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.