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In the ideal electric-dipole interaction, a vibrational transition needs a nonzero matrix element of the electric dipole moment. For a diatomic bond stretch, expand the body-axis dipole component as $d(q)=d_e+d_e^{\prime }q+\cdots$. A changing electric dipole moment, rather than a permanent value $d_e$ alone, permits the harmonic fundamental. Keeping only the linear term and harmonic wavefunctions gives $\Delta v=\pm 1$.^[ref]

The derivation is sharper than a picture of a shaking molecule. Orthogonality removes the constant term for $v^{\prime }\ne v$, and creation and annihilation operators giveHere $\delta$ is the Kronecker delta. Thus absorption raises $v$ by one when the dipole derivative and angular factor are nonzero, and its photon satisfiesA nonzero vibrational matrix element is only one factor: rotational electric-dipole selection rules, initial populations, and spectral resolution also constrain visibility.

For an isolated diatomic molecule with identical atomic nuclei, such as molecular hydrogen in its symmetric electronic ground state, stretching preserves zero electric dipole moment. By symmetry, this homonuclear stretch is infrared inactive in the ideal dipole model, although it still vibrates. Other interactions or environments can permit detection.^[ref]

A vibrational overtone excites more than one vibrational quantum: $v=0\to 2$ is the first overtone, and $0\to 3$ 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.^[ref]

For an original illustrative level model, setThen $E_1-E_0=\hslash \omega (1-2ϵ)$ and $E_2-E_0=2\hslash \omega (1-3ϵ)$. With chosen $ϵ=0.02$, the fundamental gap is $0.96\hslash \omega$ and the first-overtone gap $1.88\hslash \omega$, below twice the fundamental, $1.92\hslash \omega$. 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 ${\tilde{\nu }}_{\mathrm{harm}}$.