Anharmonic phonon behavior via irreducible derivatives: molecular dynamics and self-consistent perturbation theory

Cubic phonon interactions are now regularly computed from first principles, and the quartic interactions have begun to receive more attention. Given this realistic anharmonic vibrational Hamiltonian, the classical phonon Green's function can be precisely measured using molecular dynamics, which can then be used to rigorously assess the range of validity for self-consistent diagrammatic approaches in the classical limit. Here we use the bundled irreducible derivative approach to efficiently and precisely compute cubic and quartic phonon interactions in CaF$_2$ and ThO$_2$, systematically obtaining the vibrational Hamiltonian purely in terms of irreducible derivatives. We assess the fidelity of various bare and self-consistent diagrammatic approaches to the phonon Green's function as compared to the numerically exact solution. Specific attention is given to the phonon frequency shifts and linewidths, demonstrating that the 4-phonon bubble diagram has an important contribution to linewidth beyond $T=500$ K. Moreover, accurate results are obtained even at $T=900$ K when performing self-consistency using a 4-phonon loop and evaluating the 3-phonon and 4-phonon bubble post-self-consistency. Our demonstration of accurate computation of the classical Green's function for a realistic anharmonic vibrational Hamiltonian implies that comparison with experiment at sufficiently high temperatures can be reserved for scrutinizing the quality of the vibrational Hamiltonian and the underlying approximations to the many-electron problem.