Monte Carlo algorithms for simulating systems with adiabatic separation of electronic and nuclear degrees of freedom

Two Monte Carlo algorithms for the adiabatic sampling of nuclear and electronic degrees of freedom are introduced. In these algorithms the electronic degrees of freedom are subject to a secondary low-temperature thermostat in close analogy to the extended Lagrangian formalism used in molecular dynamics simulations. Numerical tests are carried out for two model systems of coupled harmonic oscillators, and for two more realistic systems: the water dimer and bulk liquid water. A statistical-mechanical discussion of the partition function for systems with adiabatic separation of electronic and nuclear degrees of freedom, but with finite electronic temperature, is presented. The theoretical analysis shows that the algorithms satisfy the adiabatic limit using suitable choices of the electronic temperature, T_elec, the number of electronic moves, R_elec, and the maximum step sizes used for displacements of nuclear coordinates. For quadratic coupling of the nuclear and electronic degrees of freedom, the electronic phase-space volume is independent of the nuclear coordinates. In this case, the sampling of the nuclear coordinate phase-space recovers the correct Born-Oppenheimer limit independent of T_elec, but each electronic degree of freedom contributes an offset of 0.5k_BT′_elec (with T′^-1_elec = T^-1 + T^-1_elec) to the average total energy. For nonquadratic coupling, satisfactory sampling of the nuclear coordinate phase-space requires a low T_elec to limit the ratio of the electronic phase-space volumes at T_elec and T′_elec to be close to unity.