The Spin Hamiltonian VB Theory

Coherent control is a powerful new chemical method that makes use of the availability of the twin excited state to control the course of chemical reactions by laser excitation. Thus, laser excitation from to (Fig. 17a), using two different and complementary photons causes the decay of to occur in a controlled manner either to the reactant or products. In the case where the reactants and products are two enantiomers, the twin excited state is achiral, and the coherent control approach leads to chiral resolution. 

In summary, the twin excited state plays an important role in photochemistry as well as in thermal chemistry. 

The spin-Hamiltonian VB theory uses the same approximations as the qualitative theory presented above to calculate the Hamiltonian matrix elements, but with a few simplifications. The theory is restricted to determinants having one electron per AO this restriction excludes ionic structures or molecules bearing lone pairs. As such, the theory has mainly been applied to conjugated polyenes. Another simplification is the zero-differential overlap approximation, which means that all overlaps are neglected in the formulas. 

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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The spin-Hamiltonian VB theory rests on the same principles as the qualitative theory presented in Chapter 3, with some further simplifying assumptions. This chapter describes the method and focuses on its qualitative applications. 

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. 

A simple principle of the spin-Hamiltonian VB theory, first formulated by Ovchinnikov (13), applies to alternant conjugated molecules, that is, those molecules that possess fully spin-alternant determinants. The rule is stated as follows ... 

The spin-Hamiltonian VB theory is a very simple and easy-to-use semiempi-rical tool that is based on the molecular graph. It is consistent with the VB theory described in Chapter 3, albeit with some simplifying assumptions and a more limited domain of application. Typically, this theory deals with the neutral ground or excited states of conjugated molecules or other homonuclear assemblies with one electron per site. For large systems, it reproduces the results of PPP full Cl, while dealing with a much smaller Hamiltonian matrix. 

A simple principle of the spin-Hamiltonian VB theory is that the lowest state of a molecule will have the multiplicity associated with the S value of its MSAD, that is, it will be a singlet if in the MSAD, a doublet if =... 

From the conceptual point of view, there are two general approaches to the molecular structure problem the molecular orbital (MO) and the valence bond (VB) theories. Technical difficulties in the computational implementation of the VB approach have favoured the development and the popularization of MO theory in opposition to VB. In a recent review [3], some related issues are raised and clarified. However, there still persist some conceptual pitfalls and misinterpretations in specialized literature of MO and VB theories. In this paper, we attempt to contribute to a more profound understanding of the VB and MO methods and concepts. We briefly present the physico-chemical basis of MO and VB approaches and their intimate relationship. The VB concept of resonance is reformulated in a physically meaningful way and its point group symmetry foundations are laid. Finally it is shown that the Generalized Multistructural (GMS) wave function encompasses all variational wave functions, VB or MO based, in the same framework, providing an unified view for the theoretical quantum molecular structure problem. Throughout this paper, unless otherwise stated, we utilize the non-relativistic (spin independent) hamiltonian under the Bom-Oppenheimer adiabatic approximation. We will see that even when some of these restrictions are removed, the GMS wave function is still applicable. 

In the special case where the spin-orbitals are orthonormal and the trial functions are Slater determinants the expressions for the projective reduction coefficients are both simple and limited, given by Slaters rules to be discussed in detail in later chapters in this work. With such a choice there are Hamiltonian matrix elements between functions that differ from each other only in two or fewer orbitals and Mu = 6u- The expressions for these coefficients when the orbitals are not orthogonal involve the overlap integrals S, j between all the orbitals in the functions and there is no limitation on orbital differences between the functions and Mu is not the unit matrix. Every electronic permutation must be considered in their evaluation. For non-orthogonal orbitals it is thus much more difficult to consider systems with more than a few electrons and, because atomic orbitals on different centers are not orthogonal, this difficulty has hindered the development of VB theory in a quantitative manner until very recently. An account of modern VB developments forms a later part of this handbook. Usually LCAO MOs are developed so as to be orthogonal so that given... 

In Section 1.4, we discussed the history and foundations of MO theory by comparison with VB theory. One of the important principles mentioned was the orthogonality of molecular wave functions. For a given system, we can write down the Hamiltonian H as the sum of several terms, one for each of the interactions which will determine the energy E of the system the kinetic energies of the electrons, the electron-nucleus attraction, the electron-electron and nucleus-nucleus repulsion, plus sundry terms like spin-orbit coupling and, where appropriate, other perturbations such as an applied external magnetic or electric field. We now seek a set of wave functions P, W2,... which satisfy the Schrodinger equation ... 

An even more quantitative application of VB theory can be developed from the realization that the nearest-neighbor VB model as developed, for example, by Pauling [10], can be mapped exactly onto a Heisenberg spin Hamiltonian [17]. The Heisenberg spin Hamiltonian has long been used to study the interaction between magnetic atoms in transition metal compounds and other paramagnetic substances [18], and can be written most simply as... 

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