Dynamical Steps that Bridge Piecewise Adiabatic Shapes in the Exact Time-Dependent Potential Energy Surface

We study the exact time-dependent potential energy surface (TDPES) in the presence of strong nonadiabatic coupling between the electronic and nuclear motion. The concept of the TDPES emerges from the exact factorization of the full electron-nuclear wave function [A. Abedi, N. T. Maitra, and E. K. U. Gross, Phys. Rev. Lett. 105, 123002 (2010)]. Employing a one-dimensional model system, we show that the TDPES exhibits a dynamical step that bridges between piecewise adiabatic shapes. We analytically investigate the position of the steps and the nature of the switching between the adiabatic pieces of the TDPES.

Figure 1

Left: The first two BOPESs (indicated in the figure) together with the third BOPES (black dashed line) and the initial nuclear wave function (black solid line). Right: Adiabatic electronic conditional densities as indicated in the figures: the $x$ axis represents the parametric coordinate $R$, the $y$ axis is the electronic coordinate $r$, and along the $z$ axis we plot the conditional probability of finding the electron in $r$ given that the ion is in $R$.

Figure 2

First panel (top): The gauge independent part of the TDPES (black solid line) plotted at four different times (indicated), ${ϵ}_{\mathrm{BO}}^{(1)}$ (red or dark gray dashed line), and ${ϵ}_{\mathrm{BO}}^{(2)}$ (green or light gray dashed line). Second panel (from the top): The gauge-dependent part of the TDPES is plotted at the same times. Third panel (from the top): The exact nuclear density (black dashed line) is shown together with $|F_1(R,t){|}^2$ (red or dark gray solid line) and $|F_2(R,t){|}^2$ (green or light gray solid line). Lowest panel: The exact time-dependent electronic conditional density $|{\Phi }_R(r,t){|}^2$ is plotted. The color range is the same as Fig. 1.

Figure 3

Top: A snapshot of the gauge-invariant part of the TDPES (solid black line) at the $t=31.87\mathrm{fs}$. For reference, ${ϵ}_{\mathrm{BO}}^{(1)}$ (red or dark gray dashed line) and ${ϵ}_{\mathrm{BO}}^{(2)}$ (green or light gray dashed line) are shown. Bottom: Expansion coefficients (indicated in the figure) of the (two states) adiabatic expansion of the full wave function and the exact electronic conditional wave function (see the text) at $t=31.87\mathrm{fs}$.