Orthogonalized valence bond method

The valence-bond approach plays a very important role in the qualitative discussion of chemical bonding. It provides the basis for the two most important semi-empirical methods of calculating potential energy surfaces (LEPS and DIM methods, see below), and is also the starting point for the semi-theoretical atoms-in-molecules method. This latter method attempts to use experimental atomic energies to correct for the known atomic errors in a molecular calculation. Despite its success as a qualitative theory the valence-bond method has been used only rarely in quantitative applications. The reason for this lies in the so-called non-orthogonality problem, which refers to the difficulty of calculating the Hamiltonian matrix elements between valence-bond structures. 

The generalised valence bond (GVB) method, developed by Goddard in 1970, is one of the simplest and oldest valence bond methods that use flexible orbitals in a general way. The generalised Coulson-Fischer theory for the hydrogen molecule mentioned above is used to describe every electron pair in a molecule. The orbitals for each electron pair are expanded in terms of the full basis set and are non-orthogonal. Orbitals from different pairs are forced to be orthogonal. This condition simplifies the calculations but may lead to some difficulties [160,161],... 

One widely used valence bond theory is the generalised valence bond (GVB) method of Goddard and co-workers [Bobrowicz and Goddard 1977]. In the simple Heitler-London treatment of the hydrogen molecule the two orbitals are the non-orthogonal atomic orbitals on the two hydrogen atoms. In the GVB theory the analogous wavefunction is written ... 

However, using the newly developed nonorthogonal Valence Bond SCF (VBSCF) method these VB structures can be constructed directly from purely carbene localized orbitals, without the uncertainty introduced by the orthogonality tails15,16. The orthogonal LMO analysis described above (OVB) is more convenient computationally, but a limited number of real VB calculations need to be carried out on actual heteronuclear doublebond systems to compare with and to validate the LMO results. This analysis has been carried out here using ab initio VBSCF computer codes. 

Because the n-networks of benzenoid hydrocarbons are the classical prototypical example of delocalized bonding, they provide a crucial test for chemical-bonding theories. Here there is revealed a systematic organization for valence-bond views to describe such n-bonding. With an initiation near the ab initio realm a sequence of semiempirical valence-bond models is identified and characterized. The refinement from one model to the next proceeds via either a (perturbative) restriction to a smaller model space or orthogonalization of a suitable natural basis for the model space. The known properties of the models are indicated, and possible methods of solution are mentioned. A great diversity of work is outlined, related, systematized and extended. New research is suggested. 

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]. 

Therefore, the dependence on the coefficients does not enter the gradient expression not for fixed orbitals, which is the classical Valence Bond approach and not for optimised orbitals, irrespective of whether they are completely optimised or if they are restricted to extend only over the atomic orbitals of one atom. If the wavefimction used in the orbital optimisation differs, additional work is required. This would apply to a multi-reference singles and doubles VB (cf. [20,21]). Then we would require a yet unimplemented coupled-VBSCF procedure. Note that the option to fix the orbitals is not available in orthogonal (MO) methods, due to the orthonormality restriction. 

J.H. van Lenthe and G.G. Balint-Kurti, VBSCF The optimisation of non-orthogonal orbitals in a general (Valence Bond) wavefunction, in 5th seminar on Computational Methods in Quantum Chemistry (Groningen, 1981). 

The latter was developed as a single-configuration method, complemented by non-orthogonal configuration-interaction calculations exploiting virtual orbitals in the Spin-Coupled Valence Bond (SCVB) approach [5] (for a review, see e.g. Ref. [6]). More recently, the use of perturbative virtuals has been introduced, giving rise to the SCVB variant of the method [7] [8]. There are further accounts of SCVB in this volume. 

One popular wavefunction of this form is the generalized valence bond (GVB) expansion " . With this method an N-electron molecule is described in terms of N non-orthogonal orbitals and an energy-optimized spin function as... 

The purpose of this review is to give an account of approaches of this type. That is to say we examine methods where non-orthogonal orbitals enter directly into the wavefunctions. The fundamental prototype is of course the classical valence bond (VB) theory and accordingly we begin with a survey of the description it provides of molecular electronic structure, and of its important conceptual role in the description of many fundamental molecular processes. 

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