The Configuration Integral

We start by assuming that the internal motions of the molecules are independent of density, which allows us to separate the partition function of a real fluid into two parts one that involves the intermolecular forces -i.e. the potential energy of the fluid - and is expressed in terms of the configuration integral and one that involves the internal motions in the ideal gas state. 

We proceed then with a second assumption, that of pairwise additivity, according to which the potential energy of the fluid can be approximated by the sum of interactions between all molecular pairs. This assumption allows us to determine the thermodynamic properties of a real fluid in terms of the intermolecular potential energies and the radial distribution junction, which describes the position of the molecules in the space they occupy. 

Unfortunately, the radial distribution function cannot be determined with sufficient accuracy at all densities. For low density fluids, however, solution is possible through a Taylor s series expansion around zero density, which will lead to the virial equation. 

We turn then to dense fluids and outline the main approaches used for them, which we classify into three main categories Molecular Simulation, Perturbation, and Semiempirical. More emphasis is given to the last one which, using results from the other two approaches, leads to semiempirical equations of state. 

Consider a fluid consisting of N molecules at temperature T and volume V. Following Reed and Gubbins (1973), we assume that the total energy of the fluid is the sum of two independent contributions  

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The integral over the positions is often referred to as the configurational integral,... 

In an ideal gas there are no interactions between the particles and so the potential ener function, 1 ), equals zero. exp(- f (r )/fcBT) is therefore equal to 1 for every gas partic in the system. The integral of 1 over the coordinates of each atom is equal to the volume, ai so for N ideal gas particles the configurational integral is given by (V = volume). T1 leads to the following result for the canonical partition function of an ideal gas ... 

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

To obtain thermodynamic perturbation or integration formulas for changing q, one must go back and forth between expressions of the configuration integral in Cartesian coordinates and in suitably chosen generalized coordinates [51]. This introduces Jacobian factors... 

Here, the configuration integral Q has been split into tliree integrals the integration j is over all conformations where 71 is in the ith rotameric state. F is the configurational... 

These are exact expressions for the configuration integrals. Alternatively, we can write the partition function as... 

The angle bracket denotes that the configurational integral is taken over the initial state. The conformational sampling indicated by Equation 4 is generated according to the Boltzmann probability associated with the initial state potential. As discussed in Section 2.1, convergence of conformational... 

Pu depends on the quotient flj, / TT, the calculation of the configurational integral Z(N,V,T) is avoided. The change in potential energy of the system due to the trial move determines if the attempted new configuration is accepted. 

For a classical system of N point particles enclosed in a volume V,at a temperature T, the canonical partition function can be decomposed in two factors. The first one (Qt) comes from the integration over the space of momenta of the kinetic term of the classical Hamiltonian, which represents the free motion of noninteracting particles. The second one, which introduces the interactions between the particles and involves integration over the positions, is the configuration integral. This way, equation (30)... 

The difficulty arises from the fact that the one-step transition probabilities of the Markov chain involve only ratios of probability densities, in which Z(N,V,T) cancels out. This way, the Metropolis Markov chain procedure intentionally avoids the calculation of the configurational integral, the Monte Carlo method not being able to directly apply equation (31). 

If we consider a symmetric salt of N cations and N anions then the configurational integral is... 

One begins with the configurational integral for the "comparison salt ... 

A prime on Z denotes the configurational integral without the factorial coefficients. This notation is introduced to simplify expressions which appear later. 

To determine A we need to implement the path integration as a numerical path summation. The path integral in Eqs. 23-24 is in fact isomorphic to the configuration integral of a flexible polymer which interacts with the external potential V(r(P)). This analogy can be made more explicit by considering a discrete approximation to the path integral [71]. If the path is cut up into P... 

MSE.9. 1. Prigogine et P. Janssens, Une generalisation de la methode de Lennard-Jones-Devonshire pour le calcul de I integrale de configuration (A generalization of the Lennard-Jones and Devonshire method foir the calculation of the configuration integral), Physica 16, 895-906 (1950). 

The configuration integral and all conformational averages are then evaluated using... 

Given an approximation to V( 0s ) acceptable for the purposes at hand, one can proceed to compute equilibrium, i.e., statistically mechanically averaged, values

for properties P( 0s ) of interest using standard procedures which weight each conformation of the carbohydrate molecule by the Boltzmann factor of V( 0s ) normalized by the configuration integral given in eqn. (9). 

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