Natural Bond Orbital Analysis Tutorial1

Tutorial Example For Methylamine

H3H7
\ /
H4 - C1 - N2
/ \
H5H6
The full output for this example is methylamine.

Natural Bond Orbital Analysis

The Lewis structure that is closest to your structure is determined. This process generates "natural bond orbitals," or NBOs. These orbitals are localized electron pair orbitals for bonding pairs and lone pairs. The hybridization of the atoms and the weight of each atom in each localized electron pair bond is calculated in this idealized Lewis structure. The output for methylamine is given in the table below as as example. 

Best Lewis Structure


Hybridization in the Best Lewis Structure

1. A bonding orbital for C1-N2 with 1.9979 electrons
__has 40.75% C 1 character in a sp2.68 hybrid
__has 59.25% N 2 character in a sp2.20 hybrid

2. A bonding orbital for C1-H3 with 1.9931 electrons
__has 58.59% C 1 character in a s0.95 p3 hybrid
__has 41.41% H 3 character in a s orbital

3. A bonding orbital for C1-H4 with 1.9973 electrons
__has 57.68% C 1 character in a s0.99 p3 hybrid
__has 42.32% H 4 character in a s orbital

4. A bonding orbital for C1-H5 with 1.9931 electrons
__has 58.59% C 1 character in a s0.95 p3 hybrid
__has 41.41% H 5 character in a s orbital

5. A bonding orbital for N2-H6 with 1.9919 electrons
__has 67.28% N 2 character in a s0.96 p3 hybrid
__has 32.72% H 6 character in a s orbital

6. A bonding orbital for N2-H7 with 1.9920 electrons
__has 67.28% N 2 character in a s0.96 p3 hybrid
__has 32.72% H 7 character in a s orbital

9. A lone pair orbital for N2 with 1.9695 electrons
__made from a s0.77 p3 hybrid

-With core pairs on: C 1 N 2 -

The first line for each orbital gives the type of orbital and the occupancy (between 0 and 2.0000 electrons). The type can be "bonding," "lone pair," and "antibonding." A normal Lewis structure wouldn't have any antibonding orbitals, so the presence of antibonding orbitals shows deviations from normal Lewis structures. Antibonding localized orbitals are called non-Lewis NBOs. An example is given below where antibonding orbitals are encountered. For methylamine, however, no antibonding orbitals are listed so that the stucture is adequately explained by normal Lewis electron pair orbitals. For example, the first NBO in the sample output is the 2-center bond with 1.9979 electrons between carbon (atom 1) and nitrogen (atom 2), the SigmaCN bond. If the occupancy is not 2.0, then there are deviations from an ideal Lewis structure. Methylamine shows some small deviations, but otherwise is well-approximated using Lewis structures. The next lines summarize the natural atomic hybrids of which the NBO is composed, giving the percentage of the NBO on each hybrid, the atom label, and a hybrid label showing the splambda composition (the amount of s-character, p-character, etc.). For example, the SigmaCN NBO is formed from an sp2.68 hybrid on carbon interacting with an sp2.20 hybrid on nitrogen:

SigmaCN = 0.638 (sp2.68)C + 0.770 (sp2.20)N

The sp2.68 hybrid on carbon has 72.8% p-character. The sp2.20 hybrid on nitrogen has 68.8% p-character. An idealized sp3 hybrid has 75% p-character. The SigmaCN bond then corresponds roughly to the qualitative concept of interacting sp3 hybrids. The two coefficients, 0.638 and 0.770, in the SigmaCN equation above are called polarization coefficients. The sizes of these coefficients show the importance of the two hybrids in the formation of the bond. They are easily calculated given that the percentage of the NBO on each hybrid is (100|cA|2). Nitrogen has the larger percentage of this NBO, at 59.25%, and gives the larger polarization coefficient of 0.770, because it has the higher electronegativity.

In the methylamine 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), and the N lone pair (NBO 9) of the expected Lewis structure. At the end of the table are listed the C and N core pairs, which are the 1s electrons for each atom. In this example, it is interesting to note the slight asymmetry of the three SigmaCH NBOs. The C-H bond lying trans to the nitrogen lone pair has slightly more electron density. This different orbital is NBO 3, which is for the C1-H4 bond. This difference can be explored using Perturbation Theory Energy Analysis to calculate Donor Acceptor Interactions among the electron pairs.

Donor Acceptor Interactions: Perturbation Theory Energy Analysis

The localized orbitals in your best Lewis structure can interact strongly. A filled bonding or lone pair orbital can act as a donor and an empty or filled bonding, antibonding, or lone pair orbital can act as an acceptor. These interactions can strengthen and weaken bonds. For example, a lone pair donor -> antibonding acceptor orbital interaction will weaken the bond associated with the antibonding orbital. Conversely, an interaction with a bonding pair as the acceptor will strengthen the bond. Strong electron delocalization in your best Lewis structure will also show up as donor-acceptor interactions. Only interactions greater than 20 kJ/mol are listed. 

Donor Acceptor Interactions in the Best Lewis Structure

The interaction of lone pair donor orbital, 9, for N2 with the antibonding
acceptor orbital, 75, for C1-H4 is 44.3 kJ/mol.

This calculation is done 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 natural Lewis structure.

In the methylamine example above, the lone pair donor orbital, nN -> Sigma*CH interaction between the nitrogen lone pair (NBO 9) and the antiperiplanar C1 -H4 antibond (NBO 75) is seen to give a strong stabilization, 44.3 kJ/mol.

Resonance

Systems that have multiple resonance structures or extensive delocalization cannot be described by a single Lewis structure. The failure of the Lewis approach is chemically interesting and NBO analysis readily highlights any deviations from pure localized electron pair bonds. Systems with: help highlight deviations from idealized Lewis bonding. An interesting example is ozone. Two resonance structures are necessary to describe ozone, because the molecule has a delocalized three-center bond. The Best Lewis Structure Section of the output for ozone is given below: 

Hybridization in the Best Lewis Structure

1. A bonding orbital for O1-O2 with 1.9954 electrons
__has 60.78% O 1 character in a s0.95 p3 hybrid
__has 39.22% O 2 character in a p3 hybrid

2. A bonding orbital for O1-O3 with 1.9985 electrons
__has 44.41% O 1 character in a p-pi orbital ( 99.81% p)
__has 55.59% O 3 character in a p-pi orbital ( 99.85% p)

3. A bonding orbital for O1-O3 with 1.9954 electrons
__has 60.76% O 1 character in a s0.95 p3 hybrid
__has 39.24% O 3 character in a p3 hybrid

7. A lone pair orbital for O1 with 1.9974 electrons
__made from a sp0.89 hybrid

8. A lone pair orbital for O2 with 1.9988 electrons
__made from a s orbital

9. A lone pair orbital for O2 with 1.9682 electrons
__made from a p3 hybrid

10. A lone pair orbital for O2 with 1.3836 electrons
__made from a p-pi orbital ( 99.89% p)

11. A lone pair orbital for O3 with 1.9988 electrons
__made from a s orbital

12. A lone pair orbital for O3 with 1.9682 electrons
__made from a p3 hybrid

71. A antibonding orbital for O1-O3 with 0.6110 electrons
__has 55.59% O 1 character in a p-pi orbital ( 99.81% p)
__has 44.41% O 3 character in a p-pi orbital ( 99.85% p)

O1 is the central atom and forms a single bond to O2 and a double bond to O3 (NBOs 2 and 3). Note that the Sigma bonds have occupancies that are a little low. Note that one of the lone pairs on the singly bonded outer oxygen is quite small at 1.384 electrons. The biggest indication that simple Lewis structures can't account for the bonding is the antibonding orbital with a very large occupancy of 0.611 electrons.

A quick look at the Donor Acceptor Interactions output shows how the NBO analysis deals with the delocalization: 

Donor Acceptor Interactions in the Best Lewis Structure

The interaction of bonding donor orbital, 2, for O1-O3 with the
third lone pair acceptor orbital, 10, for O2 is 65.3 kJ/mol. 

The interaction of bonding donor orbital, 2, for O1-O3 with the
antibonding acceptor orbital, 71, for O1-O3 is 35.6 kJ/mol. 

The interaction of the second lone pair donor orbital, 9, for O2
with the second antibonding acceptor orbital, 72, for O1-O3 is 66.4 kJ/mol. 

The interaction of the third lone pair donor orbital, 10, for O2
with the antibonding acceptor orbital, 71, for O1-O3 is 1394 kJ/mol. 

The interaction of the second lone pair donor orbital, 12, for O3
with the antibonding acceptor orbital, 70, for O1-O2 is 66.4 kJ/mol.

The program can't list two resonance structures, so it builds the delocalization by strong donor acceptor interactions. The largest interaction is between the third lone pair donor orbital, 10, for O2 and the antibonding acceptor orbital, 71, for O1-O3 at 1394 kJ/mol. This interaction builds a partial pi bond between O1 and O2 and weakens the pi bond between O1 and O3 by donation of electrons into the pi-antibonding orbital. This picture of the bonding in ozone may appear awkward, but the NBO analysis quickly shows the strengths and weaknesses inherent in the Lewis formalism and the more exact molecular orbital approach. Lewis structures just aren't a good model for delocalized systems. As you can see from our methylamine and ozone examples, the Donor Acceptor Interaction approach is most useful for small deviations from idealized Lewis structures 4.

References and Notes

1. Reworked from the Natural Bond Orbital Analysis Program Manual to fit the output style used by the Molecular Structure Calculations Web pages: Natural Bond Orbital Program Manual, 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
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.
4. The RESONANCE keyword is always used for the NBO analysis.

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.