  The microscopic origin of the large negative thermal expansion of cubic
scandium trifluorides (ScF3) is investigated by performing a set of anharmonic
free-energy calculations based on density functional theory. We demonstrate
that the conventional quasiharmonic approximation (QHA) completely breaks down
for ScF3 and the quartic anharmonicity, treated nonperturbatively by the
self-consistent phonon theory, is essential to reproduce the observed
transition from negative to positive thermal expansivity and the hardening of
the R4+ soft mode with heating. In addition, we show that the contribution from
the cubic anharmonicity to the vibrational free energy, evaluated by the
improved self-consistent phonon theory, is significant and as important as that
from the quartic anharmonicity for robust understandings of the temperature
dependence of the thermal expansion coefficient. The first-principles approach
of this study enables us to compute various thermodynamic properties of solids
in the thermodynamic limit with the effects of cubic and quartic
anharmonicities. Therefore, it is expected to solve many known issues of the
QHA-based predictions particularly noticeable at high temperature and in
strongly anharmonic materials.
