Theory of Nuclear Quantum Dynamics Simulations

In Chap. 2, we have seen that the theoretical study of a molecular system is, in a vast majority of cases, separated in two steps. In a first step, the electronic structure of the system is studied by solving the electronic Schrodinger equation with fixed nuclei. This approach, combined with geometry optimization techniques, allows one to locate the important features of the various potential energy surfaces (PESs) of the electronic states of interest. In the context of photochemistry, as seen in Chap. 3, this approach allows one to characterize the various decay pathways of the molecule after photoexcitation. This information can then be used to interpret the various decay time constants obtained from time-resolved spectroscopic measurements. However, in most cases, including aniline studied in Chap. 3, various decay mechanisms are in competition, and it is often difficult to infer, from this static information, the relative importance of the various decay mechanisms. It is thus often necessary to study the dynamics of the nuclei in the manifold of the excited states of interest to obtain a deeper insight into the photophysics and photochemistry of the system of interest. 

We have seen in Chap. 2 that, in a vast majority of case, the calculation of the electronic structure of molecular systems is performed in a time-independent formalism, through the solution of the time-independent electronic Schrodinger equation. In contrast, in nuclear quantum dynamics, a majority of problems can be treated either in a time-independent or in a time-dependent formalism [2, 3]. In this thesis, the time-dependent formalism was used. Therefore, in this chapter, the theoretical tools for the smdy of the quantum dynamics of the nuclei in molecular systems will be introduced from a time-dependent perspective. 

Quantum Dynamics and Laser Comrol for Photochemistry, 

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For larger systems, various approximate schemes have been developed, called mixed methods as they treat parts of the system using different levels of theory. Of interest to us here are quantuin-seiniclassical methods, which use full quantum mechanics to treat the electrons, but use approximations based on trajectories in a classical phase space to describe the nuclear motion. The prefix quantum may be dropped, and we will talk of seiniclassical methods. There are a number of different approaches, but here we shall concentrate on the few that are suitable for direct dynamics molecular simulations. An overview of other methods is given in the introduction of [21]. 

Each of the semi-classical trajectory surface hopping and quantum wave packet dynamics simulations has its pros and cons. For the semi-classical trajectory surface hopping, the lack of coherence and phase of the nuclei, and total time per trajectory are cons whereas inclusion of all nuclear degrees of freedom, the use of potentials direct from electronic structure theory, the ease of increasing accuracy by running more trajectories, and the ease of visualization of results are pros. For the quantum wave packet dynamics, the complexity of setting up an appropriate model Hamiltonian, use of approximate fitted potentials, and the... 

Hwang et al.131 were the first to calculate the contribution of tunneling and other nuclear quantum effects to enzyme catalysis. Since then, and in particular in the past few years, there has been a significant increase in simulations of QM-nuclear effects in enzyme reactions. The approaches used range from the quantized classical path (QCP) (e.g., Refs. 4,57,136), the centroid path integral approach,137,138 and vibrational TS theory,139 to the molecular dynamics with quantum transition (MDQT) surface hopping method.140 Most studies did not yet examine the reference water reaction, and thus could only evaluate the QM contribution to the enzyme rate constant, rather than the corresponding catalytic effect. However, studies that explored the actual catalytic contributions (e.g., Refs. 4,57,136) concluded that the QM contributions are similar for the reaction in the enzyme and in solution, and thus, do not contribute to catalysis. 

Modem first principles computational methodologies, such as those based on Density Functional Theory (DFT) and its Time Dependent extension (TDDFT), provide the theoretical/computational framework to describe most of the desired properties of the individual dye/semiconductor/electrolyte systems and of their relevant interfaces. The information extracted from these calculations constitutes the basis for the explicit simulation of photo-induced electron transfer by means of quantum or non-adiabatic dynamics. The dynamics introduces a further degree of complexity in the simulation, due to the simultaneous description of the coupled nuclear/electronic problem. Various combinations of electronic stmcture/ excited states and nuclear dynamics descriptions have been applied to dye-sensitized interfaces [54—57]. In most cases these approaches rely either on semi-empirical Hamiltonians [58, 59] or on the time-dependent propagation of single particle DFT orbitals [60, 61], with the nuclear dynamics being described within mixed quantum-classical [54, 55, 59, 60] or fuUy quantum mechanical approaches [61]. Real time propagation of the TDDFT excited states [62] has... 

The conceptual framework for the - semiclassical simulation of ultrafast spectroscopic observables is provided by the Wigner representation of quantum mechanics [2, 3]. Specifically, for the ultrafast pump-probe spectroscopy using classical trajectories, methods based on the semiclassical limit of the Liouville-von Neumann equation for the time evolution of the vibronic density matrix have been developed [4-8]. Our approach [4,6-8] is related to the Liouville space theory of nonlinear spectroscopy developed by Mukamel et al. [9]. It is characterized by the ability to approximately describe quantum phenomena such as optical transitions by averaging over the ensemble of classical trajectories. Moreover, quantum corrections for the nuclear dynamics can be introduced in a systematic manner, e.g. in the framework of the entangled trajectory method [10,11]. Alternatively, these effects can be also accounted for in the framework of the multiple spawning method [12]. In general, trajectory-based methods require drastically less computational effort than full quantum mechanical calculations and provide physical insight in ultrafast processes. Additionally, they can be combined directly with quantum chemistry methods for the electronic structure calculations. 

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