Factorized electron-nuclear dynamics with effective complex potential: on-the-fly implementation for H$_2^+$ in a laser field

TL;DR

The paper develops an on-the-fly implementation of Factorized Electron-Nuclear Dynamics (FENDy) with a complex time-dependent nuclear potential $V_d(q,t)=V_r(q,t)+\imath V_i(q,t)$ to model electron-nuclear dynamics beyond the Born-Oppenheimer framework. Nuclear dynamics are propagated via a quantum-trajectory ensemble, with the electronic state represented in a basis without requiring electronic eigenstates, and the nuclear and electronic subsystems coupled through derivative terms $\hat{D}^{(1)}$, $\hat{D}^{(2)}$ and a norm-preserving $V_i$. The authors apply the method to $H_2^+$ in a femtosecond laser field, demonstrating that FENDy can reproduce exact two-state dynamics when the electronic basis is efficiently projected (e.g., via Fourier expansions for $\mathbf{C}$ and basis projections for $p_\psi$, $r_\psi$). They also identify numerical challenges related to nonlocal electronic coefficients and electronic boundary conditions, and discuss strategies such as unweighted fits and low-pass filtering, outlining paths to improve accuracy and scalability for larger systems.

Abstract

Conventional theoretical and computational approaches to fully coupled quantum molecular dynamics, i.e. when both the electrons and nuclei are treated as quantum-mechanical particles, are impractical for all but the smallest chemical systems. In this paper we describe the formalism and implementation of the Factorized Electron Nuclear Dynamics (FENDy) with effective complex potential [J. Chem. Theory Comput. 19 (2023), pp 1393-1408], which goes beyond the established framework of the Born-Oppenheimer approximation or Born-Huang expansion of the molecular wavefunction. This method is based on the exact factorization of the molecular wavefunction, with the nuclei evolving under a complex time-dependent potential which captures the key features of dynamics in the nuclear subspace. The complementary electronic component of the molecular wavefunction is normalized to one for all nuclear configurations. We implement and employ FENDy to model the dynamics of H$_2^+$ molecular ion under a femtosecond laser pulse. The electronic wavefunction is represented within the standard electronic structure bases without referencing the electronic eigenstates. The nuclear wavefunction is represented as a quantum-trajectory ensemble, which in principle circumvents the exponential scaling of the numerical cost with the system size. The challenging evaluation of the gradients on unstructured grids is performed by projection on auxiliary bases.

Figures (9)

• Figure 1: (a) The excited state populations, $P_1$, obtained with FENDy for $N_p=[4,8,12]$, shown as red dash, blue dot-dash and green long dash, respectively, compared to the exact result (black solid line) as functions of time $t$. (b) The total nuclear momentum for the molecular wavefunction $\Psi$ ($\langle p\rangle=\langle\Psi| \hat{p}|\Psi\rangle$) and its electronic ($\langle\psi|\overline{p_\phi}|\psi\rangle_q$) and nuclear ($\langle\psi| p_\psi|\psi\rangle_q$) components, shown as red dash, blue dot-dash and green long dash, respectively, as functions of time $t$. The total momentum $\langle p\rangle$ from the exact dynamics is represented as black solid line.
• Figure 2: Time-evolution of the nuclear wavefunction. (a) Positions $\{q_t\}$ of the quantum trajectory ensemble representing the nuclear wavefunction $\psi(q,t)$ within the XF representation as functions of time. Every other trajectory is shown for clarity. (b) The 'exact' ground state amplitudes, $|\psi_0(q,t)|$, as functions of the nuclear coordinate $q$, obtained from the conventional two-state nonadiabatic dynamics at times indicated in the legend.
• Figure 3: Properties of the excited state dynamics (computed exactly): (a) average position of the ground (red dashed line) and excited (blue solid line) nuclear wavefunctions; (b) excited state population; (c) the electric field of the laser pulse.
• Figure 4: Snapshots of (a) $V_r$ (dashed lines) and (b) the Ehrenfest surfaces (dot dashed lines) projected into the basis. In both panels the solid lines are the ground (black) and excited (purple) BO surfaces. Note the appearance of a "hill" centered around $q=2.5$ in both panels.
• Figure 5: The Ehrenfest surface (dot dash) and $V_r$ (dashed) at $t=80$ a.u. The BO surfaces are given for reference.