The vibrational partition function[1] 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
![{\displaystyle Q_{\text{vib}}(T)=\prod _{j}{\sum _{n}{e^{-{\frac {E_{j,n}}{k_{\text{B}}T}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a4b2d56eac08238c9c92876a17143dc0144a684)
where
![{\displaystyle T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0)
is the
absolute temperature of the system,
![{\displaystyle k_{B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70f38f7b73e53fd7b5d9ca64bec3a1438cc0eade)
is the
Boltzmann constant, and
![{\displaystyle E_{j,n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cd2d5a2ea0074240f32dc5662ae558b42a5b6fa)
is the energy of the
j-th mode when it has vibrational quantum number
![{\displaystyle n=0,1,2,\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a19cb2cfd4f9ebdbc8e5cbb9b92ecb9ace85cab)
. For an isolated molecule of
N atoms, the number of
vibrational modes (i.e. values of
j) is 3
N − 5 for linear molecules and 3
N − 6 for non-linear ones.
[2] In crystals, the vibrational normal modes are commonly known as
phonons.
Approximations
Quantum harmonic oscillator
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.[1] A quantum harmonic oscillator has an energy spectrum characterized by:
![{\displaystyle E_{j,n}=\hbar \omega _{j}\left(n_{j}+{\frac {1}{2}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93a67cfa3eb5770022ffbd85bd534c3524fc728d)
where
j runs over vibrational modes and
![{\displaystyle n_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2c97b31dca4eeefe57123a12e69e6ea73f3dcd2)
is the vibrational quantum number in the
j-th mode,
![{\displaystyle \hbar }](https://wikimedia.org/api/rest_v1/media/math/render/svg/de68de3a92517953436c93b5a76461d49160cc41)
is
Planck's constant,
h, divided by
![{\displaystyle 2\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06)
and
![{\displaystyle \omega _{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c0dc871d61e141c38b6984d044cd4d19f33b89f)
is the angular frequency of the
j'th mode. Using this approximation we can derive a closed form expression for the vibrational partition function.
![{\displaystyle Q_{\text{vib}}(T)=\prod _{j}{\sum _{n}{e^{-{\frac {E_{j,n}}{k_{\text{B}}T}}}}}=\prod _{j}e^{-{\frac {\hbar \omega _{j}}{2k_{\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}}{2k_{\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}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a119e5512ea7679d8ac9a0ace76b4f12a0f7aad)
where
is total vibrational zero point energy of the system.
Often the wavenumber,
with units of cm−1 is given instead of the angular frequency of a vibrational mode[2] and also often misnamed frequency. One can convert to angular frequency by using
where c is the speed of light in vacuum. In terms of the vibrational wavenumbers we can write the partition function as
![{\displaystyle Q_{\text{vib}}(T)=e^{-{\frac {E_{\text{ZP}}}{k_{\text{B}}T}}}\prod _{j}{\frac {1}{1-e^{-{\frac {hc{\tilde {\nu }}_{j}}{k_{\text{B}}T}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e41425cd1b7f660b9ee138378f75dd002cf7d41a)
It is convenient to define a characteristic vibrational temperature
![{\displaystyle \Theta _{i,{\text{vib}}}={\frac {h\nu _{i}}{k_{\text{B}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4990f56f2da281e64c5271ee0b878f92880866d8)
where
![{\displaystyle \nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468)
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
![{\displaystyle Q_{\text{vib}}(T)=\prod _{i=1}^{f}{\frac {1}{1-e^{-\Theta _{{\text{vib}},i}/T}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc8921c5870dd4dc186a917c29c5472ee018b29a)
References
- ^ a b Donald A. McQuarrie, Statistical Mechanics, Harper & Row, 1973
- ^ a b G. Herzberg, Infrared and Raman Spectra, Van Nostrand Reinhold, 1945
See also