Qualitative VB theory

A VB calculation is just a configuration interaction in a space of AO or FO determinants, which are in general nonorthogonal to each other. It is therefore essential to derive some basic rules for calculating the overlaps and Hamiltonian matrix elements between determinants. The fully general rules have been described in detail elsewhere. Examples will be given here for commonly encovmtered simple cases. 

The overlap between the (unnormalized) determinants jaabbj and jccdd I is given by Eq. [28]  

One then integrates Eq [28] electron by electron, leading to Eq. [30] for the overlap between [aabb] and [ccdd]  

Generalization to different types of determinants is trivial. As an appU-cation, let us obtain the overlap of a VB determinant with itself, and calculate the normalization factor N of the determinant in Eq. [26]  

Generally, normalization factors for determinants are larger than unity. The exception is those VB determinants that do not have more than one spin orbital of each spin variety (e.g., the determinants that compose the HL wave function). For these latter determinants the normalization factor is unity (i.e., N= 1). 

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Note that the rules and formulas that are expressed above in the framework of qualitative VB theory are independent of the type of orbitals... 

Goddard et al. (1) and subsequently the present authors (4) also provided a simple VB explanation for the choice of the ground state. Let us reiterate this explanation based on our qualitative VB theory, outlined in Chapter 3. 

The question is whether or not these reliable predictions of quantitative VB theory may also arise from a qualitative VB theory. Early semiempirical HLVB calculations by Wheland (14,15) and for that matter any VB calculations with only HL structures, incorrectly predict that CBD has resonance energy larger than that of benzene. Wheland, who analyzed the CBD problem, concluded that ionic structures play an important role, and that their inclusion would probably correct the VB predictions. Indeed the above mentioned successful ab initio VB calculations implicitly include ionic structures due to the use of CF orbitals in the VB descriptions of benzene, CBD, and COT. As will be immediately seen, ionic structures are indeed essential for understanding the difference between aromatic and antiaromatic species, such as benzene, CBD, and COT. Furthermore, the inclusion of ionic structures bring in some novel insight into other features of these molecules, such as ring currents, and so on (see Exercise 5.4). 

This chapter was dedicated to demonstrations that all the so-called failures of VB theory are in fact not real. It was shown that in each such failure , one could use a simple VB theory, based on the principles outlined in Chapter 3, and arrive at the correct predictions—results. In so doing, this chapter also provided the reader with an opportunity to apply qualitative VB theory to some classical problems in bonding. Having done so, the reader is now more prepared for the material in Chapter 6, where VB theory is applied to chemical reactivity. 

One of the major problems in applying quantum chemical calculations to excited states is the restricted ability to interpret the calculations in large Cl expansions, such as CIS and CASPT2. This limitation often does not exist in VB theory, which in many cases can assign a few chemical structures to describe a given excited state. As such, the major goal of this chapter is to teach a conceptual VB approach to excited states, based on the qualitative VB theory discussed throughout Chapters 1—6. 

A difference between the qualitative VB theory, discussed in Chapter 3, and the spin-Hamiltonian VB theory is that the basic constituent of the latter theory is the AO-based determinant, without any a priori bias for a given electronic coupling into bond pairs like those used in the Rumer basis set of VB structures. The bond coupling results from the diagonalization of the Hamiltonian matrix in the space of the determinant basis set. The theory is restricted to determinants having one electron per AO. This restriction does not mean, however, that the ionic structures are neglected since their effect is effectively included in the parameters of the theory. Nevertheless, since ionicity is introduced only in an effective manner, the treatment does not yield electronic states that are ionic in nature, and excludes molecules bearing lone pairs. Another simplification is the zero-differential overlap approximation, between the AOs. 

The hybridizations used in this qualitative VB theory represent idealized hybridizations that are derived inductively from observed geometries."... 

The analysis of the origins of these flaws showed that two of these, (a) and (d), should be looked upon as myths of uncertain origins, while (h) and (c) are the result of the misuse of simple resonance theory that just counts resonance structures. As Shaik and Hiberty have demonstrated, in each of the four cases the proper use of relatively simple qualitative VB theory leads to correct predictions which are just as convincing as those made by MO theory. 

The central theme of this chapter is to show how the many attractive physical concepts of qualitative VB theory can be incorporated into a more general framework (the spin-coupled valence bond theory, Section III) which leads to a computational procedure that provides quantitative descriptions of the ground-state and many excited-state potential energy surfaces of molecular systems. 

It is elear from Eq. (19) that P involves a bonding interaetion [28,52] between Ha and He and will be lowered by the bending mode that brings Ha and He together. Furthermore, the expression for the avoided erossing interaetion B (Eq. (20)), based on qualitative VB theory [28], shows that this quantity will shrink to zero in an equilateral triangular... 

For recent attempts toward the development of a qualitative VB theory, see, inter alia ... 

Note that the rules and formulas that are expressed above in the framework of qualitative VB theory are independent of the type of orbitals that are used in the VB determinants purely localized AOs, fragment orbitals or Coulson-Fischer semilocalized orbitals. Depending on the kind of orbitals that are chosen, the h and S integrals take different values, but the formulas remain the same. 

In qualitative VB theory, it is customary to take the average value of the orbital energies as the origin for various quantities. With this convention, and using some simple algebra, the monoelectronic Hamiltonian between two orbitals becomes Pab, the familiar reduced resonance integral ... 

By application of the qualitative VB theory, Eq. [41] expresses the HL-bond energy of two electrons in atomic orbitals a and b, which belong to the atomic centers A d B. The binding energy is defined relative to the quasiclassical state ab or to the energy of the separate atoms, which are equal within the approximation scheme. 

Here, p is the reduced resonance integral that we have just defined and S is the overlap between orbitals a and b. Note that if instead of using purely localized AOs for a and b, we use semilocalized Coulson-Fischer orbitals, Eq. [41] will not be the simple HL-bond energy but would represent the bonding energy of the real A B bond that includes its optimized covalent and ionic components. In this case, the origins of the energy would still correspond to the QC determinant with the localized orbitals. Unless otherwise specified, we will always use qualitative VB theory with this latter convention. 

Apart from these simplifying assumptions, a fundamental difference between qualitative VB theory and spin-Hamiltonian VB theory is that the basic constituent of the latter theory is the AO determinant, without any a priori bias for a given electronic coupling into bond pairs. Instead of an interplay between VB structures, a molecule is viewed then as a collective spinordering The electrons tend to occupy the molecular space (i.e., the various atomic centers) in such a way that an electron of a spin will be surrounded by as many p spin electrons as possible, and vice versa. Determinants having this property, called the most spin-alternated determinants (MSAD) have the lowest energies (by virtue of the VB rules, in Qualitative VB Theory) and play the major role in electronic structure. As a reminder, the reader should recall from our discussion above that the unique spin-alternant determinant, which we called the quasiclassical state, is used as a reference for the interaction energy. 

We now briefly describe the principles of the method and simple rules for the construction of the Hamiltonian matrix. For the sake of consistency, rather than the original formulation of Malrieu, " we use here a formu-lation that is in harmony with the qualitative VB theory above. The method can be summarized with a few principles ... 

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