5.5.1 Introduction

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, ⁸³ Becke A. D.
J. Chem. Phys.
(1988), 88, pp. 2547. Link though without the “atomic size adjustments” of Ref.  83 Becke A. D.
J. Chem. Phys.
(1988), 88, pp. 2547. Link . The atomic integrals are then evaluated through standard one-center numerical techniques. Thus, the exchange-correlation energy is obtained as

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 ${𝐫}_{Ai}={𝐑}_A+{𝐫}_i$, so this approach requires only the choice of a suitable one-center integration grid (to define the ${𝐫}_i$), 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. ⁵⁸² Johnson B. G., Gill P. M. W., Pople J. A.
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., ⁸⁶⁶ Murray C. W., Handy N. C., Laming G. J.
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. ²¹⁰ Chien S.-H., Gill P. M. W.
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 ⁸⁵⁵ Mitani M.
Theor. Chem. Acc.
(2011), 130, pp. 645. Link ^{, 854} Mitani M., Yoshioka Y.
Theor. Chem. Acc.
(2012), 131, pp. 1169. Link is used for the largest standard grids in Q-Chem, ⁸⁵⁵ Mitani M.
Theor. Chem. Acc.
(2011), 130, pp. 645. Link ^{, 854} Mitani M., Yoshioka Y.
Theor. Chem. Acc.
(2012), 131, pp. 1169. Link namely SG-2 and SG-3. ²⁶⁹ Dasgupta S., Herbert J. M.
J. Comput. Chem.
(2017), 38, pp. 869. Link (See Section 5.5.3.)