There are now eight different spatial orbitals, 0 four of which are essentially carbon sp3-hybrid orbitals, with the other four being close to atomic hydrogen s-orbitals. The expansion of each of the VB-orbitals in terms of all the basis functions located on all the nuclei allows the orbitals to distort from the pure atomic shape. The SCVB wave function is variationally optimized, both with respect to the VB-orbital coefficients cai and the spin coupling coefficients a. The result is that a complete set of optimum "distorted" atomic orbitals is determined together with the weight of the different spin couplings. Each spin coupling term (in the so-called Rumer basis) is closely related to the concept of a resonance structure used in organic chemistry textbooks. An SCVB calculation of CH4 gives as a result that one of the spin coupling schemes completely dominates the wave function, namely that corresponding to the electron pair in each of the C—H bonds being singlet coupled. This is the quantum mechanical analogue of the graphical representation of CH4 shown in Figure 7.2.

Each of the lines represents a singlet-coupled electron pair between two orbitals that strongly overlap to form a bond, and the drawing in Figure 7.2 is the only important "resonance" form.

Figure 7.2 A representation of the dominating spin coupling in CH4

Consider now the n-system in benzene. The MO approach will generate linear combinations of the atomic p-orbitals, producing six n-orbitals delocalized over the whole molecule with four different orbital energies (two sets of degenerate orbitals).

The stability of benzene can be attributed to the large gap between the HOMO and LUMO orbitals.

A SCVB calculation considering only the coupling of the six n-electrons, gives a somewhat different picture. The VB n-orbitals are strongly localized on each carbon, resembling p-orbitals that are slightly distorted in the direction of the nearest neighbour atoms. It is now found that five spin coupling combinations are important, these

Figure 7.3 Molecular orbital energies in benzene

Figure 7.4 Representations of important spin coupling schemes in benzene are shown in Figure 7.4, where a bold line indicates two electrons coupled into a singlet pair.

Each of the two first VB structures contributes ~40% to the wave function, and each of the remaining three contributes ~6%.3 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 calculated from a VB wave function omitting certain spin coupling functions.

The MO wave function for CH4 may be improved by adding configurations corresponding to excited determinants, i.e. replacing occupied MOs with virtual MOs. Allowing all excitations in the minimal basis valence space and performing the full optimization corresponds to an [8,8]-CASSCF wave function (Section 4.6). Similarly, the SCVB wave function in eq. (7.12) may be improved by adding ionic VB structures such as CH37H+ and CH3+/H-, and this corresponds to exciting an electron from one of the singly occupied VB orbitals into another VB orbital, thereby making it doubly occupied. The importance of these excited/ionic terms can again be determined by the variational principle. If all such ionic terms are included, the fully optimized SCVB+CI wave function is for all practical purposes identical to that obtained by the MO-CASSCF approach (the only difference is a possible slight difference in the description of the carbon 1s-core orbital).

Both types of wave function provide essentially the same total energy, and thus include the same amount of electron correlation. The MO-CASSCF wave function attributes the electron correlation to interaction of 1764 configurations, the Hartree-Fock reference and 1763 excited configurations, with each of the 1763 configurations providing only a small amount of the correlation energy. The SCVB wave function (which includes only one resonance structure), however, contains 90+% of the correlation energy, and only a few percent is attributed to "excited" structures. The ability of 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 the best wave function that can be constructed in terms of prod ucts 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 coupling functions.

There are a number of technical complications associated with optimizing the SCVB wave function due to the non-orthogonal orbitals. The MO-CI or MO-CASSCF approaches simplify considerably owing to the orthogonality of the MOs, and thereby also of the Slater determinants. Computationally, the optimization of an SCVB wave function, where N electrons are coupled in all possible ways, is similar to that required for constructing an [N,N]-CASSCF wave function. This effectively limits the size of SCVB wave functions to coupling of 12-16 electrons. The actual optimization of the wave function is usually done by a second-order expansion of the energy in terms of orbital and spin coupling coefficients, and employing a Newton-Raphson type scheme, analogously to MCSCF methods (Section 4.6). The non-orthogonal orbitals have the disadvantage that it is difficult to add dynamical correlation on top of an SCVB wave function by perturbation or coupled cluster theory, although (non-orthogonal) CI methods are straightforward. SCVB+CI approaches may also be used to describe excited states, analogously to MO-CI methods.

It should be emphasized again that the results obtained from an [N,N]-CASSCF and a corresponding N-electron SCVB wave function (or SCVB+CI and MRCI) are virtually identical. The difference is in the way the results can be analyzed. Molecules in the SCVB picture are composed of atoms held together by bonds, where bonds are formed by (singlet) coupling of the electron spins between (two) overlapping orbitals. These orbitals are strongly localized, usually on a single atom, and are basically atomic orbitals slightly distorted by the presence of the other atoms in the molecule. The VB description of a bond as the result of two overlapping orbitals is in contrast to the MO approach where a bond between two atoms arises as a sum over (small) contributions from many delocalized molecular orbitals. Furthermore, the weight of the different ways spin couplings in an SCVB wave function carries a direct analogy with chemical concepts such as "resonance" structures.

The SCVB method is a valuable tool for providing insight into the problem. This is to a certain extent also possible from an MO type wave function by localizing the orbitals or by analyzing the natural orbitals (see Sections 9.4 and 9.5 for details). However, there is no unique method for producing localized orbitals, and different methods may give different orbitals. Natural orbitals are analogous to canonical orbitals delocalized over the whole molecule. The SCVB orbitals, in contrast, are uniquely determined by the variational procedure, and there is no freedom to further transforming them by making linear combinations without destroying the variational property.

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 thousands of Slater determinants, while a similar-quality VB wave function may be written as only a handful of "resonating" VB structures. Furthermore, the VB orbitals, and spin couplings, of a C—H bond in say propane and butane are very similar, in contrast to the vastly different MO descriptions of the two systems. The VB picture is thus much closer to the traditional descriptive language used with molecules composed of functional groups. The widespread availability of programs for performing CASSCF calculations, and the fact that CASSCF calculations are computationally more efficient owing to the orthogonality of the MOs, have prompted developments of schemes for transforming CASSCF wave functions to VB structures, denoted CASVB.3 A corresponding procedure using orthogonal orbitals (which introduce large weights of ionic structures) has also been reported.4

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