Temperature-dependent classical phonons from efficient non-dynamical simulations

We describe a rigorous approach to the calculation of classic lattice-dynamical quantities from simulations that do not require an explicit consideration of the time evolution. We focus on the temperature-dependent vibrational spectrum. We start from the usual moment expansion of the relevant time correlation function (position-position or velocity-velocity) for a many-body system, and show that it can be conveniently split into one-body-like contributions by using a basis in which the low-order terms are diagonal. This allows us to compute the main spectral features (e.g., position {\em and} width of the phonon peaks) from thermal averages readily available from any statistical simulation. We demonstrate our method with an application to a model system that presents a structural transition and strongly temperature-dependent phonons. Our theory justifies and clarifies the status of previous heuristic schemes to estimate phonon frequencies in a computationally efficient way.