Atomic Integrals Containing r23λ r31μ r12v

Abstract

S states of atoms may be calculated to high precision by means of expansions containing correlation factors of the form rijν. Resulting integrals may be expressed as sums of auxiliary functions which are evaluated by algebraic methods. Explicit expressions, suitable for computer programing, are given for such integrals for three-electron atoms; these include integrals containing correlation factors with linked indices, r23λ r31μ r12ν, with any set of (integral) values λ, μ, ν. Expressions are based on an expansion of rijν in terms of Legendre polynomials in cosθij; coefficients of this expansion are represented by a conveniently evaluated formula and some numerical values are given. The expressions given here have been found quite convenient in preliminary calculations of lithium ground state.

Improved Hylleraas calculations for ground state energies of lithium ISO–electronic sequence

“…Larsson used a method of trial and error to determine the most effective basis set such that the convergence rate was the fastest. In the present calculation, the first 60 terms (shown in Table I The necessary integrals involved in this work are solved using the method of Perkins [15], and some numerical examples have been published elsewhere [16]. Most integrals have led to exact results suitable for mechanine computations except for the integrals in which the powers of the three interelectronic coordinates are all odd.…”

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).…”

“…Scrinzi [22] also derived the matrix eigenproblem Eqs. (23)(24)(25) Table I1 to determine p . The assignment of the eigenvalues p k to the lower bounds Ej-) is made with Eq.…”

Compact and accurate variational wave functions of three-electron atomic systems constructed from semi-exponential radial basis functions

“…Our algorithms used in calculations of all required matrix elements is based on the Perkins formula for three-electron integrals [8] in relative coordinates. Note also that all calculations in this work have been performed with the use of standard quadruple precision accuracy (30 decimal digits per computer word).…”