Rovibrational coupling
Encyclopedia
Rovibrational coupling is a coupled rotational and vibrational excitation of a molecule
. It is different from rovibronic coupling
, which involves a change in all of electronic, vibrational, and rotational states simultaneously.
s. Consequently, it is possible to observe both rotational and vibrational transitions in the vibrational spectrum. Although many methods are available for observing vibrational spectra, the two most common methods are infrared spectroscopy
and Raman spectroscopy
.
The energy of rotational transitions is on the order of
J whereas vibrational transitions have energies on the order of 
J. Therefore, highly-resolved vibrational spectra will contain fine structure
corresponding to the rotational transitions that occur at the same time as a vibrational transition. Although molecular vibrations and rotations do have some effect on one other, this interaction is usually small. Consequently, the rotational and vibrational contributions to the energy of the molecule can be considered independently to a first approximation:
where
is the vibrational quantum number
,
is the rotational quantum number
, h is Planck's constant,
is the frequency of the vibration, 
is the speed of light
, and
is the rotational constant
.
. This is because of the vector addition properties of quantum mechanical angular momenta, and because light particles (photons) have angular momenta of 1. In spectroscopy, the transitions where 
are referred to as the P-branch. Transitions with 
are referred to as Q-branch, and 
as R-branch.
For linear molecules the most commonly observed case is that only transitions with
are observed. This is only possible when the molecule has a "singlet" ground state, that is there are no unpaired electron spins in the molecule. For molecules that do have unpaired electrons, Q branches (see below) are commonly observed.
The gap between the R- and P-branches is known as the Q-branch. A peak would appear here for a vibrational transition in which the rotational energy did not change (
). However, according to the quantum mechanical rigid rotor
model upon which rotational spectroscopy is based, there is a spectroscopic selection rule
that requires that
. This selection rule explains why the P- and R-branches are observed, but not the Q-branch (as well as branches for which 
, 
, etc.).
The positions of the peaks in the spectrum can be predicted using the rigid rotor
model. One prediction of the rigid rotor
model is that the space between each of the peaks should be
where 
is the rotational constant
for a given molecule. Experimentally, it is observed that the spacing between the R-branch peaks decreases as the frequency increases. Similarly, the spacing between the P-branch peaks increases as the frequency decreases. This variation in the spacing results from the bonds between the atoms in a molecule not being entirely rigid.
This variation can be mostly accounted for using a slightly more complex model that takes into account the variation in the rotational constant as the vibrational energy changes. Using this model, the positions of the R-branch peaks are predicted to be at:
where
is the rotational constant
for the
vibrational level and 
is the rotational constant
for the
vibrational level. Likewise, the P-branch peaks are predicted to be at:
Rotational-vibrational spectra will also show some fine structure due to the presence of different isotopes in the spectrum. In the spectrum shown above, all of the rotational peaks are slightly split into two peaks. One peak corresponds to 35Cl and the other to 37Cl. The ratio of the peak intensities corresponds to the natural abundance
of these two isotopes.
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
. It is different from rovibronic coupling
Rovibronic coupling
Rovibronic coupling denotes the simultaneous interactions between rotational, vibrational, and electronic degrees of freedom in a molecule. When a rovibronic transition occurs, the rotational, vibrational, and electronic states change simultaneously, unlike in rovibrational coupling...
, which involves a change in all of electronic, vibrational, and rotational states simultaneously.
Rotational-Vibrational Spectroscopy
Generally vibrational transitions occur in conjunction with rotational transitionRotational transition
A rotational transition is an abrupt change in angular momentum in quantum physics. Like all other properties of a quantum particle, angular momentum is quantized, meaning it can only equal certain discrete values, which correspond to different rotational energy states. When a particle loses...
s. Consequently, it is possible to observe both rotational and vibrational transitions in the vibrational spectrum. Although many methods are available for observing vibrational spectra, the two most common methods are infrared spectroscopy
Infrared spectroscopy
Infrared spectroscopy is the spectroscopy that deals with the infrared region of the electromagnetic spectrum, that is light with a longer wavelength and lower frequency than visible light. It covers a range of techniques, mostly based on absorption spectroscopy. As with all spectroscopic...
and Raman spectroscopy
Raman spectroscopy
Raman spectroscopy is a spectroscopic technique used to study vibrational, rotational, and other low-frequency modes in a system.It relies on inelastic scattering, or Raman scattering, of monochromatic light, usually from a laser in the visible, near infrared, or near ultraviolet range...
.
The energy of rotational transitions is on the order of
J whereas vibrational transitions have energies on the order of J. Therefore, highly-resolved vibrational spectra will contain fine structureFine structure
In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to first order relativistic corrections.The gross structure of line spectra is the line spectra predicted by non-relativistic electrons with no spin. For a hydrogenic atom, the gross structure energy...
corresponding to the rotational transitions that occur at the same time as a vibrational transition. Although molecular vibrations and rotations do have some effect on one other, this interaction is usually small. Consequently, the rotational and vibrational contributions to the energy of the molecule can be considered independently to a first approximation:
where
is the vibrational quantum numberQuantum number
Quantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities. This is distinguished from classical mechanics where the values can range continuously...
,
is the rotational quantum numberQuantum number
Quantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities. This is distinguished from classical mechanics where the values can range continuously...
, h is Planck's constant,
is the frequency of the vibration, is the speed of lightSpeed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
, and
is the rotational constantRigid rotor
The rigid rotor is a mechanical model that is used to explain rotating systems.An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space three angles are required. A special rigid rotor is the linear rotor which isa 2-dimensional object, requiring...
.
Evaluating Spectra
The strict selection rule for the absorption of dipole radiation (the strongest component of light) is that. This is because of the vector addition properties of quantum mechanical angular momenta, and because light particles (photons) have angular momenta of 1. In spectroscopy, the transitions where are referred to as the P-branch. Transitions with are referred to as Q-branch, and as R-branch.For linear molecules the most commonly observed case is that only transitions with
are observed. This is only possible when the molecule has a "singlet" ground state, that is there are no unpaired electron spins in the molecule. For molecules that do have unpaired electrons, Q branches (see below) are commonly observed.The gap between the R- and P-branches is known as the Q-branch. A peak would appear here for a vibrational transition in which the rotational energy did not change (
). However, according to the quantum mechanical rigid rotorRigid rotor
The rigid rotor is a mechanical model that is used to explain rotating systems.An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space three angles are required. A special rigid rotor is the linear rotor which isa 2-dimensional object, requiring...
model upon which rotational spectroscopy is based, there is a spectroscopic selection rule
Selection rule
In physics and chemistry a selection rule, or transition rule, formally constrains the possible transitions of a system from one state to another. Selection rules have been derived for electronic, vibrational, and rotational transitions...
that requires that
. This selection rule explains why the P- and R-branches are observed, but not the Q-branch (as well as branches for which , , etc.).The positions of the peaks in the spectrum can be predicted using the rigid rotor
Rigid rotor
The rigid rotor is a mechanical model that is used to explain rotating systems.An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space three angles are required. A special rigid rotor is the linear rotor which isa 2-dimensional object, requiring...
model. One prediction of the rigid rotor
Rigid rotor
The rigid rotor is a mechanical model that is used to explain rotating systems.An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space three angles are required. A special rigid rotor is the linear rotor which isa 2-dimensional object, requiring...
model is that the space between each of the peaks should be
where is the rotational constantRigid rotor
The rigid rotor is a mechanical model that is used to explain rotating systems.An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space three angles are required. A special rigid rotor is the linear rotor which isa 2-dimensional object, requiring...
for a given molecule. Experimentally, it is observed that the spacing between the R-branch peaks decreases as the frequency increases. Similarly, the spacing between the P-branch peaks increases as the frequency decreases. This variation in the spacing results from the bonds between the atoms in a molecule not being entirely rigid.
This variation can be mostly accounted for using a slightly more complex model that takes into account the variation in the rotational constant as the vibrational energy changes. Using this model, the positions of the R-branch peaks are predicted to be at:
where
is the rotational constantRigid rotor
The rigid rotor is a mechanical model that is used to explain rotating systems.An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space three angles are required. A special rigid rotor is the linear rotor which isa 2-dimensional object, requiring...
for the
vibrational level and is the rotational constantRigid rotor
The rigid rotor is a mechanical model that is used to explain rotating systems.An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space three angles are required. A special rigid rotor is the linear rotor which isa 2-dimensional object, requiring...
for the
vibrational level. Likewise, the P-branch peaks are predicted to be at:Rotational-vibrational spectra will also show some fine structure due to the presence of different isotopes in the spectrum. In the spectrum shown above, all of the rotational peaks are slightly split into two peaks. One peak corresponds to 35Cl and the other to 37Cl. The ratio of the peak intensities corresponds to the natural abundance
Natural abundance
In chemistry, natural abundance refers to the abundance of isotopes of a chemical element as naturally found on a planet. The relative atomic mass of these isotopes is the atomic weight listed for the element in the periodic table...
of these two isotopes.