Classical Statistical Mechanics

The derivation of the partition function that we have discussed is based on a quantum mechanical description of molecular energy levels (electronic, vibrational, rotational, and translational). We have seen that thermodynamic properties for an ideal gas can calculated from the partition function. To use a partition function for real gases, liquids and solids we require a means to include the intermolecular potential in the description of the system. While this is possible from a quantum mechanical method, the trajectory determined in a computer simulation is based on a classical force field. It is not practical to calculate the dynamics of large systems using quantum mechanics. Here we explore the derivation of a classical partition function and show how it can be used to determine thermodynamic properties.

The molecular partition function we have seen previously is

This is a sum-over-states partition function. It has the form of a sum over exp{-b(energy)} where the energy levels represent all quantum states. Since the corresponding classical energy function is continuous we require an integral over all space instead of a sum. The classical hamiltonian is expressed in terms of the momenta p_i and positions q_i of the particles in the system. Using the classical description we can assume that

The differential dpdq denotes all of the coordinates on which H depends. That is dpdq = dp₁dp₂…dp_sdq₁dq₂…dq_s where s is the number of positions or momenta needed to specify the motion and position of the particle. We show below explicitly that s=3 for translation, s=2 for rotation, and s=1 vibrational modes. The ps and qs are general positions and momenta and can be Cartesian coordinates, polar coordinates or internal coordinates (e.g. for vibrations they can also be normal coordinates).

Translation

For an ideal monatomic gas the hamiltonian for translational motion is H = 1/2m(p_xj² + p_yj² + p_zj²) + U(x_j,y_j,z_j) summed over all j particles in the system.

For an ideal gas the potential energy is zero and we can neglect the second term in the above hamiltonian. If we substitute this kinetic energy term into the integral above we have

which can be written as

The first integral is the the volume, V and thus we have

We recognize that this result differs from the quantum mechanical result by a factor of h³.

Rotation

The hamiltonian for a rigid rotor is

where I is the moment of inertia of the molecule. According to the classical partition function above we have

In the following we show how this integral is evaluated.

The hamiltonian H contains no f spatial dependence. Therefore the f integral results in a factor 2p. This results in two Gaussian integrals.

Performing all but the integration over q have

We factor out the constant and get

The quantum mechanical version of the rotational partition function has an extra factor of 1/h².

Vibration

For the classical harmonic oscillator

When this hamiltonian is substituted into the classical partition function we find

These are both Gaussian integrals and so we have

where the denominator is equal to angular frequency, w.

We can write

The quantum mechanical version of the vibrational partition function has an extra factor of h in the denominator (in the high temperature limit). Including this factor the vibrational partition function can be expressed in the ways shown below in the high temperature limit.

The classical partition function

Our assumption based on these comparisons will be that we can express the classical molecular partition function as

In general we can express the hamiltonian as

H(p,q) = T(p) + U(q)

Where T is the kinetic energy we have considered above and U is the potential energy. In the above representation the kinetic energy is expressed as a function of the momenta only and the intermolecular potential is expressed as a function of positions only. Expressing the system partition function in these general terms we have

We have made use of the notation

The factor s is the dimensionality of the integral over positions and momenta. This dimensionality is 3 for translational motion, 2 for rotational motion, and 1 for vibrational motion. The idea that the dimensionality is 1 for vibrational motion must be discussed in the context of polyatomic molecules where there is more than one vibrational mode. Since the dimensionality is 1 for a diatomic where the nuclei move only along the internuclear vector, it turns our that each vibrational mode of a polyatomic molecule will have a dimensionality of 1 if the coordinate system is the normal coordinate system. In the normal coordinate system all of the atoms move in phase for a particular vibrational mode. Therefore, they can be expressed as a single coordinate even though their motions have different amplitudes in Cartesian coordinate space. This concept is discussed further when we discuss the normal coordinate system of polyatomic molecules.

For a monatomic gas, for example, the hamiltonian is

If we substitute this hamiltonian into the above equation for the system partition function we can do the momentum integrals. The result is

where

is called the classical configuration integral. This integral is a 3N dimensional integral over spatial coordinates with an exponential function of the N-body potential. The classical approach to the partition function shows us how the intermolecular potential contributes to the partition function. The configuration integral Z_N is not soluble analytically. This difficulty is a major motivation for the development of approximate techniques and molecular dynamics.

There are a few points worth making concerning the above equations.

If the potential energy is zero U(x₁,y₁,z₁,…,z_N) = 0 and e^-U/kT = e⁰ = 1. For an ideal gas the integral becomes

and

There is no potential for translation or rotation. However, there is an intramolecular potential for vibrational modes. The quantum mechanical origin of the intermolecular potential is the bonding interactions between molecules. This appears in the derivation of q_vib but does not contribute to U as described in the configuration integral.

The expression for the intermolecular potential in Z_N is a N body potential where N is the number of molecules in the system. As we have seen, practical calculation of the potential involves expanding the potential in a series of two-body, three-body, etc. potentials. As we have seen the intermolecular potential can be expressed as a sum of Lennard-Jones and Coulomb terms. For any pair of molecules separated by a distance R we can write the two-body potential as U(R) = 4e[(s/R)¹² - (s/R)⁶] + q₁q₂/4pe₀R.

This is form of the potential used in molecular dynamics simulations. The two-body potential can also be used in conjunction with the radial distribution function as an approximate method for the calculation of the configuration integral.

Calculation of average properties

The average energy is

Where U is the average potential energy given by

The average pressure is given by

In general this must be solved by an approximate technique. We shall show in this course that the radial distribution function can provide an approximate method for the evaluation of the pressure equation. Here we simply note that for an ideal gas Z_N = V^N and the equation becomes the ideal gas equation of state P = NkT/V.