Monte Carlo Sampling from Different Ensembles

9 Monte Carlo Sampling from Different Ensembles 

A Monte Carlo simulation traditionally samples from the constant NVT (canonical) ensemble, but the teclmique can also be used to sample from different ensembles. A common alternative is the isothermal-isobaric, or constant NPT, ensemble. To simulate from this ensemble, it is necessary to have a scheme for changing the volume of the simulation cell in order to keep the pressure constant. This is done by combining random displacements of the particles with random changes in the volume of the simulation cell. The size of each volume change is governed by the maximum volume change, V ax-Thus a new volume is generated from the old volume as follows  

The advantage of using scaled coordinates is that they are independent of the size of the simulation box. Thus the energy of the configuration in a different-sized box (with side 

The criterion used to accept or reject a new configuration is slightly different for the iso-thermal-isobaric simulation than for a simulation in the canonical ensemble. The following quantity is used  

If AH is negative then the move is accepted otherwise, exp(—AH/lcgT) is compared to a random number between 0 and 1 and the move accepted according to  

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The molecular dynamics and Monte Carlo simulation methods differ in a variety of ways. The most obvious difference is that molecular dynamics provides information about the time dependence of the properties of the system whereas there is no temporal relationship between successive Monte Carlo configurations. In a Monte Carlo simulation the outcome of each trial move depends only upon its immediate predecessor, whereas in molecular dynamics it is possible to predict the configuration of the system at any time in the future - or indeed at any time in the past. Molecular dynamics has a kinetic energy contribution to the total energy whereas in a Monte Carlo simulation the total energy is determined directly from the potential energy function. The two simulation methods also sample from different ensembles. Molecular dynamics is traditionally performed under conditions of constant number of particles (N), volume (V) and energy (E) (the microcanonical or constant NVE ensemble) whereas a traditional Monte Carlo simulation samples from the canonical ensemble (constant N, V and temperature, T). Both the molecular dynamics and Monte Carlo techniques can be modified to sample from other ensembles for example, molecular dynamics can be adapted to simulate from the canonical ensemble. Two other ensembles are common ... 

The remaining aspect of a trajectory simulation is choosing the initial momenta and coordinates. These initial conditions are chosen so that the results from an ensemble of trajectories may be compared with experiment and/or theory, and used to make predictions about the chemical system s molecular dynamics. In this chapter Monte Carlo methods are described for sampling the appropriate distributions for initial values of the coordinates and momenta. Trajectories may be integrated in different coordinates and conjugate momenta, such as internal [7], Jacobi [8], and Cartesian. However, the Cartesian coordinate representation is most general for systems of any size and the Monte Carlo selection of Cartesian coordinates and momenta is described here for a variety of chemical processes. Many of... 

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