  Recent progress in simulation methodologies and in computer power allow first
principle simulations of condensed systems with Born-Oppenheimer electronic
energies obtained by Quantum Monte Carlo methods. Computing free energies and
therefore getting a quantitative determination of phase diagrams is one step
more demanding in terms of computer resources. In this paper we derive a
general relation to compute the free energy of an ab-initio model with
Reptation Quantum Monte Carlo (RQMC) energies from the knowledge of the free
energy of the same ab-initio model in which the electronic energies are
computed by the less demanding but less accurate Variational Monte Carlo (VMC)
method. Moreover we devise a procedure to correct transition lines based on the
use of the new relation. In order to illustrate the procedure, we consider the
liquid-liquid phase transition in hydrogen, a first order transition between a
lower pressure, molecular and insulating phase and a higher pressure, partially
dissociated and conducting phase. We provide new results along the T = 600K
isotherm across the phase transition and find good agreement between the
transition pressure and specific volumes at coexistence for the model with RQMC
accuracy between the prediction of our procedure and the values that can be
directly inferred from the observed plateau in the pressure-volume curve along
the isotherm. This work paves the way for future use of VMC in first principle
simulations of high pressure hydrogen, an essential simplification when
considering larger system sizes or quantum proton effects by Path Integral
Monte Carlo methods.
