Double Asymptotic Expansion of Three-Center Electron Repulsion Integrals 2nd Derivatives

The calculation of the skeleton Hessian in the framework of auxiliary density functional theory (ADFT) requires the computation of the second derivatives of three-center electron repulsion integrals (ERIs). Because the evaluations of ERIs and their derivatives involve the calculations of the nonanalytic Boys functions, this calculation step can become a computational bottleneck for large molecular systems. To avoid this bottleneck the double asymptotic expansion of three-center ERIs was proposed. In this work, we extend this expansion to analytic second derivatives of these ERIs. To this end, the working equations for the double asymptotic expansion of three-center ERI second derivatives are derived. They are implemented in the analytic second energy derivative calculation in the ADFT branch of deMon2k. To assess their computational performance skeleton Hessian matrix benchmark calculations of linear alkane chains, carbon fullerenes, DNA fragments and hydrogen saturated mobil-5 type (ZSM-5) zeolite fragments with up to 43,000 basis functions are presented.