  In the last few years we have been developing a Monte Carlo simulation method
to cope with systems of many electrons and ions in the Born-Oppenheimer (BO)
approximation, the Coupled Electron-Ion Monte Carlo Method (CEIMC). Electronic
properties in CEIMC are computed by Quantum Monte Carlo (QMC) rather than by
Density Functional Theory (DFT) based techniques. CEIMC can, in principle,
overcome some of the limitations of the present DFT based ab initio dynamical
methods. Application of the new method to high pressure metallic hydrogen has
recently appeared. In this paper we present a new sampling algorithm that we
have developed in the framework of the Reptation Quantum Monte Carlo (RQMC)
method chosen to sample the electronic degrees of freedom, thereby improving
its efficiency. Moreover, we show here that, at least for the case of metallic
hydrogen, variational estimates of the electronic energies lead to an accurate
sampling of the proton degrees of freedom.
