On the evaluation of four-electron correlated integrals with a Slater basis: some simplifications and some closed form examples

Abstract

Four-electron correlated integrals with explicit rij dependence and derived from a Slater-type basis set are discussed. Attention is directed to the more complex cases where the powers on the rij factors are odd. Particular effort is focused on the case where there are five factors of rij raised to odd powers, a case that has considerable practical significance. Several integral cases involving three and four rij factors raised to odd powers are evaluated in closed form.

Analytic evaluation of some three- and four- electron atomic integrals involving s STO's and exponential correlation with unlinked $r_{ij}$'s

“…which is identical with equation( 41) of [52]. It is also observed in equation( 90) that if λ 12 → λ 1 , the right hand side expression assumes 0/0 form.…”

“…The respective generating integral is given by (52) and is denoted, in short, by I g 41 (λ 1 , λ 2 , λ 3 , λ 4 ; λ 12 , λ 13 , λ 14 ). Similarly from equation( 50) we can express the generating integral as…”

“…It is worth pointing out here that a closed-form expression for the integral on the left hand side of equation( 61) was obtained by King [52] directly, by employing expansion formula of Sack [71] and of Perkins [72] for r ν ij . The corresponding expression contains a minor typographical error which is corrected by changing the signs before the fourth and the fifth terms within the curly brackets on the right hand side of equation( 44) of the reported paper [52], which has been pointed out very recently in [59].…”

“…It is worth pointing out here that a closed-form expression for the integral on the left hand side of equation( 61) was obtained by King [52] directly, by employing expansion formula of Sack [71] and of Perkins [72] for r ν ij . The corresponding expression contains a minor typographical error which is corrected by changing the signs before the fourth and the fifth terms within the curly brackets on the right hand side of equation( 44) of the reported paper [52], which has been pointed out very recently in [59]. Incorporating these minor corrections, it is easy to show that the corrected expression becomes identical with the one given on the right hand side of equation( 61) obtained, as a special case, from the generating integral I g 42 defined in equation( 53) and evaluated and reported earlier in paper I.…”

“…Frolov et al [48] have investigated bound state spectra and properties of the doublet states in Li atom and some Li-like ions employing Hy-CI and CI methods. The basic four-electron atomic correlated integral with integrand involving all the six eletron-electron separation coordinates was reduced to a sum of several auxiliary functions X (denoted as W 4 ) [49][50][51], and later reevaluated by King [52] to increase the effectiveness in computation and reported closed-form expressions for some integrals as special cases. Analytical expressions for the auxialiary functions A, V, W and X (redenoted as A 1 , A 2 , A 3 and A 4 , respectively) along with their highly accurate values have been reported in [53,54].…”