Convergence Analysis of Cauchy Integrals in First-Principles Calculations and Associated Numerical Deviations

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Abstract
Coulomb/exchange integrals in first-principles-calculation are Cauchy-integrals. This paper presents a study on this issue. First, to overcome the long computation-time caused by evaluating multiple Riemann-sums for Coulomb/exchange integrals, we show that their convergence is same as the equivalent-Coulomb/exchange integrals of a point-charge with an electron-cloud and a mixed-electron-one, respectively. This reduction transforms the original six-dimensional Riemann-sums into three/two-dimensional ones, drastically cutting the computation-time and allowing the integrand-singularities to be approached as possible. Using the representative hydrogen-wave-functions as the initial-ones, we calculate the corresponding the equivalent-Coulomb/exchange integrals, and show that the integrals are convergent, but a general initial-wave-functions does not guarantee the convergence. Therefore, verifying convergence for selected initial-wave-functions is a prerequisite for ensuring the reliability of first-principles-calculations. Furthermore, as the step-size of Riemann-sum increases, the relative-deviations increase significantly. Hence, the selection of step-sizes and the associated cumulative-deviation analysis are another key factor for the credibility of the computational results.

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