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(June 30, 2021)

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.
^{
74
}
J. Chem. Phys.

(1988),
88,
pp. 2547.
Link
This atomic quadrature
scheme is predicated on the definition of atomic cell functions ${P}_{a}(\text{\mathbf{r}})$,
that define smoothed Voronoi polyhedra centered about each atom. These cell
functions are products of switching functions that define the atomic cell of atom
$a$, and fall rapidly from $\approx 1$ near the nucleus of $a$, to $\approx 0$
near any other nucleus. The integration weights provided by this scheme are
multiplied into the Lebedev quadrature weights in any practical DFT calculation:

$${w}_{n}(\mathbf{r})=\frac{{P}_{n}(\mathbf{r})}{\sum _{m}{P}_{m}(\mathbf{r})}$$ | (5.70) |

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.

POP_BECKE

Controls the printing of atomic Becke populations.

TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

TRUE
Print atomic Becke populations.
FALSE
Do not print atomic Becke populations.

RECOMMENDATION:

None

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
criterion.
^{
74
}
J. Chem. Phys.

(1988),
88,
pp. 2547.
Link
In shifting the intersect of neighboring atomic
cell functions, the point at which the Becke weights begin to fall from
$\approx 1$ to $\approx 0$ 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
^{
1010
}
J. Chem. Phys.

(1964),
41,
pp. 3199.
Link
and the ab initio based weights of
Pacios.
^{
828
}
J. Comput. Chem.

(1995),
16,
pp. 133.
Link

BECKE_SHIFT

Controls atomic cell shifting in determination of Becke weights.

TYPE:

STRING

DEFAULT:

BRAGG_SLATER

OPTIONS:

UNSHIFTED
Use the original weighting scheme of Becke (bisection point).
BRAGG_SLATER
Use the empirically derived Bragg-Slater radii.
UNIVERSAL_DENSITY
Use the ab initio derived Pacios radii.

RECOMMENDATION:

If interested in the partitioning of the default atomic quadrature, use UNSHIFTED.
If using for physical interpretation, choose BRAGG_SLATER or UNIVERSAL_DENSITY.
All cDFT calculations and calculations where POP_BECKE=TRUE
will default to BRAGG_SLATER radii, otherwise the default grid is UNSHIFTED.

A much less arbitrary scheme with which to count electrons comes from the fragment-based Hirshfeld
partition.
^{
928
}
J. Chem. Theory Comput.

(2015),
11,
pp. 528.
Link
^{,}
^{
442
}
J. Phys. Chem. A

(2021),
125,
pp. 1243–1256.
Link
The fragment-based Hirshfeld (FBH) partition uses weights constructed from isolated fragment densities in the form,

$${w}_{n}(\text{\mathbf{r}})=\frac{{\rho}_{n}(\text{\mathbf{r}})}{\sum _{m}{\rho}_{m}(\text{\mathbf{r}})},$$ | (5.71) |

where ${\rho}_{n}(\text{\mathbf{r}})$ is the density of the isolated fragment, $n$.
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.

CDFT_POP

Sets the charge partitioning scheme for cDFT or cDFT-CI jobs.

TYPE:

STRING

DEFAULT:

BECKE

OPTIONS:

BECKE
Linear combination of atomic Becke functions
FBH
Fragment-based Hirshfeld partition

RECOMMENDATION:

None