Orbitally-Resolved Mechanical Properties of Solids from Maximally Localized Wannier Functions

We present a technique for partitioning the total energy from a semi-local density functional theory calculation into contributions from individual electronic states in a localized Wannier basis. We use our technique to reveal the key role played by the $s$ and $p$ orbitals of the apical oxygen atoms in a curious elastic anomaly exhibited by ferroelectric PbTiO${}_3$ under applied stress, which has so far gone unexplained. Our technique enables new insights into the chemical origins of the mechanical properties of materials, or any property given by an energy derivative.