166     0.00003    -0.38524     0.15510    -0.53641     0.57206
9   0.00007    -0.60187     0.00001     0.00000     0.00000     0.00000
10   0.53307     0.00006     0.55757     0.08893     0.39803     0.36997
11   0.27165     0.00003     0.38524    -0.15509    -0.53637    -0.57209
12   0.00007    -0.60187     0.00001     0.00000     0.00000     0.00000

ROOT NO.    7           8           9          10          11          12
   -0.00019    -0.00044    -0.00016     3.38368     2.03661    -0.76725
1   0.25401     0.00000     0.00000     0.00000     0.00000     0.00000
2   0.00000    -0.25401     0.00000     0.00000     0.00000    -0.17792

3   0.00000     0.00000    -0.25401     0.00000    -0.19832     0.00000
4   0.25401     0.00000     0.00000     0.00000     0.00000     0.00000
5   0.00000    -0.25401     0.00000     0.00000     0.00000     0.17731
6   0.00000     0.00000    -0.25401     0.00000     0.19764     0.00000
7   0.25401     0.00000     0.00000     0.00000     0.00000    -0.26930
8   0.00000    -0.25401     0.00000     0.00000     0.00000     0.35565
9   0.00000     0.00000    -0.25401     0.70572     0.39642     0.00000
10   0.25401     0.00000     0.00000     0.00000     0.00000     0.26930
11   0.00000    -0.25401     0.00000     0.00000     0.00000     0.35565
12   0.00000     0.00000    -0.25401    -0.70572     0.39642     0.00000
      MASS-WEIGHTED COORDINATE ANALYSIS

                                                               Note 6
ROOT NO.    1           2           3           4           5           6
 1209.90331  1214.67040  1490.52685  2114.53841  3255.93651  3302.12319
1   0.00000     0.00000    -0.16877    -0.66231     0.00000     0.00271
2   0.26985     0.00003     0.00000     0.00000     0.01649     0.00000
3   0.00002    -0.15005     0.00000     0.00000     0.00000     0.00000
4   0.00000     0.00000    -0.13432     0.73040     0.00001    -0.22013
5  -0.46798    -0.00005     0.00000     0.00000     0.29524     0.00001
6  -0.00006     0.51814     0.00000     0.00000     0.00000     0.00000
7  -0.53018    -0.00005     0.56805     0.05871    -0.40255     0.37455
8   0.27018     0.00003    -0.39249     0.10238    -0.54246     0.57918
9   0.00007    -0.59541     0.00001     0.00000     0.00000     0.00000
10   0.53018     0.00006     0.56806     0.05871     0.40252     0.37457
11   0.27018     0.00003     0.39249    -0.10238    -0.54242    -0.57922
12   0.00007    -0.59541     0.00001     0.00000     0.00000     0.00000

                                                               Note 7
ROOT NO.    7           8           9          10          11          12
   -0.00025    -0.00022    -0.00047     3.38368     2.03661    -0.76725
1   0.72996     0.00000     0.00000     0.00000     0.00000     0.00000
2   0.00000    -0.72996     0.00000     0.00000     0.00000    -0.62774
3   0.00000     0.00000    -0.72996     0.00000    -0.66681     0.00000
4   0.63247     0.00000     0.00000     0.00000     0.00000     0.00000
5   0.00000    -0.63247     0.00000     0.00000     0.00000     0.54204
6   0.00000     0.00000    -0.63247     0.00000     0.57578     0.00000
7   0.18321     0.00000     0.00000     0.00000     0.00000    -0.23848
8   0.00000    -0.18321     0.00000     0.00000     0.00000     0.31495
9   0.00000     0.00000    -0.18321     0.70711     0.33455     0.00000
10   0.18321     0.00000     0.00000     0.00000     0.00000     0.23848
11   0.00000    -0.18321     0.00000     0.00000     0.00000     0.31495
12   0.00000     0.00000    -0.18321    -0.70711     0.33455     0.00000
     DESCRIPTION OF VIBRATIONS

VIBRATION   1            ATOM PAIR     ENERGY CONTRIBUTION          RADIAL
FREQ.     1209.90       C 2 --  H 3           42.7% ( 79.4%)          12.6%
T-DIPOLE   0.8545       C 2 --  H 4           42.7%                   12.6%
TRAVEL     0.1199       O 1 --  C 2           14.6%                    0.0%
RED. MASS  1.9377
VIBRATION   2            ATOM PAIR     ENERGY CONTRIBUTION          RADIAL
FREQ.     1214.67       C 2 --  H 3           45.1% ( 62.3%)           0.0%
T-DIPOLE   0.1275       C 2 --  H 4           45.1%                    0.0%
TRAVEL     0.1360       O 1 --  C 2            9.8%                    0.0%
RED. MASS  1.5004
VIBRATION   3            ATOM PAIR     ENERGY CONTRIBUTION          RADIAL
FREQ.     1490.53       C 2 --  H 4           49.6% ( 61.5%)           0.6%
T-DIPOLE   0.3445       C 2 --  H 3           49.6%                    0.6%
TRAVEL     0.1846       O 1 --  C 2            0.9%                  100.0%
RED. MASS  0.6639
VIBRATION   4            ATOM PAIR     ENERGY CONTRIBUTION          RADIAL
FREQ.     2114.54       O 1 --  C 2           60.1% (100.5%)         100.0%
T-DIPOLE   3.3662       C 2 --  H 4           20.0%                   17.7%
TRAVEL     0.0484       C 2 --  H 3           20.0%                   17.7%
RED. MASS  6.7922
VIBRATION   5            ATOM PAIR     ENERGY CONTRIBUTION          RADIAL
FREQ.     3255.94       C 2 --  H 3           49.5% ( 72.2%)          98.1%
T-DIPOLE   0.7829       C 2 --  H 4           49.5%                   98.1%
TRAVEL     0.1174       O 1 --  C 2            1.0%                    0.0%
RED. MASS  0.7508
VIBRATION   6            ATOM PAIR     ENERGY CONTRIBUTION          RADIAL
FREQ.     3302.12       C 2 --  H 4           49.3% ( 69.8%)          95.5%
T-DIPOLE   0.3478       C 2 --  H 3           49.3%                   95.5%
TRAVEL     0.1240       O 1 --  C 2            1.4%                  100.0%
RED. MASS  0.6644

         SYSTEM IS A GROUND STATE

  FORMALDEHYDE, MNDO ENERGY = -32.8819  See Manual.
  DEMONSTRATION OF MOPAC - FORCE AND THERMODYNAMICS CALCULATION

         MOLECULE IS NOT LINEAR
         THERE ARE  6 GENUINE VIBRATIONS IN THIS SYSTEM
         THIS THERMODYNAMICS CALCULATION IS LIMITED TO
         MOLECULES WHICH HAVE NO INTERNAL ROTATIONS
                                                                   Note 8
                   CALCULATED THERMODYNAMIC PROPERTIES

                                      *
TEMP. (K)  PARTITION FUNCTION   H.O.F.    ENTHALPY   HEAT CAPACITY  ENTROPY
                           KCAL/MOL   CAL/MOLE    CAL/K/MOL   CAL/K/MOL

298  VIB.         1.007                    23.39484    0.47839    0.09151
 ROT.     709.                        888.305      2.981     16.026
 INT.     714.                        911.700      3.459     16.117
 TRA.    0.159E+27                   1480.509      4.968     36.113
 TOT.                       -32.882  2392.2088     8.4274    52.2300
* NOTE: HEATS OF FORMATION ARE RELATIVE TO THE
         ELEMENTS IN THEIR STANDARD STATE AT 298K

TOTAL CPU TIME:            32.26 SECONDS
== MOPAC DONE ==

Note 1: All three words, ROT, FORCE, and THERMO are necessary in order to obtain thermodynamic properties. In order to obtain results for only one temperature, THERMO has the first and second arguments identical. The symmetry number for the C_{2v} point-group is 2.

Note 2: Internal coordinate derivatives are in kcal/Angstrom or kcal/radian. Values of less than about 0.2 are quite acceptable.

Note 3: In larger calculations, the time estimates are useful. In practice they are pessimistic, and only about 70% of the time estimated will be used, usually. The principal moments of inertia can be directly related to the microwave spectrum of the molecule. They are simple functions of the geometry of the system, and are usually predicted with very high accuracy.

Note 4: Zero point energy is already factored into the MNDO parameterization. Force constant data are not printed by default. If you want this output, specify LARGE in the keywords.

Note 5: Normal coordinate analysis has been extensively changed. The first set of eigenvectors represent the ‘normalized’ motions of the atoms. The sum of the speeds (not the velocities) of the atoms adds to unity. This is verified by looking at the motion in the ‘z’ direction of the atoms in vibration 2. Simple addition of these terms, unsigned, adds to 1.0, whereas to get the same result for mode 1 the scalar of the motion of each atom needs to be calculated first.

Users might be concerned about reproducibility. As can be seen from the vibrational frequencies from Version 3.00 to 6.00 given below, the main difference over earlier FORCE calculations is in the trivial frequencies.

                        Real Frequencies of Formaldehyde
 Version 3.00  1209.96   1214.96   1490.60   2114.57   3255.36   3301.57
 Version 3.10  1209.99   1215.04   1490.59   2114.57   3255.36   3301.58
 Version 4.00  1209.88   1214.67   1490.52   2114.52   3255.92   3302.10
 Version 5.00  1209.89   1214.69   1490.53   2114.53   3255.93   3302.10
 Version 6.00  1209.90   1214.67   1490.53   2114.54   3255.94   3302.12
                       Trivial Frequencies of Formaldehyde
                 T(x)      T(y)      T(z)       R(x)      R(y)      R(z)

 Version 3.00  -0.00517  -0.00054  -0.00285   57.31498  11.59518   9.01619
 Version 3.10  -0.00557   0.00049  -0.00194   87.02506  11.18157  10.65295
 Version 4.00  -0.00044  -0.00052  -0.00041   12.99014  -3.08110  -3.15427
 Version 5.00   0.00040  -0.00044  -0.00062   21.05654   2.80744   3.83712
 Version 6.00  -0.00025  -0.00022  -0.00047    3.38368   2.03661  -0.76725

Note 6: Normal modes are not of much use in assigning relative importance to atoms in a mode. Thus in iodomethane it is not obvious from an examination of the normal modes which mode represents the C–I stretch. A more useful description is provided by the energy or mass-weighted coordinate analysis. Each set of three coefficients now represents the relative energy carried by an atom. (This is not strictly accurate as a definition, but is believed (by JJPS) to be more useful than the stricter definition.)

Note 7: The following description of the coordinate analysis is given without rigorous justification. Again, the analysis, although difficult to understand, has been found to be more useful than previous descriptions.

On the left-hand side are printed the frequencies and transition dipoles. Underneath these are the reduced masses and idealized distances traveled which represent the simple harmonic motion of the vibration. The mass is assumed to be attached by a spring to an infinite mass. Its displacement is the travel.

The next column is a list of all pairs of atoms that contribute significantly to the energy of the mode. Across from each pair (next column) is the percentage energy contribution of the pair to the mode, calculated according to the formula described below.

Formula for energy contribution

The total vibrational energy, T, carried by all pairs of bonded atoms in a molecule is first calculated. For any given pair of atoms, A and B, the relative contribution, R(A, B), as a percentage, is given by the energy of the pair, P (A, B), times 100 divided by T, i.e.,

R(A, B) = \frac{100 × P (A, B)}{T}

As an example, for formaldehyde the energy carried by the pair of atoms (C,O) is added to the energy of the two (C,H) pairs to give a total, T. Note that this total cannot be related to anything which is physically meaningful (there is obvious double-counting), but it is a convenient artifice. For mode 4, the C=O stretch, the relative contribution of the carbonoxygen pair is 60.1%. It might be expected to be about 100% (after all, we envision the C=O bond as absorbing the photon); however, the fact that the carbon atom is vibrating implies that it is changing its position relative to the two hydrogen atoms. If the total vibrational energy, E_{v} (the actual energy of the absorbed photon, as distinct from T), were carried equally by the carbon and oxygen atoms, then the relative contributions to the mode would be C=O, 50% ; C–H, 25% ; C–H, 25%, respectively. This leads to the next entry, which is given in parentheses.

For the pair with the highest relative contribution (in mode 4, the C=O stretch), the energy of that pair divided by the total energy of the mode, E_{v}, is calculated as a percentage. This is the absolute contribution, A as a percentage, to the total energy of the mode.

A(A, B) = \frac{100 × P (A, B)}{E_{v}}

Now the C=O is seen to contribute 100.5 percent of the energy. For this sort of partitioning only the sum of all A’s must add to 100%, each pair can contribute more or less than 100%.

5.3 Example of reaction path with symmetry

In the case of a free rotator, e.g. ethane, the A of any specific bonded pair to the total energy can be very high (several hundred percent). It may be easier to view P/E_{v} as a contribution to the total energy of the mode, E_{v}. In this case the fact that P/E_{v} can be greater than unity can be explained by the fact that there are other relative motions within the molecule which make a negative contribution to E_{v}. From the R’s an idea can be obtained of where the energy of the mode is going; from the A value the significance of the highest contribution can be inferred. Thus, in mode 4 all three bonds are excited, but because the C=O bond carries about 100% of the energy, it is clear that this is really a C=O bond stretch mode, and that the hydrogens are only going along for the ride. In the last column the percentage radial motion is printed. This is useful in assigning the mode as stretching or bending. Any non-radial motion is de-facto tangential or bending. To summarize: The new analysis is more difficult to understand, but is considered by the author (JJPS) to be the easiest way of describing what are often complicated vibrations.

Note 8: In order, the thermodynamic quantities calculated are:

1. The vibrational contribution,

2. The rotational contribution,

3. The sum of (1) and (2), this gives the internal contribution,

4. The translational contribution.

For partition functions the various contributions are multiplied together.

A new quantity is the heat of formation at the defined temperature. This is intended for use in calculating heats of reaction. Because of a limitation in the data available, the H.o.F. at T Kelvin is defined as “The heat of formation of the compound at T Kelvin from it’s elements in their standard state at 298 Kelvin”. Obviously, this definition of heat of formation is incorrect, but should be useful in calculating heats of reaction, where the elements in their standard state at 298 Kelvin drop out.

5.3 Example of reaction path with symmetry

In this example, one methyl group in ethane is rotated relative to the other and the geometry is optimized at each point. As the reaction coordinate involves three hydrogen atoms moving, symmetry is imposed to ensure equivalence of all hydrogens.

Line  1:          SYMMETRY   T=600
Line  2:    ROTATION OF METHYL GROUP IN ETHANE
Line  3:    EXAMPLE OF A REACTION PATH CALCULATION
Line  4:    C
Line  5:    C    1.479146 1
Line  6:    H    1.109475 1  111.328433 1
Line  7:    H    1.109470 0  111.753160 0  120.000000 0   2  1  3
Line  8:    H    1.109843 0  110.103163 0  240.000000 0   2  1  3
Line  9:    H    1.082055 0  121.214083 0   60.000000 -1  1  2  3
Line 10:    H    1.081797 0  121.521232 0  180.000000 0   1  2  3
Line 11:    H    1.081797 0  121.521232 0  -60.000000 0   1  2  3
Line 12:    0    0.000000 0    0.000000 0    0.000000 0   0  0  0
Line 13:    3 1 4 5 6 7 8
Line 14:    3 2 4 5 6 7 8
Line 15:    6 7 7
Line 16:    6 11 8
Line 17:
Line 18:     70 80 90 100 110 120 130 140 150

Points to note:

1. The dihedrals of the second and third hydrogens are not marked for optimization: the dihedrals follow from point-group symmetry.

2. All six C–H bond lengths and H–C–C angles are related by symmetry: see lines 13 and 14.

3. The dihedral on line 9 is the reaction coordinate, while the dihedrals on lines 10 and 11 are related to it by symmetry functions on lines 15 and 16. The symmetry functions are defined by the second number on lines 13 to 16 (see SYMMETRY for definitions of functions 1, 2, 7, and 11).

4. Symmetry data are ended by a blank line.

5. The reaction coordinate data are ended by the end of file. Several lines of data are allowed.

6. Whenever symmetry is used in addition to other data below the geometry definition it will always follow the “blank line” immediately following the geometry definition. The other data will always follow the symmetry data.

Background

6.1 Introduction

While all the theory used in MOPAC is in the literature, so that in principle one could read and understand the algorithm, many parts of the code involve programming concepts or constructions which, while not of sufficient importance to warrant publication, are described here in order to facilitate understanding.

6.2 AIDER

AIDER will allow gradients to be defined for a system. MOPAC will calculate gradients, as usual, and will then use the supplied gradients to form an error function. This error function is: (supplied gradients − initial calculated gradients), which is then added to the computed gradients, so that for the initial SCF, the apparent gradients will equal the supplied gradients.

A typical data-set using AIDER would look like this:

 PM3 AIDER AIGOUT GNORM=0.01 EF
Cyclohexane
X
X     1    1.0
C     1    CX    2  CXX
C     1    CX    2  CXX    3  120.000000
C     1    CX    2  CXX    3 -120.000000
X     1    1.0   2  90.0   3    0.000000
X     1    1.0   6  90.0   2  180.000000
C     1    CX    7  CXX    3  180.000000
C     1    CX    7  CXX    3   60.000000
C     1    CX    7  CXX    3  -60.000000
H     3    H1C   1  H1CX   2    0.000000
H     4    H1C   1  H1CX   2    0.000000
H     5    H1C   1  H1CX   2    0.000000
H     8    H1C   1  H1CX   2  180.000000
H     9    H1C   1  H1CX   2  180.000000
H    10    H1C   1  H1CX   2  180.000000
H     3    H2C   1  H2CX   2  180.000000
H     4    H2C   1  H2CX   2  180.000000
H     5    H2C   1  H2CX   2  180.000000
H     8    H2C   1  H2CX   2    0.000000

Background

H     9    H2C   1  H2CX   2    0.000000
H    10    H2C   1  H2CX   2    0.000000
CX      1.46613
H1C     1.10826
H2C     1.10684
CXX    80.83255
H1CX  103.17316
H2CX  150.96100
AIDER
  0.0000
 13.7589   -1.7383
 13.7589   -1.7383    0.0000
 13.7589   -1.7383    0.0000
  0.0000    0.0000    0.0000
  0.0000    0.0000    0.0000
 13.7589   -1.7383    0.0000
 13.7589   -1.7383    0.0000
 13.7589   -1.7383    0.0000
-17.8599   -2.1083    0.0000
-17.8599   -2.1083    0.0000
-17.8599   -2.1083    0.0000
-17.8599   -2.1083    0.0000
-17.8599   -2.1083    0.0000
-17.8599   -2.1083    0.0000
-17.5612   -0.6001    0.0000
-17.5612   -0.6001    0.0000
-17.5612   -0.6001    0.0000
-17.5612   -0.6001    0.0000
-17.5612   -0.6001    0.0000
-17.5612   -0.6001    0.0000

Each supplied gradient goes with the corresponding internal coordinate. In the example given, the gradients came from a 3-21G calculation on the geometry shown. Symmetry will be taken into account automatically. Gaussian prints out gradients in atomic units; these need to be converted into kcal/mol/Angstrom or kcal/mol/radian for MOPAC to use. The resulting geometry from the MOPAC run will be nearer to the optimized 3-21G geometry than if the normal geometry optimizers in Gaussian had been used.

6.3 Correction to the peptide linkage

The residues in peptides are joined together by peptide linkages, –HNCO–. These linkages are almost flat, and normally adopt a trans configuration; the hydrogen and oxygen atoms being on opposite sides of the C–N bond. Experimentally, the barrier to interconversion in N-methyl acetamide is about 14 kcal/mole, but all four methods within MOPAC predict a significantly lower barrier, PM3 giving the lowest value.

The low barrier can be traced to the tendency of semiempirical methods to give pyramidal nitrogens. The degree to which pyramidalization of the nitrogen atom is preferred can be seen in the following series of compounds.

     Compound      MINDO/3    MNDO    AM1    PM3     Exp
Ammonia             Py          Py     Py     Py      Py
Aniline             Py          Py     Py     Py      Py

6.4 Level of precision within MOPAC

  Formamide           Py          Py     Flat   Py      Py
  Acetamide           Flat        Py     Flat   Py      Flat
  N-methyl formamide  Flat        Py     Flat   Py      Flat
  N-methyl acetamide  Flat        Flat   Flat   Py      Flat

To correct this, a molecular-mechanics correction has been applied. This consists of identifying the -R-HNCO- unit, and adding a torsion potential of form:

k × sin θ^{2}

where θ is the X–N–C–O angle, X=R or H, and k varies from method to method. This has two effects: there is a force constraining the nitrogen to be planar, and HNCO barrier in N–methyl acetamide is raised to 14.00 kcal/mole. When the MM correction is in place, the nitrogen atom for all methods for the last three compounds shown above is planar. The correction should be user-transparent.

Cautions

1. This correction will lead to errors of 0.5–1.5 kcal/mole if the peptide linkage is made or broken in a reaction calculation.

2. If the correction is applied to formamide the nitrogen will be flat, contrary to experiment.

3. When calculating rotation barriers, take into account the rapid rehybridization which occurs. When the dihedral is 0 or 180 degrees the nitrogen will be planar (sp2), but at 90 degrees the nitrogen should be pyramidal, as the partial double bond is broken. At that geometry the true transition state involves motion of the nitrogen substituent so that the nitrogen in the transition state is more nearly sp2. In other words, a simple rotation of the HNCO dihedral will not yield the activation barrier, however it will be within 2 kcal/mole of the correct answer. The 14 kcal barrier mentioned earlier refers to the true transition state.

4. Any job involving a CONH group will require either the keyword NOMM or MMOK. If you do not want the correction to be applied, use the keyword “NOMM” (NO Molecular Mechanics).

6.4 Level of precision within MOPAC

Several users have criticised the tolerances within MOPAC. The point made is that significantly different results have been obtained when different starting conditions have been used, even when the same conformer should have resulted. Of course, different results must be expected — there will always be small differences — nonetheless any differences should be small, e.g. heats of formation (HoF) differences should be less than about 0.1 kcal/mole. MOPAC has been modified to allow users to specify a much higher precision than the default when circumstances warrant it.

Reasons for low precision

There are several reasons for obtaining low quality results. The most obvious cause of such errors is that for general work the default criteria will result in a difference in HoF of less than 0.1 kcal/mole. This is only true for fairly rigid systems, e.g. formaldehyde and benzene. For systems with low barriers to rotation or flat potential surfaces, e.g. aniline or water dimer, quite large HoF errors can result.

Various precision levels

In normal (non-publication quality) work the default precision of MOPAC is recommended. This will allow reasonably precise results to be obtained in a reasonable time. Unless this precision proves unsatisfactory, use this default for all routine work.

The best way of controlling the precision of the geometry optimization and gradient minimization is by specifying a gradient norm which must be satisfied. This is done via the keyword GNORM=. Altering the GNORM automatically disables the other termination tests resulting in the gradient norm dominating the calculation. This works both ways: a GNORM of 20 will give a very crude optimization while a GNORM of 0.01 will give a very precise optimization. The default GNORM is 1.0.

When the highest precision is needed, such as in exacting geometry work, or when you want results which cannot be improved, then use the combination keywords GNORM=0.0 and SCFCRT=1.D-NN; NN should be in the range 2–15. Increasing the SCF criterion (the default is SCFCRT=1.D-4) helps the line search routines by increasing the precision of the heat of formation calculation; however, it can lead to excessive run times, so take care. Also, there is an increased chance of not achieving an SCF when the SCF criterion is excessively increased.

Superficially, requesting a GNORM of zero might seem excessively stringent, but as soon as the run starts, it will be cut back to 0.01. Even that might seem too stringent. The geometry optimization will continue to lower the energy, and hopefully the GNORM, but frequently it will not prove possible to lower the GNORM to 0.01. If, after 10 cycles, the energy does not drop then the job will be stopped. At this point you have the best geometry that MOPAC, in its current form, can give.

If a slightly less than highest precision is needed, such as for normal publication quality work, set the GNORM to the limit wanted. For example, for a flexible system, a GNORM of 0.1 to 0.5 will normally be good enough for all but the most demanding work.

If higher than the default, but still not very high precision is wanted, then use the keyword PRECISE. This will tighten up various criteria so that higher than routine precision will be given.

If high precision is used, so that the printed GNORM is 0.000, and the resulting geometry resubmitted for one SCF and gradients calculation, then normally a GNORM higher than 0.000 will result. This is NOT an error in MOPAC: the geometry printed is only precise to six figures after the decimal point. Geometries need to be specified to more than six decimals in order to drive the GNORM to less than 0.000.

If you want to test MOPAC, or use it for teaching purposes, the GNORM lower limit of 0.01 can be overridden by specifying LET, in which case you can specify any limit for GNORM. However, if it is too low the job may finish due to an irreducible minimum in the heat of formation being encountered. If this happens, the “STATIONARY POINT” message will be printed.

Finally there is a full analytical derivative function within MOPAC. These use STO-6G Gaussian wavefunctions because the derivatives of the overlap integral are easier to calculate in Gaussians than in STO’s. Consequently, there will be a small difference in the calculated HoFs when analytical derivatives are used. If there is any doubt about the accuracy of the finite derivatives, try using the analytical derivatives. They are a bit slower than finite derivatives but are more precise (a rough estimate is 12 figures for finite difference, 14 for analytical).

Some calculations, mainly open shell RHF or closed shell RHF with C.I. have untracked errors which prevent very high precision. For these systems GNORM should be in the range 1.0 to 0.1.

How large can a gradient be and still be acceptable?

A common source of confusion is the limit to which the GNORM should be reduced in order to obtain acceptable results. There is no easy answer, however a few guidelines can be given.

First of all reducing the GNORM to an arbitarily small number is not sensible. If the keywords GNORM=0.000001, LET, and EF are used, a geometry con be obtained which is precise to about 0.000001 Å. If ANALYT is also used, the results obtained will be slightly different. Chemically, this change is meaningless, and no significance should be attached to such numbers. In addition, 6.5 Convergence tests in subroutine ITER

any minor change to the algorithm, such as porting it to a new machine, will give rise to small changes in the optimized geometry. Even the small changes involved in going from MOPAC 5.00 to MOPAC 6.00 caused small changes in the optimized geometry of test molecules.

As a guide, a GNORM of 0.1 is sufficient for all heat-of-formation work, and a GNORM of 0.01 for most geometry work. If the system is large, you may need to settle for a GNORM of 1.0–0.5.

This whole topic was raised by Dr. Donald B. Boyd of Lilly Research Laboratories, who provided unequivocal evidence for a failure of MOPAC and convinced me of the importance of increasing precision in certain circumstances.

6.5 Convergence tests in subroutine ITER

Self–consistency test

The SCF iterations are stopped when two tests are satisfied. These are (1) when the difference in electronic energy, in eV, between any two consecutive iterations drops below the adjustable parameter, SELCON, and the difference between any three consecutive iterations drops below ten times SELCON, and (2) the difference in density matrix elements on two successive iterations falls below a preset limit, which is a multiple of SELCON.

SELCON is set initially to 0.0001 kcal/mole; this can be made 100 times smaller by specifying PRECISE or FORCE. It can be over-ridden by explicitly defining the SCF criterion via

SCFCRT=1.D-12.

SELCON is further modified by the value of the gradient norm, if known. If GNORM is large, then a more lax SCF criterion is acceptable, and SCFCRT can be relaxed up to 50 times it’s default value. As the gradient norm drops, the SCF criterion returns to its default value.

The SCF test is performed using the energy calculated from the Fock matrix which arises from a density matrix, and not from the density matrix which arises from a Fock. In the limit, the two energies would be identical, but the first converges faster than the second, without loss of precision.

6.6 Convergence in SCF calculation

A brief description of the convergence techniques used in subroutine ITER follows.

ITER, the SCF calculation, employs six methods to achieve a self-consistent field. In order of usage, these are:

1. Intrinsic convergence by virtue of the way the calculation is carried out. Thus a trial Fock gives rise to a trial density matrix, which in turn is used to generate a better Fock matrix.

This is normally convergent, but many exceptions are known. The main situations when the intrinsic convergence does not work are:

(a) A bad starting density matrix. This normally occurs when the default starting density matrix is used. This is a very crude approximation, and is only used to get the calculation started. A large charge is generated on an atom in the first iteration, the second iteration overcompensates, and an oscillation is generated.

(b) The equations are only very slowly convergent. This can be due to a long-lived oscillation or to a slow transfer of charge.

2. Oscillation damping. If, on any two consecutive iterations, a density matrix element changes by more than 0.05, then the density matrix element is set equal to the old element shifted by 0.05 in the direction of the calculated element. Thus, if on iterations 3 and 4 a certain density matrix element was 0.55 and 0.78, respectively, then the element would be set to 0.60 (=0.55+0.05) on iteration 4. The density matrix from iteration 4 would then be used in the construction of the next Fock matrix. The arrays which hold the old density matrices are not filled until after iteration 2. For this reason they are not used in the damping before iteration 3.

3. Three-point interpolation of the density matrix. Subroutine CNVG monitors the number of iterations, and if this is exactly divisible by three, and certain other conditions relating to the density matrices are satisfied, a three-point interpolation is performed. This is the default converger, and is very effective with normally convergent calculations. It fails in certain systems, usually those where significant charge build-up is present.

4. Energy-level shift technique. The virtual M.O. energy levels are normally shifted to more positive energy. This has the effect of damping oscillations, and intrinsically divergent equations can often be changed to intrinsically convergent form. With slowly-convergent systems the virtual M.O. energy levels can be moved to a more negative value.

The precise value of the shift used depends on the behavior of the iteration energy. If it is dropping, then the HOMO-LUMO gap is reduced, if the iteration energy rises, the gap is increased rapidly.

5. Pulay’s method. If requested, when the largest change in density matrix elements on two consecutive iterations has dropped below 0.1, then routine CNVG is abandoned in favor of a multi-Fock matrix interpolation. This relies on the fact that the eigenvectors of the density and Fock matrices are identical at self-consistency, so [P.F]=0 at SCF. The extent to which this condition does not occur is a measure of the deviance from self-consistency. Pulay’s method uses this relationship to calculate that linear combination of Fock matrices which minimize [P.F]. This new Fock matrix is then used in the SCF calculation.

Under certain circumstances, Pulay’s method can cause very slow convergence, but sometimes it is the only way to achieve a self-consistent field. At other times the procedure gives a ten-fold increase in speed, so care must be exercised in its use. (invoked by the keyword PULAY)

6. The Camp-King converger. If all else fails, the Camp-King converger is just about guaranteed to work every time. However, it is time-consuming, and therefore should only be invoked as a last resort.

It evaluates that linear combination of old and current eigenvectors which minimize the total energy. One of its strengths is that systems which otherwise oscillate due to charge surges, e.g. CHO–H, the C–H distance being very large, will converge using this very sophisticated converger.

6.7 Causes of failure to achieve an SCF

In a system where a biradical can form, such as ethane decomposing into two CH3 units, the normal RHF procedure can fail to go self-consistent. If the system has marked biradicaloid character, then BIRADICAL or UHF and TRIPLET can often prove successful. These options rely on the assumption that two unpaired electrons can represent the open shell part of the wave-function.

Consider H–Cl, with the interatomic distance being steadily increased. At first the covalent bond will be strong, and a self-consistent field is readily obtained. Gradually the bond will become more ionic, and eventually the charge on chlorine will become very large. The hydrogen, meanwhile, will become very electropositive, and there will be an increased energy advantage to any one electron to transfer from chlorine to hydrogen. If this in fact occurred, the hydrogen would suddenly become very electron-rich and would, on the next iteration, lose its extra electron to the chlorine. A sustained oscillation would then be initiated. To prevent this, if BIRADICAL is specified, exactly one electron will end up on hydrogen. A similar result can be obtained by specifying TRIPLET in a UHF calculation.

6.8 Torsion or dihedral angle coherency

6.8 Torsion or dihedral angle coherency

MOPAC calculations do not distinguish between enantiomers, consequently the sign of the dihedrals can be multiplied by −1 and the calculations will be unaffected. However, if chirality is important, a user should be aware of the sign convention used.

The dihedral angle convention used in MOPAC is that defined by Klyne and Prelog in Experientia 16, 521 (1960). In this convention, four atoms, AXYB, with a dihedral angle of 90 degrees, will have atom B rotated by 90 degrees clockwise relative to A when X and Y are lined up in the direction of sight, X being nearer to the eye. In their words, “To distinguish between enantiomeric types the angle ‘tau’ is considered as positive when it is measured clockwise from the front substituent A to the rear substituent B, and negative when it is measured anticlockwise.” The alternative convention was used in all earlier programs, including QCPE 353.

6.9 Vibrational analysis

Analyzing normal coordinates is very tedious. Users are normally familiar with the internal coordinates of the system they are studying, but not familiar with the cartesian coordinates. To help characterize the normal coordinates, a very simple analysis is done automatically, and users are strongly encouraged to use this analysis first, and then to look at the normal coordinate eigenvectors.

In the analysis, each pair of bonded atoms is examined to see if there is a large relative motion between them. By bonded is meant within the Van der Waals’ distance. If there is such a motion, the indices of the atoms, the relative distance in Angstroms, and the percentage radial motion are printed. Radial plus tangential motion adds to 100%, but as there are two orthogonal tangential motions and only one radial, the radial component is printed.

6.10 A note on thermochemistry

By

Tsuneo Hirano Department of Synthetic Chemistry Faculty of Engineering University of Tokyo Hongo, Bunkyo-ku, Tokyo, Japan

6.10.1 Basic Physical Constants

Taken from: “Quantities, Units and Symbols in Physical Chemistry,” Blackwell Scientific Publications Ltd, Oxford OX2 0EL, UK, 1987 (IUPAC, based on CODATA of ICSU, 1986). pp 81–82.

Speed of light, c = 2.99792458 × 10^{10} cm/s (Definition)

Boltzmann constant, k = R/Na = 1.380658 × 10^{−23} J/K = 1.380658 × 10^{−16} erg/K Planck constant, h = 6.6260755 × 10^{−34} J s = 6.6260755 × 10^{−27} erg s Gas constant, R = 8.314510 J/mol/K = 1.987216 cal/mol/K Avogadro number, N_{a} = 6.0221367 × 10^{23} /mol Volume of 1 mol of gas, V_{0} = 22.41410 l/mol (at 1 atm, 25 C) 1 J = 1. × 10^{7} erg 1 kcal = 4.184 kJ (Definition) 1 eV = 23.0606 kcal/mol 1 a.u. = 27.21135 eV/mol = 627.509 6 kcal/mol 1 cm^{−1} = 2.859144 cal/mol = N_{a}hc/4.184^{7} 1 atm = 1.01325 × 10^{5} Pa = 1.01325 × 10^{6} dyn/cm^{2} (Definition)

Moment of inertia: I 1 amu angstrom^{2} = 1.660540 × 10^{−40} g cm^{2}.

Rotational constants: A, B, and C (e.g. A = h/(8π^{2}I)) With I in amu angstroms^{2} then: A (in MHz) = 5.053791 × 10^{5}/I A (in cm^{−1}) = 5.053791 × 10^{5}/cI = 16.85763/I

6.10.2 Thermochemistry from ab initio MO methods

Ab initio MO methods provide total energies, E_{eq}, as the sum of electronic and nuclear-nuclear repulsion energies for molecules, isolated in vacuum, without vibration at 0 K.

E_{eq} = E_{el} + E_{nuclear−nuclear}  (6.1)

¿From the 0 K potential surface and using the harmonic oscillator approximation, we can calculate the vibrational frequencies, ν_{i}, of the normal modes of vibration. Using these, we can calculate vibrational, rotational and translational contributions to the thermodynamic quantities such as the partition function and heat capacity which arise from heating the system from 0 to T K.

Q: partition function, E: energy, S: entropy, and C: heat capacity.

[Vibration]

Q_{vib} = ∑_{i} \frac{1}{[1 − exp(−hν_{i}/kT)]}  (6.2)

E_{vib}, for a molecule at the temperature T as:

E_{vib} = ∑_{i} \{\frac{hν_{i}}{2} + \frac{hν_{i} exp(−hν_{i}/kT)}{[1 − exp(−hν_{i}/kT)]}\}  (6.3)

where h is the Planck constant, ν_{i} the i–th normal vibration frequency, and k the Boltzmann constant. For 1 mole of molecules, E_{vib} should be multiplied by the Avogadro number N_{a} = R/k. Thus:

E_{vib} = N_{a} ∑_{i} \{\frac{hν_{i}}{2} + \frac{hν_{i} exp(−hν_{i}/kT)}{[1 − exp(−hν_{i}/kT)]}\}  (6.4)

Note that the first term in equation (6.4) is the Zero-point vibration energy. Hence, the second term in eq. (6.4) is the additional vibrational contribution due to the temperature increase from 0 K to T K. Namely,

E_{vib} = E_{zero} + E_{vib}(0 → T)  (6.5)
E_{zero} = N_{a} ∑_{i} \frac{hν_{i}}{2}  (6.6)

E_{vib}(0 → T) = N_{a} ∑_{i} \frac{hν_{i} exp(−hν_{i}/kT)}{[1 − exp(−hν_{i}/kT)]}  (6.7)

The value of E_{vib} from GAUSSIAN 82 and 86 includes E_{zero} as defined by Eqs. (6.4,6.7).

S_{vib} = R ∑_{i} \{\frac{(hν_{i}/kT) exp(−hν_{i}/kT)}{[1 − exp(−hν_{i}i/kT)]} − ln[1 − exp(−hν_{i}/kT)]\}  (6.8)

C_{vib} = R ∑_{i} \{\frac{(hν_{i}/kT)^{2} exp(−hν_{i}/kT)}{[1 − exp(−hν_{i}/kT)]^{2}}\}  (6.9)

At temperature T > 0 K, a molecule rotates about the x, y, and z-axes and translates in x, y, and z-directions. By assuming the equipartition of energy, energies for rotation and translation, E_{rot} and E_{tr}, are calculated.

6.10 A note on thermochemistry

[Rotation]

σ is symmetry number. I is moment of inertia. I_{A}, I_{B}, and I_{C} are moments of inertia about A, B, and C axes.

Linear molecule

Q_{rot} = \frac{8π^{2}IkT}{σh^{2}}  (6.10)
E_{rot} = (2/2)RT  (6.11)
S_{rot} = R ln [\frac{8π^{2}IkT}{σh^{2}} ] + R  (6.12)
= R ln I + R ln T − R ln σ − 4.349203

where −4.349203 = R ln[8 × 10^{−16}π^{2}k/(N_{a}h^{2})] + R.

C_{rot} = (2/2)R  (6.13)

Non-linear molecule

Q_{rot} = (\frac{\sqrt{π}}{σ} ) [\frac{8π^{2}kT}{h^{2}} ]^{3/2} \sqrt{I_{A}I_{B}I_{C}}
= (\frac{\sqrt{π}}{σ} ) [(\frac{8π^{2}cI_{A}}{h} ) (\frac{8π^{2}cI_{B}}{h} ) (\frac{8π^{2}cI_{C}}{h} )]^{1/2} (\frac{kT}{hc} )^{3/2}  (6.14)
E_{rot} = (3/2)RT  (6.15)
S_{rot} = \frac{R}{2} ln \{(\frac{π}{\sqrt{σ}} ) (\frac{8π^{2}cI_{A}}{h} ) (\frac{8π^{2}cI_{B}}{h} ) (\frac{8π^{2}cI_{C}}{h} ) (\frac{kT}{hc} )^{3}\} + (3/2)R (6.16)
= (R/2) ln (I_{A}I_{B}I_{C}) + (3/2)R ln T − R ln σ − 5.3863921

Here, −5.3863921 is calculated as:

R ln \{\frac{1}{h^{3}} (\frac{10^{−16}}{N_{a}} )^{3/2} \sqrt{(3 × 2^{9} × π^{7} × k)}\} + (3/2)R.

C_{rot} = (3/2)R  (6.17)

[Translation]

M is Molecular weight.

Q_{tra} = (\frac{\sqrt{2πMkT/N_{a}}}{h} )^{3}  (6.18)
E_{tra} = (3/2)RT  (6.19)
S_{tra} = R \{\frac{5}{2} + \frac{3}{2} ln (\frac{2πk}{h^{2}} ) + ln k + \frac{3}{2} ln (\frac{M}{N_{a}} ) + \frac{5}{2} ln T − ln p\}  (6.20)
= (5/2)R ln T + (3/2)R ln M − R ln p − 2.31482  (6.21)
C_{tra} = (5/2)R  (6.22)

or H_{tra} = (5/2)RT due to the pV term (cf. H = U + pV). The internal energy U at T is:

U = E_{eq} + [E_{vib} + E_{rot} + E_{tra}]  (6.23)

or

U = E_{eq} + [(E_{zero} + E_{vib}(0 → T)) + E_{rot} + E_{tra}]  (6.24)

Enthalpy H for one mole of gas is defined as

H = U + pV  (6.25)

Assumption of an ideal gas (i.e., pV = RT) leads to

H = U + pV = U + RT  (6.26)

Thus, Gibbs free energy G can be calculated as:

G = H − T S(0 → T)  (6.27)

Thermochemistry in MOPAC

It should be noted that MO parameters for MINDO/3, MNDO, AM1 and PM3 are optimized so as to reproduce the experimental heat of formation (i.e., standard enthalpy of formation or the enthalpy change to form a mole of compound at 25 degrees C from its elements in their standard state) as well as observed geometries (mostly at 25 degrees C), and not to reproduce the E_{eq} and equilibrium geometry at 0 K.

In this sense, E_{scf} (defined as Heat of formation), force constants, normal vibration frequencies etc are all related to the values at 25 degree C, not to 0 K!!!!! Therefore, the E_{zero} calculated in FORCE is not the true E_{zero}. Its use as E_{zero} should be made at your own risk, bearing in mind the situation discussed above.

Since E_{scf} is standard enthalpy of formation (at 25 degree C):

E_{scf} = E_{eq}+E_{zero}+E_{vib}(0 → 298.15)+E_{rot}+E_{tra}+pV +∑ [−E_{elec}(atom) + ΔH_{f}(atom)] (6.28)

To avoid the complication arising from the definition of E_{scf}, within the thermodynamics calculation the Standard Enthalpy of Formation, ΔH, is calculated by

ΔH = E_{scf} + (H_{T} − H_{298})  (6.29)

Here, E_{scf} is the heat of formation (at 25 degree C) given in the output list, and H_{T} and H_{298} are the enthalpy contributions for the increase of the temperature from 0 K to T and 298.15, respectively. In other words, the enthalpy of formation is corrected for the difference in temperature from 298.15 K to T. The method of calculation for T and H_{298} will be given below.

In MOPAC, the variables defined below are used:

C_{1} = \frac{hc}{kT}  (6.30)

The wavenumber, ω_{i}, in cm^{−1}:

ν_{i} = ω_{i}c  (6.31)
E_{WJ} = exp(−hν_{i}/kT) = exp(−ω_{i}hc/kT) = exp(−ω_{i}C_{1})  (6.32)

The rotational constants A, B, and C in cm^{−1}:

A = \frac{h}{(8π^{2}cI_{A})}  (6.33)

Energy and Enthalpy in cal/mol, and Entropy in cal/mol/K. Thus, eqs. (6.2–6.27) can be written as follows.

6.10 A note on thermochemistry [Vibration]

Q_{vib} = π ∑_{i} \frac{1}{(1 − E_{WJ})}  (6.34)
E_{0} = \frac{0.5N_{a}hc}{4.184 × 10^{7}} ∑_{i} ω_{i}  (6.35)
= 1.429572 ∑_{i} ω_{i}  (6.36)
E_{vib}(0 → T) = N_{a}hc ∑_{i} \frac{ω_{i}E_{WJ}}{1 − E_{WJ}} = (R/k)hc ∑_{i} \frac{W_{i}E_{WJ}}{1 − E_{WJ}}  (6.37)
S_{vib} = R(hc/kT) ∑_{i} \{\frac{ω_{i}E_{WJ}}{(1 − E_{WJ})}\} − R ∑_{i} ln(1 − E_{WJ})
= RC_{1} ∑_{i} \{\frac{ω_{i}E_{WJ}}{(1 − E_{WJ})}\} − R ∑_{i} ln(1 − E_{WJ})  (6.38)
C_{vib} = R(hc/kT)^{2} ∑_{i} \{\frac{ω_{i}^{2}E_{WJ}}{(1 − E_{WJ})^{2}} \}
= RC_{1}^{2} ∑_{i} \{\frac{ω_{i}^{2}E_{WJ}}{(1 − E_{WJ})^{2}} \}  (6.39)

[Rotation]

Linear molecule

Q_{rot} = (1/σ)(1/A)(kT/hc) = \frac{1}{σAC_{1}}  (6.40)
E_{rot} = (2/2)RT  (6.41)
S_{rot} = R ln (\frac{kT}{σhcA}) + R = R ln (\frac{1}{σAC_{1}} ) + R = R ln (\frac{kT}{σhcA}) + R  (6.42)
C_{rot} = (2/2)R  (6.43)

Non-linear molecule

Q_{rot} = \frac{1}{σ} [\frac{π}{(ABCC_{1}^{3})}]^{1/2}  (6.44)
E_{rot} = (3/2)RT  (6.45)
S_{rot} = \frac{R}{2} ln \{\frac{π}{σ^{2}ABC} (\frac{kT}{hc} )^{3}\} + (3/2)R
= 0.5R3 ln(kT/hc) − 2 ln σ + ln (\frac{π}{ABC} ) + 3  (6.46)
= 0.5R−3 ln C_{1} − 2 ln σ + ln (\frac{π}{ABC} ) + 3
C_{rot} = (3/2)R  (6.47)

[Translation]

Q_{tra} = (\frac{\sqrt{2πMkT/N_{a}}}{h} )^{3} = (\frac{\sqrt{1.660540 ×^{−24} ×2πMkT}}{h} )^{3}  (6.48)

E_{tra} = (3/2)RT  (6.49)
H_{tra} = (3/2)RT + pV = (5/2)RT cf. pV = RT  (6.50)
S_{tra} = (R/2)[5 ln T + 3 ln M] − 2.31482 cf. p = 1atm
= 0.993608[5 ln T + 3 ln M] − 2.31482  (6.51)

In MOPAC:

H_{vib} = E_{vib}(0 → T)  (6.52)

(Note: E_{zero} is not included in H_{vib} ω_{i} is not derived from force-constants at 0 K) and for T:

H_{T} = [H_{vib} + H_{rot} + H_{tra}]  (6.53)

while for T = 298.15 K:

H_{298} = [H_{vib} + H_{rot} + H_{tra}]  (6.54)

Note that H_{T} (and H_{298}) is equivalent to:

(E_{vib} − E_{zero}) + E_{rot} + (E_{tra} + pV)  (6.55)

except that the normal frequencies are those obtained from force constants at 25 degree C, or at least not at 0 K.

Thus, Standard Enthalpy of Formation, ΔH, can be calculated according to Eqs. (6.24,6.25) and (6.28), as shown in Eq. (6.29);

ΔH = E_{scf} + (H_{T} − H_{298})  (6.56)

Note that E_{zero} is already counted in E_{scf}, see Eq. (6.28).

By using Eq. (6.26), Standard Internal Energy of Formation, ΔU, can be calculated as:

ΔU = ΔH − R(T − 298.15)  (6.57)

Standard Gibbs Free-Energy of Formation, ΔG, can be calculated by taking the difference from that for the isomer or that at different temperature:

ΔG = [ΔH − T S] (for the state under consideration) − [ΔH − T S] (for reference state) (6.58)

Taking the difference is necessary to cancel the unknown values of standard entropy of formation for the constituent elements.

6.11 Reaction coordinates

The Intrinsic Reaction Coordinate method pioneered and developed by Mark Gordon has been incorporated in a modified form into MOPAC. As this facility is quite complicated all the keywords associated with the IRC have been grouped together in this section.

DRC

The Dynamic Reaction Coordinate is the path followed by all the atoms in a system assuming conservation of energy, i.e., as the potential energy changes the kinetic energy of the system changes in exactly the opposite way so that the total energy (kinetic plus potential) is a constant. If started at a ground state geometry, no significant motion should be seen. Similarly, starting at a transition state geometry should not produce any motion - after all it is a stationary point and during the lifetime of a calculation it is unlikely to accumulate enough momentum to travel far from the starting position.

In order to calculate the DRC path from a transition state, either an initial deflection is necessary or some initial momentum must be supplied.

Because of the time-dependent nature of the DRC the time elapsed since the start of the reaction is meaningful, and is printed.

6.11 Reaction coordinates

Description

The course of a molecular vibration can be followed by calculating the potential and kinetic energy at various times. Two extreme conditions can be identified: (a) gas phase, in which the total energy is a constant through time, there being no damping of the kinetic energy allowed, and (b) liquid phase, in which kinetic energy is always set to zero, the motion of the atoms being infinitely damped.

All possible degrees of damping are allowed. In addition, the facility exists to dump energy into the system, appearing as kinetic energy. As kinetic energy is a function of velocity, a vector quantity, the energy appears as energy of motion in the direction in which the molecule would naturally move. If the system is a transition state, then the excess kinetic energy is added after the intrinsic kinetic energy has built up to at least 0.2 kcal/mole.

For ground-state systems, the excess energy sometimes may not be added; if the intrinsic kinetic energy never rises above 0.2kcal/mole then the excess energy will not be added.

Equations used

Force acting on any atom:

g(i) + g′(i)t + g′′(i)t^{2} = \frac{dE}{dx(i)} + \frac{d^{2}E}{dx(i)^{2}} + \frac{d^{3}E}{dx(i)^{3}}

Acceleration due to force acting on each atom:

a(i) = \frac{1}{M(i)}(g(i) + g′(i)t + g′′(i)t^{2})

New velocity:

V (o) + \frac{1}{M(i)} (Dtg(i) + (1/2)Dt^{2}g′(i) + (1/3)Dt^{3}g′′(i))

or:

V (i) = V (i) + V′(i)t + V′′(i)t^{2} + V′′′(i)t^{3}

That is, the change in velocity is equal to the integral over the time interval of the acceleration. New position of atoms:

X(i) = X(o) + V (o)t + (1/2)V′t^{2} + (1/3)V′′t^{3} + (1/4)V′′′t^{4}

That is, the change in position is equal to the integral over the time interval of the velocity.

The velocity vector is accurate to the extent that it takes into account the previous velocity, the current acceleration, the predicted acceleration, and the change in predicted acceleration over the time interval. Very little error is introduced due to higher order contributions to the velocity; those that do occur are absorbed in a re-normalization of the magnitude of the velocity vector after each time interval.

The magnitude of Dt, the time interval, is determined mainly by the factor needed to renormalize the velocity vector. If it is significantly different from unity, Dt will be reduced; if it is very close to unity, Dt will be increased.

Even with all this, errors creep in and a system, started at the transition state, is unlikely to return precisely to the transition state unless an excess kinetic energy is supplied, for example 0.2kcal/mole.

The calculation is carried out in cartesian coordinates, and converted into internal coordinates for display. All cartesian coordinates must be allowed to vary, in order to conserve angular and translational momentum.

IRC

The Intrinsic Reaction Coordinate is the path followed by all the atoms in a system assuming all kinetic energy is completely lost at every point, i.e., as the potential energy changes the kinetic energy generated is annihilated so that the total energy (kinetic plus potential) is always equal to the potential energy only.

The IRC is intended for use starting with the transition state geometry. A normal coordinate is chosen, usually the reaction coordinate, and the system is displaced in either the positive or negative direction along this coordinate. The internal modes are obtained by calculating the mass-weighted Hessian matrix in a force calculation and translating the resulting cartesian normal mode eigenvectors to conserve momentum. That is, the initial cartesian coordinates are displaced by a small amount proportional to the eigenvector coefficients plus a translational constant; the constant is required to ensure that the total translational momentum of the system is conserved as zero. At the present time there may be small residual rotational components which are not annihilated; these are considered unimportant.

General description of the DRC and IRC

As the IRC usually requires a normal coordinate, a force constant calculation normally has to be done first. If IRC is specified on its own a normal coordinate is not used and the IRC calculation is performed on the supplied geometry.

A recommended sequence of operations to start an IRC calculation is as follows:

1. Calculate the transition state geometry. If the T/S is not first optimized, then the IRC calculation may give very misleading results. For example, if NH3 inversion is defined as the planar system but without the N–H bond length being optimized the first normal coordinate might be for N–H stretch rather than inversion. In that case the IRC will relax the geometry to the optimized planar structure.

2. Do a normal FORCE calculation, specifying ISOTOPE in order to save the FORCE matrices. Do not attempt to run the IRC directly unless you have confidence that the FORCE calculation will work as expected. If the IRC calculation is run directly, specify ISOTOPE anyway: that will save the FORCE matrix and if the calculation has to be re-done then RESTART will work correctly.

3. Using IRC=n and RESTART run the IRC calculation. If RESTART is specified with IRC=n then the restart is assumed to be from the FORCE calculation. If RESTART is specified without IRC=n, say with IRC on its own, then the restart is assumed to be from an earlier IRC calculation that was shut down before going to completion.

A DRC calculation is simpler in that a force calculation is not a prerequisite; however, most calculations of interest normally involve use of an internal coordinate. For this reason IRC=n can be combined with DRC to give a calculation in which the initial motion (0.3kcal worth of kinetic energy) is supplied by the IRC, and all subsequent motion obeys conservation of energy. The DRC motion can be modified in three ways:

1. It is possible to calculate the reaction path followed by a system in which the generated kinetic energy decays with a finite half-life. This can be defined by DRC=n.nnn, where n.nnn is the half-life in femtoseconds. If n.nn is 0.0 this corresponds to infinite damping simulating the IRC. A limitation of the program is that time only has meaning when DRC is specified without a half-life.

2. Excess kinetic energy can be added to the calculation by use of KINETIC=n.nn. After the kinetic energy has built up to 0.2kcal/mole or if IRC=n is used then n.nn kcal/mole of kinetic energy is added to the system. The excess kinetic energy appears as a velocity vector in the same direction as the initial motion.

3. The RESTART file <filename>.RES can be edited to allow the user to modify the velocity vector or starting geometry. This file is formatted.

Frequently DRC leads to a periodic, repeating orbit. One special type — the orbit in which the direction of motion is reversed so that the system retraces its own path — is sensed for and if detected the calculation is stopped after exactly one cycle. If the calculation is to be continued, the keyword GEO-OK will allow this check to be by-passed.

Due to the potentially very large output files that the DRC can generate extra keywords are provided to allow selected points to be printed. After the system has changed by a preset amount the following keywords can be used to invoke a print of the geometry.

          KeyWord         Default             User Specification
          X-PRIO      0.05 Angstroms             X-PRIORITY=n.nn
          T-PRIO      0.10 Femtoseconds          T-PRIORITY=n.nn
          H-PRIO      0.10 kcal/mole             H-PRIORITY=n.nn

Option to allow only extrema to be output

In the geometry specification, if an internal coordinate is marked for optimization then when that internal coordinate passes through an extremum a message will be printed and the geometry output.

Difficulties can arise from the way internal coordinates are processed. The internal coordinates are generated from the cartesian coordinates, so an internal coordinate supplied may have an entirely different meaning on output. In particular the connectivity may have changed. For obvious reasons dummy atoms should not be used in the supplied geometry specification. If there is any doubt about the internal coordinates or if the starting geometry contains dummy atoms then run a 1SCF calculation specifying XYZ. This will produce an ARC file with the “ideal” numbering — the internal numbering system used by MOPAC. Use this ARC file to construct a data file suitable for the DRC or IRC.

Notes:

1. Any coordinates marked for optimization will result in only extrema being printed.

2. If extrema are being printed then kinetic energy extrema will also be printed.

Keywords for use with the IRC and DRC

1. Setting up the transition state: NLLSQ SIGMA TS.

2. Constructing the FORCE matrix: FORCE or IRC=n, ISOTOPE, LET.

3. Starting an IRC: RESTART and IRC=n, T-PRIO, X-PRIO, H-PRIO.

4. Starting a DRC: DRC or DRC=n.nn, KINETIC=n.nn.

5. Starting a DRC from a transition state: (DRC or DRC=n) and IRC=n, KINETIC=n.

6. Restarting an IRC: RESTART and IRC.

7. Restarting a DRC: RESTART and (DRC or DRC=n.nn).

8. Restarting a DRC starting from a transition state: RESTART and (DRC or DRC=n.nn).

Other keywords, such as T=nnn or GEO-OK can be used anytime.

Examples of DRC/IRC data

Use of the IRC/DRC facility is quite complicated. In the following examples various ‘reasonable’ options are illustrated for a calculation on water. It is assumed that an optimized transition-state geometry is available.

Example 1: A Dynamic Reaction Coordinate, starting at the transition state for water inverting, initial motion opposite to the transition normal mode, with 6kcal of excess kinetic energy added in. Every point calculated is to be printed (Note all coordinates are marked with a zero, and T-PRIO, H-PRIO and X-PRIO are all absent). The results of an earlier calculation using the same keywords is assumed to exist. The earlier calculation would have constructed the force matrix. While the total cpu time is specified, it is in fact redundant in that the calculation will run to completion in less than 600 seconds.

    KINETIC=6 RESTART  IRC=-1 DRC T=600
        WATER
      H    0.000000  0    0.000000  0    0.000000  0   0  0  0
      O    0.911574  0    0.000000  0    0.000000  0   1  0  0
      H    0.911574  0  180.000000  0    0.000000  0   2  1  0
      0    0.000000  0    0.000000  0    0.000000  0   0  0  0

Example 2: An Intrinsic Reaction Coordinate calculation. Here the restart is from a previous IRC calculation which was stopped before the minimum was reached. Recall that RESTART with IRC=n implies a restart from the FORCE calculation. Since this is a restart from within an IRC calculation the keyword IRC=n has been replaced by IRC. IRC on its own (without the “=n”) implies an IRC calculation from the starting position — here the RESTART position — without initial displacement.

       RESTART  IRC  T=600
        WATER
      H    0.000000  0    0.000000  0    0.000000  0   0  0  0
      O    0.911574  0    0.000000  0    0.000000  0   1  0  0
      H    0.911574  0  180.000000  0    0.000000  0   2  1  0
      0    0.000000  0    0.000000  0    0.000000  0   0  0  0

Output format for IRC and DRC

The IRC and DRC can produce several different forms of output. Because of the large size of these outputs, users are recommended to use search functions to extract information. To facilitate this, specific lines have specific characters. Thus, a search for the “%” symbol will summarize the energy profile while a search for “AA” will yield the coordinates of atom 1, whenever it is printed. The main flags to use in searches are:

           SEARCH FOR                    YIELDS
            '% '         Energies for all points calculated,
                         excluding extrema
            '%M'         Energies for all turning points
            '%MAX'       Energies for all maxima
            '%MIN'       Energies for all minima
            '%'          Energies for all points calculated
            'AA*'        Internal coordinates for atom 1 for every point
            'AE*'        Internal coordinates for atom 5 for every point
            '123AB*'     Internal coordinates for atom 5 for point 123

As the keywords for the IRC/DRC are interdependent, the following list of keywords illustrates various options.

    KEYWORD                 RESULTING ACTION
    DRC                     The Dynamic Reaction Coordinate is calculated.
                            Energy is conserved, and no initial impetus.
    DRC=0.5                 In the DRC kinetic energy is lost with a half-
                            life of 0.5 femtoseconds.
    DRC=-1.0                Energy is put into a DRC with an half-life of
                            -1.0 femtoseconds, i.e., the system gains
                            energy.
    IRC                     The Intrinsic Reaction Coordinate is
                            calculated.  No initial impetus is given.
                            Energy not conserved.
    IRC=-4                  The IRC is run starting with an impetus in the
                            negative of the 4th normal mode direction. The
                            impetus is one quantum of vibrational energy.
    IRC=1 KINETIC=1         The first normal mode is used in an IRC, with
                            the initial impetus being 1.0kcal/mole.
    DRC KINETIC=5           In a DRC, after the velocity is defined, 5 kcal
                            of kinetic energy is added in the direction of
                            the initial velocity.
    IRC=1 DRC KINETIC=4     After starting with a 4 kcal impetus in the
                            direction of the first normal mode, energy is
                            conserved.
    DRC VELOCITY KINETIC=10 Follow a DRC trajectory which starts with an
                            initial velocity read in, normalized to a
                            kinetic energy of 10 kcal/mol.

Instead of every point being printed, the option exists to print specific points determined by the keywords T-PRIORITY, X-PRIORITY and H-PRIORITY. If any one of these words is specified, then the calculated points are used to define quadratics in time for all variables normally printed. In addition, if the flag for the first atom is set to T then all kinetic energy turning points are printed. If the flag for any other internal coordinate is set to T then, when that coordinate passes through an extremum, that point will be printed. As with the PRIORITYs, the point will be calculated via a quadratic to minimize non-linear errors.

N.B.: Quadratics are unstable in the regions of inflection points, in these circumstances linear interpolation will be used. A result of this is that points printed in the region of an inflection may not correspond exactly to those requested. This is not an error and should not affect the quality of the results.

Test of DRC—verification of trajectory path

Introduction: Unlike a single-geometry calculation or even a geometry optimization, verification of a DRC trajectory is not a simple task. In this section a rigorous proof of the DRC trajectory is presented; it can be used both as a test of the DRC algorithm and as a teaching exercise. Users of the DRC are asked to follow through this proof in order to convince themselves that the DRC works as it should.

Part 1: The nitrogen molecule

For the nitrogen molecule and using MNDO, the equilibrium distance is 1.103802 Å, the heat of formation is 8.276655 kcal/mole and the vibrational frequency is 2739.6 cm^{−1}. For small displacements, the energy curve versus distance is parabolic and the gradient curve is approximately linear, as is shown in the following table. A nitrogen molecule is thus a good approximation to a harmonic oscillator.

STRETCHING CURVE FOR NITROGEN MOLECULE
N--N DIST     HoF            GRADIENT
(Angstroms)   (kcal/mole) (kcal/mole/Angstrom)
1.1180       8.714564        60.909301
1.1170       8.655723        56.770564
1.1160       8.601031        52.609237
1.1150       8.550512        48.425249
1.1140       8.504188        44.218525
1.1130       8.462082        39.988986
1.1120       8.424218        35.736557
1.1110       8.390617        31.461161
1.1100       8.361303        27.162720
1.1090       8.336299        22.841156
1.1080       8.315628        18.496393
1.1070       8.299314        14.128353
1.1060       8.287379         9.736959
1.1050       8.279848         5.322132
1.1040       8.276743         0.883795
1.1030       8.278088        -3.578130
1.1020       8.283907        -8.063720
1.1010       8.294224       -12.573055
1.1000       8.309061       -17.106213
1.0990       8.328444       -21.663271
1.0980       8.352396       -26.244309
1.0970       8.380941       -30.849404
1.0960       8.414103       -35.478636
1.0950       8.451906       -40.132083
1.0940       8.494375       -44.809824
1.0930       8.541534       -49.511939
1.0920       8.593407       -54.238505
1.0910       8.650019       -58.989621
1.0900       8.711394       -63.765330

Period of vibration

The period of vibration (time taken for the oscillator to undertake one complete vibration, returning to its original position and velocity) can be calculated in three ways. Most direct is the calculation from the energy curve; using the gradient constitutes a faster, albeit less direct, method, while calculating it from the vibrational frequency is very fast but assumes that the vibrational spectrum has already been calculated.

1. From the energy curve. For a simple harmonic oscillator the period r is given by:

r = 2π\sqrt{\frac{m}{k}}

where m is the reduced mass and k is the force constant. The reduced mass (in amu) of a nitrogen molecule is 14.0067/2 = 7.00335, and the force-constant can be calculated from:

E − c = (1/2)k(R − R_{o})^{2}

Given R_{o} = 1.1038, R = 1.092, c = 8.276655 and E = 8.593407 kcal/mol then:

= 4548.2 kcal/mole/A^{2}
= 4545 × 4.184 × 10^{3} × 10^{7} × 10^{16} ergs/cm^{2}
= 1.9029 × 10^{30} ergs/cm^{2}

Therefore:

r = 2 × 3.14159 × \sqrt{\frac{7.0035}{1.9029 × 10^{30}}} seconds = 12.054 × 10^{−15} s = 12.054 fs

2. From the gradient curve. The force constant is the derivative of the gradient wrt distance:

k = \frac{dG}{dx}

Since we are using discrete points, the force constant is best obtained from finite differences:

k = \frac{(G_{2} − G_{1})}{(x_{2} − x_{1})}

For x_{2} = 1.1100, G_{2} = 27.163 and for x_{1} = 1.0980, G_{1} = −26.244, giving rise to k = 4450 kcal/mole/A^{2} and a period of 12.186 fs.

3. From the vibrational frequency. Given a “frequency” (wavenumber) of vibration of N_{2} of ν̄ = 2739.6 cm^{−1}, the period of oscillation, in seconds, is given directly by:

r = \frac{1}{cν̄} = \frac{1}{2739.6 × 2.998 × 10^{10}}

or as 12.175 femtoseconds.

Summarizing, by three different methods the period of oscillation of N_{2} is calculated to be 12.054, 12.186 and 12.175 fs, average 12.138 fs.

Initial dynamics of N_{2} with N–N distance = 1.094 Å

A useful check on the dynamics of N_{2} is to calculate the initial acceleration of the two nitrogen atoms after releasing them from a starting interatomic separation of 1.094 Å.

At R(N-N) = 1.094 Å, G = −44.810 kcal/mole/Å or −18.749 × 10^{19} erg/cm. Therefore acceleration, f = −18.749 × 10^{19}/14.0067 cm/sec/sec or −13.386 × 10^{18} cm/s^{2} which is −13.386 × 10^{15}× Earth surface gravity!

Distance from equilibrium = 0.00980 Å. After 0.1 fs, velocity is 0.110^{−15}(−13.38610^{18}) cm/sec or 1338.6 cm/s.

In the DRC the time-interval between points calculated is a complicated function of the curvature of the local surface. By default, the first time-interval is 0.105fs, so the calculated velocity at this time should be 0.105 × 1338.6 = 1405.6 cm/s, in the DRC calculation the predicted velocity is 1405.6 cm/s.

The option is provided to allow sampling of the system at constant time-intervals, the default being 0.1 fs. For the first few points the calculated velocities are as follows.

      TIME   CALCULATED    LINEAR       DIFF.
              VELOCITY    VELOCITY    VELOCITY
      0.000        0.0       0.0        0.0
      0.100     1338.6    1338.6        0.0
      0.200     2673.9    2677.2       -3.3

      0.300     4001.0    4015.8      -14.8
      0.400     5317.3    5354.4      -37.1
      0.500     6618.5    6693.0      -74.5
      0.600     7900.8    8031.6     -130.8

As the calculated velocity is a fourth-order polynomial of the acceleration, and the acceleration, its first, second and third derivatives, are all changing, the predicted velocity rapidly becomes a poor guide to future velocities.

For simple harmonic motion the velocity at any time is given by:

v = v_{0} sin(2πt/r)

By fitting the computed velocities to simple harmonic motion, a much better fit is obtained:

             Calculated   Simple Harmonic       Diff
      Time    Velocity    25316.Sin(0.529t)
     0.000        0.0           0.0              0.0
     0.100     1338.6        1338.6              0.0
     0.200     2673.9        2673.4             +0.5
     0.300     4001.0        4000.8             +0.2
     0.400     5317.3        5317.0             +0.3
     0.500     6618.5        6618.3             +0.2
     0.600     7900.8        7901.0             -0.2

The repeat-time required for this motion is 11.88 fs, in good agreement with the three values calculated using static models. The repeat time should not be calculated from the time required to go from a minimum to a maximum and then back to a minimum – only half a cycle. For all real systems the potential energy is a skewed parabola, so that the potential energy slopes are different for both sides; a compression (as in this case) normally leads to a higher force-constant, and shorter apparent repeat time (as in this case). Only the addition of the two half-cycles is meaningful.

Conservation of normal coordinate

So far this analysis has only considered a homonuclear diatomic. A detailed analysis of a large polyatomic is impractical, and for simplicity a molecule of formaldehyde will be studied.

In polyatomics, energy can transfer between modes. This is a result of the non-parabolic nature of the potential surface. For small displacements the surface can be considered as parabolic. This means that for small displacements interconversion between modes should occur only very slowly. Of the six normal modes, mode 1, at 1204.5 cm^{−1}, the in-plane C–H asymmetric bend, is the most unsymmetric vibration, and is chosen to demonstrate conservation of vibrational purity.

Mode 1 has a frequency corresponding to 3.44 kcal/mole and a predicted vibrational time of 27.69 fs. By direct calculation, using the DRC, the cycle time is 27.55 fs. The rate of decay of this mode has an estimated half-life of a few thousands femtoseconds.

Rate of decay of starting mode

For trajectories initiated by an IRC=n calculation, whenever the potential energy is a minimum the current velocity is compared with the supplied velocity. The square of the cosine of the angle between the two velocity vectors is a measure of the intensity of the original mode in the current vibration.

Half-Life for decay of initial mode

Vibrational purity is assumed to decay according to zero’th order kinetics. The half-life is thus −0.6931472t/ log(ψ^{2}) fs, where ψ^{2} is the square of the overlap integral of the original vibration 6.12 Sparkles

with the current vibration. Due to the very slow rate of decay of the starting mode, several half-life calculations should be examined. Only when successive half-lives are similar should any confidence be placed in their value.

DRC print options

The amount of output in the DRC is controlled by three sets of options. These sets are:

• Equivalent Keywords H-PRIORITY, T-PRIORITY, and X-PRIORITY

• Potential Energy Turning Point option.

• Geometry Maxima Turning Point options.

If T-PRIORITY is used then turning points cannot be monitored. Currently H-PRIORITY and X-PRIORITY are not implemented, but will be as soon as practical.

To monitor geometry turning points, put a “T” in place of the geometry optimization flag for the relevant geometric variable.

To monitor the potential energy turning points, put a “T” for the flag for atom 1 bond length (Do not forget to put in a bond-length (zero will do)!).

The effect of these flags together is as follows.

1. No options: All calculated points will be printed. No turning points will be calculated.

2. Atom 1 bond length flagged with a “T”: If T-PRIO, etc. are NOT specified, then potential energy turning points will be printed.

3. Internal coordinate flags set to “T”: If T-PRIO, etc. are NOT specified, then geometry extrema will be printed. If only one coordinate is flagged, then the turning point will be displayed in chronologic order; if several are flagged then all turning points occuring in a given time-interval will be printed as they are detected. In other words, some may be out of chronologic order. Note that each coordinate flagged will give rise to a different geometry: minimize flagged coordinates to minimize output.

4. Potential and geometric flags set: The effect is equivalent to the sum of the first two options.

5. T-PRIO set: No turning points will be printed, but constant time-slices (by default 0.1 fs) will be used to control the print.

6.12 Sparkles

Four extra ‘elements” have been put into MOPAC. These represent pure ionic charges, roughly equivalent to the following chemical entities:

    Chemical Symbol          Equivalent to
          +                 Tetramethyl ammonium radical, Potassium
                            atom or Cesium atom.
          ++                Barium atom.
          -                 Borohydride radical, Halogen, or
                            Nitrate radical
          --                Sulfate, oxalate.

For the purposes of discussion these entities are called ‘sparkles’: the name arises from consideration of their behavior.

Behavior of sparkles in MOPAC

Sparkles have the following properties:

1. Their nuclear charge is integer, and is +1, +2, −1, or −2; there are an equivalent number of electrons to maintain electroneutrality, +1, +2, −1, and −2 respectively. For example, a ‘+’ sparkle consists of a unipositive nucleus and an electron. The electron is donated to the quantum mechanics calculation.

2. They all have an ionic radius of 0.7 Å. Any two sparkles of opposite sign will form an ion-pair with a interatomic separation of 1.4 Å.

3. They have a zero heat of atomization, no orbitals, and no ionization potential.

They can be regarded as unpolarizable ions of diameter 1.4Å. They do not contribute to the orbital count, and cannot accept or donate electrons.

Since they appear as uncharged species which immediately ionize, attention should be given to the charge on the whole system. For example, if the alkaline metal salt of formic acid was run, the formula would be: HCOO+ where ‘+’ is the unipositive sparkle. The charge on the system would then be zero.

A water molecule polarized by a positive sparkle would have the formula H_{2}O^{+}, and the charge on the system would be +1.

At first sight, a sparkle would appear to be too ionic to be a point charge and would combine with the first charge of opposite sign it encountered.

This representation is faulty, and a better description would be of an ion, of diameter 1.4Å, and the charge delocalized over its surface. Computationally, a sparkle is an integer charge at the center of a repulsion sphere of form exp(−αr). The hardness of the sphere is such that other atoms or sparkles can approach within about 2Å quite easily, but only with great difficulty come closer than 1.4Å.

Uses of Sparkles

1. They can be used as counterions, e.g. for acid anions or for cations. Thus, if the ionic form of an acid is wanted, then the moieties H·X, H·−, and +·X could be examined.

2. Two sparkles of equal and opposite sign can form a dipole for mimicking solvation effects. Thus water could be surrounded by six dipoles to simulate the solvent cage. A dipole of value D can be made by using the two sparkles + and −, or using ++ and --. If + and − are used, the inter-sparkle separation would be D/4.803Å. If ++ and -- are used, the separation would be D/9.606Å. If the inter-sparkle separation is less than 1.0Å (a situation that cannot occur naturally) then the energy due to the dipole on its own is subtracted from the total energy.

3. They can operate as polarization functions. A controlled, shaped electric field can easily be made from two or more sparkles. The polarizability in cubic Angstroms of a molecule in any particular orientation can then easily be calculated.

6.13 Mechanism of the frame in FORCE calculation

The FORCE calculation uses cartesian coordinates, and all 3N modes are calculated, where N is the number of atoms in the system. Clearly, there will be 5 or 6 “trivial” vibrations, which represent the three translations and two or three rotations. If the molecule is exactly at a stationary point, then these “vibrations” will have a force constant and frequency of precisely zero. If the force calculation was done correctly, and the molecule was not exactly at a stationary point, then the three translations should be exactly zero, but the rotations would be non-zero. The extent to which the rotations are non-zero is a measure of the error in the geometry.

6.14 Configuration interaction

If the distortions are non-zero, the trivial vibrations can interact with the low-lying genuine vibrations or rotations, and with the transition vibration if present.

To prevent this the analytic form of the rotations and vibrations is calculated, and arbitrary eigenvalues assigned; these are 500, 600, 700, 800, 900, and 1000 millidynes/angstrom for Tx, Ty, Tz, Rx, Ry and Rz (if present), respectively. The rotations are about the principal axes of inertia for the system, taking into account isotopic masses. The “force matrix” for these trivial vibrations is determined, and added on to the calculated force matrix. After diagonalization the arbitrary eigenvalues are subtracted off the trivial vibrations, and the resulting numbers are the “true” values. Interference with genuine vibrations is thus avoided.

6.14 Configuration interaction

MOPAC contains a very large Multi-Electron Configuration Interaction calculation, MECI, which allows almost any configuration interaction calculation to be performed. Because of its complexity, two distinct levels of input are supported; the default values will be of use to the novice while an expert has available an exhaustive set of keywords from which a specific C.I. can be tailored.

A MECI calculation involves the interaction of microstates representing specific permutations of electrons in a set of M.O.’s. Starting with a set electronic configuration, either closed shell or open shell, but unconditionally restricted Hartree-Fock, the first step in a MECI calculation is the removal from the M.O.’s of the electrons to be used in the C.I.

Each microstate is then constructed from these empty M.O.’s by adding in electrons according to a prescription. The energy of the configuration is evaluated, as is the energy of interaction with all previously-defined configurations. Diagonalization then results in state functions. From the eigenvectors the expectation value of s^{2} is calculated, and the spin-states of the state functions calculated.

General overview of keywords

Keywords associated with the operations of MECI are:

     SINGLET                DOUBLET               EXCITED
     TRIPLET                QUARTET               BIRADICAL
     QUINTET                SEXTET                ESR
     OPEN(n1,n2)            C.I.=n                MECI
     ROOT=n

Each keyword may imply others; thus TRIPLET implies an open-shell system, therefore OPEN(2,2), and C.I.=2 are implied, if not user specified.

Starting electronic configuration

MECI is restricted to RHF calculations, but with that single restriction any starting configuration will be supported. Examples of starting configurations would be

    System               KeyWords used       Starting Configuration
   Methane                 <none>             2.00 2.00 2.00 2.00 2.00
   Methyl Radical          <none>             2.00 2.00 2.00 2.00 1.00
   Twisted Ethylene        TRIPLET            2.00 2.00 2.00 1.00 1.00
   Twisted Ethylene        OPEN(2,2)          2.00 2.00 2.00 1.00 1.00
   Twisted Ethylene Cation OPEN(1,2)          2.00 2.00 2.00 0.50 0.50
   Methane Cation          CHARGE=1 OPEN(5,3) 2.00 2.00 1.67 1.67 1.67

Choice of starting configuration is important. For example, if twisted ethylene, a groundstate triplet, is not defined using TRIPLET or OPEN(2,2), then the closed-shell ground-state structure will be calculated. Obviously, this configuration is a legitimate microstate, but from the symmetry of the system a better choice would be to define one electron in each of the two formally degenerate pi-type M.O.’s. The initial SCF calculation does not distinguish between OPEN(2,2) and TRIPLET since both keywords define the same starting configuration. This can be verified by monitoring the convergence using PL, for which both keywords give the same SCF energy.

Removal of electrons from starting configuration

For a starting configuration of alpha M.O. occupancies O(i), O(i) being in the range 0.0 to 1.0, the energies of the M.O.’s involved in the MECI can be calculated from:

E(i) = ∑_{j} {[2J(i, j) − K(i, j)]O(j)}

where J(i, j) and K(i, j) are the coulomb and exchange integrals between M.O.’s i and j. The M.O. index j runs over those M.O.’s involved in the MECI only. Most MECI calculations will involve between 1 and 5 M.O.’s, so a system with about 30 filled or partly filled M.O.’s could have M.O.’s 25–30 involved. The resulting eigenvalues correspond to those of the cationic system resulting from removal of n electrons, where n is twice the sum of the orbital occupancies of those M.O.’s involved in the C.I.

The arbitrary zero of energy in a MECI calculation is the starting ground state, without any correction for errors introduced by the use of fractional occupancies. In order to calculate the energy of the various configurations, the energy of the vacuum state (i.e., the state resulting from removal of the electrons used in the C.I.) needs to be evaluated. This energy is defined by:

GSE = ∑_{i} [E(i)O(i) + J(i, i) × O(i) × O(i) + ∑_{j<i}{2[2J(i, j) − K(i, j)] × O(i) × O(j)}]

Formation of microstate configuration

Microstates are particular electron configurations. Thus if there are 5 electrons in 5 levels, then various microstates could be as follows:

               Microstates for 5 electrons in 5 M.O.'s
      Electron Configuration               Electron Configuration
        Alpha       Beta       M(s)          Alpha       Beta        M(s)
      1 2 3 4 5  1 2 3 4 5                 1 2 3 4 5  1 2 3 4 5
1     1,1,1,0,0  1,1,0,0,0    1/2      4    1,1,1,1,1  0,0,0,0,0     5/2
2     1,1,0,0,0  1,1,1,0,0   -1/2      5    1,1,0,1,0  1,1,0,0,0     1/2
3     1,1,1,0,0  0,0,0,1,1    1/2      6    1,1,0,1,0  1,0,1,0,0     1/2

For 5 electrons in 5 M.O.’s there are 252 microstates (10!/(5!5!)), but as states of different spin do not mix, we can use a smaller number. If doublet states are needed then 100 states (5!/(2!3!)(5!/3!2!) are needed. If only quartet states are of interest then 25 states (5!/(1!4!)(5!/4!1!) are needed and if the sextet state is required, then only one state is calculated.

In the microstates listed, state 1 is the ground-state configuration. This can be written as (2,2,1,0,0), meaning that M.O.’s 1 and 2 are doubly occupied, M.O. 3 is singly occupied by an alpha electron, and M.O.’s 4 and 5 are empty. Microstate 1 has a component of spin of 1/2, and is a pure doublet. By Kramer’s degeneracy — sometimes called time-inversion symmetry — microstate 2 is also a doublet, and has a spin of 1/2 and a component of spin of −1/2.

Microstate 3, while it has a component of spin of 1/2, is not a doublet, but is in fact a component of a doublet, a quartet and a sextet. The coefficients of these states can be calculated from the Clebsch-Gordon 3-J symbol. For example, the coefficient in the sextet is 1/\sqrt{(}5).

Microstate 4 is a pure sextet. If all 100 microstates of component of spin = 1/2 were used in a C.I., one of the resulting states would have the same energy as the state resulting from microstate 4.

Microstate 5 is an excited doublet, and microstate 6 is an excited state of the system, but not a pure spin-state.

By default, if n M.O.’s are included in the MECI, then all possible microstates which give rise to a component of spin = 0 for even electron systems, or 1/2 for odd electron systems, will be used.

              Permutations of Electrons among Molecular Orbitals
     (0,1) =   0      (2,4) = 1100   (3,5) = 11100   (2,5) = 11000
                              1010           11010           10100
     (1,1) =   1              1001           11001           10010
                              0110           10110           10001
     (0,2) =   0              0101           10101           01100
                              0011           10011           01010
     (1,2) =  10                             01110           01001
              01      (1,4) = 1000           01101           00110
                              0100           01011           00101
     (1,3) = 100              0010           00111           00011
             010              0001
             001
     (2,3) = 110
             101
             011
              Sets of Microstates for Various MECI Calculations
           Odd Electron Systems        Even Electron Systems
             Alpha   Beta   No. of        Alpha   Beta   No. of
                            Configs.                     Configs.
      C.I.=1 (1,1) * (0,1)  =   1          (1,1) * (1,1) =    1
           2 (1,2) * (0,2)  =   2          (1,2) * (1,2) =    4
           3 (2,3) * (1,3)  =   9          (2,3) * (2,3) =    9
           4 (2,4) * (1,4)  =  24          (2,4) * (2,4) =   36
           5 (3,5) * (2,5)  = 100          (3,5) * (3,5) =  100

Multi electron configuration interaction

The numbering of the M.O.’s used in the MECI is standard, and follows the Aufbau principle. The order of filling is in order of energy, and alpha before beta. This point is critically important in deciding the sign of matrix elements. For a 5 M.O. system, then, the order of filling is:

(1)(¯1)(2)(¯2)(3)(¯3)(4)(¯4)(5)(¯5)

A triplet state arising from two microstates, each with a component of spin = 0, will thus be the positive combination.

(¯1)(2) + (1)(¯2)

This is in variance with the sign convention used in earlier programs for running MNDO. This standard sign convention was chosen in order to allow the signs of the microstate coefficients to conform to those resulting from the spin step-down operator.

Matrix elements between all pairs of microstates are calculated in order to form the secular determinant. Many elements will be identically zero, due to the interacting determinants differing by more than two M.O.’s. For the remaining interactions the following types can be identified.

1. The two determinants are identical: No permutations are necessary in order to calculate the sign of the matrix element. E(p, p) is given simply by:

where: O_{α}(i, p) is the occupancy of α M.O. i in microstate p and O_{β}(i, p) is the occupancy of β M.O. i in microstate p.

2. Determinants differing by exactly one M.O.: The differing M.O. can be of type α or β. It is sufficient to evaluate the case in which both M.O.’s are of alpha type, the beta form is obtained in like manner.

E(p, q) = ∑_{k} {(⟨ij|kk⟩ − ⟨ik|jk⟩)[Occa(k) − Occg(k)] + (⟨ij|kk⟩)[Occb(k) − Occg(k)]}

E(p, q) may need to be multiplied by −1, if the number of two electron permutations required to bring M.O.’s i and j into coincidence is odd.

Where Occa(k) is the alpha molecular orbital occupancy in the configuration interaction.

3. Determinants differing by exactly two M.O.’s: The two M.O.’s can have the same or opposite spins. Three cases can be identified:

(a) Both M.O.’s have alpha spin: For the first microstate having M.O.’s i and j, and the second microstate having M.O.’s k and l, the matrix element connecting the two microstates is given by:

Q(p, q) = ⟨ik|jl⟩ − ⟨il|jk⟩

E(p, q) may need to be multiplied by −1, if the number of two electron permutations required to bring M.O. i into coincidence with M.O. k and M.O. j into coincidence with M.O. l is odd.

(b) Both M.O.’s have beta spin: The matrix element is calculated in the same manner as in the previous case.

(c) One M.O. has alpha spin, and one beta spin: For the first microstate having M.O.’s alpha(i) and beta(j), and the second microstate having M.O.’s alpha(k) and beta(l), the matrix element connecting the two microstates is given by:

Q(p, q) = ⟨ik|jl⟩

E(p, q) may need to be multiplied by −1, if the number of two electron permutations required to bring M.O. i into coincidence with M.O. k and M.O. j into coincidence with M.O. l is odd.

States arising from various calculations

Each MECI calculation invoked by use of the keyword C.I.=n normally gives rise to states of quantized spins. When C.I. is used without any other modifying keywords, the following states will be obtained.

No. of M.O.'s      States Arising            States Arising From
     From Odd Electron Systems    Even Electron Systems
in MECI     Doublets                     Singlets Triplets
1            1                            1
2            2                            3        1
3            8         1                  6        3
4           20         4                 20       15        1
5           75        24       1         50       45        5

These numbers of spin states will be obtained irrespective of the chemical nature of the system.

6.15 Reduced masses in a force calculation

Calculation of spin-states

In order to calculate the spin-state, the expectation value of S2 is calculated.

where Ne is the no. of electrons in C.I., C(i, k) is the coefficient of microstate i in State k, N_{α}(i) is the number of alpha electrons in microstate i, N_{β}(i) is the number of beta electrons in microstate i, O_{α}(l, k) is the occupancy of alpha M.O. l in microstate k, O_{β}(l, k) is the occupancy of beta M.O. l in microstate k, S(+) is the spin shift up or step up operator, S(−) is the spin shift down or step down operator, the Kroneker delta is 1 if the two terms in brackets following it are identical.

The spin state is calculated from:

S = (1/2)[\sqrt{(}1 + 4S2) − 1]

In practice, S is calculated to be exactly integer, or half integer. That is, there is insignificant error due to approximations used. This does not mean, however, that the method is accurate. The spin calculation is completely precise, in the group theoretic sense, but the accuracy of the calculation is limited by the Hamiltonian used, a space-dependent function.

Choice of state to be optimized

MECI can calculate a large number of states of various total spin. Two schemes are provided to allow a given state to be selected. First, ROOT=n will, when used on its own, select the n’th state, irrespective of its total spin. By default n=1. If ROOT=n is used in conjunction with a keyword from the set SINGLET, DOUBLET, TRIPLET, QUARTET, QUINTET, or SEXTET, then the n’th root of that spin-state will be used. For example, ROOT=4 and SINGLET will select the 4th singlet state. If there are two triplet states below the fourth singlet state then this will mean that the sixth state will be selected.

Calculation of unpaired spin density

Starting with the state functions as linear combinations of configurations, the unpaired spin density, corresponding to the alpha spin density minus the beta spin density, will be calculated for the first few states. This calculation is straightforward for diagonal terms, and only those terms are used.

6.15 Reduced masses in a force calculation

Reduced masses for a diatomic are given by:

\frac{m_{1} × m_{2}}{m_{1} + m_{2}}

For a Hydrogen molecule the reduced mass is thus 0.5; for heavily hydrogenated systems, e.g. methane, the reduced mass can be very low. A vibration involving only heavy atoms , e.g. a C–N in cyanide, should give a large reduced mass.

For the ‘trivial’ vibrations the reduced mass is ill-defined, and where this happens the reduced mass is set to zero.

6.16 Use of SADDLE calculation

A SADDLE calculation uses two complete geometries, as shown on the following data file for the ethyl radical hydrogen migration from one methyl group to the other.

Line  1:            UHF  SADDLE
Line  2:         ETHYL RADICAL HYDROGEN MIGRATION
Line  3:
Line  4:     C    0.000000 0    0.000000 0    0.000000 0   0  0  0
Line  5:     C    1.479146 1    0.000000 0    0.000000 0   1  0  0
Line  6:     H    1.109475 1  111.328433 1    0.000000 0   2  1  0
Line  7:     H    1.109470 1  111.753160 1  120.288410 1   2  1  3
Line  8:     H    1.109843 1  110.103163 1  240.205278 1   2  1  3
Line  9:     H    1.082055 1  121.214083 1   38.110989 1   1  2  3
Line 10:     H    1.081797 1  121.521232 1  217.450268 1   1  2  3
Line 11:     0    0.000000 0    0.000000 0    0.000000 0   0  0  0
Line 12:     C    0.000000 0    0.000000 0    0.000000 0   0  0  0
Line 13:     C    1.479146 1    0.000000 0    0.000000 0   1  0  0
Line 14:     H    1.109475 1  111.328433 1    0.000000 0   2  1  0
Line 15:     H    1.109470 1  111.753160 1  120.288410 1   2  1  3
Line 16:     H    2.109843 1   30.103163 1  240.205278 1   2  1  3
Line 17:     H    1.082055 1  121.214083 1   38.110989 1   1  2  3
Line 18:     H    1.081797 1  121.521232 1  217.450268 1   1  2  3
Line 19:     0    0.000000 0    0.000000 0    0.000000 0   0  0  0
Line 20:

Details of the mathematics of SADDLE appeared in print in 1984, (M. J. S. Dewar, E. F. Healy, J. J. P. Stewart, J. Chem. Soc. Faraday Trans. II , 3, 227, (1984)) so only a superficial description will be given here.

The main steps in the saddle calculation are as follows:

1. The heats of formation of both systems are calculated.

2. A vector R of length 3N −6 defining the difference between the two geometries is calculated.

3. The scalar P of the difference vector is reduced by some fraction, normally about 5 to 15 percent.

4. Identify the geometry of lower energy; call this G.

5. Optimize G, subject to the constraint that it maintains a constant distance P from the other geometry.

6. If the newly-optimized geometry is higher in energy then the other geometry, then go to 1. If it is higher, and the last two steps involved the same geometry moving, make the other geometry G without modifying P, and go to 5.

7. Otherwise go back to 2.

The mechanism of 5 involves the coordinates of the moving geometry being perturbed by an amount equal to the product of the discrepancy between the calculated and required P and the vector R.

As the specification of the geometries is quite difficult, in that the difference vector depends on angles (which are, of necessity ill-defined by 360 degrees) SADDLE can be made to run in cartesian coordinates using the keyword XYZ. If this option is chosen then the initial steps of the calculation are as follows:

1. Both geometries are converted into cartesian coordinates.

2. Both geometries are centered about the origin of cartesian space.

3. One geometry is rotated until the difference vector is a minimum — this minimum is within 1 degree of the absolute bottom.

4. The SADDLE calculation then proceeds as described above.

6.17 How to escape from a hilltop

Limitations:

The two geometries must be related by a continuous deformation of the coordinates. By default, internal coordinates are used in specifying geometries, and while bond lengths and bond angles are unambiguously defined (being both positive), the dihedral angles can be positive or negative. Clearly 300 degrees could equally well be specified as −60 degrees. A wrong choice of dihedral would mean that instead of the desired reaction vector being used, a completely incorrect vector was used, with disastrous results.

To correct this, ensure that one geometry can be obtained from the other by a continuous deformation, or use the XYZ option.

6.17 How to escape from a hilltop

A particularly irritating phenomenon sometimes occurs when a transition state is being refined. A rough estimate of the geometry of the transition state has been obtained by either a SADDLE or reaction path or by good guesswork. This geometry is then refined by SIGMA or by NLLSQ, and the system characterized by a force calculation. It is at this point that things often go wrong. Instead of only one negative force constant, two or more are found. In the past, the recommendation has been to abandon the work and to go on to something less masochistic. It is possible, however, to systematically progress from a multiple maximum to the desired transition state. The technique used will now be described.

If a multiple maximum is identified, most likely one negative force constant corresponds to the reaction coordinate, in which case the objective is to render the other force constants positive. The associated normal mode eigenvalues are complex, but in the output are printed as negative frequencies, and for the sake of simplicity will be described as negative vibrations. Use DRAW-2 to display the negative vibrations, and identify which mode corresponds to the reaction coordinate. This is the one we need to retain.

Hitherto, simple motion in the direction of the other modes has proved difficult. However the DRC provides a convenient mechanism for automatically following a normal coordinate. Pick the largest of the negative modes to be annihilated, and run the DRC along that mode until a minimum is reached. At that point, refine the geometry once more using SIGMA and repeat the procedure until only one negative mode exists.

To be on the safe side, after each DRC+SIGMA sequence do the DRC+SIGMA operation again, but use the negative of the initial normal coordinate to start the trajectory. After both stationary points are reached, choose the lower point as the starting point for the next elimination. The lower point is chosen because the transition state wanted is the highest point on the lowest energy path connecting reactants to products. Sometimes the two points will have equal energy: this is normally a consequence of both trajectories leading to the same point or symmetry equivalent points.

After all spurious negative modes have been eliminated, the remaining normal mode corresponds to the reaction coordinate, and the transition state has been located.

This technique is relatively rapid, and relies on starting from a stationary point to begin each trajectory. If any other point is used, the trajectory will not be even roughly simple harmonic. If, by mistake, the reaction coordinate is selected, then the potential energy will drop to that of either the reactants or products, which, incidentally, forms a handy criterion for selecting the spurious modes: if the potential energy only drops by a small amount, and the time evolution is roughly simple harmonic, then the mode is one of the spurious modes. If there is any doubt as to whether a minimum is in the vicinity of a stationary point, allow the trajectory to continue until one complete cycle is executed. At that point the geometry should be near to the initial geometry.

Superficially, a line-search might appear more attractive than the relatively expensive DRC. However, a line-search in cartesian space will normally not locate the minimum in a mode. An obvious example is the mode corresponding to a methyl rotation.

Keyword Sequences to be Used

1. To locate the starting stationary point given an approximate transition state:- SIGMA

2. To define the normal modes:- FORCE ISOTOPE

At this point, copy all the files to a second filename, for use later.

3. Given vibrational frequencies of −654, −123, 234, and 456, identify via DRAW–2 the normal coordinate mode, let’s say that is the −654 mode. Eliminate the second mode by:

IRC=2 DRC T=30M RESTART LARGE

Use is made of the FORCE restart file.

4. Identify the minimum in the potential energy surface by inspection or using the VAX

SEARCH command, of form: SEARCH <Filename>.OUT %

5. Edit out of the output file the data file corresponding to the lowest point, and refine the geometry using: SIGMA

6. Repeat the last three steps but for the negative of the normal mode, using the copied files. The keywords for the first of the two jobs are: IRC=-2 DRC T=30M RESTART LARGE

7. Repeat the last four steps as often as there are spurious modes.

8. Finally, carry out a DRC to confirm that the transition state does, in fact, connect the reactants and products. The drop in potential energy should be monotonic. If you are unsure whether this last operation will work successfully, do it at any time you have a stationary point. If it fails at the very start, then we are back where we were last year – give up and go home!!

6.17.1 EigenFollowing

Description of the EF and TS function by Dr Frank Jensen Department of Chemistry Odense University 5230 Odense Denmark

The current version of the EF optimization routine is a combination of the original EF algorithm of Simons et al. (J. Phys. Chem. 89, 52) as implemented by Baker (J. Comp. Chem. 7,

385) and the QA algorithm of Culot et al. (Theo. Chim. Acta 82, 189), with some added features for improving stability.

The geometry optimization is based on a second order Taylor expansion of the energy around the current point. At this point the energy, the gradient and some estimate of the Hessian are available. There are three fundamental steps in determining the next geometry based on this information:

• finding the “best” step within or on the hypersphere with the current trust radius.

• possibly reject this step based on various criteria.

• update the trust radius.

1. For a minimum search the correct Hessian has only positive eigenvalues. For a Transition State (TS) search the correct Hessian should have exactly one negative eigenvalue, and the corresponding eigenvector should be in the direction of the desired reaction coordinate. The geometry step is parameterized as g/(s − H), where s is a shift factor which ensure that the step-length is within or on the hypersphere. If the Hessian has the correct structure, a pure Newton-Raphson step is attempted. This corresponds to setting the shift factor to zero. If this step is longer than the trust radius, a P-RFO step is attempted. If this is also too long, then the best step on the hypersphere is made via the QA formula. This three step procedure is the default. The pure NR step can be skipped by giving the keyword NONR. An alternative to the QA step is to simply scale the P-RFO step down to the trust radius by a multiplicative constant, this can be accomplished by specifying RSCAL.

2. Using the step determined from 1), the new energy and gradient are evaluated. If it is a TS search, two criteria are used in determining whether the step is “appropriate”. The ratio between the actual and predicted energy change should ideally be 1. If it deviates substantially from this value, the second order Taylor expansion is no longer accurate. RMIN and RMAX (default values 0 and 4) determine the limits on how far from 1 the ratio can be before the step is rejected. If the ratio is outside the RMIN and RMAX limits, the step is rejected, the trust radius reduced by a factor of two and a new step is determined. The second criteria is that the eigenvector along which the energy is being maximized should not change substantially between iterations. The minimum overlap of the TS eigenvector with that of the previous iteration should be larger than OMIN, otherwise the step is rejected. Such a step rejection can be recognized in the output by the presence of (possibly more) lines with the same CYCLE number. The default OMIN value is 0.8, which allows fairly large changes to occur, and should be suitable for most uncomplicated systems. See below for a discussion of how to use RMIN, RMAX and OMIN for difficult cases. The selection of which eigenvector to follow towards the TS is given by MODE=n, where n is the number of the Hessian eigenvector to follow. The default is MODE=1. These features can be turned off by giving suitable values as keywords, e.g. RMIN=-100 RMAX=100 effectively inhibits step rejection. Similarly setting OMIN=0 disables step rejection based on large changes in the structure of the TS mode. The default is to use mode following even if the TS mode is the lowest eigenvector. This means that the TS mode may change to some higher mode during the optimization. To turn of mode following, and thus always follow the mode with lowest eigenvalue, set MODE=0. If it is a minimum search the new energy should be lower than the previous.

The acceptance criteria used is that the actual/predicted ratio should be larger than RMIN, which for the default value of RMIN=0 is equivalent to a lower energy. If the ratio is below RMIN, the step is rejected, the trust radius reduced by a factor of two and a new step is predicted. The RMIN, RMAX and OMIN features has been introduced in the current version of EF to improve the stability of TS optimizations. Setting RMIN and RMAX close to one will give a very stable, but also very slow, optimization. Wide limits on RMIN and RMAX may in some cases give a faster convergence, but there is always the risk that very poor steps are accepted, causing the optimization to diverge. The default values of 0 and 4 rarely rejects steps which would lead to faster convergence, but may occasionally accept poor steps. If TS searches are found to cause problems, the first try should be to lower the limits to 0.5 and 2. Tighter limits like 0.8 and 1.2, or even 0.9 and 1.1, will almost always slow the optimization down significantly but may be necessary in some cases.

In minimum searches it is usually desirable that the energy decreases in each iteration. In certain very rigid systems, however, the initial diagonal Hessian may be so poor that the algorithm cannot find an acceptable step larger than DDMIN, and the optimization terminates after only a few cycles with the “TRUST RADIUS BELOW DDMIN” warning long before the stationary point is reached. In such cases the user can specify RMIN to some negative value, say -10, thereby allowing steps which increases the energy.

The algorithm has the capability of following Hessian eigenvectors other than the one with the lowest eigenvalue toward a TS. Such higher mode following are always much more difficult to make converge. Ideally, as the optimization progresses, the TS mode should at some point become the lowest eigenvector. Care must be taken during the optimization, however, that the nature of the mode does not change all of a sudden, leading to optimization to a different TS than the one desired. OMIN has been designed for ensuring that the nature of the TS mode only changes gradually, specifically the overlap between to successive TS modes should be higher than OMIN. While this concept at first appears very promising, it is not without problems when the Hessian is updated. As the updated Hessian in each step is only approximately correct, there is a upper limit on how large the TS mode overlap between steps can be. To understand this, consider a series of steps made from the same geometry (e.g. at some point in the optimization), but with steadily smaller step-sizes. The update adds corrections to the Hessian to make it a better approximation to the exact Hessian. As the step-size become small, the updated Hessian converges toward the exact Hessian, at least in the direction of the step. The old Hessian is constant, thus the overlap between TS modes thus does not converge toward 1, but rather to a constant value which indicate how well the old approximate Hessian resembles the exact Hessian. Test calculations suggest a typical upper limit around 0.9, although cases have been seen where the limit is more like 0.7. It appears that an updated Hessian in general is not of sufficient accuracy for reliably rejecting steps with TS overlaps much greater than 0.80. The default OMIN of 0.80 reflects the typical use of an updated Hessian. If the Hessian is recalculated in each step, however, the TS mode overlap does converge toward 1 as the step-size goes toward zero, and in this cases there is no problems following high lying modes. Unfortunately setting RECALC=1 is very expensive in terms of computer time, but used in conjecture with OMIN=0.90 (or possibly higher), and maybe also tighter limits on RMIN and RMAX, it represents an option of locating transitions structures that otherwise might not be possible. If problems are encountered with many step rejections due to small TS mode overlaps, try reducing OMIN, maybe all the way down to 0. This most likely will work if the TS mode is the lowest Hessian eigenvector, but it is doubtful that it will produce any useful results if a high lying mode is followed. Finally, following modes other than the lowest toward a TS indicates that the starting geometry is not “close” to the desired TS. In most cases it is thus much better to further refined the starting geometry, than to try following high lying modes. There are cases, however, where it is very difficult to locate a starting geometry which has the correct Hessian, and mode following may be of some use here

6.17.2 Franck-Condon considerations

This section was written based on discussions with

Victor I. Danilov Department of Quantum Biophysics Academy of Sciences of the Ukraine Kiev 143 Ukraine

The Frank-Condon principle states that electronic transitions take place in times that are very short compared to the time required for the nuclei to move significantly. Because of this, care must be taken to ensure that the calculations actually do reflect what is wanted.

Examples of various phenomena which can be studied are:

Photoexcitation If the purpose of a calculation is to predict the energy of photoexcitation, then the ground-state should first be optimized. Once this is done, then a C.I. calculation can be carried out using 1SCF. With the appropriate keywords (MECI C.I.=n etc.), the energy of photoexcitation to the various states can be predicted.

6.18 Outer Valence Green’s Function

A more expensive, but more rigorous, calculation, would be to optimize the geometry using all the C.I. keywords. This is unlikely to change the results significantly, however.

Fluorescence If the excited state has a sufficiently long lifetime, so that the geometry can relax, then if the system returns to the ground state by emission of a photon, the energy of the emitted photon will be less (it will be red-shifted) than that of the exciting photon. To do such a calculation, proceed as follows:

• Optimize the ground-state geometry using all the keywords for the later steps, but specify the ground state, e.g. C.I.=3 EF GNORM=0.01 MECI.

• Optimize the excited state, e.g. C.I.=3 ROOT=2 EF GNORM=0.01 MECI.

• Calculate the Franck-Condon excitation energy, using the results of the ground-state calculation only.

• Calculate the Franck-Condon emission energy, using the results of the excited state calculation only.

• If indirect emission energies are wanted, these can be obtained from the ΔH_{f} of the optimized excited and optimized ground-state calculations.

In order for fluorescence to occur, the photoemission probability must be quite large, therefore only transitions of the same spin are allowed. For example, if the ground state is S_{0}, then the fluorescing state would be S_{1}.

Phosphorescence If the photoemission probability is very low, then the lifetime of the excited state can be very long (sometimes minutes). Such states can become populated by S_{1} → T_{1} intersystem crossing. Of course, the geometry of the system will relax before the photoemission occurs.

Indirect emission If the system relaxes from the excited electronic, ground vibrational state to the ground electronic, ground vibrational state, then a more complicated calculation is called for. The steps of such a calculation are:

• Optimize the geometry of the excited state.

• Using the same keywords, except that the ground state is specified, optimize the geometry of the ground state.

• Take the difference in ΔH_{f} of the optimized excited and optimized ground-state calculations.

• Convert this difference into the appropriate units.

Excimers An excimer is a pair of molecules, one of which is in an electronic excited state. Such systems are usually stabilized relative to the isolated systems. Optimization of the geometries of such systems is difficult. Suggestions on how to improve this type of calculation would be appreciated.

6.18 Outer Valence Green’s Function

This section is based on materials supplied by

Dr David Danovich The Fritz Haber Research Center for Molecular Dynamics The Hebrew University of Jerusalem 91904 Jerusalem Israel

The OVGF technique was used with the self-energy part extended to include third order perturbation corrections, [?]. The higher order contributions were estimated by the renormalization procedure. The actual expression used to calculate the self-energy part, ∑_{pp}(w), chosen in the

diagonal form, is given in equation (6.59), where ∑_{pp}^{(2)}(w) and ∑_{pp}^{(3)}(w) are the second- and thirdorder corrections, and A is the screening factor accounting for all the contributions of higher orders.

∑_{pp}(w) = ∑_{pp}^{(2)}(w) + (1 − A)^{−1} ∑_{pp}^{3}(w)  (6.59)

The particular expression which was used for the second-order corrections is given in equation (6.60).

∑_{pp}^{(2)}(w) = ∑_{a} ∑_{i,j} \frac{(2V_{paij} − V_{paji})V_{paij}}{w + e_{a} − e_{i} − e_{j}} + ∑_{a,b} ∑_{i} \frac{(2V_{piab} − V_{piba})V_{piab}}{w + e_{i} − e_{a} − e_{b}}  (6.60)

where

V_{pqrs} = ∫ ∫ ϕ_{p}^{∗}(1)ϕ_{q}^{∗}(2)(1/r_{12})ϕ_{r}^{∗}(1)ϕ_{s}^{∗}(2)dτ_{1}dτ_{2}

In equation (6.60), i and j denote occupied orbitals, a and b denote virtual orbitals, p denotes orbitals of unspecified occupancy, and e denotes an orbital energy. The equations are solved by an iterative procedure, represented in equation (6.61).

w_{p}^{i+1} = e_{p} + ∑_{pp}(w^{i})  (6.61)

The SCF energies and the corresponding integrals, which were calculated by one of the semiempirical methods (MNDO, AM1, or PM3), were taken as the zero’th approximation and all M.O.s may be included in the active space for the OVGF calculations.

The expressions used for ∑_{pp}^{(3)} and A are given in [?].

The OVGF method itself, is described in detail in [?].

6.18.1 Example of OVGF calculation

Because Danovich’s OVGF method is new to MOPAC, users will want to see how well it works. The data-set test green.dat will calculate the first 8 I.P.s for dimethoxy-s-tetrazine. This calculation is discussed in detail in [?]. The experimental and calculated I.P.s are shown in Table 6.1.

Table 6.1: OVGF Calculation, Comparison with Experiment

M.O. Expt* PM3 Error OVGF(PM3) Error
n_{1}  9.05 10.15 1.10  9.46  0.41
π_{1}  9.6 10.01 0.41  9.65  0.05
n_{2}  11.2 11.96 0.76  11.13  -0.07
π_{2}  11.8 12.27 0.47  11.43  -0.37

*: R. Gleiter, V. Schehlmann, J. Spanget-Larsen, H. Fischer and F. A. Neugebauer, J. Org. Chem., 53, 5756 (1988).

From this, we see that for PM3 the average error is 0.69eV, but after OVGF correction, the error drops to 0.22eV. This is typical of nitrogen heterocycle calculations.

6.19 COSMO (Conductor-like Screening Model)

6.19 COSMO (Conductor-like Screening Model)

This section was written based on material provided by:

Andreas Klamt Bayer AG Q18, D-5090 Leverkusen-Bayerwerk Germany

Unlike the Self-Consistent Reaction Field model [?], the Conductor-like Screening Model (COSMO) is a new continuum approach which, while more complicated, is computationally quite efficient. The expression for the total screening energy is simple enough to allow the first derivatives of the energy with respect to atomic coordinates to be easily evaluated.

Details of the procedure have been submitted for publication: A. Klamt and G. Schuurmann, COSMO: A New Approach to Dielectric Screening in Solvents with Explicit Expressions for the Screening Energy and its Gradient, J. Chem. Soc., Perkin Trans. 2, 1993. (in press).

The COSMO procedure generates a conducting polygonal surface around the system (ion or molecule), at the van der Waals’ distance. By introducing a ε-dependent correction factor,

f(ε) = \frac{(ε − 1)}{(ε + \frac{1}{2})},

into the expressions for the screening energy and its gradient, the theory can be extended to finite dielectric constants with only a small error.

The accuracy of the method can be judged by how well it reproduces known quantities, such as the heat of solution in water (water has a dielectric constant of 78.4 at 25^{∘}C), Table 6.2. Here, the keywords used were

NSPA=60 GRADIENTS 1SCF EPS=78.4 AM1 CHARGE=1

From the Table we see that the glycine zwitterion becomes the stable form in water, while the neutral species is the stable gas-phase form.

The COSMO method is easy to use, and the derivative calculation is of sufficient precision to allow gradients of 0.1 to be readily achieved.

Table 6.2: Calculated and Observed Hydration Energies

Compound Method ΔH_{f} (kcal/mol) Hydration
gas phase solution phase ΔH(calc.) Enthalpy(exp.) †
NH_{4}^{+}  AM1  150.6  59.5  91.1  88.0
N(Me)_{4}^{+}  AM1  157.1  101.1  56.0  59.9
N(Et)_{4}^{+}  AM1  132.1  84.2  47.9  57.0

Glycine

neutral  AM1  -101.6  -117.3  15.7  −−
zwitterion AM1  -59.2  -125.6  66.4  −−

†: Y. Nagano, M. Sakiyama, T. Fujiwara, Y. Kondo, J. Phys. Chem., 92, 5823 (1988).

6.20 Solid state capability

Currently MOPAC can only handle up to one-dimensional extended systems. As the solid-state method used is unusual, details are given at this point.

If a polymer unit cell is large enough, then a single point in k-space, the Gamma point, is sufficient to specify the entire Brillouin zone. The secular determinant for this point can be constructed by adding together the Fock matrix for the central unit cell plus those for the adjacent unit cells. The Born–von Karman cyclic boundary conditions are satisfied, and diagonalization yields the correct density matrix for the Gamma point.

At this point in the calculation, conventionally, the density matrix for each unit cell is constructed. Instead, the Gamma-point density and one-electron density matrices are combined with a “Gamma-point-like” Coulomb and exchange integral strings to produce a new Fock matrix. The calculation can be visualized as being done entirely in reciprocal space, at the Gamma point.

Most solid-state calculations take a very long time. These calculations, called “Cluster” calculations after the original publication, require between 1.3 and 2 times the equivalent molecular calculation.

A minor ‘fudge’ is necessary to make this method work. The contribution to the Fock matrix element arising from the exchange integral between an atomic orbital and its equivalent in the adjacent unit cells is ignored. This is necessitated by the fact that the density matrix element involved is invariably large.

The unit cell must be large enough that an atomic orbital in the center of the unit cell has an insignificant overlap with the atomic orbitals at the ends of the unit cell. In practice, a translation vector of more that about 7 or 8Å is sufficient. For one rare group of compounds a larger translation vector is needed. Polymers with delocalized π–systems, and polymers with very small band-gaps will require a larger translation vector, in order to accurately sample k-space. For these systems, a translation vector in the order of 15-20 Angstroms is needed.

Program

The logic within MOPAC is best understood by use of flow-diagrams.

There are two main sequences, geometric and electronic. These join only at one common subroutine COMPFG. It is possible, therefore, to understand the geometric or electronic sections in isolation, without having studied the other section.

7.1 Main geometric sequence

                   ______
                  |      |
                  | MAIN |
                  |      |
                  |______|
 _____________________|______________________________
|      |         ___|_______|_____|   ___|___
|   ___|___|       |     |          |   |  |       |
|  |IRC/DRC|    | FORCE |     |  REACT1  |   |  | PATHS |
|  |  or   |    |       |     |          |   |  |       |
|  |  EF   |    |_______|___|__________|   |  |_______|
|  |  or   |     |    |     |      |         |      |
|  | POLAR |     |    |     |      |         |      |
|  |_______|     |    |__|      |_________|______|
_|_____|      _|____|  |             ___|___
| NLLSQ |  |     |      |  |  |____________|       |
|  and  |  |     | FMAT |  |               | FLEPO |
| POWSQ |  |     |      |  |               |       |
|_______|  |     |______|  |               |_______|
____|___|   |         |     |          _____|__|
| SEARCH | |   |         |     |         |        |  |
|   or   | |   |         |     |         | LINMIN |  |
| LOCMIN | |   |         |     |         |        |  |
|________| |   |         |     |         |________|  |
|______|___|_________|_____|______________|______|
                  ____|___
                 |        |
                 | COMPFG |  (See ELECTRONIC SEQUENCE)
                 |        |
                 |________|

\underline{Program}

7.2 Main electronic flow

             ________
            |        |
            | COMPFG |  (See GEOMETRIC SEQUENCE)
            |        |
            |________|
 _______________|____________________
___|______|______|____|
|       |       |       |    |        |  |
| HCORE |_______| DERIV |____| GMETRY |  |
|       |       |       |    |        |  |
|_______|       |_______|    | SYMTRY |  |
|            |     |     |        |  |
|        ____|__|     |________|  |
|       |       |  |                 |
|       | DCART |  |                 |
|       |       |  |                 |
|       |_______|  |_____________|
|         __|___|__|__________
|        |     |        |      |    |         |
|        | DHC |    ____| ITER |____|   RSP   |
|        |     |   |    |      |    |         |
|        |_____|   |    |______|    |_________|
|         |   |    |        | |
|_____|   |    |        | |       ________
   |    |     |    |        | |      |        |
  _|____|_|____|_| |______| DENSIT |
 |        |  |       |      |        |        |
 | ROTATE |  | FOCK1 |      |        |  CNVG  |
 |        |  |       |      |        | PULAY  |
 | H1ELEC |  | FOCK2 |      |        |________|
 |        |  |       |      |
 |________|  |_______|      |_____
     |                     |      |
  ___|__| MECI |
 |      |                  |      |
 | DIAT |                  |______|
 |      |
 |______|
     |
   __|_
  |    |
  | SS |
  |____|

7.3 Control within MOPAC

Almost all the control information is passed via the single datum “KEYWRD”, a string of 80 characters, which is read in at the start of the job.

Each subroutine is made independent, as far as possible, even at the expense of extra code or calculation. Thus, for example, the SCF criterion is set in subroutine ITER, and nowhere else. Similarly, subroutine DERIV has exclusive control of the step size in the finite-difference 7.3 Control within MOPAC

calculation of the energy derivatives. If the default values are to be reset, then the new value is supplied in KEYWRD, and extracted via INDEX and READA. The flow of control is decided by the presence of various keywords in KEYWRD.

When a subroutine is called, it assumes that all data required for its operation are available in either common blocks or arguments. Normally no check is made as to the validity of the data received. All data are “owned” by one, and only one, subroutine. Ownership means the implied permission and ability to change the data. Thus MOLDAT “owns” the number of atomic orbitals, in that it calculates this number, and stores it in the variable NORBS. Many subroutines use NORBS, but none of them is allowed to change it. For obvious reasons no exceptions should be made to this rule. To illustrate the usefulness of this convention, consider the eigenvectors, C and CBETA. These are owned by ITER. Before ITER is called, C and CBETA are not calculated, after ITER has been called C and CBETA are known, so any subroutine which needs to use the eigenvectors can do so in the certain knowledge that they exist.

Any variables which are only used within a subroutine are not passed outside the subroutine unless an overriding reason exists. This is found in PULAY and CNVG, among others where arrays used to hold spin-dependent data are used, and these cannot conveniently be defined within the subroutines. In these examples, the relevant arrays are “owned” by ITER.

A general subroutine, of which ITER is a good example, handles three kinds of data: First, data which the subroutine is going to work on, for example the one and two electron matrices; second, data necessary to manipulate the first set of data, such as the number of atomic orbitals; third, the calculated quantities, here the electronic energy, and the density and Fock matrices.

Reference data are entered into a subroutine by way of the common blocks. This is to emphasize their peripheral role. Thus the number of orbitals, while essential to ITER, is not central to the task it has to perform, and is passed through a common block.

Data the subroutine is going to work on are passed via the argument list. Thus the one and two electron matrices, which are the main reason for ITER’s existence, are entered as two of the four arguments. As ITER does not own these matrices it can use them but may not change their contents. The other argument is EE, the electronic energy. EE is owned by ITER even though it first appears before ITER is called.

Sometimes common block data should more correctly appear in an argument list. This is usually not done in order to prevent obscuring the main role the subroutine has to perform. Thus ITER calculates the density and Fock matrices, but these are not represented in the argument list as the calling subroutine never needs to know them; instead, they are stored in common.

7.3.1 Subroutine GMETRY

Description for programmers

GMETRY has two arguments, GEO and COORD. On input GEO contains either (a) internal coordinates or (b) cartesian coordinates. On exit COORD contains the cartesian coordinates.

The normal mode of usage is to supply the internal coordinates, in which case the connectivity relations are found in common block GEOKST.

If the contents of NA(1) is zero, as required for any normal system, then the normal internal to cartesian conversion is carried out.

If the contents of NA(1) is 99, then the coordinates found in GEO are assumed to be cartesian, and no conversion is made. This is the situation in a FORCE calculation.

A further option exists within the internal to cartesian conversion. If STEP, stored in common block REACTN, is non-zero, then a reaction path is assumed, and the internal coordinates are adjusted radially in order that the “distance” in internal coordinate space from the geometry specified in GEO is STEPP away from the geometry stored in GEOA, stored in REACTN.

During the internal to cartesian conversion, the angle between the three atoms used in defining a fourth atom is checked to ensure that it is not near to 0 or 180 degrees. If it is near to these angles, then there is a high probability that a faulty geometry will be generated and to prevent this the calculation is stopped and an error message printed.

Note:

1. If the angle is exactly 0 or 180 degrees, then the calculation is not terminated: This is the normal situation in a high-symmetry molecule such as propyne.

2. The check is only made if the fourth atom has a bond angle which is not zero or 180 degrees.

Error messages produced by MOPAC

MOPAC produces several hundred messages, all of which are intended to be self-explanatory. However, when an error occurs it is useful to have more information than is given in the standard messages.

The following alphabetical list gives more complete definitions of the messages printed.

AN UNOPTIMIZABLE GEOMETRIC PARAMETER . . .

When internal coordinates are supplied, six coordinates cannot be optimized. These are the three coordinates of atom 1, the angle and dihedral on atom 2 and the dihedral on atom 3. An attempt has been made to optimize one of these. This is usually indicative of a typographic error, but might simply be an oversight. Either way, the error will be corrected and the calculation will not be stopped here.

ATOM NUMBER nn IS ILLDEFINED

The rules for definition of atom connectivity are:

1. Atom 2 must be connected to atom 1 (default - no override)

2. Atom 3 must be connected to atom 1 or 2, and make an angle with 2 or 1.

3. All other atoms must be defined in terms of already-defined atoms: these atoms must all be different. Thus atom 9 might be connected to atom 5, make an angle with atom 6, and have a dihedral with atom 7. If the dihedral was with atom 5, then the geometry definition would be faulty.

If any of these rules is broken, a fatal error message is printed, and the calculation stopped.

ATOMIC NUMBER nn IS NOT AVAILABLE . . .

An element has been used for which parameters are not available. Only if a typographic error has been made can this be rectified. This check is not exhaustive, in that even if the elements are acceptable there are some combinations of elements within MINDO/3 that are not allowed. This is a fatal error message.

ATOMIC NUMBER OF nn ?

An atom has been specified with a negative or zero atomic number. This is normally caused by forgetting to specify an atomic number or symbol. This is a fatal error message.

Error messages produced by MOPAC

ATOMS nn AND nn ARE SEPARATED BY nn.nnnn ANGSTROMS

Two genuine atoms (not dummies) are separated by a very small distance. This can occur when a complicated geometry is being optimized, in which case the user may wish to continue. This can be done by using the keyword GEO-OK. More often, however, this message indicates a mistake, and the calculation is, by default, stopped.

ATTEMPT TO GO DOWNHILL IS UNSUCCESSFUL . . .

A quite rare message, produced by Bartel’s gradient norm minimization. Bartel’s method attempts to minimize the gradient norm by searching the gradient space for a minimum. Apparently a minimum has been found, but not recognized as such. The program has searched in all (3N − 6) directions, and found no way down, but the criteria for a minimum have not been satisfied. No advice is available for getting round this error.

BOTH SYSTEMS ARE ON THE SAME SIDE . . .

A non-fatal message, but still cause for concern. During a SADDLE calculation the two geometries involved are on opposite sides of the transition state. This situation is verified at every point by calculating the cosine of the angle between the two gradient vectors. For as long as it is negative, then the two geometries are on opposite sides of the T/S. If, however, the cosine becomes positive, then the assumption is made that one moiety has fallen over the T/S and is now below the other geometry. That is, it is now further from the T/S than the other, temporarily fixed, geometry. To correct this, identify geometries corresponding to points on each side of the T/S. (Two geometries on the output separated by the message "SWAPPING...") and make up a new data-file using these geometries. This corresponds to points on the reaction path near to the T/S. Run a new job using these two geometries, but with BAR set to a third or a quarter of its original value, e.g. BAR=0.05. This normally allows the T/S to be located.

C.I. NOT ALLOWED WITH UHF

There is no UHF configuration interaction calculation in MOPAC. Either remove the keyword that implies C.I. or the word UHF.

CALCULATION ABANDONED AT THIS POINT

A particularly annoying message! In order to define an atom’s position, the three atoms used in the connectivity table must not accidentally fall into a straight line. This can happen during a geometry optimization or gradient minimization. If they do, and if the angle made by the atom being defined is not zero or 180 degrees, then its position becomes ill-defined. This is not desirable, and the calculation will stop in order to allow corrective action to be taken. Note that if the three atoms are in an exactly straight line, this message will not be triggered. The good news is that the criterion used to trigger this message was set too coarsely. The criterion has been tightened so that this message now does not often appear. Geometric integrity does not appear to be compromized.

CARTESIAN COORDINATES READ IN, AND CALCULATION . . .

If cartesian coordinates are read in, but the calculation is to be carried out using internal coordinates, then either all possible geometric variables must be optimized, or none can be optimized. If only some are marked for optimization then ambiguity exists. For example, if the “X” coordinate of atom 6 is marked for optimization, but the “Y” is not, then when the conversion to internal coordinates takes place, the first coordinate becomes a bond-length, and the second an angle. These bear no relationship to the “X” or “Y” coordinates. This is a fatal error.

CARTESIAN COORDINATES READ IN, AND SYMMETRY . . .

If cartesian coordinates are read in, but the calculation is to be carried out using internal coordinates, then any symmetry relationships between the cartesian coordinates will not be reflected in the internal coordinates. For example, if the “Y” coordinates of atoms 5 and 6 are equal, it does not follow that the internal coordinate angles these atoms make are equal. This is a fatal error.

ELEMENT NOT FOUND

When an external file is used to redefine MNDO, AM1, or PM3 parameters, the chemical symbols used must correspond to known elements. Any that do not will trigger this fatal message.

ERROR DURING READ AT ATOM NUMBER . . .

Something is wrong with the geometry data. In order to help find the error, the geometry already read in is printed. The error lies either on the last line of the geometry printed, or on the next (unprinted) line. This is a fatal error.

FAILED IN SEARCH, SEARCH CONTINUING

Not a fatal error. The McIver-Komornicki gradient minimization involves use of a line-search to find the lowest gradient. This message is merely advice. However, if SIGMA takes a long time, consider doing something else, such as using NLLSQ, or refining the geometry a bit before resubmitting it to SIGMA.

<<<<----**** FAILED TO ACHIEVE SCF. ****---->>>>

The SCF calculation failed to go to completion; an unwanted and depressing message that unfortunately appears every so often.

To date three unconditional convergers have appeared in the literature: the SHIFT technique, Pulay’s method, and the Camp-King converger. It would not be fair to the authors to condemn their methods. In MOPAC all sorts of weird and wonderful systems are calculated, systems the authors of the convergers never dreamed of. MOPAC uses a combination of all three convergers at times. Normally only a quadratic damper is used.

If this message appears, suspect first that the calculation might be faulty, then, if you feel confident, use PL to monitor a single SCF. Based on the SCF results either increase the number of allowed iterations, default: 200, or use PULAY, or Camp-King, or a mixture.

If nothing works, then consider slackening the SCF criterion. This will allow heats of formation to be calculated with reasonable precision, but the gradients are likely to be imprecise.

GEOMETRY TOO UNSTABLE FOR EXTRAPOLATION . . .

In a reaction path calculation the initial geometry for a point is calculated by quadratic extrapolation using the previous three points.

If a quadratic fit is likely to lead to an inferior geometry, then the geometry of the last point calculated will be used. The total effect is to slow down the calculation, but no user action is recommended.

** GRADIENT IS TOO LARGE TO ALLOW . . .

Before a FORCE calculation can be performed the gradient norm must be so small that the third and higher order components of energy in the force field are negligible. If, in the system under examination, the gradient norm is too large, the gradient norm will first be reduced using FLEPO, unless LET has been specified. In some cases the FORCE calculation may be run only to decide if a state is a ground state or a transition state, in which case the results have only two interpretations. Under these circumstances, LET may be warranted.

GRADIENT IS VERY LARGE . . .

In a calculation of the thermodynamic properties of the system, if the rotation and translation vibrations are non-zero, as would be the case if the gradient norm was significant, then these ‘vibrations’ would interfere with the low-lying genuine vibrations. The criteria for THERMO are much more stringent than for a vibrational frequency calculation, as it is the lowest few genuine vibrations that determine the internal vibrational energy, entropy, etc.

ILLEGAL ATOMIC NUMBER

An element has been specified by an atomic number which is not in the range 1 to 107. Check the data: the first datum on one of the lines is faulty. Most likely line 4 is faulty.

IMPOSSIBLE NUMBER OF OPEN SHELL ELECTRONS

The keyword OPEN(n1,n2) has been used, but for an even-electron system n1 was specified as odd or for an odd-electron system n1 was specified as even. Either way, there is a conflict which the user must resolve.

IMPOSSIBLE OPTION REQUESTED

A general catch-all. This message will be printed if two incompatible options are used, such as both MINDO/3 and AM1 being specified. Check the keywords, and resolve the conflict.

INTERNAL COORDINATES READ IN, AND CALCULATION . . .

If internal coordinates are read in, but the calculation is to be carried out using cartesian coordinates, then either all possible geometric variables must be optimized, or none can be optimized. If only some are marked for optimization, then ambiguity exists. For example, if the bond-length of atom 6 is marked for optimization, but the angle is not, then when the conversion to cartesian coordinates takes place, the first coordinate becomes the ‘X’ coordinate and the second the ‘Y’ coordinate. These bear no relationship to the bond length or angle. This is a fatal error.

INTERNAL COORDINATES READ IN, AND SYMMETRY . . .

If internal coordinates are read in, but the calculation is to be carried out using cartesian coordinates, then any symmetry relationships between the internal coordinates will not be reflected in the cartesian coordinates. For example, if the bond-lengths of atoms 5 and 6 are equal, it does not follow that these atoms have equal values for their ‘X’ coordinates. This is a fatal error.

JOB STOPPED BY OPERATOR

Any MOPAC calculation, for which the SHUTDOWN command works, can be stopped by a user who issues the command "$SHUT <filename>, from the directory which contains <filename>.DAT.

MOPAC will then stop the calculation at the first convenient point, usually after the current cycle has finished. A restart file will be written and the job ended. The message will be printed as soon as it is detected, which would be the next time the timer routine is accessed.

**** MAX. NUMBER OF ATOMS ALLOWED: . . .

At compile time the maximum sizes of the arrays in MOPAC are fixed. The system being run exceeds the maximum number of atoms allowed. To rectify this, modify the file DIMSIZES.DAT to increase the number of heavy and light atoms allowed. If DIMSIZES.DAT is altered, then the whole of MOPAC should be re-compiled and re-linked.

**** MAX. NUMBER OF ORBITALS: . . .

At compile time the maximum sizes of the arrays in MOPAC are fixed. The system being run exceeds the maximum number of orbitals allowed. To rectify this, modify the file DIMSIZES.DAT to change the number of heavy and light atoms allowed. If DIMSIZES.DAT is altered, then the whole of MOPAC should be re-compiled and re-linked.

**** MAX. NUMBER OF TWO ELECTRON INTEGRALS . . .

At compile time the maximum sizes of the arrays in MOPAC are fixed. The system being run exceeds the maximum number of two-electron integrals allowed. To rectify this, modify the file DIMSIZES.DAT to modify the number of heavy and light atoms allowed. If DIMSIZES.DAT is altered, then the whole of MOPAC should be re-compiled and re-linked.

NAME NOT FOUND

Various atomic parameters can be modified in MOPAC by use of EXTERNAL=. These comprise:

          Uss         Betas         Gp2          GSD
          Upp         Betap         Hsp          GPD
          Udd         Betad         AM1          GDD
          Zs          Gss           Expc         FN1
          Zp          Gsp           Gaus         FN2
          Zd          Gpp           Alp          FN3

Thus to change the Uss of hydrogen to −13.6 the line USS H -13.6 could be used. If an attempt is made to modify any other parameters, then an error message is printed, and the calculation terminated.

NUMBER OF PARTICLES, nn GREATER THAN . . .

When user-defined microstates are not used, the MECI will calculate all possible microstates that satisfy the space and spin constraints imposed. This is done in PERM, which permutes N electrons in M levels. If N is greater than M, then no possible permutation is valid. This is not a fatal error - the program will continue to run, but no C.I. will be done.

NUMBER OF PERMUTATIONS TOO GREAT, LIMIT 60

The number of permutations of alpha or beta microstates is limited to 60. Thus if 3 alpha electrons are permuted among 5 M.O.’s, that will generate 10 = 5!/(3!2!) alpha microstates, which is an allowed number. However if 4 alpha electrons are permuted among 8 M.O.’s, then 70 alpha microstates result and the arrays defined will be insufficient. Note that 60 alpha and 60 beta microstates will permit 3600 microstates in all, which should be more than sufficient for most purposes. (An exception would be for excited radical icosohedral systems.)

SYMMETRY SPECIFIED, BUT CANNOT BE USED IN DRC

This is self explanatory. The DRC requires all geometric constraints to be lifted. Any symmetry constraints will first be applied, to symmetrize the geometry, and then removed to allow the calculation to proceed.

SYSTEM DOES NOT APPEAR TO BE OPTIMIZABLE

This is a gradient norm minimization message. These routines will only work if the nearest minimum to the supplied geometry in gradient-norm space is a transition state or a ground state. Gradient norm space can be visualized as the space of the scalar of the derivative of the energy space with respect to geometry. To a first approximation, there are twice as many minima in gradient norm space as there are in energy space.

It is unlikely that there exists any simple way to refine a geometry that results in this message. While it is appreciated that a large amount of effort has probably already been expended in getting to this point, users should steel themselves to writing off the whole geometry. It is not recommended that a minor change be made to the geometry and the job re-submitted.

Try using SIGMA instead of POWSQ.

TEMPERATURE RANGE STARTS TOO LOW, . . .

The thermodynamics calculation assumes that the statistical summations can be replaced by integrals. This assumption is only valid above 100K, so the lower temperature bound is set to 100, and the calculation continued.

THERE IS A RISK OF INFINITE LOOPING . . .

The SCF criterion has been reset by the user, and the new value is so small that the SCF test may never be satisfied. This is a case of user beware!

THIS MESSAGE SHOULD NEVER APPEAR, CONSULT A PROGRAMMER!

This message should never appear; a fault has been introduced into MOPAC, most probably as a result of a programming error. If this message appears in the vanilla version of MOPAC (a version ending in 00), please contact JJPS as I would be most interested in how this was achieved.

THREE ATOMS BEING USED TO DEFINE . . .

If the cartesian coordinates of an atom depend on the dihedral angle it makes with three other atoms, and those three atoms fall in an almost straight line, then a small change in the cartesian coordinates of one of those three atoms can cause a large change in its position. This is a potential source of trouble, and the data should be changed to make the geometric specification of the atom in question less ambiguous.

This message can appear at any time, particularly in reaction path and saddle-point calculations.

An exception to this rule is if the three atoms fall into an exactly straight line. For example, if, in propyne, the hydrogens are defined in terms of the three carbon atoms, then no error will be flagged. In such a system the three atoms in the straight line must not have the angle between them optimized, as the finite step in the derivative calculation would displace one atom off the straight line and the error-trap would take effect.

Correction involves re-defining the connectivity. LET and GEO-OK will not allow the calculation to proceed.

- - - - - - - TIME UP - - - - - - -

The time defined on the keywords line or 3,600 seconds, if no time was specified, is likely to be exceeded if another cycle of calculation were to be performed. A controlled termination of the run would follow this message. The job may terminate earlier than expected: this is ordinarily due to one of the recently completed cycles taking unusually long, and the safety margin has been increased to allow for the possibility that the next cycle might also run for much longer than expected.

TRIPLET SPECIFIED WITH ODD NUMBER OF ELECTRONS

If TRIPLET has been specified the number of electrons must be even. Check the charge on the system, the empirical formula, and whether TRIPLET was intended.

""""""""""""""UNABLE TO ACHIEVE SELF-CONSISTENCY
See the error-message: <<<<----**** FAILED TO ACHIEVE SCF.  ****---->>>>.
UNDEFINED SYMMETRY FUNCTION USED

Symmetry operations are restricted to those defined, i.e., in the range 1–18. Any other symmetry operations will trip this fatal message.

UNRECOGNIZED ELEMENT NAME

In the geometric specification a chemical symbol which does not correspond to any known element has been used. The error lies in the first datum on a line of geometric data.

**** WARNING ****

Don’t pay too much attention to this message. Thermodynamics calculations require a higher precision than vibrational frequency calculations. In particular, the gradient norm should be very small. However, it is frequently not practical to reduce the gradient norm further, and to date no-one has determined just how slack the gradient criterion can be before unacceptable errors appear in the thermodynamic quantities. The 0.4 gradient norm is only a suggestion.

WARNING: INTERNAL COORDINATES . . .

Triatomics are, by definition, defined in terms of internal coordinates. This warning is only a reminder. For diatomics, cartesian and internal coordinates are the same. For tetra-atomics and higher, the presence or absence of a connectivity table distinguishes internal and cartesian coordinates, but for triatomics there is an ambiguity. To resolve this, cartesian coordinates are not allowed for the data input for triatomics.

Error messages produced by MOPAC

Criteria

MOPAC uses various criteria which control the precision of its stages. These criteria are chosen as the best compromise between speed and acceptable errors in the results. The user can override the default settings by use of keywords; however, care should be exercised as increasing a criterion can introduce the potential for infinite loops, and decreasing a criterion can result in unacceptably imprecise results. These are usually characterized by ‘noise’ in a reaction path, or large values for the trivial vibrations in a force calculation.

9.1 SCF criterion

Name:          SCFCRT.
Defined in     ITER.
Default value  0.0001 kcal/mole
Basic Test     Change in energy in kcal/mole on successive
               iterations is less than SCFCRT.
Exceptions:    If PRECISE is specified,        SCFCRT=0.000001
               If a polarization calculation   SCFCRT=1.D-11
               If a FORCE calculation          SCFCRT=0.0000001
               If SCFCRT=n.nnn is specified    SCFCRT=n.nnn
               If a BFGS optimization, SCFCRT becomes a function
               of the difference between the current energy and
               the lowest energy of previous SCFs.
Secondary tests: (1) Change in density matrix elements on two
                     successive iterations must be less than 0.001
                 (2) Change in energy in eV on three successive
                     iterations must be less than 10 x SCFCRT.

9.2 Geometric optimization criteria

Name:           TOLERX   "Test on X Satisfied"
Defined in      FLEPO
Default value   0.0001 Angstroms
Basic Test      The projected change in geometry is less than
                TOLERX Angstroms.
Exceptions      If GNORM is specified, the TOLERX test is not used.
Name:           DELHOF    "Herbert's Test Satisfied"

\underline{Criteria}

Defined in      FLEPO
Default value   0.001
Basic Test      The projected decrease in energy is less than
                DELHOF kcals/mole.
Exceptions      If GNORM is specified, the DELHOF test is not used.
Name:           TOLERG    "Test on Gradient Satisfied"
Defined in      FLEPO
Default value   1.0
Basic Test      The gradient norm in kcals/mole/Angstrom is less
                than TOLERG multiplied by the square root of the
                number of coordinates to be optimized.
Exceptions      If GNORM=n.nnn is specified, TOLERG=n.nnn divided
                by the square root of the number of coordinates
                to be optimized, and the secondary tests are not
                done.  If LET is not specified, n.nnn is reset to
                0.01, if it was smaller than 0.01.
                If PRECISE is specified, TOLERG=0.2
                If a SADDLE calculation, TOLERG is made a function
                of the last gradient norm.
Name:           TOLERF    "Heat of Formation Test Satisfied"
Defined in      FLEPO
Default value   0.002 kcal/mole
Basic Test      The calculated heats of formation on two successive
                cycles differ by less than TOLERF.
Exceptions      If GNORM is specified, the TOLERF test is not used.
Secondary Tests For the TOLERG, TOLERF, and TOLERX tests, a
                second test in which no individual component of the
                gradient should be larger than TOLERG must be
                satisfied.
Other Tests     If, after the TOLERG, TOLERF, or TOLERX test has been
                satisfied three consecutive times the heat of
                formation has dropped by less than 0.3kcal/mole, then
                the optimization is stopped.
Exceptions      If GNORM is specified, then this test is not performed.
Name:           TOL2
Defined in      POWSQ
Default value   0.4
Basic Test      The absolute value of the largest component of the
                gradient is less than TOL2
Exceptions      If PRECISE is specified, TOL2=0.01
                If GNORM=n.nn is specified, TOL2=n.nn
                If LET is not specified, TOL2 is reset to
                0.01, if n.nn was smaller than 0.01.

9.2 Geometric optimization criteria

Name:           TOLS1
Defined in      NLLSQ
Default Value   0.000 000 000 001
Basic Test      The square of the ratio of the projected change in the
                geometry to the actual geometry is less than TOLS1.
Name:           <none>
Defined in      NLLSQ
Default Value   0.2
Basic Test      Every component of the gradient is less than 0.2.

Criteria

Debugging

There are three potential sources of difficulty in using MOPAC, each of which requires special attention. There can be problems with data, due to errors in the data, or MOPAC may be called upon to do calculations for which it was not designed. There are intrinsic errors in MOPAC which extensive testing has not yet revealed, but which a user’s novel calculation uncovers. Finally there can be bugs introduced by the user modifying MOPAC, either to make it compatible with the host computer, or to implement local features.

For whatever reason, the user may need to have access to more information than the normal keywords can provide, and a second set, specifically for debugging, is provided. These keywords give information about the working of individual subroutines, and do not affect the course of the calculation.

10.1 Debugging keywords

A full list of keywords for debugging subroutines:

1ELEC          the one-electron matrix.                          Note 1
COMPFG         Heat of Formation.
DCART          Cartesian derivatives.
DEBUG                                                            Note 2
DEBUGPULAY     Pulay matrix, vector, and error-function.         Note 3
DENSITY        Every density matrix.                             Note 1
DERI1          Details of DERI1 calculation
DERI2          Details of DERI2 calculation
DERITR         Details of DERITR calculation
DERIV          All gradients, and other data in DERIV.
DERNVO         Details of DERNVO calculation
DFORCE         Print Force Matrix.
DIIS           Details of DIIS calculation
EIGS           All eigenvalues.
FLEPO          Details of BFGS minimization.
FMAT
FOCK           Every Fock matrix                                 Note 1
HCORE          The one electron matrix, and two electron integrals.
ITER           Values of variables and constants in ITER.
LARGE          Increases amount of output generated by other keywords.
LINMIN         Details of line minimization (LINMIN, LOCMIN, SEARCH)
MOLDAT         Molecular data, number of orbitals, "U" values, etc.
MECI           C.I. matrices, M.O. indices, etc.
PL             Differences between density matrix elements       Note 4

\underline{Debugging}

               in ITER.
LINMIN         Function values, step sizes at all points in the
               line minimization (LINMIN or SEARCH).
TIMES          Times of stages within ITER.
VECTORS        All eigenvectors on every iteration.              Note 1

Notes

1. These keywords are activated by the keyword DEBUG. Thus if DEBUG and FOCK are both specified, every Fock matrix on every iteration will be printed.

2. DEBUG is not intended to increase the output, but does allow other keywords to have a special meaning.

3. PULAY is already a keyword, so DEBUGPULAY was an obvious alternative.

4. PL initiates the output of the value of the largest difference between any two density matrix elements on two consecutive iterations. This is very useful when investigating options for increasing the rate of convergence of the SCF calculation.

Suggested procedure for locating bugs

Users are supplied with the source code for MOPAC, and, while the original code is fairly bug-free, after it has been modified there is a possibility that bugs may have been introduced. In these circumstances the author of the changes is obviously responsible for removing the offending bug, and the following ideas might prove useful in this context.

First of all, and most important, before any modifications are done a back-up copy of the standard MOPAC should be made. This will prove invaluable in pinpointing deviations from the standard working. This point cannot be over-emphasized — make a back-up before modifying MOPAC!.

Clearly, a bug can occur almost anywhere, and a logical search sequence is necessary in order to minimize the time taken to locate it.

If possible, perform the debugging with a small molecule, in order to save time (debugging is, of necessity, time consuming) and to minimize output.

The two sets of subroutines in MOPAC, those involved with the electronics and those involved in the geometrics, are kept strictly separate, so the first question to be answered is which set contains the bug. If the heats of formation, derivatives, I.P.s, and charges, etc., are correct, the bug lies in the geometrics; if faulty, in the electronics.

Bug in the Electronics Subroutines

Use formaldehyde for this test. The supplied data-file MNRSD1.DAT could be used as a template for this operation. Use keywords 1SCF, DEBUG, and any others necessary.

The main steps are:

1. Check the starting one-electron matrix and two-electron integral string, using the keyword HCORE. It is normally sufficient to verify that the two hydrogen atoms are equivalent, and that the pi system involves only pz on oxygen and carbon. Note that numerical values are not checked, but only relative values.

If an error is found, use MOLDAT to verify the orbital character, etc.

If faulty the error lies in READ, GETGEO or MOLDAT.

Otherwise the error lies in HCORE, H1ELEC or ROTATE.

If the starting matrices are correct, go on to step (2).

10.1 Debugging keywords

2. Check the density or Fock matrix on every iteration, with the words FOCK or DENSITY. Check the equivalence of the two hydrogen atoms, and the pi system, as in (1).

If an error is found, check the first Fock matrix. If faulty, the bug lies in ITER, probably in the Fock subroutines FOCK1 or FOCK2. or in the (guessed) density matrix (MOLDAT). An exception is in the UHF closed-shell calculation, where a small asymmetry is introduced to initiate the separation of the alpha and beta UHF wavefunctions.

If no error is found, check the second Fock matrix. If faulty, the error lies in the density matrix DENSIT, or the diagonalization RSP.

If the Fock matrix is acceptable, check all the Fock matrices. If the error starts in iterations 2 to 4, the error probably lies in CNVG, if after that, in PULAY, if used.

If SCF is achieved, and the heat of formation is faulty, check HELECT. If C.I. was used check MECI.

If the derivatives are faulty, use DCART to verify the cartesian derivatives. If these are faulty, check DCART and DHC. If they are correct, or not calculated, check the DERIV finite difference calculation. If the wavefunction is non-variationally optimized, check DERNVO.

If the geometric calculation is faulty, use FLEPO to monitor the optimization, DERIV may also be useful here.

For the FORCE calculation, DCART or DERIV are useful for variationally optimized functions, COMPFG for non-variationally optimized functions.

For reaction paths, verify that FLEPO is working correctly; if so, then PATHS is faulty.

For saddle-point calculations, verify that FLEPO is working correctly; if so, then REACT1 is faulty.

Keep in mind the fact that MOPAC is a large calculation, and while intended to be versatile, many combinations of options have not been tested. If a bug is found in the original code, please communicate details to the Academy, to Dr. James J. P. Stewart, Frank J. Seiler Research Laboratory, U.S. Air Force Academy, Colorado Springs, CO 80840–6528.

Debugging

Installing MOPAC

MOPAC is distributed on a magnetic tape as a set of FORTRAN-77 files, along with ancillary documents such as command, help, data and results files. The format of the tape is that of DIGITAL’S VAX computers. The following instructions apply only to users with VAX computers: users with other machines should use the following instructions as a guide to getting MOPAC up and running.

1. Put the magnetic tape on the tape drive, write protected.

2. Allocate the tape drive with a command such as $ALLOCATE MTA0:

3. Go into an empty directory which is to hold MOPAC

4. Mount the magnetic tape with the command $MOUNT MTA0: MOPAC

5. Copy all the files from the tape with the command $COPY MTA0:*.* * A useful operation after this would be to make a hard copy of the directory. You should now

have the following sets of files in the directory:

1. A file, AAAINVOICE.TXT, summarizing this list.

2. A set of FORTRAN–77 files, see Appendix A.

3. The command files COMPILE, MOPACCOM, MOPAC, RMOPAC, and SHUT.

4. A file, MOPAC.OPT, which lists all the object modules used by MOPAC.

5. Help files MOPAC.HLP and HELP.FOR

6. A text file MOPAC.MAN.

7. A manual summarizing the updates, called UPDATE.MAN.

8. Two test-data files: TESTDATA.DAT and MNRSD1.DAT, and corresponding results files, TESTDATA.OUT and MNRSD1.OUT.

Structure of command files: COMPILE

The parameter file DIMSIZES.DAT should be read and, if necessary, modified before COMPILE is run.

DO NOT RUN COMPILE AT THIS TIME!!

Installing MOPAC

COMPILE should be run once only. It assigns DIMSIZES.DAT, the block of FORTRAN which contains the PARAMETERS for the dimension sizes to the logical name “SIZES”. This is a temporary assignment, but the user is strongly recommended to make it permanent by suitably modifying LOGIN file(s). COMPILE is a modified version of Maj Donn Storch’s COMPILE for DRAW-2.

All the FORTRAN files are then compiled, using the array sizes given in DIMSIZES.DAT: these should be modified before COMPILE is run. If, for whatever reason, DIMSIZES.DAT needs to be changed, then COMPILE should be re-run, as modules compiled with different DIMSIZES.DAT will be incompatible.

The parameters within DIMSIZES.DAT that the user can modify are MAXLIT, MAXHEV, MAXTIM and MAXDMP. MAXLIT is assigned a value equal to the largest number of hydrogen atoms that a MOPAC job is expected to run, MAXHEV is assigned the corresponding number of heavy (non-hydrogen) atoms. The ratio of light to heavy atoms should not be less than 1/2. Do not set MAXHEV or MAXLIT less than 7. If you do, some subroutines will not compile correctly. Some molecular orbital eigenvector arrays are overlapped with Hessian arrays, and to prevent compilation time error messages, the number of allowed A.O.’s must be greater than, or equal to three times the number of allowed real atoms. MAXTIM is the default maximum time in seconds a job is allowed to run before either completion or a restart file being written. MAXDMP is the default time in seconds for the automatic writing of the restart files. If your computer is very reliable, and disk space is at a premium, you might want to set MAXDMP as MAXDMP=999999.

If SYBYL output is wanted, set ISYBYL to 1, otherwise set it to zero.

If you want, NMECI can be changed. Setting it to 1 will save some space, but will prevent all C.I. calculations except simple radicals.

If you want, NPULAY can be set to 1. This saves memory, but also disables the PULAY converger.

If you want, MESP can be varied. This is only meaningful if ESP is installed.

Compile MOPAC. This operation takes about 7 minutes, and should be run “on-line”, as a question and answer session is involved.

When everything is successfully compiled, the object files will then be assembled into an executable image called MOPAC.EXE. Once the image exists, there is no reason to keep the object files, and if space is at a premium these can be deleted at this time.

If you need to make any changes to any of the files, COMPILE followed by the names of the changed files will reconstruct MOPAC, provided all the other OBJ files exist. For example, if you change the version number in DIMSIZES.DAT, then READ.FOR and WRITE.FOR are affected and will need to be recompiled. This can be done using the command @COMPILE WRITE,READ

In the unlikely event that you want to link only, use the command @COMPILE LINK Sometimes the link stage will fail, and give the message

 "%LINK-E-INSVIRMEM, insufficient virtual memory for 2614711. pages
  -LINK-E-NOIMGFIL, image file not created",

or your MOPAC will not run due to the size of the image. In these cases you should ask the system manager to alter your PGFLQUO and WSEXTENT limits. Possibly the system limits, VIRTUALPAGECNT CURRENT and MAX will need to be changed. As an example, on a Microvax 3600 with 16Mb of memory:

 PGFLQUO=50000,  WSEXTENT=16000,  VIRTUALPAGECNT  CURRENT=40768,
 VIRTUALPAGECNT MAX=600000

are sufficient for the default MOPAC values of 43 heavy and 43 light atoms.

In order for users to have access to MOPAC they must insert in their individual LOGIN.COM files the line:

$@ <Mopac-directory>MOPACCOM

where <Mopac-directory> is the name of the disk and directory which holds all the MOPAC files. For example:

DRA0:[MOPAC]
thus: $@ DRA0:[MOPAC]MOPAC

MOPACCOM.COM should be modified once to accommodate local definitions of the directory which is to hold MOPAC. This change must also be made to RMOPAC.COM and to MOPAC.COM.

MOPAC

This command file submits a MOPAC job to a queue. Before use, MOPAC.COM should be modified to suit local conditions. The user’s VAX is assumed to run three queues, called QUEUE3, QUEUE2, and QUEUE1. The user should substitute the actual names of the VAX queues for these symbolic names. Thus, for example, if the local names of the queues are “TWELVEHOUR”, for jobs of length up to 12 hours, “ONEHOUR”, for jobs of less than one hour, and “30MINS” for quick jobs, then in place of “QUEUE3”, “QUEUE2”, and “QUEUE1” the words “TWELVEHOUR”, “ONEHOUR”, and “30MINS” should be inserted.

RMOPAC

RMOPAC is the command file for running MOPAC. It assigns all the data files that MOPAC uses to the channels. If the user wants to use other file-name endings than those supplied, the modifications should be made to RMOPAC.

When a long job ends, RMOPAC will also send a mail message to the user giving a brief description of the job. You may want to change the default definition of “a long job”; currently it is 12 hours. This feature was written by Dr. James Petts of Kodak Ltd Research Labs.

A recommended sequence of operations to get MOPAC up and running would be:

1. Modify the file DIMSIZES.DAT. The default sizes are 40 heavy atoms and 40 light atoms. Do not make the size less than 7 by 7.

2. Read through the COMMAND files to familiarize yourself with what is being done.

3. Edit the file MOPAC.COM to use the local queue names.

4. Edit the file RMOPAC.COM if the default file-names are not acceptable.

5. Edit MOPACCOM.COM to assign MOPACDIRECTORY to the disk and directory which will hold MOPAC.

6. Edit the individual LOGIN.COM files to insert the following line:

     $@ <Mopac-directory>MOPACCOM

Note that MOPACDIRECTORY cannot be used, as the definition of MOPACDIRECTORY is made in MOPACCOM.COM

7. Execute the modified LOGIN command so that the new commands are effective.

8. Run COMPILE.COM. This takes about 8 minutes to execute.

9. Enter the command $MOPAC

You will receive the message What file? : to which the reply should be the actual data-file name. For example, “MNRSD1”, the file is assumed to end in .DAT, e.g. MNRSD1.DAT. You will then be prompted for the queue:

     What queue? :

Any queue defined in MOPAC.COM will suffice: "SYS$BATCH"

Finally, the priority will be requested: What priority? [5]: To which any value between 1 and 5 will suffice. Note that the maximum priority is limited by the system (manager).

11.1 ESP calculation

As supplied, MOPAC will not do the ESP calculation because of the large memory requirement of the ESP. To install the ESP, make the following changes:

1. Rename ESP.ROF to ESP.FOR

2. Add to the first line of MOPAC.OPT the string “ ESP, ” (without the quotation marks).

3. Edit MNDO.FOR to uncomment the line C# CALL ESP.

4. Compile ESP and MNDO, and relink MOPAC using, e.g. @COMPILE ESP,MNDO.

5. If the resulting executable is too large, modify DIMSIZES.DAT to reduce MAXHEV and MAXLIT, then recompile everything and relink MOPAC with @COMPILE.

To familiarize yourself with the system, the following operations might be useful.

1. Run the (supplied) test molecules, and verify that MOPAC is producing “acceptable” results.

2. Make some simple modifications to the datafiles supplied in order to test your understanding of the data format

3. When satisfied that MOPAC is working, and that data files can be made, begin production runs.

Working of SHUTDOWN command

If, for whatever reason, a run needs to be stopped prematurely, the command $SHUT <jobname> can be issued. This will execute a small command-language file, which copies the data-file to form

a new file called <filename>.END.

The next time MOPAC calls function SECOND, the presence of a readable file called SHUT-DOWN, logically identified with <filename>.END, is checked for, and if it exists, the apparent elapsed CPU time is increased by 1,000,000 seconds, and a warning message issued. No further action is taken until the elapsed time is checked to see if enough time remains to do another cycle. Since an apparently very long time has been used, there is not enough time left to do another cycle, and the restart files are generated and the run stopped.

SHUTDOWN is completely machine–independent.

Specific instructions for mounting MOPAC on other computers have been left out due to limitations of space in the Manual; however, the following points may prove useful:

1. Function SECOND is machine-specific. SECOND is double-precision, and should return the CPU time in seconds, from an arbitary zero of time. If the SHUT command has been issued, the value returned by SECOND should be increased by 1,000,000.

2. On UNIX-based and other machines, on-line help can be provided by using help.f. Documentation on help.f is in help.f.

3. OPEN and CLOSE statements are a fruitful source of problems. If MOPAC does not work, most likely the trouble lies in these statements.

4. RMOPAC.COM should be read to see what files are attached to what logical channel.

11.1 ESP calculation

How to use MOPAC

The COM file to run the MOPAC can be accessed using the command “MOPAC” followed by none, one, two or three arguments. Possible options are:

   MOPAC   MYDATAFILE 120  4
   MOPAC   MYDATAFILE 120
   MOPAC   MYDATAFILE

In the latter case it is assumed that the shortest queue will be adequate. The COM file to run the MOPAC can be accessed using the command ”MOPAC” followed by none, one or two arguments. Possible options are:

   MOPAC   MYDATAFILE 120
   MOPAC   MYDATAFILE

In the latter case it is assumed that the default time (15 seconds) will be adequate.

MOPAC

In this case you will be prompted for the datafile, and then for the queue. Restarts should be user transparent. If MOPAC does make any restart files, do not change them (It would be hard to do anyhow, as they’re in machine code), as they will be used when you run a RESTART job. The files used by MOPAC are:

     File         Description                     Logical name
<filename>.DAT    Data                                FOR005
<filename>.OUT    Results                             FOR006
<filename>.RES    Restart                             FOR009
<filename>.DEN    Density matrix (in binary)          FOR010
SYS$OUTPUT        LOG file                            FOR011
<filename>.ARC    Archive or summary                  FOR012
<filename>.GPT    Data for program DENSITY            FOR013
<filename>.SYB    SYBYL data                          FOR016
SETUP.DAT         SETUP data                          SETUP

Short version

For various reasons it might not be practical to assemble the entire MOPAC program. For example, your computer may have memory limitations, or you may have very large systems to be run, or some options may never be wanted. For whatever reason, if using the entire program is undesirable, an abbreviated version, which lacks the full range of options of the whole program, can be specified at compile time.

At the bottom of the DIMSIZES.DAT file the programmer is asked for various options to be used in compiling. These options allow arrays of MECI, PULAY, and ESP to assume their correct size.

As long as no attempt is made to use the reduced subroutines, the program will function normally. If an attempt is made to use an option which has been excluded then the program will error.

Size of MOPAC

The amount of storage required by MOPAC depends mainly on the number of heavy and light atoms. As it is useful for programmers to have an idea of how large various MOPACs are, the following data are presented as a guide.

Sizes of various MOPAC Version 6.00 executables in which the number of heavy atoms is equal to the number of light atoms, assembled on a VAX computer, are:

  No. of heavy atoms     Size of Executable (Kbytes)
                        MOPAC 5.00   MOPAC 6.00     (AMPAC 2.00)
      10                    1,653      2,054           N/A
      20                    3,442      4,689          4,590
      30                    6,356      8,990          9,150
      40                   10,400     14,955         15,588
      50                   15,572     22,586         23,944
      60                   21,872     31,880         34,145
     100                   58,361     87,519
     200                  228,602    336.867
     300                  511,723    754,540

The size, S, of any given MOPAC executable, in Kbytes, may be estimated for MOPAC 5.00 as:

S = 9939 + N ∗ 9.57 + N ∗ N ∗ 5.64

and for MOPAC 6.00 as:

S = 1091 + N ∗ 13.40 + N ∗ N ∗ 8.33

The large increase in size of MOPAC was caused mainly by the inclusion of the analytical C.I. derivatives. Because they are so much more efficient and accurate than finite differences, and because computer memory is becoming more available, this increase was accepted as the lesser of two evils.

The size of MOPAC executables will vary from machine to machine, due to the different sizes of the code. For a VAX, this amounts to approximately 0.1Mb. Most machines use a 64 bit or 8 byte double precision real number, so the multipliers of N and N*N should apply to them. For large jobs, 0.1Mb is negligible, therefore the above expression should be applicable to most computers.

No. of lines in program in Version 5.00 = 22,084 = 17,718 code + 4,366 comment. Version 6.00 = 31,857 = 22,526 code + 9,331 comment.

Appendix A

Names of FORTRAN-77 files

AABABC   ANALYT   ANAVIB   AXIS     BLOCK    BONDS    BRLZON
CALPAR   CAPCOR   CDIAG    CHRGE    CNVG     COMPFG   DATIN
DCART    DELMOL   DELRI    DENROT   DENSIT   DEPVAR   DERI0
DERI1    DERI2    DERI21   DERI22   DERI23   DERITR   DERIV
DERNVO   DERS     DFOCK2   DFPSAV   DIAG     DIAT     DIAT2
DIIS     DIJKL1   DIJKL2   DIPIND   DIPOLE   DOFS     DOT
DRC      DRCOUT   EF       ENPART   EXCHNG   FFHPOL   FLEPO
FMAT     FOCK1    FOCK2    FORCE    FORMXY   FORSAV   FRAME
FREQCY   GEOUT    GEOUTG   GETGEG   GETGEO   GETSYM   GETTXT
GMETRY   GOVER    GRID     H1ELEC   HADDON   HCORE    HELECT
HQRII    IJKL     INTERP   ITER     JCARIN   LINMIN   LOCAL
LOCMIN   MAMULT   MATOUT   MATPAK   MECI     MECID    MECIH
MECIP    MNDO     MOLDAT   MOLVAL   MULLIK   MULT     NLLSQ
NUCHAR   PARSAV   PARTXY   PATHK    PATHS    PERM     POLAR
POWSAV   POWSQ    PRTDRC   QUADR    REACT1   READ     READA
REFER    REPP     ROTAT    ROTATE   RSP      SEARCH   SECOND
SETUPG   SOLROT   SWAP     SYMTRY   THERMO   TIMER    UPDATE
VECPRT   WRITE    WRTKEY   WRTTXT   XYZINT

Names of FORTRAN-77 files

Appendix B

Subroutine calls in MOPAC

A list of the program segments which call various subroutines.

SUBROUTINE            CALLS
AABABC
AABACD
AABBCD
AINTGS
ANALYT       DERS   DELRI  DELMOL
ANAVIB
AXIS         RSP
BABBBC
BABBCD
BANGLE
BFN
BINTGS
BKRSAV       GEOUT
BONDS        VECPRT MPCBDS
BRLZON       CDIAG  DOFS
CALPAR
CAPCOR
CDIAG        ME08A  EC08C   SORT
CHRGE
CNVG
COE
COMPFG       SETUPG SYMTRY  GMETRY  TIMER  HCORE  ITER
             DIHED  DERIV   MECIP
DANG
DATIN        UPDATE  MOLDAT  CALPAR
DCART        ANALYT  DHC     DIHED
DELMOL       ROTAT
DELRI
DENROT       GMETRY  COE
DENSIT
DEPVAR
DERI0
DERI1        TIMER   DHCORE  SCOPY  DFOCK2 SUPDOT MTXM   MXM
             DIJKL1  MECID   MECIH  SUPDOT TIMER
DERI2        DERI21  DERI22  MXM    OSINV  MTXM   SCOPY  DERI23

Subroutine calls in MOPAC

             DIJKL2  MECID   MECIH  SUPDOT
DERI21       MTXMC   HQRII   MXM
DERI22       MXM     MXMT    FOCK2  FOCK1  SUPDOT
DERI23       SCOPY
DERITR       SYMTRY  GMETRY  HCORE  ITER   DERIV  DERNVO  DCART
             JCARIN  MXM     GEOUT  DERITR
DERNVO       DERI0   DERI1   DERI2
DERS
DFOCK2       JAB     KAB
DFPSAV       XYZINT  GEOUT
DHC          H1ELEC  ROTATE  SOLROT  FOCK2
DHCORE       H1ELEC  ROTATE
DIAG         EPSETA
DIAGI
DIAT         COE     GOVER   DIAT2
DIAT2        SET
DIHED        DANG
DIIS         SPACE   VECPRT  MINV
DIJKL1       FORMXY
DIJKL2
DIPIND       CHRGE   GMETRY
DIPOLE
DOFS
DRC          GMETRY  COMPFG  PRTDRC
DRCOUT
EA08C        EA09C
EA09C
EC08C        EA08C
EF           BKRSAV  COMPFG  BKRSAV  UPDHES  HQRII  FORMD  SYMTRY
ENPART
EPSETA
EXCHNG
FFHPOL       COMPFG  DIPIND   VECPRT  RSP    MATOUT
FLEPO        DFPSAV  COMPFG   SCOPY   GEOUT  SUPDOT  LINMIN  DIIS
FMAT         FORSAV  COMPFG   CHRGE
FOCK2        JAB     KAB
FOCK2D
FORCE        GMETRY  COMPFG   NLLSQ   FLEPO  WRITE   XYZINT  AXIS
             FMAT    VECPRT   FRAME   RSP    MATOUT  FREQCY  MATOUT
             DRC     ANAVIB   THERMO
FORMD        OVERLP
FORMXY
FORSAV
FRAME        AXIS
FREQCY       BRLZON  FRAME    RSP
GEOUT        XYZINT  WRTTXT   CHRGE
GEOUTG       XXX
GETDAT
GETGEG       GETVAL  GETVAL   GETVAL
GETGEO       GEOUT   NUCHAR   XYZINT
GETSYM
GETTXT       UPCASE
GMETRY       GEOUT
GOVER

GRID         DFPSAV  FLEPO    GEOUT  WRTTXT
H1ELEC       DIAT
HADDON       DEPVAR
HCORE        H1ELEC  ROTATE   SOLROT VECPRT
HELECT
HQRII
IJKL         PARTXY
INTERP       HQRII   SCHMIT   SCHMIB  SPLINE
ITER         EPSETA  VECPRT   FOCK2   FOCK1  WRITE  INTERP  PULAY
             HQRII   DIAG     MATOUT  SWAP   DENSIT CNVG
JAB
JCARIN       SYMTRY  GMETRY
KAB
LINMIN       COMPFG  EXCHNG
LOCAL        MATOUT
LOCMIN       COMPFG  EXCHNG
MNDO         GETDAT  READ  MOLDAT  DATIN   REACT1  GRID    PATHS
             PATHK   FORCE DRC     NLLSQ   COMPFG  POWSQ   EF
             FLEPO   WRITE POLAR
MAMULT
MATOUT
ME08A        ME08B
ME08B
MECI         IJKL    PERM  MECIH   VECPRT  HQRII   MATOUT
MECIH
MECIP        MXM
MINV
MOLDAT       REFER  GMETRY VECPRT
MOLVAL
MPCBDS
MPCPOP
MPCSYB
MTXM
MTXMC        MXM
MULLIK       RSP    GMETRY  MULT  DENSIT  VECPRT
MULT
MXM
MXMT
NLLSQ        PARSAV COMPFG  GEOUT  LOCMIN  PARSAV
NUCHAR
OSINV
OVERLP
PARSAV       XYZINT  GEOUT
PARTXY       FORMXY
PATHK        DFPSAV  FLEPO  GEOUT  WRTTXT
PATHS        DFPSAV  FLEPO  WRITE
PERM
POLAR        GMETRY  AXIS   COMPFG  FFHPOL
POWSAV       XYZINT  GEOUT
POWSQ        POWSAV  COMPFG  VECPRT  RSP    SEARCH
PRTDRC       CHRGE   XYZINT  QUADR
PULAY        MAMULT  OSINV
QUADR
REACT1       GETGEO  SYMTRY  GEOUT  GMETRY  FLEPO  COMPFG   WRITE

READ         GETTXT  GETGEG  GETGEO DATE    GEOUT  WRTKEY  GETSYM
             SYMTRY  NUCHAR  WRTTXT GMETRY
REFER
REPP
ROTAT
ROTATE        REPP
RSP           EPSETA  TRED3  TQLRAT  TQL2  TRBAK3
SAXPY
SCHMIB
SCHMIT
SCOPY
SEARCH        COMPFG
SECOND
SET           AINTGS   BINTGS
SETUPG
SOLROT        ROTATE
SORT
SPACE
SPLINE        BFN
SUPDOT
SWAP
SYMTRY        HADDON
THERMO
TIMCLK
TIMER
TIMOUT
TQL2
TQLRAT
TRBAK3
TRED3
UPCASE
UPDATE
UPDHES
VECPRT
WRITE         DATE   WRTTXT  GEOUT  DERIV   TIMOUT SYMTRY  GMETRY GEOUT
              VECPRT MATOUT  CHRGE  BRLZON  MPCSYB DENROT  MOLVAL BONDS
              LOCAL  ENPART  MULLIK MPCPOP  GEOUTG
WRTKEY
WRTTXT
XXX
XYZGEO        BANGLE  DIHED
XYZINT        DIHED  BANGLE  XYZGEO

A list of subroutines called by various segments (the inverse of the first list)

Subroutine          Called by
AABABC     MECIH
AABACD     MECIH
AABBCD     MECIH
AINTGS     SET
ANALYT     DCART
ANAVIB     FORCE
AXIS       FORCE      FRAME      POLAR
BABBBC     MECIH
BABBCD     MECIH

BANGLE     XYZGEO     XYZINT
BFN        SPLINE
BINTGS     SET
BKRSAV     EF
BONDS      WRITE
BRLZON     FREQCY     WRITE
CALPAR     DATIN
CAPCOR     ITER
CDIAG      BRLZON
CHRGE      DIPIND     FMAT       GEOUT      PRTDRC     WRITE
CNVG       ITER
COE        DENROT     DIAT
COMPFG     DRC        EF         FFHPOL     FLEPO      FMAT
           FORCE      LINMIN     LOCMIN     MNDO       NLLSQ
           POLAR      POWSQ      REACT1     SEARCH
DANG       DIHED
DATIN      MNDO
DCART      DERITR
DELMOL     ANALYT
DELRI      ANALYT
DENROT     WRITE
DENSIT     ITER       MULLIK
DEPVAR     HADDON
DERI0      DERNVO
DERI1      DERNVO
DERI2      DERI2      DERNVO
DERI21     DERI2
DERI22     DERI2
DERI23     DERI2
DERITR     DERITR
DERNVO     DERITR
DERS       ANALYT
DFOCK2     DERI1
DFPSAV     FLEPO      GRID       PATHK      PATHS
DHC        DCART      DERI1
DHCORE     DERI1
DIAG       DERI21     ITER
DIAGI      DERI21
DIAT       DIAT       H1ELEC
DIAT2      DIAT
DIHED      COMPFG     DCART      XYZGEO     XYZINT
DIIS       FLEPO
DIJKL1     DERI1
DIJKL2     DERI2
DIPIND     FFHPOL
DIPOLE     FMAT       WRITE
DOFS       BRLZON
DRC        FORCE      MNDO
DRCOUT     PRTDRC
EA08C      EC08C
EA09C      EA08C
EC08C      CDIAG
EF         MNDO
ENPART     WRITE

EPSETA     DIAG       ITER       RSP
EXCHNG     LINMIN     LOCMIN
FFHPOL     POLAR
FLEPO      FORCE      GRID       MNDO       PATHK      PATHS
           REACT1
FMAT       FORCE
FOCK2      DERI22     DHC        ITER
FORCE      MNDO
FORMD      EF
FORMXY     DIJKL1     PARTXY
FORSAV     FMAT
FRAME      FORCE      FREQCY
FREQCY     FORCE
GEOUT      BKRSAV     DERITR     DFPSAV     FLEPO      GETGEO
           GMETRY     GRID       NLLSQ      PARSAV     PATHK
           POWSAV     REACT1     READ
WRITE      WRITE
GEOUTG     WRITE
GETDAT     MNDO
GETGEG     READ
GETGEO     REACT1     READ
GETSYM     READ
GETTXT     READ
GMETRY     COMPFG     DENROT     DERITR     DIPIND     DRC
           FORCE      JCARIN     MOLDAT     MULLIK     POLAR
           REACT1     READ       WRITE
GOVER      DIAT
GRID       MNDO
H1ELEC     DHC        DHCORE     HCORE
HADDON     SYMTRY
HCORE      COMPFG     DERITR
HELECT     DCART      DERI2      ITER
HQRII      EF         INTERP     ITER       MECI
IJKL       MECI
INTERP     ITER
ITER       COMPFG     DERITR
JAB        DFOCK2     FOCK2
JCARIN     DERITR
KAB        DFOCK2     FOCK2
LINMIN     FLEPO
LOCAL      WRITE
LOCMIN     NLLSQ
MNDO       (main segment)
MAMULT     PULAY
MATOUT     FFHPOL     FORCE      ITER       LOCAL      MECI
           WRITE
ME08A      CDIAG
ME08B      ME08A
MECI       COMPFG     DERI1      DERI2      MECI
MECIH      DERI1      DERI2      MECI
MECIP      COMPFG
MINV       DIIS
MOLDAT     DATIN      MNDO
MOLVAL     WRITE

MPCBDS     BONDS
MPCPOP     WRITE
MPCSYB     WRITE
MTXM       DERI1      DERI2      DERI21
MTXMC      DERI21
MULLIK     WRITE
MULT       MULLIK
MXM        DERI1      DERI2      DERI21     DERI22     DERITR
           MECIP      MTXMC
MXMT       DERI22
NLLSQ      FORCE      MNDO
NUCHAR     GETGEO     READ
OSINV      DERI2      PULAY
OVERLP     FORMD
PARSAV     NLLSQ
PARTXY     IJKL
PATHK      MNDO
PATHS      MNDO
PERM       MECI
POLAR      MNDO
POWSAV     POWSQ
POWSQ      MNDO
PRTDRC     DRC
PULAY      ITER
QUADR      PRTDRC
REACT1     MNDO
READ       MNDO
REFER      MOLDAT
REPP       ROTATE
ROTAT      DELMOL     DHC        DHCORE     HCORE      SOLROT
ROTATE     DHC        DHCORE     HCORE      SOLROT
RSP        AXIS       FFHPOL     FORCE      FREQCY     MULLIK
           POWSQ
SCHMIB     INTERP
SCHMIT     INTERP
SCOPY      DERI1      DERI2      DERI23     FLEPO
SEARCH     POWSQ
SECOND     DERI2      DRC        EF         ESP        FLEPO
           FMAT       FORCE      GRID       ITER       MNDO
           NLLSQ      PATHK      PATHS      POWSQ      REACT1
           TIMER      WRITE
SET        COMPFG     DIAT2
SETUPG     COMPFG
SOLROT     DHC        HCORE
SORT       CDIAG
SPACE      DIIS
SPLINE     INTERP
SUPDOT     DERI1      DERI1      DERI2      DERI22     FLEPO
SWAP       ITER
SYMTRY     COMPFG     DERITR     EF         JCARIN     REACT1
           READ       WRITE
THERMO     FORCE
TIMCLK     SECOND
TIMER      COMPFG     DERI1      DERI1

TIMOUT     WRITE
TQL2       RSP
TQLRAT     RSP
TRBAK3     RSP
TRED3      RSP
UPCASE     GETTXT
UPDATE     DATIN
UPDHES     EF
VECPRT     BONDS      DIIS       FFHPOL     FORCE      HCORE
           ITER       MECI       MOLDAT     MULLIK     POWSQ
           WRITE
WRITE      FORCE      ITER       MNDO       PATHS      REACT1
WRTKEY     READ
WRTTXT     GEOUT      GRID       PATHK      READ       WRITE
XXX        GEOUTG
XYZGEO     XYZINT
XYZINT     DFPSAV     FORCE      GEOUT      GETGEO     PARSAV
           POWSAV     PRTDRC

Appendix C

Description of subroutines

• AABABC Utility: Calculates the configuration interaction matrix element between two configurations differing by exactly one alpha M.O. Called by MECI only.

• AABACD Utility: Calculates the configuration interaction matrix element between two configurations differing by exactly two alpha M.O.’s. Called by MECI only.

• AABBCD Utility: Calculates the configuration interaction matrix element between two configurations differing by exactly two M.O.’s; one configuration has alpha M.O. “A” and beta M.O. “C” while the other configuration has alpha M.O. “B” and beta M.O. “D”. Called by MECI only.

• AINTGS Utility: Within the overlap integrals, calculates the A-integrals. Dedicated to function SS within DIAT.

• ANALYT Main Sequence: Calculates the analytical derivatives of the energy with respect to cartesian coordinates for all atoms. Use only if the mantissa is short (less than 52 bits) or out of interest. Should not be used for routine work on a VAX.

• ANAVIB Utility: Gives a brief interpretation of the modes of vibration of the molecule. The principal pairs of atoms involved in each vibration are identified, and the mode of motion (tangential or radial) is output.

• AXIS Utility: Works out the three principal moments of inertia of a molecule. If the system is linear, one moment of inertia is zero. Prints moments in units of cm^{−1} and 10^{−40} g cm^{2}.

• BABBBC Utility: Calculates the configuration interaction matrix element between two configurations differing by exactly one beta M.O. Called by MECI only.

• BABBCD Utility: Calculates the configuration interaction matrix element between two configurations differing by exactly two beta M.O.’s. Called by MECI only.

• BANGLE Utility: Given a set of coordinates, BANGLE will calculate the angle between any three atoms.

• BFN Utility: Calculates the B-functions in the Slater overlap.

• BINTGS Utility: Calculates the B-functions in the Slater overlap.

• BKRSAV Utility: Saves and restores data used by the eigenvector following subroutine. Called by EF only.

• BONDS Utility: Evaluates and prints the valencies of atoms and bond-orders between atoms. Main argument: density matrix. No results are passed to the calculation, and no data are changed. Called by WRITE only.

Description of subroutines

• BRLZON Main Sequence: BRLZON generates a band structure, or phonon structure, for high polymers. Called by WRITE and FREQCY.

• CALPAR Utility: When external parameters are read in via EXTERNAL=, the derived parameters are worked out using CALPAR. Note that all derived parameters are calculated for all parameterized elements at the same time.

• CAPCOR Utility: Capping atoms, of type Cb, should not contribute to the energy of a system. CAPCOR calculates the energy contribution due to the Cb and subtracts it from the electronic energy.

• CDIAG Utility: Complex diagonalization. Used in generating eigenvalues of complex Hermitian secular determinant for band structures. Called by BRLZON only.

• CHRGE Utility: Calculates the total number of valence electrons on each atom. Main arguments: density matrix, array of atom charges (empty on input). Called by ITER only.

• CNVG Utility: Used in SCF cycle. CNVG does a three-point interpolation of the last three density matrices. Arguments: Last three density matrices, Number of iterations, measure of self-consistency (empty on input). Called by ITER only.

• COE Utility: Within the general overlap routine COE calculates the angular coefficients for the s, p and d real atomic orbitals given the axis and returns the rotation matrix.

• COMPFG Main Sequence: Evaluates the total heat of formation of the supplied geometry, and the derivatives, if requested. This is the nodal point connecting the electronic and geometric parts of the program. Main arguments: on input: geometry, on output: heat of formation, gradients.

• DANG Utility: Called by XYZINT, DANG computes the angle between a point, the origin, and a second point.

• DATIN Utility: Reads in external parameters for use within MOPAC. Originally used for the testing of new parameters, DATIN is now a general purpose reader for parameters. Invoked by the keyword EXTERNAL.

• DCART Utility: Called by DERIV, DCART sets up a list of cartesian derivatives of the energy wrt coordinates which DERIV can then use to construct the internal coordinate derivatives.

• DELMOL Utility: Part of analytical derivates. Two-electron.

• DELRI Utility: Part of analytical derivates. Two-electron.

• DENROT Utility: Converts the ordinary density matrix into a condensed density matrix over basis functions s (sigma), p (sigma) and p (pi), i.e., three basis functions. Useful in hybridization studies. Has capability to handling “d” functions, if present.

• DENSIT Utility: Constructs the Coulson electron density matrix from the eigenvectors. Main arguments: Eigenvectors, No. of singly and doubly occupied levels, density matrix (empty on input) Called by ITER.

• DEPVAR Utility: A symmetry-defined “bond length” is related to another bond length by a multiple. This special symmetry function is intended for use in Cluster calculations. Called by HADDON.

• DERI0 Utility: Part of the analytical C.I. derivative package. Calculates the diagonal dominant part of the super-matrix.

• DERI1 Utility: Part of the analytical C.I. derivative package. Calculates the frozen density contribution to the derivative of the energy wrt cartesian coordinates, and the derivatives of the frozen Fock matrix in M.O. basis. It’s partner is DERI2.

• DERI2 Utility: Part of the analytical C.I. derivative package. Calculates the relaxing density contribution to the derivative of the energy wrt cartesian coordinates. Uses the results of DERI1.

• DERI21 Utility: Part of the analytical C.I. derivative package. Called by DERI2 only.

• DERI22 Utility: Part of the analytical C.I. derivative package. Called by DERI2 only.

• DERI23 Utility: Part of the analytical C.I. derivative package. Called by DERI2 only.

• DERITR Utility: Calculates derivatives of the energy wrt internal coordinates using full SCF’s. Used as a foolproof way of calculating derivatives. Not recommended for normal use.

• DERIV Main Sequence: Calculates the derivatives of the energy with respect to the geometric variables. This is done either by using initially cartesian derivatives (normal mode), by analytical C.I. RHF derivatives, or by full SCF calculations (NOANCI in half-electron and C.I. mode). Arguments: on input: geometry, on output: derivatives. Called by COMPFG.

• DERNVO Analytical C.I. Derivative main subroutine. Calculates the derivative of the energy wrt geometry for a non-variationally optimized wavefunction (a SCF-CI wavefunction).

• DERS Utility: Called by ANALYT, DERS calculates the analytical derivatives of the overlap matrix within the molecular frame.

• DEX2 Utility: A function called by ESP.

• DFOCK2 Utility: Part of the analytical C.I. derivative package. Called by DERI1, DFOCK2 calculates the frozen density contribution to the derivative of the energy wrt cartesian coordinates.

• DFPSAV Utility: Saves and restores data used by the BFGS geometry optimization. Main arguments: parameters being optimized, gradients of parameters, last heat of formation, integer and real control data. Called by FLEPO.

• DHC Utility: Called by DCART and calculates the energy of a pair of atoms using the SCF density matrix. Used in the finite difference derivatve calculation.

• DHCORE Utility: Part of the analytical C.I. derivative package. Called by DERI1, DHCORE calculates the derivatives of the 1 and 2 electron integrals wrt cartesian coordinates.

• DIAG Utility: Rapid pseudo-diagonalization. Given a set of vectors which almost blockdiagonalize a secular determinant, DIAG modifies the vectors so that the block-diagonalization is more exact. Main arguments: Old vectors, Secular Determinant, New vectors (on output). Called by ITER.

• DIAGI Utility: Calculates the electronic energy arising from a given configuration. Called by MECI.

• DIAT Utility: Calculates overlap integrals between two atoms in general cartesian space. Principal quantum numbers up to 6, and angular quantum numbers up to 2 are allowed. Main arguments: Atomic numbers and cartesian coordinates in Angstroms of the two atoms, Diatomic overlaps (on exit). Called by H1ELEC.

• DIAT2 Utility: Calculates reduced overlap integrals between atoms of principal quantum numbers 1, 2, and 3, for s and p orbitals. Faster than the SS in DIAT. This is a dedicated subroutine, and is unable to stand alone without considerable backup. Called by DIAT.

• DIGIT Utility: Part of READA. DIGIT assembles numbers given a character string.

• DIHED Utility: Calculates the dihedral angle between four atoms. Used in converting from cartesian to internal coordinates.

• DIIS Utility: Pulay’s Geometric Direct Inversion of the Iterative Subspace (G-DIIS) accelerates the rate at which the BFGS locates an energy minimum. (In MOPAC 6.00, the DIIS is only partially installed — several capabilities of the DIIS are not used)

• DIJKL1 Utility: Part of the analytical C.I. derivative package. Called by DERI1, DIJKL1 calculates the two-electron integrals over M.O. bases, e.g. <i,j (1r) k,l¿/.

• DIJKL2 Utility: Part of the analytical C.I. derivative package. Called by DERI2, DIJKL2 calculates the derivatives of the two-electron integrals over M.O. bases, e.g. <i,j (1r) k,l¿/, wrt cartesian coordinates.

• DIPIND Utility: Similar to DIPOLE, but used by the POLAR calculation only.

• DIPOLE Utility: Evaluates and, if requested, prints dipole components and dipole for the molecule or ion. Arguments: Density matrix, Charges on every atom, coordinates, dipoles (on exit). Called by WRITE and FMAT.

• DIST2 Utility: Called by ESP only, DIST2 works out the distance between two points in 3D space.

• DOFS Main Sequence: Calculates the density of states within a Brillouin zone. Used in polymer work only.

• DOT Utility: Given two vectors, X and Y, of length N, function DOT returns with the dot product X.Y. I.e., if X=Y, then DOT = the square of X. Called by FLEPO.

• DRC Main Sequence: The dynamic and intrinsic reaction coordinates are calculated by following the mass-weighted trajectories.

• DRCOUT Utility: Sets up DRC and IRC data in quadratic form preparatory to being printed.

• EA08C Part of the diagonalizer RSP.

• EA09C Part of the diagonalizer RSP.

• EC08C Part of the diagonalizer RSP.

• EF Main Sequence: EF is the Eigenvector Following routine. EF implements the keywords EF and TS.

• ELESP Utility: Within the ESP, ELESP calculates the electronic contribution to the electrostatic potential.

• ENPART Utility: Partitions the energy of a molecule into its monatomic and diatomic components. Called by WRITE when the keyword ENPART is specified. No data are changed by this call.

• EPSETA Utility: Calculates the machine precision and dynamic range for use by the two diagonalizers.

• ESP Main Sequence: ESP is not present in the default copy of MOPAC. ESP calculates the atomic charges which would reproduce the electrostatic potential of the nuclii and electronic wavefunction.

• ESPBLO Block Data: Used by the ESP calculation, ESPBLO fills two small arrays!

• ESPFIT Utility: Part of the ESP. ESP fits the quantum mechanical potential to a classical point charge model

• EXCHNG Utility: Dedicated procedure for storing 3 parameters and one array in a store. Used by SEARCH.

• FFHPOL Utility: Part of the POLAR calculation. Evaluates the effect of an electric field on a molecule.

• FLEPO Main Sequence: Optimizes a geometry by minimizing the energy. Makes use of the first and estimated second derivatives to achieve this end. Arguments: Parameters to be optimized, (overwritten on exit with the optimized parameters), Number of parameters, final optimized heat of formation. Called by MAIN, REACT1, and FORCE.

• FM06AS Utility: Part of CDIAG.

• FM06BS Utility: part of CDIAG.

• FMAT Main sequence: Calculates the exact Hessian matrix for a system This is done by either using differences of first derivatives (normal mode) or by four full SCF calculations (half electron or C.I. mode). Called by FORCE.

• FOCK1 Utility: Adds on to Fock matrix the one-center two electron terms. Called by ITER only.

• FOCK2 Utility: Adds on to Fock matrix the two-center two electron terms. Called by ITER and DERIV. In ITER the entire Fock matrix is filled; in DERIV, only diatomic Fock matrices are constructed.

• FOCK2D Written out of MOPAC 6.00.

• FORCE Main sequence: Performs a force-constant and vibrational frequency calculation on a given system. If the starting gradients are large, the geometry is optimized to reduce the gradient norm, unless LET is specified in the keywords. Isotopic substitution is allowed. Thermochemical quantities are calculated. Called by MAIN.

• FORMD Main Sequence: Called by EF. FORMD constructs the next step in the geometry optimization or transition state location.

• FORMXY Utility: Part of DIJKL1. FORMXY constructs part of the two- electron integral over M.O.’s.

• FORSAV Utility: Saves and restores data used in FMAT in FORCE calculation. Called by FMAT.

• FRAME Utility: Applies a very rigid constraint on the translations and rotations of the system. Used to separate the trivial vibrations in a FORCE calculation.

• FREQCY Main sequence: Final stage of a FORCE calculation. Evaluates and prints the vibrational frequencies and modes.

• FSUB Utility: Part of ESP.

• GENUN Utility: Part of ESP. Generates unit vectors over a sphere. called by SURFAC only.

• GEOUT Utility: Prints out the current geometry. Can be called at any time. Does not change any data.

• GEOUTG Utility: Prints out the current geometry in Gaussian Z-matrix format.

• GETDAT Utility: Reads in all the data, and puts it in a scratch file on channel 5.

• GETGEG Utility: Reads in Gaussian Z-matrix geometry. Equivalent to GETGEO and GETSYM combined.

• GETGEO Utility: Reads in geometry in character mode from specified channel, and stores parameters in arrays. Some error-checking is done. Called by READ and REACT1.

• GETSYM Utility: Reads in symmetry data. Used by READ.

• GETTXT Utility: Reads in KEYWRD, KOMENT and TITLE.

• GETVAL Utility: Called by GETGEG, GETVAL either gets an internal coordinate or a logical name for that coordinate.

• GMETRY Utility: Fills the cartesian coordinates array. Data are supplied from the array GEO, GEO can be (a) in internal coordinates, or (b) in cartesian coordinates. If STEP is non-zero, then the coordinates are modified in light of the other geometry and STEP. Called by HCORE, DERIV, READ, WRITE, MOLDAT, etc.

• GOVER Utility: Calculates the overlap of two Slater orbitals which have been expanded into six gaussians. Calculates the STP-6G overlap integrals.

• GRID Main Sequence: Calculates a grid of points for a 2-D search in coordinate space. Useful when more information is needed about a reaction surface.

• H1ELEC Utility: Given any two atoms in cartesian space, H1ELEC calculates the oneelectron energies of the off-diagonal elements of the atomic orbital matrix.

H(i, j) = −S(i, j)[β(i) + β(j)]/2

Called by HCORE and DERIV.

• HADDON Utility: The symmetry operation subroutine, HADDON relates two geometric variables by making one a dependent function of the other. Called by SYMTRY only.

• HCORE Main sequence: Sets up the energy terms used in calculating the SCF heat of formation. Calculates the one and two electron matrices, and the nuclear energy. Called by COMPFG.

• HELECT Utility: Given the density matrix, and the one electron and Fock matrices, calculates the electronic energy. No data are changed by a call of HELECT. Called by ITER and DERIV.

• HQRII Utility: Rapid diagonalization routine. Accepts a secular determinant, and produces a set of eigenvectors and eigenvalues. The secular determinant is destroyed.

• IJKL Utility: Fills the large two-electron array over a M.O. basis set. Called by MECI.

• INTERP Utility: Runs the Camp-King converger. q.v.

• ITER Main sequence: Given the one and two electron matrices, ITER calculates the Fock and density matrices, and the electronic energy. Called by COMPFG.

• JAB Utility: Calculates the coulomb contribution to the Fock matrix in NDDO formalism. Called by FOCK2.

• JCARIN Utility: Calculates the difference vector in cartesian coordinates corresponding to a small change in internal coordinates.

• KAB Utility: Calculates the exchange contribution to the Fock matrix in NDDO formalism. Called by FOCK2.

• LINMIN Main sequence: Called by the BFGS geometry optimized FLEPO, LINMIN takes a step in the search-direction and if the energy drops, returns. Otherwise it takes more steps until if finds one which causes the energy to drop.

• LOCAL Utility: Given a set of occupied eigenvectors, produces a canonical set of localized bonding orbitals, by a series of 2 × 2 rotations which maximize ⟨ψ^{4}⟩. Called by WRITE.

• LOCMIN Main sequence: In a gradient minimization, LOCMIN does a line- search to find the gradient norm minimum. Main arguments: current geometry, search direction, step, current gradient norm; on exit: optimized geometry, gradient norm.

• MAMULT Utility: Matrix multiplication. Two matrices, stored as lower half triangular packed arrays, are multiplied together, and the result stored in a third array as the lower half triangular array. Called from PULAY.

• MATOUT Utility: Matrix printer. Prints a square matrix, and a row-vector, usually eigenvectors and eigenvalues. The indices printed depend on the size of the matrix: they can be either over orbitals, atoms, or simply numbers, thus M.O.’s are over orbitals, vibrational modes are over numbers. Called by WRITE, FORCE.

• ME08A, ME08B Utilities: Part of the complex diagonalizer, and called by CDIAG.

• MECI Main sequence: Main function for Configuration Interaction, MECI constructs the appropriate C.I. matrix, and evaluates the roots, which correspond to the electronic energy of the states of the system. The appropriate root is then returned. Called by ITER only.

• MECID Utility: Constructs the differential C.I. secular determinant.

• MECIH Utility: Constructs the normal C.I. secular determinant.

• MECIP Utility: Reforms the density matrix after a MECI calculation.

• MINV Utility: Called by DIIS. MINV inverts the Hessian matrix.

• MNDO Main sequence: MAIN program. MNDO first reads in data using READ, then calls either FLEPO to do geometry optimization, FORCE to do a FORCE calculation, PATHS for a reaction with a supplied coordinate, NLLSQ for a gradient minimization or REACT1 for locating the transition state. Starts the timer.

• MOLDAT Main Sequence: Sets up all the invariant parameters used during the calculation, e.g. number of electrons, initial atomic orbital populations, number of open shells, etc. Called once by MNDO only.

• MOLVAL Utility: Calculates the contribution from each M.O. to the total valency in the molecule. Empty M.O.’s normally have a negative molecular valency.

• MTXM Utility: Part of the matrix package. Multiplies together two rectangular packed arrays, i.e., C = A.B.

• MTXMC Utility: Part of the matrix package. Similar to MTXM.

• MULLIK Utility: Constructs and prints the Mulliken Population Analysis. Available only for RHF calculations. Called by WRITE.

• MULT Utility: Used by MULLIK only, MULT multiplies two square matrices together.

• MXM Utility: Part of the matrix package. Similar to MTXM.

• MXMT Utility: Part of the matrix package. Similar to MTXM.

• MYWORD Utility: Called in WRTKEY, MYWORD checks for the existance of a specific string. If it is found, MYWORD is set true, and the all occurances of string are deleted. Any words not recognised will be flagged and the job stopped.

• NAICAP Utility: Called by ESP.

• NAICAP Utility: Called by ESP.

• NLLSQ Main sequence: Used in the gradient norm minimization.

• NUCHAR Takes a character string and reads all the numbers in it and stores these in an array.

• OSINV Utility: Inverts a square matrix. Called by PULAY only.

• OVERLP Utility: Part of EF. OVERLP decides which normal mode to follow.

• OVLP Utility: Called by ESP only. OVLP calculates the overlap over Gaussian STO’s.

• PARSAV Utility: Stores and restores data used in the gradient-norm minimization calculation.

• PARTXY Utility: Called by IJKL only, PARTXY calculates the partial product <i,j (1r) in ¡i,j (1/r) k,l¿/.

• PATHK Main sequence: Calculates a reaction coordinate which uses a constant step-size. Invoked by keywords STEP and POINTS.

• PATHS Main sequence: Given a reaction coordinate as a row-vector, PATHS performs a FLEPO geometry optimization for each point, the later geometries being initially guessed from a knowledge of the already optimized geometries, and the current step. Called by MNDO only.

• PDGRID Utility: Part of ESP. Calculates the Williams surface.

• PERM Utility: Permutes n1 electrons of alpha or beta spin among n2 M.O.’s.

• POLAR Utility: Calculates the polarizability volumes for a molecule or ion. Uses 19 SCF calculations, so appears after WRITE has finished. Cannot be used with FORCE, but can be used anywhere else. Called by WRITE.

• POWSAV Utility: Calculation store and restart for SIGMA calculation. Called by POWSQ.

• POWSQ Main sequence: The McIver - Komornicki gradient minimization routine. Constructs a full Hessian matrix and proceeds by line-searches Called from MAIN when SIGMA is specified.

• PRTDRC Utility: Prints DRC and IRC results according to instructions. Output can be (a) every point calculated (default), (b) in constant steps in time, space or energy.

• PULAY Utility: A new converger. Uses a powerful mathematical non-iterative method for obtaining the SCF Fock matrix. Principle is that at SCF the eigenvectors of the Fock and density matrices are identical, so [F.P] is a measure of the non-self consistency. While very powerful, PULAY is not universally applicable. Used by ITER.

• QUADR: Utility: Used in printing the IRC - DRC results. Sets up a quadratic in time of calculated quantities so that PRTDRC can select specific reaction times for printing.

• REACT1 Main sequence: Uses reactants and products to find the transition state. A hypersphere of N dimensions is centered on each moiety, and the radius steadily reduced. The entity of lower energy is moved, and when the radius vanishes, the transition state is reached. Called by MNDO only.

• READ Main sequence: Almost all the data are read in through READ. There is a lot of data-checking in READ, but very little calculation. Called by MNDO.

• READA Utility: General purpose character number reader. Used to enter numerical data in the control line as <variable>=n.nnn where <variable> is a mnemonic such as SCFCRT or CHARGE. Called by READ, FLEPO, ITER, FORCE, and many other subroutines.

• REFER Utility: Prints the original references for atomic data. If an atom does not have a reference, i.e., it has not been parameterized, then a warning message will be printed and the calculation stopped.

• REPP Utility: Calculates the 22 two-electron reduced repulsion integrals, and the 8 electronnuclear attraction integrals. These are in a local coordinate system. Arguments: atomic numbers of the two atoms, interatomic distance, and arrays to hold the calculated integrals. Called by ROTATE only.

• ROTAT Utility: Rotates analytical two-electron derivatives from atomic to molecular frame.

• ROTATE Utility: All the two-electron repulsion integrals, the electron- nuclear attraction integrals, and the nuclear-nuclear repulsion term between two atoms are calculated here. Typically 100 two- electron integrals are evaluated.

• RSP Utility: Rapid diagonalization routine. Accepts a secular determinant, and produces a set of eigenvectors and eigenvalues. The secular determinant is destroyed.

• SAXPY Utility: Called by the utility SUPDOT only!

• SCHMIB Utility: Part of Camp-King converger.

• SCHMIT Utility: Part of Camp-King converger.

• SCOPY Utility: Copies an array into another array.

• SDOT Utility: Forms the scalar of the product of two vectors.

• SEARCH Utility: Part of the SIGMA and NLLSQ gradient minimizations. The line-search subroutine, SEARCH locates the gradient minimum and calculates the second derivative of the energy in the search direction. Called by POWSQ and NLLSQ.

• SECOND Utility: Contains VAX specific code. Function SECOND returns the number of CPU seconds elapsed since an arbitrary starting time. If the SHUTDOWN command has been issued, the CPU time is in error by exactly 1,000,000 seconds, and the job usually terminates with the message “time exceeded”.

• SET Utility: Called by DIAT2, evaluates some terms used in overlap calculation.

• SETUP3 Utility: Sets up the Gaussian expansion of Slater orbitals using a STO-3G basis set.

• SETUPG Utility: Sets up the Gaussian expansion of Slater orbitals using a STO-6G basis set.

• SOLROT Utility: For Cluster systems, adds all the two-electron integrals of the same type, between different unit cells, and stores them in a single array. Has no effect on molecules.

• SORT Utility: Part of CDIAG, the complex diagonalizer.

• SPACE Utility: Called by DIIS only.

• SPCG Written out of Version 6.00.

• SPLINE Utility: Part of Camp-King converger.

• SS Utility: An almost general Slater orbital overlap calculation. Called by DIAT.

• SUPDOT Utility: Matrix mutiplication A=B.C

• SURFAC Utility: Part of the ESP.

• SWAP Utility: Used with FILL=, SWAP ensures that a specified M.O. is filled. Called by ITER only.

• SYMTRY Utility: Calculates values for geometric parameters from known geometric parameters and symmetry data. Called whenever GMETRY is called.

• THERMO Main sequence: After the vibrational frequencies have been calculated, THERMO calculates thermodynamic quantities such as internal energy, heat capacity, entropy, etc, for translational, vibrational, and rotational, degrees of freedom.

• TIMCLK Utility: Vax-specific code for determining CPU time.

• TIMER Utility: Prints times of various steps.

• TIMOUT Utility: Prints total CPU time in elegant format.

• TQL2 Utility: Part of the RSP.

• TQLRAT Utility: Part of the RSP.

• TRBAK3 Utility: Part of the RSP.

• TRED3 Utility: Part of the RSP.

• UPDATE Utility: Given a set of new parameters, stores these in their appropriate arrays. Invoked by EXTERNAL.

• UPDHES Utility: Called by EF, UPDHES updates the Hessian matrix.

• VECPRT Utility: Prints out a packed, lower-half triangular matrix. The labeling of the sides of the matrix depend on the matrix’s size: if it is equal to the number of orbitals, atoms, or other. Arguments: The matrix to be printed, size of matrix. No data are changed by a call of VECPRT.

• WRITE Main sequence: Most of the results are printed here. All relevant arrays are assumed to be filled. A call of WRITE only changes the number of SCF calls made, this is reset to zero. No other data are changed. Called by MAIN, FLEPO, FORCE.

• WRTKEY Main Sequence: Prints all keywords and checks for compatability and to see if any are not recognised. WRTKEY can stop the job if any errors are found.

• WRTTXT Main Sequence: Writes out KEYWRD, KOMENT and TITLE. The inverse of GETTXT.

• XXX Utility: Forms a unique logical name for a Gaussian Z-matrix logical. Called by GEOUTG only.

• XYZINT Utility: Converts from cartesian coordinates into internal.

• XYZGEO XYZINT sets up its own numbering system, so no connectivity is needed.

Appendix D

Heats of formation

Test MNDO, PM3 and AM1 compounds

In order to verify that MOPAC is working correctly, a large number of tests need to be done. These take about 45 minutes on a VAX 11–780, and even then many potential bugs remain undetected. It is obviously impractical to ask users to test MOPAC. However, users must be able to verify the basic working of MOPAC, and to do this the following tests for the elements have been provided.

Each element can be tested by making up a data-file using estimated geometries and running that file using MOPAC. The optimized geometries should give rise to heats of formation as shown. Any difference greater than 0.1 kcal/mole indicates a serious error in the program.

Caveats

1. Geometry definitions must be correct.

2. Heats of formation may be too high for certain compounds. This is due to a poor starting geometry trapping the system in an excited state. (Affects ICl at times)

Element     Test Compound          Heat of Formation
                             MINDO/3     MNDO    AM1   PM3
Hydrogen         CH4           -6.3    -11.9   -8.8  -13.0
Lithium          LiH                   +23.2
Beryllium        BeO                   +38.6         +53.0
Boron            BF3         -270.2   -261.0 -272.1*
Carbon           CH4           -6.3    -11.9   -8.8  -13.0
Nitrogen         NH3           -9.1     -6.4   -7.3   -3.1
Oxygen           CO2          -95.7    -75.1  -79.8  -85.0
Fluorine         CF4         -223.9   -214.2 -225.7 -225.1
Magnesium        MgF2                               -160.7
Aluminium        AlF                   -83.6  -77.9  -50.1
Silicon          SiH          +82.9    +90.2  +84.5  +94.6
Phosphorus       PH3           +2.5     +3.9  +10.2   +0.2
Sulfur           H2S           -2.6     +3.8   +1.2   -0.9
Chlorine         HCl          -21.1    -15.3  -24.6  -20.5
Zinc             ZnMe2                 +19.9  +19.8    8.2
Gallium          GaCl3                               -79.7
Germanium        GeF                   -16.4  -19.7   -3.3
Arsenic          AsH3                                +12.7
Selenium         SeCl2                               -38.0
Bromine          HBr                    +3.6  -10.5   +5.3

Heats of formation

Cadmium          CdCl2                               -48.6
Indium           InCl3                               -72.8
Tin              SnF                   -20.4         -17.5
Antimony         SbCl3                               -72.4
Tellurium        TeH2                                +23.8
Iodine           ICl                    -6.7  -4.6   +10.8
Mercury          HgCl2                 -36.9 -44.8   -32.7
Thallium         TlCl                                -13.4
Lead             PbF                   -22.6         -21.0
Bismuth          BiCl3                               -42.6
* Not an exhaustive test of AM1 boron.

Appendix E

References

On G-DIIS

“Computational Strategies for the Optimization of Equilibrium Geometry and Transition-State Structures at the Semiempirical Level”, Peter L. Cummings, Jill E. Gready, J. Comp. Chem., 10:939-950 (1989).

On Analytical C.I. Derivatives

“An Efficient Procedure for Calculating the Molecular Gradient, using SCF-CI Semiempirical Wavefunctions with a Limited Number of Configurations”, M. J. S. Dewar, D. A. Liotard, J. Mol. Struct. (Theochem), 206:123-133 (1990).

On Eigenvector Following

J. Baker, J. Comp. Chem., 7:385 (1986).

On ElectroStatic Potentials (ESP)

“Atomic Charges Derived from Semiempirical Methods”, B. H. Besler, K. M. Merz, Jr., P. A. Kollman, J. Comp. Chem., 11:431-439 (1990).

On MNDO

“Ground States of Molecules. 38. The MNDO Method. Approximations and Parameters.”, M.J.S. Dewar, W.Thiel, J. Am. Chem. Soc., 99:4899, (1977).

Original References for Elements Parameterized in MNDO:

H M.J.S. Dewar, W. Thiel, J. Am. Chem. Soc., 99, 4907, (1977).

Li Parameters taken from the MNDOC program, written by Walter Thiel, Quant. Chem. Prog. Exch. No. 438; 2:63, (1982).

Be M.J.S. Dewar, H.S. Rzepa, J. Am. Chem. Soc., 100:777, (1978).

B M.J.S. Dewar, M.L. McKee, J. Am. Chem. Soc., 99:5231, (1977).

C M.J.S. Dewar, W. Thiel, J. Am. Chem. Soc., 99:4907, (1977).

N M.J.S. Dewar, W. Thiel, J. Am. Chem. Soc., 99:4907, (1977).

O M.J.S. Dewar, W. Thiel, J. Am. Chem. Soc., 99:4907, (1977).

F M.J.S. Dewar, H.S. Rzepa, J. Am. Chem. Soc., 100:58, (1978).

\underline{References}

Al L.P. Davis, R.M. Guidry, J.R. Williams, M.J.S. Dewar, H.S. Rzepa J. Comp. Chem., 2:433, (1981).

Si (a) M.J.S. Dewar, M.L. McKee, H.S. Rzepa, J. Am. Chem. Soc., 100:3607 (1978). ‡ (c) M.J.S. Dewar, J. Friedheim, G. Grady, E.F. Healy, J.J.P. Stewart, Organometallics, 5:375 (1986).

P M.J.S. Dewar, M.L. McKee, H.S. Rzepa, J. Am. Chem. Soc., 100: 3607 (1978).

S (a) M.J.S. Dewar, M.L. McKee, H.S. Rzepa, J. Am. Chem. Soc., 100:3607 (1978). ‡ (b) M.J.S. Dewar, C. H. Reynolds, J. Comp. Chem., 7:140 (1986).

Cl (a) M.J.S. Dewar, M.L. McKee, H.S. Rzepa, J. Am. Chem. Soc., 100, 3607 (1978). ‡ (b) M.J.S. Dewar, H.S. Rzepa, J. Comp. Chem., 4, 158, (1983)

Zn M.J.S. Dewar, K. M. Merz, Organometallics, 5:1494 (1986).

Ge M.J.S. Dewar, G.L. Grady, E.F. Healy, Organometallics, 6:186 (1987).

Br M.J.S. Dewar, E.F. Healy, J. Comp. Chem., 4:542, (1983).

I M.J.S. Dewar, E.F. Healy, J.J.P. Stewart, J. Comp. Chem., 5:358, (1984).

Sn M.J.S. Dewar, G.L. Grady, J.J.P. Stewart, J. Am. Chem. Soc., 106:6771 (1984).

Hg M.J.S. Dewar, G.L. Grady, K. Merz, J.J.P. Stewart, Organometallics, 4:1964, (1985).

Pb M.J.S. Dewar, M. Holloway, G.L. Grady, J.J.P. Stewart, Organometallics, 4:1973, (1985).

N.B.: ‡— Parameters defined here are obsolete.

On MINDO/3

Part XXVI, Bingham, R.C., Dewar, M.J.S., Lo, D.H, J. Am. Chem. Soc., 97, (1975).

On AM1

“AM1: A New General Purpose Quantum Mechanical Molecular Model”, M.J.S. Dewar, E.G. Zoebisch, E.F. Healy, J.J.P. Stewart, J. Am. Chem. Soc., 107:3902–3909 (1985).

On PM3

“Optimization of Parameters for Semi-Empirical Methods I–Method”, J.J.P. Stewart, J. Comp. Chem., 10:221 (1989). “Optimization of Parameters for Semi-Empirical Methods II–Applications, J.J.P. Stewart, J. Comp. Chem., 10:221 (1989). (These two references refer to H, C, N, O, F, Al, Si, P, S, Cl, Br, and I).

“Optimization of Parameters for Semi-Empirical Methods III–Extension of PM3 to Be, Mg, Zn, Ga, Ge, As, Se, Cd, In, Sn, Sb, Te, Hg, Tl, Pb, and Bi”, J.J.P. Stewart, J. Comp. Chem. (In press, expected date of publication, Feb. 1991).

Original References for Elements Parameterized in AM1:

H M.J.S. Dewar, E.G. Zoebisch, E.F. Healy, J.J.P. Stewart, J. Am. Chem. Soc., 107:3902-3909 (1985).

B M.J.S. Dewar, C Jie, E. G. Zoebisch, Organometallics, 7:513-521 (1988).

C M.J.S. Dewar, E.G. Zoebisch, E.F. Healy, J.J.P. Stewart, J. Am. Chem. Soc., 107:3902-3909 N M.J.S. Dewar, E.G. Zoebisch, E.F. Healy, J.J.P. Stewart, J. Am. Chem. Soc., 107:3902-3909

(1985).

O M.J.S. Dewar, E.G. Zoebisch, E.F. Healy, J.J.P. Stewart, J. Am. Chem. Soc., 107:3902-3909 (1985).

F M.J.S. Dewar, E.G. Zoebisch, Theochem., 180:1 (1988).

Al M.J.S. Dewar, A.J. Holder, Organometallics, 9:508 (1990).

Si M.J.S. Dewar, C. Jie, Organometallics, 6:1486-1490 (1987).

P M.J.S. Dewar, C.Jie, Theochem., 187:1 (1989)

S (No reference)

Cl M.J.S. Dewar, E.G. Zoebisch, Theochem., 180:1 (1988).

Zn M.J.S. Dewar, K.M. Merz, Jr., Organometallics, 7:522 (1988).

Ga M.J.S. Dewar, C. Jie, Organometallics, 8:1544 (1989).

Br M.J.S. Dewar, E.G. Zoebisch, Theochem., 180:1 (1988).

I M.J.S. Dewar, E.G. Zoebisch, Theochem., 180:1 (1988).

Hg M. J. S. Dewar, C. Jie, Organometallics, 8:1547 (1989). (see also PARASOK for the use of MNDO parameters for other elements)

On Shift

“The Dynamic ‘Level Shift’ Method for Improving the Convergence of the SCF Procedure”, A. V. Mitin, J. Comp. Chem., 9:107-110 (1988).

On Half-Electron

“Ground States of Conjugated Molecules. IX. Hydrocarbon Radicals and Radical Ions”, M.J.S. Dewar, J.A. Hashmall, C.G. Venier, J.A.C.S., 90:1953 (1968). “Triplet States of Aromatic Hydrocarbons”, M.J.S. Dewar, N. Trinajstic, Chem. Comm., 646, (1970). “Semiempirical SCF-MO Treatment of Excited States of Aromatic Compounds”, M.J.S. Dewar, N. Trinajstic, J. Chem. Soc., (A), 1220, (1971).

On Pulay’s Converger

“Convergence Acceleration of Iterative Sequences. The Case of SCF Iteration”, Pulay, P., Chem. Phys. Lett., 73:393, (1980).

On Pseudodiagonalization

“Fast Semiempirical Calculations”, Stewart. J.J.P., Csaszar, P., Pulay, P., J. Comp. Chem., 3:227, (1982).

On Localization

“A New Rapid Method for Orbital Localization.”, P.G. Perkins and J.J.P. Stewart,J.C.S. Faraday (II), 77:000, (1981).

On Diagonalization

Beppu, Y., Computers and Chemistry, Vol.6 (1982).

On MECI

“Molecular Orbital Theory for the Excited States of Transition Metal Complexes”, D.R. Armstrong, R. Fortune, P.G. Perkins, and J.J.P. Stewart, J. Chem. Soc., Faraday II, 68:1839-1846 (1972).

On Broyden-Fletcher-Goldfarb-Shanno Method

Broyden, C. G., Journal of the Institute for Mathematics and Applications, Vol. 6, pp 222–231, 1970. Fletcher, R., Computer Journal, Vol. 13, pp 317–322, 1970. Goldfarb, D., Mathematics of Computation, Vol. 24, pp 23-26, 1970. Shanno, D. F., Mathematics of Computation, Vol. 24, pp 647-656 1970. See also summary in: Shanno, D. F., J. of Optimization Theory and Applications, Vol.46, No 1 pp 87-94 1985.

On Polarizability

“Calculation of Nonlinear Optical Properties of Molecules”, H. A. Kurtz, J. J. P. Stewart, K. M. Dieter, J. Comp. Chem., 11:82 (1990). See also “Semiempirical Calculation of the Hyperpolarizability of Polyenes”, H. A. Kurtz, I. J. Quant. Chem. Symp., 24, xxx (1990).

On Thermodynamics

“Ground States of Molecules. 44 MINDO/3 Calculations of Absolute Heat Capacities and Entropies of Molecules without Internal Rotations. Dewar, M.J.S., Ford, G.P., J. Am. Chem. Soc., 99:7822 (1977).

On SIGMA Method

Komornicki, A., McIver, J. W., Chem. Phys. Lett., 10:303, (1971). Komornicki, A., McIver, J. W., J. Am. Chem. Soc., 94:2625 (1971).

On Molecular Orbital Valency

“Valency and Molecular Structure”, Gopinathan, M. S., Siddarth, P., Ravimohan, C., Theor. Chim. Acta , 70:303 (1986).

On Bonds

“Bond Indices and Valency”, Armstrong, D.R., Perkins, P.G., Stewart, J.J.P., J. Chem. Soc., Dalton, 838 (1973). For a second, equivalent, description, see also: Gopinathan, M. S., and Jug, K., Theor. Chim. Acta, 63:497 (1983).

On Locating Transition States

“Location of Transition States in Reaction Mechanisms”, M.J.S. Dewar, E.F. Healy, J.J.P. Stewart, J. Chem. Soc., Faraday Trans. II, 3:227, (1984).

On Dipole Moments for Ions

“Molecular Quadrupole Moments”, A.D. Buckingham, Quarterly Reviews, 182 (1958 or 1959).

On Polymers

“MNDO Cluster Model Calculations on Organic Polymers”, J.J.P. Stewart, New Polymeric Materials, 1:53-61 (1987). “Calculation of Elastic Moduli using Semiempirical Methods”, H. E. Klei, J.J.P. Stewart, Int. J. Quant. Chem., 20:529-540 (1986).

Index

+, 13 &, 12 0SCF, 13 1ELECTRON, 13 1SCF, 13

Ab initio total energies, 72 AIDER, 13 AIGIN, 14 AIGOUT, 14 AM1, 14

elements in, 44 references, 150

ANALYT, 14 analytical derivatives, 68

BAR, 14 BIRADICAL, 14 Boltzmann constant, 71 BONDS, 15 Born–von Karman, 100 Boyd, Donald B., 69 bugs

locating, 118

C.I., 15 capped bonds, 45 CDC, 1 CHARGE, 16 cluster model, 100 command file

COMPILE, 121 MOPAC, 123 RMOPAC, 123

CONH linkage, 67 constants

physical, 71 coordinates

Cartesian, 42 examples, 46 Gaussian

example, 66 internal to Cartesian, 42 reaction, 76

COSMO, 99 CRAY–XMP, 1 CYCLE, 95

Danilof, V. I., v

data commas in, 41 for ethene, 5 for polytetrahydrofuran, 6 for polythene, 47 layout, 6 MNRSD1 input, 49 output, 50 tabs in, 41 TESTDATA input, 55 output, 56

Data General, 1 DCART, 17 DEBUG, 17 DEC, 1 definitions

Boltzmann constant, 71 velocity of light, 71

DELHOF, 113 DENOUT, 17 DENSITY, 4 DENSITY (0), 17 Dewar research group, 3 dihedral angle coherency, 71 dipole moments of ions, 152 DRAW program, 4 DRC

background, 76

conservation of momentum, 77 definition, 76 dummy atoms in, 79 keyword options, 81

Eigenvector

following, 94 elements

\underline{INDEX}

specification of, 44 energy, 72 entropy, 72 error messages, 105 ESP

installing, 124 Excimers, 97

Fluorescence, 97 force constant, 82 force constants, 1, 61 frame

description of, 86 Franck-Condon, 96

gas constant, R, 71 Gaussian coordinates, 42 geometry

flags for, 43 Gibbs free energy, 74 GMETRY

description, 103 Gordon, Mark, 76 Gould, 1 grid map, 46

H–PRIORITY, 85 heat capacity, 72 heat of formation, 74

criteria, 113

molecular standards, 147 HELP, 4 Hessian, 94 Hirano, Tsuneo, 71

IBM PC-AT, 1 internal coordinate definition, 41 ions, 1 IRC

definition, 78 example of, 80 example of restart, 80 keywords for, 79

isotopes specification of, 44

Jensen, Frank, 94

keywords debugging, 117 specification, 6

KINETIC, 81 kinetic energy

damping, 77 Klamt, Andreas, v Klyne and Prelog, 71 Korambath, Prakashan, v Kurtz, Henry, A., v

LaTeX, v liquids, 77 Localized orbitals, 2

mass-weighted coordinates, 62 MECI

description of, 87 messages, 105 MNDO

elements in, 44 MODE, 95 MOHELP, 4 molecular orbitals, 1 MOPAC

geometric structure, 101 copyright, 2 cost, 2 criteria, 113 development, 3 electronic structure, 102 error messages, 105 installing, 121 programming policy, 102 size of, 126 using, 125 version number, 52

MOSOL, 4

NONR, 94

normal coordinate analysis, 62

OMIN, 94

partition function, 72 PATH calculation, 63 Petts, Dr. J., 123 Phosphorescence, 97 Photoemission, 97 Photoexcitation energy, 96 POLAR, 28 polymers, 1

data for, 6 Program

DENSITY, 4 MOHELP, 4 MOSOL, 4

QCPE address, 2

radicals, 1

reaction coordinate

specification, 46 Red-shift, 97 reduced mass, 82 references

AM1, 150 BFGS method, 152 bonds, 152 CI derivatives, 149 diagonalization, 152 dipole moments of ions, 152 eigenvector following, 149 ESP, 149 half-electron, 151 localization, 151 MECI, 152 MINDO/3, 150 MNDO, 149 MO valency, 152 on G–DIIS, 149 PM3, 150 polarizability, 152 polymers, 153 pseudo-diagonalization, 151 Pulay’s converger, 151 SHIFT, 151 SIGMA method, 152 thermodynamics, 152 transition state location, 152

RMAX, 95 RMIN, 95 RSCAL, 94

SCFCRT, 113 SHUTDOWN, 124 sparkles, 45 subroutines

calls in MOPAC, 129 description of, 137 full list of, 127

supercomputers, 1 symmetry functions

defined, 35

Taylor expansion, 94 THERMO

example of, 55 TOL2, 113 TOLERG, 113 TOLERX, 113 TOLS1, 113 transition state, 1

UNIX on-line help, 124

VAX, 2, 4 version number

of MOPAC, 52 vibrational analysis, 62

zero point energy, 61