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To accurately describe the energetics of transition metal systems, density functional approximations (DFAs) must provide a balanced description of s- and d- electrons. One measure of this is the sd transfer error, which has previously been defined as $E\left({3\mathrm{d}}^{n-1}{4\mathrm{s}}^1\right)-E\left({3\mathrm{d}}^{n-2}{4\mathrm{s}}^2\right)$. Theoretical concerns have been raised about this definition due to its evaluation of excited-state energies using ground-state DFAs. A more serious concern appears to be strong correlation in the 4s²configuration. Here, we define a ground-state measure of the sd energy imbalance, based on the errors of s- and d-electron second ionization energies of the 3d atoms, that effectively circumvents the aforementioned problems. We find an improved performance as we move from the local spin density approximation (LSDA) to the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) to the regularized and restored Strongly Constrained and Appropriately Normed (r²SCAN) meta-GGA for first-row transition metal atoms. However, we find large (∼2 eV) ground-state sd energy imbalances when applying a Perdew–Zunger 1981 self-interaction correction. This is attributed to an “energy penalty” associated with the noded 3d orbitals. A local scaling of the self-interaction correction to LSDA results in a balance of s- and d-errors. 

We have developed an accurate and efficient deep-learning potential (DP) for graphane, which is a fully hydrogenated version of graphene, using a very small training set consisting of 1000 snapshots from a 0.5 ps density functional theory (DFT) molecular dynamics simulation at 1000 K. We have assessed the ability of the DP to extrapolate to system sizes, temperatures, and lattice strains not included in the training set. The DP performs surprisingly well, outperforming an empirical many-body potential when compared with DFT data for the phonon density of states, thermodynamic properties, velocity autocorrelation function, and stress–strain curve up to the yield point. This indicates that our DP can reliably extrapolate beyond the limit of the training data. We have computed the thermal fluctuations as a function of system size for graphane. We found that graphane has larger thermal fluctuations compared with graphene, but having about the same out-of-plane stiffness.