Issue 1: Should I care about the integration grid?
When performing calculations using KS-DFT, the xc energy is calculated as an integral over the corresponding xc energy density (which in turn depends on the local quantities mentioned above). The integration is usually performed on a three‑dimensional real-space grid obtained by partitioning the multi-center integral into atomic contributions using a nuclear weight function. The nuclear partitioning scheme of Becke142 is used by the vast majority of software in conjunction with atomic grids that are further separated into radial and angular parts. Specially constructed quadrature schemes are usually used for the angular portion, such as the Lebedev scheme.143–146 The radial part is generally treated with the Euler-Maclaurin scheme.147 The Lebedev–Euler-Maclaurin integration grids are represented by two numbers—n ,m —with n denoting the number of radial points and m the number of angular points. For example, the UltraFine grid in the Gaussian program148corresponds to a 99,590 grid, but many other options are available. The quality of the quadrature grid follows the same simple axiom discussed for basis set: “the larger, the better”. However, once again, software developers (and ultimately users) have to strike a balance between accuracy and computational cost. Many recent papers have focused on the importance of integration grids for molecular calculations using DFT.149–153 The choice of the integration grid can have a significant impact on the quality of electronic energies, as shown for some members of the Minnesota family of functionals,149–151,154 for which a grid of at least 99 radial and 590 angular points is necessary to converge some problematic cases (Figure 3 shows the dissociation curve for the argon dimer155 as calculated by the MN15-L41 functional with the def2-QZVPPD107 basis set and different integration grids). Often such grids are not the default that is automatically chosen by quantum chemistry programs, and the untrained users often don’t change the default setting. In some cases, avoiding the decision on the integration grid (and hence relying on the default choice of the selected software) can be as deleterious as choosing an inadequately small basis set. Unfortunately, though, an untrained user is more blind to this issue than the more common methodological ones discussed above. The grid issue does not only affect electronic energies but might also affect computed thermodynamic quantities such as entropies, Gibbs’ free energies and vibrational frequencies.152,156 The main message of the experiment in this section is that the 99,590 integration grid—corresponding to the UltraFine grid in the Gaussian program—is just a minimum requirement that ensures convergence of molecular calculations.149–152,156,157 When reporting computed results, it is important to indicate the integration grid used to obtain structures and energy values to make them reproducible.