Overlap integral angular dependence

Kettle and his co-workers (39—42) used a model rather similar to the AOM to discuss stereochemistry. A perturbation approach led to the proportionality of MO energies (relative to the unperturbed orbitals) to squared overlap integrals, as in the AOM. For systems where the valence shell orbitals are evenly occupied, the total stabilization energy shows no angular dependence, suggesting that steric forces determine the equilibrium geometry. 

The angular overlap model is a relatively crude method which appears to yield results at least as good as those afforded by the crystal field model. As with all simple empirical models, the AOM depends on many approximations and assumptions which cannot be expected to be even approximately correct. Thus, for example, the parameter a is assumed to depend only on the identity of the metal and the ligand, and on the internuclear distance it is independent of the stoichiometry or stereochemistry. The theoretical basis for assuming the proportionality of the AOM matrix elements to overlap integrals is closely related to the Wolfs-berg-Helmholz approximation for the off-diagonal matrix elements of the one-electron operator ... 

The difference between the various l,n-biradicals is due to the first factor in eq. (30), the structural factor. For ethylene with w = 90°, these factors become 0, 1 and 0 for the x, y and z component, respectively, and together with the sin2d proportionality of the y component, this completely describes the computational SOC results shown in Figure 2b. For trimethylene (w = 60°) the structural factors of all three component are nonzero since the overlap integral (A B) exhibits a similar dependence on the rotational angles as the z component of the angular momentum, while the x and y component are zero at the [90,90] conformation... 

FIGURE 1.4. Angular dependence of the overlap integral for some commonly encountered pairs of atomic orbitals. 

Maximal overlap will occur at some finite value of r which depends on the magnitude of the orbital exponents for the two atoms. The angular dependence of the overlap integral follows immediately from the analytic form of ) in equation I. I and expressed in Table I. I. We can often write the overlap integral as in equation 1.17 ... 

---