Projector-based efficient estimation of force constants

Estimating force constants for crystal structures is crucial for calculating various phonon-related properties. However, this task becomes particularly challenging when dealing with a large number of atoms or when third- and higher-order force constants are required. In this study, we propose an efficient approach that involves constructing a complete orthonormal basis set for the force constants. We formulate this approach using projection matrices and their eigenvectors to meet the requirements for the force constants. This basis set enables precise inference of the force constants from displacement-force data sets. Our efficient algorithms for basis-set construction and force-constant estimation are implemented in the SYMFC code. Furthermore, several applications demonstrated in this study indicate that the current approach facilitates the efficient and accurate determination of force constants.