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A fully analytical integration of properties over the 3D volume of the β sphere in topological atoms
Author(s) -
Popelier Paul L. A.
Publication year - 2018
Publication title -
journal of computational chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.907
H-Index - 188
eISSN - 1096-987X
pISSN - 0192-8651
DOI - 10.1002/jcc.25158
Subject(s) - multipole expansion , spherical harmonics , gaussian , gaussian quadrature , angular momentum , numerical integration , physics , atom (system on chip) , recursion (computer science) , classical mechanics , quantum mechanics , mathematical analysis , mathematics , nyström method , computer science , algorithm , boundary value problem , embedded system
Atomic multipole moments associated with a spherical volume fully residing within a topological atom (i.e., the β sphere) can be obtained analytically. Such an integration is thus free of quadrature grids. A general formula for an arbitrary rank spherical harmonic multipole moment is derived, for an electron density comprising Gaussian primitives of arbitrary angular momentum. The closed expressions derived here are also sufficient to calculate the electrostatic potential, the two types of kinetic energy, as well as the potential energy between atoms. Some integrals have not been solved explicitly before but through recursion and substitution are broken down to more elementary listed integrals. The proposed method is based on a central formula that shifts Gaussian primitives from one center to another, which can be derived from the well‐known plane‐wave expansion (or Rayleigh equation). © 2018 The Authors. Journal of Computational Chemistry Published by Wiley Periodicals, Inc.