Dynamics of Electrons and Nuclei

Vladimir I. Chizhik, Yuri S. Chernyshev, Alexey V. Donets +3 more

Bruce D. Popp

Yang Zhao, Satoshi Yokojima, Guan-Hua Chen

Nuclear dynamics is incorporated into an efficient density matrix formalism of electronic dynamics which has been applied to molecular systems containing thousands of atoms. The formalism for the combined dynamics of electrons and nuclei is derived from the Dirac–Frenkel variational principle. The single electron reduced density matrices and the Glauber coherent states are used for the electronic and nuclear degrees of freedom, respectively. The new formalism is applicable to simulate the dynamics of large molecular systems. As an illustration of its validity, the formalism is employed to calculate the electron and nuclei dynamics of hydrogen molecules.

Leonard M. Sander

Axel Schild

Quantum trajectory calculations for electrons are a useful tool in the field of molecular dynamics, e.g. to understand processes in ultrafast spectroscopy. They have, however, two limitation: On the one hand, such calculations are typically based on the Born-Oppenheimer approximation (BOA) and the electron dynamics for stationary nuclei is considered, thus neglecting quantum effects of the nuclei. On the other hand, even if the quantum nuclear motion would be taken into account, a BOA dynamics on a single potential energy surface would not provide any electron trajectories because the electronic part is treated as a stationary problem. By using the exact factorization method, we overcome these limitations and generalize the theory of electronic quantum trajectories to a fully quantum-mechanical treatment of the nuclei. After reviewing the time-dependent theory of quantum hydrodynamics and quantum trajectories, we show that the nuclei can be viewed as a quantum clock for the electronic motion and we develop a fully quantum-mechanical clock-dependent version of quantum hydrodynamics. This theory is used to obtain electronic trajectories for quantum nuclei, as is exemplified for a model system of a proton-coupled electron transfer dynamics. Our work generalizes the concept of quantum trajectories and lays the foundations for the development of trajectory-based simulation methods of electron dynamics beyond the BOA.

Herman J. C. Berendsen

Introduction We now move to considering the dynamics of a system of nuclei and electrons. Of course, both electrons and nuclei are subject to the laws of quantum mechanics, but since nuclei are 2000 to 200,000 times heavier than electrons, we expect that classical mechanics will be a much better approximation for the motion of nuclei than for the motion of electrons. This means that we expect a level of approximation to be valid, where some of the degrees of freedom (d.o.f.) of a system behave essentially classically and others behave essentially quantum-mechanically. The system then is of a mixed quantum/classical nature. Most often the quantum subsystem consists of system of electrons in a dynamical field of classical nuclei, but the quantum subsystem may also be a selection of generalized nuclear coordinates (e.g., corresponding to high-frequency vibrations) while other generalized coordinates are supposed to behave classically, or describe the motion of a proton in a classical environment. So, in this chapter we consider the dynamics of a quantum system in a non-stationary potential. In Section 5.2 we consider the time-dependent potential as externally given, without taking notice of the fact that the sources of the time-dependent potential are moving nuclei, which are quantum particles themselves, feeling the interaction with the quantum d.o.f. Thus we consider the time evolution of the quantum system, which now involves mixing-in of excited states, but we completely ignore the back reaction of the quantum system onto the d.o.f. that cause the time-dependent potential, i.e., the moving nuclei.