The Spin-Coupled VB

We have seen that with a system of n electrons in a spin state S there are, for n linearly independent orbitals, / (given by Eq. (18)) linearly independent spatial functions that can be constructed from these orbitals. In the present notation the SCVB wave function is written as the general 

Using familiar methods of the calculus of variations, one can set the first variation of the energy with respect to the orbitals and linear coefficients to zero. This leads to a set of Fock-like operators, one for each orbital. Gerratt, et al use a second-order stabilized Newton-Raphson algorithm for the optimization. This gives a set of occupied and virtual orbitals from each Fock operator as well as optimum 7,-s. 

The SCVB energy is, of course, just the result from this optimization. Should a more elaborate wave function be needed, the virtual orbitals are available for a more-or-less conventional, but non-orthogonal configuration interaction (Cl) that may be used to improve the SCVB result. Thus improving the basic SCVB result here may involve a wave function with many terms. 

SCVB wave functions for very simple systems appear similar to those of the GGVB method, but the orthogonality constraints in the latter have increasingly serious impacts on the results for larger systems. 

---

The ordinary unrestricted Hartree-Fock (UHF) function is not written like either of these. It is not a pure spin state (doublet) as are these functions. The spin coupled VB (SCVB) function is lower in energy than the UHF in the same basis. 

No matter whether calculated within the perfect pairing VB approach or by the spin-coupled VB approach, in both cases the CC hybrid orbital extends outside the three-membered ring as expected by the schematic representations in Figure 7, but also inside... 

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. 

We begin with some general considerations of perhaps lesser-known, but important, features of exact electronic wavefunctions. Our motive is to establish a theoretical framework together with a reasonably consistent notation in order to carry through the spin-coupled VB and other expansions of the total wavefunction. We consider an atomic or molecular system consisting of N electrons and A nuclei. We assume the Born-Oppenheimer separation and write the Hamiltonian operator for the motion of the electrons in the form ... 

Because of the use of N distinct sets, we expect the spin-coupled VB expansion to converge much faster, and indeed the results so far show that this is the case. As we discuss in more detail below, the spin-coupled function (18) by itself possesses all the correct qualitative characteristics of the ground state of a molecular system, and 200-700 terms of expansion (17) are sufficient to attain chemical accuracy ( 0.1 eV) for the first 10-15 eigenstates of a given symmetry, and spectroscopic accuracy ( 100 cm ) for the ground state. ° ... 

The CH ion is of considerable importance in interstellar chemisty, and has also been studied by MCSCF and Cl methods It is therefore well suited as a full-scale demonstration of the spin-coupled VB procedure described above. The basis set used was of modest size (18cr, 20n, 6S Slater orbitals), and is the same as that used by Green except for omission of 4f functions. However, no diffuse 3s(C) or 3p(C) functions, which would be needed to describe any Rydberg character in excited states, were included. 

The spin-coupled VB calculations comprised one-, two- and three-fold excitations from the original spin-coupled structure, and give rise to 400 spatial configurations or 592 spin-coupled structures. In Fig. 6 are shown... 

The aim of the spin-coupled VB calculations is to determine potential energy curves for sufficient states of the CH system that are of sufficient accuracy and include a uniform amount of electron correlation. For this... 

The spin-coupled VB calculations include a total of 12spatial configurations which give rise to 228 spin-coupled configurations. 

In order to include this phenomenon, two procedures are possible (i) the wavefunctions of A and B are represented by Cl expansions or (ii) the wavefunctions are determined at the spin-coupled VB level. 

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). 

One of the most useful types of constraint is the restriction of the spin coupling to just a single mode. Many molecular systems are described rather well by the perfect pairing mode of spin coupling, for example. A useful alternative, especially when this is not the case, is to base the structure coefficients on the CASSCF wavefunction in the VB orbital basis ... 

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. 

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... 

The Kekule description of benzene, as expressed in the classical VB form, appears to be much closer to reality than is a description in terms of delocalized molecular orbitals, and it provides a clear picture of the behavior of correlated electrons in this molecule. The special properties of benzene arise fundamentally from the mode of coupling of the electron spins around the carbon ring framework. Except for small but crucial distortions of the orbitals, the spin-coupled and classical VB descriptions of this molecule are very similar. 

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. 

In the second way, one can decide that all determinants will have their spins arranged in alternated order, that is, a, (3, a, etc., in which case the spin-coupling is characterized by permutations of the AOs. In such a case, the coefficients of the AO-determinants are all positive for a singlet state. Both methods are strictly equivalent, and the first one is generally employed in this book. However, the method of half-determinants is easier to apply with the second convention, which will be used in this section. Accordingly, the VB functions for structures 6 and 7 can be rewritten as in Equation 4.17 and 4.18 ... 

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. 

---