Localized orbitals from delocalized wavefunctions

The ambiguity about which non-orthogonal subunits receive credit for unaccounted density in the overlap region is the source of the many reported differences between alternative computational dissections. The associated overlap density can be assigned to the filled orbital (and counted towards steric effects) or to the unfilled orbital (and counted towards hyperconjugative charge-transfer). All methods that harbor such overlap ambiguities are expected to differ sharply from NBO-based assessments of intramolecular or inter-molecular interactions. 

After this general preface, let s describe the three popular approaches in more detail. 

Natural bond orbital (NBO) analysis The NBO analysis transforms the canonical delocalized Hartree-Fock (HF) MOs and non-orthogonal atomic orbitals (AOs) into the sets of localized natural atomic orbitals (NAOs), hybrid orbitals (NHOs), and bond orbital (NBOs). Each of these localized basis sets is complete, orthonormal, and describes the wavefunction with the minimal amount of filled orbitals in the most rapidly convergent fashion. Filled NBOs describe the hypothetical, strictly localized Lewis structure. NPA charge assignments based on NBO analysis correlate well with empirical charge measures.  

The interactions between filled and antibonding orbitals represent the deviation from the localized Lewis structure and can be used as a measure of delocalization. Since the occupancies of filled NBOs are highly condensed, the delocalizing interactions can be treated by a standard second order perturbation approach 

The NBO procedure is not the only localization technique for transforming delocalized MOs into the intuitive Lewis structure description. Foster and Boys, Edmiston and Ruedenberg, and Pipek and Mezey reported alternative localization procedures that provide additional bridges between MO and VB theories. From the organic chemist s point of view, these approaches are conceptually similar to NBO and, for the sake of brevity, will not be discussed here. 

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Block localized wavefunction (BLW) method s The electron delocalization to the cationic carbon and neutral boron center can be estimated by removing the vacant p-orbitals from the expansion space of molecular orbitals. Although this simple orbital deletion procedure (ODP) technique is limited to the analysis of positive hyperconjugation in carbocations and boranes, it has been generalized and extended to the block localized wavefunction (BLW) method. ... 

The unpaired electron has the complication that it is not localized on a single point but, in general, is delocalized on the entire molecule. So, in every point of space where the molecular orbital (MO) containing the unpaired electron has a non-zero value, the average electron magnetic moment sensed by the nucleus is different from zero and is proportional to (Sz) times the fraction of unpaired electron present at that point. Such a fraction is called spin density p, which for a single electron is given by the square of its wavefunction at that point. 

The determinant (= total molecular wavefunction T) just described will lead to (remainder of Section 5.2) n occupied, and a number of unoccupied, component spatial molecular orbitals i//. These orbitals i// from the straightforward Slater determinant are called canonical (in mathematics the word means in simplest or standard form ) molecular orbitals. Since each occupied spatial ip can be thought of as a region of space which accommodates a pair of electrons, we might expect that when the shapes of these orbitals are displayed ( visualized Section 5.5.6) each one would look like a bond or a lone pair. However, this is often not the case for example, we do not find that one of the canonical MOs of water connects the O with one H, and another canonical MO connects the O with another H. Instead most of these MOs are spread over much of a molecule, i.e. delocalized (lone pairs, unlike conventional bonds, do tend to stand out). However, it is possible to combine the canonical MOs to get localized MOs which look like our conventional bonds and lone pairs. This is done by using the columns (or rows) of the Slater T to create a T with modified columns (or rows) if a column/row of a determinant is multiplied by k and added to another column/row, the determinant remains kD (Section 4.3.3). We see that if this is applied to the Slater determinant with k = 1, we will get a new determinant corresponding to exactly the same total wavefunction, i.e. to the same molecule, but built up from different component occupied MOs i//. The new T and the new i// s are no less or more correct than the previous ones, but by appropriate manipulation of the columns/rows the i// s can be made to correspond to our ideas of bonds and lone pairs. These localized MOs are sometimes useful. 

Confusion often arises between the virtual orbitals 0, of SCF-MO theory and the localized valence antibonds Xb)-Although the spaces spanned by these sets overlap to a considerable extent, expansion of one set in terms of the other (e.g., in LCNBO-MO form) shows that they are far from identical. Indeed, the virtual orbitals are by their nature completely unoccupied, making no physical contribution to the SCF wavefunction or measurable properties. In contrast, the valence antibonds contribute irreducibly to the energy lowering and density shifts associated with electron delocalization, and their non-zero occupancies reflect the important physical effects of delocalization on the wavefunction and molecular properties. 

It is interesting to compare the Ce(CgH6)2 ground state wavefunction with a corresponding one of H2 in its S+ ground state at a stretched bond distance of 2 A (2.7 R<,), as shown in Figurelfi.lO on the left side. Here the two electrons are treated in an active space built by the ag and linear combinations of the Is orbitals on each of the two atoms. The orbitals a and b now represent any orthonormalized pair of orbitals built from these Is orbitals. The natural orbitals (rotation angle d> = 0°) correspond to the ag and orbitals, delocalized between the two atoms. A rotation by = 45° leads to the Is orbitals localized on the two... 

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