5 Density Functional Theory

5.5 DFT Numerical Quadrature

(June 30, 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 Becke A. D.
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

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. 513 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., 775 Murray C. W., Handy N. C., Laming G. J.
Mol. Phys.
(1993), 78, pp. 997.
Link
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. 194 Chien S.-H., Gill P. M. W.
J. Comput. Chem.
(2003), 24, pp. 732.
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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 765 Mitani M.
Theor. Chem. Acc.
(2011), 130, pp. 645.
Link
, 764 Mitani M., Yoshioka Y.
Theor. Chem. Acc.
(2012), 131, pp. 1169.
Link
is used for the largest standard grids in Q-Chem, 765 Mitani M.
Theor. Chem. Acc.
(2011), 130, pp. 645.
Link
, 764 Mitani M., Yoshioka Y.
Theor. Chem. Acc.
(2012), 131, pp. 1169.
Link
namely SG-2 and SG-3. 244 Dasgupta S., Herbert J. M.
J. Comput. Chem.
(2017), 38, pp. 869.
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(See Section 5.5.2.)