New Techniques in Numerical Integration: The Computation of Molecular Integrals over Exponential-Type Functions

Abstract

The numerical evaluation of challenging integrals is a topic of interest in applied mathematics. We investigate molecular integrals in the B function basis, an exponentially decaying basis with a compact analytical Fourier transform. The Fourier property allows analytical expressions for molecular integrals to be formulated in terms of semi-infinite highly oscillatory integrals with limited exponential decay. The semi-infinite integral representations in terms of nonphysical variables stand as the bottleneck in the calculation. To begin our numerical experiments, we conduct a comparative study of the most popular numerical steepest descent methods, extrapolation methods and sequence transformations for computing semi-infinite integrals. It concludes that having asymptotic series representations for integrals and applying sequence transformations leads to the most efficient algorithms. For three-center nuclear attraction integrals, we find an analytical expression for the semi-infinite integrals. Numerical experiments show the resulting algorithm is approximately 10 times more efficient than the state-of-the-art. For the four-center two-electron Coulomb integrals, we take a different approach. The integrand has singularities in the complex plane that can be near the path of integration, making standard quadrature routines unreliable. The trapezoidal rule with double exponential variable transformations has been shown to have very promising properties as a general-purpose integrator. We investigate the use of conformal maps to maximize the convergence rate, resulting in a nonlinear program for the optimized variable transformation.