5 Density Functional Theory

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

EXC=AatomsiApointswAif(𝐫Ai), (5.15)

where the function f is the aforementioned XC integrand and the quantities wAi are the quadrature weights. The sum over i runs over grid points belonging to atom A, which are located at positions 𝐫Ai=𝐑A+𝐫i, so this approach requires only the choice of a suitable one-center integration grid (to define the 𝐫i), which is independent of nuclear configuration. These grids are implemented in Q-Chem in a way that ensures that the EXC 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,) 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 quadrature662, 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.)