On Certain Integrals Useful in Studies of Electronic Structure

Abstract

Formulas are given for those two-center one-electron integrals, over Slater basis functions, of the type 〈χA|1/rAirB j|χB〉, 〈χB|1/rAirB j|χB〉, and one-center two-electron integrals 〈χχ′|1/r122|χ″χ″′〉, which arise in the application of the generalized variational method.

Calculation of some integrals for the atomic three-electron problem

“…The results were checked by direct calculation using multiple precision arithmetic. Liichow and Kleindienst [18]have just published an investigation of some of the properties of the functions 6 and F. Equation (10) can be simplified to 1 -1 g r '+ "r' " G(l, tr) dr, dr dr x=0 (10) 4~(m/2) ( n-/2) I(i j,k, 2-, m, n, a,p, y)= -32m3 g g a ", g a,[Ri(ij,k,m, n, w, s, t, a, p, y) p (2w +1)[(1/2) ],=p, =p +R2(i j, k, m, n, w, s, t, a, p, y }], where I Ri = g F(l, a)[ Wt (i+2+21+2s+2t j +1+n -2l -2s+2a, k+1+m 2t -2', a-,p, y) l x=0 + WL (i +2+2l +2s+2t, k+1+m 2t 2tc, j +-1+n -2l -2s+2s, a, y, p) + WL (k +1+21+2t -2s. , j+1+2s+2s,i+2+m +n -21 -2s 2t, y, p,a) 2 + Wt (j+1+2s+2tr, k+1+2l+2t 21-i~+2+m +n -21 2t -2s, p-, y, a) + WL (j+1+2s +21', i +2+n +2t -2s, k+1+m 2t 2-a, p, a, y-) + WL (k+1+2l +2t 2', i -+2+m +2s 2t j +-1+n -2l -2s+2 i', ap)], I -1 R2=g G(l, s)[ W(i +2+2l +2t+2s, j+2+n -21 -2s +2m.…”

Evaluation of some three-body variational integrals

“…But then, one needs as starting point the integral with the highest wanted l. As it can be derived from Eqs. (8)(9)(10), that integral can be obtained through the computation of the quadrature…”

“…[3] and [6] respectively. For γ = 0 much work has been done [2,[7][8][9][10][11], also including explicitly the coupling of the angular momentum of the two dynamical particles [12]. Some work has been devoted to the analogous integrals for four-or more-body problems [6,7,11,13,14].…”