Partition function, classical limit

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

The classical bath sees the quantum particle potential as averaged over the characteristic time, which - if we recall that in conventional units it equals hjk T- vanishes in the classical limit h- Q. The quasienergy partition function for the classical bath now simply turns into an ordinary integral in configuration space. 

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 final expression is the classical limit, valid above a certain critical temperature, which, however, in practical cases is low (i.e. 85 K for H2, 3 K for CO). For a homonuclear or a symmetric linear molecule, the factor a equals 2, while for a het-eronuclear molecule cr=l (Tab. 3.1). This symmetry factor stems from the indistinguishable permutations the molecule may undergo due to the rotation and actually also involves the nuclear partition function. The symmetry factor can be estimated directly from the symmetry of the molecule. 

Because the frequency of a weakly bonded vibrating system is relatively small, i.e. kBT hu we may approximate its partition function by the classical limit k T/hv, and arrive at the rate expression in transition state theory ... 

Nonlinear polyatomic molecules require further consideration, depending on their classification, as given in Section 9.2.2. In the classical, high-temperature limit, the rotational partition function for a nonlinear molecule is given by... 

With neglect of the quantum effects that arise from the exchange of identical particles [147], (8.66) gives the exact quantum partition function in the limit P — oo. For finite P, Qp((3) is the canonical partition function of a classical system composed of ring polymers. Each quantum particle corresponds to a ring polymer of P beads in which neighboring beads are connected by harmonic springs with force... 

The discussion will be limited to systems at temperatures which are high enough so that we need only consider the classical partition function Q in calculating the Helmholtz free energy A, where, of course... 

In 1933, J.G. Kirkwood explicitly showed that the canonical partition function Q for a system of N monatomic particles reduces to an integral over phase space in the limit of high temperature (Equation 4.81). The result corresponds to classical mechanics (i.e. the spacing between energy levels is small compared to kT)... 

The classical vibrational partition function can be found by letting the temperature go to infinity. This means that we take the limit of Equation 4.91 as u -> 0(u = hv/kT). 

The importance of understanding isotope effects in the high temperature (classical) limit has been stressed before. In the limit of infinite temperature, the reduced isotopic partition function ratios all go to unity and k /k2 also goes to unity. The kinetic isotope effect becomes... 

It has been previously noted that the first quantum correction to the classical high temperature limit for an isotope effect on an equilibrium constant is interesting. Each vibrational frequency makes a contribution c[>(u) to RPFR and this contribution can be expanded in powers of u with the first non-vanishing term proportional to u2/24, the so called first quantum correction. Similarly, for rates one introduces the first quantum correction for the reduced partition function ratios, includes the Wigner correction for k /k2 and makes use of relations like Equation 4.103 for small x and small y, to find a value for the rate constant isotope effect (omitting the noninteresting symmetry number term)... 

It is easily shown that, in the classical limit, Eqs. (41) and (42) are consistent with the thermal capture rate constants for the oscillator model of charge-permanent dipole capture. The relevant part of the activated complex partition function, instead of Eq. (11), can be written as... 

In the following we elaborate VTST expressions for various charge-dipole potentials. For demonstrative purposes, we further consider the isotropic locked permanent-dipole case where SACM and PST are identical. We also consider the real anisotropic permanent-dipole case in the quantum low-temperature and classical high-temperature oscillator limits. Finally we show comparisons for real permanent and induced-dipole cases. We always employ explicit adiabatic channel eigenvalues for calculating partition functions or numbers of states. 

It should be noted that the result in Eq. (7.59) is strictly valid only in the classical high-temperature limit which, except for very high temperatures, is not well satisfied for typical vibrational frequencies. Qualitatively, a similar result will also be obtained when the exact vibrational partition functions are used in Eq. (7.58). Rotational contributions were also neglected, but the moments of inertia associated with the activated complex are often larger than the moments of inertia of the reactant. Thus, we have very often that > Q, and large pre-exponential factors may often arise due to a loose transition state as well as due to a substantial change in geometry between the reactant and the activated complex. 

It is often impossible to obtain the quantized energies of a complicated system and therefore the partition function. Fortunately, a classical mechanical description will often suffice. Classical statistical mechanics is valid at sufficiently high temperatures. The classical treatment can be derived as a limiting case of the quantum version for cases where energy differences become small compared with ksT. 

Finally, the classical partition function for N interacting molecules, that is, the classical limit of Eq. (A.2), takes the form... 

Here U N) is the interaction potential energy for the complete system at a specific configuration, uniformly the same quantity that has been discussed above. U(TV) is an ejfective potential designed to be used in classical-limit partition function calculations, e.g. Eq. (3.17), p. 40, in order to include quantum mechanical effects approximately. We will call this /(TV) the quadratic Feynman-Hibbs (QFH) model. In Eq. (3.67), Mj is the mass of atom j, and V is the Laplacian of the... 

This can be found after transforming the partition function into a Gauss integral.23 Then the value of det (1) is required, not its roots. The classical limit might be important in physiology. 

Neglecting rotational motion, and assuming that the vibrational modes are all harmonic, the classical limit of the vibrational partition function of a molecule... 

From the discussion up to this point the reader will surely appreciate that a rigorous, first-principles calculation of the partition function for a macroscopic system is generally precluded even in the classical hmit with the exception of rather simple models of limited usefulness (see Chapter 3). However, the problem of calculating the partition function (or the configuration integral) in closed form becomes tractable if we introduce as a key assiunptiou that correlations between molecules arc entirely negligible. 

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. 

Note that the expression (1.60) is not purely classical since it contains two corrections of quantum mechanical origin the Planck constant h and the M. Therefore, Q defined in (1.60) is actually the classical limit of the quantum mechanical partition function in (1.59). The purely classical partition function consists of the integral expression on the rhs of (1.60) without the factor (h3NM). This partition function fails to produce the correct form of the chemical potential or of the entropy of the system. 

The dependence of both p and S on the density p through In p is confirmed by experiment. We note here that had we used the purely classical partition function [i.e., the integral excluding the factors h3NM in (1.60)], we would not have obtained such a dependence on the density. This demonstrates the necessity of using the correction factors h3NM even in the classical limit of the quantum mechanical partition function. 

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