Time-dependent mixed quantum classical approaches

Koppel [180] has performed exact time-dependent quantum wave-packet propagations for this model, the results of which are depicted in Fig. 2A. He showed that the initially excited C state decays irreversibly into the X state within 250 fs. The decay is nonexponential and exhibits a pronounced beating of the C and B state populations. This model will allow us to test mixed quantum-classical approaches for multistate systems with several conical intersections. 

A mixed quantum classical description of EET does not represent a unique approach. On the one hand side, as already indicated, one may solve the time-dependent Schrodinger equation responsible for the electronic states of the system and couple it to the classical nuclear dynamics. Alternatively, one may also start from the full quantum theory and derive rate equations where, in a second step, the transfer rates are transformed in a mixed description (this is the standard procedure when considering linear or nonlinear optical response functions). Such alternative ways have been already studied in discussing the linear absorbance of a CC in [9] and the computation of the Forster-rate in [10]. 

The standard translation of the full quantum formula of the absorbance, Eq. (31), to a mixed quantum classical description (see, e.g., [16-18]) is similar to what is the essence of the (electronic ground-state) classical path approximation introduced in the foregoing section. One assumes that all involved nuclear coordinates behave classically and their time-dependence is obtained by carrying out MD simulations in the systems electronic ground state. This approach when applied to the absorbance is known as the dynamical classical limit (DCL, see, for example, Res. [17]). 

Recently [8-11] an alternative treatment to mix quantum mechanics with classical mechanics, based on Bohmian quantum trajectories was proposed. Briefly, the quantum subsystem is described by a time-dependent Schrodinger equation that depends parametrically on classical variables. This is similar to other approaches discussed above. The difference comes from the way the classical trajectories are calculated. In our approach, which was called mixed quantum-classical Bohmian (MQCB) trajectories, the wave packet is used to define de Broglie-Bohm quantum trajectories [12] which in turn are used to calculate the force acting on the classical variables. 

Presumably the most straightforward approach to chemical dynamics in intense laser fields is to use the time-independent or time-dependent adiabatic states [352], which are the eigenstates of field-free or field-dependent Hamiltonian at given time points respectively, and solve the Schrodinger equation in a stepwise manner. However, when the laser field is approximately periodic, one can also use a set of field-dressed periodic states as an expansion basis. The set of quasi-static states in a periodic Hamiltonian is derived by a Floquet type analysis and is often referred to as the Floquet states [370]. Provided that the laser field is approximately periodic, advantages of using the latter basis set include (1) analysis and interpretation of the electron dynamics is clearer since the Floquet state population often vary slowly with the timescale of the pulse envelope and each Floquet state is characterized as a field-dressed quasi-stationary state, (2) under some moderate conditions, the nuclear dynamics can be approximated by mixed quantum-classical (MQC) nonadiabatic dynamics on the field-dressed PES. The latter point not only provides a powerful clue for interpretation of nuclear dynamics but also implies possible MQC formulation of intense field molecular dynamics. 

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... 

In the adiabatic bend approximation (ABA) for the same reaction,18 the three radial coordinates are explicitly treated while an adiabatic approximation was used for the three angles. These reduced dimensional studies are dynamically approximate in nature, but nevertheless can provide important information characterizing polyatomic reactions, and they have been reviewed extensively by Clary,19 and Bowman and Schatz.20 However, quantitative determination of reaction probabilities, cross-sections and thermal reaction rates, and their relation to the internal states of the reactants would require explicit treatment of five or the full six degrees-of-freedom in these four-atom reactions, which TI methods could not handle. Other approximate quantum approaches such as the negative imaginary potential method16,21 and mixed classical and quantum time-dependent method have also been used.22... 

A description, which is somewhat intermediate between a full quantum model and a classical propagation scheme is the Gaussian wave packet approach for the classical-like mode. This picture was first introduced by Heller [44] and adapted by Billing [18,45,46] to the mixed classical quantum system. Therein one mode is described as a time dependent Gaussian wave packet [18,15]... 

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