Rotational–vibrational spectroscopy is a branch of molecular
spectroscopy concerned with
infrared
Infrared (IR), sometimes called infrared light, is electromagnetic radiation (EMR) with wavelengths longer than those of visible light. It is therefore invisible to the human eye. IR is generally understood to encompass wavelengths from around ...
and
Raman spectra of
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
s in the
gas phase
In the physical sciences, a phase is a region of space (a thermodynamic system), throughout which all physical properties of a material are essentially uniform. Examples of physical properties include density, index of refraction, magnetiza ...
. Transitions involving changes in both
vibrational and
rotational
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
states can be abbreviated as rovibrational (or ro-vibrational) transitions. When such transitions emit or absorb
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
s (
electromagnetic radiation
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) li ...
), the
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
is proportional to the difference in energy levels and can be detected by certain kinds of
spectroscopy. Since changes in rotational
energy level
A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The t ...
s 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
Rotational spectroscopy is concerned with the measurement of the energies of transitions between quantized rotational states of molecules in the gas phase. The spectra of polar molecules can be measured in absorption or emission by microwave ...
gives the rotational
quantum numbers, energy levels, and
selection rule
In physics and chemistry, a selection rule, or transition rule, formally constrains the possible transitions of a system from one quantum state to another. Selection rules have been derived for electromagnetic transitions in molecules, in atoms, in ...
s. 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
Water (chemical formula ) is an Inorganic compound, inorganic, transparent, tasteless, odorless, and Color of water, nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living ...
is important because of the presence of water vapor in the atmosphere.
Overview
Ro-vibrational spectroscopy concerns molecules in the
gas phase
In the physical sciences, a phase is a region of space (a thermodynamic system), throughout which all physical properties of a material are essentially uniform. Examples of physical properties include density, index of refraction, magnetiza ...
. 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''.
[Hollas p101]
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.
[Traditionally, infrared spectra are shown with the wavenumber scale decreasing from left to right, corresponding to increasing wavelength. More modern texts may show the wavenumber scale increasing from left to right. The P-branch is always at lower wavenumbers than the Q-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
In rotordynamics, the rigid rotor is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rig ...
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
In rotordynamics, the rigid rotor is a mechanical model of Rotation, rotating systems. An arbitrary rigid rotor is a 3-dimensional Rigid body, rigid object, such as a top. To orient such an object in space requires three angles, known as Euler an ...
.
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 (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
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 (or
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
) 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
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the res ...
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. (See
Appendix)
Similarly the difference ''R''(''J'') − ''P''(''J'') depends only on the constants ''B''′ and ''D''′ for the excited vibrational state (''v'' = 1), and ''B''′ can be used to determine the internuclear distance in that state (which is inaccessible to pure rotational spectroscopy).
:
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
,
[Term value is directly related to energy by ] 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
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
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 (in the
Born–Oppenheimer approximation
In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the best-known mathematical approximation in molecular dynamics. Specifically, it is the assumption that the wave functions of atomic nuclei and elect ...
) by combining the expressions for vibration and rotation.
: