Semilocalized orbitals

Mueller, C. R., J. Chem. Phys. 19, 1498, Scmilocalized orbitals. II. A comparison of atomic, molecular and semilocalized orbital methods for diatomic hydrogenfluoride. ... 

Since the CH bond orbitals are relatively distant, their interaction is very weak and the six crCH semilocalized orbitals are very close in energy. On the other hand the bond orbitals interact strongly so that the energies of the six [Pg.24]

We nowr go one step further and allow for the mixing of [Pg.24]

The electronic interactions between the MMe3 substituents and the sulphur rm orbital were analysed121 on the basis of the semilocalized orbitals approximation in two series of the structures S(MMe3)2 and MeSMMe3 (M = C, Si, Ge, Sn, Pb). 

VALENCE BOND METHODS BASED ON SEMILOCALIZED ORBITALS 239... 

All the VB methods that deal with semilocalized orbitals use a generalization of the Coulson—Fischer idea (12), whereby a bond is described as a singlet coupling between two electrons in nonorthogonal orbitals that possess small delocalization tails resulting from the variational orbital optimization. Albeit formally covalent, this description implicitly involves some optimal contributions of ionic terms, as a decomposition of the wave function in terms of pure AO determinants would show (see Eqs. 3.5 and 3.6). For a polyatomic... 

The generalized valence bond (GVB) approach was one of the first methods where semilocalized orbitals, as developed by Coulson and Fischer, have been employed to polyatomic molecules [66-72]. GVB is typically applied within a restricted formulation introducing two simplifications for the construction of the VB structures (for a simple diatomic molecule AB, there are two VB structures the so-called ionic structures, and A B+, and a covalent structure, A-—B). In... 

Note that the rules and formulas that are expressed above in the framework of qualitative VB theory are independent of the type of orbitals that are used in the VB determinants purely localized AOs, fragment orbitals or Coulson-Fischer semilocalized orbitals. Depending on the kind of orbitals that are chosen, the h and S integrals take different values, but the formulas remain the same. 

Itoh, T., and Yoshizumi, H., J. Phys. Soc. Japan 10. 201, "Application of the alternant orbital method to benzene." Construction of a determinant using the semilocalized MO and selection of the singlet component of it. 

Morita, A., and Takahashi, K., Progr. Theoret. Phys. [Kyoto) 19, 257, Theory of cohesive energy of LiH crystal—the method of semilocalized crystalline orbitals."... 

Here, a and b are purely localized AOs, while Coulson—Fischer orbitals energy minimization, are generally not very delocalized (e < 1), and as such they can be viewed as distorted orbitals that remain atomic-like in nature. However, minor as this may look, the slight delocalization renders the Coulson—Fischer wave function equivalent to the VB-full wave function (Eq. 3.4a) with the three classical structures. A straightforward expansion of the Coulson—Fischer wave function leads to the linear combination of the classical structures in Equation 3.6. 

Note that if instead of using purely localized AOs for a and b, we use semilocalized Coulson-Fischer orbitals, Equation 3.37 will no more be the... 

This equivalence clearly projects that the MO—VB rivalry, discussed in Chapter 1, is unfortunate and senseless. Both VB and MO theories are not so diametrically different that they exclude each other, but rather two representations of reality, which are mathematically equivalent. The best approach is to use these two representations jointly and benefit from their complementary insight. In fact, from the above discussion of how to write a VB wave function, it is apparent that there is a spectrum of orbital representations that stretches between the fully local VB representations through semilocalized CF orbitals, to the use of delocalized fragment orbitals VB (FO—VB), and all... 

CF orbitals Coulson-Fisher-type orbitals. These are semilocalized AOs, also called overlap enhanced orbitals (OEOs) which are localized on a given center, but have small delocalization tails on other centers. Special cases of CF AOs are BDOs (see above). 

AU these efforts point out how the GGA form is severely limited, and in practice some compromise must be done between molecular applications, which require more nonlocal functionals [57], and soUd-state uses, for which local functionals always provide good results [58]. One general way to go beyond the GGA is to construct a fiUly nonlocal density functional. This goal is quite ambitious and trials might lead to functionals useless for practical purpose [59]. A more practical way is to construct a functional casting additional semilocal information. This can be done, for instance, including the kinetic energy density of the occupied KS orbitals ... 

It must be emphasized that this type of topological theory is not limited to three-center bonds. Its extension to other types of semilocal-ized multi-centered molecular orbitals might well prove useful in an extensive study of valence in certain intermetallic compounds in which... 

The main issue involved in using DFT and the KS scheme pertains to construction of expressions for the XC functional, Exc[n], containing the many-body aspects of the problems (1.38). The main approaches to this issue are (a) local functionals the Thomas Fermi (TF) and LDA, (b)semilocal or gradient-dependent functionals the gradient-expansion approximation (GEA) and generalized gradient approximation (GGA), and (c) nonlocal functionals hybrids, orbital functionals, and SIC. For detailed discussions the reader is referred to the reviews [257,260-272]. 

The analytical forms of the modern PPs used today have little in common with the formulas we obtain by a strict derivation of the theory (Dolg 2000). Formally, the pseudo-orbital transformation leads to nodeless pseudovalence orbitals for the lowest atomic valence orbitals of a given angular quantum number l (one-component) or Ij (two-component). The simplest and historically the first choice is the local ansatz for A VCy in Equation (3.4). However, this ansatz turned out to be too inaccurate and therefore was soon replaced by a so-called semilocal form, which in two-component form may be written as... 

This chapter is intended to give the reader an idea of what types of functionals exist, and to describe what their main features are, separately for local functionals (TF, LDA and Xa Sec. 5.1), semilocal, or gradient-dependent, functionals (GEA and GGA Sec. 5.2), and nonlocal functionals (hybrids, orbital functionals such as meta-GGAs, EXX and SIC, and integral-dependent functionals such as ADA Sec. 5.3). This chapter does deal only most superficially with the actual construction of these functionals. For more... 

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