Vibrational partition function: Difference between revisions

Revision as of 05:44, 28 September 2010

Revision as of 04:15, 13 February 2011

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The '''vibrational partition function''' 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==

===Quantum Harmonic Oscillator===

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 degree of freedom of molecules towards thermodynamic variables. A quantum harmonic oscillator has an energy spectrum characterized by:

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 degree of freedom of molecules towards thermodynamic variables. A quantum harmonic oscillator has an energy spectrum characterized by:

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

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

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