Vibrational partition function explained

The vibrational partition function^[1] traditionally refers to the component of the canonical partition function resulting from the vibrational degrees of freedom of a system. The vibrational partition function is only well-defined in model systems where the vibrational motion is relatively uncoupled with the system's other degrees of freedom.

Definition

For a system (such as a molecule or solid) with uncoupled vibrational modes the vibrational partition function is defined by$$Q\_\backslash text(T)\; =\; \backslash prod\_j$$where

T is the absolute temperature of the system, k _{B} is the Boltzmann constant, and E _{j,n} is the energy of the j-th mode when it has vibrational quantum number n = 0 , 1 , 2 , \ldots . For an isolated molecule of N atoms, the number of vibrational modes (i.e. values of j) is 3N − 5 for linear molecules and 3N − 6 for non-linear ones.^[2] In crystals, the vibrational normal modes are commonly known as phonons.

Approximations

Quantum harmonic oscillator

The most common approximation to the vibrational partition function uses a model in which the vibrational eigenmodes or normal modes of the system are considered to be a set of uncoupled quantum harmonic oscillators. It is a first order approximation to the partition function which allows one to calculate the contribution of the vibrational degrees of freedom of molecules towards its thermodynamic variables.^[1] A quantum harmonic oscillator has an energy spectrum characterized by:$$E\_\; =\; \backslash hbar\backslash omega\_j\backslash left(n\_j\; +\; \backslash frac\backslash right)$$where j runs over vibrational modes and

n _{j} is the vibrational quantum number in the j-th mode, \hbar is Planck's constant, h, divided by 2 \pi and \omega _{j} is the angular frequency of the jth mode. Using this approximation we can derive a closed form expression for the vibrational partition function.

$$Q\_\backslash text(T)\; =\backslash prod\_j=\; \backslash prod\_j\; e^\; \backslash sum\_n\; \backslash left(e^\; \backslash right)^n=\; \backslash prod\_j\; \backslash frac=\; e^\; \backslash prod\_j\; \backslash frac$$

where $E\_\backslash text\; =\; \backslash frac\; \backslash sum\_j\; \backslash hbar\; \backslash omega\_j$ is total vibrational zero point energy of the system.

Often the wavenumber,

\tilde{\nu} with units of cm⁻¹ is given instead of the angular frequency of a vibrational mode and also often misnamed frequency. One can convert to angular frequency by using \omega = 2 \pi c \tilde{\nu} where c is the speed of light in vacuum. In terms of the vibrational wavenumbers we can write the partition function as$$Q\_\backslash text(T)\; =\; e^\; \backslash prod\_j\; \backslash frac$$

It is convenient to define a characteristic vibrational temperature $$\backslash Theta\_\; =\; \backslash frac$$where

\nu is experimentally determined for each vibrational mode by taking a spectrum or by calculation. By taking the zero point energy as the reference point to which other energies are measured, the expression for the partition function becomes$$Q\_\backslash text(T)\; =\; \backslash prod\_^f\; \backslash frac$$

See also

• Partition function (mathematics)

Notes and References

1. Donald A. McQuarrie, Statistical Mechanics, Harper & Row, 1973
2. G. Herzberg, Infrared and Raman Spectra, Van Nostrand Reinhold, 1945