Cooper orbital energies

From a conceptual point of view, it appears that polymer quantum chemistry is an ideal field for cooperation between condensed matter physicists and molecular quantum chemists. There exists a common interpretation in the discussions concerning orbital energies, orbital symmetry, and gross charges by chemists and solid-state physicists. These physicists use terms less familiar to the chemist, such as first Brillouin zone, dependence of wave function with respect to wave vector k (the one-electron wave function is called an orbital by the chemist), Fermi surfaces, Fermi contours, and density of states (DOS). 

The analysis of the regioselective reactivity of olefins in identical topochemical environments by three computational methods concludes that both steric factors (cavity and potential energy) and electronic factors (perturbation energy from orbital interactions) play important cooperative roles in determining which C—C double bond in a molecule reacts first in [2-1-2] photodimerization. The steric factor is considered to be effective in the movement of olefins at an early stage of the reaction, whereas the electronic factors are effective in the adduction of olefins at a later stage of the reaction. 

Fig. 15 Predicted cooperative effects on activation energies (in kcal/mol) at the B3LYP/ 6-31G level for model enediynes ( push and pull denote through-space repulsive (steric) and attractive (H-bonding) interactions of ort/zo-substituents with in-plane 71-orbitals of an adjacent acetylene moeity).

For orbitals with a radial node, there is a minimum in the cross section, known as a Cooper minimum. At some photon energy, the contributions from the inner and outer parts of the orbital will tend to cancel one another (Figure 7(d)). Figure 7(e) illustrates the occurrence of such a minimum in the cross section of the 3p electrons of the elements Si to Cl. Figure 7(f) shows the cross section of the ti band of TiCfr, which shows a Cooper minimum at 40 eV " by symmetry, this is pure Cl 3p in character (see also Section 4.2.4). If an orbital has more than one radial node, there may be a corresponding number of Cooper minima in the cross section. The minima may occur just above threshold, as is the case for the 3s orbital of Na, or several hundred eV, as is found for 4d and 5f electrons of transition elements. Rules for Cooper minima, which have been arrived at by experimental and theoretical investigations, are as follows ... 

With such minima, the photon energy may be tuned to minimize the contribution from certain types of orbitals to a PE spectrum. The angular parameter, p, can vary substantially in the region of a Cooper minimum as the contribution from the / - / -F 1 wave is small, leading to domination of p by the / — 1 wave. 

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