Integrators integration time step

In this article we describe an extension of SISM to a system of molecules for which it can be assumed that both bond stretching and angle bending describe satisfactorily all vibrational motions of the molecule. The SISM presented here allows the use of an integration time step up to an order of magnitude larger than possible with other methods of the same order and complexity. 

The analytical treatment of high frequency terms in the Hamiltonian proposed here allows to use SISM significantly longer integration time step than can be used by other methods of the same order and complexity. 

A second idea to save computational time addresses the fact that hydrogen atoms, when involved in a chemical bond, show the fastest motions in a molecule. If they have to be reproduced by the simulation, the necessary integration time step At has to be at least 1 fs or even less. This is a problem especially for calculations including explicit solvent molecules, because in the case of water they do not only increase the number of non-bonded interactions, they also increase the number of fast-moving hydrogen atoms. This particular situation is taken into account... 

T(f) corresponds to the actual temperature at the time t, At is the integration time step, and the relaxation time represents the strength of the coupling (smaller values mean stronger coupling to the bafli). If the coupling is too strong (r smaller... 

In the region where > 1 then if the interparticle force is assumed to be constant over thi integration time step the following result is obtained [van Gunsteren et al. 1981] ... 

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. 

In this expression. Ait is the size of the integration time step, Xj is a characteristic relaxation time, and T is the instantaneous temperature. In the simulation of water, they found a relaxation time of Xj = 0.4 ps to be appropriate. However, this method does not correspond exactly to the canonical ensemble. 

According to the namre of the empirical potential energy function, described in Chapter 2, different motions can take place on different time scales, e.g., bond stretching and bond angle bending vs. dihedral angle librations and non-bond interactions. Multiple time step (MTS) methods [38-40,42] allow one to use different integration time steps in the same simulation so as to treat the time development of the slow and fast movements most effectively. 

With the propagator written in this way, the equation of motion can be integrated by a multiple time step algorithm in Cartesian coordinates because At and At are different integration time steps (At > At when n> 1). As an example, the force terms are separated into two components... 

This technique is quite simple and easy to visualize and program. It is not very rapid in converging to the correct solution, but it is rock-bottom stable (it won t blow up on you numerically). It works well in dynamic simulations because the step size can be adjusted to correspond approximately to the rate at which the variable is changing with time during the integration time step. 

In a typical (classical) molecular system, the fastest motion is bond vibration which, for a heavy-atom-hydrogen bond has a period of about 10 " s. Thus, for a system containing such bonds, an integration time step At should not much exceed 0.1 fs. This rather short time... 

Note that in die leapfrog method, position depends on the velocities as computed one-half time step out of phase, dins, scaling of the velocities can be accomplished to control temperature. Note also that no force-deld calculations actually take place for the fractional time steps. Eorces (and thus accelerations) in Eq. (3.24) are computed at integral time steps, halftime-step-forward velocities are computed therefrom, and these are then used in Eq. (3.23) to update the particle positions. The drawbacks of the leapfrog algorithm include ignoring third-order terms in the Taylor expansions and the half-time-step displacements of the position and velocity vectors - both of these features can contribute to decreased stability in numerical integration of the trajectoiy. 

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). 

In the numerical integration of a particle trajectory, the selection of the integral time step is important. A typical way of choosing the integral time step is based on the interacting duration between the turbulent eddy and the particle. This interacting duration Tj may be determined by... 

Thus, integration of the equations of motion introduces another basic parameter common for any MD simulation, i.e. the integration time-step, At. 

---