  Ab initio calculations based on the Density Functional Theory are used to
show that the Debye frequency is a linear function of density to a high
accuracy for several elemental solids at pressures (at least) up to 360 GPa.
This implies that the ratio of density over the (Debye-frequency-based)
vibrational Gr\"{u}neisen parameter is a linear function of density in this
region. Numerical data from first principles calculations for several systems
at temperatures up to 2000K suggest that this is also true for the thermal
Gr\"{u}neisen parameter in the same range of pressure. Our analytical form of
the vibrational Gr\"{u}neisen parameter is applied to an implementation of the
Lindemann's melting criterion to obtain a simple extrapolation formula for the
melting temperatures of materials at higher densities. This prediction is
tested against available experimental and numerical data for several elemental
solids.
