Generalized Valence Bond GVB Method

A number of types of calculations begin with a HF calculation and then correct for correlation. Some of these methods are Moller-Plesset perturbation theory (MPn, where n is the order of correction), the generalized valence bond (GVB) method, multi-conhgurational self-consistent held (MCSCF), conhgu-ration interaction (Cl), and coupled cluster theory (CC). As a group, these methods are referred to as correlated calculations. 

An MCSCF calculation in which all combinations of the active space orbitals are included is called a complete active space self-consistent held (CASSCF) calculation. This type of calculation is popular because it gives the maximum correlation in the valence region. The smallest MCSCF calculations are two-conhguration SCF (TCSCF) calculations. The generalized valence bond (GVB) method is a small MCSCF including a pair of orbitals for each molecular bond. 

The generalized valence bond (GVB) method was the earliest important generalization of the Coulson—Fischer idea to polyatomic molecules (13,14). The method uses OEOs that are free to delocalize over the whole molecule during orbital optimization. Despite its general formulation, the GVB method is usually used in its restricted form, referred to as GVB SOPP, which introduces two simplifications. The first one is the perfect-pairing (PP) approximation, in which only one VB structure is generated in the calculation. The wave function may then be expressed in the simple form of Equation 9.1, as a product of so-called geminal two-electron functions ... 

Although the emphasis in ab initio calculations in recent years has been on calculations using the MO approach, alternative methods have not been neglected, particularly Valence Bond (VB) methods of various types. Particularly promising has been the work of Goddard and co-workers with the so-called Generalized Valence Bond (GVB) method.13... 

Finally we discuss the Generalized Valence Bond (GVB) method as proposed by Hunt, Hay and Goddard./180,181,182/ This is a method aimed at improving some of the HF procedures, for instance to obtain the correct dissociation limit, and to partially include correlation effects. The basic idea is that the type... 

Generalized Gradient Approximation (GGA), in density functional theory, 184 Generalized frequencies, 314 Generalized Valence Bond (GVB) method, 202... 

The generalized valence bond (GVB) method developed by Gaddard and coworkers does not employ a full Cl in the valence shell, but includes only the configurations needed to describe proper dissociation of a chemical bond. As will be demonstrated later, such a wavefunction represents a restricted form of the CASSCF wavefunction, where the active subspace is partitioned into subsets with a fixed number of electrons occupying the orbitals in each subset (acutally the GVB function is further restricted by allowing only specific spin couplings within each subset). 

THE MULTICONnCURATION SELF-CONSISTENT FIELD (MCSCF) AND THE GENERALIZED VALENCE BOND (GVB) METHODS... 

At the Hartree-Fock level (Hartree 1928 Fock 1930), the energies for closed-shell systems are evaluated using the restricted Hartree-Fock (RHF) method (Hall and Lennard-Jones 1951 Roothaan 1951). For the open-shell molecules, there are several methods that are available in most programs the unrestricted Hartree-Fock (UHF) method (Pople and Nesbet 1954), several variants of the restricted open-shell Hartree-Fock (ROHF) method (Hsu et al. 1976 McWeeny and Diercksen 1968), and the generalized valence bond (GVB) method (Bobrowicz and Schaefer 1977). 

This wavefunction is a linear combination of two configurations, and their expansion coefficients, Cl and C2, are obtained as variational parameters. Each configuration places both electrons in the same spatial orbital, and c )i and c )2 are orthogonal orbitals. This representation of a valence bond wavefunction proves particularly helpful in computer calculations of the wavefunctions as in the generalized valence bond (GVB) method. 

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