The Fundamental Time Step

The time step should be at least an order of magnitude lower than the shortest period of internal motion. If hydrogens are present in a system near room temperature a value of 0.5 femtoseconds (0.0005 ps) is usually appropriate. 

---

To integrate the equations of motion in a stable and reliable way, it is necessary that the fundamental time step is shorter than the shortest relevant timescale in the problem. The shortest events involving whole atoms are C-H vibrations, and therefore a typical value of the time step is 2fs (10-15s). This means that there are up to one million time steps necessary to reach (real-time) simulation times in the nanosecond range. The ns range is sufficient for conformational transitions of the lipid molecules. It is also sufficient to allow some lateral diffusion of molecules in the box. As an iteration time step is rather expensive, even a supercomputer will need of the order of 106 s (a week) of CPU time to reach the ns domain. 

A dynamics simulation requires a set of initial coordinates and velocities, and an interaction potential (energy function). For a short time step, the interaction may be considered constant, allowing a set of updated positions and velocities to be estimated, at which point the new interaction can be calculated. By taking a (large) number of (small) time steps, the time behaviour of the system can be obtained. Since the phase space is huge, and the fundamental time step is short, the simulation will only explore... 

Electron-Nuclear Dynamics (END) method, " where both the orbitals describing the electronic wave function and the nuclear degrees of freedom are described by expansion into a Gaussian basis set, which moves along with the nuclei. Such an approach in principle allows a complete solution of the combined nuclear-electron Schrodinger equation without having to invoke approximations beyond those imposed by the basis set. Inclusion of the electronic parameters in the dynamics, however, means that the fundamental time step is short, and this results in a high computational cost for even quite short simulations and simple wave functions. 

HyperChem includes a number of time periods associated with a trajectory. These include the basic time step in the integration of Newton s equations plus various multiples of this associated with collecting data, the forming of statistical averages, etc. The fundamental time period is Atj s At, the integration time step stt in the Molecular Dynamics dialog box. 

Assume a principle timescale, T, has been identified for study dictated by the principle phenomenological effect of interest (e.g., a thermal response study). Once the value of this timescale is known, the integration time step, At, is then chosen so as to provide enough information (solution data) to allow an accurate picture of the transient, e.g., 1/100th of T. Once the value for At is known, then the type of formulation (quasi-steady formula versus the fundamental dynamic formula) of each individual physical phenomena can be selected (see Table 9.2). 

Since its inception, r-RESPA has proven to be indispensable for molecular dynamics simulations of large systems. Using r-RESPA to break down the force calculations into different time scales has given rise to the ability to take significantly larger fundamental time steps without suffering a decrease in the energy conservation of the simulation. 

The squared step length divided by the fundamental time scale r necessary for one step of movement.) The number 6 comes from the space dimensions d multiplied by 2 for the directions. The diffusion of a marked particle obtained in such a way is the self-diffusion constant or marker diffusion constant. 

All these considerations lead to the need of a new B-bar operator if the fundamental conservation laws of energy and momentum are to be preserved. The new operator needs to account not only for the discrete finite element interpolations in space, but also the discrete structure in time of the EDMC time-stepping algorithms, as presented in this paper. 

These pose a nontrivial problem and demand some fundamental thought. We have (Fig. 4.2) a cell containing oxidised and reduced species at concentrations c and Cg, both functions of x (and t). We have just completed a calculation of all c values, including for x=0 (c q and Cg q) and are now ready for the next time step. This introduces a new electrode potential parameter p (if the potential varies with time, this will be a new value). Reversibility implies that equilibrium appropriate to the new p is established immediately, so the Nernst equation applies we can rewrite it, using our dimensionless p ... 

In terms of writing a basic time step integration code segment, flic code could be implemented to return a solution set after each fundamental time step or it could be implemented to return with a solution set only after incrementing the time by an amount set by the time desired for saving a data set This later approach will be used here to implement a fundamental time step algoriflnn since the intermediate time step solutions are to be eventually discarded anyway. The fundamental time integration routine will then need time information on 1) the initial time, 2) the final time and 3) the number of fundamental time increments between the initial and final time. 

In addition to functions and time information, a code segment will need as input an array of spatial values on which the solution is desired (the spatial grid may be uniform or nonuniform), plus initial values of the solution variables at the initial time value and initial derivative values if the equation is second order in the time derivative. A possible calling sequence for such a fundamental time step integration routine is then pdebivbv(eqsub, tvals, x, u, ut, utt), where eqsub is a table containing the names of the functions (defining equations and boundary func-... 

The method presented in the next section is an attempt to overcome the barrier due to the highest frequencies whatever their origin. Although it has been implemented and tested for unconstrained dynamics only, there is no fundamental reason why it cannot be applied to overcome the less restrictive time step barrier arising in constrained dynamics. 

An examination of the autocorrelation function (0(0) <2(0) annucleophilic attack step in the catalytic reaction of subtilisin is presented in Fig. 9.4. As seen from the figure, the relaxation times for the enzymatic reaction and the corresponding reference reaction in solution are not different in a fundamental way and the preexponential factor t 1 is between 1012 and 1013 sec-1 in both cases. As long as this is the case, it is hard to see how enzymes can use dynamical effects as a major catalytic factor. 

Trace-gas Lifetimes. The time scales for tropospheric chemical reactivity depend upon the hydroxyl radical concentration [HO ] and upon the rate of the HO/trace gas reaction, which generally represents the slowest or rate-determining chemical step in the removal of an individual, insoluble, molecular species. These rates are determined by the rate constant, e,g. k2s for the fundamental reaction with HO, a quantity that in general must be determined experimentally. The average lifetime of a trace gas T removed solely by its reaction with HO,... 

---