  High pressure and high temperature properties of AB (A = $^6$Li, $^7$Li; B =
H, D, T) are investigated with first-principles method comprehensively. It is
found that the H$^{-}$ sublattice features in the low-pressure electronic
structure near the Fermi level of LiH are shifted to that dominated by the
Li$^{+}$ sublattice in compression. The lattice dynamics is studied in
quasi-harmonic approximation, from which the phonon contribution to the free
energy and the isotopic effects are accurately modelled with the aid of a
parameterized double-Debye model. The obtained equation of state (EOS) matches
perfectly with available static experimental data. The calculated principal
Hugoniot is also in accordance with that derived from shock wave experiments.
Using the calculated principal Hugoniot and the previous theoretical melting
curve, we predict a shock melting point at 56 GPa and 1923 K. In order to
establish the phase diagram for LiH, the phase boundaries between the B1 and B2
solid phases are explored. The B1-B2-liquid triple point is determined at about
241 GPa and 2413 K. The remarkable shift in the phase boundaries by isotopic
effect and temperature reveal the significant role played by lattice
vibrations. Furthermore, the Hugoniot of the static-dynamic coupling
compression is assessed. Our EOS suggests that a precompression of the sample
to 50 GPa will allow the shock Hugoniot passing through the triple point and
entering the B2 solid phase. This transition leads to a discontinuity with 4.6%
volume collapse, about four times greater than the same B1-B2 transition at
zero temperature.
