Vibrational partition function

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.

For a system (such as a molecule or solid) with uncoupled vibrational modes the vibrational partition function is defined by Q vib ( T ) = ∏ j ∑ n e − E j , n k B T {\displaystyle Q_{\text{vib}}(T)=\prod _{j}{\sum _{n}{e^{-{\frac {E_{j,n}}{k_{\text{B}}T}}}}}} where T {\displaystyle T} is the absolute temperature of the system, k B {\displaystyle k_{B}} is the Boltzmann constant, and E j , n {\displaystyle E_{j,n}} is the energy of the jth mode when it has vibrational quantum number n = 0 , 1 , 2 , … {\displaystyle n=0,1,2,\ldots } . For an isolated molecule of N atoms, the number of vibrational modes (i.e. values of j) is 3N − 5 for linear molecules and 3N − 6 for non-linear ones.^[2] In crystals, the vibrational normal modes are commonly known as phonons.

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: E j , n = ℏ ω j ( n j + 1 2 ) {\displaystyle E_{j,n}=\hbar \omega _{j}\left(n_{j}+{\frac {1}{2}}\right)} where j runs over vibrational modes and n j {\displaystyle n_{j}} is the vibrational quantum number in the jth mode, ℏ {\displaystyle \hbar } is the Planck constant, h, divided by 2 π {\displaystyle 2\pi } and ω j {\displaystyle \omega _{j}} is the angular frequency of the jth mode. Using this approximation we can derive a closed form expression for the vibrational partition function. Q vib ( T ) = ∏ j ∑ n e − E j , n k B T = ∏ j e − ℏ ω j 2 k B T ∑ n ( e − ℏ ω j k B T ) n = ∏ j e − ℏ ω j 2 k B T 1 − e − ℏ ω j k B T = e − E ZP k B T ∏ j 1 1 − e − ℏ ω j k B T {\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}}}}}} where E ZP = 1 2 ∑ j ℏ ω j {\textstyle E_{\text{ZP}}={\frac {1}{2}}\sum _{j}\hbar \omega _{j}} is total vibrational zero point energy of the system.

Often the wavenumber, ν ~ {\displaystyle {\tilde {\nu }}} 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 ω = 2 π c ν ~ {\displaystyle \omega =2\pi c{\tilde {\nu }}} where c is the speed of light in vacuum. In terms of the vibrational wavenumbers we can write the partition function as Q vib ( T ) = e − E ZP k B T ∏ j 1 1 − e − h c ν ~ j k B T {\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}}}}}}

It is convenient to define a characteristic vibrational temperature Θ i , vib = h ν i k B {\displaystyle \Theta _{i,{\text{vib}}}={\frac {h\nu _{i}}{k_{\text{B}}}}} where ν {\displaystyle \nu } 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 Q vib ( T ) = ∏ i = 1 f 1 1 − e − Θ vib , i / T {\displaystyle Q_{\text{vib}}(T)=\prod _{i=1}^{f}{\frac {1}{1-e^{-\Theta _{{\text{vib}},i}/T}}}}

• Partition function (mathematics)