A nuclear quantum effect with pure anharmonicity and the anomalous   thermal expansion of silicon
  Despite the widespread use of silicon in modern technology, its peculiar thermal expansion is not well-understood. Adapting harmonic phonons to the specific volume at temperature, the quasiharmonic approximation, has become accepted for simulating the thermal expansion, but has given ambiguous interpretations for microscopic mechanisms. To test atomistic mechanisms, we performed inelastic neutron scattering experiments from 100-1500K on a single-crystal of silicon to measure the changes in phonon frequencies. Our state-of-the-art ab initio calculations, which fully account for phonon anharmonicity and nuclear quantum effects, reproduced the measured shifts of individual phonons with temperature, whereas quasiharmonic shifts were mostly of the wrong sign. Surprisingly, the accepted quasiharmonic model was found to predict the thermal expansion owing to a fortuitous cancellation of contributions from individual phonons. 
D.S. Kim, O. Hellman, J. Herriman, H.L. Smith, J.Y.Y. Lin, N.   Shulumba, J.L. Niedziela, C.W. Li, D.L. Abernathy, B. Fultz
cond-mat.mtrl-sci