Interelectron repulsion matrix

Fig. 2. This figure shows the electronic energy of the ground state of H2 molecule, calculated in a crude approximation using only one configuration. The benchmark calculation of Kolos and Wolniewicz is exhibited for comparison. Accuracy can be seen to be improved by using more atomic orbitals even when a rough approximation is used for the interelectron repulsion matrix element.

In this appendix we shall try to show that interelectron repulsion matrix element T l j of equation (78) does not depend independently on = kg and R, but depends only on their product, 5 = kgR If we take the Fourier transform of equation (79), we obtain ... 

Since the building-blocks from which it composed are independent of pK, the interelectron repulsion matrix I), v is also independent of pK and hence independent of energy. The energy-independent interelectron repulsion matrix I), v consists of pure numbers (in atomic units) which can be evaluated once and for all and stored. 

Table 1 Roots of the ground state 77-block of the interelectron repulsion matrix for the Li-like, Be-like, B-like and C-like isoelec-tronic series...

Table 2 Roots of the ground state 7 -block of the interelectron repulsion matrix Tv, v for the N-like, O-like, F-like, and Ne-like isoelectronic series...

Since only Coulomb potentials are involved, the matrix T v, v turns out to be energy independent. Its elements are pure numbers that depend only on N, the number of electrons, and are independent of the nuclear charge Z. The roots lK of the energy-independent interelectron repulsion matrix T v, v are also pure numbers (Table 1). In the large-Z approximation, the generalized Sturmian secular equation (41) reduces to the requirement ... 

This chapter is a review, and most of what is reported here can be found in our own books and papers and those of the authors whose works are cited. There are, however, some results that do not appear elsewhere. Among these are (42)-(47), that demonstrate that T v v, the interelectron repulsion matrix based on Goscinskian congurations, is energy independent and consists of pure numbers when expressed in atomic units. Other new results include Table 3, and much of Sect. 4.4. Sects. 5.2-5.4 and much of the Appendix were previously reported only in the Ph.D. thesis of one of us and in works that are now in press. 

In the united-atom hmit, s = 0, this gives = 1/2 while in the separated-atom limit, s 00 we have —> 1. Beginning with a series of s-values, we can now immediately generate the corresponding values of bg andN, as shown in Table 1. Interestingly, the interelectron repulsion matrix element, T n, which seems at first sight to depend independently on the two parameters kg and R, can be shown to depend only on their product, s = kgR, (Appendix 1). The approximate functional dependence of this matrix element on s is given [19] by... 

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 six elements can be used to generate the n-electron ligand field energies, to which interelectronic repulsion and spin-orbit coupling, which are already defined on the dxy, dX2, dyz, dx2 y2, dz2 basis, are added. Alternatively, the five eigenvalues can be used, but an additional term is necessary to define the correct linear combinations of the dx2 y2 and dz2 orbitals on which to evaluate interelectronic repulsion and spin-orbit coupling. Either way, there are still six total, or five spectroscopically independent, terms in the ligand field potential matrix. 

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... 

The contribution of interelectron repulsion to diagonal matrix elements of the type shown in equation (56) is more complex. However, it turns out that in the case of spherical symmetry one can get rid of the m-summations by making use of the sum rule for spherical harmonics. The result is as follows ... 

The matrix representing the interelectronic repulsion is taken from pubhshed tables (2,11). For configurations both references use Slater-... 

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