Linear response for machine-learned interatomic potentials: phonons at arbitrary wave vectors from the primitive cell

Abstract

Phonon calculations with machine-learned interatomic potentials are almost exclusively performed with finite displacements in supercells, an approach that is exact only at wave vectors commensurate with the supercell and whose error depends on the range of the force constants. In message-passing networks this range grows with the number of layers, and reaches 140 angstrom for the deepest universal models. Here we show that the dynamical matrix of a message-passing potential can be obtained exactly in the primitive cell, at any wave vector, as the linear response of the forces to a displacement modulated by a Bloch phase, in the spirit of density-functional perturbation theory. The phase enters only where the features of neighboring atoms are gathered, and the trained model is used without modification. One further derivative yields group velocities and mode Grüneisen parameters analytically. We apply the method to NequIP and Prophet, and reproduce the Hessians of supercells to machine precision. Using these results as a reference, we find that the supercells commonly employed in high-throughput calculations lead to errors of several tenths of a THz in the phonon frequencies, and that these errors decrease slowly with the size of the supercell, in particular for deep models. For a low-symmetry silicate, the supercell predicts a dynamical instability that the model does not have. Quantities that depend on derivatives of the dispersion, such as the group velocities and the Grüneisen parameter at low temperature, are considerably more affected than the frequencies themselves. When both approaches are used to obtain the same force constants, their computational cost is comparable for crystals with symmetry.