Vibrational partition function

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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.

[edit] Approximations

[edit] Quantum Harmonic Oscillator

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:

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

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:

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

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