Cramer C.J. Essentials of Computational Chemistry Theories and Models
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APPENDIX D |
maximally ‘distinct’ orbitals. It should be obvious that an infinite number of choices might be made for how best to combine the HF orbitals to make new orbitals, with goodness being determined only by how useful the final orbitals prove to be in a given qualitative or semi-quantitative empirical chemical model.
D.2 Natural Bond Orbital Analysis
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localization algorithm that hews particularly closely |
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and Weinhold 1988; for latest generation code and additional technical discussion see www.chem.wisc.edu/ nbo5/). NBO localization is a multistep process the details of which are sufficiently complicated that they do not warrant specific identification here – however, a general overview of the procedure can be described fairly simply. In an initial step, orbitals that are associated almost entirely with a single atom, e.g., core orbitals and lone pairs, are localized as so-called natural atomic orbitals (NAOs). Next, orbitals involving bonding (or antibonding) between pairs of atoms are localized by using only the basis set AOs of those atoms. Finally, the remaining Rydberg-like orbitals are identified, and all orbitals are made orthogonal to one another. The result is that, except for very small contributions from other AOs to ensure orthogonality, all NAOs and Rydberg orbitals are described using the basis-set AOs of a single atom and all NBOs are described using the basis-set AOs of two atoms (in cases where resonance or other delocalization effects require orbitals delocalized over more than two atoms, additional work is required). Thus, NBO analysis provides an orbital picture that is as close as possible to a classical Lewis structure for a molecule.
This localization scheme permits the assignment of hybridization both to the atomic lone pairs and to each atom’s contributions to its bond orbitals. Hybridization is a widely employed and generally useful chemical concept even though it has no formal basis in the absence of high-symmetry constraints. With NBO analysis, the percent s and p character (and d, f, etc.) is immediately evident from the coefficients of the AO basis functions from which the NAO or NBO is formed. In addition, population analysis can be carried out using the NBOs to derive partial atomic charges (NPA, see Section 9.1.3.2).
Another useful chemical concept is hyperconjugation, which rationalizes certain chemical phenomena in terms of filled-orbital–empty-orbital interactions (see Section 2.2.3). Consider for instance the torsional potential about the C(3)–O(4) bond in 2,4-dioxaheptane (Figure D.2). Delocalization of high-energy oxygen lone pair density into the low-energy σ C(3)–O(2) antibonding orbital is expected to stabilize torsional conformations that maximize the overlap between such orbitals (in an oxahydrocarbon case like this, the phenomenon is referred to as the anomeric effect).
NBO analysis can be used to quantify this phenomenon. Since the NBOs do not diagonalize the Fock operator (or the Kohn–Sham operator, if the analysis is carried out for DFT instead of HF), when the Fock matrix is formed in the NBO basis, off-diagonal elements will in general be non-zero. Second-order perturbation theory indicates that these off-diagonal elements between filled and empty NBOs can be interpreted as the stabilization energies
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Figure D.2 The torsional potential for 2,4-dioxaheptane ( , left ordinate) exhibits a two-fold periodicity suggesting a large influence from hyperconjugative effects. The sum of the NBO interaction energies for the two O(4) lone pairs delocalizing into the C(3)– O(2) σ orbital (- - - - - -, right ordinate) shows the same periodicity, with an amplitude of about 11 kcal mol−1. The smaller amplitude of the full torsional potential reflects primarily the presence of other influences as well as the approximate nature of the NBO analysis. See Cramer, Kelterer, and French 2001
deriving from hyperconjugation. Figure D.2 illustrates how the minima in the torsional potential energy curve for 2,4-dioxaheptane are influenced by the changes in hyperconjugation between the O(4) lone pairs and the C(3)–O(2) antibonding σ orbital as quantified by NBO analysis (a similar analysis may be carried out for filled-orbital–filled-orbital repulsive interactions, which are the electronic component of steric interactions). This approach is a nice energetic complement to other analyses focusing on structural changes or changes in partial atomic charges (cf. Figure 9.3) in the investigation of hyperconjugative effects. Nevertheless, it must be stressed that the NBO procedure is only a conceptual model, since it is based on orbitals, and limitations in its utility may be expected in cases where chemical species are poorly represented as Lewis structures.
References
Cramer, C. J., Kelterer, A.-M., and French, A. D. 2001. J. Comput. Chem., 22, 1194. Reed, A. E., Curtiss, L. A., and Weinhold, F. 1988. Chem. Rev., 88, 899.
