Molecular integrals from Fast Fourier Transforms (FFT) instead of recurrences: The McMurchie-Davidson case.

We report a closed formula expressing the McMurchie-Davidson (MD) key intermediates {[r]; r + r + r ≤ L} directly in terms of the set of basic integrals {[0]; m ≤ L}, without any recurrences. This formula can be evaluated at O(L) cost per output [r] with dense matrix multiplications and Fast Fourier Transforms (FFT). Key to this is the fact that the transformation that builds Cartesian angular momentum from the basic integrals, {[0]}↦{[l]} (κ ∈ {x, y, z}), can be phrased as a circulant-matrix/vector product, which is susceptible to FFTs.

Fast Evaluation of Two-Center Integrals over Gaussian Charge Distributions and Gaussian Orbitals with General Interaction Kernels.

We present efficient algorithms for computing two-center integrals and integral derivatives, with general interaction kernels (), over Gaussian charge distributions of general angular momenta . While formulated in terms of traditional ab initio integration techniques, full derivations and required secondary information, as well as a reference implementation, are provided to make the content accessible to other fields. Concretely, the presented algorithms are based on an adaption of the McMurchie-Davidson Recurrence Relation (MDRR) combined with analytical properties of the solid harmonic transformation; this obviates all intermediate recurrences except the adapted MDRR itself, and allows it to be applied to fully contracted auxiliary kernel integrals.