  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 the cubic
and quartic phonon interactions of CaF$_2$, systematically obtaining the
vibrational Hamiltonian purely in terms of irreducible derivatives. non
frequency shifts and linewidths, We demonstrate that the 4-phonon sunset
diagram has an important contribution to the optical phonon linewidths beyond
$T=500$ K. Reasonable results are obtained even at $T=900$ K when performing
self-consistency using the 4-phonon loop diagram and evaluating the 3-phonon
bubble and 4-phonon sunset diagrams post self-consistency. Further improvements
are obtained by performing quasiparticle perturbation theory, where both the
4-phonon loop and the real part of the 3-phonon bubble are employed during
self-consistency. Our irreducible derivative approach to self-consistent
perturbation theory is a robust tool for studying anharmonic phonons in both
the quantum and classical regimes.
