Unifying Decoherence and Phase Evolution in Mixed Quantum-Classical Dynamics through Exact Factorization

Abstract

We propose new mixed quantum-classical equations of motion within the exact factorization framework to describe electronic coherence and phase evolution simultaneously. The derivation reveals that not only a projected quantum momentum correction, recently identified by Arribas and Maitra [Phys. Rev. Lett. 133, 233201 (2024)], but also a phase-correction appears within the order of $\hbar$, providing a unified and rigorous account of electronic coherence and phase evolution as well as their effect on the nuclear force. Benchmark tests on one- and two-dimensional model systems demonstrate that the new formulations capture key nonadiabatic features with high accuracy.

Figures (5)

• Figure 1: Transmission probability for the lower state ($T_1$) as a function of initial momentum $k_0$ for various dynamics methods. $T_1$ is obtained as $T_1 = \int_0^{\infty} dR \; |\chi_1(R,t_f)|^2$ for the exact wave-packet dynamics and $T_1 = \sum_{I \in \{R^{(I)}(t_f)>0\}} |C_1^{(I)}(t_f)|^2/N_\mathrm{traj}$ for the trajectory-based dynamics. $R^{(I)}(t)$ and $t_f$ denote the trajectory as a function of time $t$ and the final time of a simulation, respectively, with the total number of trajectories $N_\mathrm{traj}$.
• Figure 2: Time evolution of BO population $\langle|C_1|^2\rangle(t)$ (upper) and coherence $\langle|C_1C_2|^2\rangle(t)$ (lower) with different initial momenta $k_0 = 30$ and $40$ a.u.
• Figure 3: Spatial distribution of nuclear wave packets ($|\chi_i(R,t)|^2$) and BO populations ($|C_i(R,t)|^2$) at different time steps, $t=1300$ a.u. (a,c,e,g) and $1900$ a.u. (b,d,f,h) with the initial momentum $k_0=50.0$ a.u. for CT-based approaches (CT/CTv2) (a-d) and SHXF-based approaches (SHXF/SHXFv2) (e-h). The BO potential energies $\epsilon_{1/2}$ and the NAC $d_{12}$ are depicted to interpret wave-packet dynamics (a,b,e,f).
• Figure 4: Spatial distribution of nuclear wave packets (a,b) and BO populations (c,d) at $t=2200$ a.u. with the initial momentum $k_0=30.0$ a.u. for CTv2 (a,c) and CTv2 with the QM $\mathbf{\mathcal{P}}_\nu$ and the PQM $\mathbf{\mathcal{Q}}_\nu$ obtained from quantum dynamics (b,d). BO potentials and NAC are depicted as in Fig. \ref{['fig:spatial_k0_50']}.
• Figure 5: Transmission probability for the lower state ($T_1$) for the 2DNS model, as in Fig. \ref{['fig:RT_all']}.