Boys localization method

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 ... 

The local self-consistent field (LSCF) or fragment SCF method has been developed for treating large systems [105,134-139], in which the bonds at the QM/MM junction ( frontier bonds ) are described by strictly localized bond orbitals. These frozen localized bond orbitals are taken from calculations on small models, and remain unchanged in the QM/MM calculation. The LSCF method has been applied at the semiempirical level [134-137], and some developments for ab initio calculations have been made [139]. Gao et al. have developed a similar Generalized Hybrid Orbital method for semiempirical QM/MM calculations, in which the semiempirical parameters of atoms at the junction are modified to enhance the transferability of the localized bond orbitals [140]. Recent developments for ab initio QM/MM calculations include the method of Phillip and Friesner [141], who use Boys-localized orbitals in ab initio Hartree-Fock QM/MM calculations. These orbitals are again taken from calculations on small model systems, and kept frozen in QM/MM calculations. 

Interestingly, and probably due to a very exciting connection between the Fermi-hole and the localized orbitals [28], various localization methods result in rather similar localized orbitals, except for the description of double bonds by a o- and 7r-orbital-pair or two equivalent r (banana) bonds. Boys localization gives r orbitals, while the Edmiston-Ruedenberg and the popula-... 

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... 

A recently often used practical method is that of proposed by Pipek and Mezey [26], Their intrinsic localization is based on a special mathematical measure of localization. It uses no external criteria to. define a priori orbitals. The method is similar to the Edmiston-Ruedenberg s localization method in the a-n separation of the orbitals while it works as economically as the Boys procedure. For the application of their localization algorithm, the knowledge of atomic overlap integrals is sufficient. This feature allows the adoption of their algorithm for both ab initio and semiempirical methods. The implementation of die procedure in existing program systems is easy, and this property makes the Pipek-Mezey s method very attractive for practical use. 

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. 

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. 

The first linear scaling QMC method to use Gaussian basis functions is due to Manten and Luchow [162]. Their method truncated Boys-localized orbitals [164] by neglecting basis functions centered on atoms more than three bond lengths away from the centroid of the LMO. The deletion of basis functions simultaneously reduces both the number of LMOs that must be evaluated and the number of basis functions that must be evaluated and transformed. 

Advantages of the method the method is universal and can practically be applied for any system the method is efficient and can be automated to apply in computer code [62]. Disadvantages of the method in the method the criterion of localization is fixed as the Boys localization criterion, therefore there is no opportunity to receive WF as much as possible localized concerning any another criterion the symmetry use is not included, in some cases the number of iterations strongly depends on the initial approximation the use of a sufficiently dense grid of wavevectors is necessary to obtain a good accuracy in the gradient calculation. 

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 exploitation of localized orbitals for dispersion energy calculations has already been proposed since the early works on local correlation methods [41 5]. In classical and semiclassical models most often the atoms are selected as force centers only a few works exploit the advantages related to the use of two-center localized orbitals and lone pairs. A notable exception is the recent work of Silvestrelh and coworkers [46-50], who adapted the Tkatchenko-Scheffler model [16] for maximally localized Wannier functions (MLWF), which are essentially Boys localized orbitals for solids. It is worthwhile to mention that one of the very first use of the bond polarizabilities as interacting units for the description of London dispersion forces has been suggested as early as in 1969 by Claverie and Rein [51] see also [52],... 

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 ... 

Table 1 Molecular Cg coefficients from the dipolar oscillator orbited method, using LDA, PBE, RHF and sr-LDAdr-RHF deteminrmts in aug-cc-pVTZ basis set and Boys localized orbitals ...

Epoi also relies on a local picture as it uses polarizabilities distributed at the Boys LMOs centroids [44] on bonds and lone pairs using a method due to Garmer et al. [35], In this framework, polarizabilities are distributed within a molecular fragment an therefore, the induced dipoles do not need to interact together (like in the Appleq-uist model) within a molecule as their value is only influenced by the electric fields from the others interacting molecules. 

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. 

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. 

We have successively used the Magnasco-Perico (1967) external criterion and the Boys (1960) internal criterion. The results obtained by the Magnasco-Perico procedure allowed us (Leroy and Peeters, 1975) to study the transferable properties of localized orbitals and to elaborate a simple parametric method to construct wave functions for saturated hydrocarbons (Degand et al., 1973), unsaturated hydrocarbons (Leroy and Peeters, 1974), heteroatomic aliphatic compounds (Clarisse et al., 1976), and polymers (Peeters et al., 1980). Furthermore, we have been able to analyze the concept of bond energy in terms of localized orbitals (Leroy et al., 1975). A careful review on the utilization of transferability in MO theory has been realized by O Leary et al. (1975). 

Various methods of constructing a unitary matrix V with localizing properties have been proposed [for a review, see Ref.45>]. For the present analysis we have adopted Boys method4 ), as being the simplest intrinsic method (for a definition of intrinsic versus external methods, see Ruedenberg47)]. 

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