A fully analytical integration of properties over the 3D volume of the β sphere in topological atoms

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.

Promising insights in parallel grid‐based algorithms for quantum chemical topology

“…Practically, a value of 0.001 has been used in this work. Note that this minimal radius r μ is similar to the β‐sphere concept introduced in the QTAIM framework as a spherical volume centered on an attractor that is enclosed within its basin volume 34 …”

“…Note that this minimal radius r μ is similar to the β-sphere concept introduced in the QTAIM framework as a spherical volume centered on an attractor that is enclosed within its basin volume. 34 As shown by both Equations ( 1) and ( 2), the computational effort depends of the number of primitives locally involved at a given grid point for the computation of density functions ρ r ð Þor η r ð Þ: We notice that the contribution of a given primitive varies according to the distance between the grid point and the location of its center. For distances larger than r μ , the contribution of the primitive becomes negligible and can be smoothly neglected from the calculation of the one-density functions.…”

Non-covalent interactions from a Quantum Chemical Topology perspective

“…The advantages listed so far need to be balanced by the disadvantage of computational expense and algorithmic difficulties that QCT introduces. A brief literature review on QCT algorithms has been given before, in both Introductions of two previous papers [36,37] on new algorithms. The computational expense of calculating atomic properties has slackened the uptake of QCT over the decades but this infelicity dwindles by the year through the automatic advent of improved hardware.…”

Quantum chemical topology from tight augmented core densities

“…As a reminder, a β‐sphere is a spherical volume that is fully enclosed within the topological QTAIM basin volume at whose nucleus it is centered. [ 30,31 ] Most of the time, the β‐sphere is defined as the largest sphere that is fully contained within the atomic basin. Thus, its radius is enough large, around 90% of the distance between the nucleus and the nearest BCP.…”