740 18-4. Most Molecules Are in the Ground Vibrational State at Room Temperature In this section, we will evaluate the vibrational part of the partition function of a diatomic molecule under the harmonic-oscillator approximation. If we measure the vibrational energy levels relative to the bottom of the internuclear potential well, the energies are given by (Section 5-4) 8, = (u + ; ) hu v = 0, 1, 2, ... (18.22) with v = (k/u) /2/2n, where k is the force constant of the molecule and u is its reduced mass. The vibrational partition function quib becomes 9 vib ( T ) = [e- BE. = [e-B ( U+1 )hu = e-Bhu/2 e -Bhuv U=0 This summation can be evaluated easily by recognizing it to be a geometric series (MathChapter I): with x = e-Bhu