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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,^{52} though without the “atomic
size adjustments” of Ref. 52. 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.^{440}

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.*,^{672} 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.^{159} 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^{662, 661} is used
for the largest standard grids in Q-Chem,^{662, 661}
namely SG-2 and SG-3.^{202} (See
Section 5.5.2.)