Lattice dynamics and structural phase stability of group-IV elemental solids with the r$^2$SCAN functional

The strongly constrained and appropriately normed (SCAN) meta-generalized gradient approximation (meta-GGA) functional is a milestone achievement of electronic structure theory. Recently, a revised and restored form (r$^2$SCAN) has been suggested as a replacement for SCAN in high-throughput applications. Here, we assess the accuracy and reliability of the r$^2$SCAN meta-GGA functional for the group-IV elemental solids carbon (C), silicon (Si), germanium (Ge), and tin (Sn). We show that the r$^2$SCAN functional agrees closely with its parent functional SCAN for elastic constants, bulk moduli, and phonon dispersions, but the numerical stability of r$^2$SCAN is superior. Both meta-GGA functionals outperform standard GGA (Perdew-Burke-Ernzerhof) in terms of accuracy and approach the level of common hybrid functionals (Heyd-Scuseria-Ernzerhof). However, we find that r$^2$SCAN performs much worse than SCAN for the $α\leftrightarrow β$ phase transition of both Ge and Sn, yielding larger phase energy differences and transition pressures.