Nonlinear Optimal Control of Electron Dynamics within Hartree-Fock Theory

TL;DR

This paper develops a method to optimize electric fields for controlling electron dynamics in molecules using time-dependent Hartree-Fock theory, making the problem computationally feasible for larger systems.

Contribution

It introduces a novel optimization framework with an adjoint state method for nonlinear TDHF equations, enabling efficient control of molecular electron states.

Findings

Successfully controls molecular states with small neural network controls

Achieves target states within acceptable constraints

Demonstrates applicability to multiple molecular systems

Abstract

Consider the problem of determining the optimal applied electric field to drive a molecule from an initial state to a desired target state. For even moderately sized molecules, solving this problem directly using the exact equations of motion -- the time-dependent Schr\"odinger equation (TDSE) -- is numerically intractable. We present a solution of this problem within time-dependent Hartree-Fock (TDHF) theory, a mean field approximation of the TDSE. Optimality is defined in terms of minimizing the total control effort while maximizing the overlap between desired and achieved target states. We frame this problem as an optimization problem constrained by the nonlinear TDHF equations; we solve it using trust region optimization with gradients computed via a custom-built adjoint state method. For three molecular systems, we show that with very small neural network parametrizations of the control, our method yields solutions that achieve desired targets within acceptable constraints and tolerances.