Fully Quantum Description of the Zundel Ion: Combining Variational Quantum Monte Carlo with Path Integral Langevin Dynamics

We introduce a novel approach for a fully quantum description of coupled electron ion systems from first principles. It combines the variational quantum Monte Carlo solution of the electronic part with the path integral formalism for the quantum nuclear dynamics. On the one hand, the path integral molecular dynamics includes nuclear quantum effects by adding a set of fictitious classical particles (beads) aimed at reproducing nuclear quantum fluctuations via a harmonic kinetic term. On the other hand, variational quantum Monte Carlo can provide Born-Oppenheimer potential energy surfaces with a precision comparable to the most advanced post-Hartree-Fock approaches, and with a favorable scaling with the system size. In order to cope with the intrinsic noise due to the stochastic nature of quantum Monte Carlo methods, we generalize the path integral molecular dynamics using a Langevin thermostat correlated according to the covariance matrix of quantum Monte Carlo nuclear forces. The variational parameters of the quantum Monte Carlo wave function are evolved during the nuclear dynamics, such that the Born-Oppenheimer potential energy surface is unbiased. Statistical errors on the wave function parameters are reduced by resorting to bead grouping average, which we show to be accurate and well-controlled. Our general algorithm relies on a Trotter breakup between the dynamics driven by ionic forces and the one set by the harmonic interbead couplings. The latter is exactly integrated, even in the presence of the Langevin thermostat, thanks to the mapping onto an Ornstein-Uhlenbeck process. This framework turns out to be also very efficient in the case of noiseless (deterministic) ionic forces. The new implementation is validated on the Zundel ion (H5O2+) by direct comparison with standard path integral Langevin dynamics calculations made with a coupled cluster potential energy surface. Nuclear quantum effects are confirmed to be dominant over thermal effects well beyond room temperature, giving the excess proton an increased mobility by quantum tunneling.