Partition function methods

To reiterate a point that we made earlier, these problems of accurately calculating the free energy and entropy do not arise for isolated molecules that have a small number of well-characterised minima which can all be enumerated. The partition function for such systems can be obtained by standard statistical mechanical methods involving a summation over the mini mum energy states, taking care to include contributions from internal vibrational motion. 

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

Let us now turn to the case T -> 0. First of all, somewhat suspicious is the combination of the continuous integral (2.1) with the discrete partition function Zq (2.13), usual for CLTST. This serious deficiency cannot be circumvented in the framework of this CLTST-based formalism, and a more rigorous reasoning is needed to describe the quantum situation. Introduction of adequate methods will be the objective of the next sections devoted to the path-integral formalism, so here we... 

Following the Im F method, one looks for the partition function of the system... 

Except for the nonlocal last term in the exponent, this expression is recognized as the average of the one-dimensional quantum partition function over the static configurations of the bath. This formula without the last term has been used by Dakhnovskii and Nefedova [1991] to handle a bath of classical anharmonic oscillators. The integral over q was evaluated with the method of steepest descents leading to the most favorable bath configuration. 

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. 

To introduce the transfer matrix method we repeat some well-known facts for a 1-D lattice gas of sites with nearest neighbor interactions [31]. Its grand canonical partition function is given by... 

For a given Hamiltonian the calculation of the partition function can be done exactly in only few cases (some of them will be presented below). In general the calculation requires a scheme of approximations. Mean-field approximation (MFA) is a very popular approximation based on the steepest descent method [17,22]. In this case it is assumed that the main contribution to Z is due to fields which are localized in a small region of the functional space. More crudely, for each kind of particle only one field is... 

E. Evaluation of the Cell Partition Function by the Method of Lennard-Jones and Devonshire... 

The theory introduced by Lennard-Jones and Devonshire13 17 for the study of liquids provides a powerful method for the quantitative evaluation of the partition function of a solute molecule within its cavity.51 Because the application of this method to the present problem has been described in detail,62 we shall restrict ourselves to its most essential features. 

Several methods have been developed for the quantitative description of such systems. The partition function of the polymer is computed with the help of statistical thermodynamics which finally permits the computation of the degree of conversion 0. In the simplest case, it corresponds to the linear Ising model according to which only the nearest segments interact cooperatively149. The second possibility is to start from already known equilibrium relations and thus to compute the relevant degree of conversion 0. 

Again, therefore, all thermodynamic properties of a system in quantum statistics can be derived from a knowledge of the partition function, and since this is the trace of an operator, we can choose any convenient representation in which to compute it. The most fruitful application of this method is probably to the theory of imperfect gases, and is well covered in the standard reference works.23... 

Chymotrypsin, 170,171, 172, 173 Classical partition functions, 42,44,77 Classical trajectories, 78, 81 Cobalt, as cofactor for carboxypeptidase A, 204-205. See also Enzyme cofactors Condensed-phase reactions, 42-46, 215 Configuration interaction treatment, 14,30 Conformational analysis, 111-117,209 Conjugated gradient methods, 115-116. See also Energy minimization methods Consistent force field approach, 113 Coulomb integrals, 16, 27 Coulomb interactions, in macromolecules, 109, 123-126... 

In the following, the MO applications will be demonstrated with two selected equilibrium reactions, most important in radical chemistry disproportionation and dimerization. The examples presented will concern MO approaches of different levels of sophistication ab initio calculations with the evaluation of partition functions, semiempirical treatments, and simple procedures employing the HMO method or perturbation theory. 

When the log /J/pH measurement of a peptide is performed by the shake-flask or the partition chromatography method (using hydrophilic buffers to control pH), usually the shape of the curve is that of a parabola (see Ref. 371 and Fig. 1 in Ref. 282), where the maximum log I) value corresponds to the pH at the isoelectric point (near pH 5-6). Surprisingly, when the potentiometric method is used to characterize the same peptide [275], the curve produced is a step function, as indicated by the thick line in Fig. 4.5 for dipeptide Trp-Phe. 

We note that the calculation of At/ will depend primarily on local information about solute-solvent interactions i.c., the magnitude of A U is of molecular order. An accurate determination of this partition function is therefore possible based on the molecular details of the solution in the vicinity of the solute. The success of the test-particle method can be attributed to this property. A second feature of these relations, apparent in Eq. (4), is the evaluation of solute conformational stability in solution by separately calculating the equilibrium distribution of solute conformations for an isolated molecule and the solvent response to this distribution. This evaluation will likewise depend on primarily local interactions between the solute and solvent. For macromolecular solutes, simple physical approximations involving only partially hydrated solutes might be sufficient. 

Return now to the assertions of the Introduction. The explanation of assertion (1) was pointed out previously. Assertion (2), that the PDT is a practical computational tool, was the subject of Sect. 9.2.3. See especially the discussion "the general computational tricks work for the PDT also, emphasizing the general statistical methods of stratification and importance weighting, and their correspondence to the natural theoretical analyses of the PDT partition function. 

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