Vibrational partition function: Difference between revisions

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The '''vibrational partition function'''<ref name="McQuarrie">Donald A. McQuarrie, ''Statistical Mechanics'', Harper & Row, 1973</ref> 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.

==Approximations Definition ==

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

===Quantum Harmonic Oscillator===

<math display="block">Z_Q_\text{vib}(T) = \prod_j{\sum_isum_n{e^{-\frac{E_{j,in}}{kTk_\text{B} T}}}} </math>

where <math> T </math> is the [[Thermodynamic temperature|absolute temperature]] of the system, <math> k_B </math> is the [[Boltzmann constant]], and <math> E_{j,n} </math> is the energy of the ''j''th mode when it has vibrational quantum number <math> n = 0, 1, 2, \ldots </math>. For an isolated molecule of ''N'' atoms, the number of [[Molecular vibration|vibrational modes]] (i.e. values of ''j'') is {{nowrap|3''N'' − 5}} for linear molecules and {{nowrap|3''N'' − 6}} for non-linear ones.<ref name="Herzberg">G. Herzberg, ''Infrared and Raman Spectra'', Van Nostrand Reinhold, 1945</ref> In crystals, the vibrational normal modes are commonly known as [[phonon]]s.

== Approximations ==

The most common approximation to the vibrational partition function uses a model in which the [[vibrational eigenmodes]] or [[vibrational 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 degree of freedom of molecules towards thermodyanmic variables. A quantum harmonic oscillator has an energy spectrum characterized by:

=== Quantum harmonic oscillator ===

<math>E_{j,i}=\hbar\omega_j(i+\frac{1}{2})</math>

The most common approximation to the vibrational partition function uses a model in which the [[vibrational eigenmodes]] or [[vibrational normal modesmode]]s of the system are considered to be a set of uncoupled [[quantum harmonic oscillatorsoscillator]]s. It is a first order approximation to the partition function which allows one to calculate the contribution of the vibrational degreedegrees of freedom of molecules towards thermodyanmicits thermodynamic variables.<ref name="McQuarrie"/> A quantum harmonic oscillator has an energy spectrum characterized by:

<math display="block">E_{j,in} = \hbar\omega_j\left(in_j + \frac{1}{2}\right)</math>

where ''j'' runs over vibrational modes and <math> n_j </math> is the vibrational quantum number in the ''j''th mode, <math> \hbar </math> is the [[Planck constant]], ''h'', divided by <math> 2 \pi </math> and <math> \omega_j </math>

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

<math display="block">Q_\text{vib}(T) =\prod_j{\sum_n{e^{-\frac{E_{j,n}}{k_\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}} }</math>

where <math display="inline"> E_\text{ZP} = \frac{1}{2} \sum_j \hbar \omega_j </math> is total vibrational zero point energy of the system.

Often the [[wavenumber]], <math>\tilde{\nu}</math> with units of cm⁻¹ is given instead of the angular frequency of a vibrational mode<ref name="Herzberg" /> and also often misnamed frequency. One can convert to angular frequency by using <math> \omega = 2 \pi c \tilde{\nu} </math> where ''c'' is the [[speed of light]] in vacuum. In terms of the vibrational wavenumbers we can write the partition function as

where j is an index representing the vibrational mode, and i is the quantum number for each energy level of the jth vibrational mode. The vibrational partition function is then calculated as:

<math display="block">Q_\text{vib}(T) = e^{- \frac{E_\text{ZP}}{k_\text{B} T}} \prod_j \frac{1}{ 1 - e^{-\frac{ h c \tilde{\nu}_j}{k_\text{B} T}} }</math>

It is convenient to define a characteristic vibrational temperature

<math display="block"> \Theta_{i,\text{vib}} = \frac{h \nu_i}{k_\text{B}} </math>

where <math> \nu </math> 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

<math display="block"> Q_\text{vib}(T) = \prod_{i=1}^f \frac{1}{1-e^{-\Theta_{\text{vib},i}/T}} </math>

== References ==

{{reflist}}

== See also ==

* [[Partition function (mathematics)]]

<math>Z_{vib}=\prod_j{\sum_i{e^{-\frac{E_{j,i}}{kT}}}}</math>

{{Statistical mechanics topics}}

{{Physics-stub}}

[[Category:ArticlesPartition lacking sources (Erik9bot)functions]]