Slater determinants valence bond theory

Previously, we have treated orbitals as covering the molecule as a whole, and have not from the start restricted the orbitals to any one atom. Many molecular orbitals can be approximated as linear combinations of atomic orbitals. Another way to consider molecular wavefunctions is in terms of products of atomic orbitals. This is valence bond theory, and ultimately it is very useful for describing the structures of molecules. Valence bond (or VB) theory dates from 1927, when W. Heftier and F. W. London constructed the first successful quantum-mechanical approximation of the hydrogen molecule, H2. It was developed further by J. C. Slater (of Slater determinant fame) and Linus Pauling. 

In practice, each CSF is a Slater determinant of molecular orbitals, which are divided into three types inactive (doubly occupied), virtual (unoccupied), and active (variable occupancy). The active orbitals are used to build up the various CSFs, and so introduce flexibility into the wave function by including configurations that can describe different situations. Approximate electronic-state wave functions are then provided by the eigenfunctions of the electronic Flamiltonian in the CSF basis. This contrasts to standard FIF theory in which only a single determinant is used, without active orbitals. The use of CSFs, gives the MCSCF wave function a structure that can be interpreted using chemical pictures of electronic configurations [229]. An interpretation in terms of valence bond sti uctures has also been developed, which is very useful for description of a chemical process (see the appendix in [230] and references cited therein). 

Most of the commonly used electronic-structure methods are based upon Hartree-Fock theory, with electron correlation sometimes included in various ways (Slater, 1974). Typically one begins with a many-electron wave function comprised of one or several Slater determinants and takes the one-electron wave functions to be molecular orbitals (MO s) in the form of linear combinations of atomic orbitals (LCAO s) (An alternative approach, the generalized valence-bond method (see, for example, Schultz and Messmer, 1986), has been used in a few cases but has not been widely applied to defect problems.)... 

In 1927, Burrau calculated the energy of Hj and Heitler and London treated the hydrogen molecule. In 1928, the Heitler-London or valence bond method was applied to many electron systems, and simultaneously Hund and Mulliken started the development of the molecular orbital theory. In 1931, Slater expressed the v/avefunctions of complex molecules in terms of Slater determinants made up of linear combinations of atomic orbitals. Thus, the Golden Age was born. 

Slater determinants are usually constructed from molecular spinorbitals. If, instead, we use atomic spinorbitals and the Ritz variational method (Slater determinants as the expansion functions), we would get the most general formulation of the valence bond (VB) method. The beginning of VB theory goes back to papers by Heisenbeig, the first application was made by Heitler and London, and later theory was generalized by Hurley, Lennard-Jones, and Pople. The essence of the VB method can be explained by an example. Let us take the hydrogen molecule with atomic spinorbitals of type liaO and Vst (abbreviated as aa and b ) centered at two nuclei. Let us construct from them several (non-normalized) Slater determinants, for instance ... 

The centrality of the FNA has spawned considerable research into improvement of the approach. The strategies for obtaining better nodes are numerous. Canonical HF orbitals, Kohn-Sham orbitals from density functional theory (DFT), and natural orbitals from post-HF methods have been used. The latter do not necessarily yield better nodes than single configuration wave functions [39-41]. More success has been found with alternative wave function forms that include correlation more directly than sums of Slater determinants. These include antisymmetrized geminal power functions [42,43], valence-bond [44,45] and Pfaffian [46] forms as well as... 

PHF methods can, in turn, be classified as the variational and nonvariational ones. In the former gronp of methods the coefficients in linear combination of Slater determinants and in some cases LCAO coefficients in HF MOs are optimized in the PHF calculations, in the latter such an optimization is absent. To the former group of PHF methods one refers different versions of the configuration interaction (Cl) method, the multi-configuration self-consistent field (MCSCF) method, the variational coupled cluster (CC) approach and the rarely used valence bond (VB) and generaUzed VB methods. The nonvariational PHF methods inclnde the majority of CC reaUza-tions and many-body perturbation theory (MBPT), called in its molecular realization the MoUer-Plessett (MP) method. In MP calculations not only RHF but UHF MOs are also used [107]. 

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