The Partition Function

The derivation of the transition state theory expression for the rate constant requires some ideas from statistical mechanics, so we will develop these in a digression. Consider an assembly of molecules of a given substance at constant temperature T and volume V. The total number N of molecules is distributed among the allowed quantum states of the system, which are determined by T, V, and the molecular structure. Let , be the number of molecules in state i having energy e,- per molecule. Then , is related to e, by Eq. (5-17), which is known as theBoltzmann distribution. 

Let us now ask what fraction of the total number of molecules is in the quantum state having energy Ej. We define this fraction by 

The fraction fj may also be interpreted as the probability that a molecule will be in the state having energy Ej. 

The denominator in Eq. (5-18) is extremely important, because it represents the distribution of molecules over all of the states available to them. We therefore distinguish it with the symbol Q and the name partition function. 

The term partition function conveys the idea of distribution over states the German word is Zustandsumme, sum of states. From the above relationships, we have N = KQ and p. = ArT In (N/Q). 

The partition function turns out to be a very useful quantity in our calculations, and it is important that we understand its properties. As we said earlier, the name comes from the realization that r is a measure of the distribution of energy among excited states, as can be seen by writing r 

As we discussed earlier, at low temperatures, kT et. ct.. .., each e,/kT is large, and all exp( - e,jkT) terms are negligibly small, so that r = go. With increasing temperature, the exponential terms become larger and r increases. In the limit of 

We can write an expression for Z ,b, the partition function for the combination of these two harmonic oscillators. It must take into account all of the possible 

Since a and b are identical, we can drop the subscripts and write that for two identical, but distinguishable, harmonic oscillators 

We can generalize this expression to write that for N identical units Z = -A . 

The Partition Function. — The basis for all the calculations is the evaluation of the partition function. For an assembly of N non-localized 

Partington, An advanced Treatise on Physical Chemistry , Longmans, Green, London, 1949, vol. 1. 

All the thermodynamic functions can be expressed in terms of the partition function and written in terms of q. They are (where L is Avogadro s constant) 

The only completely general method of evaluating the partition function is by summation of the separately calculated terms in equation (2). If q cannot be expressed in a closed form, then for numerical evaluation of the thermodynamic functions equations (3), (7), and (9) are most convenient they contain sums related to q and defined by 

The orientational distribution function of rods with long axis a is designated by f(a). pf(a)dQa is the number of rods which point to the solid angle dQ 

First of all, we examine the partition function Z — an important function in thermodynamics and statistics, and calculate the free energy of the system according to the formula 

The total energy of the system, U, is the sum of the kinetic energy of each rigid rod T (translation and rotation) and potential energy V). 

The rotational kinetic energy of a rigid rod is more complicated and we adopt the form from the literature as follows 

The potential energy of rigid rods is not only a function of the relative distance (r ) of the mass centers of two rigid rods but is also associated with the orientations of two molecules (a and aj). It is expressed as 

As an example, evaluate the molecular translational partition function per unit volume for Ar atoms at 1000 K. The mass of one Ar atom is 6.634 x 10-26 kg. So the translational partition function per unit volume is 

To calculate the molecular rotational partition function for an asymmetric, linear molecule, we use Eq. 8.16 for the energy level of rotational state /, and Eq. 8.18 for its degeneracy. As discussed in Section 8.2, rotational energy levels are very closely spaced compared to k/jT unless the molecule s moment of inertia is very small. Therefore, for most molecules, replacing the summation in Eq. 8.50 with an integral introduces little error. Thus the 

This integral is easily found by noting that d [ j (j + 1)] = (2y + Wj - 

We noted in Section 8.2 that only half the values of j are allowed for homonuclear diatomics or symmetric linear polyatomic molecules (only the even-y states or only the odd- y states, depending on the nuclear symmetries of the atoms). The evaluation of qmt would be the same as above, except that only half of the j s contribute. The result of the integration is exactly half the value in Eq. 8.64. Thus a general formula for the rotational partition function for a linear molecule is 

Consider the molecular rotational partition function for the CO molecule, a linear diatomic molecule. The moment of inertia of CO is / = 1.4498 x 10-46 kg-m2, and its rotational symmetry number is a = 1. Thus, evaluating Eq. 8.65 at T = 300 K, we find the rotational partition function to be 

It follows from Example 16.3 that if two systems of constant mass and volume are in equilibrium with a heat reservoir, the probability of any of their quantum states is an exponential function of its energy  

Cj and Cj are constants characteristic of the first and second system respectively b is also a constant, but the same for both systems, if Eq. 16.6.2b is to be satisfied. In addition b must be negative, otherwise the probability P, would become infinite for infinite values of the energy of the q.s. Finally, we can speculate that since both systems are in equilibrium with the heat reservoir, b must be related to the temperature of it. This is indeed demonstrated by Denbigh (1981, p.350), who shows that  

Returning now to the system under consideration - and using C instead of Cl - we note that since EP,- = 1.0  

The probability, thus, of quantum state i with energy Ej is given by  

The summation is taken over all the available quantum states, and Q is called the partition function (p.f.) of the system at constant volume and temperature. 

This is one of the most important parameters in statistical mechanics since it is directly related to the thermodynamic properties of a system. The summation in (6.10) is made over all energy states, so z is a function of the partitioning of all particles among all energies for the equilibrium configuration. 

To simplify matters, let us say that there is only one unique ground state with a degeneracy, go = 1. Then 

Because the partition function is related to the number of particles occupying energy levels above the ground state, it can be used to calculate the average internal energy, e, of a particle. From equations (6.6) and (6.7) the average energy is 

Substituting (6.10) into (6.9) gives the more common form of the Boltzmann 

The constant volume restriction is imposed because the dependence of i on volume is not taken into account in this formulation. 

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Since translational and internal energy (of rotation and vibration) are independent, the partition function for the gas can be written... 

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. 

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... 

Finally, the generalization of the partition function q m transition state theory (equation (A3.4.96)) is given by... 

These equations lead to fomis for the thermal rate constants that are perfectly similar to transition state theory, although the computations of the partition functions are different in detail. As described in figrne A3.4.7 various levels of the theory can be derived by successive approximations in this general state-selected fomr of the transition state theory in the framework of the statistical adiabatic chaimel model. We refer to the literature cited in the diagram for details. 

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. 

All molecules in the second and subsequent layers are assumed to behave similarly to a liquid, in particular to have the same partition fimction. This is assumed to be different to the partition function (A2.2) of molecules adsorbed into the first layer. 

In the present study we try to obtain the isotherm equation in the form of a sum of the three terms Langmuir s, Henry s and multilayer adsorption, because it is the most convenient and is easily physically interpreted but, using more a realistic assumption. Namely, we take the partition functions as in the case of the isotherm of d Arcy and Watt [20], but assume that the value of V for the multilayer adsorption appearing in the (5) is equal to the sum of the number of adsorbed water molecules on the Langmuir s and Henry s sites ... 

Note that there is not a unique separation of the partition function as Zq = trans vib jjowever, using the result for the ideal gas translational partition function... 

This definition is based on and proportional to the g-expectation value. However, it is more useful since it is not necessary to evaluate the partition function to compute an average. 

Mciny of the theories used in molecular modelling involve multiple integrals. Examples include tire two-electron integrals formd in Hartree-Fock theory, and the integral over the piriitii >ns and momenta used to define the partition function, Q. In fact, most of the multiple integrals that have to be evaluated are double integrals. 

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. 

The thermodynamic properties that we have considered so far, such as the internal energy, the pressure and the heat capacity are collectively known as the mechanical properties and can be routinely obtained from a Monte Carlo or molecular dynamics simulation. Other thermodynamic properties are difficult to determine accurately without resorting to special techniques. These are the so-called entropic or thermal properties the free energy, the chemical potential and the entropy itself. The difference between the mechanical emd thermal properties is that the mechanical properties are related to the derivative of the partition function whereas the thermal properties are directly related to the partition function itself. To illustrate the difference between these two classes of properties, let us consider the internal energy, U, and the Fielmholtz free energy, A. These are related to the partition function by ... 

LS now consider the problem of calculating the Helmholtz free energy of a molecular 1. Our aim is to express the free energy in the same functional form as the internal that is as an integral which incorporates the probability of a given state. First, we itute for the partition function in Equation (6.21) ... 

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. 

For translational, rotational and vibrational motion the partition function Ccin be calculated using standard results obtained by solving the Schrodinger equation ... 

IS thermodynamic properties can be calculated from the partition function. Here we j state some of the most common ... 

By combining Equations (8.4) and (8.6) we can see that the partition function for a re system has a contribution due to ideal gas behaviour (the momenta) and a contributii due to the interactions between the particles. Any deviations from ideal gas behaviour a due to interactions within the system as a consequence of these interactions. This enabl us to write the partition function as ... 

A consequence of writing the partition function as a product of a real gas and an ideal g part is that thermod)mamic properties can be written in terms of an ideal gas value and excess value. The ideal gas contributions can be determined analytically by integrating o the momenta. For example, the Helmholtz free energy is related to the canonical partitii function by ... 

To calculate the partition function for a system of N atoms using this simple Monte Car integration method would involve the following steps ... 

The difference between an MM calculation of the enthalpy of formation and a bond energy scheme comes in the steric energy, which was shown in Eile 4-3. The sum of compression, bending, etc. energies is the steric energy, E = 2.60 kcal mol in Eile 4-3. This is added to BE, as is the partition function energy contribution (see below), PCE = 2.40 kcal moP, to yield... 

Another way of formulating this problem is to use derivatives of the partition function without a weight function. This is done with the following relationships ... 

The quantities and Q that appear in equation 48 are approximations for the complete partition function. For highest accuracy, above about 9000 K, the partition functions should be used. 

Whereas this two-parameter equation states the same conclusion as the van der Waals equation, this derivation extends the theory beyond just PVT behavior. Because the partition function, can also be used to derive aH the thermodynamic functions, the functional form, E, can be changed to describe this data as weH. Corresponding states equations are typicaHy written with respect to temperature and pressure because of the ambiguities of measuring volume at the critical point. 

The total partition function may be approximated to the product of the partition function for each contribution to the heat capacity, that from the translational energy for atomic species, and translation plus rotation plus vibration for the diatomic and more complex species. Defining the partition function, PF, tlrrough the equation... 

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