Free energy methods in coupled electron ion Monte Carlo

Abstract

Recent progress in simulation methodologies and in computer power allow first-principles 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 = 600 K 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-principles simulations of high-pressure hydrogen, an essential simplification when considering larger system sizes or quantum proton effects by Path Integral Monte Carlo methods.

Keywords:

• Monte Carlo methods
• quantum Monte Carlo
• high-pressure hydrogen
• free energies and phase diagrams

Acknowledgments

CP is supported by the Italian Institute of Technology (IIT) under the SEED project grant number 259 SIMBEDD – Advanced Computational Methods for Biophysics, Drug Design and Energy Research. DMC is supported by DOE grant DE-FG52-09NA29456. This work was performed in part under the auspices of the US DOE by LLNL under Contract DE-AC52-07NA27344. Financial support from the Erasmus Mundus Program-Atosim is acknowledged. Computer resources were provided by CASPUR (Italy) within the Competitive HPC Initiative, grant number cmp09-837, and by the DEISA Consortium (www.deisa.eu), co-funded through the EU FP6 project RI-031513 and the FP7 project RI-222919, through the DEISA Extreme Computing Initiative (DECI 2009).

Notes

Notes

1. In case of fermions one must employ the fixed node approximation to avoid the sign problem and E ₀ is the fixed node ground state energy which is an upper bound of the true ground state energy 2.

2. We can gain an idea of the relative amount of the free energy corrections by comparing the difference mH with the VMC free energy difference between phase I and II at the same thermodynamic points, mH: the contribution of the correction is then quite small as expected, about 1.5% of the VMC free energy differences.