Valence bond theory matrix elements

Bond strengths are essentially controlled by valence ionization potentials. In the well established extended Hiickel theory (EHT) products of atomic orbital overlap integrals and valence ionization potentials are used to construct the non-diagonal matrix elements which then appear in the energy eigenvalues. The data in Table 1 fit our second basic rule perfectly. 

The valence-bond approach plays a very important role in the qualitative discussion of chemical bonding. It provides the basis for the two most important semi-empirical methods of calculating potential energy surfaces (LEPS and DIM methods, see below), and is also the starting point for the semi-theoretical atoms-in-molecules method. This latter method attempts to use experimental atomic energies to correct for the known atomic errors in a molecular calculation. Despite its success as a qualitative theory the valence-bond method has been used only rarely in quantitative applications. The reason for this lies in the so-called non-orthogonality problem, which refers to the difficulty of calculating the Hamiltonian matrix elements between valence-bond structures. 

The extracted Natural Hybrid Orbitals (NHOs) are therefore not simply encoded forms of the molecular shape, as envisioned in valence shell electron pair repulsions (VSEPR)-type caricatures of hybridization theory. Instead, the NHOs represent optimal fits to the ESS-provided electronic occupancies (first-order density matrix elements cf. V B, p. 21ff) in terms of known angular properties of basis AOs. Thus, the NHOs predict preferred directional characteristics of bonding from angular patterns of electronic occupancy, and the deviations (if any) between NHO directions and the actual directions of bonded nuclei give important clues to bond strain or bending that are important descriptors of molecular stability and function. 

---