Quantum Dynamics with Electronic Friction

A theory of electronic friction is developed using the exact factorization of the electronic-nuclear wave function. No assumption is made regarding the electronic bath, which can be made of independent or interacting electrons, and the nuclei are treated quantally. The ensuing equation of motion for the nuclear wave function is a nonlinear Schrödinger equation including a friction term. The resulting friction kernel agrees with a previously derived mixed quantum-classical result by Dou et al., [Phys. Rev. Lett. 119, 046001 (2017)], except for a pseudomagnetic contribution in the latter that is here removed. More specifically, it is shown that the electron dynamics generally washes out the gauge fields appearing in the adiabatic dynamics. However, these are fully re-established in the typical situation where the electrons respond rapidly on the slow time scale of the nuclear dynamics (Markov limit). Hence, we predict Berry’s phase effects to be observable also in the presence of electronic friction. Application to a model vibrational relaxation problem proves that the proposed approach represents a viable way to account for electronic friction in a fully quantum setting for the nuclear dynamics.

Figure 1

(a) Schematics illustrating the model vibrational relaxation problem described in the text: a hydrogen atom (H) bound to a metallic surface (M) via the Morse potential $v(x)=D_e(1-e^{-\alpha x}{)}^2$ is displaced from its equilibrium position (green arrow) and relaxes under the action of the electronic friction $\gamma$. Also shown are the initial wave packet profile $|\psi (x){|}^2$ and the mean relaxation path of the system. Panels (b) and (c) display the evolution of the average position (b) and of the average system energy (c) for $D_e=1.55\mathrm{eV}$, $\alpha =1.238$ $a_0^{-1}$, and ${\tau }_{\text{relax}}=M{\gamma }^{-1}=50\mathrm{fs}$. Green lines are the results of the 1D nonlinear equation with uniform $\gamma$, red lines are numerically exact results from high-dimensional wave packet calculations using an explicit bath model (see main text for details).