Vibrational partition function

Definition

For a system (such as a molecule or solid) with uncoupled vibrational modes the vibrational partition function is defined by

Q v i b ( T ) = ∏ j ∑ n e − E j , n k B T

where T is the absolute temperature of the system, k B is the Boltzmann constant, and E j , n is the energy of j'th mode when it has vibrational quantum number n = 0 , 1 , 2 , … . 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. In crystals, the vibrational normal modes are commonly known as phonons.

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. A quantum harmonic oscillator has an energy spectrum characterized by:

E j , n = ℏ ω j ( n j + 1 2 )

where j runs over vibrational modes and n j is the vibrational quantum number in the j 'th mode, ℏ is Planck's constant, h, divided by 2 π and ω j is the angular frequency of the j'th mode. Using this approximation we can derive a closed form expression for the vibrational partition function.

Q v i b ( T ) = ∏ j ∑ n e − E j , n k B T = ∏ j e − ℏ ω j 2 k B T ∑ n ( e − ℏ ω j k B T ) n = ∏ j e − ℏ ω j 2 k B T 1 − e − ℏ ω j k B T = e − E Z P k B T ∏ j 1 1 − e − ℏ ω j k B T

where E Z P = 1 2 ∑ j ℏ ω j is total vibrational zero point energy of the system.

Often the wavenumber, ν ~ 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 ω = 2 π c ν ~ where c is the speed of light in vacuum. In terms of the vibrational wavenumbers we can write the partition function as

Q v i b ( T ) = e − E Z P k B T ∏ j 1 1 − e − h c ν ~ j k B T