Spin-coupled valence bond theory

The generalization of a Coulson-Fischer type wave function to the molecular case with an arbitrary-size basis set is known as Spin-Coupled Valence Bond (SCVB) theory.  

It is again instructive to compare with the traditional MO approach, taking the CHt molecule as an example. The MO single-determinant description (RHF, which is identical to UHF near the equilibrium geometry) of the valence orbitals is in terms of four delocalized orbitals, each occupied by two electrons with opposite spin. The C—H 

Here A is the usual antisymmetrizer (eq. (3.21)) and a bar above a MO indicates that the electron has a P spin function, no bar indicates an a spin function. 

The Qfi symbol is used to designate the rth combination of spin functions couphng N electrons to give an overall spin of 5, and there are fs number of ways of doing this. The value oifs is given by eq. (7.11). 

For a singlet wave function (5 = 0), the number of coupling schemes for N electrons is given in Table 7.1. 

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Another approach is spin-coupled valence bond theory, which divides the electrons into two sets core electrons, which are described by doubly occupied orthogonal orbitals, and active electrons, which occupy singly occupied non-orthogonal orbitals. Both types of orbital are expressed in the usual way as a linear combination of basis functions. The overall wavefunction is completed by two spin fimctions one that describes the coupling of the spins of the core electrons and one that deals with the active electrons. The choice of spin function for these active electrons is a key component of the theory [Gerratt ef al. 1997]. One of the distinctive features of this theory is that a considerable amount of chemically significant electronic correlation is incorporated into the wavefunction, giving an accuracy comparable to CASSCF. An additional benefit is that the orbitals tend to be... 

T orbital for benzene obtained from spin-coupled valence bond theory. (Figure redrawn from Gerratt ], D L oer, P B Karadakov and M Raimondi 1997. Modem valence bond theory. Chemical Society Reviews 87 100.) figure also shows the two Kekule and three Dewar benzene forms which contribute to the overall wavefunction Kekuleform contributes approximately 40.5% and each Dewar form approximately 6.4%. 

Resonance between three 7t-complex structures might lead to stabilization of 1 in the sense of 7t-aromatic stabilization involving the six CC bond electrons. Therefore, Dewar8 has discussed the stability of 1 in terms of a u-aromatic stabilization (Section V). However, spin-coupled valence bond theory clearly shows that 1 cannot be considered as the aromatic benzene51. The 7t-complex description of 1 is a (very formal) model description, which should be discarded as soon as it leads to conflicting descriptions of the properties of 1. This will be discussed in Section V. 

FIGURE 17. Schematic representation of the symmetry-unique spin-coupling patterns in cyclopropane (above) and benzene (below). In the case of cyclopropane, carbon hybrid orbitals and, in the case of benzene, carbon p n orbitals are shown. For each structure, Gallup-Norbeck occupation numbers as determined by spin-coupled valence bond theory are given. All data from Reference 51... 

It is not appropriate here to present a detailed account of the formalism of spin-coupled valence bond theory, and of its computational implementation. Instead, we provide a brief, almost entirely qualitative overview of our method, concentrating on the most interesting results for benzenoid aromatic molecules. Further details may be found in the literature cited and in various recent reviews [1,2,3]. 

The extent to which these heterobenzenes are actually aromatic has been the topic of extensive computational work spin-coupled valence bond theory suggesting that borabenzene has an aromatic sextet <1997IJQ441>, and hybrid... 

D. L. Cooper, J. Gerratt, M. Raimondi, Chem. Rev. 91, 929 (1991). Applications of Spin-Coupled Valence Bond Theory. 

This process of constructing functions for the various resonant formulae, followed by an adequate combination of them, is mathematically more complex than the mathematics of molecular orbital theory. It is therefore understandable that, after the initial preference of chemists for the v.b. bond theory which has a closer relation to Lewis structures - especially due to the contribution of Linus Pauling - m.o. theory became increasingly popular. In addition, m.o. theory leads directly, not only to fundamental states (through the occupied m.o.), but also to excited states (through vacant m.o.) of molecules. In recent years, however, a new form of valence-bond theory has been developed that is more amenable to computation (spin-coupled valence-bond theory) in which the molecular wavefunction is expressed as a linear combination of all the coupling schemes of the various electrons corresponding to the same resultant spin (ref. 97). 

IV. Survey of Results of Spin-coupled Valence Bond 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. 

IV. SURVEY OF RESULTS OF SPIN-COUPLED VALENCE BOND THEORY... 

Recent Applications of Spin-Coupled Valence Bond Theory to Charge Transfer Collisions... 

Study of the Electronic States of the Benzene Molecule Using Spin-Coupled Valence Bond Theory. 

Texts on alternative types of valence-bond theory include those of references 1 and 2. Reference 3 provides a review of modern ab initio methods and classical valence-bond approaches to electronic structure. References 4 and 5 provide reviews of generalised and spin-coupled valence-bond theory. Both of these theories can use delocalised orbitals that involve more than two atomic centres to accommodate the active-space electrons. In references 6-8, overviews are presented of aspects of increased-valence theory. 

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