Path-integral techniques dynamics

We have considered the stochastic dynamics of a particle interacting with its environment of two-level systems in the presence of an external potential field. The treatment is based on the canonical quantization procedure. This approach directly yields the dissipative term and the noise operators. It may be pertinent to mention that, although the calculation of dissipative effects is straightforward, the treatment of noise is not simple as far as the path integral techniques are concerned. 

The dynamics of PAH and PAH related systems have been studied in the past mostly in the ground state, usually to compute vibrational spectra, and without considering the effects of nuclei quantum delocalisation. We have however seen in this review that a variety of methods can now to be used to include these effects. In particular, the path integral molecular dynamics approach, which has received a growing interest in the recent years and whose first applications to PAH like systems have recently appeared in the literature, could become a standard method in the next years. Explicit dynamics in the excited states would also be interesting to go beyond the search for conical interstection techniques to analyse relaxation of an excited system. 

So far, one can be much more successful in calculating a rate constant when one knows in advance that it exists, than in answering the question of whether it exists. A considerable breakthrough in this area was the solution of the spin-boson problem, which, however, has only limited relevance to any problem in chemistry because it neglects the effects of intrawell dynamics (vibrational relaxation) and does not describe thermally activated transitions. A number of attempts have been made to go beyond the two-level system approximation, but the basic question of how vibrational relaxation affects the transition from coherent oscillations to exponential decay awaits a quantitative solution. Such a solution might be obtained by numerical computation of real-time path integrals for the density matrix using the influence functional technique. 

Memory effects play an important role for the description of dynamical effects in open quantum systems. As mentioned above, Meier and Tannor [32] developed a time-nonlocal scheme employing the numerical decomposition of the spectral density. The TL approach as discussed above as well as the approaches by Yan and coworkers [33-35] use similar techniques. Few systems exist for which exact solutions are available and can serve as test beds for the various theories. Among them is the damped harmonic oscillator for which a path-integral solution exists [1], In the simple model of an initially excited... 

Additionally, non-Hamiltonian d5mamics can be used in applications/ methodologies such as Path-integral MD, replica-exchange methods, variable transformation techniques, free energy dynamics methods, and other new applications. Generating these alternative statistical ensembles from simulation requires the use of extended systems or extended phase space [9]. In these systems, the simulations do not only include the N coordinate and momentum vectors that are needed to describe a classical Ai-particle system, but they also include a set of additional control or extended variables that are used to drive the fluctuations required by the ensemble of interest. 

Variations on this surface hopping method that utilize Pechukas [106] formulation of mixed quantum-classical dynamics have been proposed [107,108]. Surface hopping algorithms [109-111] for non-adiabatic dynamics based on the quantum-classical Liouville equation [109,111-113] have been formulated. In these schemes the dynamics is fully prescribed by the quantum-classical Liouville operator and no additional assumptions about the nature of the classical evolution or the quantum transition probabilities are made. Quantum dynamics of condensed phase systems has also been carried out using techniques that are not based on surface hopping algorithms, in particular, centroid path integral dynamics [114] and influence functional methods [115]. 

Makri N (2008) Equilibrium and dynamical path integral methods in bacterial photosynthesis. In Aartsma TJ, Matysik J (eds) Biophysical techniques in photosynthesis. Series advances in photosynthesis and respiration, vol 2. Springer, Dordrecht... 

The information available from the femtosecond optical techniques will be illustrated by recent results on electron trapping and solvation in aqueous media at room temperature. In pure liquid water, the existence of a precursor of hydrated state have been obtained by femtosecond absorption spectroscopy. These studies provide unique experimental basis for testing recent theoretical results obtained by newly developed techniques (Monte Carlo, molecular dynamics simulations, path integral) and to permit the obtention of a consistent picture of electron trapping and solvation in aqueous solutions. 

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