Rotational–vibrational spectroscopy

Rotational–vibrational spectroscopy is a branch of molecular spectroscopy that is 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. The ro-vibrational spectrum of the asymmetric rotor water is important because of the presence of water vapor in the atmosphere.

Overview

Ro-vibrational spectroscopy concerns molecules in the gas phase. There are sequences of quantized rotational levels associated with both the ground and excited vibrational states. The spectra are often resolved into lines due to transitions from one rotational level in the ground vibrational state to one rotational level in the vibrationally excited state. The lines corresponding to a given vibrational transition form a band.
In the simplest cases the part of the infrared spectrum involving vibrational transitions with the same rotational quantum number in ground and excited states is called the Q-branch. On the high frequency side of the Q-branch the energy of rotational transitions is added to the energy of the vibrational transition. This is known as the R-branch of the spectrum for ΔJ = +1. The P-branch for ΔJ = −1 lies on the low wavenumber side of the Q branch. The appearance of the R-branch is very similar to the appearance of the pure rotation spectrum, and the P-branch appears as a nearly mirror image of the R-branch. The Q branch is sometimes missing because of transitions with no change in J being forbidden.
The appearance of rotational fine structure is determined by the symmetry of the molecular rotors which are classified, in the same way as for pure rotational spectroscopy, into linear molecules, spherical-, symmetric- and asymmetric- rotor classes. The quantum mechanical treatment of rotational fine structure is the same as for pure rotation.
The strength of an absorption line is related to the number of molecules with the initial values of the vibrational quantum number ν and the rotational quantum number, and depends on temperature. Since there are actually states with rotational quantum number, the population with value increases with initially, and then decays at higher. This gives the characteristic shape of the P and R branches.
A general convention is to label quantities that refer to the vibrational ground and excited states of a transition with double prime and single prime, respectively. For example, the rotational constant for the ground state is written as and that of the excited state as
Also, these constants are expressed in the molecular spectroscopist's units of cm⁻¹. so that in this article corresponds to in the definition of rotational constant at Rigid rotor.

Method of combination differences

Numerical analysis of ro-vibrational spectral data would appear to be complicated by the fact that the wavenumber for each transition depends on two rotational constants, and. However combinations which depend on only one rotational constant are found by subtracting wavenumbers of pairs of lines which have either the same lower level or the same upper level. For example, in a diatomic molecule the line denoted P is due to the transition → , and the line R is due to the transition →. The difference between the two wavenumbers corresponds to the energy difference between the and levels of the lower vibrational state and is denoted by since it is the difference between levels differing by two units of J. If centrifugal distortion is included, it is given by
where means the frequency of the given line. The main term, comes from the difference in the energy of the rotational state, and that of the state,
The rotational constant of the ground vibrational state B′′ and centrifugal distortion constant, D′′ can be found by least-squares fitting this difference as a function of J. The constant B′′ is used to determine the internuclear distance in the ground state as in pure rotational spectroscopy.
Similarly the difference R − P depends only on the constants B′ and D′ for the excited vibrational state, and B′ can be used to determine the internuclear distance in that state.

Linear molecules

Heteronuclear diatomic molecules

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, ω_e is the harmonic wavenumber and χ_e 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, B_v depends on the moment of inertia of the molecule, I_v, which varies with the vibrational quantum number, v
where m_A and m_B 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 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 : 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 B_ν 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 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 r_eq = 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.

Homonuclear diatomic molecules

The quantum mechanics for homonuclear diatomic molecules such as dinitrogen, N₂, and fluorine, F₂, is qualitatively the same as for heteronuclear diatomic molecules, but the selection rules governing transitions are different. Since the electric dipole moment of the homonuclear diatomics is zero, the fundamental vibrational transition is electric-dipole-forbidden and the molecules are infrared inactive. However, a weak quadrupole-allowed spectrum of N₂ can be observed when using long path-lengths both in the laboratory and in the atmosphere. The spectra of these molecules can be observed by Raman spectroscopy because the molecular vibration is Raman-allowed.
Dioxygen is a special case as the molecule is paramagnetic so magnetic-dipole-allowed transitions can be observed in the infrared. The unit electron spin has three spatial orientations with respect to the molecular rotational angular momentum vector, N, so that each rotational level is split into three states with total angular momentum , J = N + 1, N, and N - 1, each J state of this so-called p-type triplet arising from a different orientation of the spin with respect to the rotational motion of the molecule. Selection rules for magnetic dipole transitions allow transitions between successive members of the triplet so that for each value of the rotational angular momentum quantum number N there are two allowed transitions. The ¹⁶O nucleus has zero nuclear spins angular momentum, so that symmetry considerations demand that N may only have odd values.