Anharmonic Grüneisen theory based on self-consistent phonon theory: Impact of phonon-phonon interactions neglected in the quasiharmonic theory

We formulate a theory of thermal expansion based on the self-consistent phonon (SCP) theory, which nonperturbatively considers anharmonic effect. show that Gr\"unseisen formula holds within SCP by replacing frequency frequency. By comparing it with quasiharmonic approximation (QHA), we derive explicit formulas correction to QHA result. anharmonicity gives small $O(\langle{\hat{U}_4\rangle}/\langle{\hat{U}_2\rangle})$ coefficient $\alpha$, where $\hat{U}_2$ and $\hat{U}_4$ are harmonic quartic terms potential energy surface. On other hand, two temperature ($T$)-dependent shift comparable original term. In strongly materials such as NaCl MgO, these tend cancel out each other, explains why sometimes reasonable values for $T$-dependent while fails expansion.