Interelectron repulsion integrals

Craig, D. P., Proc. Roy. Soc. [London) A202, 498, Electronic levels in simple conjugated systems. I. Configuration interaction in cyclobutadiene. (ii) All the interelectron repulsion integrals, three- and four-centered atomic integrals, are included. 

The problem of evaluating matrix elements of the interelectron repulsion part of the potential between many-electron molecular Sturmian basis functions has the degree of difficulty which is familiar in quantum chemistry. It is not more difficult than usual, but neither is it less difficult. Both in the present method and in the usual SCF-CI approach, the calculations refer to exponential-type orbitals, but for the purpose of calculating many-center Coulomb and exchange integrals, it is convenient to expand the ETO s in terms of a Cartesian Gaussian basis set. Work to implement this procedure is in progress in our laboratory. 

The approximations in the SHM are its peremptory treatment of the overlap integrals S (Section 4.3.4, discussion in connection with Eqs. 4.55), its drastic truncation of the possible values of the Fock matrix elements into just a, jl and 0 (Section 4.3.4, discussion in connection with Eqs. 4.61), its complete neglect of electron spin, and its glossing over (although not exactly ignoring) interelectronic repulsion by incorporating this into the a and jl parameters. 

The two-electron matrix G, the electron repulsion matrix (Eq. 5.104), is calculated from the two-electron integrals (Eqs. 5.110) and the density matrix elements (Eq. 5.81). This is intuitively plausible since each two-electron integral describes one interelectronic repulsion in terms of basis functions (Fig. 5.10) while each density matrix element represents the electron density on (the diagonal elements of P in Eq. 5.80) or between (the off-diagonal elements of P) basis functions. To calculate the matrix elements Grs (Eqs. 5.106-5.108) we need the appropriate integrals (Eqs. 5.110) and density matrix elements. These latter are calculated from... 

All the necessary elements can be calculated in terms of integrals over the atomic orbitals y, integrals which involve either the nuclear attraction operators H(v) or the interelectronic repulsions 1 jr v. 

The nephelauxetic effect is described quantitatively by parameter, f, which is equal to the ratio of the interelectron repulsion parameters (either Slater s integrals, Fk, or Racah parameters, Ek) in the complex and in the free ion. We may write... 

Keywords Exponential-type orbitals Generalized Sturmians Hyperspherical harmonics Interelectron repulsion integrals Isoenergetic configurations Momentum space Quantum theory Sturmians... 

Interelectron Repulsion Integrals for Molecular Sturmians from... 

We will now show that when the densities are produced by products of Coulomb Sturmians, interelectron repulsion integrals of the type shown in (165) and (167) can be readily evaluated using Fock s relationship and the properties of hyper-spherical harmonics. Suppose that... 

The integrals over dp in (182) are simple enough to be evaluated by Mathematica and they can conveniently be stored as functions kR in the form of interpolation functions. Notice that the integrals depend only on n and /, and there are therefore fewer of them than there would be if they also depended on m. The first 105 of these functions are shown in Fig. 3. Equations (173), (180), and (182) give us a very rapid and convenient way of evaluating integrals of the form shown in (173), where the densities are formed from products of Coulomb Sturmian basis functions located respectively on the two centers, a and a. They constitute the largest contribution to the effects of interelectron repulsion. 

In this expansion, the coefficients r nJj and a, are universals that can be calculated once and for all, and that never have to be recalculated. When the basis functions scale with changing values of k, the expansion scales automatically too. Because the coefficients are universals, we can use many terms in the expansion and thus obtain especially good accuracy. The fact that the interelectron repulsion integrals divided by k are independent of k can be shown by arguments similar to those shown in (42)-(47). When divided by k, the interelectron repulsion integrals are pure functions of the parameters s = kx and Sa = kXa. Therefore, they scale automatically with changes of scale of the basis functions. The independence from k also implies that the molecular-Sturmian-based interelectron repulsion integrals can be pre-evaluated and stored. 

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