Multipole expansion applied to the Fock matrix

Let us first concentrate on C g N). As seen from eq. (9.77) there are two type of interactions to calculate the nuclear attraction integrals Vp A ) and the electron repulsion integrals qs). In the second term, we may use the multipole expansion of jL given in the Appendix X on p. 1039. In the first term, we will do the same, 

A p(l) is anchored at cell 0, and such a distribution decays exponentially when cell j goes away from 

Similar reasoning may be used for Cp(N). The interacting objects are of the type i.e. 

Electronic Motion in the Mean Field Periodic Systems 

Let us note that in the square parentheses we have nothing but a multipole moment of unit cell h. Indeed, the first term represents the multipole moment of all the nuclei of cell h, while the second term is the multipole moment of electrons of unit cell h. The later can best be seen if we recall the normalization condition (9.68) TUrJZ N E = In, with 2 o denoting 

---

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],... 

---