H2 and localized orbitals

H2 and localized orbitals Table 2.1. A TZ2P1D basisforH2 calculations. 

The reader will recall that in Chapter 2 we gave examples of H2 calculations in which the orbitals were restricted to one or the other of the atomic centers and in Chapter 3 the examples used orbitals that range over more than one nuclear center. The genealogies of these two general sorts of wave functions can be traced back to the original Heitler-London approach and the Coulson-Fisher[15] approach, respectively. For the purposes of discussion in this chapter we will say the former approach uses local orbitals and the latter, nonlocal orbitals. One of the principal differences between these approaches revolves around the occurrence of the so-called ionic structures in the local orbital approach. We will describe the two methods in some detail and then return to the question of ionic stmctures in Chapter 8. 

This LBO-based wave function is not a VB wave function. Nevertheless, it represents a Lewis structure, and hence also a pictorial analogue of a perfectpairing VB wave function. The difference between the LBO and VB wave functions is that the latter involves electron correlation while the former does not. As such, in a perfectly paired VB wave function, based on CF orbitals, each localized Be—H bond would involve an optimized covalent—ionic combination as we demonstrated above for H2 and generalized for other 2e bonds. In contrast, the LBOs in Equation 3.65 possess some constrained combination of these components, with exaggeration of the bond ionicity. 

The complete active space valence bond (CASVB) method is an approach for interpreting complete active space self-consistent field (CASSCF) wave functions by means of valence bond resonance structures built on atom-like localized orbitals. The transformation from CASSCF to CASVB wave functions does not change the variational space, and thus it is done without loss of information on the total energy and wave function. In the present article, some applications of the CASVB method to chemical reactions are reviewed following a brief introduction to this method unimolecular dissociation reaction of formaldehyde, H2CO — H2+CO, and hydrogen exchange reactions, H2+X — H+HX (X=F, Cl, Br, and I). 

The density functional theory calculations of primary KIE and secondary deuterium kinetic isotope effects (SKIE) did not reproduce satisfactorily all the experimentally determined KIE and deuterium (4,4- H2)- and 6,6- H2-SKIE, though the non-local DFT methods provide transition state energies on a par with correlated molecular orbital theory. ... 

The symmetry projection of the wavefunction is equivalent to a particular orbital transformation among the occupied orbitals of the wavefunction. If the CSF expansion is full within these sets of symmetry-related orbitals, no new CSFs will be generated by this orbital transformation. This type of wavefunction could have been computed directly in terms of symmetry orbitals with no loss of generality. (In fact, the CSF expansion expressed in terms of symmetry orbitals will usually result in fewer expansion terms because the symmetry blocking of the individual CSFs allows those of the incorrect symmetries to be deleted from the expansion.) However, if the CSF expansion is not full within these orbital sets, it is possible that the symmetry transformation of the orbitals will generate new CSF expansion terms. The coefficients of these new CSF expansion terms are determined by the old expansion coefficients and the symmetry transformation coefficients. For example, consider the case of two H2 molecules, described in terms of localized orbitals, separated by a reflection plane. Assume that the localized description of the two H2 molecules is of the form... 

The 2bi, 5aj, and 2b2 orbital cross sections of H2S shown in Figure 15 are seen to decrease rapidly with photon energy relative to the H2O Ib, 3aj and lb2 counterparts (33). By contrast, the 6ai(y ) and 4a] (y ) orbitals In these compounds are localized in discrete spectral regions as bound states. 

As an illustration of the above formalism we calculate the CEPA correlation energy of our dimer of two noninteracting H2 molecules using both localized and delocalized orbitals. Since with localized orbitals there is no coupling between different pairs in this model (i.e., = 0... 

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