The Hylleraas‐CI method in molecular calculations: Two‐electron integrals

Abstract

In the Hylleraas-CI method, first proposed by Sims and Hagstrom, correlation factors of the type r& are included into the configurations of a CI expansion. The computation of the matrix elements requires the evaluation of different two-, three-, and four-electron integrals. In this article we present formulas for the two-electron integrals over Cartesian Gaussian functions, the most used basis functions in molecular calculations. Most of the integrals have been calculated analytically in closed form (some of them in terms of the incomplete Gamma function), but in one case a numerical integration is required, although the interval for the integration is finite and the integrand well-behaved. We have also reported on partial and preliminary computations for the H1 molecule using our four-center general formulas; a basis set of s-andp-type functions yielded at R = 1.4001 A an energy of -1.174380 8.u. to be compared with Kolos and Wolniewicz value of -1.174475.

Shorter Expansions of Molecular Integrals in Hylleraas-CI Calculations

“…Clementi and his co-workers have recently derived the necessary formulas for molecular integrals over Cartesian Gaussian orbitals' 3,4 ' and have carried out excellent numerical work on some two-electron systems' [5][6][7][8][9] !. Their results indicate that Cartesian Gaussian functions can be efficiently used in Hylleraas-CI calculations to obtain highly accurate energies of molecular systems.…”

“…The Hylleraas-CI method' [1][2][3][4] ' is an attractive and, in principle, general method to include correlation effects in wave functions. In this method, the wave function is expanded as a superposition of configurations…”

Molecular integrals in the generalized hylleraas–CI method

“…The length of this expansion in F,(Z) is z + 1, where = Xf=, (Li + M i + Ni). An alternative method that can be used to solve the basic integrals is similar to what was used by Clementi and his co-workers [3,4], i.e., to perform integrations directly. The expanding length in F,(Z), however, is + 4 in that approach.…”

“…Details of the evaluation depend on the concrete form of the correlation factor. Takingf: as powers of the interelectronic distance ro results in the Hylleraas-cI method [l-41, for which Clementi and his co-workers have recently derived the formulas of molecular integrals over Cartesian Gaussian orbitals [3,4] and have carried out an excellent numerical work on some two-electron systems [ 5 ] . In the most complicated cases of their formulas, a two-dimensional integration is needed.…”

On the use of explicitly correlated functions in variational computations for small molecules

“…The Hy-cr wave function has been applied successfully by for two-electron molecules: H2, HeH+, and H: . Two-electron integrals appearing in these calculations are relatively simple and require at most the one-dimensional numerical integration[44]. The extension of this approach to many-electron molecules leads to the necessity to evaluate a large number of cumbersome three-and fourelectron integrals involving two-dimensional integration[45] and, in consequence, to discouraging results, as was demonstrated recently byClementi et al [46] for the three-electron H3 molecule.…”