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.

{{Unreferenced|auto=yes|date=December 2009}}

The '''vibrational partition function''' <ref>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.

== Definition ==

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

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

<math>Q_{vib}(T) =\prod_j{\sum_n{e^{-\frac{E_{j,n}}{k_B T}}}} </math>

=== Quantum harmonic oscillator ===

where <math> T </math> is the [[Thermodynamic temperature|absolute temperature]] of the system, <math> k_B </math> is the [[Boltzmann constant]],

The most common approximation to the vibrational partition function uses a model in which the vibrational eigenmodes or [[normal mode|vibrational normal modes]]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 degrees of freedom of molecules towards its thermodynamic variables.<ref>Donald A. name="McQuarrie, ''ibid''<"/ref> A quantum harmonic oscillator has an energy spectrum characterized by:

and <math> E_{j,n} </math> is the energy of j'th mode when it has vibrational quantum number <math> n= 0, 1, 2, \ldots </math>.

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

For an isolated molecule of N atoms the number of vibrational modes (i.e. values of j) equals 3N-5 or 3N-6 dependent upon whether the molecule

where ''j'' runs over vibrational modes and n<math> aren_j as</math> describedis abovethe vibrational quantum number in the ''j''th mode, <math> \hbar </math> is the [[Plancks Constant | Planck's Constantconstant]], ''h'', divided by <math> 2 \pi </math> and <math> \omega_j </math>

is linear or nonlinear respectively.<ref> G. Herzberg, ''Infrared and Raman Spectra'', Van Nostrand Reinhold, 1945 </ref>.

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 \frac{e^{-\frac{\hbar \omega_j}{2 k_Bk_\text{B} T}}}{ 1 -\sum_n \left( e^{-\frac{\hbar \omega_j}{k_Bk_\text{B} T}} } \right)^n

= \prod_j \frac{e^{- \frac{E_{ZP}\hbar \omega_j}{k_B2 k_\text{B} T}} \prod_j \frac{1}{ 1 - e^{-\frac{\hbar \omega_j}{k_Bk_\text{B} T}} }

Q_{vib}(T) = e^{- \frac{E_\text{ZP}}{k_Bk_\text{B} T}} \prod_j \frac{1}{ 1 - e^{-\frac{ h c\hbar \tildeomega_j}{k_\nutext{B}_j}{k_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

<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

==Approximations==

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

===Quantum Harmonic Oscillator===

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

The most common approximation to the vibrational partition function uses a model in which the vibrational eigenmodes or [[normal mode|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 degrees of freedom of molecules towards its thermodynamic variables.<ref>Donald A. McQuarrie, ''ibid''</ref> A quantum harmonic oscillator has an energy spectrum characterized by:

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

== References ==

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

{{reflist}}

== See also ==

where j and n are as described above, <math> \hbar </math> is [[Plancks Constant | Planck's 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>Q_{vib}(T) =\prod_j{\sum_n{e^{-\frac{E_{j,n}}{k_B T}}}} = \prod_j e^{-\frac{\hbar \omega_j}{2 k_B T}} \sum_n \left( e^{-\frac{\hbar \omega_j}{k_B T}} \right)^n

= \prod_j \frac{e^{-\frac{\hbar \omega_j}{2 k_B T}}}{ 1 - e^{-\frac{\hbar \omega_j}{k_B T}} }

= e^{- \frac{E_{ZP}}{k_B T}} \prod_j \frac{1}{ 1 - e^{-\frac{\hbar \omega_j}{k_B T}} }

</math>

where

<math> E_{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 <math> cm^{-1} ,</math> is given instead of the angular frequency of a vibrational mode <ref> G. Herzberg, ''ibid'' </ref>

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

<math>

Q_{vib}(T) = e^{- \frac{E_{ZP}}{k_B T}} \prod_j \frac{1}{ 1 - e^{-\frac{ h c \tilde{\nu}_j}{k_B T}} }

</math>

== References ==

<references />

==See also==

* [[Partition function (mathematics)]]

{{Statistical mechanics topics}}

{{DEFAULTSORT:Vibrational Partition Function}}

{{Physics-stub}}

[[Category:Partition functions]]