The Generalized Valence-Bond Method

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  

Each geminal function is a singlet-coupled GVB pair ( p a, p b) that is associated with a particular bond or lone pair in the molecule. For example, CH4 will have the familiar Lewis structure and its wave function will involve a product of four geminal functions, each corresponding to a C-H bond. 

The second simplification, which is introduced for computational convenience, is the strong orthogonality (SO) constraint, by which all the orbitals in 

Equation 9.1 are required to be orthogonal to each other unless they belong to the geminal pair, that is, 

This strong orthogonality constraint, while seemingly a restriction, is usually not a serious one, since it applies to orbitals that are not expected to overlap significantly. On the other hand, the orbitals ( p a, p b) that are coupled together in the same GVB pair, of course, display a strong overlap. The term SOPP GVB then describes a perfectly paired GVB wave function generated under the constraint of zero-overlap between the orbitals of different pairs. 

The Heitler-London VB wave function for ground-state Ha is [Eq. (13.101)] ls (l)lsj(2) ls (2)lsj(l) multiphed by a normalization constant and a spin function. The GVB ground-state Ha wave function replaces this spatial function by/(l)g(2) + /(2)g(l), where the functions/and g are found by minimization of the variational integral. To find / and g, one expands each of them in terms of a basis set of AOs and finds the expansion coefficients by iteratively solving one-electron equations that resemble the equations of the SCF MO method. 

Clearly, the GVB method will give a lower energy than the simple VB wave function. The GVB method allows for the change in the AOs that occurs on molecule formation by solving variationally for / and g. In the VB method, this change is allowed for by adding to the wave function terms that correspond to ionic and other resonance structures. The GVB wave function is thus much simpler than a VB wave function with resonance structures and the calculations are simpler. 

The GVB method gives a of 4.12 eV for ground-state H2, as compared with 3.15 eV for the Heitler-London VB function, 3.78 eV for the Heitler-London-Wang function with an optimized orbital exponent, 4.03 eV for the Weinbaum function (13.110) that includes an ionic term, and 4.75 eV for the experimental value. At very large intemuclear distance R, the GVB functions / and g approach the atomic orbitals s and Hj,. Thus, like the VB (but unlike the MO ), the GVB wave function shows the correct behavior on dissociation. At intermediate distances, / is a linear combination of AOs that has its most important contribution fi om but that has a significant contribution from and lesser contributions fi om the other AOs (these contributions reflect the polarization of the AO that occurs on molecule formation). 

In the MO method, there is one orbital (an MO) for each electron pair. In the GVB method, there are two orbitals (/and g in the H2 example) for each electron pair. 

For CH4 the VB wave function with resonance structures omitted is (15.158). For CH4 the GVB wave function is 

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Ionova I V and Carter E A 1995 Crbital-based direct inversion in the iterative subspace for the generalized valence bond method J. Chem. Phys. 102 1251... 

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.)... 

The precise quantum cluster calculations of the electronic structure of SC ceramics were performed in Refs. [13,17,21]. Guo et al. [13] used the generalized valence bond method, Martin and Saxe [17] and Yamamoto et al. [21] performed calculations at the configuration interaction level. But in these studies the calculations were carried out for isolated clusters, the second aspect of the ECM scheme, see above, was not fulfilled. The influence of crystal surrounding may considerably change the results obtained. 

Another, quite independent theoretical assessment, based on ab initio calculations and the generalized valence bond method, found that the barrier height for cis,trans isomerization of cyclopropane is essentially the same (calculated value, 60.5 kcal mol-1) whether one or both of the thermal CH2 groups are rotated after opening of the CC bond 243. Thus, in 1972, there seemed to be general agreement among theoreticians that the stereomutations of cyclopropane should take place with k, about equal to kl2. 

The generalized valence bond method [55] can be considered as a special APSG technique, where each geminal consists only of two spatial orbitals. 

The choice of reference space for MRCI calculations is a complex problem. First, a multieonfigurational Hartree-Fock (MCSCF) approach must be chosen. Common among these are the generalized valence-bond method (GVB) and the complete active space SCF (CASSCF) method. The latter actually involves a full Cl calculation in a subspace of the MO space—the active space. As a consequence of this full Cl, the number of CSFs can become large, and this can create very long Cl expansions if all the CASSCF CSFs are used as reference CSFs. This problem is exacerbated when it becomes necessary to correlate valence electrons in the Cl that were excluded from the CASSCF active space. It is very common to select reference CSFs, usually by their weight in the CASSCF wave function. Even more elaborate than the use of a CASSCF wave function as the reference space is the seeond-order Cl, in which the only restriction on the CSFs is that no more than two electrons occupy orbitals empty in the CASSCF wave function. Such expansions are usually too long for practical calculations, and they seldom produce results different from a CAS reference space MRCI. 

Chapter 4 discusses configuration interaction (Cl) and is the first of the four chapters that deal with approaches incorporating electron correlation. One-electron density matrices, natural orbitals, the multiconfiguration self-consistent-field approximation, and the generalized valence bond method are... 

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