Microcanonical ensemble, Monte Carlo

By far the most common methods of studying aqueous interfaces by simulations are the Metropolis Monte Carlo (MC) technique and the classical molecular dynamics (MD) techniques. They will not be described here in detail, because several excellent textbooks and proceedings volumes (e.g., [2-8]) on the subject are available. In brief, the stochastic MC technique generates microscopic configurations of the system in the canonical (NYT) ensemble the deterministic MD method solves Newton s equations of motion and generates a time-correlated sequence of configurations in the microcanonical (NVE) ensemble. Structural and thermodynamic properties are accessible by both methods the MD method provides additional information about the microscopic dynamics of the system. 

There are many excellent reviews on the standard molecular dynamics method dealing with calculations in the microcanonical ensemble as well as on the Monte Carlo method involving calculations in the canonical, isothermal isobaric, and grand canonical ensemble (< ). In the present article, we shall limit ourselves exclusively to those developments that have taken place since the work of Andersen (4). In the molecular dynamics method, the developments are the constant-pressure, constant-temperature, constant-temperature-constant-pressure, variable shape simulation cell MD, and isostress calculations in the Monte Carlo method, it is the variable shape simulation cell calculation. 

Hase, W. L. Buckowski, D. G. Monte Carlo sampling of a microcanonical ensemble of classical harmonic oscillators, Chem. Phys. Lett. 1980, 74, 284-287. 

The Monte Carlo method is easily carried out in any convenient ensemble since it simply requires the construction of a suitable Markov chain for the importance sampling. The simulations in the original paper by Metropolis et al. [1] were carried out in the canonical ensemble corresponding to a fixed number of molecules, volume and temperature, N, V, T). By contrast, molecular dynamics is naturally carried out in the microcanonical ensemble, fixed (N, V, E), since the energy is conserved by Newton s equations of motion. This implies that the temperature of an MD simulation is not known a priori but is obtained as an output of the calculation. This feature makes it difficult to locate phase transitions and, perhaps, gave the first motivation to generalize MD to other ensembles. 

Fi is the force on particle i caused by the other particles, the dots indicate the second time derivative and m is the molecular mass. The forces on particle i in a conservative system can be written as the gradient of the potential energy, V, C/, with respect to the coordinates of particle /. In most simulation studies, U is written as a sum of pairwise additive interactions, occasionally also three-particle and four-particle interactions are employed. The integration of Eq. (1) has to be done numerically. The simulation proceeds by repeated numerical integration for tens or hundreds of thousands of small time steps. The sequence of these time steps is a set of configurations, all of which have equal probability. The completely deterministic MD simulation scheme is usually performed for a fixed number of particles, iV in a fixed volume V. As the total energy of a conservative system is a constant of motion, the set of configurations are representative points in the microcanonical ensemble. Many variants of these two basic schemes, particularly of the Monte Carlo approach exist (see, e.g.. Ref. 19-23). 

In the last section we have assumed that we perform our simulation for a fixed number, N, of particles at constant temperature, T, and volume, V, the canonical ensemble. A major advantage of the Monte Carlo technique is that it can be easily adapted to the calculation of averages in other thermodynamic ensembles. Most real experiments are performed in the isobaric-isothermal (constant- ) ensemble, some in the grand-canonical (constant-pFT) ensemble, and even fewer in the canonical ensemble, the standard Monte Carlo ensemble, and near to none in the microcanonical (constant-NFE) ensemble, the standard ensemble for molecular-dynamics simulations. 

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 ... 

Anharmonic corrections have also been determined for unimolecular rate constants using classical mechanics. In a classical trajectory (Bunker, 1962, 1964) or a classical Monte Carlo simulation (Nyman et al., 1990 Schranz et al., 1991) of the unimolecular decomposition of a microcanonical ensemble of states for an energized molecule, the initial decomposition rate constant is that of RRKM theory, regardless of the molecule s intramolecular dynamics (Bunker, 1962 Bunker, 1964). This is because a... 

Simulations of water in contact with metal surfaces have been performed in a number of ensembles, including the microcanonical (NVE), canonical (NVT), and grand canonical (p-VT) ensembles. The implementation of these ensembles differs for molecular dynamics and for Monte Carlo techniques. The NVT ensemble is convenient because the temperature of the system is maintained along with the number of particles and the volume. However, with the NVE and NVT ensembles care must be exercised to ensure that the density of the water in the system is consistent with the desired equilibrium state. For... 

To obtain Monte Carlo averages in the microcanonical ensemble, one can radially project the velocities Vp onto the hypersphere of constant energy. 

If bottlenecks restrict intramolecular vibrational energy redistribution," the unimolecular dissociation is not random and not in accord with equation (4). There is considerable interest in identifying which unimolecular reactions do not obey equation (4). In this section Monte Carlo sampling schemes are described for exciting A randomly with a micro-canonical ensemble of states and nonrandomly with mode selective excitation. For pedagogical purposes, selecting a microcanonical ensemble for a normal mode Hamiltonian is described first. 

---