Abstract
We prove that for a combined system of classical and quantum particles, it is possible to describe a dynamics for the classical particles that incorporates in a natural way the Boltzmann equilibrium population for the quantum subsystem. In addition, these molecular dynamics (MD) do not need to assume that the electrons immediately follow the nuclear motion (in contrast to any adiabatic approach) and do not present problems in the presence of crossing points between different potential energy surfaces (conical intersections or spin-crossings). A practical application of this MD to the study of the effect of temperature on molecular systems presenting (nearly) degenerate states—such as the avoided crossing in the ring-closure process of ozone—is presented.
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GENERAL SCIENTIFIC SUMMARY Introduction and background. Many interesting physical and chemical systems can be modeled and simulated using a mixed quantum-classical description in which, typically, the electrons, and possibly the lightest nuclei, are considered to be quantum, while the remaining (or all) nuclei are treated as classical particles. There are a number of methods for making this practical, such as the Car–Parrinello technique, Ehrenfest dynamics, surface hopping and Born–Oppenheimer molecular dynamics, each based on different assumptions and with different properties. However, it has proved to be difficult to find a simple quantum-classical dynamics that generates appropriate Boltzmann weights for the electronic part in equilibrium; a property that is regarded as desirable.
Main results. In this paper, we present a mixed quantum-classical dynamics that naturally extends ground-state Born–Oppenheimer molecular dynamics to the non-zero temperature case, and which presents, in equilibrium, the correct Boltzmann weights for the quantum subsystem, at fixed values of the classical variables. We show how, in the ring opening of the ozone molecule, the non-zero temperature correction significantly affects the effective potential seen by the classical part, and thus the properties of the system.
Wider implications. The correct description of the equilibrium properties of the quantum subsystem in a mixed quantum-classical model, achieved using the new dynamics presented here, has wide applications to many phenomena in which such mixed models must be used; in particular, in the case of conical intersections or spin-crossings, where the excited quantum states contribute more strongly to the resulting temperature-dependent effective potential. This has implications for transition-state theory, since the non-zero temperature potential presents lower barriers, and also in situations in which one wants to discriminate between the possible quantum-like behaviour of nuclei and simple classical phenomena.
Figure. Small ground-state effective potential for the ring opening of the ozone molecule (E1) and its modification at different non-zero temperatures according to the new scheme. One can see how, due to the influence of the first excited surface (E2), the barrier between the closed and open states significantly decreases at high temperature.