# A Statistical mechanics

# A Statistical mechanics

The main objective of *statistical mechanics* is to calculate macroscopic (thermodynamic) properties from a knowledge of microscopic information like quantum mechanical energy levels. The purpose of the present appendix is merely to present a selection^{1} of the results that are most relevant in the context of reaction dynamics.

Consider a macroscopic system with *N* particles, a volume given by *V*, and an energy given by *E*. This is the so-called *microcanonical* system. There will be an enormous number of quantum states consistent with the fixed macroscopic properties *N, V*, and *E*. The energy *E* must be one of the eigenvalues of the *N*-body Hamiltonian *H* _{N}, and the number of states of energy *E* is equal to the degeneracy, which we denote by Ω(*N, V, E*). This number is also called the microcanonical *partition function*. A basic postulate of statistical mechanics is that all states at the same energy are equally probable (the assumption of *equal a priori probabilities*). That is, the probability of finding the system in any of the states is *P* = 1/Ω(*N, V, E*).

For most practical applications, we consider not isolated systems but systems in which the temperature is fixed. This is the *canonical* system consistent with the fixed macroscopic properties (*N, V, T*), where *T* is the absolute temperature.

We consider a canonical system of *N interacting* particles described by the *N*-body Hamiltonian *H* _{N}. In such a system the energy is not fixed. We shall not give the detailed derivation here but merely state that the probability that the system is found in the *i*th energy state ε_{i}, given as an eigenvalue of the *N*-body Hamiltonian, is

_{i}is given by

*P*(ε

_{i}) = Ω(

*N, V, ε*

_{i})

*P*

_{i}.

# (p.292) A.1 A system of non-interacting molecules

Now, at sufficiently high temperatures, for *N non-interacting* identical and *indistinguishable* particles (e.g., an ideal gas), the partition function can be written in the form

*Q*is the partition function of the individual particles. In the following it will be referred to as the molecular partition function. When Eq. (A.3) is valid, the molecules are said to obey Boltzmann statistics. All results that are given below fall within this ‘ideal gas’ limit (except when it is explicitly stated that the equation is valid for interacting molecules).

The probability of finding a molecule in the energy level *E* _{i} is

*Q*is the molecular partition function,

*E*

_{i}and ω

_{i}are the energy and the degeneracy (the microcanonical partition function) of the

*i*th quantum level. Note that the summation here runs over all the energy levels. The probability distribution in Eq. (A.4) is referred to as the

*Boltzmann distribution*. If the energy is continuous (e.g., the free particle), then the probability that a molecule has energy in the energy interval

*E*→

*E*+

*dE*is

*N*(

*E*)

*dE*is the number of states in the energy interval

*E → E*+

*dE*(the microcanonical partition function). Thus,

*N*(

*E*) is the density of states.

The Boltzmann distribution is illustrated in Fig. 1.2.1 to 1.2.3 of Chapter 1.

## A.1.1 The molecular partition function

The evaluation of the molecular partition function can be simplified by noting that the total energy of the molecule may be written as a sum of the center-of-mass translational energy and the internal energy, *E* = *E* _{trans} + *E* _{int}, which implies

*E*

_{int}∼

*E*

_{vib}+

*E*

_{rot}+

*E*

_{elec}, the partition function for the internal degrees of freedom can be written in the form

These partition functions can be evaluated quite readily. In the following it turns out to be convenient to choose the zero of energy such that it coincides with the zero-point level of the quantized energy levels. To that end, we note that, if we subtract
(p.293)
the zero-point energy *e* _{0} from all energy levels, the partition function Eq. (A.5) takes the form

Here the sum is the partition function with the energy measured relative to the zero-point level and *Q* is obtained after multiplication by exp(*–E* _{0}/*k* _{B} *T*). The Boltzmann distribution Eq. (A.4) can be written in the form

Thus, the partition functions differ by the factor exp(*–E* _{0}/*k* _{B} *T*), whereas the Boltzmann distribution is invariant to such a shift of the energy scale in the standard expressions for the energy levels.

For the *translational partition function*, we first consider a particle in a one-dimensional box of length *l*. The energy levels, with the zero of energy as the zero-point level, are *E* _{n} = (*n* ^{2} – 1)*h* ^{2}/(8*ml* ^{2}), with *n =* 1,2,…, and degeneracy ω_{n} = 1. The partition function takes the form

This sum cannot be evaluated analytically. However, when the energy difference between subsequent levels can be considered as small, then the sum can be replaced by an integral. Thus, when *E* _{n+1} *– E* _{n} *=* (2*n* + 1)*h* ^{2}/(8*ml* ^{2}) ≪ *k* _{B} *T*, that is, at high temperatures,^{2} we have

The energy levels for a particle in a three-dimensional box are given as the sum of the energies for each dimension, and the partition function for the three-dimensional box is simply a product of the partition functions for each dimension; that is,

*V = l*

^{3}is the volume of the box.

In order to evaluate the *vibrational partition function* we consider a single harmonic oscillator. The energy levels, with the zero of energy as the zero-point level, are *E* _{n} *= hν* _{s} *n*, with *n =* 0,1,…, and degeneracy ω_{n} = 1. The partition function takes the form
(p.294)

*x*+

*x*

^{2}+ … = (1–

*x*)–

^{1}. For a set of

*s*harmonic oscillators (e.g., normal modes, see Appendix E), the total energy is the sum of the energies for each oscillator and the partition function becomes, accordingly, a product of partition functions of the form given in Eq. (A.15); that is,

For the *rotational partition function*, we first consider a linear rigid rotor. The energy levels are *E* _{J} *= J*(*J* + *1)ħ* ^{2}/(2*I*), with *J* = 0,1, …, and *I* is the moment of inertia. Each energy level has a degeneracy of *m* _{J} = 2*J* + 1. The partition function takes the form

This sum cannot be evaluated analytically. However, when the energy difference between subsequent levels can be considered as small (*E* _{J+1} – *E* _{J} *= ħ* ^{2}(*J*+*1*)/*I* ≪ *k* _{B} *T*, that is, at high temperatures) then the sum can be replaced by an integral:

The last integral is easily evaluated, and

This is the correct expression for the rotational partition function of a heteronuclear diatomic molecule. For a homonuclear diatomic molecule, however, it must be taken into account that the total wave function must be either symmetric or antisymmetric under the interchange of the two identical nuclei: symmetric if the nuclei have integral spins or antisymmetric if they have half-integral spins. The effect on *Q* _{rot} is that it should be replaced by *Q* _{rot}/σ, where σ is a *symmetry number* that represents the number of indistinguishable orientations that the molecule can have (i.e., the number of ways the molecule can be rotated ‘into itself). Thus, *Q* _{rot} in Eq. (A.19) should be replaced by *Q* _{rot}/σ, where *σ =* 1 for a heteronuclear diatomic molecule and *σ =* 2
(p.295)
for a homonuclear diatomic molecule. When Eq. (A.19) is extended to a non-linear molecule with moments of inertia *I* _{a}, *I* _{b}, and *I* _{c} about its principal axes, the rotational partition function takes the form

Finally, the *electronic partition function* is considered. The zero of energy is chosen as the electronic ground-state energy. The spacings between the electronic energy levels are, normally, large and only the first term in the partition function will make a significant contribution; that is,

Thus, the partition function is simply the degeneracy of the electronic ground state.

## A.1.2 Macroscopic properties

When we know the partition function, we can calculate thermodynamic quantities from a knowledge of the quantum mechanical energy levels. Consider, as an example, the energy *E*. A basic postulate of statistical mechanics is that such an energy is the average value of all the quantum mechanical energy levels with the weights given by the Boltzmann distribution; that is,

In a similar manner, the equilibrium constant of a chemical reaction can be related to the quantum mechanical energy levels of the reactants and products. Consider, as an example, a mixture of A and B molecules in equilibrium:

The equilibrium constant is

According to the Boltzmann distribution, Eq. (A.4), the probability of finding an A molecule in the energy level *E* _{a} is

*n*

_{a}is the number of A molecules in the energy level

*E*

_{a},

*N =*

*n*

_{ A }+

*N*

_{ B }is the total number of molecules, and

*Q*is the total partition function. Similarly,

*E*

_{0}is the difference between the zero-point levels of the products and the reactants. Thus,

*E*

_{b}denotes the energy levels of molecule B measured relative to the zero-point level of the molecule. Note that $Q={\displaystyle {\sum}_{a}{\omega}_{a}\mathrm{exp}(-{E}_{a}/{k}_{B}T)+{\displaystyle {\sum}_{b}{\omega}_{b}\mathrm{exp}(-{{E}^{\prime}}_{b}/{k}_{B}T)}}$. Now,

For a general equilibrium of the form

*νs*are stoichiometric coefficients, the result is

*Q*

_{A}is the partition function for molecule A,

*E*

_{0}is the difference between the zero-point levels of the products and the reactants, and the partition functions are evaluated such that the zero of energy is the zero-point level.

# (p.297) A.2 Classical statistical mechanics

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 *k* _{B} *T*.

The state of a classical system can be completely described by coordinates and momenta, i.e., a point in phase space; see Fig. 1.2.1. For a system with *s* degrees of freedom (that is, *s* coordinates are required to completely describe its position), the phase space has the dimension 2*s*. When the system evolves in time, its dynamics is described by the trajectory of the phase-space point through phase space. The trajectory is given by Hamilton’s equations of motion, Eq. (4.63), and the total energy for the molecule is given by the classical Hamiltonian *H*. The summations over states (or energy levels) in quantum statistical mechanics are replaced by integrations over regions of phase space.

The problem now is how to count the number of states in this continuous description. For proper counting, the use of quantum mechanics (Planck’s constant) is unavoidable. For a system with one degree of freedom, the number of states in the area element *dqdp* is

*h*contains one state. The

*sum of states G*(

*E*), that is, the total number of states having energy in the range from 0 to

*E*, is then the total phase-space area occupied at the energy

*E*divided by

*h*. For a multidimensional system with

*s*degrees of freedom, a phase-space cell of volume

*h*

^{s}contains one state (see Fig. 1.2.1), and the sum of states is then given by

*H*(

*q*

_{1},

*…, q*

_{s},

*p*

_{1},

*…, p*

_{s}) lies between 0 and

*E*. The

*density of states N*(

*E*) is given by

*G*(

*E*+

*dE*)

*– G*(

*E*) is a multidimensional volume of phase space enclosed by the two hypersurfaces defined by

*H*=

*E*and

*H*=

*E*+

*dE*, and this integral can be expressed as the area of the hypersurface for

*H*=

*E*multiplied by

*dE*. The condition that the range of integration is restricted to this hypersurface is in the last integral expressed in terms of a delta function. Note that the definition in Eq. (A.34) implies

The Boltzmann statistics of particles described by classical mechanics is obtained from Eqs (A.6) and (A.34):

The expectation value of the energy in Eq. (A.22) is given by

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

*H*

_{N}(

*q, p*) is the classical

*N*-body Hamiltonian with each molecule described by

*s*coordinates in configuration space. The probability of finding the system at the point

*q*

_{1},

*…, q*

_{sN},

*p*

_{1},

*…, p*

_{sN}in a volume element

*dq*

_{1}

*· · · d*

_{qsN}

*dp*

_{1}

*· · · d*

_{psN}is given by

# Further reading/references

Bibliography references:

D.A. McQuarrie, *Statistical mechanics* (University Science Books, 2000)

## Notes:

(1) For a detailed discussion of this subject see, e.g., D.A. McQuarrie, Statistical mechanics (University Science Books, 2000).

(2) The inequality cannot hold as n increases; but, by the time n is large enough to contradict this, the terms are so small that they make no contribution to the sum