Similarity of equations of motion for the classical and quantum trajectories is used to introduce a friction term dependent on the wavefunction phase into the time-dependent Schrödinger equation. The term describes irreversible energy loss by the quantum system. The force of friction is proportional to the velocity of a quantum trajectory. The resulting Schrödinger equation is nonlinear, conserves wavefunction normalization, and evolves an arbitrary wavefunction into the ground state of the system (of appropriate symmetry if applicable). Decrease in energy is proportional to the average kinetic energy of the quantum trajectory ensemble. Dynamics in the high friction regime is suitable for simple models of reactions proceeding with energy transfer from the system to the environment. Examples of dynamics are given for single and symmetric and asymmetric double well potentials.

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It has been recently shown that in the presence of an applied voltage, hydrogen and deuterium nuclei can be separated from one another using graphene membranes as a nuclear sieve, resulting in a 10-fold enhancement in the concentration of the lighter isotope. While previous studies, both experimental and theoretical, have attributed this effect mostly to differences in vibrational zero point energy (ZPE) of the various isotopes near the membrane surface, we propose that multi-dimensional quantum mechanical tunneling of nuclei through the graphene membrane influences this proton permeation process in a fundamental way. We perform ring polymer molecular dynamics calculations in which we include both ZPE and tunneling effects of various hydrogen isotopes as they permeate the graphene membrane and compute rate constants across a range of temperatures near 300 K. While capturing the experimentally observed separation factor, our calculations indicate that the transverse motion of the various isotopes across the surface of the graphene membrane is an essential part of this sieving mechanism. An understanding of the multi-dimensional quantum mechanical nature of this process could serve to guide the design of other such isotopic enrichment processes for a variety of atomic and molecular species of interest.

The Boltzmann evolution of a wavefunction can be recast as imaginary-time dynamics of the quantum trajectory ensemble. The quantum effects arise from the momentum-dependent quantum potential -computed approximately to be practical in high-dimensional systems -influencing the trajectories in addition to the external classical potential [S. Garashchuk, J. Chem. Phys. 132, 014112 (2010)]. For a nodeless wavefunction represented as ψ(x, t) = exp(−S(x, t)/¯) with the trajectory momenta defined by ∇ S(x, t), analysis of the Lagrangian and Eulerian evolution shows that for bound potentials the former is more accurate while the latter is more practical because the Lagrangian quantum trajectories diverge with time. Introduction of stationary and time-dependent components into the wavefunction representation generates new Lagrangian-type dynamics where the trajectory spreading is controlled improving efficiency of the trajectory description. As an illustration, different types of dynamics are used to compute zero-point energy of a strongly anharmonic well and low-lying eigenstates of a high-dimensional coupled harmonic system.

The quantum-mechanical tunneling is often important in low-energy reactions, which involve motion of light nuclei, occurring in condensed phase. The potential energy profile for such processes is typically represented as a double-well potential along the reaction coordinate. In a potential of this type defining reaction probabilities, rigorously formulated only for unbound potentials in terms of the scattering states with incoming/outgoing scattering boundary conditions, becomes ambiguous. Based on the analysis of a rectangular double-well potential, a modified expression for the reaction probabilities and rate constants suitable for arbitrary double-(or multiple-) well potentials is developed with the goal of quantifying tunneling. The proposed definition involves energy eigenstates of the bound potential and exact quantum-mechanical transmission probability through the barrier region of the corresponding scattering potential. Applications are given for several model systems, including proton transfer in a HO-H-CH 3 model, and the differences between the quantum-mechanical and quasiclassical tunneling probabilities are examined.