The vibrational partition function for a molecule which can be described as a simple harmonic oscillator with fundamental frequency $$\nu $$ is given by

The vibrational partition function describes how molecules are distributed among different vibrational energy levels at thermal equilibrium.

For a molecule that behaves as a quantum mechanical simple harmonic oscillator, the vibrational energy levels are given by E_v = (v + 1/2)hν, where v = 0, 1, 2, ....

When the zero-point energy is treated separately, the vibrational partition function becomes:

q_vib = 1 / [1 − exp(−hν/k_BT)]

This expression is equivalent to:

q_vib = [1 − exp(−hν/k_BT)]⁻¹

Therefore, Option B is the correct expression for the vibrational partition function.

Why the other options are incorrect:

Option A: It contains only the exponential term and is not the complete partition function.

Option C: It includes an unnecessary extra exponential factor, so it does not represent the standard vibrational partition function.

Option D: It contains the factor exp(−hν/2k_BT), which accounts for the zero-point energy. This form is obtained only when the zero-point energy is explicitly included in the partition function and is not the standard expression generally used in engineering chemistry and statistical thermodynamics.