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]⁽⁰⁾; r_x + r_y + r_z ≤ L} directly in terms of the set of basic integrals {[0]^(m); m ≤ L}, without any recurrences. This formula can be evaluated at O(L) cost per output [r]⁽⁰⁾ 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_κ]^(m+m}↦{[l_κ]^(m)} (κ ∈ {x, y, z}), can be phrased as a circulant-matrix/vector product, which is susceptible to FFTs. After simplification, a simple formula yields the final [r]⁽⁰⁾ in one step, as contraction of four auxiliary vectors over a common Fourier index k-one vector for the [0]^(m) and one for each Cartesian axis. Similar transformations occur in many integral approaches beside MD, making this idea potentially broadly applicable. The simple resulting code and data structures may make it attractive for novel hardware platforms.