## Natural Bond Orbital Analysis Tutorial1

E. D. Glendening, A. E. Reed2, J. E. Carpenter3, and F. Weinhold
Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706

## 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

at the end of the file. This is an empty' NBO keylist, specifying 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 and storage info:

----------------------output file----------------------------------------

*******************************************************************************
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)


Note that all NBO output is formatted to a maximum 80-character width for convenient display on a computer terminal. The NBO heading echoes any requested keywords (none for the present default case) and shows an estimate of the memory requirements (in double precision words) for the separate steps of the NBO process, compared to the total allocated memory available through your ESS process. Increase the memory allocated to your ESS process if the estimated NBO requests exceed the available storage.

#### 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


For each of the 28 NAO functions, this table lists the atom to which NAO is attached (in the numbering scheme of the ESS program), the angular momentum type lang' (s, px, etc., in the coordinate system of the ESS program), the orbital type (whether core, valence, or Rydberg, and a conventional hydrogenic-type label), the orbital occupancy (number of electrons, or natural population' of the orbital), and the orbital energy (in the favored units of the ESS program, in this case atomic units: 1 a.u. = 627.5 kcal/mol). [For example, NAO 4 (the highest energy C orbital of the NMB set) is the valence shell 2px orbital on carbon, occupied by 0.8909 electrons, whereas NAO 5 is a Rydberg 3px orbital with only 0.0014 electrons.] Note that the occupancies of the Rydberg (Ryd) NAOs are typically much lower than those of the core (Cor) plus valence (Val) NAOs of the natural minimum basis set, reflecting the dominant role of the NMB orbitals in describing molecular properties.

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.

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


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:

This table succinctly describes the molecular charge distribution in terms of NPA charges. [For example, the carbon atom of methylamine is asSigned a net NPA at this level; note also the slightly less positive charge on H(3) than on the other two methyl hydrogens: +0.184 vs. +0.217.]

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)
Na
--------------------------------------------------------


This exhibits the high percentage contribution (typically, > 99%) of the NMB set to the molecular charge distribution. [In the present case, for example, the 13 Rydberg orbitals of the NRB set contribute only 0.07% of the electron density, whereas the 15 NMB functions account for 99.93% of the total.]

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 in the above table is most nearly described by an idealized 1s22s12p3 electron configuration.]

#### 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

Normally, there is but one cycle of the NBO search (cf. the "RESONANCE" keyword, Section B.6.6). 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 from a nominal estimate (NAO Wiberg bond index) for the structure. [If the latter exceeds 0.1, additional NBO searches are initiated (indicated by the parenthesized number under Cycle') for alternative Lewis structures.] The Lewis structure is accepted if all 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) relative to the extra-valence orbitals (the 13 Rydberg NBOs 10-22) in the slight departures from a localized Lewis structure model. (In this case, 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 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


For each NBO (1-28), the first line of printout shows the occupancy (between 0 and 2.0000 electrons) and unique label of the NBO. This label gives the type ("BD" for 2-center bond, "CR" for 1-center core pair, "LP" for 1-center valence lone pair, "RY*" for 1-center Rydberg, and "BD*" for 2-center antibond, the unstarred and starred labels corresponding to Lewis and non-Lewis NBOs, respectively), a serial number (1, 2,... if there is a single, double,... bond between the pair of atoms), and the atom(s) to which the NBO is affixed. [For example, the first NBO in the sample output is the 2-center bond (with 1.99858 electrons) between carbon (atom 1) and nitrogen (atom 2), the Sigma CN bond.] The next lines summarize the natural atomic hybrids hA of which the NBO is composed, giving the percentage (100|cA|2) of the NBO on each hybrid (in parentheses), the polarization coefficient cA, the atom label, and a hybrid label showing the splambda composition (percentage s-character, p-character, etc.) of each hA. [For example, the Sigma CN NBO is formed from an sp3.61 hybrid (78.3% p-character) on carbon interacting with an sp2.24 hybrid (69.1% p-character) on nitrogen,

Sigma CN = 0.633(sp3.61)C + 0.774(sp.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 hC of the Sigma CN bond has largest coefficients for the 2nd and 4th NAOs, corresponding to the approximate description

hC = - 0.4653(2s)C - 0.8808(2px)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 Sigma CH NBOs, and the slightly higher occupancy (0.01569 vs. 0.0077 electrons) in the Sigma*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   --       --     --    --


The direction' of a hybrid is specified in terms of the polar (\theta ) and azimuthal (\phi ) angles (in the ESS coordinate system) of the vector describing its p-component. The hybrid direction is compared with the direction of the line of centers between the two nuclei to determine the bending' of the bond, expressed as the deviation angle ("Dev," in degrees) between these two directions. For example, in the CH3NH2 case shown above, the nitrogen NHO of the Sigma CN bond (NBO 1) is bent away from the line of C-N centers by 3.0\degree , whereas the carbon NHO is approximately aligned with the C-N axis (within the 1.0\degree threshold for printing). The N-H bonds (NBOs 5, 6) are bent even further (4.4\degree ). 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 nitrogen bond kinks' found in the tetrahedral Pople-Gordon geometry).

#### 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


This 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 loss of occupancy from the localized NBOs of the idealized Lewis structure into the empty non-Lewis orbitals (and thus, to departures from the idealized Lewis structure description), they are referred to as 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

E(2) = Delta Eij = qi / [ F(i,j)2(epsilon j - epsilon i)]

where qi is the donor orbital occupancy, epsilon i, epsilon j are diagonal elements (orbital energies) and F(i,j) is the off-diagonal NBO Fock matrix element. [In the example above, the nN -> Sigma*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  (CH4N)
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 NLMO associated with the parent donor NBO. [For example, in the table above, the nitrogen lone pair (NBO 9) is seen to be the lowest-occupancy (1.97795 electrons) and highest-energy (- 0.44592 a.u.) Lewis NBO, and to be primarily delocalized into antibonds 24, 25, 26 (the vicinal Sigma*CH 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).]

### References

1. Excerpted from the Natural Bond Orbital Analysis Program Manual.
2. Present address: Bayer AG, Abteilung AV-IM-AM, 5090 Leverkusen, Bayerwerk, Federal Republic of Germany.
3. Present address: Department of Chemistry, University of California-Irvine, Irvine, California 92717.
Principal references to the development and applications of NAO/NBO/NLMO methods 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.