In parameterizing and validating the electron force field, it is necessary to assemble reference data from experiments, and theoretical methods that are applicable to ground states and excited states. Most of the theoretical methods are well known, e.g., Hartree-Fock for computing uncorrelated energies, CCSD(T) for high-accuracy correlated energies. In two cases, though, we have studied systems beyond the scope of those methods: the uniform electron gas and dense hydrogen plasma under high pressure.
For those cases, we have used reference data obtained from stochastic methods, which are very expensive computationally but very accurate and in principle general to any system. Diffusion Monte Carlo  is a method for computing high-accuracy energies for ground states. We write the time-independent Schrodinger equation as a diffusion equation in imaginary time:
Path integral Monte Carlo  is a useful method to compute thermodynamic averages of quantum operators at finite temperature. We write the position density matrix operator as an integral over successive paths: