Discussion
The expression for the second overtone frequency in the vibrational absorption spectra of a diatomic molecule in terms of the harmonic frequency \(\omega_{e}\) and anharmonicity constant \(x_{e}\) is
Answer: D
The second overtone frequency is \(3\omega_{e}(1 - 4x_{e})\).
The anharmonic oscillator levels are \(G(v) = \omega_{e}(v + \tfrac{1}{2}) - \omega_{e}x_{e}(v + \tfrac{1}{2})^{2}\). The second overtone is the transition \(v = 0 \to 3\):
\[ \begin{aligned} G(3) - G(0) &= 3\omega_{e} - \omega_{e}x_{e}\left(\tfrac{49}{4} - \tfrac{1}{4}\right) \\ &= 3\omega_{e} - 12\omega_{e}x_{e} = 3\omega_{e}(1 - 4x_{e}) \end{aligned} \]
The fundamental (\(0 \to 1\)) is \(\omega_{e}(1 - 2x_{e})\) and the first overtone (\(0 \to 2\)) is \(2\omega_{e}(1 - 3x_{e})\), which is the distractor for those who count the fundamental as the first overtone. \(3\omega_{e}(1 - 2x_{e})\) multiplies the fundamental by three, ignoring the growing anharmonic correction, and \(2\omega_{e}(1 - x_{e})\) fits no transition.
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