The polyatomic ideal gas

We have discussed separation of variables into center-of-mass coordinates and relative coordinates. The same procedure applies to polyatomic ideal gases as to diatomic ideal gases. The translation partition function for polyatomic ideal gases has the same exact form as that for diatomic ideal gas or the monatomic ideal gas.

The rigid-rotator approximation can be applied as was the case for the diatomic ideal gas. For linear molecules there are two axes for rotation and two degrees of freedom associated with rotation. For non-linear molecules there are the axes and the moments of inertia. In a polyatomic molecule there is more than one vibrational mode. Thus, the number of degrees of freedom needs to be considered.

For a molecule with n atoms there will be 3n degrees of freedom. These can be attributed as follows:

Treatment of the vibrational potential

We can expand the vibrational potential energy function for a polyatomic molecule in a Taylor's series as was done for the diatomic molecule. However, there will be 3n coordinates.

The harmonic approximation makes the assumption that terms higher than second order are zero. The first derivative term is zero at the minimum of the potential energy surface.

Internal coordinates

We have treated the motion of atomic masses thus far using Cartesian coordinates. It is convenient to transform from Cartesian coordinates to internal coordinates. Internal coordinates consist of bond stretching, valence angle bending, out-of-plane wagging and torsions. These have the following forms

The transformation from Cartesian to internal coordinates is convenient because we often find it easier to define the force constants for stretches and bends than to try and determine their value in Cartesian space.

The parameters in typical forcefields are also given in terms of the internal coordinates specified above. These comprise the intramolecular portion of the forcefield.

Normal coordinates

In the normal coordinate system each vibrational frequency is associated with a collective motion of the atoms in the molecule. Each normal coordinate is orthogonal to all of the others. In a normal mode of vibration the atoms all move with the same frequency and phase, however, the amplitudes and directions of their motions differ. We can exemplify this with water. Since there are 3 atoms in water and it is a non-linear molecule, there are 3 vibrational degrees of freedom. These result in three normal modes of vibration shown below.

The potential energy contribution of the each of the internal coordinates to the normal mode can be computed. This is known as the potential energy distribution (PED). For example, for the symmetric stretch the PED of the mode is 50% O-H1 stretch and 50% O-H2 stretch. Within this approximation we can separate the hamiltonian into terms for each vibrational normal mode.

The solutions are

The vibrational partition function for a polyatomic molecule becomes the product of partition functions for each vibrational normal mode.

We can define the vibrational temperature

The partition function can be expressed in terms of the vibrational temperature.

Calculation of average quantities from the vibrational partition function

The average energy can be calculated from

We insert the vibrational partition function to obtain

The vibrational entropy is

which after a little algebra becomes

The heat capacity can be calculated from

Recognizing that the average energy is the energy calculated above, E_vib.

There is a great deal of utility for thermodynamic functions calculated from the vibrational normal modes of a molecule. The vibrational energy and entropy depend on the shape a multidimensional potential energy surface. If one performs a conformational search of macromolecule it is one obtains energies and structures but little direct information concerning the shape of the potential energy surface for each conformation. The vibrational entropy gives a means determining whether there are significant entropic differences in the structures and therefore whether certain conformations will be favored based on the entropy.

The rotatational partition function

For a linear triatomic molecule the problem is the same as for a diatomic molecule. The energy levels are

The degeneracy is

For a linear molecule the moment of inertia is

where d_j is the distance of the jth mass from the center of mass.

The center of mass is given by

where x_j, y_j, and z_j are the Cartesian coordinates in an arbitrary coordinate system.

The rotational partition function for a linear polyatomic molecule can be written as

where the rotational temperature is

 For a non-linear polyatomic molecule there are three rotation axes and three values of the moment of inertia. In the general case the rotational partition function is

The rotational energy and entropy are

Hindered rotation

If a rotation with a molecule is restricted then is called a hindered rotation. For example, a methyl group in ethane can be in an eclipsed or staggered configuration. As the methyl group rotates it moves from a region of low potential energy (staggered) to a region of high potential energy (eclipsed). We can represent the potential energy surface using a sinusoidal function as shown below.

At sufficiently high temperatures the molecule moves easily from one minimum to another. However, when kT << V₀ then the molecule remains in one of the potential wells. The motion then resembles the vibration of a torsional internal coordinate. This is frequently the case in common organic molecules. For example, V₀ = 2.7 - 3.0 kcal/mole in ethane while kT = 0.6 kcal/mole at 300 K. A partition function can be calculated for hindered rotation from the solution of the Schrödinger equation for this potential function.

The contribution of hindered rotation to the heat capacity is one observation that allows these considerations to be connected with experiment.