Multipole expansion applied to the total energy

Electronic Motion in the Mean Reid Periodic Systems 

Do the Fock matrix elements and the total energy per cell represent finite values  

If the Fock matrix elements were infinite, then we could not manage to carry out the Hartree-Fock-Roothaan self-consistent procedure. iEj were infinite, the pe- 

For any finite system there is no problem the results are always finite. The only danger, therefore, is the summation to infinity ( lattice sums ), which always ends with the interaction of a part or whole unit cell with an infinite number of distant cells. Let us take such an example in the simplest case of a single atom per cell. Let us assume that the atoms interact by the Lennard-Jones pairwise potential (p. 284)  

These conditions are more severe for the Fock matrix elements because each of the terms represent the interaction of a charge with complete distant unit cells. The convergence depends on the asymptotic interaction energy of the potential. In the case of the multipole-multipole interaction, we know what the asymptotic behaviour looks like, it is = R + where R stands for the intercell 

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Fock Matrix Corrections Total Energy Corrections Multipole Expansion Applied to the Fock Matrix Multipole Expansion Applied to the Total Energy... 

In Eqs. (4), (6) and (11), there are infinite summations in the overlap matrix Sk, the Fock matrix Fk, and the total energy per unit cell E JF. There are also infinite summations in T, V, Jlpq, and Kln. The infinite lattice summations in Sk, Tl and K converge themselves while V , J and the internuclear interactions have to be summed together to get converged results [20-22], In real calculations, cutoffs for the lattice summations have been imposed and multipole expansion techniques have been applied to hasten the convergence [22],... 

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