  We present a new set of three-body interaction models based on the
Bruch-McGee (BM) potential that are suitable for the study of the energy,
structural and elastic properties of solid 4He at high pressure. Our ab initio
three-body potentials are obtained from the fit to total energies and atomic
forces computed with the van der Waals density functional theory method due to
Grimme, and represent an improvement with respect to previously reported
three-body interaction models. In particular, we show that some of the
introduced BM parametrizations reproduce closely the experimental equation of
state and bulk modulus of solid helium up to a pressure of ~ 60 GPa, when used
in combination with standard pairwise interaction models in diffusion Monte
Carlo simulations. Importantly, we find that recent predictions reporting a
surprisingly small variation of the kinetic energy and Lindeman ratio on
quantum crystals under increasing pressure are likely to be artifacts produced
by the use of incomplete interaction models. Also, we show that the
experimental variation of the shear modulus, C44, at P < 25 GPa can be
quantitatively described with the new set of three-body BM potentials. At
higher pressures, however, the agreement between our C44 results and
experiments deteriorates and thus we argue that higher order many-body terms in
the expansion of the atomic interactions probably are necessary in order to
better describe elasticity in very dense solid 4He.
