Initial value

C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffe, N.J, (1971). 

Walton A R and Manolopoulos D E 1996 A new semiclassical initial value method for Franck-Condon spectra Mol. Phys. 87 961 

A second recent development has been the application 46 of the initial value representation 47 to semiclassically calculate A3.8.13 (and/or the equivalent time integral of the flux-flux correlation fiinction). While this approach has to date only been applied to problems with simplified hannonic baths, it shows considerable promise for applications to realistic systems, particularly those in which the real solvent bath may be adequately treated by a fiirther classical or quasiclassical approximation. 

Kay K G 1994 Semiclassical propagation for multidimensional systems by an initial value method J. Chem. Phys. 101 2250 

Campolieti G and Brumer P 1994 Semiclassical propagation phase indices and the initial-value formalism Phys. Rev. A 50 997 

The data from Table 7.4 are presented graphically in Fig. 7.11. The optimal is at 10°C, confirming the initial value used for this problem in Chap. 6. 

The index J can label quantum states of the same or different chemical species. Equation (A3.13.20) corresponds to a generally stiff initial value problem [42, 43]. In matrix notation one may write  

In the pure BO model, this discontinuity will be ignored. Let the initial values be given by 

Sun X, Wang H B and Miller W H 1998 Semiclassical theory of electronically nonadiabatic dynamics Results of a linearized approximation to the initial value representation J. Chem. Phys. 109 7064 

Extending time scales of Molecular Dynamics simulations is therefore one of the prime challenges of computational biophysics and attracted considerable attention [2-5]. Most efforts focus on improving algorithms for solving the initial value differential equations, which are in many cases, the Newton s equations of motion. 

We further comment that reactive trajectories that successfully pass over large barriers are straightforward to compute with the present approach, which is based on boundary conditions. The task is considerably more difficult with initial value formulation. 

In Fig. 8.3, the only cost forcing the optimal conversion hack from high values is that of the reactor. Hence, for such simple reaction systems, a high optimal conversion would he expected. This was the reason in Chap. 2 that an initial value of reactor conversion of 0.95 was chosen for simple reaction systems. 

Off-line analysis of stored data review of the stored data, organize data in different presentation windows, plot AE and plant parameters data so as to enable comparison and coirelation with the possibility to present data (histogram of AE events vs position, plant parameters and/or AE parameters vs time) conditioned in terms of time interval (initial time, final time) and/or position interval (defined portion of the component = initial coordinate, final coordinate) and/or plant parameters intervals (one or more plant parameters = initial value, final value). 

Notice that the solution is not identical to J but an approximation of it. The evolution of a and S in time may conveniently be described via the following classical Newtonian equations of motion Given the initial values 

Consider, at t = 0, some non-equilibrium ensemble density P g(P. q°) on the constant energy hypersurface S, such that it is nonnalized to one. By Liouville s theorem, at a later time t the ensemble density becomes ((t) t(p. q)), where q) is die function that takes die current phase coordinates (p, q) to their initial values time (0 ago the fimctioii ( ) is uniquely detemiined by the equations of motion. The expectation value of any dynamical variable ilat time t is therefore 

Caustics The above formulae can only be valid as long as Eq. (9) describes a unique map in position space. Indeed, the underlying Hamilton-Jacobi theory is only valid for the time interval [0,T] if at all instances t [0, T] the map (QOi4o) —> Q t, qo,qo) is one-to-one, [6, 19, 1], i.e., as long as trajectories with different initial data do not cross each other in position space (cf. Fig. 1). Consequently, the detection of any caustics in a numerical simulation is only possible if we propagate a trajectory bundle with different initial values. Thus, in pure QCMD, Eq. (11), caustics cannot be detected. 

Another difference is related to the mathematical formulation. Equation (1) is deterministic and does not include explicit stochasticity. In contrast, the equations of motion for a Brownian particle include noise. Nevertheless, similar algorithms are adopted to solve the two differential equations as outlined below. The most common approach is to numerically integrate the above differential equations using small time steps and preset initial values. 

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