Construction of the Partition Function

For an isolated molecule in the rigid rotor, harmonic oscillator approximation, the (quantum) energy states are sufficiently regular to allow an explicit construction of the partition function, as discussed in Chapter 12. For a collection of many particles the... 

We describe here only briefly the construction of the partition function PF) of this system. More details are given in Ben-Naim (1992, 2008c). 

Applications of the SCVJ propagator have been reported and show its reliability [70, 80, 81], In this connection, one notes that the P beads play a role in the construction of the partition function. However, depending on the computational scheme chosen (thermodynamic, operator) to fix properties, all of them may or may not be involved in the definition of the quantities to be averaged. For example,... 

Our point of view is that the evaluation of the partition function (9.5) can be done by using any available tool, specifically including computer simulation. If that computer simulation evaluated the mechanical pressure, or if it simulated a system under conditions of specified pressure, then /C,x would have been determined at a known value of p. With temperature, composition, and volume also known, (9.2) and (9.1) permit the construction of the full thermodynamic potential. This establishes our first assertion that the potential distribution theorem provides a basis for the general theory of solutions. 

We shall treat more compUcated cases, such as systems with a larger number of identical or different sites, and also cases of more than one type of ligand. But the general rules of constructing the canonical PF, and hence the GPF, are the same. The partition functions, either Q or have two important properties that make the tool of statistical thermodynamics so useful. One is that, for macroscopic systems, each of the partition functions is related to a thermodynamic potential. For the particular PFs mentioned above, these are... 

Again, it must be noted that evaluating the rotational components of U and. S requires relatively little in the way of molecular information. All that is required is the principal moments of inertia, which derive only from the molecular structure. Thus, any methodology capable of predicting accurate geometries should be useful in the construction of rotational partition functions and the thermodynamic variables computed therefrom. Also again, the units chosen for quantities appearing in the partition function must be consistent so as to render q dimensionless. 

Once a histogram of visits to each state has been constructed, the ratio of probabilities appearing in Eq. (4.6) can be calculated, thereby permitting evaluation of the ratio of the partition functions for any two x and y ... 

It is necessary to know the form of the partition functions to construct transition probabilities that properly sample the ensemble. For instructional purposes we record here the (semiclassical) partition functions that correspond to the osmotic and isomolar semigrand ensembles described above. In both ensembles one must average over moles of the Legendre-transformed species. Thus the osmotic semigrand ensemble partition function is... 

The statistical weight matrix constructed from the relative statistical weights permits the calculation of the partition function of a given sequence by matrix multiplication. The partition functions contribute to the evaluation of the conformational sequence probabilities P (i/n/ q> /), i.e. the probability to find a sequence of n residues in a specific conformational state q> starting at the i-th position of the chain. The results of these calculations can graphically be presented in the form of conformational probability profiles. 

The partition function 3 (Eq. 4) and the resulting expressions for the properties of the clathrate contain the cell partition functions hjt as the only unknowns. If it were possible to construct these parti-... 

Below we present the rules for constructing the partition function (PF) of a binding system. 

The partition function of a molecule also contains torsional motions and the construction of such a function requires the knowledge of molecular mass, moments of inertia, and constants describing normal vibration modes. Several of these data may be acquired from infrared and Raman spectra (67SA(A)891 85JST( 126)25), but the procedure has not yet been extensively applied owing to experimental limitations. To characterize the barrier one also needs to know more than one constant, and these are often not available from... 

The partition function, ZN, of an a-helix-forming polypeptide molecule is then constructed by assigning the following statistical weights to the central units of seven possible joint conformations of three consecutive residues ... 

With the introduction of the lattice structure and electroneutrality condition, one has to define two elementary SE units which do not refer to chemical species. These elementary units are l) the empty lattice site (vacancy) and 2) the elementary electrical charge. Both are definite (statistical) entities of their own in the lattice reference system and have to be taken into account in constructing the partition function of the crystal. Structure elements do not exist outside the crystal and thus do not have real chemical potentials. For example, vacancies do not possess a vapor pressure. Nevertheless, vacancies and other SE s of a crystal can, in principle, be seen , for example, as color centers through spectroscopic observations or otherwise. The electrical charges can be detected by electrical conductivity. 

Unfortunately, the simple importance sampling as described above cannot be used to sample multidimensional integrals over configuration space as in Eq. (1.1). The reason is simply that we do not know how to construct the transformation in Eq. (1.7) that will enable us to generate points in configuration space with a probability density as given by the Boltzmann factor. In fact, in order to do so, we must be able to compute analytically the partition function of the system. If we could do that, there would hardly be any need for computer simulations ... 

The traditional apparatus of statistical physics employed to construct models of physico-chemical processes is the method of calculating the partition function [17,19,26]. The alternative method of correlation functions or distribution functions [75] is more flexible. It is now the main method in the theory of the condensed state both for solid and liquid phases [76,77]. This method has also found an application for lattice systems [78,79]. A new variant of the method of correlation functions - the cluster approach was treated in the book [80]. The cluster approach provides a procedure for the self-consistent calculation of the complete set of probabilities of particle configurations on a cluster being considered. This makes it possible to take account of the local inhomogeneities of a lattice in the equilibrium and non-equilibrium states of a system of interacting particles. In this section the kinetic equations for wide atomic-molecular processes within the gas-solid systems were constructed. 

An a-helix bundle may become a second-order cooperative folding unit if the interaction energy terms are such that the intermediate terms in the partition function become negligibly small [Eq. (14)] and the entire partition function reduces to a two-state partition function (i.e., a partition function of the form 1 + e G/RT). If such is the case, the a-helix bundle will be either completely folded or unfolded. Higher order cooperative folding units can be constructed from lower order ones following the same rules. The most immediate application of this approach is to proteins exhibiting pure a-helical structural motifs. 

The cell model is a commonly used way of reducing the complicated many-body problem of a polyelectrolyte solution to an effective one-particle theory [24-30]. The idea depicted in Fig. 1 is to partition the solution into subvolumes, each containing only a single macroion together with its counterions. Since each sub-volume is electrically neutral, the electric field will on average vanish on the cell surface. By virtue of this construction different sub-volumes are electrostatically decoupled to a first approximation. Hence, the partition function is factorized and the problem is reduced to a singleparticle problem, namely the treatment of one sub-volume, called cell . Its shape should reflect the symmetry of the polyelectrolyte. Reviews of the basic concepts can be found in [24-26]. 

The partition function of the quantum gas can then be constructed from the single-particle repulsion partition function, qTep and the contribution of attraction, Q ... 

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