Effective Partition Function

Therefore, the associated periodic propagator (4.34) becomes (Kleinert, 2004) 

At the same time there is clear that the periodic path condition (4.518) is not arbitrarily but a compulsory step since characteristic in passing from density matrix to partition function and then to the real (measurable or workable) canonical and A-particle density, according with the density matrix algorithm (4.243)-(4.248). Therefore, the resulting partition function built from the un-normalized canonical density (4.520) assumes the simple form 

Quantum Nanochemistry-Volume I Quantum Theory and Observability 

With these considerations there appears as natural the generalization of the classical partition form Eq. (4.521) into the more comprehensive one known as the effective classical partition function (Kleinert, 1986 Giachetti et al., 1986 Janke Cheng, 1988 Voth, 1991 Cuccoli et al., 1992) 

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Calculation of the integrals in Equation 8 is obviously impossible in full. Hence, an effective partition function for a chosen chain has to be introduced. Muthukumar chooses such a function G in terms of the variational principle. 

The term [G(4>)—O] contains the fluctuations of the partition function about its effective partition function, and f dp e p(pG) contributes to the free energy of n effective chains. 

Clenerally, proves to be a very complicated function of A and reflects the non-Markov nature of the process. Therefore, one can expect that the effective partition function for the system has the form... 

We now return to the effective partition function (8.6.26) and assume that 5A can be written as pairwise additive quantities and, in addition, we assume that there exists a reference configuration for which the M particles are independently solvated. Then... 

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. 

Once the partition function is evaluated, the contributions of the internal motion to thennodynamics can be evaluated. depends only on T, and has no effect on the pressure. Its effect on the heat capacity can be... 

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. 

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). 

In CLTST there appears a kinetic isotope effect owing to the difference in partition functions in the initial state [see eq.(2.12)], and at 2Pf < o > I5... 

Substituting (5.34) and (5.35) for (5.8) and dropping in Z the constant partition function of unperturbed harmonic oscillator we get the nonlocal effective action derived by Feynman (see also Caldeira and Leggett [1983]),... 

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. 

The special case where only rotators are present, Np = 0, is of particular interest for the analysis of molecular crystals and will be studied below. Here we note that in the other limit, where only spherical particles are present, Vf = 0, and where only symmetrical box elongations are considered with boxes of side length S, the corresponding measure in the partition function (X Qxp[—/3Ep S, r )], involving the random variable S, can be simplified considerably, resulting in the effective Hamiltonian... 

We assume that exploring all possible forms for the fields corresponds to exploring the overall usual phase space. To determine the partition function Z the contributions from all the p+ r) and P- r) distributions are summed up with a statistical weight, dependent on p+ r) and p (r), put in the form analogous to the Boltzmann factor exp[—p (F)]], where the effective Hamiltonian p (F)] is a functional of the fields. The... 

Current use of statistical thermodynamics implies that the adsorption system can be effectively separated into the gas phase and the adsorbed phase, which means that the partition function of motions normal to the surface can be represented with sufficient accuracy by that of oscillators confined to the surface. This becomes less valid, the shorter is the mean adsorption time of adatoms, i.e. the higher is the desorption temperature. Thus, near the end of the desorption experiment, especially with high heating rates, another treatment of equilibria should be used, dealing with the whole system as a single phase, the adsorbent being a boundary. This is the approach of the gas-surface virial expansion of adsorption isotherms (51, 53) or of some more general treatment of this kind. 

Under most circumstances the equations given in Table 10.4 accurately calculate the thermodynamic properties of the ideal gas. The most serious approximations involve the replacement of the summation with an integral [equations (10.94) and (10.95)] in calculating the partition function for the rigid rotator, and the approximation that the rotational and vibrational partition functions for a gas can be represented by those for a rigid rotator and harmonic oscillator. In general, the errors introduced by these approximations are most serious for the diatomic molecule." Fortunately, it is for the diatomic molecule that corrections are most easily calculated. It is also for these molecules that spectroscopic information is often available to make the corrections for anharmonicity and nonrigid rotator effects. We will summarize the relationships... 

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... 

We now calculate the density of the phonon scattering states. Since we have effectively isolated the transition amplitude issue, the fact of equally strong coupling of all transitions to the lattice means that the scattering density should directly follow from the partition function of a domain via the... 

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. 

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