Partition function Monte Carlo techniques

The thermodynamic properties that we have considered so far, such as the internal energy, the pressure and the heat capacity are collectively known as the mechanical properties and can be routinely obtained from a Monte Carlo or molecular dynamics simulation. Other thermodynamic properties are difficult to determine accurately without resorting to special techniques. These are the so-called entropic or thermal properties the free energy, the chemical potential and the entropy itself. The difference between the mechanical emd thermal properties is that the mechanical properties are related to the derivative of the partition function whereas the thermal properties are directly related to the partition function itself. To illustrate the difference between these two classes of properties, let us consider the internal energy, U, and the Fielmholtz free energy, A. These are related to the partition function by ... 

The computation of quantum many-body effects requires additional effort compared to classical cases. This holds in particular if strong collective phenomena such as phase transitions are considered. The path integral approach to critical phenomena allows the computation of collective phenomena at constant temperature — a condition which is preferred experimentally. Due to the link of path integrals to the partition function in statistical physics, methods from the latter — such as Monte Carlo simulation techniques — can be used for efficient computation of quantum effects. 

The partition function Z is given in the large-P limit, Z = limp co Zp, and expectation values of an observable are given as averages of corresponding estimators with the canonical measure in Eq. (19). The variables and R ( ) can be used as classical variables and classical Monte Carlo simulation techniques can be applied for the computation of averages. Note that if we formally put P = 1 in Eq. (19) we recover classical statistical mechanics, of course. 

Schrodinger equation. When the molecule is too large and difficult for quantum mechanical calculations, or the molecule interacts with many other molecules or an external field, we turn to the methods of molecular mechanics with empirical force fields. We compute and obtain numerical values of the partition functions, instead of precise formulas. The computation of thermodynamic properties proceeds by using a number of techniques, of which the most prominent are the molecular dynamics and the Monte Carlo methods. 

Another procedure to overcome the inefficiency of Metropolis Monte Carlo is adaptive importance sampling.194-196 In this technique, the partition function (and quantities derived from it, such as the probability of a given conformation) is evaluated by continually upgrading the distribution function (ultimately to the Boltzmann distribution) to concentrate the sampling in the region (s) where the probabilities are highest. 

Previous work on the thermodynamic properties of clusters used a number of schemes to evaluate the partition function required in Eq. (3.14). In the normal-mode method, " described in the Introduction, the partition function is constructed from the standard partitioning of a polyatomic gas into classical translational and rotational terms and quantum vibrational contributions. In Monte Carlo studies it is usual to employ a state integration technique.In the state integration method Eq. (3.5) is integrated with respect to temperature to obtain... 

Recent years have seen the extensive application of computer simulation techniques to the study of condensed phases of matter. The two techniques of major importance are the Monte Carlo method and the method of molecular dynamics. Monte Carlo methods are ways of evaluating the partition function of a many-particle system through sampling the multidimensional integral that defines it, and can be used only for the study of equilibrium quantities such as thermodynamic properties and average local structure. Molecular dynamics methods solve Newton s classical equations of motion for a system of particles placed in a box with periodic boundary conditions, and can be used to study both equilibrium and nonequilibrium properties such as time correlation functions. 

Of course, no simulation technique actually samples all of phase space only a representative sample that has the distribution of the relevant ensemble is required. Similarly, we can obtain a representative sample of minima by performing minimizations from a set of points generated by a Monte Carlo (MC) or MD simulation. To eompensate for the incompleteness, we weight the density of states or partition function for each known minimum by g the number of minima of energy Ei for which the minimum i is representative [136]. Hence,... 

Another technique which has gained prominence in recent years is the Quantum Monte Carlo (QMC) technique. This technique maps a d-dimensional quantum model onto a d - - 1 dimensional classical model via a Trotter decomposition of the partition function or the ground state projection operator [56, 57]. The quantum model is then studied by performing a Monte Carlo sampling procedure on the classical model in higher dimension. For fermions, the mapping of the interacting quantum model system to the classical system could... 

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