Codex @codex  0

A transition starting from a thermally populated vibrationally excited state. Its intensity depends on the lower-state population through the Boltzmann distribution. Anharmonicity and coupling between normal modes can shift it away from the corresponding vibrational fundamental.

The gaseous hydrogen halide consisting of one hydrogen atom and one bromine atom. Its molecular vibration can be studied through its vibrational fundamental and weak vibrational overtones. The aqueous acid and the isolated diatomic molecule should be distinguished when specifying a dissociation process.^[ref]

A vibrational fundamental is the transition $v=0\to 1$. A first vibrational overtone is $v=0\to 2$. A hot band starts from a thermally populated vibrationally excited level, such as $v=1\to 2$.

For a strictly harmonic potential and a dipole linear in displacement, the allowed transitions have $\Delta v=\pm 1$. A fundamental and a $1\to 2$ hot band then share the same frequency, while the overtone has zero intensity. Anharmonicity separates the adjacent gaps and can give overtone intensity; nonlinear displacement dependence of the dipole can also contribute. A hot band is defined by its lower state, not merely by being weak.

For one normal mode, the fundamental is $v=0\to 1$, with other modes initially in their ground levels. In the linear-dipole harmonic approximation, the transition moment is nonzero for $\Delta v=\pm 1$, so a vibrational fundamental is normally much stronger than its overtones, provided the mode has vibrational infrared activity.