#HNBO 3.0 Program Manual#N #N(#INatural Bond Orbital / Natural Population Analysis / Natural Localized Molecular Orbital Programs#N) E. D. Glendening, A. E. Reed, agger J. E. Carpenter, dagger and F. Weinhold #ITheoretical Chemistry Institute and Department of Chemis- try, University of Wisconsin, Madison, Wisconsin 53706#N 92717.#N 0 The various natural localized sets can be considered to result from a sequence of transformations of the in- put atomic orbital basis set { _______________ agger #C#RNote, however, that some electronic structure packages do not make provision for calculating the spin density matrices for some types of open-shell wavefunc- tions (e.g., MCSCF wavefunctions calculated by the GUGA formalism in the GAMESS system), so that NBO analysis cannot be applied in these cases.#O dagger #C#RIf the wavefunction is not calculated in an atom-centered basis set, it would be necessary to first compute a wavefunction for each isolated atom of the molecule (in the actual basis set and geometry of the molecular calculation), then select the most highly oc- cupied natural orbitals of each atomic wavefunction to compose a final set of linearly independent atom- centered basis functions of the required dimensionali- ty. Since atom-centered basis functions are the nearly universal choice for molecular calculations, the current _________________________ July 11, 1995 - 2 - NBO program makes no provision for this step.#O Guide, Section C. +0 +lf 0>> +0 +lf 0//' |<<3________________________ -dwid +0 ipen //+0 +dht ipen //+dhlfwid +0 ipen >> input basis arr NAOs arr NHOs arr NBOs arr NLMOs _________________________ Each natural localized set forms a complete orthonormal set of one-electron functions for expanding the delo- calized molecular orbitals (MOs) or forming matrix representations of one-electron operators. The overlap of associated ``pre-orthogonal'' NAOs (PNAOs), lacking only the interatomic orthogonalization step of the NAO procedure, can be used to estimate the strength of or- bital interactions in the usual way. 0 The optimal condensation of occupancy in the natural localized orbitals leads to partitioning into high- and low-occupancy orbital types (reduction in dimensionali- ty of the orbitals having significant occupancy), as reflected in the orbital labelling. The small set of most highly-occupied NAOs, having a close correspon- dence with the effective minimal basis set of semi- empirical quantum chemistry, is referred to as the ``natural minimal basis'' (NMB) set. The NMB (core + valence) functions are distinguished from the weakly occupied ``Rydberg'' (extra-valence-shell) functions that complete the span of the NAO space, but typically make little contribution to molecular properties. Similarly in the NBO space, the highly occupied NBOs of the natural Lewis structure can be distinguished from the ``non-Lewis'' antibond and Rydberg orbitals that complete the span of the NBO space. Each pair of valence hybrids , in the NHO basis give rise to a bond (b ) and antibond (ab ) in the NBO basis, b = ab = the former a Lewis (L) and the latter a non-Lewis (NL) orbital. The antibonds (valence shell non-Lewis orbi- tals) typically play the primary role in departures (delocalization) from the idealized Lewis structure. 0 The NBO program also makes extensive provision for energetic analysis of NBO interactions, based on the availability of a 1-electron effective energy operator (Fock matrix) for the system. Estimates of energy ef- fects are based on second-order perturbation theory, or on the effect of deleting certain orbitals or matrix elements and recalculating the total energy. NBO ener- gy analysis is dependent on the specific ESS to which the NBO program is attached, as described in the Appen- July 11, 1995 - 3 - F x y t 0.0001 0.5 .05 dpx dpy 0 1 ipen //+radius +0 0// F x ym t 0.0001 0.5 .05 dpx dpy 0 1 ipen // +0 +dht ipen //+diam -dht 0//+0 +dht ipen //-radius +0 0// F x ym t 0.0001 -0.5 -0.05 dpx dpy 0 1 ipen // +radius -dhlfht 0//'ext _________________________ dix. 0 The program is provided in a core set of NBO routines that can be attached to an electronic structure system of the user's choice. In addition, specific `driver' routines are provided that facilitate the attachment to popular #Iab initio#N and semi-empirical packages (GAUSSIAN-8X, GAMESS, HONDO, AMPAC, etc.). These ver- sions are described in individual Appendices. #IA.1.2 #IStructure of the NBO Program#N 0 The overall logical structure of the NBO program and its attachment to an electronic structure system (ESS) are illustrated in the block diagram, Fig. 1. This figure illustrates how the ESS and its scratch files (in the upper part of the diagram) communicate through the interface routines RUNNBO, FEAOIN, and DELSCF with the main NBO modules and associated direct access file (in the lower part). 0 The main NBO program is represented by modules la- belled ``NBO'' and ``NBOEAN.'' These refer to the con- struction of NBOs (including natural population analysis, construction of NAOs, NLMOs, etc.) and to NBO energy analysis, respectively. Each module consists of subroutines and functions that perform the required operations. These two modules communicate with the direct-access disk file NBODAF (LFN 48, labelled ``FILE48'' elsewhere in this manual) that is created and maintained by the NBO routines. Details of the NBO and NBOEAN modules, common blocks, and direct- access file are described in the Programmer's Guide, Section C. 0 The NBO program blocks communicate with the attached ESS through three system-dependent `driver' subroutines (RUNNBO, FEAOIN, DELSCF). The purpose of these drivers is to load needed information about the wavefunction and various matrices into the FILE48 direct access file and NBO common blocks. Although the ESS is usually thought of as `driving' the NBO program, from the point of view of the NBO program the ESS is merely a `device' that provides initial input (e.g., a density matrix and label information) or other feedback (a calculated en- ergy value) upon request. Each such ESS device there- fore requires special drivers to make this feedback July 11, 1995 - 4 - ' 2//+0 +lf 0//'file' 2>> #BFigure 1:#N Schematic diagram depicting flow of informa- tion between the electronic structure system (ESS) and the NBO program, and the commun#|ication lines connecting these programs _________________________ possible. Versions of the driver subroutines are in- cluded for several popular packages. The driver rou- tines are described in more detail in the Programmer's agger #RPresent address: Bayer AG, Abteilung AV-IM-AM, 5090 Leverkusen, Bayerwerk, Federal Republic of Ger- many. dagger #RPresent address: Department of Chemistry, University of California-Irvine, Irvine, California #HTable of Contents#N #HPREFACE: HOW TO USE THIS MANUAL#N 0 The NBO manual is divided into three major sections: 0 Section A (``General Introduction and Installation'') contains general introductory and `one-time' informa- tion for the novice user: what the program does, pro- gram structure and relationship to driver electronic structure package, initial installation, `quick start' sample input data, and a brief tutorial on sample out- put. 0 Section B (``NBO User's Guide'') is for the inter- mediate user who has an installed program and general familiarity with the standard (default) options of the NBO program. This section documents the list of #Ikeywords#N that can be used to alter the standard NBO job options, with examples of the resulting output. This section is mandatory for users who wish to use the July 11, 1995 - 5 - to the ESS scratch file (called the ``dictionary file,'' ``read-write file,'' etc., in various systems) and the NBO direct access file (NBODAF). Heavier box borders mark the ESS-specific driver routines (RUNNBO, FEAOIN, _________________________ program to its full potential, to `turn off' or `turn on' various NBO options for their specialized applica- tions. 0 Section C (``NBO Programmer's Guide'') is for accom- plished programmers who are interested in program logic and the detailed layout of the source code. This sec- tion describes the relationship of the source code sub- programs to the published algorithms for NAO, NBO, and NLMO determination, providing documentation at the lev- el of individual common blocks, functions, and subrou- tines. This in turn serves as a bridge to the `micro- documentation' included as comment statements within the source code. Section C also provides guidelines for constructing `driver' routines to attach the NBO programs to new electronic structure packages. #HSection A: GENERAL INTRODUCTION AND INSTALLATION#N #BA.1 INTRODUCTION TO THE NBO PROGRAM#N #IA.1.1 #IWhat Does the NBO Program Do?#N 0 The NBO program performs the analysis of a many- electron molecular wavefunction in terms of localized electron-pair `bonding' units. The program carries out the determination of natural atomic orbitals (NAOs), natural hybrid orbitals (NHOs), natural bond orbitals (NBOs), and natural localized molecular orbitals (NLMOs), and uses these to perform natural population analysis (NPA), NBO energetic analysis, and other tasks pertaining to localized analysis of wavefunction pro- perties. The NBO method makes use of only the first- order reduced density matrix of the wavefunction, and hence is applicable to wavefunctions of general mathematical form; in the open-shell case, the analysis is performed in terms of ``different NBOs for different spins,'' based on distinct density matrices for lpha and g r This section provides a brief introduction to NBO algorithms and nomenclature. 0 NBO analysis is based on a method for optimally transforming a given wavefunction into localized form, corresponding to the one-center (``lone pair'') and two-center (``bond'') elements of the chemist's Lewis structure picture. The NBOs are obtained as local block eigenfunctions of the one-electron density ma- July 11, 1995 - 6 - DELSCF) that directly interface the ESS program. The heavy dashed lines denote calls from the NBO program `backward' to the ESS program for information needed to carry out its tasks. Otherwise, the sequential flow of program control is generally from top to bottom and from left to right in the diagram. #IA.1.3 #IInput and Output#N 0 From the user's point of view, the #_input#/ to the NBO program attached to an ESS program consists simply of one or more keywords (an NBO #Ikeylist#N) included in the ESS input file. In effect, the NBO program reads these keywords to set various job options, then interrogates the ESS program through the DELSCF and FEAOIN drivers for additional information concerning the wavefunction. The general form of NBO keylists and the specific functions associated with each keyword are detailed in the User's Guide, Section B. The method of including NBO keylists in the input file for each ESS is detailed in the specific Appendix for the ESS. 0 The following information is passed from the ESS to the NBO program (transparent to the user): The one-electron density matrix #BD#N (or density matrices in the open-shell case) in the chosen atomic orbital (AO) basis set; The AO overlap matrix #BS#N, and label information identifying the symmetry (angular momentum type) and location (number of the atom to which affixed) for each AO; Atomic number (nuclear charge) of each atom. Certain additional information is written on the FILE48 direct access file and may be used in response to specific job options, such as the AO Fock matrix #BF#N, if energy analysis is requested; the AO dipole matrix #BM#N, if dipole moment analysis is requested; or information concerning the mathematical form of the AOs (orbital exponents, contraction coefficients, etc.), if orbital _________________________ trix, and are hence ``natural'' in the sense of L mlaut owdin, having optimal convergence properties for describing the electron density. The set of high- occupancy NBOs, each taken doubly occupied, is said to represent the ``natural Lewis structure'' of the molecule. Delocalization effects appear as weak depar- tures from this idealized localized picture. July 11, 1995 - 7 - plotting information is requested to be saved as input for a contour plotting program. 0 The principal #_output#/ from the NBO program con- sists of the tables and summaries describing the results of NBO analysis, included in the ESS output file. Sample NBO output is described in Section A.2.4 below. If requested, the NBO program may also write out transformation matrices or other data to disk files. The NBO program also creates or updates two files, the direct-access file (FILE48) and the `archive' file (FILE47) that can be used to repeat NBO analysis with different options, without running the ESS program to recalculate the wavefunction. Necessary details of these files are given in Section B.7 and the Programmer's Guide, Section C. #IA.1.4 #IGeneral Capabilities and Restrictions#N 0 Principal capabilities of the NBO program are: Natural population, natural bond orbital, and natural localized molecular orbital analysis of SCF, MCSCF, CI, and M0t oller-Plesset wavefunctions (main subroutine: NBO); For RHF closed-shell and UHF wavefunctions only, ener- getic analysis of the wavefunction in terms of the interactions (Fock matrix elements) between NBOs (main subroutine: NBOEAN); Localized analysis of molecular dipole moment in terms of NLMO and NBO bond moments and their interactions (main subroutine: DIPANL). 0 A highly transportable subset of standard FORTRAN 77 is employed, with no special compiler extensions of any vendor, and all variable names of six characters or less. Common abbreviations used in naming subprograms, variables, and keywords are: = overlap matrix = density matrix (or D) = Fock matrix = dipole matrix (or DXYZ, or DX, DY, DZ) = Natural Population Analysis = Natural Atomic Orbital = Natural Bond Orbital = Natural Localized Molecular Orbital = pre-orthogonal NAO (i.e., omit interatomic orthogonali- zation) hsp 0 Most of the NBO storage is allocated dynamically, to conform to the minimum required for the molecular sys- tem under study. However, certain NBO common blocks of fixed dimensionality are used for integer storage. These are currently dimensioned to accomodate up to 99 July 11, 1995 - 8 - atoms and 500 basis functions. Section C.3 describes how these restrictions can be altered. The program is not set up to handle complex wavefunctions, but can treat any real RHF, ROHF, UHF, MCSCF (including GVB), CI, or M0t oller-Plesset-type wavefunction (i.e., any form of wavefunction for which the requisite density matrices are available) for ground or excited states of general open- or closed-shell molecules. Effective core poten- tials (``pseudo#|potentials'') can be handled, includ- ing complete neglect of core electrons as assumed in semi-empirical treatments. The atomic orbital basis functions (up to #If#N orbitals in angular symmetry) may be of general Slater-type, contracted Gaussian- type, or other general composition, including the ``effective'' ortho#|normal valence-shell AOs of semi- empirical treatments. AO basis functions are assumed to be normalized, but in general non-orthogonal. #IA.1.5 #IReferences and Relationship to Previous Versions#N 0 This program (``version 3.0'') is an extension of previous versions of the NBO method incorporated in the semi-empirical program #IBONDO#N [F. Weinhold, #IQuan- tum Chemistry Program Exchange No. 408#N (1980); ``ver- sion 1.0''] and in a GAUSSIAN-82 implementation [A. E. Reed and F. Weinhold, #IQCPE Bull. #B5#N, 141 (1985); ``version 2.0''], and should be considered to supplant those versions. Version 3.0 also supplants the various specific versions (``the GAMESS version,'' ``the AMPAC version,'' etc.) that have been informally created and distributed to individual users outside the QCPE frame- work. Principal contributors to the development of the NBO methods and programs (1975-1990) are Principal references to the development and applica- tions of NAO/NBO/NLMO methods are: J. P. Foster and F. Weinhold, #IJ. Am. Chem. Soc. #B102#N, 7211-7218 (1980). A. E. Reed and F. Weinhold, #IJ. Chem. Phys. #B78#N, 4066-4073 (1983); A. E. Reed, R. B. Weinstock, and F. Weinhold, #IJ. Chem. Phys. #B83#N, 735-746 (1985). A. E. Reed and F. Weinhold, #IJ. Chem. Phys. #B83#N, July 11, 1995 - 9 - 1736-1740 (1985). J. E. Carpenter and F. Weinhold, #IJ. Molec. Struct. (Theochem) #B169#N, 41-62 (1988); J. E. Carpenter, #IPh. D. Thesis#N, University of Wisconsin, Madison, 1987. A. E. Reed, L. A. Curtiss, and F. Weinhold, #IChem. Rev. #B88#N, 899-926 (1988); F. Weinhold and J. E. Car- penter, in, R. Naaman and Z. Vager (eds.), ``The Struc- ture of Small Molecules and Ions,'' (Plenum, New York, 1988), pp. 227-236. 0 The principal enhancements of version 3.0 include: #IGeneralized Program Interface.#N Overall program organization (Fig. 1) has been modified to standardize communication with the main ESS program. This insures that all special ESS ``versions'' of the NBO program now have consistent options and capabilities (as long as the option is meaningful in the context of the ESS), and enables the program to be offered in a greater number of specialized ESS versions than were previously available. #INAO/NPA Summary Table.#N New tables give improved display of NAOs and natural populations, including the ``natural electron configuration'' of each atom (i.e., the occupancy and type of NAOs describing the atomic electron configuration of each atom). The new NAO sum- mary tables (Section A.3.2) include an SCF atomic orbi- tal energy (if available), a conventional atomic orbi- tal label (1#Is#N, 2#Is#N, 2#Ip#N, etc., in accordance with the labelling in isolated atoms), and a shell designation (Cor = core, Val = valence, or Ryd = Ryd- berg) to aid characterization of the NAO. #INBO Summary Table.#N A new NBO summary table (Section A.3.6) has been provided to summarize the energetics and delocalization patterns of the principal NBOs. This succinctly combines the most important information from the full NBO table, diagonal NBO Fock matrix ele- ments, and 2nd-order energy analysis. #IBond Bending Analysis.#N The program includes a new analysis of hydrid directionality and bond ``bending'' (keyword BEND, Section A.3.4). #IDipole Moment Analysis.#N The program includes new optional provision (keyword DIPOLE, Section B.6.3) for analysis of the molecular dipole moment in terms of July 11, 1995 - 10 - localized NLMOs and NBOs. #IPrint options.#N The program offers new structured printing options (Section B.2.4) that give greater con- venience and flexibility in controlling printed output, with improved provision for printing matrices or basis transformations involving general NAO, NHO, NBO, NLMO or pre-orthogonal (PNAO, PNHO, PNBO, PNLMO) basis sets. #IOrbital Contour Info.#N The program makes optional provision (keyword PLOT, Section B.2.5) for writing out files that can be used by an orbital plotting program (available separately through QCPE) to draw contour diagrams of the NBOs or other natural localized orbi- tals. #IEffective Core Potentials.#N The program now handles effective core potentials (pseudo#|potentials), or the complete neglect of core levels characteristic of semi-empirical wavefunctions (Section B.6.12). The program also includes three changes to correct problems of the previous version (which may have affected a small number of users): #IUnpolarized Cores.#N NAOs identified as ``core'' orbitals are now auto#|matically carried over as unhy- bridized 1-center core NBOs (Section B.3). This has virtually no effect on the form or occupancy of a core NBO, but averts the (rare) problem of unphysical mixing between core and valence lone pairs when the occupan- cies are `accidentally' degenerate (usually, both very close to 2.000...) within the numerical machine preci- sion. A warning message is printed when the core occu- pancy is less than 1.9990, indicating a possible ``core polarization'' effect of physical significance. #IExcited State Antibond Labels.#N The program now directly investigates the nodal structure of an NBO (by examining the overlap matrix in the PNHO basis) before assigning it a label as a ``bond'' (unstarred) or ``antibond'' (starred) NBO. In previous versions, these labels were assigned on the basis of the presumed higher occupancy of the in-phase bond combination, which was generally true for ground states, but not for excited states. The program now prints a warning mes- sage whenever it encounters the ``anomalous'' situation of an out-of-phase antibond NBO having higher occupancy than the corresponding in-phase bond NBO, indicative of an excited-state configuration. [WARNING: the overlap test cannot be applied to semi-empirical methods with orthogonal AOs (e.g., AMPAC), so antibond labels for these methods are assigned, as in previous versions, on July 11, 1995 - 11 - the basis of occupancy.] #IAlternative Resonance Structures.#N The program now institutes a search for alternative Lewis (`resonance') structures when two or more structures may be competi- tive, and returns the structure of lowest non-Lewis occupancy. This corrects a possible dependence on atomic numbering in cases of strong delocalization. Despite these changes and extensions, version 3.0 has been designed to be upward compatible with v. 2.0, as nearly as possible. Previous users of NBO 2.0 should find that their jobs run similarly (i.e., most keywords continue to function as in previous versions). Thus, experienced NBO users should find little difficulty in adapting to, and experimenting with, the new capabili- ties of the program. #BA.2 INSTALLING THE NBO PROGRAM#N 0 The NBO programs and manual are provided on a distri- bution tape. The tape contains three files: the TechSet code of this manual (file NBO.MAN), a file con- taining the core NBO source routines and supporting driver routines (file NBO.SRC), and the Fortran ``ena- bler'' program (file ENABLE.FOR). 0 In overview, the installation procedure involves the following steps (the details of each step being depen- dent on your operating system): #IEnabling the NBO routines.#N Copy the contents of the distribution tape onto your system. Using your system Fortran 77 compiler, compile and link the ena- bler program to create the ENABLE.EXE executable; for example, the VMS commands to create ENABLE.EXE are #T FOR ENABLE LINK ENABLE #NNow, run the ENABLE program (e.g., type ``RUN ENABLE'' in a VMS system), and answer the prompt #T NBO program version to enable? #Nby selecting from the available offerings. Each ESS package is associated with a 3-letter identifier (``G88'' for GAUSSIAN-88, ``GMS'' for GAMESS, ``AMP'' for AMPAC, etc.). The ENABLE program will create a file #IXXX#NNBO.FOR (where `#IXXX#N' is the identifier) that incorporates the appropriate drivers for your ESS. July 11, 1995 - 12 - #ICompiling the NBO routines.#N Using your system For- tran 77 compiler, compile the #IXXX#NNBO.FOR file to an object code file (say, #IXXX#NNBO.OBJ). [Compiler errors (if any) should be fixed before proceeding. Please notify the authors if you encounter undue diffi- culties in this step.] #IModifying the ESS routines.#N In general, the ESS source Fortran code must be modified to call the NBO routines near the point where the ESS performs Mulliken Population Analysis or evaluates properties of the final wavefunction. The modification generally con- sists of inserting a single statement (viz., ``CALL RUNNBO'') in one subroutine of your ESS system. See the appropriate Appendix of this Manual for detailed information on exactly how to modify the ESS code for your chosen system. #IRebuilding the integrated ESS/NBO program.#N Re- compile your modified ESS programs and link the result- ing object file (say, ESS.OBJ) with the #IXXX#NNBO.OBJ file to form the final ESS.EXE executable. In general, this step will closely follow the initial installation procedure for your ESS, with the exception that the #IXXX#NNBO.OBJ file must be included in the link state- ment (or deposited in one of the libraries accessed by the linker, etc.). Note that installation of the NBO programs into your ESS system in no way affects the way your system processes standard input files. The only change involves enabling the reading of NBO keylists (if detected in your input file), performance of the tasks requested in the keylist, and return of control to the parent ESS program in the state in which the NBO call was encountered. 0 If you are interfacing the NBO programs to a new ESS package (not represented in the driver routines pro- vided with this distribution), see Section C for gui- dance on how to create drivers for your ESS to provide the necessary information. Alternatively, see Section B.7 for a description of the input file to GENNBO, the stand-alone version of the NBO program. 0 The TechSet-coded version of this manual, NBO.MAN, can be printed on an HP LaserJet printer (`F' car- tridge) with the TECHSET technical typesetting program [ACS Software, American Chemical Society, Marketing Communications Dept., 1155 Sixteenth Street, N.W., Washington, D.C. 20036]. #BA.3 TUTORIAL EXAMPLE FOR METHYLAMINE#N July 11, 1995 - 13 - #IA.3.1 Running the Example#N 0 This section provides an introductory `quick start' tutorial on running a simple NBO job and interpreting the output. The example chosen is that of methylamine (CH#d3#uNH#d2#u) in Pople-Gordon idealized geometry, treated at the #Iab initio#N RHF/3-21G level. This simple split-valence basis set consists of 28 AOs (nine each on C and N, two on each H), extended by 13 AOs beyond the minimal basis level. 0 Input files to run this job (or its nearest equivalent) with each ESS are given in the Appendix. (The output shown below was created with the GAMESS system.) In most cases, you can modify the standard ESS input file to produce NBO output by simply includ- ing the line #T $NBO $END #Nat the end of the file. This is an `empty' NBO keyl- ist, specifying that NBO analysis should be carried out at the #Idefault#N level. 0 The default NBO output produced by this example is shown below, just as it appears in your output file. The start of the NBO section is marked by a standard header and storage info: ******************************************************************************* N A T U R A L A T O M I C O R B I T A L A N D N A T U R A L B O N D O R B I T A L A N A L Y S I S ******************************************************************************* Job title: Methylamine...RHF/3-21G//Pople-Gordon stan- dard geometry Storage needed: 2505 in NPA, 2569 in NBO ( 750000 available) #T @seg #NNote that all NBO output is formatted to a maximum 80-character width for convenient display on a computer terminal. The NBO heading echoes any requested key- words (none for the present default case) and shows an estimate of the memory requirements (in double preci- sion words) for the separate steps of the NBO process, compared to the total allocated memory available through your ESS process. Increase the memory allo- cated to your ESS process if the estimated NBO requests exceed the available storage. #IA.3.2 Natural July 11, 1995 - 14 - Population Analysis#N #N0 The next four NBO output segments summarize the results of natural population analysis (NPA). The first segment is the main NAO table, as shown below: NATURAL POPULATIONS: Natural atomic orbital occupan- cies NAO Atom # lang Type(AO) Occupancy Energy ---------- ----------------------------------------------- 1 C 1 s Cor( 1s) 1.99900 -11.04184 2 C 1 s Val( 2s) 1.09038 -0.28186 3 C 1 s Ryd( 3s) 0.00068 1.95506 4 C 1 px Val( 2p) 0.89085 -0.01645 5 C 1 px Ryd( 3p) 0.00137 0.93125 6 C 1 py Val( 2p) 1.21211 -0.07191 7 C 1 py Ryd( 3p) 0.00068 1.03027 8 C 1 pz Val( 2p) 1.24514 -0.08862 9 C 1 pz Ryd( 3p) 0.00057 1.01801 10 N 2 s Cor( 1s) 1.99953 -15.25950 11 N 2 s Val( 2s) 1.42608 -0.71700 12 N 2 s Ryd( 3s) 0.00016 2.75771 13 N 2 px Val( 2p) 1.28262 -0.18042 14 N 2 px Ryd( 3p) 0.00109 1.57018 15 N 2 py Val( 2p) 1.83295 -0.33858 16 N 2 py Ryd( 3p) 0.00190 1.48447 17 N 2 pz Val( 2p) 1.35214 -0.19175 18 N 2 pz Ryd( 3p) 0.00069 1.59492 19 H 3 s Val( 1s) 0.81453 0.13283 20 H 3 s Ryd( 2s) 0.00177 0.95067 21 H 4 s Val( 1s) 0.78192 0.15354 22 H 4 s Ryd( 2s) 0.00096 0.94521 23 H 5 s Val( 1s) 0.78192 0.15354 24 H 5 s Ryd( 2s) 0.00096 0.94521 25 H 6 s Val( 1s) 0.63879 0.20572 26 H 6 s Ryd( 2s) 0.00122 0.99883 27 H 7 s Val( 1s) 0.63879 0.20572 28 H 7 s Ryd( 2s) 0.00122 0.99883 #T @seg #NFor each of the 28 NAO functions, this table lists the atom to which NAO is attached (in the numbering scheme of the ESS program), the angular July 11, 1995 - 15 - momentum type `lang' (#Is#N, #Ip#dx#u#N, etc., in the coor- dinate system of the ESS program), the orbital type (whether core, valence, or Rydberg, and a conventional hydrogenic-type label), the orbital occupancy (number of electrons, or `natural popu- lation' of the orbital), and the orbital energy (in the favored units of the ESS program, in this case atomic units: 1 a.u. = 627.5 kcal/mol). [For example, NAO 4 (the highest energy C orbital of the NMB set) is the valence shell 2#Ip#N#dx#u orbital on carbon, occupied by 0.8909 electrons, whereas NAO 5 is a Rydberg 3#Ip#N#dx#u orbital with only 0.0014 electrons.] Note that the occu- pancies of the Rydberg (Ryd) NAOs are typically much lower than those of the core (Cor) plus valence (Val) NAOs of the natural minimum basis set, reflecting the dominant role of the NMB orbi- tals in describing molecular properties. 0 The principal quantum numbers for the NAO labels (1#Is#N, 2#Is#N, 3#Is#N, etc.) are assigned on the basis of the energy order if a Fock matrix is available, or on the basis of occupancy otherwise. A message is printed warning of a `population inversion' if the occupancy and energy ordering do not coincide. Summary of Natural Population Analysis: Natural Population Natural --------- -------------------------------------- Atom # Charge Core Valence Rydberg Total ---------- ------------------------------------------------------------- C 1 -0.44079 1.99900 4.43848 0.00331 6.44079 N 2 -0.89715 1.99953 5.89378 0.00384 7.89715 H 3 0.18370 0.00000 0.81453 0.00177 0.81630 H 4 0.21713 0.00000 0.78192 0.00096 0.78287 H 5 0.21713 0.00000 0.78192 0.00096 0.78287 H 6 0.35999 0.00000 0.63879 0.00122 0.64001 H 7 0.35999 0.00000 0.63879 0.00122 0.64001 ======================================================================= * Total * 0.00000 3.99853 13.98820 0.01328 18.00000 July 11, 1995 - 16 - #NThe next segment is an atomic summary showing the natural atomic charges (nuclear charge minus summed natural popula- tions of NAOs on the atom) and total core, valence, and Ryd- berg populations on each atom: #T @seg #NThis table succinctly describes the molecular charge distribution in terms of NPA charges. [For example, the carbon atom of methylamine is assigned a net NPA charge of minus 0.441 at this level; note also the slightly less positive charge on H(3) than on the other two methyl hydrogens: +0.184 vs. +0.217.] Natural Population -------- ------------------------------------------------ Core 3.99853 ( 99.9632% of 4) Valence 13.98820 ( 99.9157% of 14) Natural Minimal Basis 17.98672 ( 99.9262% of 18) Natural Rydberg Basis 0.01328 ( 0.0738% of 18) --- ----------------------------------------------------- #NNext follows a summary of the NMB and NRB populations for the composite system, summed over atoms: #T @seg #NThis exhibits the high percentage contribution (typically, > 99%) of the NMB set to the molecular charge distribution. [In the present case, for example, the 13 Rydberg orbitals of the NRB set contribute only 0.07% of the electron den- sity, whereas the 15 NMB functions account for 99.93% of the total.] #NFinally, the natural populations are summarized as an effective valence electron configuration (``natural electron configuration'') for each atom: Atom # Natural Electron Configuration --------- - ------------------------------------------------------------------ C 1 [core]2s( 1.09)2p( 3.35) N 2 [core]2s( 1.43)2p( 4.47) H 3 1s( 0.81) H 4 1s( 0.78) H 5 1s( 0.78) H 6 1s( 0.64) H 7 1s( 0.64) #T @seg July 11, 1995 - 17 - #NAlthough the occupancies of the atomic orbitals are non-integer in the molecular environment, the effective atomic configurations can be related to idealized atomic states in `promoted' configura- tions. [For example, the carbon atom in the above table is most nearly described by an idealized 1s#u2#d2s#u1#d2p#u3#d electron configuration.] #IA.3.3 Natural Bond Orbital Analysis#N #N0 The next segments of the output summarize the results of NBO analysis. The first segment reports on details of the search for an NBO natural Lewis structure: NATURAL BOND ORBITAL ANALYSIS: Occupancies Lewis Structure Low High Occ. ------------------- ----------------- occ occ Cycle Thresh. Lewis Non-Lewis CR BD 3C LP (L) (NL) Dev ============================================================================= 1(1) 1.90 17.95048 0.04952 2 6 0 1 0 0 0.02 ---------- ------------------------------------------------------------------- Structure accepted: No low occupancy Lewis orbitals #T @seg #NNormally, there is but one cycle of the NBO search (cf. the ``RESONANCE'' keyword, Section B.6.6). The table sum- marizes a variety of information for each cycle: the occu- pancy thresh#|old for a `good' pair in the NBO search; the total populations of Lewis and non-Lewis NBOs; the number of core (CR), 2-center bond (BD), 3-center bond (3C), and lone pair (LP) NBOs in the natural Lewis structure; the number of low-occupancy Lewis (L) and `high-occupancy' (> 0.1e) non- Lewis (NL) orbitals; and the maximum deviation (`Dev') of any formal bond order from a nominal estimate (NAO Wiberg bond index) for the structure. [If the latter exceeds 0.1, additional NBO searches are initiated (indicated by the parenthesized number under `Cycle') for alternative Lewis structures.] The Lewis structure is accepted if all orbi- tals of the formal Lewis structure exceed the occupancy thresh#|old (default, 1.90 electrons). 0 #NNext follows a more detailed breakdown of the Lewis and non-Lewis occupancies into core, valence, and Rydberg shell contributions: WARNING: 1 low occupancy (<1.9990e) core orbital found on July 11, 1995 - 18 - C 1 -------------------------------------------------------- Core 3.99853 ( 99.963% of 4) Valence Lewis 13.95195 ( 99.657% of 14) ================== ============================ Total Lewis 17.95048 ( 99.725% of 18) ----------------------------------------------------- Valence non-Lewis 0.03977 ( 0.221% of 18) Rydberg non-Lewis 0.00975 ( 0.054% of 18) ================== ============================ Total non-Lewis 0.04952 ( 0.275% of 18) ----- --------------------------------------------------- #T @seg #NThis shows the general quality of the natural Lewis struc- ture description in terms of the percentage of the total electron density (e.g., in the above case, about 99.7%). The table also exhibits the relatively important role of the valence non-Lewis orbitals (i.e., the six valence antibonds, NBOs 23-28) relative to the extra-valence orbitals (the 13 Rydberg NBOs 10-22) in the slight departures from a local- ized Lewis structure model. (In this case, the table also includes a warning about a carbon core orbital with slightly less than double occupancy.) (Occupancy) Bond orbital/ Coefficients/ Hybrids ----- ----- --------------------------------------------------------------------- 1. (1.99858) BD ( 1) C 1- N 2 ( 40.07%) 0.6330* C 1 s( 21.71%)p 3.61( 78.29%) -0.0003 -0.4653 -0.0238 -0.8808 -0.0291 -0.0786 -0.0110 0.0000 0.0000 ( 59.93%) 0.7742* N 2 s( 30.88%)p 2.24( 69.12%) -0.0001 -0.5557 0.0011 0.8302 0.0004 0.0443 -0.0098 0.0000 0.0000 2. (1.99860) BD ( 1) C 1- H 3 ( 59.71%) 0.7727* C 1 s( 25.78%)p 2.88( 74.22%) -0.0002 -0.5077 0.0069 0.1928 0.0098 0.8396 -0.0046 0.0000 0.0000 ( 40.29%) 0.6347* H 3 s(100.00%) -1.0000 -0.0030 July 11, 1995 - 19 - 3. (1.99399) BD ( 1) C 1- H 4 ( 61.02%) 0.7812* C 1 s( 26.28%)p 2.80( 73.72%) 0.0001 0.5127 -0.0038 -0.3046 -0.0015 0.3800 -0.0017 0.7070 -0.0103 ( 38.98%) 0.6243* H 4 s(100.00%) 1.0000 0.0008 4. (1.99399) BD ( 1) C 1- H 5 ( 61.02%) 0.7812* C 1 s( 26.28%)p 2.80( 73.72%) 0.0001 0.5127 -0.0038 -0.3046 -0.0015 0.3800 -0.0017 -0.7070 0.0103 ( 38.98%) 0.6243* H 5 s(100.00%) 1.0000 0.0008 5. (1.99442) BD ( 1) N 2- H 6 ( 68.12%) 0.8253* N 2 s( 25.62%)p 2.90( 74.38%) 0.0000 0.5062 0.0005 0.3571 0.0171 -0.3405 0.0069 -0.7070 -0.0093 ( 31.88%) 0.5646* H 6 s(100.00%) 1.0000 0.0020 6. (1.99442) BD ( 1) N 2- H 7 ( 68.12%) 0.8253* N 2 s( 25.62%)p 2.90( 74.38%) 0.0000 0.5062 0.0005 0.3571 0.0171 -0.3405 0.0069 0.7070 0.0093 ( 31.88%) 0.5646* H 7 s(100.00%) 1.0000 0.0020 7. (1.99900) CR ( 1) C 1 s(100.00%)p 0.00( 0.00%) 1.0000 -0.0003 0.0000 -0.0002 0.0000 0.0001 0.0000 0.0000 0.0000 8. (1.99953) CR ( 1) N 2 s(100.00%)p 0.00( 0.00%) 1.0000 -0.0001 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 9. (1.97795) LP ( 1) N 2 s( 17.85%)p 4.60( 82.15%) 0.0000 0.4225 0.0002 0.2360 -0.0027 0.8749 -0.0162 0.0000 0.0000 July 11, 1995 - 20 - 10. (0.00105) RY*( 1) C 1 s( 1.57%)p62.84( 98.43%) 0.0000 -0.0095 0.1248 -0.0305 0.7302 -0.0046 0.6710 0.0000 0.0000 11. (0.00034) RY*( 2) C 1 s( 0.00%)p 1.00(100.00%) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0146 0.9999 12. (0.00022) RY*( 3) C 1 s( 56.51%)p 0.77( 43.49%) 0.0000 -0.0023 0.7517 -0.0237 0.3710 -0.0094 -0.5447 0.0000 0.0000 13. (0.00002) RY*( 4) C 1 s( 41.87%)p 1.39( 58.13%) 14. (0.00116) RY*( 1) N 2 s( 1.50%)p65.53( 98.50%) 0.0000 -0.0062 0.1224 0.0063 0.0371 0.0197 0.9915 0.0000 0.0000 15. (0.00044) RY*( 2) N 2 s( 0.00%)p 1.00(100.00%) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0132 0.9999 16. (0.00038) RY*( 3) N 2 s( 33.38%)p 2.00( 66.62%) 0.0000 0.0133 0.5776 0.0087 -0.8150 -0.0121 -0.0405 0.0000 0.0000 17. (0.00002) RY*( 4) N 2 s( 65.14%)p 0.54( 34.86%) 18. (0.00178) RY*( 1) H 3 s(100.00%) -0.0030 1.0000 19. (0.00096) RY*( 1) H 4 s(100.00%) -0.0008 1.0000 20. (0.00096) RY*( 1) H 5 s(100.00%) -0.0008 1.0000 21. (0.00122) RY*( 1) H 6 s(100.00%) -0.0020 1.0000 22. (0.00122) RY*( 1) H 7 s(100.00%) -0.0020 1.0000 23. (0.00016) BD*( 1) C 1- N 2 ( 59.93%) 0.7742* C 1 s( 21.71%)p 3.61( 78.29%) -0.0003 -0.4653 July 11, 1995 - 21 - -0.0238 -0.8808 -0.0291 -0.0786 -0.0110 0.0000 0.0000 ( 40.07%) -0.6330* N 2 s( 30.88%)p 2.24( 69.12%) -0.0001 -0.5557 0.0011 0.8302 0.0004 0.0443 -0.0098 0.0000 0.0000 24. (0.01569) BD*( 1) C 1- H 3 ( 40.29%) 0.6347* C 1 s( 25.78%)p 2.88( 74.22%) 0.0002 0.5077 -0.0069 -0.1928 -0.0098 -0.8396 0.0046 0.0000 0.0000 ( 59.71%) -0.7727* H 3 s(100.00%) 1.0000 0.0030 25. (0.00769) BD*( 1) C 1- H 4 ( 38.98%) 0.6243* C 1 s( 26.28%)p 2.80( 73.72%) -0.0001 -0.5127 0.0038 0.3046 0.0015 -0.3800 0.0017 -0.7070 0.0103 ( 61.02%) -0.7812* H 4 s(100.00%) -1.0000 -0.0008 26. (0.00769) BD*( 1) C 1- H 5 ( 38.98%) 0.6243* C 1 s( 26.28%)p 2.80( 73.72%) -0.0001 -0.5127 0.0038 0.3046 0.0015 -0.3800 0.0017 0.7070 -0.0103 ( 61.02%) -0.7812* H 5 s(100.00%) -1.0000 -0.0008 27. (0.00426) BD*( 1) N 2- H 6 ( 31.88%) 0.5646* N 2 s( 25.62%)p 2.90( 74.38%) 0.0000 -0.5062 -0.0005 -0.3571 -0.0171 0.3405 -0.0069 0.7070 0.0093 ( 68.12%) -0.8253* H 6 s(100.00%) -1.0000 -0.0020 28. (0.00426) BD*( 1) N 2- H 7 ( 31.88%) 0.5646* N 2 s( 25.62%)p 2.90( 74.38%) 0.0000 -0.5062 -0.0005 -0.3571 -0.0171 0.3405 -0.0069 -0.7070 -0.0093 ( 68.12%) -0.8253* H 7 s(100.00%) July 11, 1995 - 22 - -1.0000 -0.0020 #NNext follows the main listing of NBOs, displaying the form and occupancy of the complete set of NBOs that span the input AO space: #T @seg #NFor each NBO (1-28), the first line of printout shows the occupancy (between 0 and 2.0000 electrons) and unique label of the NBO. This label gives the type (``BD'' for 2-center bond, ``CR'' for 1-center core pair, ``LP'' for 1-center valence lone pair, ``RY*'' for 1-center Rydberg, and ``BD*'' for 2-center antibond, the unstarred and starred labels corresponding to Lewis and non-Lewis NBOs, respectively), a serial number (1, 2,... if there is a single, double,... bond between the pair of atoms), and the atom(s) to which the NBO is affixed. [For example, the first NBO in the sam- ple output is the 2-center bond (with 1.99858 electrons) between carbon (atom 1) and nitrogen (atom 2), the gma #dCN#u bond.] The next lines summarize the natural atomic hybrids #Ih#N#dA#u of which the NBO is composed, giving the percentage (100|#Ic#N#dA#u|#u2#d) of the NBO on each hybrid (in parentheses), the polarization coefficient #Ic#N#dA#u, the atom label, and a hybrid label showing the #Isp#N#u #d composition (percentage #Is#N-character, #Ip#N-character, etc.) of each #Ih#N#dA#u. [For example, the gma #dCN#u NBO is formed from an #Isp#N#u3.61#d hybrid (78.3% #Ip#N- character) on carbon interacting with an #Isp#N#u2.24#d hybrid (69.1% #Ip#N-character) on nitrogen, gma #dCN#u = 0.633(#Isp#N#u3.61#d)#dC#u + 0.774(#Isp#N#u2.24#d)#dN#u corresponding roughly to the qualitative concept of interacting #Isp#N#u3#d hybrids (75% #Ip#N-character) and the higher electronegativity (larger polarization coeffi- cient) of N.] Below each NHO label is the set of coeffi- cients that specify how the NHO is written explicitly as a linear combination of NAOs on the atom. The order of NAO coefficients follows the numbering of the NAO tables. [For example, in the first NBO entry, the carbon hybrid #Ih#N#dC#u of the gma #dCN#u bond has largest coefficients for the 2#und#d and 4#uth#d NAOs, corresponding to the approximate description #Ih#N#dC#u ~= minus 0.4653(2#Is#N)#dC#u minus 0.8808(2#Ip#N#dx#u)#dC#u in terms of the valence NAOs of the carbon atom.] In the CH#d3#uNH#d2#u example, the NBO search finds the C-N bond July 11, 1995 - 23 - (NBO 1), three C-H bonds (NBOs 2, 3, 4), two N-H bonds (NBOs 5, 6), N lone pair (NBO 9), and C and N core pairs (NBOs 7, 8) of the expected Lewis structure. NBOs 10-28 represent the residual non-Lewis NBOs of low occupancy. In this example, it is also interesting to note the slight asymmetry of the three gma #dCH#u NBOs, and the slightly higher occupancy (0.01569 #Ivs.#N 0.0077 electrons) in the gma *#<#dC#d1#uH#d3#u#u antibond (NBO 24) lying #Itrans#N to the nitrogen lone pair. #IA.3.4 NHO Directional Analysis#N 0 The next segment of output summarizes the angular proper- ties of the natural hybrid orbitals: NHO Directionality and "Bond Bending" (deviations from line of nuclear centers) [Thresholds for printing: angular deviation > 1.0 degree] hybrid p-character > 25.0% orbital occupancy > 0.10e Line of Centers Hybrid 1 Hybrid 2 --------------- ------------------- ------------------ NBO Theta Phi Theta Phi Dev Theta Phi Dev =============================================================================== 1. BD ( 1) C 1- N 2 90.0 5.4 -- -- -- 90.0 182.4 3.0 3. BD ( 1) C 1- H 4 35.3 130.7 34.9 129.0 1.0 -- -- -- 4. BD ( 1) C 1- H 5 144.7 130.7 145.1 129.0 1.0 -- -- -- 5. BD ( 1) N 2- H 6 144.7 310.7 145.0 318.3 4.4 -- -- -- 6. BD ( 1) N 2- H 7 35.3 310.7 35.0 318.3 4.4 -- -- -- 9. LP ( 1) N 2 -- -- 90.0 74.8 -- -- -- -- #T @seg #NThe `direction' of a hybrid is specified in terms of the polar (heta ) and azimuthal (hi ) angles (in the ESS coordinate system) of the vector describing its #Ip#N-component. The hybrid direction is compared with the direction of the line of centers between the two nuclei to determine the `bending' of the bond, expressed as the deviation angle (``Dev,'' in degrees) July 11, 1995 - 24 - between these two directions. For example, in the CH#d3#uNH#d2#u case shown above, the nitrogen NHO of the gma #dCN#u bond (NBO 1) is bent away from the line of C-N centers by 3.0 egree , whereas the carbon NHO is approximately aligned with the C-N axis (within the 1.0 egree threshold for print- ing). The N-H bonds (NBOs 5, 6) are bent even further (4.4 egree ). The information in this table is often useful in anticipating the direction of geometry changes resulting from geometry optimization (viz., likely reduced pyramidali- zation of the -NH#d2#u group to relieve the nitrogen bond `kinks' found in the tetrahedral Pople-Gordon geometry). #IA.3.5 Perturbation Theory Energy Analysis#N 0 The next segment summarizes the second-order perturbative estimates of `donor-acceptor' (bond-antibond) interactions in the NBO basis: Second Order Perturbation Theory Analysis of Fock Matrix in NBO Basis Threshold for printing: 0.50 kcal/mol E(2) E(j)-E(i) F(i,j) Donor NBO (i) Acceptor NBO (j) kcal/mol a.u. a.u. =============================================================================== within unit 1 2. BD ( 1) C 1- H 3 / 14. RY*( 1) N 2 0.84 2.18 0.038 3. BD ( 1) C 1- H 4 / 26. BD*( 1) C 1- H 5 0.52 1.39 0.024 3. BD ( 1) C 1- H 4 / 27. BD*( 1) N 2- H 6 3.03 1.37 0.057 4. BD ( 1) C 1- H 5 / 25. BD*( 1) C 1- H 4 0.52 1.39 0.024 4. BD ( 1) C 1- H 5 / 28. BD*( 1) N 2- H 7 3.03 1.37 0.057 5. BD ( 1) N 2- H 6 / 10. RY*( 1) C 1 0.56 1.78 0.028 5. BD ( 1) N 2- H 6 / 25. BD*( 1) C 1- H 4 2.85 1.51 0.059 6. BD ( 1) N 2- H 7 / 10. RY*( 1) C 1 0.56 1.78 0.028 6. BD ( 1) N 2- H 7 / 26. BD*( 1) C 1- H 5 2.85 1.51 0.059 7. CR ( 1) C 1 / 16. RY*( 3) N 2 0.61 13.11 0.080 7. CR ( 1) C 1 / 18. RY*( 1) H 3 1.40 11.99 0.116 7. CR ( 1) C 1 / 19. RY*( 1) H 4 1.55 11.99 0.122 July 11, 1995 - 25 - 7. CR ( 1) C 1 / 20. RY*( 1) H 5 1.55 11.99 0.122 8. CR ( 1) N 2 / 10. RY*( 1) C 1 1.51 16.23 0.140 8. CR ( 1) N 2 / 12. RY*( 3) C 1 0.84 16.77 0.106 8. CR ( 1) N 2 / 21. RY*( 1) H 6 0.61 16.26 0.089 8. CR ( 1) N 2 / 22. RY*( 1) H 7 0.61 16.26 0.089 9. LP ( 1) N 2 / 24. BD*( 1) C 1- H 3 8.13 1.13 0.086 9. LP ( 1) N 2 / 25. BD*( 1) C 1- H 4 1.46 1.14 0.037 9. LP ( 1) N 2 / 26. BD*( 1) C 1- H 5 1.46 1.14 0.037 #T @seg #NThis is carried out by examining all possible interactions between `filled' (donor) Lewis-type NBOs and `empty' (accep- tor) non-Lewis NBOs, and estimating their energetic impor- tance by 2nd-order perturbation theory. Since these interactions lead to loss of occupancy from the localized NBOs of the idealized Lewis structure into the empty non- Lewis orbitals (and thus, to departures from the idealized Lewis structure description), they are referred to as `delo- calization' corrections to the zeroth-order natural Lewis structure. For each donor NBO (#Ii#N) and acceptor NBO (#Ij#N), the stabilization energy E(2) associated with delo- calization (``2e-stabilization'') #Ii arr j#N is estimated as E(2) = Delta E#dij#u = q#di#u quo <> where #Iq#N#di#u is the donor orbital occupancy, \psilon #di#u, \psilon #dj#u are diagonal elements (orbital ener- gies) and F(i,j) is the off-diagonal NBO Fock matrix ele- ment. [In the example above, the #In#N#dN#u arr gma *#<#dCH#u interaction between the nitrogen lone pair (NBO 8) and the antiperiplanar C#d1#u-H#d3#u antibond (NBO 24) is seen to give the strongest stabilization, 8.13 kcal/mol.] As the heading indicates, entries are included in this table only when the interaction energy exceeds a default threshold of 0.5 kcal/mol. #IA.3.6 NBO Summary#N 0 Next appears a condensed summary of the principal NBOs, showing the occupancy, orbital energy, and the qualitative pattern of delocalization interactions associated with each: Natural Bond Orbitals (Summary): Princi- pal Delocalizations July 11, 1995 - 26 - NBO Occupancy Energy (geminal,vicinal,remote) =============================================================================== Molecular unit 1 (CH5N) 1. BD ( 1) C 1- N 2 1.99858 -0.89908 2. BD ( 1) C 1- H 3 1.99860 -0.69181 14(v) 3. BD ( 1) C 1- H 4 1.99399 -0.68892 27(v),26(g) 4. BD ( 1) C 1- H 5 1.99399 -0.68892 28(v),25(g) 5. BD ( 1) N 2- H 6 1.99442 -0.80951 25(v),10(v) 6. BD ( 1) N 2- H 7 1.99442 -0.80951 26(v),10(v) 7. CR ( 1) C 1 1.99900 -11.04131 19(v),20(v),18(v),16(v) 8. CR ( 1) N 2 1.99953 -15.25927 10(v),12(v),21(v),22(v) 9. LP ( 1) N 2 1.97795 -0.44592 24(v),25(v),26(v) 10. RY*( 1) C 1 0.00105 0.97105 11. RY*( 2) C 1 0.00034 1.02120 12. RY*( 3) C 1 0.00022 1.51414 13. RY*( 4) C 1 0.00002 1.42223 14. RY*( 1) N 2 0.00116 1.48790 15. RY*( 2) N 2 0.00044 1.59323 16. RY*( 3) N 2 0.00038 2.06475 17. RY*( 4) N 2 0.00002 2.25932 18. RY*( 1) H 3 0.00178 0.94860 19. RY*( 1) H 4 0.00096 0.94464 20. RY*( 1) H 5 0.00096 0.94464 21. RY*( 1) H 6 0.00122 0.99735 22. RY*( 1) H 7 0.00122 0.99735 23. BD*( 1) C 1- N 2 0.00016 0.57000 24. BD*( 1) C 1- H 3 0.01569 0.68735 25. BD*( 1) C 1- H 4 0.00769 0.69640 26. BD*( 1) C 1- H 5 0.00769 0.69640 27. BD*( 1) N 2- H 6 0.00426 0.68086 28. BD*( 1) N 2- H 7 0.00426 0.68086 ------------------------------- Total Lewis 17.95048 ( 99.7249%) Valence non-Lewis 0.03977 ( 0.2209%) Rydberg non-Lewis 0.00975 ( 0.0542%) ------------------------------- Total unit 1 18.00000 (100.0000%) Charge unit 1 0.00000 #T @seg #NThis table allows one to quickly identify the principal delocalizing acceptor orbitals associated with each donor NBO, and their topological relationship to this NBO, i.e., whether attached to the same atom (geminal, ``g''), to an July 11, 1995 - 27 - adjacent bonded atom (vicinal, ``v''), or to a more remote (``r'') site. These acceptor NBOs will generally correspond to the principal `delocalization tails' of the NLMO associated with the parent donor NBO. [For example, in the table above, the nitrogen lone pair (NBO 9) is seen to be the lowest- occupancy (1.97795 electrons) and highest-energy (minus 0.44592 a.u.) Lewis NBO, and to be primarily delocalized into antibonds 24, 25, 26 (the vicinal gma *#<#dCH#u NBOs). The summary at the bottom of the table shows that the Lewis NBOs 1-9 describe about 99.7% of the total electron density, with the remaining non-Lewis density found primarily in the valence-shell antibonds (particularly, NBO 24).] #HSection B: NBO USER'S GUIDE#N #BB.1 INTRODUCTION TO THE NBO USER'S GUIDE AND NBO KEYLISTS#N 0 Section B constitutes the general user's guide to the NBO program. It assumes that the user has an installed elec- tronic structure system (ESS) with attached NBO program, a general idea of what the NBO method is about, and some acquaintance with standard NBO terminology and output data. If you are completely inexperienced in these areas, read Section A (General Introduction and Installation) for the necessary background to this Section. 0 The User's Guide describes how to use the NBO program by modifying your input file to the ESS program to get some NBO output. The modification consists of adding a list of #Ikeywords#N in a prescribed #Ikeylist#N format. Four dis- tinct keylist ($KEY) types are recognized ($NBO, $CORE, $CHOOSE, and $DEL), and these will be described in turn in Sections B.2-B.5. Some of the details of inserting NBO keylists into the input file depend on the details of your ESS method, and are described in the appropriate Appendix for the ESS. However, the general form of NBO keylists and the meaning and function of each keyword are identical for all versions (insofar as the option is meaningful for the ESS), and are described herein. 0 The four keylist types have common rules of syntax: Keyl- ist delimiters are identified by a ``$'' prefix. Each keyl- ist begins with the parent keylist name (e.g., ``$NBO''), followed by any number of keywords, and ended with the word ``$END''; for example, #T July 11, 1995 - 28 - $NBO keyword1 keyword2 . . . $END !comment #N(The allowed keyword entries for each type of keylist are described in Sections B.2-B.5.) The keylist is ``free for- mat,'' with keywords separated by commas or any number of spaces. An NBO option is activated by simply including its keyword in the appropriate keylist. The order of keywords in the principal $NBO keylist does not matter, but multiple keylists must be given in the order (1) $NBO, (2) $CORE, (3) $CHOOSE, (4) $DEL of presentation in Sections B.2-B.5. Key- words may be typed in upper or lower case, and will be echoed near the top of the NBO output. A $KEY list can be continued to any number of lines, but all the entries of a $KEY list must appear in a distinct set of lines, starting with the $KEY name on the first line and ending with the closing $END on the last line (i.e., no two $KEY lists should share parts of the same line). As the above example indicates, any line in the keylist input may terminate with an exclamation point (!) followed by `comment' of your choice; the ``!'' is considered to terminate the line, and the trailing `comment' is ignored by the program. #BB.2 THE $NBO KEYLIST#N #IB.2.1 Overview of $NBO keywords#N 0 The $NBO keylist is the principal means of specifying NBO job options and controlling output, and must precede any other keylist ($CORE, $CHOOSE, or $DEL) in your input file. The allowed keywords that can appear in a $NBO keylist are grouped as follows: #IJob Control Keywords:#N #IJob Threshold Keywords:#N #IMa- trix Output Keywords:#N #IOther Output Control Keywords:#N #IPrint Level Control:#N PRINT=n Keywords are first listed and described according to these formal groupings in Sections B.2.2-B.2.6. Section B.6 illustrates the effect of commonly used $NBO keywords (as well as other $KEY lists) on the successive stages of NAO/NBO/NLMO transformation and subsequent energy or dipole analysis, with sample output for these keyword options. 0 Some keywords of the $NBO keylist require (or allow) numerical values or other parameters to specify their exact function. In this case, the numerical value or parameter must immediately follow the keyword after an equal sign (=) or any number of blank spaces. Examples: #T E2PERT=2.5 LFNPR 16 NBOMO=W25 #N(The equal sign is recommended, and will be used in the remaining examples.) July 11, 1995 - 29 - [0 Although the general user's interaction with the NBO pro- grams is usually through the documented keywords of Sections B.2.2-B.2.6, some additional `semi-documented' keywords are listed in Section B.2.7 which may be of interest to the spe- cialist.] #IB.2.2 Job Control Keywords#N 0 The keywords in this group activate or deactivate basic tasks to be performed by the NBO programs, or change the way the NBO search is conducted. Each keyword is described in terms of the option it activates (together with an indica- tion of where the option is useful): #IOPTION DESCRIPTION#N Request Natural Population Analysis and printing of NPA sum- mary tables (Section A.3.2). This keyword also activates calculation of NAOs, except for semi-empirical ESS methods. Request calculation of NBOs and printing of the main NBO table (Section A.3.3). Request printing of the NBO summary table (Section A.3.6). This combines elements of the NBO table and 2nd-order per- turbation theory analysis table (see below) in a convenient form for recognizing the principal delocalization patterns. Request search for highly delocalized structures (Section B.6.6). The NBO search normally aborts when one or more Lewis NBOs has less than the default occupancy threshold of 1.90 electrons for a `good' electron pair. When the RESO- NANCE keyword is activated, this threshold is successively lowered in 0.10 decrements to 1.50, and the NBO search repeated to find the best Lewis structure within each occu- pancy threshold. The program returns with the best overall Lewis structure (lowest total non-Lewis occupancy) found in these searches. (Useful for benzene and other highly delo- calized molecules.) Request that no bonds (2-center NBOs) are to be formed in the NBO procedure (Section B.6.7). The resulting NBOs will then simply be 1-center atomic hybrids. (Useful for highly ionic species.) Request search for 3-center bonds (Section B.6.8). The nor- mal default is to search for only 1- and 2-center NBOs. (Useful for diborane and other electron-deficient `bridged' species.) Skip the computation of NBOs, i.e., only determine NAOs and perform natural population analysis. (Useful when only NPA is desired.) Compute and print out the summary table of Natural Localized Molecular Orbitals (Section B.6.2). NLMOs are similar to July 11, 1995 - 30 - Boys or Edmiston-Ruedenberg LMOs, but more efficiently calcu- lated. (Useful for `semi-localized' description of an SCF or correlated wavefunction.) Activated automatically by all keywords that pertain to NLMOs (e.g., AONLMO, SPNLMO, DIPOLE). Note that the SKIPBO keyword has higher precedence than other keywords in this list, so that keywords with which it is implicitly in conflict (e.g., NBO, 3CBOND, NLMO) will be ignored if SKIPBO is included in the $NBO keylist. #IB.2.3 Job Threshold Keywords#N 0 The keywords in this group also activate new tasks to be performed by the NBO program, but these keywords may be modified by one or more parameters (thresholds) that control the precise action to be taken. (In each case the keywords may also be used without parameters, accepting the default values [in brackets].) #IOPTION DESCRIPTION#N Request the NHO Directional Analysis table (Section A.3.4). The three parameters [and default values] have the following significance: = threshold angular deviation for printing = threshold percentage #Ip#N-character for printing = threshold NBO occupancy for printing #NParameter values may be separated by a space or a comma. Example:#T BEND=2,10,1.9 #NThis example specifies that the bond-bending table should only include entries for angular deviations of at least 2 egree (ang), hybrids of at least 10% #Ip#N-character (pct), and NBOs of occupancy at least 1.9 electrons (occ). Request the Perturbation Theory Energy Analysis table (Sec- tion A.3.5), where = threshold energy (in kcal/mol) for printing Entries will be printed for NBO donor-acceptor interaction energies that exceed the `eval' threshold. Example:#T E2PERT=5.0 #NThis example would print only interactions of at least 5 kcal/mol (i.e., only the single entry for the 8.13 kcal/mol #In#N#dN#u arr gma *#<#dCH#u interaction in the output of Section A.3.5). July 11, 1995 - 31 - Request the Molecular Dipole Moment Analysis table (Section B.6.3), where = threshold dipole moment (Debye) for printing The program will carry out a decomposition of the total molecular dipole moment in terms of localized NLMO and NBO contributions, including all terms whose contribution (in vector norm) exceeds the `dval' threshold. Example:#T DIPOLE=0.1 #NThis example would print out dipole contributions of all NBOs (and their delocalization interactions) of magnitude ge 0.1hsp D. #NBoth the BEND and E2PERT keywords are activated by default at the standard PRINT level option (see Section B.2.6); to get an example of dipole moment analysis, include the keyl- ist #T $NBO DIPOLE $END #Nin your input file. Note that the DIPOLE keyword leads to an analysis in terms of both NBOs and NLMOs, so that the NLMO keyword (Section B.2.2) is automatically activated in this case. #IB.2.4 Matrix Output Keywords#N 0 The keywords in this group activate the printing of vari- ous matrices to the output file, or their writing to (or reading from) external disk files. The large number of key- words in this group provide great flexibility in printing out the details of the successive transformations, eps1 eps2 or the matrices of various operators in the natural local- ized basis sets. This ordered sequence of transformations forms the basis for naming the keywords. #_Keyword Names#/ 0 The keyword for printing the matrix for a particular basis transformation, IN arr OUT, is constructed from the corresponding acronymns for the two sets in the generic form ``INOUT''. For example, the transformation AO arr NBO is keyed as ``AONBO'', while that from NBOs to NLMOs is correspondingly ``NBONLMO''. The transformations are always specified in the ordered sequence shown above (i.e., ``AONBO'' is allowed, but ``NBOAO'' is an unrecognized `backward' keyword). Keywords are recognized for #Iall possible#N transformations from the input AOs to other sets July 11, 1995 - 32 - (NAO, NHO, NBO, NLMO, MO, or the pre-orthogonal PNAO, PNHO, PNBO, PNLMO sets) in the overall sequence leading to canonical MOs, i.e., AONAO AONHO AONBO AONLMO AOMO AOPNAO AOPNHO AOPNBO AOPNLMO and from each of the orthonormal natural localized sets to sets lying to the right in the sequence, i.e., NAONHO NAONBO NAONLMO NAOMO NHONBO NHONLMO NHOMO NBONLMO NBOMO NLMOMO The matrix T#dIN,OUT#u for a specified IN arr OUT transform has rows labelled by the IN set and columns labelled by the OUT set. 0 One can also print out the matrix representations of the Fock matrix (F), density matrix (DM), or dipole moment matrix (DI) in the input AO set or any of the natural local- ized sets (NAO, NHO, NBO, or NLMO). The corresponding key- word is constructed by combining the abbreviation (M) for the operator with that for the set (SET) in the generic form ``MSET''. For example, to print the Fock matrix (F) in the NBO set, use the keyword ``FNBO'', or to print the dipole matrix in the NLMO basis, use ``DINLMO''. (For the dipole matrix keywords, all three vector components will be printed.) One can also print out elements of the overlap matrix (S) in the input AO basis or any of the `pre- orthogonal' sets (PNAO, PNHO, PNBO, or PNLMO), using, e.g., ``SPNAO'' for the overlap matrix in the PNAO basis. The complete set of allowed keywords for operator matrices is: FAO FNAO FNHO FNBO FNLMO DMAO DMNAO DMNHO DMNBO DMNLMO DIAO DINAO DINHO DINBO DINLMO SAO SPNAO SPNHO SPNBO SPNLMO Other desired transformations can be readily obtained from the keyword transformations by matrix multiplication. #_Keyword Parameters#/ 0 Each generic matrix keyword (``MATKEY'') can include a July 11, 1995 - 33 - parameter that specifies the output operation to be performed on the matrix. The allowed MATKEY parameters are of two types (three for AONAO, NAONBO; see below): (print out the matrix in the standard output file, 'c' columns) (write out the matrix to disk file #In#N) #NThe first (P[c]) parameter is used to control output to the standard output file. When the MATKEY keyword is inserted in the $NBO keylist with no parameters, the matrix is by default printed (in its entirety) in the standard out- put file. Thus, ``MATKEY=P'' would be equivalent to ``MAT- KEY'', with no parameters. The complete `P[c]' form of the print parameter serves to truncate the printed matrix output to a specified number of columns [c]. For example, to print out only the first 16 columns of a matrix, use the form #T MATKEY=P16 (print 16 columns) #NFor certain matrices, one can also restrict printing to only the valence (VAL) or Lewis (LEW) columns with modified `[c]' specifiers. For the transformations to MOs, use the form #T MATKEY=PVAL (print core + valence MO columns only) #Nwhere ``MATKEY'' is AOMO, NAOMO, NHOMO, NBOMO, or NLMOMO (only). This will print out only the occupied MOs and the lowest few unoccupied MOs, e.g., the six lowest virtual MOs of the methylamine example (Section A.3), though not neces- sarily those with pre#|dominant valence character. Simi- larly, for the transformations to NBOs or NLMOs, use the form #T MATKEY=PLEW (print Lewis orbital columns only) #Nwhere ``MATKEY'' is AONBO, NHONBO, NAONBO, AONLMO, NAONLMO, NHONLMO, NBONLMO (or AOMO, NAOMO, NHOMO, NBOMO, NLMOMO). This prints out the Lewis NBOs or occupied MOs only, e.g., only the nine occupied NBOs or MOs of the methy- lamine example. Judicious use of these print parameters keeps printed output within reasonable bounds in calcula- tions with large basis sets. #NThe second type of MATKEY parameter (W[n]) is used to write the matrix (in its entirety) to a specified disk file [n]. By default, each keyword transformation matrix is associated with a particular logical file number (LFN) in the range 31-49, as shown in the table below: July 11, 1995 - 34 - #NWhen the ``MATKEY=Wn'' keyword is inserted in the $NBO keylist with no `n' specifier, the matrix is by default written out (in its entirety) to this LFN. Thus, ``MATKEY=W'' is equivalent to ``MATKEY=Wn'' if ``n'' is the default LFN for that keyword. Use the ``Wn'' parameter to direct output to any non-default LFN disk file. For exam- ple, the keyword #T AONBO=W22 #Nwould write out the AO arr NBO transformation to LFN = 22 (rather than the default LFN = 37). 0 The format of the printed output under the print `P' parameter differs from that written to an external file under the `W' parameter. The `P' output (intended for a human reader) includes an identifying label for each row, and gives the numerical entries to somewhat lesser precision (F8.4 format) than the corresponding `W' output (F15.9 for- mat), which is usually intended as input to another program. Use the ``MATKEY=W6'' keyword to route the more precise `W' form of the matrix to the standard output file, LFN 6. 0 For the AONAO, NAONBO matrices (only), one can also include a read parameter (R), #T AONAO=Rn NAONBO=Rn #Nwhich causes the matrix to be input to the program from LFN #In#N. This parameter has the effect of `freezing' orbitals to a set prescribed in the input file (thus bypass- ing the NBO optimization of these orbitals for the molecular system). For example, the keyword ``NAONBO=R44'' would have the effect of freezing the NAO arr NBO transformation coef- ficients to the form specified in LFN 44 (perhaps written with the ``NAONBO=W44'' keyword in a previous calculation on isolated molecules, and now to be used in a calculation on a molecular complex). Similarly, the keyword ``AONAO=R45'' could be used to force the analysis of an excited state to be carried out in terms of the NAOs of the ground state (previously written out with the ``AONAO=W45'' keyword). #IB.2.5 Other Output Control Keywords#N 0 The keywords in this group also help to control the I/O produced by a specified set of job options, and thus supple- ment the keywords of the previous section. However, the keywords of this section `steer' the flow of information that is routinely produced by the NBO program (or can be passed through from the ESS program) without materially affecting the actual jobs performed by the NBO program. The options associated with each keyword are tabulated below: #IOPTION DESCRIPTION#N July 11, 1995 - 35 - Set the logical file number (LFN) for NBO program output. The default LFN is #In#N = 6, the usual LFN for output from the ESS program. This option can be used to steer the NBO section of the job output to a desired file. Example:#T LFNPR=25 (re-direct NBO output to LFN 25) #N Request additional details of the NBO search. This option (primarily for programming and debugging purposes) records details of the NBO loops over atoms and atom pairs, enroute to the final NBOs. Request print-out of the NAO-Wiberg Bond Index array and related valency indices (Section B.6.5). The elements of this array are the sums of squares of off-diagonal density matrix elements between pairs of atoms in the NAO basis, and are the NAO counterpart of the Wiberg bond index [K. Wiberg, Tetrahedron #B24#N, 1083-1096 (1968)]. (This bond index is routinely used to `screen' atom pairs for possible bonding in the NBO search, but the values are not printed unless the BNDIDX keyword is activated.) Request writing of information concerning the AO basis set (geometrical positions, orbital exponents, contraction coef- ficients, etc.) to an external file, LFN 31. This is a por- tion of the information needed by the ORB#|PLOT orbital con- tour plotting programs (cf. ``PLOT'' keyword below.) 1. Request writing of #Iall#N files required by orbital contour plotting programs ORB#|PLOT. This activates the AOINFO keyword, as well as all the necessary matrix output keywords (AONBO=W37, etc.) that could be required for ORBPLOT. Request writing the FILE47 `archive' file to external disk file LFN = #In#N (or, if ``=n'' is not present, to the default LFN = 47). This file can serve as the input file to run the GENNBO program in stand-alone mode, to repeat the NBO analysis (possibly with new job options) without repeat- ing the calculation of the wavefunction (Section B.7). Request writing the NBO direct access file (DAF) to external disk file LFN = #In#N (or, if ``=n'' is not present, to the default LFN =48). #IB.2.6 Print Level Keywords#N 0 The keyword ``PRINT=n'' (#In#N = 0-4) can be used to give convenient, flexible control of all NBO output in terms of a specified print level #In#N. This keyword activates groups of keywords in a heirarchical manner, and thus incrementally increases the volume of output, ranging from #Ino#N NBO out- put (PRINT=0) to a considerable volume of detail (PRINT=4). The keywords associated with each print level are tabulated below [default value, PRINT=2]: July 11, 1995 - 36 - For each print level #In#N, the NBO output will include items activated by the listed keywords, as well as all items from lower print levels. 0 When additional keywords are included with a ``PRINT=n'' keyword in the $NBO keylist, the NBO output includes the additional keyword items as well as those implied by the print level. This can be used to tailor the NBO output to virtually any selection of output items. For example, the keylist #T $NBO PRINT=2 NLMO FNBO=P NAOMO=P11 $END #Nwould add to the standard methylamine output file of Sec- tion A.3 an NLMO summary table, the Fock matrix in the NBO basis, and the transformation coefficients for the first 11 molecular orbitals in terms of NAOs. Similarly, to produce the NPA listing only, one could use #T $NBO PRINT=1 SKIPBO $END #Nor #T $NBO PRINT=0 NPA $END #N[There is actually a slight difference between the two examples: The NBOs are determined by default (once the $NBO keylist is encountered), even if all output is suppressed with PRINT=0; in the first example, the keyword SKIPBO bypasses NBO determination, whereas in the second example the NBOs are still determined `in background.'] #IB.2.7 Semi-Documented Additional Keywords#N 0 Some additional keywords are listed below that may of use to specialists or program developers: #IOPTION DESCRIPTION#N Set the threshold of orbital occupancy desired for bond orbital selection. If this is not included, the default occupancy [1.90] will be used (or values decreasing from 1.90 to 1.50 by 0.10 steps, if the RESONANCE keyword is included). Set the projection threshold [default 0.20] to determine if a `new' hybrid orbital has too high overlap with hybrids previously found. Print total gross Mulliken populations by atom. Print gross Mulliken populations, by orbital and atom. Revises PAO to PNAO transformation matrix by post- multiplying by #BT#N#dRyd#u and #BT#N#dred#u [see the NPA July 11, 1995 - 37 - paper: A. E. Reed, R. B. Weinstock, and F. Weinhold, J. Chem. Phys. #B83#N, 735-746 (1985)]. Input or output of pure AO (PAO) to pre-NAO (PNAO) transfor- mation. The PAOs are AOs of pure angular momentum symmetry (rather than cartesian gaussians). This keyword can be used with read (`R'), write (`W', default LFN 43) or print (`P') parameters. Print out the bond-order matrix (Fock-Dirac density matrix) in the basis set of input AOs. This keyword can be used with write (`W', default LFN 49) or print (`P') parameters. #BB.3 THE $CORE LIST#N 0 In the Lewis structure picture, the inner `core' electron pairs are pictured as occupying orbitals having essentially isolated atomic orbital character. In NBO parlance, these core orbitals correspond to 1-center unhybridized NAOs of near-maximum occupancy, which are isolated on each center before the main NBO search begins for localized valence electron pairs. A warning message is printed if the occu- pancy of a presumed closed-shell core NBO falls below 1.9990 electrons (or 0.9990 in the open-shell case), indicative of a possible core-valence mixing effect of physical signifi- cance. 0 [In previous versions of the NBO program, core orbitals having the expected pure atomic character are found in essentially all cases, except where an `accidental' degen- eracy in occupancy of core and valence lone pairs leads to undesirable core-valence mixing; the present version expli- citly isolates core pairs as unhybridized NAOs prior to the main NBO search to prevent this unphysical effect.] 0 The NBO program contains a table giving the nominal number of core orbitals to be isolated on each type of atom (e.g., 1#Is#N for first-row atoms Li-Ne, 1#Is#N, 2#Is#N, 2#Ip#N for second-row atoms Na-Ar, etc.). At times, however, it is interesting to examine the effect of allowing core orbitals to mix into the bonding hybrids, or to hybridize (polarize) among themselves. This can be accomplished by including a $CORE keylist to specify the number of core orbitals to be isolated on each atomic center, thus modifying the nominal core table. Unlike other NBO keylists, the $CORE list includes only integers (rather than keywords) to specify the core modifications, but the rules are otherwise similar to those for other keylists. The $CORE list (if included) must follow the $NBO keylist and precede the $CHOOSE or $DEL keylists. 0 The format of the $CORE modification list is: July 11, 1995 - 38 - The keyword ``$CORE'' Pairs of integers, one pair for each center. The first integer indicates the atomic center (in the numbering of the main ESS) and the second is the number of core orbitals to be isolated on that atom. Note that atomic centers not included in the CORE list are assigned default cores. The keyword ``$END'', to indicate the end of core input. The entire list may also be condensed to a single line, but the word ``$CORE'' must occur as the first word of the line and ``$END'' as the last word; that is, the core modifica- tion keylist cannot continue on a line that contains other keylist information. 0 The core orbitals are isolated by occupancy, the most occupied NAOs being first selected, and full subshells are isolated at a time. Thus, for example, to select the five orbitals of the #In#N = 1 and #In#N = 2 shells as core orbi- tals, it would make no difference to select ``3'' or ``4'' (instead of ``5''), since all three of these choices would specify a core containing a 1#Is#N, 2#Is#N, and all three 2#Ip#N orbitals. The $CORE modification list is read only once, and applies to both lpha and p n-shell calcula- tion. An example, appropriate for Ni(1)-C(2)-O(3) with the indi- cated numbering of atoms, is shown below: #T $CORE 1 5 $END #NThis would direct the NBO program to isolate only 5 core orbitals on Nickel (atom 1), rather than the nominal 9 core orbitals. In other words, only 1#Is#N, 2#Is#N, and 2#Ip#N orbitals will be considered as core orbitals in the search for NBOs of NiCO, allowing the 3#Is#N and 3#Ip#N orbitals to mix with valence NAOs in bond formation. Since the carbon and oxygen atoms were not included in the modification list, the nominal set of core orbitals (1#Is#N only) is isolated on each of these atoms. [The alternative example #T $CORE 1 0 2 0 3 0 $END #N(no cores) would allow all NAOs to be included in the NBO search; this would be equivalent to the default treatment in the earlier version of the program (see Section A.1.5).] #BB.4 THE $CHOOSE KEYLIST (DIRECTED NBO SEARCH)#N 0 A $CHOOSE keylist requests that the NBO search be directed July 11, 1995 - 39 - to find a particular Lewis structure (`resonance structure') chosen by the user. (This is useful for testing the accu- racy of alternative resonance structure representations of the wavefunction, relative to the optimal Lewis structure returned in a free NBO search.) In the $CHOOSE list, a resonance structure is specified by indicating where lone pairs and bonds (including multiple bonds) are to be found in the molecule. In some cases, the user may wish to specify only the location of bonds, letting the NBO algo- rithm seek the best location for lone pairs, but it is usu- ally safest to completely specify the resonance structure, both lone pairs and bonds. 0 The format of the $CHOOSE list is: The keyword ``$CHOOSE'' The keyword ``ALPHA'' (only for open-shell wavefunction) If one-center (`lone') NBOs are to be searched for, type the keyword ``LONE'' followed by a list of pairs of numbers, the first number of each pair being the atomic center and the second the number of valence lone pairs on that atom. Ter- minate the list with ``END''. (Note that only the occupied #Ivalence#N lone pairs should be entered, since the number of core orbitals on each center is presumed known.) If two-center (`bond') NBOs are to be searched for, type the keyword ``BOND'', followed by the list of bond specifiers, and terminated by ``END''. Each bond specifier is one of the letters single bond double bond triple bond quadruple bond followed by the two atomic centers of the bond (e.g., ``D 9 16'' for a double bond between atoms 9 and 16). If three-center NBOs are to be searched for, type the key- word ``3CBOND'', followed by the list of 3-c bond specif- iers, and terminated by ``END''. Each 3-c bond specifier is again one of the letters ``S'' (single), ``D'' (double), ``T'' (triple), or ``Q'' (quadruple), followed by three integers for the three atomic centers (e.g., ``S 4 8 10'' for a single three-center bond 4-8-10). (Note that the 3CBOND keyword of the $NBO keylist is implicitly activated if 3-c bonds are included in a $CHOOSE list.) The word ``END'' to signal the end of the lpha spin list. The keyword ``BETA'' (for open-shell wavefunctions) The input for m format as above. The overall $CHOOSE list should always end with the ``$END'' keyword. July 11, 1995 - 40 - Two examples will serve to illustrate the $CHOOSE format (each is rather artificial, inasmuch as the specified $CHOOSE structure corresponds to the `normal' structure that would be found by the NBO program): The closed-shell H-bonded complex FH ots CO, with atom numbering F(1)-H(2) ots C(3)-O(4), might be specified as #T $CHOOSE LONE 1 3 3 1 4 1 END BOND S 1 2 T 3 4 END $END #NThis would direct the NBO program to search for three lone pairs on atom F(1), one lone pair on atom C(3), one lone pair on atom O(4), one bond between F(1)-H(2), and three bonds between C(3)-O(4). The open-shell FH ots O#d2#u complex, with atom numbering F(1)-H(2)#+...#-O(3)-O(4), and with the unpaired electrons on O#d2#u being of lpha spin, might be specified as #T $CHOOSE ALPHA LONE 1 3 3 3 4 3 END BOND S 1 2 S 3 4 END END BETA LONE 1 3 3 1 4 1 END BOND S 1 2 T 3 4 END END $END #NNote that this example incorporates the idea of ``dif- ferent Lewis structures for different spins,'' with a dis- tinct pattern of localized 1-c (`lone') and 2-c (`bond') functions for lpha and . 0 As with other keylists, the $CHOOSE keylist can be condensed to a smaller number of lines, as long as no line is shared with another keylist. The order of keywords within the $CHOOSE keylist should be as shown above (i.e., ALPHA before BETA, LONE before BOND, etc.), but the order of entries within a LONE or BOND list is immaterial. A $CORE keylist (if present) must precede the $CHOOSE list. #BB.5 THE $DEL KEYLIST (NBO ENERGETIC ANALYSIS)#N July 11, 1995 - 41 - #IB.5.1 Introduction to NBO Energetic Analysis#N 0 The fourth and final type of keylist is a `deletions' ($DEL) keylist, to activate NBO energetic analysis. This analysis is performed by (1) deleting specified elements (or blocks of elements) from the NBO Fock matrix, (2) diagonal- izing this new Fock matrix to obtain a new density matrix, and (3) passing this density matrix to the SCF routines for a single pass through the SCF energy evaluator. The differ- ence between this `deletion' energy and the original SCF energy provides a useful measure of the energy contribution of the deleted terms. Since a Fock matrix is required, the energetic analysis is performed for RHF and UHF wavefunc- tions only. 0 Input for the NBO energetic analysis is through the $DEL keylist, which specifies the deletions to be performed. Multiple analyses (deletions) can be performed during a sin- gle job, with each deletion included in the overall $DEL keylist. The nine distinct types of deletions input are described in Section B.5.2 below. 0 The deletions keylist begins with the ``$DEL'' keyword. For the analysis of UHF wavefunctions, the deletions for the lpha and b separately specified (see Section B.5.3). Otherwise, the input for closed shells RHF and UHF is ident- ical. The input is free format and the input for a single deletion can be spread over as many lines as desired. The desired deletions should be listed one after the other. After the last deletion, the word ``$END'' signals the end of the keylist. |<<__________________________ #BWARNING#N If symmetry is used, one must be careful to only do dele- tions that will preserve the symmetry of the electronic wavefunction!! If this is not done, the energy of the dele- tion will be incorrect because the assumption is made in evaluating the energy that the original symmetry still exists, and the variational principle may be violated. (For example, if symmetry is used for ethane, is is permissible to do a ``NOSTAR'' deletion, but not the deletion of a sin- gle C-H antibond.) The remedy is not to use symmetry in the SCF calculation. 0 In describing the deletion types, use is made of the terms ``molecular unit'' and ``chemical fragment.'' The NBO pro- gram looks at the chemical bonding pattern produced by the bonding NBOs and identifies the groups of atoms that are linked together in distinct ``molecular units'' (usually July 11, 1995 - 42 - synonymous with ``molecules'' in the chemical sense). The first atom that is not in molecular unit 1 will be in molecular unit 2, and so forth. For example, if the list of atoms is C(1), H(2), F(3), O(4), and bonding NBOs are found between C(1)- O(4) and H(2)-F(3), then molecular unit 1 will be CO and molecular unit 2 will be HF. A ``chemical fragment'' is taken to be any subset of the atoms, usually (but not neces- sarily) in the same molecular unit, and usually (but not necessarily) connected by bond NBOs. Typically, a chemical fragment might be specified to be a single atom, the four atoms of a methyl group, or any other `radical' of a molecu- lar unit, identified by giving the atom numbers of which the fragment consists. #IB.5.2 The Nine Deletion Types#N 0 The keywords and format to specify each of the nine allowed deletion types are described below: #_(1) Deletion of entire orbitals.#/ This is called for by typing ``DELETE'', then the number of orbitals to be deleted, then the keyword ``ORBITAL'' (or ``ORBITALS''), then the list of the orbitals to be deleted. Example: #TDELETE 3 ORBITALS 15 18 29 #N[See also deletion types (4) and (7) for deleting sets of orbitals.] |<<__________________________ #BWARNING#N The ``single-pass'' method of evaluating deletion energies is appropriate only for deletions of #Ilow#N-occupancy (non-Lewis) orbitals, for which the loss of self-consistency in the Coulomb and exchange potentials (due to redistribu- tion of the electron density of deleted orbitals) is small compared to the net energy change of deletion. It is funda- mentally erroneous to delete #Ihigh#N-occupancy (Lewis) orbitals by this procedure. #_(2) Deletion of specific Fock matrix elements.#/ This is called for by typing ``DELETE'', then the number of elements to be deleted, then the keyword ``ELEMENT'' (or ``ELEMENTS''), then the list of the elements to be deleted (each as a pair of integers). Example: #TDELETE 3 ELEMENTS 1 15 3 19 23 2 July 11, 1995 - 43 - #NThis example would result in the zeroing of the following Fock matrix elements: (1,15), (15,1), (3,19), (19,3), (23,2), (2,23). [See also deletion types (3), (5), (6), (8), (9) for deleting sets of elements.] #_(3) Deletion of off-diagonal blocks of the Fock matrix.#/ Each block is specified by two sets of orbitals, and all Fock matrix elements in common between these two sets are set to zero. This is called for by typing ``ZERO'', then the number of off-diagonal blocks to be zeroed, and then, for each block, the following: (1) the dimensions of the block, separated by the word ``BY'' (e.g., ``6 BY 3'' if the first set has 6 orbitals and the second set has 3 orbitals); (2) the list of orbitals in the first set; (3) the list of orbitals in the second set. An example is shown below: #T ZERO 2 BLOCKS 2 BY 5 3 4 9 10 11 14 19 3 BY 2 1 2 7 20 24 #NThis will set the following Fock matrix elements to zero: (3,9), (3,10), (3,11), (3,14), (3,19), (9,3), (10,3), (11,3), (14,3), (19,3), (4,9), (4,10), (4,11), (4,14), (4,19), (9,4), (10,4), (11,4), (14,4), (19,4), (1,20), (1,24), (2,20), (2,24), (7,20), (7,24) (20,1), (24,1), (20,2), (24,2), (20,7), (24,7) [Usually, in studying the total delocalization from one molecular unit to another, it is much easier to use deletion type (8) below. Similarly, in studying the total delocali- zation from one chemical fragment to another, it is easier to use deletion type (9).] #_(4) Deletion of all Rydberg and antibond orbitals.#/ The Rydberg and antibond orbitals are the non-Lewis NBO orbitals that have stars in their labels (RY*, BD*) in the NBO analysis output. To delete all these orbitals, simply enter ``NOSTAR''. The result of this deletion is the energy of the idealized NBO natural Lewis structure, with all Lewis NBOs doubly occupied. (Unlike other deletions, in which July 11, 1995 - 44 - there is a slight loss of variational self-consistency due to the redistributed occupancy of the deleted orbitals, the result of a ``NOSTAR'' deletion corresponds rigorously to the vari- ational expectation value of the determinant of doubly occu- pied Lewis NBOs). #_(5) Deletion of all vicinal delocalizations.#/ To delete all Fock matrix elements between Lewis NBOs and the vicinal non-Lewis NBOs, simply enter ``NOVIC''. #_(6) Deletion of all geminal delocalizations.#/ To delete all Fock matrix elements between Lewis NBOs and the geminal non-Lewis NBOs, simply enter ``NOGEM''. #_(7) Deletion of all starred (antibond/Rydberg) orbitals on a particular molecular unit.#/ This is called for by typing ``DESTAR'', then the number of molecular units to be de#|starred, then the keyword ``UNIT'' (or ``UNITS''), then the list of units. Example: #TDESTAR 2 UNITS 3 4 #N #_(8) Zeroing all delocalization from one molecular unit to another.#/ This is called for by typing ``ZERO'', then the number of delocalizations to zero, then the keyword ``DELOCALIZATION'' (can be abbreviated to ``DELOC''), and then, for each delo- calization, the word ``FROM'', the number of the donor unit, the word ``TO'', and the number of the acceptor unit. Example: #TZERO 2 DELOC FROM 1 TO 2 FROM 2 TO 1 #NThe above example would zero #Iall#N intermolecular delo- calizations between units 1 and 2 (i.e., both 1 arr 2 and 2 arr 1). The effect is to remove all Fock matrix elements between high-occupancy (core/lone pair/bond) NBOs of the donor unit to the low-occupancy (antibond/Rydberg) NBOs of the acceptor unit. The donor and acceptor units may be the same. #_(9) Zeroing all delocalization from one chemical fragment to another.#/ This is called for by typing ``ZERO'', then the number of July 11, 1995 - 45 - inter-fragment delocalizations to be zeroed, then the words ``ATOM BLOCKS'', and then, for each delocalization, the following: (1) the number of atoms in the two fragments, separated by the word ``BY'' (e.g., ``6 BY 3'' if the first fragment has 6 atoms and the second has 3 atoms); (2) the list of atoms in the first fragment; (3) the list of atoms in the second fragment. For example, to zero all delocalizations between the frag- ments defined by atoms (1,2) and by atoms (3,4,5), the $DEL entries would be #T ZERO 2 ATOM BLOCKS 2 BY 3 1 2 3 4 5 3 BY 2 3 4 5 1 2 #NIn this example, the first block removes the (1,2) arr (3,4,5) delocalizations, while the second removes the (3,4,5) arr (1,2) delocalizations. 0 For additional examples of $DEL input, see Section B.6.10. #IB.5.3 Input for UHF Analysis#N 0 Deletions of the alpha and beta Fock matrices can be done independently. The deletions are input as above (Section B.5.2) for RHF closed shell, but they must be specified separately for alpha and beta in the UHF case. 0 The deletion to be done on the alpha Fock matrix must be preceded by the keyword ``ALPHA'', and the deletion of the beta Fock matrix must be preceded by the keyword ``BETA''. (Actually, only the first letter ``A'' or ``B'' is searched for by the program.) The ALPHA deletion must precede the BETA deletion. The BETA deletion may be the same as the ALPHA deletion, or different. 0 NOTE: The types of the lpha NBOs often differ from those of the d tions lists are generally required. For example, O#d2#u (triplet) has one bond in the lpha system and three in the t m, if the unpaired electrons are in the lpha sys- tem. 0 Here are three examples to illustrate UHF open-shell dele- tions: Example 1: July 11, 1995 - 46 - #T ALPHA ZERO 1 DELOC FROM 1 TO 2 BETA NOSTAR #NExample 2: #T ALPHA ZERO 1 DELOC FROM 1 TO 2 BETA ZERO 0 DELOC #NExample 3: #T ALPHA DELETE 0 ORBITALS BETA DELETE 1 ORBITAL 8 #NIf no deletion is done, this must be specified using zero (0) with one of the deletion input formats, as shown in Examples 2,3 above. #BB.6 NBO KEYLIST ILLUSTRATIONS#N #IB.6.1 Introduction#N 0 This section illustrates the output produced by several important keyword options of the NBO keylists ($NBO, $CHOOSE, $DEL, $CORE lists), supplementing the illustrations of Section A.3. Excerpts are provided rather than full out- put, since, e.g., NPA analysis is unaffected by keywords that modify the NBO search. Keywords of general applicabil- ity are illustrated with the methylamine example (RHF/3-21G, Pople-Gordon geometry) of Section A.3, which should be con- sulted for further information. More specialized keywords (RESONANCE, 3CBOND, etc.) are illustrated with prototype molecules (benzene, diborane, etc.) chosen for the keyword. 0 Sections B.6.2-B.6.8 illustrate representative examples from the $NBO keyword groups, including NLMO, DIPOLE, BNDIDX, RESONANCE, NOBOND, 3CBOND, and matrix output key- words. Section B.6.9 and B.6.10 similarly illustrate the use of the $CHOOSE and $DEL keylists. Section B.6.11 illus- trates the output for open-shell UHF cases, emphasizing features associated with the ``different Lewis structures for different spins'' representation of lpha and S ction B.6.12 shows the effect of using effective core potentials for Cu#d2#u, also illustrating aspects of the inclusion of #Id#N functions. #IB.6.2 NLMO Keyword#N 0 When the NLMO keyword is activated, the program computes the NLMOs and prints out three tables summarizing their form. For the RHF/3-21G methylamine example (cf. Section A.3), the principal NLMO table is shown below: NATURAL LOCALIZED MOLECULAR ORBITAL (NLMO) ANALYSIS: Maximum off-diagonal element of DM in NLMO basis: 0.00000 Hybridization/Polarization Analysis of NLMOs in NAO Basis: July 11, 1995 - 47 - NLMO/Occupancy/Percent from Parent NBO/ Atomic Hybrid Contributions ---------- --------------------------------------------------------------------- 1. (2.00000) 99.9290% BD ( 1) C 1- N 2 40.039% C 1 s( 21.54%)p 3.64( 78.46%) 59.891% N 2 s( 30.98%)p 2.23( 69.02%) 0.015% H 3 s(100.00%) 0.021% H 6 s(100.00%) 0.021% H 7 s(100.00%) 2. (2.00000) 99.9301% BD ( 1) C 1- H 3 59.675% C 1 s( 25.44%)p 2.93( 74.56%) 0.040% N 2 s( 1.99%)p49.22( 98.01%) 40.258% H 3 s(100.00%) 3. (2.00000) 99.6996% BD ( 1) C 1- H 4 60.848% C 1 s( 25.25%)p 2.96( 74.75%) 0.093% N 2 s( 13.08%)p 6.65( 86.92%) 0.014% H 3 s(100.00%) 38.861% H 4 s(100.00%) 0.017% H 5 s(100.00%) 0.158% H 6 s(100.00%) 4. (2.00000) 99.6996% BD ( 1) C 1- H 5 60.848% C 1 s( 25.25%)p 2.96( 74.75%) 0.093% N 2 s( 13.08%)p 6.65( 86.92%) 0.014% H 3 s(100.00%) 0.017% H 4 s(100.00%) 38.861% H 5 s(100.00%) 0.158% H 7 s(100.00%) 5. (2.00000) 99.7206% BD ( 1) N 2- H 6 0.113% C 1 s( 5.15%)p18.41( 94.85%) 67.929% N 2 s( 25.82%)p 2.87( 74.18%) 0.137% H 4 s(100.00%) 0.014% H 5 s(100.00%) 31.793% H 6 s(100.00%) 6. (2.00000) 99.7206% BD ( 1) N 2- H 7 0.113% C 1 s( 5.15%)p18.41( 94.85%) 67.929% N 2 s( 25.82%)p 2.87( 74.18%) 0.014% H 4 s(100.00%) 0.137% H 5 s(100.00%) 31.793% H 7 s(100.00%) 7. (2.00000) 99.9499% CR ( 1) C 1 99.951% C 1 s(100.00%)p 0.00( 0.00%) July 11, 1995 - 48 - 0.013% H 3 s(100.00%) 0.013% H 4 s(100.00%) 0.013% H 5 s(100.00%) 8. (2.00000) 99.9763% CR ( 1) N 2 0.010% C 1 s( 22.30%)p 3.48( 77.70%) 99.980% N 2 s(100.00%)p 0.00( 0.00%) 9. (2.00000) 98.8972% LP ( 1) N 2 0.440% C 1 s( 1.05%)p94.15( 98.95%) 98.897% N 2 s( 17.85%)p 4.60( 82.15%) 0.489% H 3 s(100.00%) 0.085% H 4 s(100.00%) 0.085% H 5 s(100.00%) #T @seg #NFor each of the nine occuplied NLMOs, the table shows first the NLMO occupancy (necessarily 2.0000 at SCF level, as in the present example), the percentage of the total NLMO composition represented by this parent NBO (usually > 99%), and the label of the `parent' NBO. Below this, there fol- lows an NAO decomposition of the NLMO, showing the percen- tage of the NLMO on each atom and the hybrid composition ratios (effective #Isp#N#u #d character and percentage #Is- #N and #Ip#N-character) of the NAOs. For example, NLMO 9 is the most delocalized NLMO of the table, having only about a 98.9% contribution from the localized N(2) parent lone pair NBO, with `delocalization tails' composed primarily of con- tributions (~0.4% each) from C(1) and H(3), and smaller con- tributions (~0.09%) from H(4) and H(5). This corresponds to what might have been anticipated from the NBO summary table (Section A.3.6) or perturbation theory energy analysis table (Section A.3.5), which showed that the N(2) lone pair, NBO 9, is principally delocalized onto NBO 24, the vicinal C(1)-H(3) antibond [with lesser delocalizations onto NBOs 25, 26, the C(1)-H(4) and C(1)-H(5) antibonds]. #IB.6.3 DIPOLE Keyword#N 0 The DIPOLE keyword activates the NBO/NLMO analysis of the molecular dipole moment, as shown below for the example of RHF/3-21G methylamine (cf. Section A.3): Dipole moment analysis: [Print threshold: Net dipole > 0.02 Debye] NLMO bond dipole NBO bond dipole July 11, 1995 - 49 - ------------------------- ------------------------ Orbital x y z Total x y z Total =============================================================================== 1. BD ( 1) C 1- N 2 -0.76 -0.08 0.00 0.76 -0.76 -0.09 0.00 0.77 2. BD ( 1) C 1- H 3 0.49 1.90 0.00 1.96 0.50 1.90 0.00 1.97 deloc 14: 0.03 -0.01 0.00 0.03 deloc 25: -0.01 0.00 0.02 0.02 deloc 26: -0.01 0.00 -0.02 0.02 3. BD ( 1) C 1- H 4 0.67 -0.77 -1.50 1.81 0.71 -0.79 -1.50 1.84 deloc 27: -0.05 0.00 0.00 0.05 deloc 26: -0.02 0.03 -0.03 0.04 deloc 24: -0.01 -0.02 0.00 0.02 4. BD ( 1) C 1- H 5 0.67 -0.77 1.50 1.81 0.71 -0.79 1.50 1.84 deloc 28: -0.05 0.00 0.00 0.05 deloc 25: -0.02 0.03 0.03 0.04 deloc 24: -0.01 -0.02 0.00 0.02 5. BD ( 1) N 2- H 6 -0.45 0.44 0.86 1.06 -0.50 0.44 0.89 1.11 deloc 25: 0.06 -0.01 -0.02 0.06 6. BD ( 1) N 2- H 7 -0.45 0.44 -0.86 1.06 -0.50 0.44 -0.89 1.11 deloc 26: 0.06 -0.01 0.02 0.06 7. CR ( 1) C 1 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 8. CR ( 1) N 2 0.00 0.01 0.00 0.01 0.00 0.00 0.00 0.00 9. LP ( 1) N 2 -0.63 -2.85 0.00 2.91 -0.88 -2.93 0.00 3.06 deloc 24: 0.16 0.09 0.00 0.18 July 11, 1995 - 50 - deloc 25: 0.03 0.01 0.01 0.03 deloc 26: 0.03 0.01 -0.01 0.03 deloc 10: 0.02 -0.02 0.00 0.03 ---------- ------------------------------------------ Net dipole moment -0.45 -1.67 0.00 1.73 -0.71 -1.82 0.00 1.95 Delocalization correction 0.27 0.14 0.00 0.30 ---------- ------------------------------------------ Total dipole moment -0.45 -1.67 0.00 1.73 -0.45 -1.67 0.00 1.73 #T @seg #NThe bottom line of the table shows the individual (x,y,z) vector components (minus 0.45,minus 1.67,0.00) and length (1.73 D) of the total molecular dipole moment, in the coor- dinate system of the ESS program. This is decomposed in the main body of the table into the individual contributions of ``NLMO bond dipoles'' (which strictly add to give the net molecule dipole at the SCF level) and ``NBO bond dipoles'' (which must be added with their off-diagonal `deloc' contri- butions to give the net molecular moment). Each NLMO or NBO bond dipole vector t mu #dAB #u is evaluated as mab = mabe + mabn where mabe = 2#Ie#N a t r#Nar b t is the electronic dipole expectation value for an electron pair in the b NLMO or NBO, and mabn is the nuclear contribution of compensating unit positive charges at the positions of nuclei A and B (or both on A for a 1-center NBO). The `deloc' contributions below each NBO bond dipole show the off-diagonal corrections to an additive bond dipole approximation (i.e., the correc- tions to localized NBO bond dipoles to get the NLMO bond dipoles) to account for the delocalization from parent NBO #Ii#N onto other (primarily, non-Lewis) NBOs #Ij#N; in terms of the expansion of an NLMO in the set { } of NBOs, nlmo = this correction is (for each electron, lpha or ) .LM+10 + s <> where the primes on the summation denote omission of terms #Ik#N equal to #Ii#N or #Ij#N. For example, in the above table the largest individual contribution to t mu is from the nitrogen lone pair, table entry 9, which has an NLMO dipole of 2.91 Debye or NBO dipole of 3.06. The latter July 11, 1995 - 51 - has also the largest off-diagonal delocalization correction in the table, a 0.18 D correction due to the #In#N#dN#u arr gma *#<#dCH#u delocalization into the vicinal C(1)-H(3) antibond, NBO 24. 0 For a post-SCF (correlated) calculation, the dipole table would also include an additional line for the correction due to non-additivity of the NLMO bond dipoles. For an ionic species, there would also be an additional line for the ``residual nuclear charge'' contribution; here, one must be aware that the dipole moment is calculated with respect to the origin of the cartesian coordinate system chosen by the ESS program (since the dipole moment is origin-dependent in this case). 0 Note that the amount of detail in the dipole table can be altered by using the ``DIPOLE=thr'' form of the keyword to alter the threshold dipole (`thr') for printing [default: 0.02 D]. #IB.6.4 Matrix Output Keywords#N 0 Two simple examples will be given to illustrate the for- matting of output for operators or basis set transformation matrices using the matrix output keywords of Section B.2.4. For the RHF/3-21G methylamine example of Section A.3, the keyword ``FNHO'' would cause the Fock matrix in the NHO basis to be printed out. Shown below is a reproduction of the first eight columns (out of 28) of this output: NHO Fock matrix: NHO 1 2 3 4 5 6 7 8 ---------- ------- ------- ------- ------- ------- --- ---- ------- ------- 1. C1 ( N2 ) -0.0208 -0.7203 -0.0571 -0.0665 0.0438 0.0672 0.0438 0.0672 2. N2 ( C1 ) -0.7203 -0.3083 -0.0773 -0.0627 0.0835 0.0646 0.0835 0.0646 3. C1 ( H3 ) -0.0571 -0.0773 -0.1394 -0.6758 0.0638 0.0746 0.0638 0.0746 4. H3 ( C1 ) -0.0665 -0.0627 -0.6758 0.1349 0.0740 0.0672 0.0740 0.0672 5. C1 ( H4 ) 0.0438 0.0835 0.0638 0.0740 -0.1466 -0.6761 -0.0548 -0.0759 6. H4 ( C1 ) 0.0672 0.0646 0.0746 0.0672 -0.6761 0.1541 -0.0759 -0.0697 7. C1 ( H5 ) 0.0438 0.0835 0.0638 0.0740 -0.0548 -0.0759 -0.1466 -0.6761 8. H5 ( C1 ) 0.0672 0.0646 0.0746 0.0672 -0.0759 -0.0697 -0.6761 0.1541 9. N2 ( H6 ) 0.0926 0.1499 0.0240 -0.0113 0.0912 -0.0078 -0.0349 0.0134 10. H6 ( N2 ) 0.1083 0.0826 -0.0010 0.0232 -0.0118 -0.0242 0.0017 -0.0224 July 11, 1995 - 52 - 11. N2 ( H7 ) 0.0926 0.1499 0.0240 -0.0113 -0.0349 0.0134 0.0912 -0.0078 12. H7 ( N2 ) 0.1083 0.0826 -0.0010 0.0232 0.0017 -0.0224 -0.0118 -0.0242 13. C1 (cr) 0.3962 0.4168 0.4400 0.3893 -0.4447 -0.3869 -0.4447 -0.3869 14. N2 (cr) 0.6147 0.7083 0.0039 0.0249 -0.0130 -0.0251 -0.0130 -0.0251 15. N2 (lp) 0.0762 0.0955 -0.1043 0.0254 -0.0386 0.0160 -0.0386 0.0160 16. C1 (ry*) -0.1320 0.0924 0.0705 -0.0815 0.0022 -0.0037 0.0022 -0.0037 17. C1 (ry*) 0.0000 0.0000 0.0000 0.0000 0.0719 -0.0910 -0.0719 0.0910 18. C1 (ry*) -0.1023 0.0764 -0.0643 0.0795 -0.0074 0.0105 -0.0074 0.0105 19. C1 (ry*) 0.0266 -0.0213 0.0019 -0.0057 0.0667 -0.0788 0.0667 -0.0788 20. N2 (ry*) 0.0151 -0.0177 -0.0351 -0.0172 -0.0179 -0.0146 -0.0179 -0.0146 21. N2 (ry*) 0.0000 0.0000 0.0000 0.0000 -0.0158 -0.0249 0.0158 0.0249 22. N2 (ry*) 0.1799 -0.1440 -0.0064 0.0295 0.0038 -0.0289 0.0038 -0.0289 23. N2 (ry*) 0.0183 -0.0136 -0.0051 0.0213 0.0032 -0.0095 0.0032 -0.0095 24. H3 (ry*) 0.0253 -0.0038 0.2834 -0.3497 -0.0248 0.0047 -0.0248 0.0047 25. H4 (ry*) 0.0223 -0.0071 0.0211 -0.0068 -0.2789 0.3553 -0.0227 0.0069 26. H5 (ry*) 0.0223 -0.0071 0.0211 -0.0068 -0.0227 0.0069 -0.2789 0.3553 27. H6 (ry*) 0.0124 0.0172 -0.0067 0.0219 -0.0080 0.0097 0.0057 -0.0222 28. H7 (ry*) 0.0124 0.0172 -0.0067 0.0219 0.0057 -0.0222 -0.0080 0.0097 #T @seg #N 0 The NHO labels on each row identify the atom to which the NHO belongs, and (in parentheses) the atom toward which the hybrid is pointed, if a bond hybrid, or a 1-center label (cr, lp, lp*, or ry*), if a non-bonded hybrid. Thus, ``C 1 (N 2)'' (NHO 1) is the bonding hybrid on C(1) directed toward N(2), ``N 2(lp)'' (NBO 15) is a non-bonded (lone pair) hybrid on N(2), etc. This label allows one to find the precise form of the NHO in the main listing of NBOs. The FNHO matrix shows, for example, that the (1,2) Fock matrix element between the directly interacting NHOs forming the C-N bond NBO is -0.7203 a.u., whereas the (1,9) matrix July 11, 1995 - 53 - element, between the C(1) hybrid pointing toward N(2) and the N(2) hybrid pointing toward H(6), is 0.0926 a.u. 0 As a second example, the keyword ``NBOMO=PVAL'' would print out the core + valence columns of the NBO arr MO transformation, as reproduced below: MOs in the NBO basis: NBO 1 2 3 4 5 6 7 8 ---------- ------- ------- ------- ------- ------- --- ---- ------- ------- 1. C1 - N2 -0.0661 -0.0574 0.6288 -0.1243 0.0000 -0.3239 0.6816 0.0000 2. C1 - H3 -0.0018 -0.0578 0.2061 -0.4716 0.0000 0.7747 0.1386 0.0000 3. C1 - H4 0.0023 0.0579 -0.1836 0.4908 0.3813 0.2304 0.3921 0.5940 4. C1 - H5 0.0023 0.0579 -0.1836 0.4908 -0.3813 0.2304 0.3921 -0.5940 5. N2 - H6 0.0570 0.0000 -0.4742 -0.3567 -0.5937 -0.1954 0.3035 0.3814 6. N2 - H7 0.0570 0.0000 -0.4742 -0.3567 0.5937 -0.1954 0.3035 -0.3814 7. C1 (cr) -0.0021 0.9931 0.0692 -0.0920 0.0000 0.0006 0.0019 0.0000 8. N2 (cr) 0.9935 -0.0019 0.1048 0.0348 0.0000 -0.0131 0.0022 0.0000 9. N2 (lp) 0.0432 -0.0037 -0.1676 -0.1219 0.0000 0.3312 0.1525 0.0000 10. C1 (ry*) -0.0088 -0.0005 0.0114 0.0089 0.0000 -0.0016 -0.0086 0.0000 11. C1 (ry*) 0.0000 0.0000 0.0000 0.0000 0.0109 0.0000 0.0000 -0.0070 12. C1 (ry*) -0.0063 0.0001 -0.0050 -0.0035 0.0000 -0.0030 0.0026 0.0000 13. C1 (ry*) 0.0020 -0.0002 -0.0003 -0.0003 0.0000 -0.0009 0.0002 0.0000 14. N2 (ry*) -0.0041 -0.0003 -0.0006 0.0016 0.0000 0.0192 0.0107 0.0000 15. N2 (ry*) 0.0000 0.0000 0.0000 0.0000 0.0080 0.0000 0.0000 0.0124 16. N2 (ry*) 0.0035 -0.0060 -0.0039 0.0102 0.0000 -0.0023 0.0040 0.0000 17. N2 (ry*) -0.0018 0.0023 -0.0007 0.0013 0.0000 -0.0007 0.0005 0.0000 18. H3 (ry*) -0.0008 -0.0094 -0.0103 0.0146 0.0000 0.0017 -0.0021 0.0000 19. H4 (ry*) -0.0008 -0.0100 -0.0061 0.0119 0.0062 0.0004 -0.0054 -0.0098 20. H5 (ry*) -0.0008 -0.0100 -0.0061 0.0119 -0.0062 0.0004 -0.0054 0.0098 21. H6 (ry*) -0.0052 -0.0013 -0.0147 -0.0018 -0.0027 July 11, 1995 - 54 - -0.0016 -0.0097 -0.0159 22. H7 (ry*) -0.0052 -0.0013 -0.0147 -0.0018 0.0027 -0.0016 -0.0097 0.0159 23. C1 - N2 * -0.0019 -0.0035 -0.0026 0.0025 0.0000 0.0043 0.0049 0.0000 24. C1 - H3 * -0.0013 -0.0024 0.0059 -0.0018 0.0000 -0.0349 -0.0139 0.0000 25. C1 - H4 * 0.0009 0.0028 -0.0138 0.0033 -0.0408 -0.0188 0.0061 0.0148 26. C1 - H5 * 0.0009 0.0028 -0.0138 0.0033 0.0408 -0.0188 0.0061 -0.0148 27. N2 - H6 * -0.0010 0.0051 -0.0047 0.0182 0.0179 0.0122 0.0154 0.0322 28. N2 - H7 * -0.0010 0.0051 -0.0047 0.0182 -0.0179 0.0122 0.0154 -0.0322 NBO 9 10 11 12 13 14 15 ---------- ------- ------- ------- ------- ------- --- ---- ------- 1. C1 - N2 0.1062 -0.0143 0.0006 0.0000 0.0049 0.0000 -0.0061 2. C1 - H3 -0.3343 -0.0044 0.0015 0.0000 0.0007 0.0000 -0.0080 3. C1 - H4 -0.1186 -0.0186 0.0103 0.0258 -0.0048 -0.0272 -0.0104 4. C1 - H5 -0.1186 -0.0186 0.0103 -0.0258 -0.0048 0.0272 -0.0104 5. N2 - H6 -0.1167 -0.0024 -0.0145 -0.0293 -0.0162 -0.0253 0.0040 6. N2 - H7 -0.1167 -0.0024 -0.0145 0.0293 -0.0162 0.0253 0.0040 7. C1 (cr) 0.0037 -0.0134 -0.0082 0.0000 0.0008 0.0000 -0.0008 8. N2 (cr) -0.0189 -0.0055 0.0030 0.0000 -0.0026 0.0000 0.0035 9. N2 (lp) 0.9007 -0.0144 0.0055 0.0000 0.0925 0.0000 0.0130 10. C1 (ry*) -0.0128 -0.0993 0.0553 0.0000 0.0536 0.0000 0.3301 11. C1 (ry*) 0.0000 0.0000 0.0000 0.0836 0.0000 0.1845 0.0000 12. C1 (ry*) -0.0039 -0.0612 0.0748 0.0000 -0.1160 0.0000 0.1213 13. C1 (ry*) -0.0018 0.0936 0.0192 0.0000 0.1022 0.0000 -0.1516 14. N2 (ry*) -0.0086 -0.0232 0.0071 0.0000 -0.0461 0.0000 -0.0178 15. N2 (ry*) 0.0000 0.0000 0.0000 0.0176 0.0000 -0.0856 0.0000 16. N2 (ry*) 0.0006 0.0395 -0.0836 0.0000 0.0221 0.0000 -0.1565 17. N2 (ry*) 0.0003 0.0614 -0.0222 0.0000 0.0114 July 11, 1995 - 55 - 0.0000 0.0584 18. H3 (ry*) -0.0218 -0.2483 -0.2232 0.0000 0.4827 0.0000 0.0001 19. H4 (ry*) 0.0060 -0.1973 -0.3224 -0.3372 -0.2069 -0.2151 -0.0483 20. H5 (ry*) 0.0060 -0.1973 -0.3224 0.3372 -0.2069 0.2151 -0.0483 21. H6 (ry*) 0.0027 -0.2869 0.2132 0.2297 -0.0372 -0.3543 -0.1737 22. H7 (ry*) 0.0027 -0.2869 0.2132 -0.2297 -0.0372 0.3543 -0.1737 23. C1 - N2 * -0.0031 -0.2357 0.2598 0.0000 -0.1096 0.0000 0.8051 24. C1 - H3 * -0.0799 -0.3214 -0.2654 0.0000 0.6687 0.0000 0.1133 25. C1 - H4 * -0.0369 0.2559 0.3890 0.4699 0.2968 0.3193 -0.0477 26. C1 - H5 * -0.0369 0.2559 0.3890 -0.4699 0.2968 -0.3193 -0.0477 27. N2 - H6 * -0.0031 0.4339 -0.3112 -0.3280 0.0474 0.4519 0.2168 28. N2 - H7 * -0.0031 0.4339 -0.3112 0.3280 0.0474 -0.4519 0.2168 #T @seg #N 0 In this transformation matrix, rows correspond to NBOs and columns to MOs (in the ordering used elesewhere in the pro- gram), and each basis NBO is further identified with a row label. The print parameter ``PVAL'' specified that only 15 MOs (the number of core + valence orbitals) should be printed, corresponding to the nine occupied MOs 1-9 and the lowest six virtual MOs 10-15. The matrix allows one to see the composition of each canonical MO in terms of localized bond NBOs. For example, MOs 5 and 8 can be approximately described as hi #d5#u ~= minus 0.594n + 0.381c hi #d8#u ~= 0.381n + 0.594c whereas hi #d6#u is primarily the C-H(3) NBO and hi #d9#u the N lone pair NBO. #IB.6.5 BNDIDX Keyword#N 0 The BNDIDX keyword activates the printing of several types of `bond order' and valency indices, based on different assumptions and formulas, but all having some connection to the NAO/NBO/NLMO formalism. We illustrate these bond order July 11, 1995 - 56 - tables for the example of RHF/3-21G methylamine (Section A.3). 0 The first segment of BNDIDX output shows the Wiberg bond index (the sum of squares of off-diagonal density matrix elements between atoms), as formulated in terms of the NAO basis set: Wiberg bond index matrix in the NAO basis: Atom 1 2 3 4 5 6 7 ---- ------ ------ ------ ------ ------ ------ -- ---- 1. C 0.0000 0.9964 0.9472 0.9394 0.9394 0.0020 0.0020 2. N 0.9964 0.0000 0.0208 0.0052 0.0052 0.8611 0.8611 3. H 0.9472 0.0208 0.0000 0.0004 0.0004 0.0002 0.0002 4. H 0.9394 0.0052 0.0004 0.0000 0.0009 0.0079 0.0005 5. H 0.9394 0.0052 0.0004 0.0009 0.0000 0.0005 0.0079 6. H 0.0020 0.8611 0.0002 0.0079 0.0005 0.0000 0.0003 7. H 0.0020 0.8611 0.0002 0.0005 0.0079 0.0003 0.0000 Wiberg bond index, Totals by atom: Atom 1 ---- ------ 1. C 3.8265 2. N 2.7499 3. H 0.9691 4. H 0.9544 5. H 0.9544 6. H 0.8720 7. H 0.8720 #T @seg #N 0 This index is intrinsically a positive quantity, making no distinction between net bonding or antibonding character of the density matrix elements. 0 The next segment tabulates the ``overlap-weighted NAO bond order,'' as shown below: July 11, 1995 - 57 - Atom-atom overlap-weighted NAO bond order: Atom 1 2 3 4 5 6 7 ---- ------ ------ ------ ------ ------ ------ -- ---- 1. C 0.0000 0.7815 0.7614 0.7633 0.7633 -0.0103 -0.0103 2. N 0.7815 0.0000 -0.0225 -0.0097 -0.0097 0.6688 0.6688 3. H 0.7614 -0.0225 0.0000 -0.0039 -0.0039 -0.0019 -0.0019 4. H 0.7633 -0.0097 -0.0039 0.0000 0.0024 0.0038 -0.0032 5. H 0.7633 -0.0097 -0.0039 0.0024 0.0000 -0.0032 0.0038 6. H -0.0103 0.6688 -0.0019 0.0038 -0.0032 0.0000 -0.0069 7. H -0.0103 0.6688 -0.0019 -0.0032 0.0038 -0.0069 0.0000 Atom-atom overlap-weighted NAO bond order, Totals by atom: Atom 1 ---- ------ 1. C 3.0488 2. N 2.0772 3. H 0.7273 4. H 0.7527 5. H 0.7527 6. H 0.6503 7. H 0.6503 #T @seg #N 0 This index corresponds to a sum of off-diagonal NAO den- sity matrix elements between atoms, each multiplied by the corresponding PNAO overlap integral. 0 Another type of BNDIDX output appears if the NLMO keyword is included, summarizing a formal ``NLMO/NPA bond order'' that can be associated with each NLMO: Individual LMO bond orders greater than 0.002 in magnitude, with the overlap between the hybrids in the NLMO given: Atom I / Atom J / NLMO / Bond Order / Hybrid Overlap / 1 2 1 0.8007741 0.7314361 1 2 5 0.0022694 0.1796696 1 2 6 0.0022694 0.1796696 July 11, 1995 - 58 - 1 2 9 0.0088061 0.3053730 1 3 2 0.8051647 0.7862263 1 3 9 -0.0088061 -0.5762575 1 4 3 0.7772179 0.7874312 1 4 5 -0.0022694 -0.5396947 1 5 4 0.7772179 0.7874312 1 5 6 -0.0022694 -0.5396947 1 6 3 -0.0031652 -0.0920524 1 6 5 0.0022694 0.0852070 1 7 4 -0.0031652 -0.0920524 1 7 6 0.0022694 0.0852070 2 3 9 -0.0097841 -0.0930204 2 4 5 -0.0027437 -0.0701717 2 5 6 -0.0027437 -0.0701717 2 6 5 0.6358512 0.7286061 2 7 6 0.6358512 0.7286061 4 6 3 0.0031652 0.0429202 4 6 5 0.0027437 0.0399352 5 7 4 0.0031652 0.0429202 5 7 6 0.0027437 0.0399352 #T @seg #NThis NLMO bond order is calculated by the method described by A. E. Reed and P. v.R. Schleyer [#IInorg. Chem. #B27#N, 3969-3987 (1988); #IJ. Am. Chem. Soc.#N (to be published)], based on the shared occupancies and hybrid overlaps (last column) of NAOs composing the NLMO. In the above table, for example, NLMO 1 occurs only in the first line, contributing a bond of formal order 0.801 between C(1) and N(2), whereas NLMO 9 (the nitrogen lone pair) contributes a slight strengthening (+0.0088) of the C(1)-N(2) bond, a weakening (-0.0088) of the vicinal C(1)-H(3) bond, and a slight nega- t