  The tin-selenide and tin-sulfide classes of materials undergo multiple
structural transitions under high pressure leading to periodic lattice
distortions, superconductivity, and topologically non-trivial phases, yet a
number of controversies exist regarding the structural transformations in these
systems. We perform first-principles calculations within the framework of
density functional theory and a careful comparison of our results with
available experiments on SnSe$_2$ reveals that the apparent contradictions
among high-pressure results can be attributed to differences in experimental
conditions. We further demonstrate that under hydrostatic pressure a $\sqrt{3}
\times \sqrt{3} \times 1$ superstructure can be stabilized above 20 GPa in
SnS$_2$ via a periodic lattice distortion as found recently in the case of
SnSe$_2$, and that this pressure-induced phase transition is due to the
combined effect of Fermi surface nesting and electron-phonon coupling at a
momentum wave vector $\mathbf{q}$ = $(1/3, 1/3, 0)$. In addition, we
investigate the contribution of nonadiabatic corrections on the calculated
phonon frequencies, and show that the quantitative agreement between theory and
experiment for the high-energy $A_{1g}$ phonon mode is improved when these
effects are taken into account. Finally, we examine the nature of the
superconducting state recently observed in SnSe$_2$ under nonhydrostatic
pressure and predict the emergence of superconductivity with a comparable
critical temperature in SnS$_2$ under similar experimental conditions.
Interestingly, in the periodic lattice distorted phases, the critical
temperature is found to be reduced by an order of magnitude due to the
restructuring of the Fermi surface.
