Rotational Spectroscopy

Rotational spectrum from a molecule (to first order) requires that the molecule have a Dipole Moment , that is a difference between the center of Charge and the Center Of Mass , or equivalently a separation between two unlike charges. It is this dipole moment that enables the Electric Field of the Light ( Microwave ) to exert a Torque on the molecule causing it to rotate more quickly (in Excitation ) or slowly (in de-excitation). Diatomic Molecules such as Oxygen (O₂) , Hydrogen (H₂) , etc. do not have a dipole moment and hence no pure-rotational spectrum. However, electronic excitation can lead to asymmetric charge distribution and thus providing a net dipole moment to the molecule. Under such circumstances, these molecules will exhibit a rotational spectrum.

Amongst the for a non-linear molecule. As the number of atoms increases the spectrum becomes more complex as lines due to different transitions start overlapping.

UNDERSTANDING THE ROTATIONAL SPECTRUM

In Quantum Mechanics the free rotation of a molecule is quantized, that is the Rotational Energy and the Angular Momentum can only take certain fixed values; what these values are is simply related to the Moment Of Inertia ,

I

, of the molecule. In general for any molecule, there are three moments of inertia viz.

I_A

I_B

and

I_C

about three mutually orthogonal axes

A,B,

and

C

with the origin at the Center Of Mass of the system. A linear molecule is a special case in this regard. These molecules are cylindrically symmetric and one of the moment of inertia (

I_A

, which is the moment of inertia for a rotation taking place along the axis of the molecule) is negligible (i.e.

I_A << I_B = I_C

).

Classification of molecules based on rotational behavior

The general convention is to define the axes such that the axis

A

has the smallest moment of inertia (and hence the highest rotational frequency) and other axes such that

I_A <= I_B <= I_C

. Sometimes the axis

A

may be associated with the symmetric axis of the molecule, if any. If such is the case, then

I_A

need not be the smallest moment of inertia. To avoid confusion, we will stick with the former convention for the rest of the article. The particular pattern of Energy Level s (and hence of transitions in the rotational spectrum) for a molecule is determined by its symmetry. A convenient way to look at the molecules is to divide them into four different classes (based on the symmetry of their structure). These are,

# Linear molecules (or linear rotors)
# Symmetric tops (or symmetic rotors)
# Spherical tops (or spherical rotors) and
# Asymmetric tops

Dealing with each in turn:

# ''Linear molecules:''

As mentioned earlier, for a linear molecule $I_A << I_B = I_C$ . For most of the purposes, $I_A$ is taken to be zero. For a linear molecule, the separation of lines in the rotational spectrum can be related directly to the moment of inertia of the molecule, and for a molecule of known atomic masses, can be used to determine the Bond Length s (structure) directly. For Diatomic Molecules this process is trivial, and can be made from a single measurement of the rotational spectrum. For linear molecules with more atoms, rather more work is required, and it is necessary to measure molecules in which more than one isotope of each atom have been substituted (effectively this gives rise to a set of simultaneous equations which can be solved for the Bond Length s).

Examples or linear molecules: Oxygen (O=O) , Carbon Monoxide (O≡C^---) , Hydroxy Radical (OH) , Carbon Dioxide (O=C=O) , Hydrogen Cyanide (HC≡N) , Carbonyl Sulfide (O=C=S) , Chloroethyne (HC≡CCl) , Acetylene (HC≡CH)

A symmetric top is a molecule in which two moments of inertia are the same. As a matter of historical convenience, spectroscopists divide molecules into two classes of symmetric tops, '' Oblate symmetric tops'' (saucer or disc shaped) with $I_A = I_B < I_C$ and '' Prolate symmetric tops'' (rugby football, or cigar shaped) with $I_A < I_B = I_C$ . The spectra look rather different, and are instantly recognizable. As for linear molecules, the structure of ''symmetric tops'' ( Bond Length s and Bond Angles ) can be deduced from their spectra.

Examples of symmetric tops:

--- , Cyclobutadiene (C₄H₄) , Ammonia (NH₃)

--- , Propyne (CH₃C≡CH)

A spherical top molecule, can be considered as a special case of ''symmetric tops'' with equal moment of inertia about all three axes ( $I_A = I_B = I_C$ ).

Examples of spherical tops: Phosphorus Tetramer (P₄) , Carbon Tetrachloride (CCl₄) , Nitrogen Tetrahydride (NH₄) , Ammonium Ion (NH₄⁺) , Sulfur Hexafluoride (SF₆)

As you would have guessed a molecule is termed an asymmetric top if it has all three moments of inertia are different. Most of the larger molecules are asymmetric tops, even when they have a high degree of symmetry. Generally for such molecules a simple interpretation of the spectrum is not normally possible. Sometimes asymmetric tops have spectra that are similar to those of a linear molecule or a symmetric top, in which case the molecular structure must also be similar to that of a linear molecule or a symmetric top. For the most general case however, all that can be done is to fit the spectra to three different moments of inertia. If the molecular formula is known, then educated guesses can be made of the possible structure, and from this guessed structure, the moments of inertia can be calculated. If the calculated moments of inertia agree well with the measured moments of inertia, then the structure can be said to have been determined. For this approach to determining molecular structure, isotopic substitution is invaluable.

Examples of asymmetric tops: Anthracene (C₁₄H₁₀) , Water (H₂O) , Nitrogen Dioxide (NO₂)

Structure of rotational spectrum

Linear molecules

I_B = I_C

I_A = 0

J

The rotational energy levels (

F \left( J 
ight)

) of the molecule based on Rigid Rotor model can be expressed as,

:

F\left( J 
ight) = \bar B_{e} J \left( J+1 
ight)  \qquad  J = 0,1,2,...

where

\bar B_e

is the rotational constant of the molecule and is related to the moment of inertia of the molecule

I_B = I_C

by,

:

\bar B_e = {h \over{8\pi^2cI_B}}

Selection rules dictate that during emission or absorption the rotational quantum number has to change by unity i.e.

\Delta J = J^{\prime} - J^{\prime\prime} = \pm 1

. Thus the locations of the lines in a rotational spectrum will be given by,

:

\bar 
u_{J^{\prime}\leftrightarrow J^{\prime\prime}} = F\left( J^{\prime} 
ight) - F\left( J^{\prime\prime} 
ight) = 2 \bar B_e \left( J^{\prime\prime} + 1 
ight) \qquad  J^{\prime\prime} = 0,1,2,...

where

J^{\prime\prime}

denotes the lower energy level and

J^{\prime}

denotes higher energy level involved in the transition. The height of the lines is determined by the distribution of the molecules in the different levels and the probability of transition between two energy levels.

We observe that, for a rigid rotor, the transition lines are equally spaced in the wavenumber space. However, this is not always the case, except for the rigid rotor model. For Non-rigid Rotor Model , we need to consider changes in the moment of inertia of the molecule. Two primary reasons for this are,

Centrifugal distortion:

Centrifugal Force

\bar B_e

F\left( J 
ight) = \bar B_{e} J \left( J+1 
ight) - \bar D_{e} J^2 \left( J+1 
ight)^2 \qquad  J = 0,1,2,...

where

\bar D_e

is the centrifugal distortion constant.

Accordingly the line spacing for the rotational mode changes to,

:

\bar 
u_{J^{\prime}\leftrightarrow J^{\prime\prime}} = 2 \bar B_e \left( J^{\prime\prime} + 1 
ight)  - 4\bar D_e \left( J^{\prime\prime} +1 
ight)^3 \qquad  J^{\prime\prime} = 0,1,2,...

Effect of vibration on rotation:

Symmetric Top

The rotational motion of a symmetric top molecule can be described by two independent rotational quantum number (since two axes have equal moment of inertia, the rotational motion about these axes requires only one rotational quantum number for complete description). Instead of defining the two rotational quantum number for two independent axes, we associate one of the quantum number (

J

) with the total angular momentum of the molecule and the other quantum number (

K

) with the angular momentum of the axis which has different moment of inertia (i.e. axis

C

for oblate symmetric top and axis

A

for prolate symmetric tops). The rotational energy

F\left(J,K
ight)

of such a molecule, based on rigid rotor assumptions can be expressed in terms of the two previously defined rotational quantum number as follows,

:

F\left( J,K 
ight) = \bar B J \left( J+1 
ight) + \left( \bar A - \bar B 
ight) K^2 \qquad  J = 0,1,2,... \quad \mbox{and}\quad K = -J, -J+1, ...,-1, 0, 1, ..., J-1, J

where

\bar B = {h\over{8\pi^2cI_B}}

and

\bar A = {h\over{8\pi^2cI_A}}

for a ''prolate'' symmetric top molecule or

\bar A = {h\over{8\pi^2cI_C}}

for an ''oblate'' molecule.

Selection rule for the these molecules provide the guidelines for possible transitions. Accordingly,
:

\Delta J = \pm 1 \quad \mbox{and} \quad \Delta K = 0

.
This is so because

K

is associated with the axis about which the molecule is symmetric and hence has no net dipole moment in that direction. Thus there is no interaction of this mode with the light particles (photon).

This gives the transition wavenumbers of,

:

\bar 
u_{J^{\prime}\leftrightarrow J^{\prime\prime},K} = F\left( J^{\prime},K 
ight) - F\left( J^{\prime\prime},K 
ight) = 2 \bar B \left( J^{\prime\prime} + 1 
ight) \qquad  J^{\prime\prime} = 0,1,2,...

which is same as in case of linear molecule.

In case of non-rigid rotors, the first order centrifugal distortion correction is given by,

:

F\left( J,K 
ight) = \bar B J \left( J+1 
ight) + \left( \bar A - \bar B 
ight) K^2 - \bar D_J J^2\left(J+1
ight)^2 - \bar D_{JK}J\left(J+1
ight)K^2 - D_KK^4 \qquad  J = 0,1,2,... \quad \mbox{and}\quad K = -J,...,0, ..., J

The suffixes on the centrifugal distortion constant

D

indicate the rotational mode involved and are not a function of the rotational quantum number. The location of the transition lines on a spectrum are given by,

:

\bar 
u_{J^{\prime}\leftrightarrow J^{\prime\prime},K} = F\left( J^{\prime},K 
ight) - F\left( J^{\prime\prime},K 
ight) = 2 \bar B \left( J^{\prime\prime} + 1 
ight) -4D_J\left(J^{\prime\prime}+1
ight)^3 - 2D_{JK}\left(J^{\prime\prime}+1
ight)K^2 \qquad  J^{\prime\prime} = 0,1,2,...

Spherical Tops

Unlike other molecules, spherical top molecules have no net dipole moment, and hence they do not exhibit any rotational spectrum.

Asymmetric Tops

The spectrum for these molecules usually involves many lines due to three different rotational modes and there combinations. There is no general rule for studying the rotational spectum of these molecules.

Hyperfine interactions:

In addition to the main structure that is observed in microwave spectra that is due to the rotational motion of the molecules, a whole host of further interactions are responsible for small details in the spectra, and the study of these details provides a very deep understanding of molecular quantum mechanics. The main interactions responsible for small changes in the spectra (additional splittings and shifts of lines) are due to magnetic and electrostatic interactions in the molecule. The particular strength of such interactions differs in different molecules, but in general, the order of these effects (in decreasing significance) is:

# ''electron spin - electron spin interaction'' (this occurs in molecules with two or more unpaired electrons, and is a magnetic-dipole / magnetic-dipole interaction)
# ''electron spin - molecular rotation'' (the rotation of a molecule corresponds to a magnetic dipole, which interacts with the magnetic dipole moment of the electron)
# ''electron spin - nuclear spin interaction'' (the interaction between the magnetic dipole moment of the electron and the magnetic dipole moment of the nuclei (if present)).
# ''electric field gradient - nuclear electric quadrupole interaction'' (the interaction between the electric field gradient of the electron cloud of the molecule and the electric quadrupole moments of nuclei (if present)).
# ''nuclear spin - nuclear spin interaction'' (nuclear magnetic moments interacting with one another).

These interactions give rise to the characteristic energy levels that are probed in "magnetic resonance" spectroscopy such as NMR and ESR , where they represent the "zero field splittings" which are always present.

EXPERIMENTAL DETERMINATION OF THE SPECTRUM

APPLICATIONS

RESOURCES

''Microwave Spectroscopy'', Townes and Schawlow, Dover;

''Rotational Spectroscopy'', Harry Kroto, Dover;

''Rotational Spectroscopy of Diatomic molecules'', Brown and Carrington;

''Quantum Mechanics'', Mcquarrie, Donald A.

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Rotational Spectroscopy