Integration Grid Errors for Meta-GGA-Predicted Reaction Energies: Origin of Grid Errors for the M06 Suite of Functionals

We have assessed integration grid errors arising from the use of popular DFT quadrature schemes for a set of 34 organic reaction energies. The focus is primarily on M05-2X and the M06 suite of functionals (M06-L, M06, M06-2X, and M06-HF). M05-2X, M06, and M06-2X outperform popular older DFT functionals for the reaction energies studied and offer accuracies comparable to results from perturbative hybrid DFT functionals. However, these new functionals are more sensitive to the choice of quadrature grid than previous generations of DFT functionals. Errors in predicted reaction energies arising from the use of the popular SG-1 integration grid, which is the default in the Q-Chem package, are significant. In particular, M06-HF reaction energies computed with the SG-1 grid exhibit errors ranging from −6.7 to 3.2 kcal mol−1, relative to results computed with a very fine integration grid. This grid sensitivity is not a general problem for meta-generalized gradient approximation functionals, but is instead due to the specific functional forms used in these functionals. The large grid errors are traced to the kinetic energy density enhancement factor utilized in the exchange component of the M05-2X and the M06 functionals. This term contains empirically adjusted parameters that are of large magnitude for all of these functionals and for M06-HF in particular. The product of these large constants and modest integration errors for the kinetic energy density results in very large errors in individual contributions to the exchange energy. This gives rise to the large errors in reaction energies exhibited by these functionals for certain integration grids.