Spin-coupled VB theory

In the following section we present a general framework in which non-orthogonal orbitals are used to expand the exact wavefunction. This serves to explain the spin-coupled VB theory which is the basic motif of this chapter, and also to show how this reduces to classical VB theory on the one hand, and to the Cl expansion on the other. 

In Section IV results obtained so far by the spin-coupled VB theory are surveyed and in Section V we return somewhat briefly to classical VB theory. 

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 number of configurations required to construct a VB wavefunction of reasonable quality is considerably smaller in approaches implementing the Coul-son-Fischer idea to employ orbitals delocalised over more than one atom. Two of the more widely used modern VB methods, spin-coupled (SC) theory " and the generalised VB (GVB) approach, make full use of this idea by using orbitals constructed as LCAOs, just as in MO theory. Both of these methods do not include any ionic structures in the wavefunction for a neutral system. Other implementations... 

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. 

Modem valence bond (VB) theories such as Spin-Coupled theory, together with DFT and Molller-Plesset MO methods, and ab initio molecular dynamics, were employed to study structure/dynamics in representative carbonium ions. 

We have performed Spin-Coupled calculations on a series of selected carbonium ions (55). The Spin-Coupled calculations allow the study of chemical structure of die molecule, since chemical structure and connectivity are central features of VB theory. Spin-Coupled calculations for CHS+ in CSI show the system as bonded by the intuitively proposed 3c2e bond, which connects the carbon atom to two hydrogens, and three ordinary 2c2e bonds between the carbon and the other hydrogens, commonly called the tripod (Figure 3). 

As is well-known, modem valence-bond (VB) theory in its spin-coupled (SC) form (for a recent review, see Ref. 7) provides an alternative description of benzene [8-10] which, in qualitative terms, is no less convincing and is arguably even more intuitive than the MO picture with delocalized orbitals. The six n electrons are accommodated within a single product of six nonorthogonal orbitals, the spins of which are coupled in all five possible ways that lead to an overall six-electron singlet. The simultaneous optimization of the orbitals and of the weights of the five six-electron singlet spin... 

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 ... 

The purpose of this review is to discuss the main conclusions for the electronic structure of benzenoid aromatic molecules of an approach which is much more general than either MO theory or classical VB theory. In particular, we describe some of the clear theoretical evidence which shows that the n electrons in such molecules are described well in terms of localized, non-orthogonal, singly-occupied orbitals. The characteristic properties of molecules such as benzene arise from a profoundly quantum mechanical phenomenon, namely the mode of coupling of the spins of the n electrons. This simple picture is furnished by spin-coupled theory, which incorporates from the start the most significant effects of electron correlation, but which retains a simple, clear-cut visuality. The spin-coupled representation of these systems is, to all intents and purposes, unaltered by the inclusion of additional electron correlation into the wavefunction. 

For a system consisting of six electrons with a net spin of zero, = 5. For our present purposes, it is probably most useful to consider the modes of spin coupling in terms of the traditional basis of Rumer functions used in classical VB theory. For a discussion of different spin functions, and of the relationships between them, see... 

The spin-coupled wavefunction provides an improvement over the SCF energy of 199 kJ mol-1 (0.0758 hartree), which is considerable. This arises from the effects of electron correlation in the jr-electron system. The- distortion of the spin-coupled orbitals for benzene occurs because of the small amount of ionic character needed to describe the C — C n bonds. Ionic structures in spin-coupled theory are those in which one or more of the orbitals is allowed to be doubly occupied. Allowing for the different numbers of allowed spin functions, and including the spin-coupled configuration, a total of 175 VB structures can be generated in this way. A... 

Aromatic systems play a central role in organic chemistry, and a great deal of this has been fruitfully interpreted in terms of molecular orbital theory that is, in terms of electrons moving more-or-less independently of one another in delocalized orbitals. The spin-coupled model provides a clear and simple picture of the motion of correlated electrons in such systems. The spin-coupled and classical VB descriptions of benzene are very similar, except for the small but crucial distortions of the orbitals. The localized character of the orbitals allows the electrons to avoid one another. Nonetheless, the electrons are still able to influence one another directly because of the non-orthogonality of the orbitals. 

For one more example, a CASVB description for benzene is given in Fig. 3. See Refs. 1 and 2 for the computational details. The CASVB affords a clear view of the wave functions for the various states. The excitation process is represented in VB theory in terms of the rearrangement of spin couplings and... 

The method is referred to simply as GMCSC when a fixed basis set is used. In this case, it can be viewed as a non-orthogonal variant of the Multiconfiguration Self-Consistent Field (MCSCF) approach. However, GMCSC s methodological roots are firmly planted in the Modem VB camp, and more specifically in the late Joe Gerratt s Spin-Coupled theory [3]-[4]. 

An example of a basis of this kind is the SC x x... (x basis in which pairs of electron spins are first coupled to form singlets or triplets, the pairs then being coupled to form the desired resultant S. This is, of course, the natural basis to use when constructing pair wavefunctions (Section 4), and will be referred to as the Serber basis since it was first used by him in VB theory.12-13 It should be noted that in this basis the matrices VS(P) representing simple pair interchanges P/t-i/, even) are all diagonal,... 

The choice of a single function from either set (36) or (37) does not permit such a useful physical interpretation, and may indeed lead to difficulties as the internuclear distance is varied. Thus if one chooses just the perfectly paired function from the set (36), as -R-> 00 one finds each N atom is described by a curious non-stationary state - the so-called valence state of the atom, about which there has been so much discussion in the literature.18 The choice of the set of functions (36) in which orbitals participating in a bond are directly coupled to each other is just the VB theory as proposed by Slater and Pauling,19 whereas the set (37) formed from atoms in specific L-S coupled states corresponds to the spin-valence theory employed by Heitler.20... 

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