Mathematical and computational science issues in high precision Hylleraas-configuration interaction variational calculations: I. Three-electron integrals

The most difficult integral arising in Hylleraas-configuration interaction (Hy-CI) calculations, the three-electron triangle integral, is discussed. We focus on recursive techniques at both the double precision and quadruple precision level of accuracy while trying to minimize the use of higher precision arithmetic. Also, we investigate the use of series acceleration to overcome problems of slow convergence of certain integrals defined by infinite series. We find that a direct + tail Levin u-transformation convergence acceleration overcomes problems that arise when using other convergence acceleration techniques, and is the best method for overcoming the slow convergence of the triangle integral. The question of calibrating an acceleration method is also discussed, as well as ways to improve our work.