Testing the r$^2$SCAN density functional for the thermodynamic stability of solids with and without a van der Waals correction

A central aim of materials discovery is an accurate and numerically reliable description of thermodynamic properties, such as the enthalpies of formation and decomposition. The r$^2$SCAN revision of the strongly constrained and appropriately normed (SCAN) meta-generalized gradient approximation (meta-GGA) balances numerical stability with high general accuracy. To assess the r$^2$SCAN description of solid-state thermodynamics, we evaluate the formation and decomposition enthalpies, equilibrium volumes, and fundamental bandgaps of more than 1,000 solids using r$^2$SCAN, SCAN, and PBE, as well as two dispersion-corrected variants, SCAN+rVV10 and r$^2$SCAN+rVV10. We show that r$^2$SCAN achieves accuracy comparable to SCAN and often improves upon SCAN's already excellent accuracy. Whereas SCAN+rVV10 is often observed to worsen the formation enthalpies of SCAN, and makes no substantial correction to SCAN's cell volume predictions, r$^2$SCAN+rVV10 predicts marginally less-accurate formation enthalpies than r$^2$SCAN, and slightly more-accurate cell volumes than r$^2$SCAN. The average absolute errors in predicted formation enthalpies are found to decrease by a factor of 1.5 to 2.5 from the GGA level to the meta-GGA level. Smaller decreases in error are observed for decomposition enthalpies. For formation enthalpies r$^2$SCAN improves over SCAN for intermetallic systems. For a few classes of systems -- transition metals, intermetallics, weakly-bound solids, and enthalpies of decomposition into compounds -- GGAs are comparable to meta-GGAs. In total, r$^2$SCAN and r$^2$SCAN+rVV10 can be recommended as stable, general-purpose meta-GGAs for materials discovery.