SCVB

The converged SC orbitals satisfy orbital equations of the form [6]  

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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. ... 

For the eight valenee eleetrons in CH4 there are 14 possible spin couplings resulting in an overall singlet state. The full SCVB function may be written (again neglecting normalization) as... 

An SCVB calculation considering only the coupling of the six vr-electrons gives a somewhat different picture. The VB rr-orbitals are strongly localized on each carbon,... 

Each of the two first VB stmctures contributes 40% to the wave function, and each of the remaining three contributes 6%. The stability of benzene in the SCVB picture is due to resonance between these VB structures. It is furthermore straightforward to calculate the resonance energy by comparing the full SCVB energy with that ealeulated from a VB wave function omitting certain spin coupling functions. 

SCVB wave functions to include electron correlation is due to the fact that the VB orbitals are strongly localized, and since they are occupied by only one electron, they have the built-in feature of electrons avoiding each other. In a sense, an SCVB wave function is tte best wave function that can be constructed in terms of products of spatial orbitals. By allowing the orbitals to become non-orthogonal, the large majority (80-90%) of what is called electron correlation in an MO approach can be included in a single determinant wave function composed of spatial orbitals, multiplied by proper spin cou ing functions. 

The primary feature of SCVB is the use of non-orthogonal orbitals, which allows a much more compact representation of the wave function. An MO-CI wave function of a certain quality may involve many thousand Slater determinants, while a similar quality VB wave function may be written as only a handful of resonating VB structures. 

Figure 7.5 A representation of the SCVB wave function for diazomethane...

The way this function represents the system is strongly influenced by the dynamics of the problem, as well as the flexibility allowed. If we were to find the set of three orbitals and value of a minimizing W, we obtain essentially the SCVB wave function. What this looks like depends significantly on the potential energy function. If we are treating the n system of the allyl radical, where all three orbitals are nearly degenerate, we obtain one sort of answer. If, on the other hand, we treat a deep narrow potential like the Li atom, we would obtain two orbitals close to one another and like the traditional s orbital of self-consistent-field (SCF) theory. The third would resemble the 2s orbital, of course. 

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. 

The SCVB energy is, of eourse, just the result from this optimization. Should a more elaborate wave funetion be needed, the virtual orbitals are available for a more-or-less eonventional, but nonorthogonal. Cl that may be used to improve the SCVB result. Thus an aeeurate result here may also involve a wave function with many terms. 

The SCVB function produces a considerable portion of the correlation energy. 

In this chapter we describe four rather different three-electron systems the it system ofthe allyl radical, the HeJ ionic molecule, the valence orbitals ofthe BeHmolecule, and the Li atom. In line with the intent of Chapter 4, these treatments are included to introduce the reader to systems that are more complicated than those of Chapters 2 and 3, but simple enough to give detailed illustrations of the methods of Chapter 5. In each case we will examine MCVB results as an example of localized orbital treatments and SCVB results as an example of delocalized treatments. Of course, for Li this distinction is obscured because there is only a single nucleus, but there are, nevertheless, noteworthy points to be made for that system. The reader should refer back to Chapter 4 for a specific discussion of the three-electron spin problem, but we will nevertheless use the general notation developed in Chapter 5 to describe the results because it is more efficient. 

The SCVB method can also be used to study the tt system of the allyl radical. As we have seen already, only one of the two standard tableaux ffinctions is required because of the symmetry of the molecule. We show the results in Table 10.4, where we see that one arrives at 85% of the correlation energy from the largest MCVB calculation in Table 10.2. There is no entry in Table 10.4 for the EGSO weight, since it would be 1, of course. 

Figure 10.1. The first SCVB orbital for the allyl radical. The orbital amplitude is given in a plane parallel to the radical and 0.5 A distant.

In Fig. 10.1 we show an altitude drawing of the orbital amplitude of the first of the SCVB orbitals of the allyl n system. The third can be obtained by merely reflecting this one in the y-z plane of the molecule. It is seen to be concentrated at... 

The HeJ ion has the archetype three-electron bond originally described by Pauling [1], and this section gives a description of MCVB calculation and SCVB treatments for this system. All of these use a Huzinaga 6-G Is function split (411), a 4-G 2s function and a pz function with the scale set to 0.9605. We take up the MCVB treatment first. 

When these orbitals are optimized, the energies of the SCVB wave functions are higher, of course, than those of the full MCVB wave functions. We show the differences at the equilibrium and infinite intemuclear separations in Table 10.6. The energy curves are parallel within s 0.1 eV, but the SCVB energy is about 1.1 eV higher. 

In this section we give the results of MCVB and SCVB treatments of BeH using a conventional 6-3IG basis. Although there are some similarities to the HeJ ion, the lack of g-u symmetry in this case introduces a number of interesting... 

The allyl radical and the He ion both have end-for-end s mimetry and thus the corresponding orbital SCVB treatment is applied. Consequently, there was only one tableau function in each of those cases. BeH is different in this regard. In the... 

Figure 10.6. The first SCVB orbital for the BeH molecule and associated with the Be nucleus. This has the general appearance of an 5-p hybrid pointed toward the H atom, and we denote it the inner hybrid, hi. The orbital amplitude is given in the x-z plane, which contains the nuclei.

Table 10.10. Energy differences between SCVB andMCVB treatment of BeH.

Table 10.11. Coefficients and tableaux for standard tableaux functions and HLSP functions for SCVB treatment of BeH.

There is no added symmetry in this example to cause one of the standard tableaux functions to disappear. Thus, the SCVB wave function is... 

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