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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]

The strong band is $0\to 1$, while the weak band is the first vibrational overtone $0\to 2$. Let $w={\tilde{\omega }}_e$ and $q={\tilde{\omega }}_e{x}_e$. Subtracting the Morse oscillator levels givesTwice the fundamental is $881.2\mathrm{c}{\mathrm{m}}^{-1}$, slightly above the overtone because the higher vibrational gap is smaller. Eliminate $w$:soThe implied dimensionless anharmonicity constant is $x_e\simeq 0.004164$. The overtone's weakness follows from its being forbidden in the leading harmonic/linear-dipole model, rather than a lack of population in its lower state.

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.

The second vibrational overtone is $v=0\to 3$, with $\Delta v=3$, near three times the fundamental wavenumber. It requires higher-order departures from the linear-dipole harmonic model and is ordinarily weaker again. These are qualitative intensity trends; resonances can redistribute intensity between nearby vibrational states.

The first vibrational overtone is $v=0\to 2$, with $\Delta v=2$. It lies near twice the fundamental wavenumber, generally slightly below it because of anharmonicity. The linear-dipole harmonic model forbids it; anharmonic wavefunctions or higher terms in the dipole expansion give a weak intensity. Thus it is generally weaker than the corresponding fundamental.

There are three stretching fundamentals, although four stretching-region features are listed. The low feature at $1746\mathrm{c}{\mathrm{m}}^{-1}$ is predominantly C=O stretch ($A_1$). The two features at $2783$ and $2843\mathrm{c}{\mathrm{m}}^{-1}$ are the symmetric and antisymmetric C–H stretches; their order is not determined by this mode count alone.

The $3482\mathrm{c}{\mathrm{m}}^{-1}$ feature is consistent with the first C=O vibrational overtone, close to $2(1746)=3492\mathrm{c}{\mathrm{m}}^{-1}$ and shifted downward by anharmonicity. Its symmetry is $A_1\otimes A_1=A_1$, so symmetry permits the weak anharmonic transition. It is not a fourth stretching fundamental. The combination-band evidence in the next part distinguishes the two C–H assignments.

Excitation of one quantum of a normal mode from the vibrational ground state. For an infrared-active mode in the linear-dipole harmonic approximation, this is the allowed absorption underlying the usual fundamental band. A vibrational overtone excites two or more quanta from the ground state.

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 ${\tilde{\nu }}_{\mathrm{harm}}$.