For a fixed value of the radial spherical-polar coordinate $r$, a function $f(\mathbf{r})\equiv f(r,\theta ,\varphi )$ has an exact expansion in spherical harmonic functions,
$$f(r,\theta ,\varphi )=\sum _{\mathrm{\ell}=0}^{\mathrm{\infty}}\sum _{m=-\mathrm{\ell}}^{\mathrm{\ell}}{c}_{\mathrm{\ell}m}{\text{Y}}_{\mathrm{\ell}m}(\theta ,\varphi ).$$ | (5.16) |
Angular quadrature grids are designed to integrate $f(r,\theta ,\varphi )$ for fixed $r$, and are often characterized by their degree, meaning the maximum value of $\mathrm{\ell}$ for which the quadrature is exact, as well as by their efficiency, meaning the number of spherical harmonics exactly integrated per degree of freedom in the formula. Q-Chem supports the following two types of angular grids.
Lebedev grids. These are specially-constructed grids for quadrature on the surface of a sphere,^{Lebedev:1977, Lebedev:1975, Lebedev:1976, Lebedev:1999} based on the octahedral point group. Lebedev grids available in Q-Chem are listed in Table 5.2. These grids typically have near-unit efficiencies, with efficiencies exceeding unity in some cases. A Lebedev grid is selected by specifying the number of grid points (from Table 5.2) using the $rem keyword XC_GRID, as discussed below.
No. Points | Degree | No. Points | Degree | No. Points | Degree |
---|---|---|---|---|---|
(${\mathrm{\ell}}_{\mathrm{max}}$) | (${\mathrm{\ell}}_{\mathrm{max}}$) | (${\mathrm{\ell}}_{\mathrm{max}}$) | |||
6 | 3 | 230 | 25 | 1730 | 71 |
18 | 5 | 266 | 27 | 2030 | 77 |
26 | 7 | 302 | 29 | 2354 | 83 |
38 | 9 | 350 | 31 | 2702 | 89 |
50 | 11 | 434 | 35 | 3074 | 95 |
74 | 13 | 590 | 41 | 3470 | 101 |
86 | 15 | 770 | 47 | 3890 | 107 |
110 | 17 | 974 | 53 | 4334 | 113 |
146 | 19 | 1202 | 59 | 4802 | 119 |
170 | 21 | 1454 | 65 | 5294 | 125 |
194 | 23 |
Gauss-Legendre grids. These are spherical direct-product grids in the two spherical-polar angles, $\theta $ and $\varphi $. Integration in over $\theta $ is performed using a Gaussian quadrature derived from the Legendre polynomials, while integration over $\varphi $ is performed using equally-spaced points. A Gauss-Legendre grid is selected by specifying the total number of points, $2{N}^{2}$, to be used for the integration, which specifies a grid consisting of $2{N}_{\varphi}$ points in $\varphi $ and ${N}_{\theta}$ in $\theta $, for a degree of $2N-1$. Gauss-Legendre grids exhibit efficiencies of only 2/3, and are thus lower in quality than Lebedev grids for the same number of grid points, but have the advantage that they are defined for arbitrary (and arbitrarily-large) numbers of grid points. This offers a mechanism to achieve arbitrary accuracy in the angular integration, if desired.
Combining these radial and angular schemes yields an intimidating selection of quadratures, so it is useful to standardize the grids. This is done for the convenience of the user, to facilitate comparisons in the literature, and also for developers wishing to compared detailed results between different software programs, because the total electronic energy is sensitive to the details of the grid, just as it is sensitive to details of the basis set. Standard quadrature grids are discussed next.