Restricted ensemble

Monte Carlo C) simulations [91] are employed to equilibrate the system at very low temperature and high negative pressme (while deducing the temperature of minimum density) and restricted ensemble MC simulations for locating the spinodal at low temperatures. 

To obtain the complete phase behavior of supercooled silicon Vasisht et al. analyzed the interplay of various loci of extremal behavior, namely, the spinodal, temperature of density extrema, and temperature of compressibility extrema. Loci of temperature of maximum and minimum density (TMD and TMinD), temperature of maximum and minimum compressibility TMC and TMinC), and spinodal were evaluated by employing, in addition to the MD simulations, parallel tempering (PT) Monte Carlo simulations (at low temperature and pressures) and restricted ensemble Monte Carlo (REMC) simulations [95] (for locating the... 

Restricted ensemble Kohn-Sham DFT Alternatively one can perform standard KS-DFT calculations on a collection of determinants with different occupations and take a weighted average of the individual energies to obtain an estimate of the mul-tideterminantal situation. To avoid the independent calculation of several KS determinants, a generalization of this approach was proposed by Filatov and Shaik based on the coupling operator technique developed by Roothaan for restricted open-shell Hartree-Fock. This restricted open-shell Kohn-Sham (ROKS) approach was later extended to situations where fractional occupation numbers are not imposed by the... 

We now describe the extension of the restricted ensemble formalism developed by Penrose [1] to suit our goal. While Word(T) certainly exists for E > Eq, there is no guarantee that Wdis( ) also exists for > o- This is abundantly clear from the entropy for linear polymers in Figure 10.3. Most probably, there is an energy gap for Wdis( ). Otherwise, the energy of the disordered phase at absolute zero would also be o (we assume that TSdis Oas 0 ), the same as that of CR. This would most... 

M. Filatov and S. Shaik, Chem. Phys. Lett., 304, 429-437 (1999). A Spin-Restricted Ensemble-References Kohn-Sham Method and Its Application to Diradicaloid Situations. 

This selection process is then iterated, beginning from an initial state of the system, as defined by species populations, to simulate a chemical evolution. A statistical ensemble is generated by repeated simulation of the chemical evolution using different sequences of random numbers in the Monte Carlo selection process. Within limits imposed by computer time restrictions, ensemble population averages and relevant statistical information can be evaluated to any desired degree of accuracy. In particular, reliable values for the first several moments of the distribution can be obtained both inexpensively and efficiently via a computer algorithm which is incredibly easy to implement (21, 22), especially in comparison to now-standard techniques foF soTving the stiff ordinary differential equations (48, 49) which may arise in the deterministic description of chemical kinetics (53). Now consider briefly the essential features of a simple chemical model which illustrates well the attributes of stochastic chemical simulations. 

Removing the restriction on fixed [i], by considering the grand ensemble which sums over [ ], one has... 

Orkoulas G and Panagiotopoulos A Z 1999 Phase behavior of the restricted primitive model and square-well fluids from Monte Carlo simulations in the grand canonical ensemble J. Chem. Phys. 110 1581... 

The most serious problem with ensemble average approaches is that they introduce many more parameters into the calculation, making the parameter-to-observable ratio worse. The effective number of parameters has to be restrained. This can be achieved by using only a few confonners in the ensemble and by determining the optimum number of confonners by cross-validation [83]. A more indirect way of restraining the effective number of parameters is to restrict the conformational space that the molecule can search... 

Industrial environments expose individuals to a plethora of airborne chemical compounds in the form of vapors, aerosols, or biphasic mixtures of both. These atmospheric contaminants primarily interface with two body surfaces the respiratory tract and the skin. Between these two routes of systemic exposure to airborne chemicals (inhalation and transdermal absorption) the respiratory tract has the larger surface area and a much greater percentage of this surface exposed to the ambient environment. Or dinary work clothing generally restricts skin exposures to the arms, neck, and head, and special protective clothing ensembles further limit or totally eliminate skin exposures, but breathing exposes much of the airway to contaminants. 

There are basically two different computer simulation techniques known as molecular dynamics (MD) and Monte Carlo (MC) simulation. In MD molecular trajectories are computed by solving an equation of motion for equilibrium or nonequilibrium situations. Since the MD time scale is a physical one, this method permits investigations of time-dependent phenomena like, for example, transport processes [25,61-63]. In MC, on the other hand, trajectories are generated by a (biased) random walk in configuration space and, therefore, do not per se permit investigations of processes on a physical time scale (with the dynamics of spin lattices as an exception [64]). However, MC has the advantage that it can easily be applied to virtually all statistical-physical ensembles, which is of particular interest in the context of this chapter. On account of limitations of space and because excellent texts exist for the MD method [25,61-63,65], the present discussion will be restricted to the MC technique with particular emphasis on mixed stress-strain ensembles. 

It is shown in classical statistics that the probability of the grand ensemble is a maximum, under the restriction to given average energy and given avenge population per system, when the distribution is chosen to be... 

This restriction means that we shall consider ensembles representing equilibrium or stationary situations, and forego the discussion of quantum dynamical problems where the probabilities w(n are forced to change with time. 

On the other hand, polar molecules create a force field around them that is attractive or repulsive, depending on the relative orientation of the neighboring polar molecule. In this case, the spectrum of molecular arrangements actually explored by an ensemble of strongly polar molecules is severely restricted. It follows that these molecules display a more marked tendency to give a dimensionally unlimited ordered molecular arrangement and a limited mutual solubility with apolar solvents. 

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