Ensemble energy shell

An alternative method, proposed by Andersen [23], shows that the coupling to the heat bath is represented by stochastic impulsive forces that act occasionally on randomly selected particles. Between stochastic collisions, the system evolves at constant energy according to the normal Newtonian laws of motion. The stochastic collisions ensure that all accessible constant-energy shells are visited according to their Boltzmann weight and therefore yield a canonical ensemble. 

In the case of an equilibrium system the Hamiltonian is the same as that of an ensemble of conservative systems in statistical equilibrium. If the energy of the system is measured to lie between Ek and EK + AE, then the representative ensemble is also restricted to the energy shell [AE K. From the hypotheses of equal a priori probabilities and random a priori phases it then follows that the diagonal elements of the density matrix lying inside [AE]k are all equal and that all others vanish. The density matrix of the quantum statistical microcanonical ensemble is thereby determined as... 

In a similar study by Cleuren et al. samples from an initial microcanonical ensemble are considered (on an energy shell with energy E), but the dynamics in not isoenergetic, and therefore each trajectory moves off the energy shell. The Crooks Fluctuation relation is then given by ... 

In order to obtain a statistical measure for the ionization process we have calculated for an ensemble of trajectories the fraction of ionized orbits as a function of time. The initial internal energy was chosen to correspond to a completely chaotic phase space of the He -ion if the nuclear mass were infinite. The initial conditions for the internal motion have been selected randomly on the energy shell. In Fig. 11 we have illustrated the fraction of ionized orbits as a function of time up to T = 10 a.u. for a series of different CM energies and for a very strong laboratory field strength of B = 10 a.u.. For an initial CM energy of Ec = 0.053 a.u. which corresponds to an initial CM... 

Thus, preparing a microcanonical ensemble involves choosing points at random on the energy shell. 

According to Eq. (33), all states inside the energy shell are equally probable. In other words, the microcanonical ensemble represents a uniform distribution over the phase points belonging to the energy shell. Such a distribution is often referred to as a mathematical statement of the principle of equal a priori probabilities. According to this principle, the system spends an equal amount of time, over a long time period, in each of the available classical states. This statement is called the ergodic hypothesis. 

Otherwise, P ifm) vanishes. From these results, we see that the quantum microcanonical ensemble describes a uniform distribution over the eigenstates of the system HamiltonianH lying inside the energy shell [E,E+AE], Outside the energy shell, the distribution vanishes. 

In view of the formal identity of the expressions for the Gibbs entropy of quantum and classical microcanonical ensembles, the statistical mechanical expressions given by Eqs. (44)-(46) also apply to quantum systems. One need only reinterpret the quantity 2( , F, A AE) appearing in these expressions as the number of eigenstates of the system Hamiltonian H lying in the energy shell [E,E + AE]. 

In orthant sampling an initial condition for a microcanical ensemble is chosen by projecting a random unit vector of dimension 2n, with components Xi, onto the E = ff(p,q) energy shell ... 

So, for each snapshot of the simulation that contributes to the ensemble (by either MC or MD evaluation), we compute the energy differential for all of the atoms interacting with Hb rather than Ha. In Figure 12.1, the particular case of one of the hydrogen atoms on a first-shell water molecule is illustrated. As this is a non-bonded interaction in each case, the contribution from Hd in a simple force field might be... 

Encounter complex An intermolecular ensemble formed by molecular entities in contact or separated by a distance small compared to the diameter of solvent molecules and surrounded by several shells of solvent molecules the innermost shell is the solvent cage . If one of the species is excited, the excitation usually takes place prior to formation of the encounter complex. During the lifetime of the encounter complex the reactants can collide several times to form colHsion complexes, and then undergo structural and electronic changes. If the interaction between the reactants leads to a minimum in the potential energy and one of the entities is electronically excited, the encounter complex may represent an exciplex or excimer. 

Classical statistical mechanics is concerned with the probability distribution of phase points. In a classical microcanonical ensemble the phase space density is constant. Loosely speaking, aU phase points with the same energy are equally likely. In consequence the number of states of the classical system in a given energy range E to E + dE is proportional to the volume of the phase space shell defined by this energy range. 

In the work of Edwards et al [36], which included interactions only to the third neighbor shell (approximately 4.4 A), the problem of energy instability for a 1000 K simulation did not arise the timestep was lO s at all temperatures. Since no analysis of the effects of timestep and cutoff was presented, it is difficult to determine the reasons for the differences in the behavior observed by Edwards et al and in the present study. Some possibilities are (a) They may have employed a minimum image approach for the periodic boundary conditions, whereas we have calculated both primaryprimary and primary-image interactions for each atom, (b) Their system was much larger than ours (2048 vs 256 atoms), (c) Edwards et al used the NVE ensemble for the production simulations, in contrast to our choice of the NPT ensemble. 

The open-shell configuration of the iron cluster surface has been considered in the definition of the spin magnetic moment of the I ev - CyH C H-adsorbed ensembles [20]. The neutral benzoate binding energies (BE) were calculated according to the following equation ... 

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