Orbitals localization methods

Note that this correction has the problem that the Kohn-Sham equation is not invariant for the unitary transformation of occupied orbitals, even after the correction, differently from the Hartree-Fock equation. In the Hartree-Fock equation, the variations of the Coulomb self-interaction energy and its potential for the unitary transformations of occupied orbitals cancel out with those of the exchange self-interaction, while these are not compensated, even after the correction in the Kohn-Sham equation. Therefore, the effect of the self-interaction correction depends on the difference in occupied orbitals before and after the unitary transformation. For removing this difference, it is usual to localize the orbitals before the self-interaction correction (Johnson et al. 1994). Note, however, that there are various types of orbital localization methods, and the effect of the selfinteraction correction inevitably depends on them. Combining with the optimized effective potential (OEP) method (see Sect. 7.5) may be one of the most efficient ways to solve this problem. This combination enables us to consistently obtain localized potentials with no self-interaction error. 

The localized orbitals generation from the set of canonical MO allowed by symmetry requires additional choice of the localization criteria. The different orbital localization methods are implemented in molecular computer codes [35]. All of them are connected with the search for the coefficients connecting LMO with the canonical MO to satisfy the localization criteria. 

LORG (localized orbital-local origin) technique for removing dependence on the coordinate system when computing NMR chemical shifts LSDA (local spin-density approximation) approximation used in more approximate DFT methods for open-shell systems LSER (linear solvent energy relationships) method for computing solvation energy... 

We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

We have used the multisublattice generalization of the coherent potential approximation (CPA) in conjunction with the Linear-MufRn-Tin-Orbital (LMTO) method in the atomic sphere approximation (ASA). The LMTO-ASA is based on the work of Andersen and co-workers and the combined technique allows us to treat all phases on equal footing. To treat itinerant magnetism we have employed for the local spin density approximation (LSDA) the Vosko-Wilk-Nusair parameterization". 

The operation involved in the definition of the EPI is an exchange of atoms on sites i and j and it is a kind of localized perturbation. So the orbital peeling method provides an efficient means for obtaining the generalized phase shifts. 

Theoretical calculations were performed with the linear muffin tin orbital (LMTO) method and the local density approximation for exchange and correlation. This method was used in combination with supercell models containing up to 16 atoms to calculate the DOS. The LMTO calculations are run self consistently and the DOS obtained are combined with the matrix elements for the transitions from initial to final states as described in detail elsewhere (Botton et al., 1996a) according to the method described by Vvedensky (1992). A comparison is also made between spectra calculated for some of the B2 compounds using the Korringa-Kohn-Rostoker (KKR) method. 

Haberlen, O.D. and Rdsch, N. (1992) A scalar-relativistic extension of the linear combination of Gaussian-type orbitals local density functional method application to AuFl, AuCl and Au2. Chemical Physics Letters, 199, 491-496. 

Garmer DR, Stevens WJ (1989) Transferability of molecular distributed polarizabilities from a simple localized orbital based method. J Phys Chem 93 8263... 

Stabilization of the systems due to homoconjugation is discussed. 13C and nB NMR chemical shifts of the compounds were also calculated using the individual gauge for localized orbitals (IGLO) method <2000JOC5956>. 

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. 

To circumvent problems associated with the link atoms different approaches have been developed in which localized orbitals are added to model the bond between the QM and MM regions. Warshel and Levitt [17] were the first to suggest the use of localized orbitals in QM/MM studies. In the local self-consistent field (LSCF) method the QM/MM frontier bond is described with a strictly localized orbital, also called a frozen orbital [43]. These frozen orbitals are parameterized by use of small model molecules and are kept constant in the SCF calculation. The frozen orbitals, and the localized orbital methods in general, must be parameterized for each quantum mechanical model (i.e. energy-calculation method and basis set) to achieve reliable treatment of the boundary [34]. This restriction is partly circumvented in the generalized hybrid orbital (GHO) method [44], In this method, which is an extension of the LSCF method, the boundary MM atom is described by four hybrid orbitals. The three hybrid orbitals that would be attached to other MM atoms are fixed. The remaining hybrid orbital, which represents the bond to a QM atom, participates in the SCF calculation of the QM part. In contrast with LSCF approach the added flexibility of the optimized hybrid orbital means that no specific parameterization of this orbital is needed for each new system. 

That is, it also reduces the long-range repulsions between the orbitals as much as possible. Thus, this localization method achieves three objectives concentration of the molecular orbitals, short-range separation of different orbitals, and long-range separation of different orbitals. 

For such localizations to be effective in the present context, those orbital symmetry constraints that would prevent maximal localization in larger molecules must be abandoned. For instance, in the NCCN molecule, Ciy symmetry can be preserved during the localization process, but not left-right mirror symmetry. We have used Raffenetti s (55) version of the Edmiston-Ruedenberg localization method 54). 

It is prerequisite to define localized, diabatic state wave fimctions, representing specific Lewis resonance configurations, in a VB-like method. Although this can in principle be done using an orbital localization technique, the difficulty is that these localization methods not only include orthorgonalization tails, but also include delocalization tails, which make contribution to the electronic delocalization effect and are not appropriate to describe diabatic potential energy surfaces. We have proposed to construct the locahzed diabatic state, or Lewis resonance structure, using a strictly block-localized wave function (BLW) method, which was developed recently for the study of electronic delocalization within a molecule.(28-3 1)... 

In this paper we present preliminary results of an ab-initio study of quantum diffusion in the crystalline a-AlMnSi phase. The number of atoms in the unit cell (138) is sufficiently small to permit computation with the ab-initio Linearized Muffin Tin Orbitals (LMTO) method and provides us a good starting model. Within the Density Functional Theory (DFT) [15,16], this approach has still limitations due to the Local Density Approximation (LDA) for the exchange-correlation potential treatment of electron correlations and due to the approximation in the solution of the Schrodinger equation as explained in next section. However, we believe that this starting point is much better than simplified parametrized tight-binding like s-band models. 

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