** Present address: Department of Chemistry, University of California-Irvine, Irvine, California 92717.

PREFACE: HOW TO USE THIS MANUAL

The NBO manual is divided into three major sections:

Section A ("General Introduction and Installation") contains general introductory and 'one-time' information for the novice user: what the program does, program structure and relationship to driver electronic structure package, initial installation, 'quick start' sample input data, and a brief tutorial on sample output.

Section B ("NBO User's Guide") is for the intermediate user who has an installed program and general familiarity with the standard (default) options of the NBO program. This section documents the list of keywords 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 program to its full potential, to 'turn off' or 'turn on' various NBO options for their specialized applications.

Section C ("NBO Programmer's Guide") is for accomplished programmers who are interested in program logic and the detailed layout of the source code. This section describes the relationship of the source code subprograms to the published algorithms for NAO, NBO, and NLMO determination, providing documentation at the level of individual common blocks, functions, and subroutines. 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.

A.1 INTRODUCTION TO THE NBO PROGRAM

A.1.1 What Does the NBO Program Do?

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 properties. 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 and spin.* This section provides a brief introduction to NBO algorithms and nomenclature.

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 matrix, and are hence "natural" in the sense of Löwdin, 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 departures from this idealized localized picture.

The various natural localized sets can be considered to result from a sequence of transformations of the input atomic orbital basis set {_i},** _______________

*Note, however, that some electronic structure packages do not make provision for calculating the spin density matrices for some types of open-shell wavefunctions (e.g., MCSCF wavefunctions calculated by the GUGA formalism in the GAMESS system), so that NBO analysis cannot be applied in these cases.

**If 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 occupied natural orbitals of each atomic wavefunction to compose a final set of linearly independent atom-centered basis functions of the required dimensionality. Since atom-centered basis functions are the nearly universal choice for molecular calculations, the current NBO program makes no provision for this step.

input basis NAOs NHOs NBOs NLMOs

Each natural localized set forms a complete orthonormal set of one-electron functions for expanding the delocalized 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 orbital interactions in the usual way.

The optimal condensation of occupancy in the natural localized orbitals leads to partitioning into high- and low-occupancy orbital types (reduction in dimensionality of the orbitals having significant occupancy), as reflected in the orbital labelling. The small set of most highly-occupied NAOs, having a close correspondence 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 h_A, h_B in the NHO basis give rise to a bond (_AB) and antibond (*_AB) in the NBO basis,

_AB = c_Ah_A + c_Bh_B

*_AB = c_Bh_A - c_Ah_B

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 effects are based on second-order perturbation theory, or on the effect of deleting certain orbitals or matrix elements and recalculating the total energy. NBO energy analysis is dependent on the specific ESS to which the NBO program is attached, as described in the Appendix.

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 ab initio and semi-empirical packages (GAUSSIAN-8X, GAMESS, HONDO, AMPAC, etc.). These versions are described in individual Appendices.

A.1.2 Structure of the NBO Program

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).

The main NBO program is represented by modules labelled "NBO" and "NBOEAN." These refer to the construction 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.

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 energy value) upon request. Each such ESS device therefore requires special drivers to make this feedback possible. Versions of the driver subroutines are included for several popular packages. The driver routines are described in more detail in the Programmer's Guide, Section C.

Figure 1: Schematic diagram depicting flow of information between the electronic structure system (ESS) and the NBO program, and the communication lines connecting these programs 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, 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.

A.1.3 Input and Output

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 keylist) 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.

The following information is passed from the ESS to the NBO program (transparent to the user):

1. The one-electron density matrix D (or density matrices in the open-shell case) in the chosen atomic orbital (AO) basis set;

2. The AO overlap matrix S, and label information identifying the symmetry (angular momentum type) and location (number of the atom to which affixed) for each AO;

3. 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 F, if energy analysis is requested; the AO dipole matrix M, if dipole moment analysis is requested; or information concerning the mathematical form of the AOs (orbital exponents, contraction coefficients, etc.), if orbital plotting information is requested to be saved as input for a contour plotting program.

The principal output from the NBO program consists 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.

A.1.4 General Capabilities and Restrictions

Principal capabilities of the NBO program are:

1. Natural population, natural bond orbital, and natural localized molecular orbital analysis of SCF, MCSCF, CI, and Møller-Plesset wavefunctions (main subroutine: NBO);

2. For RHF closed-shell and UHF wavefunctions only, energetic analysis of the wavefunction in terms of the interactions (Fock matrix elements) between NBOs (main subroutine: NBOEAN);

3. Localized analysis of molecular dipole moment in terms of NLMO and NBO bond moments and their interactions (main subroutine: DIPANL).

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:

S = overlap matrix
DM = density matrix (or D)
F = Fock matrix
DI = dipole matrix (or DXYZ, or DX, DY, DZ)
NPA = Natural Population Analysis
NAO = Natural Atomic Orbital
NBO = Natural Bond Orbital
NLMO = Natural Localized Molecular Orbital
PNAO = pre-orthogonal NAO (i.e., omit interatomic orthogonalization)
PNHO, PNBO, PNLMO = pre-orthogonal NHO, etc. (formed from PNAOs)  

Most of the NBO storage is allocated dynamically, to conform to the minimum required for the molecular system under study. However, certain NBO common blocks of fixed dimensionality are used for integer storage. These are currently dimensioned to accomodate up to 99 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 Møller-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 potentials ("pseudopotentials") can be handled, including complete neglect of core electrons as assumed in semi-empirical treatments. The atomic orbital basis functions (up to f orbitals in angular symmetry) may be of general Slater-type, contracted Gaussian-type, or other general composition, including the "effective" orthonormal valence-shell AOs of semi-empirical treatments. AO basis functions are assumed to be normalized, but in general non-orthogonal.

A.1.5 References and Relationship to Previous Versions

This program ("version 3.0") is an extension of previous versions of the NBO method incorporated in the semi-empirical program BONDO [F. Weinhold, Quantum Chemistry Program Exchange No. 408 (1980); "version 1.0"] and in a GAUSSIAN-82 implementation [A. E. Reed and F. Weinhold, QCPE Bull. 5, 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 framework.

Principal contributors to the development of the NBO methods and programs (1975-1990) are

Natural Bond Orbitals:

J. P. Foster and F. Weinhold, J. Am. Chem. Soc. 102, 7211-7218 (1980).

Natural Atomic Orbitals and Natural Population Analysis:

A. E. Reed and F. Weinhold, J. Chem. Phys. 78, 4066-4073 (1983); A. E. Reed, R. B. Weinstock, and F. Weinhold, J. Chem. Phys. 83, 735-746 (1985).

Natural Localized Molecular Orbitals:

A. E. Reed and F. Weinhold, J. Chem. Phys. 83, 1736-1740 (1985).

Open-Shell NBO:

J. E. Carpenter and F. Weinhold, J. Molec. Struct. (Theochem) 169, 41-62 (1988); J. E. Carpenter, Ph. D. Thesis, University of Wisconsin, Madison, 1987.

Review Articles:

A. E. Reed, L. A. Curtiss, and F. Weinhold, Chem. Rev. 88, 899-926 (1988); F. Weinhold and J. E. Carpenter, in, R. Naaman and Z. Vager (eds.), "The Structure of Small Molecules and Ions," (Plenum, New York, 1988), pp. 227-236.

The principal enhancements of version 3.0 include:

1. Generalized Program Interface. 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.

2. NAO/NPA Summary Table. 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 summary tables (Section A.3.2) include an SCF atomic orbital energy (if available), a conventional atomic orbital label (1s, 2s, 2p, etc., in accordance with the labelling in isolated atoms), and a shell designation (Cor = core, Val = valence, or Ryd = Rydberg) to aid characterization of the NAO.

3. NBO Summary Table. 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 elements, and 2nd-order energy analysis.

4. Bond Bending Analysis. The program includes a new analysis of hydrid directionality and bond "bending" (keyword BEND, Section A.3.4).

5. Dipole Moment Analysis. The program includes new optional provision (keyword DIPOLE, Section B.6.3) for analysis of the molecular dipole moment in terms of localized NLMOs and NBOs.

6. Print options. The program offers new structured printing options (Section B.2.4) that give greater convenience 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.

7. Orbital Contour Info. 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 orbitals.

8. Effective Core Potentials. The program now handles effective core potentials (pseudopotentials), 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):

9. Unpolarized Cores. NAOs identified as "core" orbitals are now automatically carried over as unhybridized 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 occupancies are 'accidentally' degenerate (usually, both very close to 2.000...) within the numerical machine precision. A warning message is printed when the core occupancy is less than 1.9990, indicating a possible "core polarization" effect of physical significance.

10. Excited State Antibond Labels. 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 message 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 the basis of occupancy.]

11. Alternative Resonance Structures. The program now institutes a search for alternative Lewis ('resonance') structures when two or more structures may be competitive, 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 capabilities of the program.

A.2 INSTALLING THE NBO PROGRAM

The NBO programs and manual are provided on a distribution tape. The tape contains three files: the TechSet code of this manual (file NBO.MAN), a file containing the core NBO source routines and supporting driver routines (file NBO.SRC), and the Fortran "enabler" program (file ENABLE.FOR).

In overview, the installation procedure involves the following steps (the details of each step being dependent on your operating system):

1. Enabling the NBO routines. Copy the contents of the distribution tape onto your system. Using your system Fortran 77 compiler, compile and link the enabler program to create the ENABLE.EXE executable; for example, the VMS commands to create ENABLE.EXE are

     FOR ENABLE
     LINK ENABLE

     NBO program version to enable?

2. Compiling the NBO routines. Using your system Fortran 77 compiler, compile the XXXNBO.FOR file to an object code file (say, XXXNBO.OBJ). [Compiler errors (if any) should be fixed before proceeding. Please notify the authors if you encounter undue difficulties in this step.]

3. Modifying the ESS routines. 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 consists 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.

4. Rebuilding the integrated ESS/NBO program. Re-compile your modified ESS programs and link the resulting object file (say, ESS.OBJ) with the XXXNBO.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 XXXNBO.OBJ file must be included in the link statement (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.

If you are interfacing the NBO programs to a new ESS package (not represented in the driver routines provided with this distribution), see Section C for guidance 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.

The TechSet-coded version of this manual, NBO.MAN, can be printed on an HP LaserJet printer ('F' cartridge) with the TECHSET technical typesetting program [ACS Software, American Chemical Society, Marketing Communications Dept., 1155 Sixteenth Street, N.W., Washington, D.C. 20036].

A.3 TUTORIAL EXAMPLE FOR METHYLAMINE

A.3.1 Running the Example

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 (CH₃NH₂) in Pople-Gordon idealized geometry, treated at the ab initio 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.

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 including the line

     $NBO $END

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 standard geometry             

Storage needed:  2505 in NPA,  2569 in NBO ( 750000 available)

A.3.2 Natural Population Analysis

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 occupancies 
                                                         
 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

The principal quantum numbers for the NAO labels (1s, 2s, 3s, 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.

The next segment is an atomic summary showing the natural atomic charges (nuclear charge minus summed natural populations of NAOs on the atom) and total core, valence, and Rydberg populations on each atom:

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

Next follows a summary of the NMB and NRB populations for the composite system, summed over atoms:

                                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)
--------------------------------------------------------

Finally, 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)

A.3.3 Natural Bond Orbital Analysis

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

Next 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  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)
--------------------------------------------------------

Next follows the main listing of NBOs, displaying the form and occupancy of the complete set of NBOs that span the input AO space:

    (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
  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
 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 -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%)
                                       -1.0000 -0.0020

_CN = 0.633(sp^3.61)_C + 0.774(sp^2.24)_N

corresponding roughly to the qualitative concept of interacting sp³ hybrids (75% p-character) and the higher electronegativity (larger polarization coefficient) of N.] Below each NHO label is the set of coefficients 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 h_C of the _CN bond has largest coefficients for the 2^nd and 4^th NAOs, corresponding to the approximate description

h_C -0.4653(2s)_C - 0.8808(2p_x)_C

in terms of the valence NAOs of the carbon atom.] In the CH₃NH₂ example, the NBO search finds the C-N bond (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 _CH NBOs, and the slightly higher occupancy (0.01569 vs. 0.0077 electrons) in the *_C1H3 antibond (NBO 24) lying trans to the nitrogen lone pair.

A.3.4 NHO Directional Analysis

The next segment of output summarizes the angular properties 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   --       --     --    --

A.3.5 Perturbation Theory Energy Analysis

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
  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

i j

E(2) = E_ij = q_i (F(i,j)²)/(_j - _i)

where q_i is the donor orbital occupancy, _i, _j are diagonal elements (orbital energies) and F(i,j) is the off-diagonal NBO Fock matrix element. [In the example above, the n_N *_CH interaction between the nitrogen lone pair (NBO 8) and the antiperiplanar C₁-H₃ 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.

A.3.6 NBO Summary

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):

                                                    Principal Delocalizations
          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

B.1 INTRODUCTION TO THE NBO USER'S GUIDE AND NBO KEYLISTS

Section B constitutes the general user's guide to the NBO program. It assumes that the user has an installed electronic 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.

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 keywords in a prescribed keylist format. Four distinct 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.

The four keylist types have common rules of syntax: Keylist delimiters are identified by a "$" prefix. Each keylist begins with the parent keylist name (e.g., "$NBO"), followed by any number of keywords, and ended with the word "$END"; for example,

     $NBO   keyword1   keyword2   . . .   $END      !comment

B.2 THE $NBO KEYLIST

B.2.1 Overview of $NBO keywords

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:

Job Control Keywords:

Matrix Output Keywords:

Other Output Control Keywords:

Print Level Control: 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.

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:

     E2PERT=2.5   LFNPR 16  NBOMO=W25

[ Although the general user's interaction with the NBO programs 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 specialist.]

B.2.2 Job Control Keywords

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 indication of where the option is useful):

KEYWORD OPTION DESCRIPTION

NPA Request Natural Population Analysis and printing of NPA summary tables (Section A.3.2). This keyword also activates calculation of NAOs, except for semi-empirical ESS methods.

NBO Request calculation of NBOs and printing of the main NBO table (Section A.3.3).

NBOSUM Request printing of the NBO summary table (Section A.3.6). This combines elements of the NBO table and 2nd-order perturbation theory analysis table (see below) in a convenient form for recognizing the principal delocalization patterns.

RESONANCE 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 RESONANCE 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 occupancy 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 delocalized molecules.)

NOBOND 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.)

3CBOND Request search for 3-center bonds (Section B.6.8). The normal default is to search for only 1- and 2-center NBOs. (Useful for diborane and other electron-deficient 'bridged' species.)

SKIPBO Skip the computation of NBOs, i.e., only determine NAOs and perform natural population analysis. (Useful when only NPA is desired.)

NLMO Compute and print out the summary table of Natural Localized Molecular Orbitals (Section B.6.2). NLMOs are similar to Boys or Edmiston-Ruedenberg LMOs, but more efficiently calculated. (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.

B.2.3 Job Threshold Keywords

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].)

KEYWORD parameter(s) OPTION DESCRIPTION

BEND ang, pct, occ 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
pct [25] = threshold percentage p-character for printing
occ [0.1] = threshold NBO occupancy for printing

Parameter values may be separated by a space or a comma.

Example:

     BEND=2,10,1.9

E2PERT eval Request the Perturbation Theory Energy Analysis table (Section A.3.5), where

eval [0.5] = threshold energy (in kcal/mol) for printing

Entries will be printed for NBO donor-acceptor interaction energies that exceed the 'eval' threshold.

Example:

     E2PERT=5.0

DIPOLE dval Request the Molecular Dipole Moment Analysis table (Section B.6.3), where

dval [0.02] = 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:

     DIPOLE=0.1

Both 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 keylist

     $NBO  DIPOLE  $END

B.2.4 Matrix Output Keywords

The keywords in this group activate the printing of various matrices to the output file, or their writing to (or reading from) external disk files. The large number of keywords in this group provide great flexibility in printing out the details of the successive transformations,

input AOs (PNAOs) NAOs NHOs NBOs NLMOs canonical MOs

or the matrices of various operators in the natural localized basis sets. This ordered sequence of transformations forms the basis for naming the keywords.

Keyword Names

The keyword for printing the matrix for a particular basis transformation, IN OUT, is constructed from the corresponding acronymns for the two sets in the generic form "INOUT". For example, the transformation AO 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 all possible transformations from the input AOs to other sets (NAO, NHO, NBO, NLMO, MO, or the pre-orthogonal PNAO, PNHO, PNBO, PNLMO sets) in the overall sequence leading to canonical MOs, i.e.,

AO basis: 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_IN,OUT for a specified IN OUT transform has rows labelled by the IN set and columns labelled by the OUT set.

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 localized sets (NAO, NHO, NBO, or NLMO). The corresponding keyword 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

Each generic matrix keyword ("MATKEY") can include a 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):
MATKEY=P[c] (print out the matrix in the standard output file, 'c' columns)

MATKEY=W[n] (write out the matrix to disk file n)

The 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 output file. Thus, "MATKEY=P" would be equivalent to "MATKEY", 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

     MATKEY=P16         (print 16 columns)

     MATKEY=PVAL        (print core + valence MO columns only)

     MATKEY=PLEW        (print Lewis orbital columns only)

The 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:

     AONBO=W22

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 format), 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.

For the AONAO, NAONBO matrices (only), one can also include a read parameter (R),

     AONAO=Rn
     NAONBO=Rn

B.2.5 Other Output Control Keywords

The keywords in this group also help to control the I/O produced by a specified set of job options, and thus supplement 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:

KEYWORD OPTION DESCRIPTION

LFNPR=n Set the logical file number (LFN) for NBO program output. The default LFN is 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:

  LFNPR=25   (re-direct NBO output to LFN 25)
 

BNDIDX 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 24, 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.)

AOINFO Request writing of information concerning the AO basis set (geometrical positions, orbital exponents, contraction coefficients, etc.) to an external file, LFN 31. This is a portion of the information needed by the ORBPLOT orbital contour plotting programs (cf. "PLOT" keyword below.)

PLOT Request writing of all files required by orbital contour plotting programs ORBPLOT. This activates the AOINFO keyword, as well as all the necessary matrix output keywords (AONBO=W37, etc.) that could be required for ORBPLOT.

ARCHIVE=n Request writing the FILE47 'archive' file to external disk file LFN = 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 repeating the calculation of the wavefunction (Section B.7).

NBODAF=n Request writing the NBO direct access file (DAF) to external disk file LFN = n (or, if "=n" is not present, to the default LFN =48).

B.2.6 Print Level Keywords

The keyword "PRINT=n" (n = 0-4) can be used to give convenient, flexible control of all NBO output in terms of a specified print level n. This keyword activates groups of keywords in a heirarchical manner, and thus incrementally increases the volume of output, ranging from no NBO output (PRINT=0) to a considerable volume of detail (PRINT=4). The keywords associated with each print level are tabulated below [default value, PRINT=2]:

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

     $NBO  PRINT=2  NLMO  FNBO=P  NAOMO=P11  $END

     $NBO  PRINT=1  SKIPBO  $END

     $NBO  PRINT=0  NPA  $END

B.2.7 Semi-Documented Additional Keywords

Some additional keywords are listed below that may of use to specialists or program developers:

KEYWORD OPTION DESCRIPTION

THRESH=val 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).

PRJTHR=val Set the projection threshold [default 0.20] to determine if a 'new' hybrid orbital has too high overlap with hybrids previously found.

MULAT Print total gross Mulliken populations by atom.

MULORB Print gross Mulliken populations, by orbital and atom.

RPNAO Revises PAO to PNAO transformation matrix by post-multiplying by T_Ryd and T_red [see the NPA paper: A. E. Reed, R. B. Weinstock, and F. Weinhold, J. Chem. Phys. 83, 735-746 (1985)].

PAOPNAO Input or output of pure AO (PAO) to pre-NAO (PNAO) transformation. 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.

BOAO 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.

B.3 THE $CORE LIST

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 occupancy 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 significance.

[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' degeneracy in occupancy of core and valence lone pairs leads to undesirable core-valence mixing; the present version explicitly isolates core pairs as unhybridized NAOs prior to the main NBO search to prevent this unphysical effect.]

The NBO program contains a table giving the nominal number of core orbitals to be isolated on each type of atom (e.g., 1s for first-row atoms Li-Ne, 1s, 2s, 2p 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.

The format of the $CORE modification list is:

first line: The keyword "$CORE"

next lines: 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.

last line: 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 modification keylist cannot continue on a line that contains other keylist information.

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 n = 1 and n = 2 shells as core orbitals, 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 1s, 2s, and all three 2p orbitals. The $CORE modification list is read only once, and applies to both and spin manifolds in an open-shell calculation.

An example, appropriate for Ni(1)-C(2)-O(3) with the indicated numbering of atoms, is shown below:

     $CORE
       1   5
     $END

[The alternative example

     $CORE   1  0    2  0    3  0   $END

B.4 THE $CHOOSE KEYLIST (DIRECTED NBO SEARCH)

A $CHOOSE keylist requests that the NBO search be directed to find a particular Lewis structure ('resonance structure') chosen by the user. (This is useful for testing the accuracy 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 algorithm seek the best location for lone pairs, but it is usually safest to completely specify the resonance structure, both lone pairs and bonds.

The format of the $CHOOSE list is:

first line: The keyword "$CHOOSE"

next line: The keyword "ALPHA" (only for open-shell wavefunction)

next lines: 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. Terminate the list with "END". (Note that only the occupied valence 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

"S" single bond
"D" double bond
"T" triple bond
"Q" 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 keyword "3CBOND", followed by the list of 3-c bond specifiers, 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.)

next line: The word "END" to signal the end of the spin list.

next line: The keyword "BETA" (for open-shell wavefunctions)

next lines: The input for spin, same format as above. The overall $CHOOSE list should always end with the "$END" keyword.

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...CO, with atom numbering F(1)-H(2)...C(3)-O(4), might be specified as

     $CHOOSE
        LONE  1  3
              3  1
              4  1     END
        BOND  S  1  2
              T  3  4  END
     $END

(2) The open-shell FH...O₂ complex, with atom numbering F(1)-H(2)...O(3)-O(4), and with the unpaired electrons on O₂ being of spin, might be specified as

     $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

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.

B.5 THE $DEL KEYLIST (NBO ENERGETIC ANALYSIS)

B.5.1 Introduction to NBO Energetic Analysis

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) diagonalizing 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 difference 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 wavefunctions only.

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 single 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.

The deletions keylist begins with the "$DEL" keyword. For the analysis of UHF wavefunctions, the deletions for the and spin manifolds must be separately specified (see Section B.5.3). Otherwise, the input for closed shells RHF and UHF is identical. The input is free format and the input for a single deletion can be spread over as many lines as desired. The desired deletions should be listed one after the other. After the last deletion, the word "$END" signals the end of the keylist.

If symmetry is used, one must be careful to only do deletions that will preserve the symmetry of the electronic wavefunction!! If this is not done, the energy of the deletion 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 single C-H antibond.) The remedy is not to use symmetry in the SCF calculation.

In describing the deletion types, use is made of the terms "molecular unit" and "chemical fragment." The NBO program 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 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 necessarily) 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 molecular unit, identified by giving the atom numbers of which the fragment consists.

B.5.2 The Nine Deletion Types

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:

DELETE  3  ORBITALS  15  18  29

The "single-pass" method of evaluating deletion energies is appropriate only for deletions of low-occupancy (non-Lewis) orbitals, for which the loss of self-consistency in the Coulomb and exchange potentials (due to redistribution of the electron density of deleted orbitals) is small compared to the net energy change of deletion. It is fundamentally erroneous to delete high-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:

DELETE  3  ELEMENTS  1 15  3 19  23 2

(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:

     ZERO  2  BLOCKS  2  BY  5
                               3  4
                               9  10  11  14  19
                      3  BY  2
                               1  2  7
                              20  24

(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 delocalization 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 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 variational expectation value of the determinant of doubly occupied 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 destarred, then the keyword "UNIT" (or "UNITS"), then the list of units.

Example:

DESTAR  2  UNITS  3  4
 

(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 delocalization, the word "FROM", the number of the donor unit, the word "TO", and the number of the acceptor unit.

Example:

ZERO  2  DELOC  FROM 1 TO 2   FROM 2 TO 1

(9) Zeroing all delocalization from one chemical fragment to another.

This is called for by typing "ZERO", then the number of 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 fragments defined by atoms (1,2) and by atoms (3,4,5), the $DEL entries would be

     ZERO  2  ATOM BLOCKS
              2  BY  3
                       1  2
                       3  4  5
              3  BY  2
                       3  4  5
                       1  2

For additional examples of $DEL input, see Section B.6.10.

B.5.3 Input for UHF Analysis

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.

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.

NOTE: The types of the NBOs often differ from those of the NBOs, so that distinct , deletions lists are generally required. For example, O₂ (triplet) has one bond in the system and three in the system, if the unpaired electrons are in the system.

Here are three examples to illustrate UHF open-shell deletions:

Example 1:

     ALPHA  ZERO  1  DELOC  FROM  1  TO  2
     BETA   NOSTAR

     ALPHA  ZERO  1  DELOC  FROM  1  TO  2
     BETA   ZERO  0  DELOC

     ALPHA  DELETE  0  ORBITALS
     BETA   DELETE  1  ORBITAL  8

B.6 NBO KEYLIST ILLUSTRATIONS

B.6.1 Introduction

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 output, since, e.g., NPA analysis is unaffected by keywords that modify the NBO search. Keywords of general applicability are illustrated with the methylamine example (RHF/3-21G, Pople-Gordon geometry) of Section A.3, which should be consulted for further information. More specialized keywords (RESONANCE, 3CBOND, etc.) are illustrated with prototype molecules (benzene, diborane, etc.) chosen for the keyword.

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 keywords. Section B.6.9 and B.6.10 similarly illustrate the use of the $CHOOSE and $DEL keylists. Section B.6.11 illustrates the output for open-shell UHF cases, emphasizing features associated with the "different Lewis structures for different spins" representation of and spin manifolds. Section B.6.12 shows the effect of using effective core potentials for Cu₂, also illustrating aspects of the inclusion of d functions.

B.6.2 NLMO Keyword

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:
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%)
                           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%)

B.6.3 DIPOLE Keyword

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
                            -------------------------  ------------------------
         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
                                            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