Spline methods for multiconfiguration Hartree–Fock calculations

Abstract

The earlier numerical multiconfiguration Hartree-Fock atomic structure package was not designed with high-performance computers in mind. In this paper, some new algorithms based on spline-Galerkin methods are described that are appropriate for concurrentlvector architectures. The goal is to improve the level of numerical accuracy by several orders of magnitude using fewer basis functions than points in a numerical grid. Of critical importance is the robustness of the code: The most serious problems in the numerical implementation were associated with orthogonality constraints. In a spline basis approach, the orthogonality requirements can be integrated into quadratically convergent update procedures. These procedures are evaluated for a number of cases.

Implementation of exterior complex scaling in B-splines to solve atomic and molecular collision problems

“…Before giving some preliminary results on single photoionization and autoionizing states of helium, we now turn to the evaluation of those two-electron integrals. The approach we take is a version of the method used to compute these integrals in the discrete variable representation and is similar in spirit but very different in detail to more standard approaches to the computation of these integrals in a B-spline basis (Fisher and Guo 1990, 1992, Qui and Fisher 1999.…”

Accurate CI expansion in a spline basis of the helium ground‐state wave function

“…Moreover, the algorithm applies without modification to the generalized eigenvalue problem, as we have in the present case due to spline nonorthogonality. 29 The algorithm runs as follows: The matrices are blocked according to the I index. Then, Repeat step 2 until maximum variation in the energy E and coefficients cil are less than a prefixed threshold.…”

Integration by Cell Algorithm for Slater Integrals in a Spline Basis

“…An important development was finding a method for evaluating the Slater matrix elements [8,9]. In this approach, the Slater matrix elements R k (i, j; i , j ) are rewritten as…”

“…Because of the apparently large number of numerical operations needed to evaluate this integral accurately, they finally concluded that B-splines were not suitable for atomic structure calculations. However, there has recently been a renewed interest in the use of B-splines in complex atomic systems: Bottcher and Strayer [2] applied the B-splines to time-dependent problems, Johnson and co-workers to many-body perturbation theory [3][4][5], Fischer and co-workers [6][7][8][9] to Hartree-Fock calculations and continuum problems, Hansen and co-workers to orthogonal operators and Rydbergy series [10,11], Decleva and co-workers [12][13][14] to multichannel continuum problems, and van der Hart to R-matrix theory [15]. More detailed discussions can be found in a recent review by Sapirstein and Johnson [16].…”

“…The exact values of the Slater integrals are from Ref. [9]. Difference 1, the difference of the exact value and the one evaluated with the traditional method; Difference 2, the difference of the exact value and the one evaluated with the integration by cell method.…”