5.5 DFT Numerical Quadrature

5.5.2 Standard Quadrature Grids

Four different “standard grids" are available in Q-Chem, designated SG-n, for n=0,1,2, or 3; both quality and the computational cost of these grids increases with n. These grids are constructed starting from a “parent” grid (Nr,NΩ) consisting of Nr radial spheres with NΩ angular (Lebedev) grid points on each, then systematically pruning the number of angular points in regions where sophisticated angular quadrature is not necessary, such as near the nuclei where the charge density is nearly spherically symmetric and at long distance from the nuclei where it varies slowly. A large number of points are retained in the valence region where angular accuracy is critical. The SG-n grids are summarized in Table 5.3. While many electronic structure programs use some kind of procedure to delete unnecessary grid points in the interest of computational efficiency, Q-Chem’s SG-n grids are notable in that the complete grid specifications are available in the peer-reviewed literature,299, 170, 209 to facilitate reproduction of Q-Chem DFT calculations using other electronic structure programs. Just as computed energies may vary quite strongly with the choice of basis set, so too in DFT they may vary strongly with the choice of quadrature grid. In publications, users should always specify the grid that is used, and it is suggested to cite the appropriate literature reference from Table 5.3.

Pruned Ref. Parent Grid No. Grid Points Default Grid for
Grid (Nr,NΩ) (C atom)a Which Functionals?b
SG-0 170 (23, 170) 1,390 (36%) None
SG-1 299 (50, 194) 3,816 (39%) LDA, most GGAs and hybrids
SG-2 209 (75, 302) 7,790 (34%) Meta-GGAs; B95- and B97-based functionals
SG-3 209 (99, 590) 17,674 (30%) Minnesota functionals
aNumber in parenthesis is the fraction of points retained from the parent grid
bReflects Q-Chem versions since v. 4.4.2
Table 5.3: Standard quadrature grids available in Q-Chem, along with the number of grid points for a carbon atom, showing the reduction in grid points due to pruning.

The SG-0 and SG-1 grids are designed for calculations on large molecules using GGA functionals. SG-1 affords integration errors on the order of 0.2 kcal/mol for medium-sized molecules and GGA functionals, including for demanding test cases such as reaction enthalpies for isomerizations. (Integration errors in total energies are no more than a few μhartree, or 0.01 kcal/mol.) The SG-0 grid was derived in similar fashion, and affords a root-mean-square error in atomization energies of 72 μhartree with respect to SG-1, while relative energies are reproduced well.170 In either case, errors of this magnitude are typically considerably smaller than the intrinsic errors in GGA energies, and hence acceptable. As seen in Table 5.3, SG-1 retains <40% of the grid points of its parent grid, which translates directly into cost savings.

Both SG-0 and SG-1 were optimized so that the integration error in the energy falls below a target threshold, but derivatives of the energy (including such properties as (hyper)polarizabilities153) are often more sensitive to the quality of the integration grid. Special care is required, for example, when imaginary vibrational frequencies are encountered, as low-frequency (but real) vibrational frequencies can manifest as imaginary if the grid is sparse. If imaginary frequencies are found, or if there is some doubt about the frequencies reported by Q-Chem, the recommended procedure is to perform the geometry optimization and vibrational frequency calculations again using a higher-quality grid. (The optimization should converge quite quickly if the previously-optimized geometry is used as an initial guess.)

SG-1 was the default DFT integration grid for all density functionals for Q-Chem versions 3.2–4.4. Beginning with Q-Chem v. 4.4.2, however, the default grid is functional-dependent, as summarized in Table 5.3. This is a reflection of the fact that although SG-1 is adequate for energy calculations using most GGA and hybrid functionals (although care must be taken for some other properties, as discussed below), it is not adequate to integrate many functionals developed since 2005. These include meta-GGAs, which are more complicated due to their dependence on the kinetic energy density (τσ in Eq. (5.10)) and/or the Laplacian of the density (2ρσ). Functionals based on B97, along with the Minnesota suite of functionals,1052, 1053 contain relatively complicated expressions for the exchange inhomogeneity factor, and are therefore also more sensitive to the quality of the integration grid.975, 606, 209 To integrate these modern density functionals, the SG-2 and SG-3 grids were developed,209 which are pruned versions of the medium-quality (75, 302) and high-quality (99, 590) integration grids, respectively. Tests of properties known to be highly sensitive to the quality of the integration grid, such as vibrational frequencies, hyper-polarizabilities, and potential energy curves for non-bonded interactions, demonstrate that SG-2 is usually adequate for meta-GGAs and B97-based functionals, and in many cases is essentially converged with respect to an unpruned (250, 974) grid.209 The Minnesota functionals are more sensitive to the grid, and while SG-3 is often adequate, it is not completely converged in the case of non-bonded interactions.209

Note:  1. SG-0 was re-optimized for Q-Chem v. 3.0, so results may differ slightly as compared to older versions of the program. 2. The SG-2 and SG-3 grids use a double-exponential radial quadrature,209 whereas a general grid (selected by setting XC_GRID = XY, as described in Section 5.4) uses an Euler-MacLaurin radial quadrature. As such, absolute energies cannot be compared between, e.g., SG-2 and XC_GRID = 000075000302, even though SG-2 uses a pruned (75, 302) grid. However, energy differences should be quite similar between the two.