Overview of CASVB Method

We have proposed two types of CASVB method. The first one is a method where the valence bond structures are constructed from orthogonal localized molecular orbitals (LMOs) [1], and the second is one from nonorthogonal localized molecular orbitals [2]. 

The idea of CASVB is based on the fact that the densities of variational wave functions are invariant under the transformations which hold the variational space unchanged. In the CASSCF case, a complete active space (CAS) is invariant under the linear transformation of active orbitals and also that of configuration state functions (CSFs). 

We may re-define the active orbitals utilizing the invariance of the active orbital space. In the CASVB with nonorthogonal LMOs, we employ Rueden-berg s procedure of projected localized MOs [6-8] and obtain quasi-atomic CASSCF MOs that have maximal overlaps with atomic orbitals (AOs) of the free atoms. Consider an AO, Xa, centered on a nucleus A. Diagonalizing the matrix, 

The full configuration space that is spanned by all possible configurations generated from these quasi-atomic CASSCF MOs is identical to that of full Cl space that is constructed from the canonical CASSCF MOs. Thus, we use (p as orbitals from which a CASVB wave function is constructed. To obtain the corresponding VB structures, we project a canonical CASSCF wave function onto a VB wave function. The projection does not modify the original wave function but simply re-expresses it in the VB language. Let Vt/CASSCF be a CASSCF wave function, 

The occupation number (or weight) of a VB structure is calculated with 

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