5.5 DFT Numerical Quadrature

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

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.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.)