Generalized Valence Bond Theory

The SCVB wave function allows all possible spin couplings to take place and has no restrictions on the form of the orbitals. The Generalized Valence Bond (GVB) method can be considered as a reduced version of the full problem where only certain subsets of spin couplings are allowed.5 For a typical case of a singlet system, the GVB method has two (non-orthogonal) orbitals assigned to each bond, and each pair of electrons in a bond are required to couple to a singlet pair. The coupling of such singlet pairs will then give the overall singlet spin state. This is known as Perfect Pairing (PP), and is one of the many possible spin coupling schemes, and such two-electron two-orbital pairs are called geminal pairs. Just as an orbital is a wave function for one electron, a geminal is a wave function for two electrons. In order to reduce the computational problem, the Strong Orthogonality (SO) condition is normally imposed on the GVB wave function. This means that orbitals belonging to different pairs are required to be orthogonal. While the perfect pairing coupling typically is the largest contribution to the full SCVB wave function, the strong orthogonality constraint is often a quite poor approximation, and may lead to artefacts. For diazomethane, for example, the SCVB wave function is dominated (91%) by the PP coupling, leading to the conclusion that the molecule has essentially normal C=N and N=N n-bonds, perpendicular to the plane defined by the CH2 moiety.6 Taking into account also the in-plane bonding, this suggest that diazomethane is best described with a triple bond between the two nitrogens, thereby making the central nitrogen "hypervalent", as illustrated in Figure 7.5.

Figure 7.5 A representation of the SCVB wave function for diazomethane

There are strong overlaps between the VB orbitals, the smallest overlap (between the carbon and terminal nitrogen) is ~0.4, and that between the two orbitals on the central nitrogen is ~0.9. The GVB-SOPP approach, however, forces these geminal pairs to be orthogonal, leading to the conclusion that the electronic structure of diazomethane has a very strong diradical nature, as illustrated in Figure 7.6.

Figure 7.6 A representation of the GVB wave function for diazomethane

References

1. S. Shaik, P. C. Hiberty, Rev. Comp. Chem., 20 (2004), 1.

2. D. L. Cooper, J. Gerratt, M. Raimondi, Chem. Rev, 91 (1991), 929; J. Gerratt, D. L. Cooper, P. B. Karadakov, M. Raimondi, Chem. Soc. Rev., 26 (1997), 87.

3. D. L. Cooper, T. Thorsteinsson, J. Gerratt, Int. J. Quant. Chem., 65 (1997), 439.

4. K. Hirao, H. Nakano, K. Nakayama, M. Dupuis, J. Chem. Phys., 105 (1996), 9227.

5. W. A. Goddard III, L. B. Harding, Ann. Rev. Phys. Chem., 29 (1978), 363.

6. D. L. Cooper, J. Gerratt, M. Raimondi, S. C. Wright, Chem. Phys. Lett., 138 (1987), 296.

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