Abstract
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.
Highlights
Quantum chemical topology (QCT),[1] pioneered[2] by the research group of the late Richard Bader, has carved out a space among non-topological methods in extracting insight from wave functions, both in chemistry and solid state physics
We focus on the involved mathematics of deriving closed formulae for the b sphere’s contribution to atomic multipole moments of arbitrary spherical harmonic rank, the two types of kinetic energy (K and G), the electrostatic potential and the potential energy between topological atoms
We present a detailed derivation of a fully analytical 3D integration over the volume bounded by the b sphere inside a topological atom
Summary
Quantum chemical topology (QCT),[1] pioneered[2] by the research group of the late Richard Bader, has carved out a space among non-topological methods in extracting insight from wave functions, both in chemistry (including metal-metal interactions[3]) and solid state physics (including highresolution X-ray crystallography[4]). This imaginative approach started, in quantum chemistry, as (the Quantum Theory of ) Atoms in Molecules[5] (QT)(AIM) in the early 1970s. Better controlling the atomic integration over a spherical volume that resides completely within a topological atom (see below) is a sign of progress
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