Time-step

When the friction coefficient is set to zero, HyperChem performs regular molecular dynamics, and one should use a time step that is appropriate for a molecular dynamics run. With larger values of the friction coefficient, larger time steps can be used. This is because the solution to the Langevin equation in effect separates the motions of the atoms into two time scales the short-time (fast) motions, like bond stretches, which are approximated, and longtime (slow) motions, such as torsional motions, which are accurately evaluated. As one increases the friction coefficient, the short-time motions become more approximate, and thus it is less important to have a small timestep. 

Temperature is handled the same way in Langevin dynamics as it its in molecular dynamics. High temperature runs may be used to overcome potential energy barriers. Cooling a system to a low temperature in steps may result in a different stable conformation than would be found by direct geometry optimization. 

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It is possible to use the quantum states to predict the electronic properties of the melt. A typical procedure is to implement molecular dynamics simulations for the liquid, which pemiit the wavefiinctions to be detemiined at each time step of the simulation. As an example, one can use the eigenpairs for a given atomic configuration to calculate the optical conductivity. The real part of tire conductivity can be expressed as... 

The complete propagator is then constructed by piecing together A time steps, leading to... 

The later time evolution is shown in Figrne A3.13.7 between 90 and 100 fs, and m Figrne A3.13.8, between 390 and 400 fs, after the beginning of the excitation (time step t j)- Tln-ee observations are readily made first,... 

This advances the coordinates and momenta over a small time step 8 t. A piece of pseudo-code illustrates how this works ... 

Important features of the Verlet algoritlnn are (a) it is exactly time reversible (b) it is loMf order in time, hence pennitting long time steps (c) it is easy to program. 

A straightforward derivation (not reproduced here) shows that the effect of the diree successive steps embodied in equation (b3.3.7), with the above choice of operators, is precisely the velocity Verlet algorithm. This approach is particularly usefiil for generating multiple time-step methods. 

So the fast-varying forces must be computed many times at short intervals the slow-varying forces are used just before and just after this stage, and they only need be calculated once per long time step. 

The entire simulation mn consists of NSTEP long steps each step consists of nstep shorter sub-steps. DT and dt are the corresponding time steps, DT = nstep dt. 

Some people prefer to use the multiple time step approach to handle fast degrees of freedom, while others prefer to use constraints, and there are situations in which both techniques are applicable. Constraints also find an application in the study of rare events, where a system may be studied at the top of a free energy barrier (see later), or for convenience when it is desired to fix a thennodynamic order parameter or ordering direction... 

MD, one needs to use multiple time step methods to ensure proper handling of the sprmg vibrations, and there is a possible physical bottleneck in the transfer of energy between the spring system and the other degrees of freedom which must be handled properly [199]. In MC, one needs to use special methods to sample configuration space efficiently [200, 201]. 

Figure B3.4.15. A possible Feymnaim path trajectory for a ID variable as a function of time. This trajectory carries an oscillating component with it, where. S is the action of the trajectory. The trajectory is highly fluctuating its values at each time step (v(dt), etc) are not correlated.

This method has been devised as an effective numerical teclmique of computational fluid dynamics. The basic variables are the time-dependent probability distributions f x, f) of a velocity class a on a lattice site x. This probability distribution is then updated in discrete time steps using a detenninistic local rule. A carefiil choice of the lattice and the set of velocity vectors minimizes the effects of lattice anisotropy. This scheme has recently been applied to study the fomiation of lamellar phases in amphiphilic systems [92, 93]. 

Each of these operators is unitary U —t) = U t). Updating a time step with the propagator Uf( At)U At)Uf At) yields the velocity-Verlet algorithm. Concatenating the force operator for successive steps yields the leapfrog algorithm ... 

Fig. 6. Snapshot from a dynamic density functional simulation of the self-organisation of the block copolymer PL64 (containing 30 propylene oxide rmd 26 ethylene oxide units (EO)i3(PO)3o(EO)i3) in 70% aqueous solution. The simulation was carried out during 6250 time steps on a 64 x 64 x 64 grid (courtesy of B.A.C. van Vlimmeren and J.G.E.M. Praaije, Groningen).

Olender and Elber, 1996] Olender, R., and Elber, R. Calculation of classical trajectories with a very large time step Formalism and numerical examples. J. Chem. Phys. 105 (1996) 9299-9315... 

X is a matrix whose elements Xu give the mass-weighted internal displacements of each atomic coordinate i from its average position at a given time step t. N is the total number of integration steps. 

An interesting approach has recently been chosen in the MBO(N)D program ([Moldyn 1997]). Structural elements of different size varying from individual peptide planes up to protein domains can be defined to be rigid. During an atomistic molecular dynamics simulation, all fast motion orthogonal to the lowest normal modes is removed. This allows use of ca. 20 times longer time steps than in standard simulations. 

Approximate Simulation Between Two Structures with Large Time Steps... 

Related to the previous method, a simulation scheme was recently derived from the Onsager-Machlup action that combines atomistic simulations with a reaction path approach ([Oleander and Elber 1996]). Here, time steps up to 100 times larger than in standard molecular dynamics simulations were used to produce approximate trajectories by the following equations of motion ... 

Abstract. Molecular dynamics (MD) simulations of proteins provide descriptions of atomic motions, which allow to relate observable properties of proteins to microscopic processes. Unfortunately, such MD simulations require an enormous amount of computer time and, therefore, are limited to time scales of nanoseconds. We describe first a fast multiple time step structure adapted multipole method (FA-MUSAMM) to speed up the evaluation of the computationally most demanding Coulomb interactions in solvated protein models, secondly an application of this method aiming at a microscopic understanding of single molecule atomic force microscopy experiments, and, thirdly, a new method to predict slow conformational motions at microsecond time scales. 

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