Foster-Boys Method

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 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 ... 

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. 

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]. 

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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