Boys localization procedure

In our calculations Boys localization procedure was applied to localize the occupied and virtual orbitals in two separate blocks. The properties of the localized virtual orbitals of normal saturated hydrocarbons are investigated in detail in [65] and for all-trans conjugated polyenes in [68], In Fig. 1 the schematic plots of ellipsoids of selected occupied and virtual orbitals of C5H12 in the 6-31G basis set are presented in the plane of the CC bonds. Only two characteristic sets of LMOs are selected, a terminal CH bond and a CC bond. The left panel shows the occupied and the right the virtual LMOs. The plots of occupied and virtual LMOs in minimal basis are very similar to case (a) and they are not presented here. We can see that the localized orbitals are well separated and their spatial extent is restricted to a small part of the molecule even for the diffuse u2 and 0 3 LMOs. The virtual LMOs are spread along the axis of chemical bonds and they are not restricted to the region between the atoms. 

In this section representative results are presented for a normal saturated hydrocarbon (C5H12) molecule as a model system. The CC bond length is 1.526 A the CH distance is 1.094 A and the bond angles are 120°. The LMOs were created using the Boys localization procedure. 

In conclusion, the Boys localization procedure may be considered as the mathematification of the qualitative Lewis and Linnett theories. As pointed out, in this perspective a formal signification of the symbols used by these authors can be proposed these represent not localized electrons but centroids of charge of localized orbitals. Within this interpretation, the former theories agree much better with the ideas of quantum mechanics such as the wave character of electrons that prevents their localization. Moreover, the results obtained by the Boys procedure allow us to predict the Lewis and (or) Linnett structures of many chemical species without performing explicit calculations. For example, linear and bent 1,3-dipoles are described by the following general formulas ... 

One may re-define the active orbitals utilizing the invariance of the active orbital space. In the orthogonal CASVB method, the LMOs constructed by Boys localization procedure are used that is, active orbitals are transformed so as to have the minimum sum of expectation values. If the active orbitals are defined appropriately, the LMOs obtained nearly always turn out to be localized on a single atomic center with small localization tails on to neighboring atoms. In the non-orthogonal CASVB case, the atomic-like orbitals are constructed by Ruedenberg s projected localization procedure. 

FIGURE 23. Pictorial representation of the matrix elements (in eV) of the Hartree-Fock matrix of cubane 80 in a localized basis. The values have been obtained by subjecting the STO-3G ab initio results to a Foster-Boys localization procedure The crossterms refer to the LMOs indicated by bold lines and to the symmetry-equivalent pairs... 

The a posteriori localization of the subspace of the occupied orbitals is a relatively standard procedure, which can be achieved following a large variety of localization criteria (for a succinct overview, see Ref. [53]). In the context of correlation energy calculations, i.e., in various local correlation approaches , the most widely used localization methods are based either on the criterion of Foster and Boys [24] or that proposed by Pipek and Mezey [54]. For reasons which become clearer below, in the present work we will use the Foster-Boys localization criterion, which can be expressed in various equivalent forms [26]. In its the most suggestive formulation, the Foster-Boys localization procedure consists in the maximization of the squared distance between the centroids of the orbitals ... 

A single-determinant wave-function of closed shell molecular systems is invariant against any unitary transformation of the molecular orbitals apart from a phase factor. The transformation can be chosen in order to obtain LMOs. Starting from CMOs a number of localization procedures have been proposed to get LMOs the most commonly used methods are as given by the authors of (Edmiston et ah, 1963) and (Boys, 1966), while the procedures provided by (Pipek etal, 1989) and (Saebo etal., 1993) are also of interest. It could be stated that all the methods yield comparable results. Each LMO densities are found to be relatively concentrated in some spatial region. They are, furthermore, expected to be determined mainly by that part of the molecule which occupies that given region and its nearby environment rather than by the whole system. 

One more example of the CASSCF procedure will be outlined calculating the barrier to rotation around the CC double bond in ethene. Step 2, orbital localization, showed nicely localized orbitals when NBO localization was used, but the orbitals were harder to identify with Boys localization. For a CAS(2,2)/6-31G optimization the active orbitals chosen were the n and 7t MOs, and for a CAS(4,4)/6-31G optimization the n, n, cr and cr MOs. The input structures were the normal planar ethene and perpendicular (90° twisted) ethene. Optimization and frequency calculations gave a minimum for the planar and a transition state for the perpendicular structures. The energies (without ZPE, for comparison with those calculated with the GVB method by Wang and Poirier [71]) were ... 

Localized MO s have been derived for Bi6H20, using Boy s procedure, from the results of a PRDDO calculation, based on the simpler hydrides... 

Second, the topological method attempts to avoid all arbitrary decisions in assigning electrons. The choice of atomic orbitals, which clearly determines the Mulliken population, is not a factor in the topological procedure. The set of molecular orbitals one uses (e.g., the canonical orbitals or the natural orbitals or Boys-localized orbitals) alters neither the energy of the molecule nor the electron density. However, the NPA method makes use of a particular set of MOs only. 

The localization procedure developed by Boys <60MI 209-01 > was also used and As,l (Boys) was calculated as above. Values calculated by both methods correlate well with experimental aromaticity criteria although the Boys method provides better correlation. The order of aromaticity as shown by these calculations is thiophene > pyrrole > furan which is not always the case by some of the other methods. 

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. 

The local MP2 electron-correlation method for nonconducting crystals [109] is an extension to crystalline solids of the local correlation MP2 method for molecules (see Sect. 5.1.5), starting from a local representation of the occupied and virtual HF subspaces. The localized HF crystalline orbitals of the occupied states are provided in the LCAO approximation by the CRYSTAL program [23] and based on a Boys localization criterion. The localization technique was considered in Sect. 3.3.3. The label im of the occupied localized Wannier functions (LWF) Wim = Wj(r — Rm) includes the type of LWF and translation vector Rm, indicating the primitive unit cell, in which the LWF is centered (m = 0 for the reference cell). The index i runs from 1 to A i, the number of filled electron bands used for the localization procedure the correlation calculation is restricted usually to valence bands LWFs. The latter are expressed as a linear combination of the Gaussian-type atomic orbitals (AOs) Xfiif Rn) = Xfin numbered by index = 1,..., M M is the number of AOs in the reference cell) and the cell n translation vector... 

The localization procedure follows the prescription of Boys [46, 47, 48, 49] and leads to Localized Molecular Orbitals (LMO) cpi that bear physical meaning. These orbitals are centered around nuclei or bonds and are thus easily interpreted in the way of textbook chemistry. The essence of the IGLO method is the choice of an Individual Gauge for (each) Localized Molecular Orbital, which explains the abbreviation. The gauge origin R is set to the centre of charge of every LMO independently. Additionally, the stationarity of the so-... 

It is also possible to localize the MOs by localization procedures. We have shifted the emphasis from previous localization procedures by Boys and Ruedenberg to a new localization procedure by Pipek and Mezey. It maximizes the population localization and is suitable for ab initio and semiempirical methods. 

Although the localization by energy criteria (Edmiston-Ruedenberg) may be considered more fundamental than one based on distance (Boys) or atomic charge (Pipek-Mezey), the difference in computational effort means that the Boys or Pipek-Mezey procedures are often used in practice, especially since there is normally little difference in the shape of the final LMOs. 

As to the localization of occupied orbitals the conventional procedures (Edmiston-Ruedenberg (Edmiston etal., 1963), Boys (Boys, 1966) might not be the most suitable because they do not restrict the magnitude of the off-diagonal Fock matrix elements. Regarding the localization of virtual orbitals, they cannot be localized uniquely into... 

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