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 (or ro-vibrational) transitions. When such transitions emit or absorb photons (electromagnetic radiation), 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 (ΔJ = 0) 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 (but shifted to much higher wavenumbers), 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 J {\displaystyle J} , and depends on temperature. Since there are actually 2 J + 1 {\displaystyle 2J+1} states with rotational quantum number J {\displaystyle J} , the population with value J {\displaystyle J} increases with J {\displaystyle J} initially, and then decays at higher J {\displaystyle J} . 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 B ′ ′ , {\displaystyle B^{\prime \prime },} and that of the excited state as B ′ . {\displaystyle B^{\prime }.}

Also, these constants are expressed in the molecular spectroscopist's units of cm−1. so that B {\displaystyle B} in this article corresponds to B ¯ = B / h c {\displaystyle {\bar {B}}=B/hc} 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, B ′ ′ {\displaystyle B^{\prime \prime }} and B ′ {\displaystyle B^{\prime }} . However combinations which depend on only one rotational constant are found by subtracting wavenumbers of pairs of lines (one in the P-branch and one in the R-branch) which have either the same lower level or the same upper level. For example, in a diatomic molecule the line denoted P(J + 1) is due to the transition (v = 0, J + 1) → (v = 1, J) (meaning a transition from the state with vibrational quantum number ν going from 0 to 1 and the rotational quantum number going from some value J + 1 to J, with J > 0), and the line R(J − 1) is due to the transition (v = 0, J − 1) → (v = 1, J). The difference between the two wavenumbers corresponds to the energy difference between the (J + 1) and (J − 1) levels of the lower vibrational state and is denoted by Δ 2 {\displaystyle \Delta _{2}} since it is the difference between levels differing by two units of J. If centrifugal distortion is included, it is given by

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