Theory of Nonadiabatic Electron Dynamics in Nanomaterials

Abstract

Definition Nonadiabatic dynamics is an evolution of a coupled quantum-classical electron–nuclear system involving changes of electronic state in response to nuclear trajectory. Electron dynamics is a process involving changes of the quantum state of an electronic subsystem over time. The quantum state is defined by the system wave function, C(r, R, t). The wave function depends on electronic, r, and nuclear, R, coordinates of all particles belonging to the system and on time, t. The evolution of the wave function is given by the time-dependent Schrodinger equation (TD-SE): iħ @C r, R, t ð Þ @t ¼ ^ H r, R, t ð ÞC r, R, t ð Þ; (1) where ^ H(r, R, t) is the total Hamiltonian of the system. The total Hamiltonian is a sum of the kinetic energy of nuclei, ^ T nucl , and the electronic Hamiltonian, ^ H el (r, R, t) : ^ H r, R, t ð Þ¼ ^ T nucl þ ^ H el r, R, t ð Þ; (2a) ^ T nucl ¼ À X a ħ 2 2M a ∇ 2 a : (2b) Here, M i is the mass of the i-th nucleus, ∇ ! a @ @X a , @ @Y a , @ @Z a T is the gradient operator, and ħ is the reduced Planck's constant. The electronic Hamiltonian, ^ H el (r, R, t), incorporates kinetic energy of the electrons, electron–nuclear, electron–electron, and nuclear–nuclear interactions. The latter is merely an additive constant depending only on the geometry of the system. The electronic Hamiltonian may also include explicit dependence on time, e.g., via field–matter interaction terms. Solution of Eq. 1 constitutes the main goal of electronic dynamics studies. For simple low-dimensional models, the equation can be solved numerically on a grid. Such solution is considered exact, provided that