Tutorial Example for Methylamine

Running an NBO Calculation

This page provides an introductory quick-start tutorial on running an NBO calculation 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 perform this calculation are given here for Gaussian and GAMESS. The pop=nbo option of the Gaussian program requests default NBO analysis. NBO analysis is requested in the GAMESS calculation by simply including the line

 $NBO  $END

in the GAMESS input file. This "empty" NBO keylist specifies that NBO analysis should be carried out at the default level.


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, citation, job title, and storage info:

*********************************** NBO 5.0 ***********************************
            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
*******************************************************************************
 (c) Copyright 1996-2001 Board of Regents of the University of Wisconsin System
     on behalf of the Theoretical Chemistry Institute.  All Rights Reserved.

         Cite this program as:

         NBO 5.0.  E. D. Glendening, J. K. Badenhoop, A. E. Reed,
         J. E. Carpenter, J. A. Bohmann, C. M. Morales, and F. Weinhold
         (Theoretical Chemistry Institute, University of Wisconsin,
         Madison, WI, 2001)

Job title: Methylamine...RHF/3-21G//Pople-Gordon standard geometry         

Storage needed:        2562 in NPA,        3278 in NBO (     2000000 available)

Note that all NBO output is formatted to a maximum 80-character width for convenient display. The NBO heading echoes any requested keywords (none for the present default case).

Natural Population Analysis

The next four NBO output segments summarize the results of natural population analysis (NPA). The first segment is the main table of natural atomic orbitals (NAOs) 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

For each of the 28 NAO functions, this table lists the atom to which the NAO is attached, the angular momentum type (s, px, etc.), the orbital type (whether core, valence, or Rydberg, with a conventional hydrogenic-type label), the orbital occupancy (number of electrons, or "natural population" of the orbital), and the orbital energy (in atomic units: 1 a.u. = 627.5 kcal/mol). Note that the occupancies of the Rydberg (Ryd) NAOs are typically much lower than those of the core (Cor) and valence (Val) NAOs of the natural minimum basis (NMB) set, reflecting the dominant role of the NMB orbitals in describing molecular properties.

The NAOs have the expected shapes and nodal structures of hydrogen-like orbitals. For example, some s-type and p-type NAOs of carbon atom C(1) are displayed below in 3-D (surface), 2-D (contour) and 1-D (profile) orbital images:

NAOs of C(1)
3-D Surface 2-D Contour 1-D Profile

The principal quantum numbers for the NAO labels (1s, 2s, 3s, etc.) are assigned on the basis of the energy if a Fock or Kohn-Sham matrix is available, or on the basis of occupancy otherwise. A message warning of a "population inversion" is printed if the occupancy and energy ordering do not coincide (of interest, but rarely of concern).

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

This table succinctly describes the molecular charge distribution in terms of NPA charges, as shown in the diagram:

Next follows a summary of the natural minimal basis (NMB) and natural Rydberg basis (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)
--------------------------------------------------------

This reveals the high percentage contribution (typically, > 99%) of the NMB set to the molecular charge distribution.

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)

Although 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" configurations. For example, the carbon atom is seen to have valence configuration (2s)1.09(2p)3.35, approximating the idealized promoted configuration for sp3 hybridization.

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

Normally, there is but one cycle of the NBO search. The table summarizes a variety of information for each cycle: the occupancy threshold 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 for the structure from a nominal estimate (NAO Wiberg bond index). The Lewis structure is accepted if all orbitals of the formal Lewis structure exceed the occupancy threshold (default = 1.90 electrons).

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

This shows the general quality of the natural Lewis structure 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, listed below) relative to the extra-valence orbitals (the 13 Rydberg NBOs 10-22) in the slight departures from a localized Lewis structure model. The table also includes a warning about a carbon core orbital with slightly less than double occupancy.

Next follows the main listing of NBOs, displaying the form and occupancy of the complete set of orbitals 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

For each NBO (1-28), the first line of printout shows the occupancy (between 0 and 2 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. The next lines summarize the natural atomic hybrids hA of which the NBO is composed, giving the percentage (cA-squared) of the NBO on each hybrid (in parentheses), the polarization coefficient cA, the atom label, and a hybrid label showing the sp-hybridization (percentage s-character, p-character, etc.) of each hA. For example, the C-N bond (NBO 1; lower panel) and antibond (NBO 23; upper panel)

sCN = 0.663(sp3.61)C + 0.774(sp2.24)N
sCN* = 0.774(sp3.61)C - 0.633(sp2.24)N

are displayed below, together with the C(sp3.61) and N(sp2.24) bonding NHOs (middle panels) of which they are both composed (but with opposing polarizations and phases):

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.

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 sCH NBOs, and the slightly higher occupancy (0.016 vs. 0.008 electrons) in the sCH3 antibond (NBO 24) lying trans to the nitrogen lone pair.

NHO Directional Analysis

The next segment of output summarizes the angular properties of the natural hybrid orbitals (NHOs):

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   --       --     --    --

The "direction" of a hybrid is specified in terms of the polar (q) and azimuthal (f) angles (in the coordinate system of the calling program) of the vector describing its p-component. For more general spldm hybrids the hybrid direction is determined numerically to correspond to the maximum angular amplitude. The hybrid direction is then 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) between these two directions. For example, in the CH3NH2 case shown above, the nitrogen NHO of the sCN bond (NBO 1) is bent away from the line of C-N centers by 3.0°, whereas the carbon NHO is approximately aligned with the C-N axis (within the 1.0° threshold for printing). The N-H bonds (NBOs 5, 6) are bent even further (by 4.4°). The information in this table is often useful in anticipating the direction of geometry changes resulting from geometry optimization (viz., likely reduced pyramidalization of the -NH2 group to relieve the ~4° nitrogen bond bending found in the tetrahedral Pople-Gordon geometry).

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

This analysis is carried out by examining all possible interactions between "filled" (donor) Lewis-type NBOs and "empty" (acceptor) non-Lewis NBOs, and estimating their energetic importance by 2nd-order perturbation theory. Since these interactions lead to donation 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 "delocalization" corrections to the zeroth-order natural Lewis structure. For each donor NBO (i) and acceptor NBO (j), the stabilization energy E(2) associated with delocalization ("2e-stabilization") i ® j is estimated as


where qi is the donor orbital occupancy, ei, ej are diagonal elements (orbital energies) and F(i,j) is the off-diagonal NBO Fock matrix element. [In the example above, the nN ® sCH* interaction between the nitrogen lone pair (NBO 9) and the antiperiplanar C1-H3 antibond (NBO 24) is seen to give the strongest stabilization, 8.13 kcal/mol; this interaction is displayed from all angles on the NBO homepage.] As the heading indicates, entries are included in this table only when the interaction energy exceeds a default threshold of 0.5 kcal/mol.

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

This 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 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 natural localized molecular orbital (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 (1.978 electrons) and highest-energy (-0.446 a.u.) Lewis NBO, and to be primarily delocalized into antibonds 24, 25, 26 (the vicinal sCH 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).]

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