Spin-coupled VB

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... 

The first calculations on benzene using optimised orbitals were done by Cooper et al. [52], using their spin-coupled VB method [20]. A review [53] has appeared with an overview of their work on aromatic and anti-aromatic compounds. 

Alternative VB representations for the cycloaddition process are displayed and discussed in Refs. [2,10,19,34], Some of them also include the less-important increased-valence structures, such as 8-10 here, in the resonance schemes. Recent MO [35] and spin-coupled VB [36] studies for this process do not give consideration to the concerted diradical formulation. The VB studies of Ref. [37] correspond to the concerted diradical mechanism discussed in Ref. [34]. 

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. 

Fig. 5. Comparison of spin-coupled VB (0 500 structures ) and MCSCF-CI calculations ( Green et al. ) for the ground state of CH. ...

Spin-coupled VB calculations were carried out using a total of 26 orbitals six occupied, six a virtual and 14 n virtual orbitals. The final wavefunctions consisted of 500 structures of the type described in Eq. (17) formed from 286 distinct spatial configurations of S symmetry. These consist of the spin-coupled reference function and (1 -I- 2 -I- 3 -I- 4)-fold excitations. No excitations from the (ffj, ffj) core were included. About half of these structures (single plus double replacements) contribute to the ground state, the remainder improves the description of the excited states. 

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... 

In Fig. 8a the spin-coupled and final spin-coupled VB potentials are compared for the ground state. Fig. 8b shows the associated spin-coupling coefficients. It can be seen that the spin-coupled calculation by itself yields a potential energy curve with all the essential features of the final result. The spin-coupling coefficients display much of the essential chemistry of bond formation in BH. At large distances, coefficients are C2 I and 0, which shows that the two 2s-like orbitals stemming from B are coupled to a... 

Fig. 8. (a) Spin-coupled and final spin-coupled VB potential energy curves for BH (X X + H. (b) Spin-coupling coefficients for the potential curves of (a). 

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. 

We consider an N-electron system which we describe by a spin-coupled VB or classical VB expansion of the type (17). We employ a total of n orbitals which are in general non-orthogonal, and which we denote by l/ or l,2,...,n j= 1,2,...,A1). From these we form a certain finite set of structures... 

It is therefore likely that the factorized cofactor-driven algorithm, which is incorporated in the program MILANO, is optimal for a wavefunction of spin-coupled VB form (17) where the structures are derived by excitations from a reference configuration of N singly occupied orbitals. 

This technique can be incorporated in the factorized cofactor-driven algorithm and used when possible for the computation of the cofactors of order (Af, — 2) and — 2). However, it does not afford any advantage in the general spin-coupled VB case, since the orbitals in the reference configuration are singly occupied. This is because here n = -I- always increases up to... 

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. 

The GVB/spin-coupled VB wave function for a molecular system of active electrons with total spin S and projection M is ... 

In this paper we shall follow this line of reasoning, to show there are some symmetry conditions to be obeyed in order to represent the wavefunction of a molecule as a superposition of waveflinctions of resonance hybrids. Although some of the results to be presented may drastically differ from the classical VB picture, all the predictions are fiilly supported by calculations using modem VB methods, such as GVB (Generalized VB) and SCVB (Spin-Coupled VB), which consider explicit optimization of the singly-occupied orbitals, as suggested by Coulson and Fisher . 

As discussed elsewhere , the ideal IPM to be used as referenee for cal-eulating electronic correlation effects is the one which takes into account the Pauli principle and the permutation symmetry of the electrons, thus eliminating the so-called non-d)mamic correlation effects. Modem VB-type models, such as GVB (Generalized or SCVB (Spin-Coupled VB), present all the re-... 

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