VBSCF Methods

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. 

In the VBSCF method the wave function is expanded as a linear combination of the 20 structures described in Table 1. Since the orbitals are nonorthogonal, the sum of the squares of the structure expansion coefficients (C,) don t add up to one. However, a measure of the importance of each structure s wave function ( I, ) can be obtained from the formula in equation 1 for the weight, tV,l38) ... 

W. Wu, S. Shaik, W. H. Saunders, Jr., J. Phys. Chem. A 106, 11616 (2002). Comparative Study of Identity Proton Transfer Reactions Between Simple Atoms or Groups by VBSCF Methods. 

The Valence Bond Self-Consistent Field (VBSCF) method has been devised by Balint-Kurti and van Lenthe (32), and was further modified by Verbeek (6,33) who also developed an efficient implementation in a package called TURTLE (11). Basically, the VBSCF method is a multiconfiguration SCF procedure that allows the use of nonorthogonal orbitals of any type. The wave function is given as a linear combination of VB structures, (Eq. 9.7). 

The VBSCF method permits complete flexibility in the definition of the orbitals used for constructing the VB structures, d>A-. The orbitals can be allowed to delocalize freely during the orbital optimization (resulting in... 

OEOs), thereby performing GVB or SCVB calculations. The orbitals can also be defined by pairs that are allowed to delocalize on only two centers (BDOs), or they can be defined as strictly localized on a single center or fragment (see below). The VBSCF method is implemented in the TURTLE module (now being a part of GAMESS—UK) and in the XMVB package. 

The flexibility of the valence bond self-consistent field (VBSCF) method can be exploited to calculate VB wave functions based on orbitals that are purely localized on a single atom or fragment. In such a case, the VBSCF wave function takes a classical VB form, which has the advantage of giving a very detailed description of an electronic system, as, for example, the interplay between the various covalent and ionic structures in a reaction. On the other hand, since covalent and ionic structures have to be explicitly considered for... 

In its actual implementation, the VBPCM method is based on the VBSCF method (see above). Thus, the wave function is expressed in the usual manner as a linear combination of VB structures, Equation 9.8, but now these VB structures are optimized and interacting with one another in the presence of a polarizing field of the solvent, by a self-consistent procedure. Within this model, the interaction between solute and solvent is represented by an interaction potential, VR, which is treated as a perturbation to the Hamiltonian H° of the solute molecule in vacuum. The Schrodinger equation for the VB wave function now reads... 

The VBPCM procedure is not, in principle, restricted to the VBSCF method it has the potential ability to be implemented to more sophisticated methods like BOVB, VBCI, or other methods. The method is implemented in XMVB. 

It follows that the GVB calculation takes into account some radial correlation, not present at the VBSCF level, hence the higher energy of the latter. Of course, this is not a limitation of the VBSCF method, because one could have started with ionic structures that match the ones embedded in the GVB wave function. 

To run a BOVB calculation smoothly, it is advisable to start from an appropriate guess function, which may be, for example, a preliminary VBSCF wave function. In the XMVB program, the BOVB procedure sets up automatically by coding the keyword bovb in the input. As with the VBSCF method, the spectator orbitals in the BOVB method may be defined as either localized or delocalized, resulting in the L- and D-BOVB methods, respectively. 

Technically, the simultaneous optimization of orbitals and coefficients for a multistructure VB wave function can be done with the VBSCF method due to Balint-Kurti and van Lenthe [21,22], The VBSCF method has the same format as the classical VB method with an important difference. While the classical VB method uses orbitals that are optimized for the separate atoms, the VBSCF method uses a variational optimization of the atomic orbitals in the molecular wave function. In this manner the atomic orbitals adapt themselves to the molecular environment with a resulting significant improvement in the total energy and other computed properties. 

The VBSCF method [J. H. van Lenthe and G. G. BaUnt-Kurti, J. Chem. Phys., 78, 5699 (1983)] writes the molecular VB wave function as a linear combination of covalent and ionic VB structures and simultaneously optimizes the coefficients in the linear combination and the orbitals (which are expressed using a set of basis functions). The orbitals might be localized on individual atoms (as in the classical VB method of the last section) or they might be taken as sanidelocaUzed (as in the GVB method). 

The breathing-orbital VB (BOVB) method [P. C. Hiberty and S. Shaik, Theor. Chem. Acc., 108, 255 (2002) and references cited therein] differs in two ways from the VBSCF method. Each orbital in the BOVB method is always taken to be localized on an individual atom. The orbitals used in different structures are free to differ, so that each VB structure has its own set of optimized orbitals. 

VBCI A Post VBSCF Method that Involves Dynamic Correlation... 

---