Statistical mechanical partition function

This form first noted in Ref. [66] has been again obtained by Sutherland [67]. It appears much in the form of a statistical mechanical partition function, so that a variety of statistical mechanical, or graph-theoretic, techniques might be used in the evaluation of it. 

Equilibrium vapor pressure of bulk liquid Relative pressure, p/p 0 Statistical mechanical partition function Density in molecules/cubic centimeter Gas constant per mole Number of moles of adsorbed gas Number of moles of adsorbent Isosteric heat of adsorption Differential heat of adsorption Isothermal heat of adsorption Adiabatic heat of adsorption... 

For more complicated reactions, RRKM theory can be applied, using the statistical mechanical partition functions calculated within the BAC-MP4 method. ... 

High moduli, memory effects, and SANS results which are inconsistent with classical theories of rubber elasticity provoke the need for a new theory. The ideas of junction rearrangement, if correct, require that none of the models of affine deformation should be expected to apply. A statistical mechanical partition function, properly formulated for a polymeric elastomer, should yield predictions of chain deformation, and additional assinnptions relating macroscopic and molecular geometry are superfluous. 

For the SCLF model adsorption can be seen as purely partitioning behavior. Adsorption is quantified in this model by assigning a statistical mechanical partition function to each solute molecule. The partition function recognizes three major features of adsorption (1) entropy of adsorption, (2) energy of adsorption, and (3) solute-solvent interaction energy. 

In section 1.2, we introduced the quantum mechanical partition function in the T, V, N ensemble. In most applications of statistical thermodynamics to problems in chemistry and biochemistry, the classical limit of the quantum mechanical partition function is used. In this section, we present the so-called classical canonical partition function. 

The most fundamental starting point for any theoretical approach is the quantum mechanical partition function PF), and the fundamental connection between the partition function and the corresponding thermodynamic potential. Once we have a PF, either exact or approximate, we can derive all the thermodynamic quantities by using standard relationships. Statistical mechanics is a general and very powerful tool to connect between microscopic properties of atoms and molecules, such as mass, dipole moment, polarizability, and intermolecular interaction energy, on the one hand, and macroscopic properties of the bulk matter, such as the energy, entropy, heat capacity, and compressibility, on the other. 

Thus the kinetic and statistical mechanical derivations may be brought into identity by means of a specific series of assumptions, including the assumption that the internal partition functions are the same for the two states (see Ref. 12). As discussed in Section XVI-4A, this last is almost certainly not the case because as a minimum effect some loss of rotational degrees of freedom should occur on adsorption. 

The grand canonical ensemble is a set of systems each with the same volume V, the same temperature T and the same chemical potential p (or if there is more than one substance present, the same set of p. s). This corresponds to a set of systems separated by diathennic and penneable walls and allowed to equilibrate. In classical thennodynamics, the appropriate fimction for fixed p, V, and Tis the productpV(see equation (A2.1.3 7)1 and statistical mechanics relates pV directly to the grand canonical partition function... 

This is connnonly known as the transition state theory approximation to the rate constant. Note that all one needs to do to evaluate (A3.11.187) is to detennine the partition function of the reagents and transition state, which is a problem in statistical mechanics rather than dynamics. This makes transition state theory a very usefiil approach for many applications. However, what is left out are two potentially important effects, tiiimelling and barrier recrossing, bodi of which lead to CRTs that differ from the sum of step frmctions assumed in (A3.11.1831. 

The canonical ensemble is the name given to an ensemble for constant temperature, number of particles and volume. For our purposes Jf can be considered the same as the total energy, (p r ), which equals the sum of the kinetic energy (jT(p )) of the system, which depends upon the momenta of the particles, and the potential energy (T (r )), which depends upon tlie positions. The factor N arises from the indistinguishability of the particles and the factor is required to ensure that the partition function is equal to the quantum mechanical result for a particle in a box. A short discussion of some of the key results of statistical mechanics is provided in Appendix 6.1 and further details can be found in standard textbooks. 

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. 

MaxweU-Boltzmaim particles are distinguishable, and a partition function, or distribution, of these particles can be derived from classical considerations. Real systems exist in which individual particles ate indistinguishable. Eor example, individual electrons in a soHd metal do not maintain positional proximity to specific atoms. These electrons obey Eermi-Ditac statistics (133). In contrast, the quantum effects observed for most normal gases can be correlated with Bose-Einstein statistics (117). The approach to statistical thermodynamics described thus far is referred to as wave mechanics. An equivalent quantum theory is referred to as matrix mechanics (134—136). 

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. 

The key feature in statistical mechanics is the partition function Just as the wave function is the corner-stone of quantum mechanics (from that everything else can be calculated by applying proper operators), the partition function allows calculation of alt macroscopic functions in statistical mechanics. The partition function for a single molecule is usually denoted q and defined as a sum of exponential terms involving all possible quantum energy states Q is the partition function for N molecules. 

Both Flory and Huggins derived statistical mechanical expressions for aS . Their expressions are still among the best available. For this reason, Prigogine and his co-workers concentrated their efforts on revising the statistical mechanical configurational partition function which leads, among other things, toAH. ... 

Using the formalism of statistical mechanics, Giddings et al. [135] investigated the effects of molecular shape and pore shape on the equilibrium distribution of solutes in pores. The equilibrium partition coefficient is defined as the ratio of the partition function in the pore... 

Table 10.4 lists the rate parameters for the elementary steps of the CO + NO reaction in the limit of zero coverage. Parameters such as those listed in Tab. 10.4 form the highly desirable input for modeling overall reaction mechanisms. In addition, elementary rate parameters can be compared to calculations on the basis of the theories outlined in Chapters 3 and 6. In this way the kinetic parameters of elementary reaction steps provide, through spectroscopy and computational chemistry, a link between the intramolecular properties of adsorbed reactants and their reactivity Statistical thermodynamics furnishes the theoretical framework to describe how equilibrium constants and reaction rate constants depend on the partition functions of vibration and rotation. Thus, spectroscopy studies of adsorbed reactants and intermediates provide the input for computing equilibrium constants, while calculations on the transition states of reaction pathways, starting from structurally, electronically and vibrationally well-characterized ground states, enable the prediction of kinetic parameters. 

In statistical mechanics the properties of a system in equilibrium are calculated from the partition function, which depending on the choice for the ensemble considered involves a sum over different states of the system. In the very popular canonical ensemble, that implies a constant number of particles N, volume V, and temperature T conditions, the quasiclassical partition function Q is... 

The various contributions to the energy of a molecule were specified in Eq. (47). However, the fact that the electronic partition function was assumed to be equal to one should not be overlooked. In effect, the electronic energy was assumed to be equal to zero, that is, that the molecule remains in its ground electronic state. In the application of statistical mechanics to high-temperature systems this approximation is not appropriate. In particular, in the analysis of plasmas the electronic contribution to the energy, and thus to the partition function, must be included. 

Moreover, QXyz+ is a partition function from which the degree of freedom corresponding to motion along the reaction coordinate has been removed. The contribution of this motion has already been taken into account in the analysis leading to equation 4.3.27. Readers who wish to acquaint themselves with the manner in which partition functions are calculated are encouraged to consult standard texts in physical chemistry and statistical mechanics. 

In the remainder of this chapter, we review the fundamentals that underlie the theoretical developments in this book. We outline, in sequence, the concept of density of states and partition function, the most basic approaches to calculating free energies and the essential strategies for improving the efficiency of these calculations. The ideas discussed here are, most likely, known to the reader. They can also be found in classical books on statistical mechanics [132-134] and molecular simulations [135, 136]. Thus, we do not attempt to be exhaustive. On the contrary, we present the material in a way that is most directly relevant to the topics covered in the book. 

This equation forms the fundamental connection between thermodynamics and statistical mechanics in the canonical ensemble, from which it follows that calculating A is equivalent to estimating the value of Q. In general, evaluating Q is a very difficult undertaking. In both experiments and calculations, however, we are interested in free energy differences, AA, between two systems or states of a system, say 0 and 1, described by the partition functions Qo and (), respectively - the arguments N, V., T have been dropped to simplify the notation ... 

Another approach to calculating AA relies on estimating the appropriate probability density functions. The connection between the probabilities of different states and the partition function is natural in statistical mechanics. Equation (1.19) is a reflection of this connection. Similarly, the probability of observing the potential energy of the system being equal to U is ... 

---