Analysis of some integrals arising in the atomic three-electron problem

Abstract

A detailed analysis is presented for the evaluation of atomic integrals of the form f -ar& -r2 -r3 rIr/r3123 r3, r", 2e ' 'dr, dr2dr, , which arise in several contexts of the three-electron atomic problem. All convergent integrals with i & -2, j~-2, k~-2, m~-1, and n~-1 are examined.These integrals are solved by two distinct procedures. A majority of the integrals can be evaluated by a reduction of the three-electron integrals to integrals arising in the atomic two-electron integral problem. A second approach allows all integrals with the aforementioned indices to be evaluated by the use of Sack s expansion [J. Math. Phys. 5, 245 (1964)] of the interelectronic separation, which leads to a reduction of the above nine-dimensional integrals to a set of three-dimensional integrals. A discussion is given for the numerical evaluation of the three-dimensional integrals that arise.PACS number(s): 31.15.+ q, 31. 20.Di, 02.60.+ y

Accurate upper and lower bounds to the 2S states of the lithium atom

“…This is the best lower bound that can be calculated with Temple's formula for a given basis set. Positive eigenvalues pi are meaningless because of the condition Ei-) < E , < u in Temple's formula (23)(24)(25).…”

Extended Hylleraas three-electron integral

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“…In some special cases [1] this expansion can be avoided and Eq. (1) can be expressed in terms of the two-electron Hylleraas integral.…”

Gegenbauer expansions for three‐electron integrals

“…If we now introduce the addition formula for the Legendre polynomials, as illustrated by (10) where the Y j are normalized spherical harmonics, we have…”