Monte Carlo method general description

A brief description of the Monte Carlo method in general... 

This chapter makes a humble attempt to introduce some of the basic concepts of the Monte Carlo method for simulating ensembles of particles. Before we jump into the heart of the matter, we would like to direct the interested reader to a more general description of different Monte Carlo methods by Hammersley and Handscomb [1] and to the excellent textbooks on particle simulations by Allen and Tildesley [2], Binder and Heer-mann [3], and Frenkel and Smit [4]. 

We do not enter into the description of Monte Carlo of Molecular Dynamics methods, as these details are not important for our discussion. There are other more general aspects of computer simulations to consider here. 

From its inception, the combined Quantum Mechanics/Molecular Mechanics (QM/MM) method [1-3] has played an important roll in the explicit modeling of solvent [4], Whereas Molecular Mechanics (MM) methods on their own are generally only able to describe the effect of solvent on classical properties, QM/MM methods allow one to examine the effect of the solvent on solute properties that require a quantum mechanical (QM) description. In most cases, the solute, sometimes together with a few solvent molecules, is treated at the QM level of theory. The solvent molecules, except for those included in the QM region, are then treated with an MM force field. The resulting potential can be explored using Monte Carlo (MC) or Molecular Dynamics (MD) simulations. Besides the modeling of solvent, QM/MM methods have been particularly successful in the study of biochemical systems [5] and catalysis [6],... 

The above reasoning shows that the stretched exponential function (4.14), or Weibull function as it is known, may be considered as an approximate solution of the release problem. The advantage of this choice is that it is general enough for the description of drug release from vessels of various shapes, in the presence or absence of different interactions, by adjusting the values of the parameters a and b. Monte Carlo simulation methods were used to calculate the values of the parameters a and (mainly) the exponent b [87]. 

The method was generalized for an arbitrary number of atonrs [D. Blume, and C.H. Greene, Monte Carlo Hyperspherical Description of Helium Cluster Excited Stated , 2000]. 

The simulation was based on the 5-state RIS model mainly because it is, at the moment, the best statistical description of atactic poly(propylene). It should be borne in mind, however, that the ab initio simulation technique is not restricted to RIS models. Any model with sufficient predictive power may be used as a basis for the simulation The conformational (or more generally geometric) statistics may be replaced by the results of a molecular dynamics or Monte Carlo study, or any other method. Within computer time limitations, the relevant geometries may be fed into Eq. 3.15 and used as a basis for the quantum chemistry calculations. 

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