Dynamical algorithm, transition path

Recently, the string algorithm was used for finding the transition path in Alanine dipeptide. If one can apply it successfully to a large protein, then this will be the final missing link in the scheme outlined above for studying dynamics in conformation space. 

For these reasons we cannot use (7(R) as a rigid support for dynamical studies of trajectories of representative points. G(R) has to be modified, at every point of each trajectory, and these modifications depend on the nature of the system, on the microscopic properties of the solution, and on the dynamical parameters of the trajectories themselves. This rather formidable task may be simplified in severai ways we consider it convenient to treat this problem in a separate Section. It is sufficient to add here that one possible way is the introduction into G (R) of some extra coordinates, which reflect the effects of these retarding forces. These coordinates, collectively called solvent coordinates (nothing to do with the coordinates of the extra solvent molecules added to the solute ) are here indicated by S, and define a hypersurface of greater dimensionality, G(R S). To show how this approach of expanding the coordinate space may be successfully exploited, we refer here to the proposals made by Truhlar et al. (1993). Their formulation, that just lets these solvent coordinates partecipate in the reaction path, allows to extend the algorithms and concepts of the above mentioned variational transition state theory to molecules in solution. 

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]. 

Amical uses a dynamic loop scheduling algorithm [13, 12]. This algorithm is adapted to control-flow-dominated designs written in Vhdl, and is a development of the path-based approach proposed by Camposano [2]. Essentially, the scheduler reads in a Vhdl description and produces a behavioral FSM in the form of a transition table. Each transition (macro-cycle) corresponds to the execution of a control step under a given condition. A macro-cycle may need several basic cycles (clock cycles) for its execution. The top left window in figure 4 shows the transition table composed of two states and five transitions. 

The basic idea of the algorithm is to allow a path to jump with the prob-abihty estimated internally (in a self-contained maimer) with the electronic-state mixing as in the semiclassical Ehrenfest theory but the number of transitions (hops) should be minimized. Suppose we have an electronic wavepacket (t)) = X]/C /(t) >/) as in the SET, the dynamics of the corresponding density matrix, pij t) = Cj t)Cj t), is written as... 

Algorithmically, action-based methods are similar to the NEB method since in both cases a path functional is minimized. They differ, however, in the nature of the particular functional. While in the NEB method a path functional is constructed in an ad hoc way such that the path ttaverses the transition state separating reactants fi om products, the functional minimized in action-based methods corresponds, in principle, to the fully dynamical trajectories of classical mechanics. This property, however, is lost if extremely large time steps are used. In this case, the method yields a possible sequence of events that may be encountered by the system as it evolves fi om its initial to its final state, but a dynamical interpretation of such a sequence of states is not strictly permissible any more. Nevertheless, large time step trajectories that minimize the Gauss (Onsager-Machlup) action can provide possible scenarios for transitions that are computationally untreatable otherwise. 

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