The general form of the molecular partition function is an infinite sum which is open form, making it difficult to calculate, this is why the sum is approximated as a closed form which leads to an algebraic equation. The derivation of the closed form on the equation is as follows:
The open form of the vibrational partition function:
and by substituting (i.e., singly degenerate) into the summation for the resulting equation is:
The j is taken outside the brackets by the common exponent rule:
Note that is the equation that represents the energy levels of a harmonic oscillator which is used to approximate the vibrational molecular degree of freedom. The vibrational zero point energy is not negligible and must be defined at n=0.
Next, in order for the open system to be converted into the closed system, the equation must take the form of a geometric series identity, like in calculus.
This is done by first letting
this converges when x<1, giving and by replacing x with the original expression, you have:
where is the vibrational frequency of the molecule, which can by found by the following equation:
where k is the spring constant of the molecule and is the reduced mass