Microcanonical ensemble, potential energy

Chemical dynamics simulations were performed to study the unimolecular decomposition of microcanonical ensembles versus energy of the Cl — CH Br ion lipole complex.An analytic potential energy surface was used for the simulations. The complex has two unimolecular reaction paths, i.e. dissociation to C/ + CHjBr or isomerization to the CICH —Br ion iipole complex. The simulations were performed for energies of 30-80 kcal mol and the resulting non-exponential N t)IN 0) were lit by a sum of three exponentials, i.e. eqn (20.18). The resulting// and ki fitting parameters are listed in Table 20.2. 

The bulk of the infomiation about anhannonicity has come from classical mechanical calculations. As described above, the aidiannonic RRKM rate constant for an analytic potential energy fiinction may be detemiined from either equation (A3.12.4) [13] or equation (A3.12.24) [46] by sampling a microcanonical ensemble. This rate constant and the one calculated from the hamionic frequencies for the analytic potential give the aidiannonic correctiony j ( , J) in equation (A3.12.41). The transition state s aidiannonic classical sum of states is found from the phase space integral... 

The basic assumption in statistical theories is that the initially prepared state, in an indirect (true or apparent) unimolecular reaction A (E) —> products, prior to reaction has relaxed (via IVR) such that any distribution of the energy E over the internal degrees of freedom occurs with the same probability. This is illustrated in Fig. 7.3.1, where we have shown a constant energy surface in the phase space of a molecule. Note that the assumption is equivalent to the basic equal a priori probabilities postulate of statistical mechanics, for a microcanonical ensemble where every state within a narrow energy range is populated with the same probability. This uniform population of states describes the system regardless of where it is on the potential energy surface associated with the reaction. 

The molecular dynamics simulation was performed using the MOTECC suite of programs [54] in the context of a microcanonical statistical ensemble. The system considered is a cube, with periodic boundary conditions, which contains 343 water molecules. The molecular dynamic simulation of water performed at ambient conditions revealed good agreement with experimental measurements. The main contribution to the total potential energy comes from the two-body term, while the many-body polarisation term contribution amounts to 23% of the total potential energy. Some of the properties calculated during the simulation are reported in Table 3. 

For lattice systems, or for systems interacting through discontinuous potential energy functions (e.g., hard-spheres), (19) cannot be used. In such cases, one can introduce a temperature by resorting to a microcanonical ensemble formulation [22,23] of the density-of-states algorithm [18]. 

These two methods will be described by considering a collection of atoms or molecules (particles) in the canonical or microcanonical ensemble, i.e., we will assume that the number of particles, the volume and either the temperature (T) or total potential energy ( ) are held constant in the simulation. (NVT, NVE). 

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). 

The classical anharmonic RRKM rate constant for a fluxional molecule may be calculated from classical trajectories by following the initial decay of a microcanonical ensemble of states for the unimolecular reactant, as given by equation 1A3.12.41. Such a calculation has been performed for dissociation of the Alg and A1j3 clusters using a model analytic potential energy function written as a sum of Lennard-Jones and Axelrod-Teller potentials [30]. Stmctures of some of the Alg minima, for the potential function, are shown in figure A3.12.6. The deepest potential minimum has... 

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 ... 

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