Coulomb integrals, definition

This is an exchange intend, the functions on the left and right of e jrn differ from each other by an exchange of electrons one and two. The general definitions of the Coulomb integral and the exchange integral are... 

From definition (53), there is immediate recognition of the special integral / = iiifilw) as the Coulomb integral describing repulsion between two electrons with probabilities and [Pg.198]

The general definitions of the Coulomb integral and the exchange integral... 

Here q represents the coulomb energy of an electron occupying a definite p]h orbital in unsubstituted benzene its value has been estimated to be about —2.7 v. e. = —60 kcal./mole.5 /3 is a resonance integral between adjacent orbitals its value has been estimated to be about —0.85 v. e. = — 20 kcal./mole.6 Sk is a constant, the purpose of which is to allow for the different electron affinities of the different atoms. For Sk > 0, the... 

Here Zg is the number of tt electrons provided by atom is essentially an ionization potential for an electron extracted from in the presence of the part of the framework associated with atom r alone (a somewhat hypothetical quantity), is a framework resonance integral, and is the coulomb interaction between electrons in orbitals < >, and <(>,. The essential parameters, in the semi-empirical form of the theory, are cug, and and from their definition these quantities are expected to be characteristic of atom r or bond r—s, not of the particular molecule in which they occur (for a discussion see McWeeny, 1964). In the SCF calculation, solution of (95) leads to MO s from which charges and bond orders are calculated using (97) these are used in setting up a revised Hamiltonian according to (98) and (99) and this is put back into (95) which is solved again to get new MO s, the process being continued until self-consistency is achieved. It is now clear that prediction of the variation of the self-consistent E with respect to the parameters is a matter of considerable difficulty. 

Equation (4.49) indicates that for this wave function the classical Coulomb repulsion between the electron clouds in orbitals a and b is reduced by Kab, where the definition of this integral may be inferred from comparing the third equality to the fourth. This fascinating consequence of the Pauli principle reflects the reduced probability of finding two electrons of the same spin close to one another - a so-called Fermi hole is said to surround each electron. 

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