page_title: Rotational–vibrational spectroscopy
section_title: Heteronuclear diatomic molecules
hierarchical_section_title: Rotational–vibrational spectroscopy / Linear molecules / Heteronuclear diatomic molecules
caption_attribution_description: English: Part of the ro-vib spectrum of nitric oxide
context_page_description: Rotational–vibrational spectroscopy is a branch of molecular spectroscopy concerned with infrared and Raman spectra of molecules in the gas phase. Transitions involving changes in both vibrational and rotational states can be abbreviated as rovibrational transitions. When such transitions emit or absorb photons, the frequency is proportional to the difference in energy levels and can be detected by certain kinds of spectroscopy. Since changes in rotational energy levels are typically much smaller than changes in vibrational energy levels, changes in rotational state are said to give fine structure to the vibrational spectrum. For a given vibrational transition, the same theoretical treatment as for pure rotational spectroscopy gives the rotational quantum numbers, energy levels, and selection rules. In linear and spherical top molecules, rotational lines are found as simple progressions at both higher and lower frequencies relative to the pure vibration frequency. In symmetric top molecules the transitions are classified as parallel when the dipole moment change is parallel to the principal axis of rotation, and perpendicular when the change is perpendicular to that axis.
context_section_description: Diatomic molecules with the general formula AB have one normal mode of vibration involving stretching of the A-B bond. The vibrational term values , for an anharmonic oscillator are given, to a first approximation, by
  where v is a vibrational quantum number, ωₑ is the harmonic wavenumber and χₑ is an anharmonicity constant.
  When the molecule is in the gas phase, it can rotate about an axis, perpendicular to the molecular axis, passing through the centre of mass of the molecule. The rotational energy is also quantized, with term values to a first approximation given by
  where J is a rotational quantum number and D is a centrifugal distortion constant. The rotational constant, Bᵥ depends on the moment of inertia of the molecule, Iᵥ, which varies with the vibrational quantum number, v
  where mA and mB are the masses of the atoms A and B, and d represents the distance between the atoms. The term values of the ro-vibrational states are found (in the Born–Oppenheimer approximation) by combining the expressions for vibration and rotation.
  The first two terms in this expression correspond to a harmonic oscillator and a rigid rotor, the second pair of terms make a correction for anharmonicity and centrifugal distortion. A more general expression was given by Dunham.
  The selection rule for electric dipole allowed ro-vibrational transitions, in the case of a diamagnetic diatomic molecule is
  The transition with Δv=±1 is known as the fundamental transition. The selection rule has two consequences.
  Both the vibrational and rotational quantum numbers must change. The transition : (Q-branch) is forbidden
  The energy change of rotation can be either subtracted from or added to the energy change of vibration, giving the P- and R- branches of the spectrum, respectively.
  The calculation of the transition wavenumbers is more complicated than for pure rotation because the rotational constant Bν is different in the ground and excited vibrational states. A simplified expression for the wavenumbers is obtained when the centrifugal distortion constants  and  are approximately equal to each other.
  where positive m values refer to the R-branch and negative values refer to the P-branch. The term ω₀ gives the position of the (missing) Q-branch, the term  implies an progression of equally spaced lines in the P- and R- branches, but the third term,  shows that the separation between adjacent lines changes with changing rotational quantum number. When  is greater than , as is usually the case, as J increases the separation between lines decreases in the R-branch and increases in the P-branch. Analysis of data from the infrared spectrum of carbon monoxide, gives value of  of 1.915 cm⁻¹ and  of 1.898 cm⁻¹. The bond lengths are easily obtained from these constants as r₀ = 113.3 pm, r₁ = 113.6 pm. These bond lengths are slightly different from the equilibrium bond length. This is because there is zero-point energy in the vibrational ground state, whereas the equilibrium bond length is at the minimum in the potential energy curve. The relation between the rotational constants is given by
  where ν is a vibrational quantum number and α is a vibration-rotation interaction constant which can be calculated when the B values for two different vibrational states can be found. For carbon monoxide req = 113.0 pm.
  Nitric oxide, NO, is a special case as the molecule is paramagnetic, with one unpaired electron. Coupling of the electron spin angular momentum with the molecular vibration causes lambda-doubling with calculated harmonic frequencies of  1904.03 and 1903.68 cm⁻¹. Rotational levels are also split.
caption: Spectrum of R-branch of nitric oxide, NO, simulated with Spectralcalc,[6] showing λ-doubling caused by the presence of an unpaired electron in the molecule
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