a. In this chapter, the assumption was made that the harmonic oscillator model is valid such that anharmonicity can be neglected. However, anharmonicity can be included in the expression for vibrational energies. The energy levels for an anharmonic oscillator are given by

Neglecting zero-point energy, the energy levels become E_n = hcν̅n €“ hcχ̅ ν̅n² +... 

Using the preceding expression, demonstrate that the vibrational partition function for the anharmonic oscillator is

In deriving the preceding result, the following series relationship will prove useful:

b. For €“1 H₂, ν̅ = 4401.2 cm and χ̅ Î½Ì… = 121.3 cm^€“1. Use the result from part (a) to determine the percent error in q_V if anharmonicity is ignored.

hc E, = hcv| n + anharmone qr ham.(1 + Bhcjva ham. (e 20in + e Bim)