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(May 16, 2021)

In practical DFT calculations, the forms of the approximate exchange-correlation functionals used are quite complicated, such that the required integrals involving the functionals generally cannot be evaluated analytically. Q-Chem evaluates these integrals through numerical quadrature directly applied to the exchange-correlation integrand. Several standard quadrature grids are available (“SG-$n$”, $n=0,1,2,3$), with a default value that is automatically set according to the complexity of the functional in question.

The quadrature approach in Q-Chem is generally similar to that found in many
DFT programs. The multi-center XC integrals are first partitioned into
“atomic” contributions using a nuclear weight function. Q-Chem uses the
nuclear partitioning of Becke,
^{
74
}
J. Chem. Phys.

(1988),
88,
pp. 2547.
Link
though without the “atomic
size adjustments” of Ref. 74. The atomic integrals are
then evaluated through standard one-center numerical techniques. Thus, the
exchange-correlation energy is obtained as

$${E}_{\mathrm{XC}}=\sum _{A}^{\mathrm{atoms}}\sum _{i\in A}^{\mathrm{points}}{w}_{Ai}f({\mathbf{r}}_{Ai}),$$ | (5.15) |

where the function $f$ is the aforementioned XC integrand and the quantities
${w}_{Ai}$ are the quadrature weights. The sum over $i$ runs over grid points
belonging to atom $A$, which are located at positions ${\mathbf{r}}_{Ai}={\mathbf{R}}_{A}+{\mathbf{r}}_{i}$, so this approach requires only the choice
of a suitable one-center integration grid (to define the ${\mathbf{r}}_{i}$),
which is independent of nuclear configuration. These grids are implemented in
Q-Chem in a way that ensures that the ${E}_{\mathrm{XC}}$ is rotationally-invariant,
*i.e.*, that is does not change when the molecule undergoes rigid rotation in
space.
^{
513
}
Chem. Phys. Lett.

(1994),
220,
pp. 377.
Link

Quadrature grids are further separated into radial and angular parts. Within
Q-Chem, the radial part is usually treated by the Euler-Maclaurin scheme
proposed by Murray *et al.*,
^{
775
}
Mol. Phys.

(1993),
78,
pp. 997.
Link
which maps the semi-infinite
domain $[0,\mathrm{\infty})$ onto $[0,1)$ and applies the extended trapezoid rule to the
transformed integrand. Alternatively, Gill and Chien proposed a radial scheme
based on a Gaussian quadrature on the interval $[0,1]$ with a different weight
function.
^{
194
}
J. Comput. Chem.

(2003),
24,
pp. 732.
Link
This “MultiExp" radial quadrature is exact for
integrands that are a linear combination of a geometric sequence of exponential
functions, and is therefore well suited to evaluating atomic integrals.
However, the task of generating the MultiExp quadrature points becomes
increasingly ill-conditioned as the number of radial points increases, so that
a “double exponential" radial quadrature
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765
}
Theor. Chem. Acc.

(2011),
130,
pp. 645.
Link
^{,}
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764
}
Theor. Chem. Acc.

(2012),
131,
pp. 1169.
Link
is used
for the largest standard grids in Q-Chem,
^{
765
}
Theor. Chem. Acc.

(2011),
130,
pp. 645.
Link
^{,}
^{
764
}
Theor. Chem. Acc.

(2012),
131,
pp. 1169.
Link
namely SG-2 and SG-3.
^{
244
}
J. Comput. Chem.

(2017),
38,
pp. 869.
Link
(See
Section 5.5.2.)