Practical computation of the Tully–Micha force in an adiabatic electronic representation

Develop a practical computational procedure to evaluate the Tully–Micha classical nuclear force when the electronic wavefunction is expanded in the adiabatic energy eigenstate basis, enabling application of the Tully–Micha derivation in mixed quantum–classical dynamics beyond one-dimensional systems.

Background

The mean-field equations for mixed quantum–classical dynamics can be derived either via Ehrenfest’s theorem (yielding forces as gradients of the expectation value of the electronic Hamiltonian) or via the semiclassical limit of the nuclear Schrödinger equation as in the Tully–Micha derivation. While Tully argued the derivations are equivalent, the authors note the equivalence is rigorously true only for one-dimensional systems.

Because SLED is formulated in an adiabatic representation, the authors use the Ehrenfest force and report they see no practical way to compute the Tully–Micha force under an adiabatic expansion, highlighting a methodological gap for multidimensional applications.