Partition function quantum expression

The sum in Eq. (29.30) is over all the quantum states of the molecule, so q is the molecular partition function. If quantum states have the same energy, they are said to be degenerate degeneracy = gi. The terms in the partition function can be grouped according to the energy level. The states yield g equal terms in the partition function. The expression in Eq. (29.30) can be written as... 

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. 

We will use the harmonic oscillator approximation to derive an equation for the vibrational partition function. The quantum mechanical expression gives the vibrational energies as... 

In summary, Eq. (86) is a general expression for the number of particles in a given quantum state. If t = 1, this result is appropriate to Fenni-rDirac statistics, or to Bose-Einstein statistics, respectively. However, if i is equated torero, the result corresponds to the Maxwell -Boltzmann distribution. In many cases the last is a good approximation to quantum systems, which is furthermore, a correct description of classical ones - those in which the energy levels fotm a continuum. From these results the partition functions can be calculated, leading to expressions for the various thermodynamic functions for a given system. In many cases these values, as obtained from spectroscopic observations, are more accurate than those obtained by direct thermodynamic measurements. 

Specifically, following the rate expression of QTST in Eq. (4-1) and assuming the quantum transmission coefficients the dynamic frequency factors are the same, the kinetic isotope effect between two isopotic reactions L and H is rewritten in terms of the ratio of the partial partition functions at the centroid reactant and transition state... 

This expression reflects the quantum-mechanical version of equipartition. Substituting the partition functions for different modes into (6) the classical equipartition principle can readily be recovered. 

The most accurate theories of reaction rates come from statistical mechanics. These theories allow one to write the partition function for molecules and thus to formulate a quantitative description of rates. Rate expressions for many homogeneous elementary reaction steps come from these calculations, which use quantum mechanics to calculate the energy levels of molecules and potential energy surfaces over which molecules travel in the transition between reactants and products. These theories give... 

The quantum version of the partition function is obtained by replacing the phase space integral and the classical Boltzmann distribution with the trace operation of the quantum Boltzmann operator, giving the usual expression... 

The Boltzmann distribution of the populations of a collection of molecules at some temperature T was discussed in Section 8.3.2. This distribution, given by Eq. 8.46 or 8.88, was expressed in terms of the quantum mechanical energy levels and the partition function for a particular type of motion, for instance, translational, vibrational, or rotational motion. It is useful to express such population distributions in other forms, particularly to obtain an expression for the distribution of velocities. The velocity distribution function basically determines the (translational) energy available for overcoming a reaction barrier. It also determines the frequency of collisions, which directly contributes to the rate constant k. 

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. 

Note that this expression is identical to the approximate form of the quantum mechanical partition function in Eq. (A.14). The energy distribution in Eq. (A.36) becomes... 

Applications in statistical mechanics are based on constructing expressions for Q(N, V, E) (and other partition functions for various ensembles) based on the nature of the interactions of the particles in a given system. To understand how thermodynamic principles arise from statistics, however, it is not necessary to worry about how one might go about computing Q(N, V, E), or how Q might depend on N, V, and E for particular systems (classical or quantum mechanical). It is necessary simply to appreciate that the quantity Q(N, V, E) exists for an NVE system. 

The above formula for Z, the NPT partition function, was first reported by Guggenheim [74], who wrote the expression down by analogy rather than based on a detailed derivation. While this form of the partition function is thought to be broadly valid and is widely applied (for example in molecular dynamics simulation [6]), it introduces the conceptual difficulty that the meaning of the discrete volumes Vi is not clear. Discrete energy states arise naturally from quantum statistics. Yet it is not necessarily obvious what discrete volumes to sum over in Equation (12.50). In fact for most applications it makes sense to replace the discrete sum with a continuous volume integral. Yet doing so results in a partition function that has units of volume, which is inappropriate for a partition function that formally should be unitless. 

Analytic expressions are available for both the quantum harmonic vibrator partition function Q ib(T) and the classical rigid rotor partition function Q ot(T) [40] further simplifying the analysis. 

Numerical evaluation of thermodynamic projserties using either the full quantum expressions for the partition function and the density matrix or the... 

The basic problem for Monte Carlo simulations of quantum systems is that the partition function is no longer a simple sum over classical configurations as in (16) but an operator expression... 

If the expression for the translational partition function is inserted into equation (16.8), it is readily found, since tt, m, fc, h and V are all constant, that the translational contribution Et to the energy, in excess of the zero-point value, is equal to %RT per mole, which is precisely the classical value. The corresponding molar heat capacity at constant volume is thus f P. As stated earlier, therefore, translational energy may be treated as essentially classical in behavior, since the quantum theory leads to the s ame results as does the classical treatment. Nevertheless, the partition function derived above [[equation (16.16) [] is of the greatest importance in connection with other thermodynamic properties, as w ill be seen in Chapter IX. 

The statistical weight factor of each electronic level, normal or excited, is equal to 2j + 1, where 7 is the so-called resultant quantum number of the atom in the given state. The expression for the electronic factor in the partition function is then given by... 

However, as expressions for both partition functions have been derived quantum mechanically, they do not permit us per se to calculate equilibriiun properties of thermal sj stems because quantum (as well as classic) mechanics as such does not have any concept of temperature. However, an inspection of Eqs. (2.37) and (2.38) reveals that botli partition functions still contain one or more yet-to-be-determined Lagiangian multipliers. These need to be calculated in a way that the resulting expressions become consistent with thermodynamics as it was introduced in Chapter 1. 

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