Foster-Boys localization

Li, Liu and Lu investigated the electronic structures and the possible aromaticity of some 10 r-electron systems, including the dication, at the HF/6-31G level [118]. The optimised S-S bond length of is 210 pm. Based on the analysis of the bonding characteristics in terms of the canonical molecular orbital and the Foster-Boys localized molecular orbital, they concluded that is of weak aromaticity. This is due to the occupation of the weak antibonding MOs. As a consequence, the bond strengths of the 10 r-electron systems decrease with respect to their 6 r-electron counterparts. 

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 ... 

From Eq. (13) it can be inferred that in region dominated by a single orbital (e.g., after Foster-Boys localization), the fp will be close to zero. With this argumentation, ELF should show regions dominated by localized orbitals. While this can be schematically shown for nearly perfectly localized orbitals [43, 44], for real systems the tp does not manifest such evident localization dependency. To express the spatial extent of orbital localization for all m occupied orbitals of the A -electron system at once, let us define the orbital-contribution entropy Sorb ... 

In the LORG theory, occupied orbitals are localized according to the Foster-Boys criterion16. In the calculations described above they chose the LORG centroid assignment113. 

Four different Fock/Kohn-Sham operators have been applied to obtain the orbitals, which are subsequently localized by the standard Foster-Boys procedure. In addition to the local/semi-local functionals LDA and PBE, the range-separated hybrid RSHLDA [37, 56] with a range-separation parameter of /r = 0.5 a.u. as well as the standard restricted Hartree-Fock (RHF) method were used. The notations LDA[M] and LDA[0] refer to the procedure applied to obtain the matrix elanents either by the matrix algebra [M] or by the operator algebra [O] method. All calculations were done with the aug-cc-pVTZ basis set, using the MOLPRO quantum chemical program package [57]. The matrix elements were obtained by the MATROP facility of MOLPRO [57] the Cg coefficients were calculated by Mathematica. 

Numerical LMOs of this work are determined by the natural localized-molecular-orbital (NLMO) method A. E. Reed and F. Weinhold, J. Chem. Phys. 83 (1985), 1736. The LMOs determined by other methods (e.g., C. Edmiston and K. Ruedenberg, Rev. Mod. Phys. 34 [1963], 457 and J. M. Foster and S. F. Boys, Rev. Mod. Phys. 32 [1960], 300) are rather similar, and could be taken as equivalent for present purposes. 

If the wave function that one considers, is a single Slater determinant , the spin orbitals (p, from which , is constructed, are not uniquely determined, but rather there is an infinity of equivalent sets of qo, related by unitary transformations. To some extent one can make the qo, unique if one requires either that they are canonical (diagonalize the Fock operator) and are symmetry-adapted, or localized (e.g. according to the criteria of Edmiston and Ruedenberg or Foster and Boys [1-3]). The localized spin orbitals have some advantages both for the chemical interpretation and for the computation of correlation corrections. 

The earliest approach to localized orbitals developed by Foster and Boys [14-16] applies the two-electron operator... 

Let s consider for a moment the Boys-Foster localization we can rewrite... 

The elaboration of the so-called exclusive orbitals was proposed by Foster and Boys [6]. This idea was modified by Boys whose method is often used when producing LMOs. In the procedure of Boys the criterion for finding localized molecular orbitals using a convenient unitary transformation is as follows the rj = r. - iv, i.e. the distances of the centroids of charge vectors of the orbitals... 

For the calculations of the shielding tensor using the Individual Gauge for Localized Orbitals (IGLO) method, the occupied MO s were localized by the method of Foster and Boys [81]. Unless noted otherwise, we use the experimental molecular geometries [82]. 

It is obvious that the problem is the delocalized nature of the MOs. Fortunately it has been known, based on the work of Lennard-Jones [192] and worked out mainly by Edmiston and Ruedenberg [193] and also by Foster and Boys [194], that it is possible to transform the (occupied) MOs to localized orbitals (LMOs) which span the same space, and which are orthogonal to each other, but which are mainly localized in rather restricted areas of space. 

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. 

Calculation of the Edmiston-Ruedenberg energy-localized MOs is very time consuming. Boys (and Foster) proposed a method to find localized MOs that is computationally much faster than the Edmiston-Ruedenberg method and that gives similar results in most cases see D. A. Kleier, J. Chem. Phys., 61, 3905 (1974). The Boys method defines the LMOs as those that maximize the sum of the squares of the distances between the centroids of charge of all pairs of occupied LMOs. The centroid of charge of orbital is defined as the point at (xc,yc>Zc), where Xc = ff>i x tf>i), yc = (I y I ). Zc = 4>i z (t>i) Tjj is, the distance between the centroids of LMOs i and j, the Boys LMOs maximize 2j>, 2, rfj. 

The method described above is essentially that due to Edmiston and Ruedenberg (1963). Another commonly used localization criterion (Boys and Foster, 1960) is that of maximum separation of the centroids of the transformed orbitals, and other methods have also been successfully applied (see e.g. Gilbert, 1964 Magnasco and Perico, 1967 McWeeny and Del Re, 1968). 

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