Partition function quantum mechanical

It is now necessary to examine the partition function in more detail. The energy states for translation are assumed to be given by the quantum-mechanical picture of a particle in a box. For a one-dimensional box of length a. 

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. 

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 original idea of approximating the quantum mechanical partition function by a classical one belongs to Feynman [Feynman and Vernon 1963 Feynman and Kleinert 1986]. Expanding an arbitrary /S-periodic orbit, entering into the partition-function path integral, in a Fourier series in Matsubara frequencies v . 

The partition functions vib. and for use in Eq. (5-23) can be evaluated by quantum mechanical arguments. We will subsequently require vib. which is given by Eq. (5-25), where v is the vibrational frequency. 

Next, Ah and Ad are written in terms of partition functions (see Section 5.2), which are in principle calculable from quantum mechanical results together with experimental vibrational frequencies. The application of this approach to mechanistic problems involves postulating alternative models of the transition state, estimating the appropriate molecular properties of the hypothetical transition state species, and calculating the corresponding k lko values for comparison with experiment.""- " "P... 

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. 

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 order to obtain the partition function for systems of this type (where the thermal energy and potential barrier are of the same magnitude), it is necessary to have the quantum mechanical energy levels associated with the barrier. Pitzer5 has used a potential of the form... 

The model [39] was developed using three assumptions the conformers are in thermodynamic equilibrium, the peak intensities of the T-shaped and linear features are proportional to the populations of the T-shaped and linear ground-state conformers, and the internal energy of the complexes is adequately represented by the monomer rotational temperature. By using these assumptions, the temperature dependence of the ratio of the intensities of the features were equated to the ratio of the quantum mechanical partition functions for the T-shaped and linear conformers (Eq. (7) of Ref. [39]). The ratio of the He l Cl T-shaped linear intensity ratios were observed to decay single exponentially. Fits of the decays yielded an approximate ground-state binding... 

This is the translational partition function for any particle of mass m, moving over a line of length I, in one dimension. Please note that this result is exactly the same as that we calculated from quantum mechanics for a particle in a one-dimensional box. For a particle moving over an area A on a surface, the partition function of translation is... 

Finally, the rotational partition function of a diatomic molecule follows from the quantum mechanical energy level scheme ... 

Most students are introduced to quantum mechanics with the study of the famous problem of the particle in a box. While this problem is introduced primarily for pedagogical reasons, it has nevertheless some important applications. In particular, it is the basis for the derivation of the translational partition function for a gas (Section 10.8.1) and is employed as a model for certain problems in solid-state physics. 

A first step toward quantum mechanical approximations for free energy calculations was made by Wigner and Kirkwood. A clear derivation of their method is given by Landau and Lifshitz [43]. They employ a plane-wave expansion to compute approximate canonical partition functions which then generate free energy models. The method produces an expansion of the free energy in powers of h. Here we just quote several of the results of their derivation. 

The partition function is the extension of the Boltzmann distribution to a complete set of allowed energies. Because of their different quantum-mechanical structures, different energy modes have different partition functions. 

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. 

Only for a special class of compound with appropriate planar symmetry is it possible to distinguish between (a) electrons, associated with atomic cores and (7r) electrons delocalized over the molecular surface. The Hiickel approximation is allowed for this limited class only. Since a — 7r separation is nowhere perfect and always somewhat artificial, there is the temptation to extend the Hiickel method also to situations where more pronounced a — ix interaction is expected. It is immediately obvious that a different partitioning would be required for such an extension. The standard HMO partitioning that operates on symmetry grounds, treats only the 7r-electrons quantum mechanically and all a-electrons as part of the classical molecular frame. The alternative is an arbitrary distinction between valence electrons and atomic cores. Schemes have been devised [98, 99] to handle situations where the molecular valence shell consists of either a + n or only a electrons. In either case, the partitioning introduces extra complications. The mathematics of the situation [100] dictates that any abstraction produce disjoint sectors, of which no more than one may be non-classical. In view if the BO approximation already invoked, only the valence sector could be quantum mechanical9. In this case the classical remainder is a set of atomic cores in some unspecified excited state, called the valence state. One complication that arises is that wave functions of the valence electrons depend parametrically on the valence state. 

Fig. 6.1 An aspartate amino acid partitioned into quantum and classical (MM) regions. The functional group of the side chain, involved in the chemical reaction, lies within the quantum region and the backbone atoms are treated by using a molecular mechanics force field.

Finally some other developments are briefly mentioned for the sake of completeness. These include the work of Sposito and Babcock 184> which bears close resemblance to the partitioned potential methods. Their idea is to solve the quantum mechanical energy spectrum of the complex using only the empirical potential function as the potential operator in the Hamiltonian. This spectrum then leads to calculations of temperature-dependent energies of formation. 

The second product is over the 3N—6(3N—5) normal mode frequencies of the ideal gas harmonic molecule to which Equation 4.78 applies. Thus the product over vibrations Equation 4.90 is indeed the quantum mechanical contribution to the molecular partition function for the ideal gas. 

The result (Equation 4.90) could have been derived more simply. It has been emphasized that the quantum mechanical contribution to the partition function ratio arises from the quantization of vibrational energy levels. For the molecular translations and rotations quantization has been ignored because the spacing of translational and rotational energy levels is so close as to be essentially continuous (As/kT 1). 

From the quantum mechanical standpoint the appearance of the factor 1/2 = 1/s for the diatomic case means the configurations generated by a rotation of 180° are identical, so the number of distinguishable states is only one-half the classical total. Thus the classical value of the partition function must be divided by the symmetry number which is 1 for a heteronuclear diatomic and 2 for a homonuclear diatomic molecule. 

Equation 4.117 makes complete sense. One of the first things one learns in dealing with phase space integrals is to be careful and not over-count the phase space volume as has already been repeatedly pointed out. In quantum mechanics equivalent particles are indistinguishable. The factor n ni is exactly the number of indistinguishable permutations, while A accounts for multiple minima in the BO surface. It is proper that this factor be included in the symmetry number. Since the BO potential energy surface is independent of isotopic substitution it follows that A is also independent of isotope substitution and cannot affect the isotopic partition function ratio. From Equation 4.116 it follows... 

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