= Vibrational excitation
Vibrational excitation raises the energy associated with internal nuclear motion in a <molecule>. It can occur within one <electronic state>; changing vibrational energy does not necessarily promote an electron. For a <diatomic molecule>, the simplest coordinate is a change in the separation of its two <atomic nuclei>.
Near a stable equilibrium bond length, the <quantum harmonic oscillator> model has
$$
E_v=\hbar\omega\left(v+\frac12\right),\qquad v=0,1,2,\ldots,
$$
with adjacent spacing $\hbar\omega$. The $v=0$ state already has <zero-point energy>; vibrational excitation refers to raising the level above that ground state.<University Physics Volume 3 by Samuel J. Ling and others>[Section 7.5, harmonic-oscillator energies and molecular vibration]{ref} Anharmonicity changes the spacing and becomes important away from the potential minimum.
For a chosen vibrational spacing $\widetilde\nu=1000\,\mathrm{cm^{-1}}$, $\Delta E=hc\widetilde\nu\approx0.124\,\mathrm{eV}$, corresponding to wavelength $10\,\mathrm{\mu m}$. In thermal equilibrium, the population ratio of adjacent nondegenerate harmonic levels is
$$
\frac{N_1}{N_0}=\exp\left(-\frac{hc\widetilde\nu}{k_BT}\right).
$$
At $300\,\mathrm K$ it is about $0.0083$. This stipulated example illustrates why a high-frequency mode can be mostly in its vibrational ground state even though the sample is not at zero temperature.
A motivating apparatus is the <University of Florida HCl and DCl FTIR experiment>: an infrared transmission spectrum is measured using <Fourier transform spectroscopy> and a matched background. The observed rotational-vibrational bands constrain stretching energies and isotope dependence.<Rotation Vibration Spectrum of the HCl Molecule, University of Florida laboratory>[Printed pages 10–12, matched scans, isotope assignments and fitted band origin]{ref} Whether a vibrational transition absorbs infrared light also depends on <vibrational infrared activity>, not just its energy spacing.
A short excitation pulse can create a nonequilibrium vibrational population. Subsequent <vibrational relaxation> redistributes that energy through other modes or the surroundings. Measuring a population change, a transient spectrum and energy deposited as heat are related but distinct experimental observations.
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