Boys localization

The Boys localization scheme uses the square of the distance between two electrons as the operator, and minimizes the expectation value. 

The operator used in the Boys localization scheme can be expanded as... 

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. 

This energy difference should be a measure of the C-C bond energy in cyclopentane. These calculations used NBO localization (the result of Boys localization was messy when visualized) and CASSCF(2,2)/6-31G. ... 

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-... 

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. 

S. F. Boys, Localized orbitals and localized adjustment functions, in Quantum Theory of Atoms, Molecules, and the Solid State, edited by P.-O. Lowdin, pages 253-262. Academic Press, New York, 1966. 

The idea of distributed dipole moments has also been transferred to the dynamic domain and we shall discuss recent work from our laboratory in this section in more detail. With the help of maximally localized Wannier functions local dipoles and charges on atoms can be derived. The Wannier functions are obtained by Boys localization scheme [217]. Thus, Wannier orbitals [218] are the condensed phase analogs of localized molecular orbitals known from quantum chemistry. Access to the electronic structure during a CPMD simulation allows the calculation of electronic properties. Through an appropriate unitary transformation U of the canonical Kohn-Sham orbitals maximally localized Wannier functions (MLWFs)... 

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 ... 

The decomposition mechanisms were studied by using B3LYP/6-31 l+G(d,p)//B3LYP/6-31+G(d).3 The SOC matrix elements between singlet and triplet states were estimated by using CASCI wave functions based on Boys localized orbitals with respect to the full Pauli-Breit SOC operator see ref 1 for details. 

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. 

Hi) Three Q bonds (banana bonds). There are neither a nor n bonds, bnt the P-0 bond is a formal triple bond with the three curved regions of electron density between P and O in a symmetrical fashion. The remaining lone pair on the oxygen atom points away from phosphoras along the P-0 axis. The banana bond description originates from a Boys localization scheme [1,29-31] and from generalized valence bond calculations [6,32-34] for the phosphine oxide derivatives. 

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