Finite-temperature electronic simulations without the Born-Oppenheimer constraint

The adiabatic approximation, typically assumed when performing standard Born-Oppenheimer (BO) molecular dynamics, can become unreliable at finite temperature, and specifically when the temperature is larger than the electronic energy gap between the ground state and the low-lying excited states. In this regime, relevant for many important chemical processes, the non-adiabatic couplings between the electronic energy states can produce finite temperature effects in several molecular properties, such as the geometry, the vibrational frequencies, the binding energy, and several chemical reactions. In this work, we introduce a novel finite-temperature non-adiabatic molecular dynamics based on a novel covariant formulation of the electronic partition function. In this framework, the nuclei are not constrained to move in a specific electronic potential energy surface. Then, by using a rigorous variational upper bound to the free energy, we are led to an approximate partition function that can be evaluated numerically. The method can be applied to any technique capable to provide an energy value over a given wave function ansatz depending on several variational parameters and atomic positions. In this work, we have applied the proposed method within a quantum Monte Carlo (QMC) scheme. In particular, we consider in this first application only classical ions, but we explicitly include an electronic correlation (Jastrow) term in the wave function, by extending in this way the standard variational QMC method, from ground state to finite temperature properties. We show that our approximation reduces correctly to the standard ground-state Born-Oppenheimer (gsBO) at zero temperature and to the correct high temperature limit. Moreover, at temperatures large enough, this method improves the upper bound of the free energy obtained with a single BO energy surface, since within our approach it is possible to estimate the electron entropy of a correlated ansatz in an efficient way. We test this new method on the simple hydrogen molecule, where at low temperature we recover the correct gsBO low temperature limit. Moreover, we show that the dissociation of the molecule is possible at a temperature much smaller than the one corresponding to the gsBO energy surface, in good agreement with experimental evidence. Several extensions of the proposed technique are also discussed, as for instance the inclusion of quantum effects for ions and the calculation of critical (magnetic, superconducting) temperatures.