Repulsive matrix

The accepted wisdom is that the Hiickel Hamiltonian matrix should be identified with the matrix (h where G is the electron repulsion matrix of Chapter 6. The basis for this belief is that that the matrix (h has eigenvalues that do sum correctly to the electronic energy. 

Within our approach the entire molecnlar symmetry is exploited to increase the efficiency of the code in every step of the calcnlation. For a molecule belonging to a group G of order G, only 7v /(8 G ) symmetry-distinct two-electron integrals over a basis set of J f Ganssian atomic fnnctions are calcnlated and processed at each iteration within SCF, first- and second-order CHF procednres. A skeleton Conlomb repulsion matrix is obtained by processing the non-rednndant list of nniqne two-electron integrals, then the actual repulsion matrices G , a < /7, are obtained via the equation... 

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 ... 

Table 2.1. Numerical values for overlap, kinetic energy, nuclear attraction, and electron repulsion matrix elements in the two-state calculation.

Let us construct the Hamiltonian matrix by adding the CF matrix to the electron repulsion matrix ... 

Each element of the electron repulsion matrix G has eight 2-electron repulsion integrals, and of these 32 there appear to be 14 different ones ... 

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... 

From the G values based on the initial guess c s the initial-guess electron repulsion matrix is... 

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. 

The zero subscripts in Eqs (5.127) and (5.128) emphasize that the initial-guess c s, with no iterative refinement, were used to calculate G in the subsequent iterations of the SCF procedure H will remain constant while G will be refined as the c s, and thus the P s, change from SCF cycle to cycle. The change in the electron repulsion matrix G corresponds to that in the molecular wavefiinction as the c s change (recall the LCAO expansion) it is the wavefunction (squared) which represents the time-averaged electron distribution and thus the electron/charge cloud repulsion (sections 5.2.3.2,... 

Electron-electron repulsion matrix elements have the form... 

Using this R matrix, form the electron-repulsion matrix G from the stored repulsion integrals. 

The subroutine scfGR takes the matrix in R, which is m by m, and, using the repulsion integrals assumed to be available on a file nfile, forms and adds the repulsion matrix G to the matrix in G. In the firagment of code above the one-electron Hamiltonian is put into HF before calling scfGR. 

G Input/Output this routine adds (repeat adds) the electron repulsion matrix G R) to the contents on input of G. Thus, the matrix G will normally contain a one-electron Hamiltonian on... 

The operation of the new subroutine Gof R has been very slightly generalised from the code of the closed-shell case it is now possible to calculate a more general electron-repulsion matrix ... 

Use the partial" HF repulsion matrix in H and the permutations in perms to form the full repulsion matrix. 

Use perms to transform the partial electron repulsion matrix in H into the full SCF repulsion matrix. 

Add this repulsion matrix to the one-electron Hamiltonian matrix to give a Hartree-Fock matrix defined over the (non-orthogonal) basis orbitals. ... 

In the same way as we treated the formation of the electron-repulsion matrix G, we simply assume the existence of a subroutine which does the work ... 

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