Vibrational partition function

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

$$Q\_\backslash text\{vib\}(T)\; =\; \backslash prod\_j\{\backslash sum\_n\{e^\{-\backslash frac\{E\_\{j,n\}\}\{k\_\backslash text\{B\}\; T\}\}\}\}$$

where $T$ is the absolute temperature of the system, $k\_B$ is the Boltzmann constant, and $E\_\{j,n\}$ is the energy of the jth mode when it has vibrational quantum number $n\; =\; 0,\; 1,\; 2,\; \backslash ldots$. For an isolated molecule of N atoms, the number of vibrational modes (i.e. values of j) is {{nowrap|3N − 5}} for linear molecules and {{nowrap|3N − 6}} for non-linear ones.G. Herzberg, Infrared and Raman Spectra, Van Nostrand Reinhold, 1945 In crystals, the vibrational normal modes are commonly known as phonons.

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:

$$E\_\{j,n\}\; =\; \backslash hbar\backslash omega\_j\backslash left(n\_j\; +\; \backslash frac\{1\}\{2\}\backslash right)$$

where j runs over vibrational modes and $n\_j$ is the vibrational quantum number in the jth mode, $\backslash hbar$ is the Planck constant, h, divided by $2\; \backslash pi$ and $\backslash omega\_j$

is the angular frequency of the jth mode. Using this approximation we can derive a closed form expression for the vibrational partition function.

$$Q\_\backslash text\{vib\}(T)\; =\backslash prod\_j\{\backslash sum\_n\{e^\{-\backslash frac\{E\_\{j,n\}\}\{k\_\backslash text\{B\}\; T\}\}\}\}$$

= \prod_j e^{-\frac{\hbar \omega_j}{2 k_\text{B} T}} \sum_n \left( e^{-\frac{\hbar \omega_j}{k_\text{B} T}} \right)^n

= \prod_j \frac{e^{-\frac{\hbar \omega_j}{2 k_\text{B} T}}}{ 1 - e^{-\frac{\hbar \omega_j}{k_\text{B} T}} }

= e^{- \frac{E_\text{ZP}}{k_\text{B} T}} \prod_j \frac{1}{ 1 - e^{-\frac{\hbar \omega_j}{k_\text{B} T}} }

where $E\_\backslash text\{ZP\}\; =\; \backslash frac\{1\}\{2\}\; \backslash sum\_j\; \backslash hbar\; \backslash omega\_j$ is total vibrational zero point energy of the system.

Often the wavenumber, $\backslash tilde\{\backslash nu\}$ 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 $\backslash omega\; =\; 2\; \backslash pi\; c\; \backslash tilde\{\backslash nu\}$ where c is the speed of light in vacuum. In terms of the vibrational wavenumbers we can write the partition function as

$$Q\_\backslash text\{vib\}(T)\; =\; e^\{-\; \backslash frac\{E\_\backslash text\{ZP\}\}\{k\_\backslash text\{B\}\; T\}\}\; \backslash prod\_j\; \backslash frac\{1\}\{\; 1\; -\; e^\{-\backslash frac\{\; h\; c\; \backslash tilde\{\backslash nu\}\_j\}\{k\_\backslash text\{B\}\; T\}\}\; \}$$

It is convenient to define a characteristic vibrational temperature

$$\backslash Theta\_\{i,\backslash text\{vib\}\}\; =\; \backslash frac\{h\; \backslash nu\_i\}\{k\_\backslash text\{B\}\}$$

where $\backslash nu$ 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

$$Q\_\backslash text\{vib\}(T)\; =\; \backslash prod\_\{i=1\}^f\; \backslash frac\{1\}\{1-e^\{-\backslash Theta\_\{\backslash text\{vib\},i\}/T\}\}$$

{{reflist}}

• Partition function (mathematics)

{{Statistical mechanics topics}}

Category:Partition functions