Delley partitioning

[susilehtola]

Delley's paper on numerical atomic orbital calculations, J. Chem. Phys. 92, 508 (1990), states employing a partition function (eq. 3)

$$ p_A = \frac { g_A({\bf r} - {\bf R}_A) } { \sum_B g_B ({\bf r} - {\bf R}_B) } $$

where "preferred choices are the functions"

• $gA = \rho{A}^2(r)$,
• $gA = \rho{A}^2(r) / r^2$, and
• $gA = \rho{A}(r) [\exp(r_0/r) - 1 - r_0/r] $

where $\rho_A$ is the atom's electron density.

Delley writes

These partition functions automatically depend on atom sizes via $p_A$ and have given consistently good results for all sorts of compounds including heteronuclear compounds containing heavy transition metals and light elements including hydrogen. The last function leads to continuous vanishing of all partition functions and all their derivatives on all nuclei where they are not centered.

These schemes are clearly similar to Becke's, and should be quite easy to implement. The only thing necessary are just the atomic electron densities. The adaptive Molpro grid https://github.com/wavefunction91/GauXC/issues/51 implementation of J. Chem. Phys. 157, 234106 (2022) employed Slater atomic densities:

In Ref. 79, these are defined for elements up to $n = 6$ with effective principal numbers of $n^\star = 3.7$, 4.0, 4.2 for $n = 4, 5, 6$. The implementation in Molpro uses $n^\star = 4.4$ for seventh row elements based on a simple linear extrapolation of the $n^\star$ values given by Leach.79 The advantage of using Slater’s densities over tabulated numerical densities is that they are given analytically and can be easily computed at any given grid point.

[susilehtola]

The Baker et al 1994 paper defines the following weighting

in this scheme the angular quadrature weights are derived from superpositions of spherical atomic densities for all the atoms that make up the system being studied

$$ w_A({\bf r}) = \rhoA^n({\bf r}) / \sum{B} \rho_B^n({\bf r}) $$

where $n$ is a parameter; $n=2$ was used in that paper.