In density functional theory calculations, the integration over the total density is
evaluated on a molecular grid that is systematically broken up into interlocking
multi-center atomic quadrature grids.
J. Chem. Phys.
(1988), 88, pp. 2547. This atomic quadrature scheme is predicated on the definition of atomic cell functions , that define smoothed Voronoi polyhedra centered about each atom. These cell functions are products of switching functions that define the atomic cell of atom , and fall rapidly from near the nucleus of , to near any other nucleus. The integration weights provided by this scheme are multiplied into the Lebedev quadrature weights in any practical DFT calculation:
In some cases, it may be useful to print out the atomic Becke populations that are defined by these atomic cell functions. Becke population analysis may be requested by setting POP_BECKE to TRUE in the input file.
The default quadrature scheme uses atomic cell functions that intersect
precisely at bond midpoints. Consequently, the default atomic cell functions
will yield physically meaningless atomic populations. However, it is possible
to shift the intersect of the atomic cell functions using an atomic radius
J. Chem. Phys.
(1988), 88, pp. 2547. In shifting the intersect of neighboring atomic cell functions, the point at which the Becke weights begin to fall from to changes depending on the atomic radius of each atom. While the choice of atomic radius is arbitrary, these atomic cell shifts introduce a physical basis for the partitioning of the underlying atomic quadrature. Two choices for atomic radii exist in Q-Chem for use with Becke weights, namely the empirically derived radii introduced by Bragg and Slater 1042 J. Chem. Phys.
(1964), 41, pp. 3199. and the ab initio-derived weights due to Pacios. 852 J. Comput. Chem.
(1995), 16, pp. 133.
A much less arbitrary scheme with which to count electrons comes from the fragment-based Hirshfeld
J. Chem. Theory Comput.
(2015), 11, pp. 528. , 453 J. Phys. Chem. A
(2021), 125, pp. 1243–1256. The fragment-based Hirshfeld (FBH) partition uses weights constructed from isolated fragment densities in the form,
where is the density of the isolated fragment, . Note that unlike the atomic Becke partition, the FBH partition is not constructed from linear combinations of atomic weights, but is instead built from whole fragment densities. The FBH partition comes directly from the densities of the isolated fragments, which are not as arbitrary as the choosing the effective atomic radii in the Becke partition. In order to apply FBH partitioning, one must define fragments within the $molecule section to host the constraints, but the input for the $cdft section remains unchanged and still applies constraints on a per-atom basis.