TY - JOUR

T1 - Molecular integrals from Fast Fourier Transforms (FFT) instead of recurrences

T2 - The McMurchie-Davidson case

AU - Peels, Mieke

AU - Knizia, Gerald

N1 - Funding Information:
This work was supported by a start-up grant from the Pennsylvania State University.
Publisher Copyright:
© 2020 Author(s).

PY - 2020/6/21

Y1 - 2020/6/21

N2 - We report a closed formula expressing the McMurchie-Davidson (MD) key intermediates {[r](0); rx + ry + rz ≤ 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, {[0κ](m+m′)}→{[lκ](m)} (κ ϵ {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.

AB - We report a closed formula expressing the McMurchie-Davidson (MD) key intermediates {[r](0); rx + ry + rz ≤ 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, {[0κ](m+m′)}→{[lκ](m)} (κ ϵ {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.

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U2 - 10.1063/5.0002880

DO - 10.1063/5.0002880

M3 - Article

C2 - 32571074

AN - SCOPUS:85086934453

VL - 152

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 23

M1 - 231103

ER -