The Partition Function of a System

At this point, we have determined the complete partition function Q for a molecule. It is 

We start off by suggesting that the total energy of a system is the sum of the individual types of energy a molecule can have electronic, translational, vibrational, and so on. The total energy of the system is the sum of the energies of the individual particles. Thus, due to the original definition of the partition function, the overall partition function for the system is the product of the individual partition functions of N molecules  

If the system is composed of only one kind of molecule, then all of the individual Qj values are the same, and we simply have Q being multiplied by itself N times. To write this another way, we have 

However, this doesn t account for the fact that the individual molecules in the system are indistinguishable at the macroscopic level. Recall the examples of the balls in boxes at the beginning of Chapter 17. We found that there were fewer possible unique arrangements when the two balls were the same color. Another way to consider this is that there will be fewer possible arrangements if we cannot tell which gas molecules are which that is, if they are indistinguishable. (We do know, however, that the molecules are the same compound. We just can t tell, say, one molecule of water apart from any other molecule of water in our system.) Similarly, the above expression for is overvalued. Statistics can show that the value is too high by a multiplicative factor of N N factorial). N, recall, is the 

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Quantum Corrections. The obvious way to introduce quantum corrections in eq. 10 would be to interpret Za and Z as quantum partition functions however, this neglects tunneling (Z+, being the partition function of a system constrained to the top of the activation bar-... 

What is the importance of the partition function of a system What simplifications occur if the molecules of the system interact only weakly ... 

Once the partition function of a system is known, the thermodynamic functions are readily found. The total partition function of the system can be written in this case as... 

The partition function of a system with energy levels e(J) and degeneracies g(J) is... 

Each term A(T, P, Nu Nl) in the sum on the right-hand side of (2.4.8) is the partition function of a system in which the conversion reaction L H has been frozen in. In other words, the system is viewed as a mixture of two components L and... 

Thus, the partition function of a system with one additional water molecule at Rq is simply the sum over all the states of this... 

Knowing the partition function of a system containing N molecules occupying an area of A, the chemical potential of a molecule is ... 

Statistical mechanics. Statistical mechanics provides the tools required to deal with the physics of materials at high temperature. There are many excellent introductory texts for the interested reader (Hill 1956 Landau and Lifshitz 1980 McQuarrie 1976 Reif 1965). We focus on equilibrium thermodynamic properties because these play such an important role in our understanding of the Earth s interior. For the purposes of this discussion, we require a single result of statistical mechanics the partition function of a system of N atoms... 

In Section HI, Equation (III-53) we have derived an expression for the partition function of a system of interacting rigid hnear rotators in the approximation of small quantum corrections. 

This equation is equivalent to the statement that the partition function of a system with one additional s particle at a fixed position is the sum of the partition function of the same system with one A particle at a fixed position and the partition function of the same system with one B particle at a fixed position. 

Another approach for obtaining an equation of state is based on the partition function of a system derived firom statistical mechanics. One of these models is the Perturbed Hard-Chain Theory (PHCT) proposed by Beret and Prausnitz [6]. It was subsequently extended and modified by Cotterman et al. [7] and Morris et al. [8] as the Perturbed Soft-Chain Theory (PSCT). 

The partition function of a system plays a central role in statistical thermodynamics. The concept was first introduced by Boltzmann, who gave it the German name Zustandssumme, i.e., a sum over states. The partition function is an important tool because it enables the calculation of the energy and entropy of a molecule, as well as its equilibrium. Rate constants of reactions in which the molecule is involved can even be predicted. The only input for calculating the partition function is the molecule s set of characteristic energies, ,-, as determined by spectroscopic measurements or by a quantum mechanical calculation. In the next section the entropy and energy of an ideal monoatomic gas and a diatomic molecule is computed. 

The part of the chemical potential due to solvent-solute van der Waals interaction is customarily obtained by a coupling process, in which the interaction is turned on gradually by means of a coupling parameter I. The technique consists in the evaluation of the change in the partition function of a system of N solvent molecules after adding a solute molecule at a fixed position, and leads to the following expression for Gvdw ... 

The distribution of cluster sizes can be derived microscopically from statistical mechanics. The derivation is based on Refs. [10, 101, 9]. The partition function of a system containing N particles in a volume V at temperature T is given by... 

To calculate the partition function for a system of N atoms using this simple Monte Car integration method would involve the following steps ... 

The partition function for a system of Na adsorbed and Nf free polymer molecules is given by... 

The only systematic method available for calculating properties of an interacting many-body system consists in some kind of expansion in powers of the interaction. Consider, for instance, the partition function of a single chain, which in our model takes the form... 

According to statistical mechanics, the partition function for a system of volume V, at the temperature T, containing N independent (non-interacting) indistinguishable... 

For the PMF (m>0, = q3m), becomes simply Z, a configurational partition function of a system of N particles. In a special case, if m = 2, then q3m for the pairwise potential V may be denoted as R - the distance between two chosen particles. Using this for calculating the intergral of Eq. (3-12) denoted in parentheses, leads to the following form ... 

The partition function of a solute-solvent system for a given electronic state, where for the N solutes we use as (classical) molecular coordinates the center of mass position rG, the eulerian angles 0, , molecular frame and the internal coordinates Xjn providing the atom positions in the molecular frame, is [27,28]... 

The product krKx can now be written in terms of the partition functions of A, B,. . . and X, while the activity coefficients/a,/b,. . . may be either estimated from various theories of solution or measured experimentally. The value of fx, like its partition function, can only be assumed from more or less reasonable models for the reaction and theories of the solvation of X in the system. 

These relations will be established specifically in Chapter 3. j3 = kT, where k is the Boltzmann constant, and A is the thermal de Broglie wavelength. Q Ua = 1) = ISa is the canonical ensemble partition function of a system... 

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. 

In Section IV.A, we have shown that the quantum partition function in D dimensions looks like a classical partition function of a system in (D+ 1) dimensions, with the extra dimension being the time. With this mapping and allowing the space and time variables to have discrete values, we turn the quantum problem into an effective classical lattice problem. 

Since simulations of quantum systems suffer from the same problems as classical simulations, the extension of these generalized sampling schemes to quantum systems is highly desired. The extension is not immediately obvious since the partition function of a quantum system cannot be cast in the classical form... 

Let us consider first a system of N molecules of a pure r-mer fluid at temperature T and external pressure P. The molecules are considered arranged on a quasi-lattice of sites, Nq of which are empty. The empty sites, however, are not considered distributed randomly throughout the volume of the system. In a general way we may consider that the partition function of our system can be written as follows ... 

Consider the grand partition function of a system of spherical particles exposed to an external potential i/r ... 

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