Electron repulsion integrals definitions

Here h are the one-electron integrals including the electron kinetic energy and the electron-nuclear attraction terms, and gjjkl are the two-electron repulsion integrals defmed by (3 19). The summations in (3 24) are over the molecular orbital basis, and the definition is, of course, only valid as long as we work in this basis. Notice that the number of electrons does not appear in the defmition of the Hamiltonian. All such information is found in the Slater determinant basis. This is true for all operators in the second quantization formalism. 

The definition of the exchange matrix precludes its calculation as a simple matrix product involving F. The K matrix involves the same R matrix but with different electron-repulsion integrals ... 

Electron repulsion integrals may be evaluated by a straightforward generalization of the McMurchie-Davidson algorithm [34], using the definition of the two-spinor charge operator. The Coulomb interaction involves only the Eg-coefficients for q = 0, and results in G-spinor integrals of the form... 

The widely used notation of (13.162) should not be misinterpreted as an overlap integral. Other notations, some mutually contradictory, are used for electron repulsion integrals, so it is always wise to check an author s definition. 

In addition to the conventional overlap energy term there is a second term that depends on the bond-order Pq and the repulsive integral between an electron in adsorbate orbital 0 and an electron in orbital I on the neighboring atom, is equal to /fjj, Ek).(2.282), and the second term in follows from the first term of (2.283a) when using definition Eq.(2.35c) for the bond-order Pi2 and the zero-differential overlap approximation Eq.(2.284). In section 2 Eq.(2.40b) related to Eq.(2.309) was derived. It was shown to lead to a reduction of the interaction energy of two electrons with different spins in the same orbital (see Eq.(2.42,II)). The result implies that Ucff is not independent of the bond-order. An improved expression for would be ... 

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. 

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]

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