= Vibrational overtone
{title2=$\Delta v=2,3,\ldots$}
A <vibrational overtone> excites more than one vibrational quantum: $v=0\to2$ is the first overtone, and $0\to3$ the second. Such transitions vanish in the combined harmonic-potential and linear-dipole approximation. <Anharmonicity> changes <wavefunctions> and level spacing; nonlinear terms in the <electric dipole moment> can also supply otherwise missing transition <matrix elements>.<14: Harmonic Oscillators and IR Spectroscopy, UCD Chem 110A by Delmar S. Larsen>[Lecture 14, overtone section, Table 1 and following potential-expansion discussion]{ref}
For an original illustrative level model, set
$$
E_v-V_0=\hbar\omega\left[(v+1/2)-\epsilon(v+1/2)^2\right],\qquad 0<\epsilon\ll1.
$$
Then $E_1-E_0=\hbar\omega(1-2\epsilon)$ and $E_2-E_0=2\hbar\omega(1-3\epsilon)$. With chosen $\epsilon=0.02$, the fundamental gap is $0.96\hbar\omega$ and the first-overtone gap $1.88\hbar\omega$, below twice the fundamental, $1.92\hbar\omega$. This low-level illustration does not determine transition intensity or justify extrapolation to arbitrarily large $v$. It shows why even a well-resolved fundamental band origin need not equal the harmonic curvature scale $\widetilde\nu_{\rm harm}$.
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