Ring Polymer Molecular Dynamics with Electronic Transitions

Full quantum dynamics of molecules and materials is of fundamental importance, which requires a faithful description of simultaneous quantum motions of the electron and nuclei. A new scheme is developed for nonadiabatic simulations of coupled electron-nuclear quantum dynamics with electronic transitions based on the Ehrenfest theorem and ring polymer molecular dynamics. Built upon the isomorphic ring polymer Hamiltonian, time-dependent multistate electronic Schrödinger equations are solved self-consistently with approximate equation of motions for nuclei. Each bead bears a distinct electronic configuration and thus moves on a specific effective potential. This independent-bead approach provides an accurate description of the real-time electronic population and quantum nuclear trajectory, maintaining a good agreement with the exact quantum solution. Implementation of first-principles calculations enables us to simulate photoinduced proton transfer in ${\mathrm{H}}_2\mathrm{O}\text{−}{\mathrm{H}}_2{\mathrm{O}}^{+}$ where we find a good agreement with experiment.

Figure 1

Schematic diagrams for (a) centroid approximation, (b) bead approximation, and (c) independent-bead approximation approaches. The solid black lines represent the effective potential curves. Adjacent beads are connected by spring interaction (wave lines).

Figure 2

(a) Two adiabatic potential energy curves (black lines) and the nonadiabatic coupling strength (green line) as a function of position $R$ for the avoided crossing model. (b) The real-time electronic population in the excited state obtained by different methods.

Figure 3

Comparisons between spatial distribution of beads from RPMD-IB simulation and the wave packet profile from exact solution. All beads are divided into two states according to their electronic occupation. The wave packet is plotted as ${|{\chi }_i|}^2$ ($i=0$, 1), where 0 and 1 refer to the ground and excited states, respectively. The blue bar denotes the beads population in ground state in (a), (c), and (e), and the beads population in the excited state is represented by the red bar in (b), (d), and (f). We compared these two kinds of distributions at $t=10\mathrm{fs}$ [(a) and (b)], $t=14\mathrm{fs}$ [(c) and (d)], and $t=17\mathrm{fs}$ [(e) and (f)], respectively.

Figure 4

(a) The distance difference of the hydrogen atom (H) and its two nearest oxygen atoms (O1, O2). Shadows indicate the quantum fluctuation of all beads. Schematic setting is shown in the right panel. (b) The snapshots at 0 and 47 fs in the excited-state dynamics. In the RPMD-IB simulation, the configurations of all beads are presented. At 47 fs, the whole nuclear wave packet of split hydrogen atom approaches the oxygen atom (O1) of the first water molecule, indicating the formation of hydronium (${\mathrm{H}}_3{\mathrm{O}}^{+}$) and OH radical.

Figure 5

(a) The time evolution of the Hirshfeld electron occupation for two hydrogen atoms in the second water molecule obtained from the Ehrenfest and RPMD-IB simulations. We averaged the Hirshfeld charges over all beads in the RPMD-IB simulation. (b) The atomic mechanism for the formation of hydronium and OH radical. (c) The projected density of states on the two water molecules (upper panel) and the population of each energy level (lower panel) at 5, 20, and 45 fs. We choose a typical bead to represent the evolution of electronic structure.