Analysis of some integrals arising in the atomic four‐electron problem

Kinetic-energy matrix elements for atomic Hylleraas-CI wave functions

Hylleraas-CI is a superposition-of-configurations method in which each configuration is constructed from a Slater-type orbital (STO) product to which is appended (linearly) at most one interelectron distance rij. Computations of the kinetic energy for atoms by this method have been difficult due to the lack of formulas expressing these matrix elements for general angular momentum in terms of overlap and potential-energy integrals. It is shown here that a strategic application of angular-momentum theory, including the use of vector spherical harmonics, enables the reduction of all atomic kinetic-energy integrals to overlap and potential-energy matrix elements. The new formulas are validated by showing that they yield correct results for a large number of integrals published by other investigators.

UNSYMMETRICAL AND SYMMETRICAL ONE-RANGE ADDITION THEOREMS FOR SLATER TYPE ORBITALS AND COULOMB–YUKAWA-LIKE CORRELATED INTERACTION POTENTIALS OF INTEGER AND NONINTEGER INDICES

Using one-center expansion relations for the Slater type orbitals (STOs) of noninteger principal quantum numbers in terms of integer nSTOs derived in this study with the help of ψa-exponential type orbitals (ψa-ETOs, a = 1, 0, -1, -2,…), the general formulas through the integer nSTOs are established for the unsymmetrical and symmetrical one-range addition theorems for STOs and Coulomb–Yukawa-like correlated interaction potentials (CIPs) with integer and noninteger indices. The final results are especially useful for the computations of arbitrary multicenter multielectron integrals that arise in the Hartree–Fock–Roothaan (HFR) approximation and also in the correlated methods based upon the use of STOs as basis functions.

Comment on “Analysis of some integrals arising in the atomic four-electron problem” [J. Chem. Phys. 99, 3622 (1993)]

A paper by F. W. King, "Analysis of some integrals arising in the atomic four-electron problem", contains expressions which it is shown can be further simplified to an extent making the formulation significantly more efficient. Two errors in one of the equations are also identified.

Improvements on the application of convergence accelerators for the evaluation of some three-electron atomic integrals

Convergence accelerator methods are employed to analyze some of the most difficult three-electron integrals that arise in atomic calculations. These integrals have an explicit dependence on the interelectronic coordinates, and take the form integral r(i)(1)r(j)(2)r(k)(3)r(l)(23)r(m)(31)r(n)(12) exp((-alpha(r1)-beta(r2)-gamma(r3))dr(1)dr(2)dr(3). The focus of the present investigation are the most difficult cases of the parameter set [i, j, k, l, m, n]. Several convergence accelerator techniques are studied, and a comparison presenting the relative effectiveness of each technique is reported. When the convergence accelerator approach is combined with specialized numerical quadrature methods, we find that the overall technique yields high-precision results and is fairly efficient in terms of computational resources. Difficulties associated with the standard numerical precision loss of convergence accelerator techniques are circumvented.