In Case You Were Wondering Just What Microwave Spectroscopy Is

8/13/2019 In Case You Were Wondering Just What Microwave Spectroscopy Is

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In case you were wondering just what microwave spectroscopy is, I hope that some of the

information on this page can help you to understand how we interpret the data that we take and

get interesting and useful chemical results.

When we put a sample into our machine a portion of the sample is in the gas phase, and with a

buffer gas we send this portion into the cavity of the spectrometer. Inside the cavity is a standing

mode of microwave radiation that we can change by adjusting the separation of the mirrors and

the frequency of light input into the chamber.

Figure 1. The Microwave Cavity

Click here to view an image of our spectrometer cavity

We then detect an 'echo' signal on a computer that informs the scanner whether or not themolecule is absorbing that frequency. If there is no absorbance we view noise but if there is

absorbance we view what is called a free induction decay signal.

Figure 2. A Free Induction Decay.

http://www.chem.arizona.edu/kukolich/research/mwspec/spectra/canpump.JPGhttp://www.chem.arizona.edu/kukolich/research/mwspec/spectra/canpump.JPGhttp://www.chem.arizona.edu/kukolich/research/mwspec/spectra/canpump.JPG


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Depending on how many spectral lines are in the region being scanned these FID's can be

simple, a singly oscillating (sinusoidal or cosinusoidal) function with a superimposed exponential

decay. However when more than one line is present in the region being scanned we often see two

or more different frequencies in the FID pattern (such as the one above). These frequencies

correspond to differences between the natural absorbance frequency of the molecule and the

frequency of the light in the cavity. We can accurately determine the natural frequency of themolecule by doing a Fourier Transform of the FID and adding (or subtracting) the difference

frequency from the frequency of the light in the cavity. The Fourier Transform of this FID isshown below.


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Figure 3. The Fourier Transform of the FID above.

After we take data on a few lines we move around the frequency spectrum to find more of themin an effort to decipher the pattern and simoultaneously assign the line their respective quanta.

The two lines shown here are part of a quadruplet caused by the I = 3/2 quadrupole nucleus of theChlorine atom which is in the molecule that was being studied.


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Figure 4. A quadrupole quadruplet pattern in the microwave spectrum of chloroferrocene.

In this case the quadrupole structure is a fine (small scale) part of the spectrum and on a larger

scale the spectrum reveals that the molecule is an asymmetric top. Because asymmetric tops have

three unique axes of rotation there are many more rotational transitions available in complarisonto a symmetric top molecule. This is revealed by the presence of K-states, in which K is a unit of

angular momentum projected onto the molecular axes (in a symmetric top all K states are

equivalent).

Figure 5. A series of K-state transitions in the J=5--4 region of the microwave spectrum of

chloroferrocene.


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The large scale structure of the microwave spectrum reveals that most of the transitions are

grouped into regions, each of these regions is a set of different J level transitions. J is one quanta

of angular momentum of the molecule about it's major rotation axes. Thus a molecule in its' J = 2

state is spinning twice as fast as a molecule in the J = 1 state.

Figure 6. The entire microwave spectrum of chloroferrocene in the 4-11 GHz frequency

range.


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Rotational spectroscopy

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Part of the rotational spectrum of trifluoroiodomethane, CF3I.[notes 1] Each rotational transition is

labeled with the quantum numbers, J , of the final and initial states, and is extensively split by theeffects of nuclear quadrupole coupling with the 127I nucleus.

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 moleculescan be measured in absorption or emission by microwave spectroscopy[1] or by far infrared

spectroscopy. The rotational spectra of non-polar molecules cannot be observed by thosemethods, but can be observed and measured by Raman spectroscopy. Rotational spectroscopy is

http://en.wikipedia.org/wiki/Rotational_spectroscopy#mw-navigationhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#mw-navigationhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#mw-navigationhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#p-searchhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#p-searchhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#p-searchhttp://en.wikipedia.org/wiki/Trifluoroiodomethanehttp://en.wikipedia.org/wiki/Trifluoroiodomethanehttp://en.wikipedia.org/wiki/Trifluoroiodomethanehttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-1http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-1http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-1http://en.wikipedia.org/wiki/Nuclear_quadrupole_resonancehttp://en.wikipedia.org/wiki/Nuclear_quadrupole_resonancehttp://en.wikipedia.org/wiki/Nuclear_quadrupole_resonancehttp://en.wikipedia.org/wiki/Spectroscopyhttp://en.wikipedia.org/wiki/Spectroscopyhttp://en.wikipedia.org/wiki/Spectroscopyhttp://en.wikipedia.org/wiki/Moleculehttp://en.wikipedia.org/wiki/Moleculehttp://en.wikipedia.org/wiki/Moleculehttp://en.wikipedia.org/wiki/Gas_phasehttp://en.wikipedia.org/wiki/Gas_phasehttp://en.wikipedia.org/wiki/Gas_phasehttp://en.wikipedia.org/wiki/Chemical_polarityhttp://en.wikipedia.org/wiki/Chemical_polarityhttp://en.wikipedia.org/wiki/Chemical_polarityhttp://en.wikipedia.org/wiki/Absorption_%28optics%29http://en.wikipedia.org/wiki/Absorption_%28optics%29http://en.wikipedia.org/wiki/Absorption_%28optics%29http://en.wikipedia.org/wiki/Emission_%28electromagnetic_radiation%29http://en.wikipedia.org/wiki/Emission_%28electromagnetic_radiation%29http://en.wikipedia.org/wiki/Emission_%28electromagnetic_radiation%29http://en.wikipedia.org/wiki/Microwavehttp://en.wikipedia.org/wiki/Microwavehttp://en.wikipedia.org/wiki/Microwavehttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-2http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-2http://en.wikipedia.org/wiki/Far_infraredhttp://en.wikipedia.org/wiki/Far_infraredhttp://en.wikipedia.org/wiki/Far_infraredhttp://en.wikipedia.org/wiki/Raman_spectroscopyhttp://en.wikipedia.org/wiki/Raman_spectroscopyhttp://en.wikipedia.org/wiki/Raman_spectroscopyhttp://en.wikipedia.org/wiki/File:CF3I_spectrum2.pnghttp://en.wikipedia.org/wiki/File:CF3I_spectrum2.pnghttp://en.wikipedia.org/wiki/Raman_spectroscopyhttp://en.wikipedia.org/wiki/Far_infraredhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-2http://en.wikipedia.org/wiki/Microwavehttp://en.wikipedia.org/wiki/Emission_%28electromagnetic_radiation%29http://en.wikipedia.org/wiki/Absorption_%28optics%29http://en.wikipedia.org/wiki/Chemical_polarityhttp://en.wikipedia.org/wiki/Gas_phasehttp://en.wikipedia.org/wiki/Moleculehttp://en.wikipedia.org/wiki/Spectroscopyhttp://en.wikipedia.org/wiki/Nuclear_quadrupole_resonancehttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-1http://en.wikipedia.org/wiki/Trifluoroiodomethanehttp://en.wikipedia.org/wiki/Rotational_spectroscopy#p-searchhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#mw-navigation


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sometimes referred to as pure rotational spectroscopy to distinguish it from rotational-vibrational

spectroscopy where changes in rotational energy occur together with changes in vibrational

energy, and also from ro-vibronic spectroscopy (or just vibronic spectroscopy) where rotational,

vibrational and electronic energy changes occur simultaneously.

For rotational spectroscopy, molecules are classified according to symmetry into spherical top,

linear and symmetric top; analytical expressions can be derived for the rotational energy terms of

these molecules. Analytical expressions cannot be derived for the fourth category, asymmetric

top, but spectra can be fitted using numerical methods. The rotational energies are derived

theoretically by considering the molecules to be rigid rotors and then applying extra terms to

account for centrifugal distortion, fine structure, hyperfine structure and Coriolis coupling. Fitting

the spectra to the theoretical expressions gives numerical values of the angular moments of inertia

from which very precise values of molecular bond lengths and angles can be derived in favorablecases. In the presence of an electrostatic field there is Stark splitting which allows molecular

electric dipole moments to be determined.

An important application of rotational spectroscopy is in exploration of the chemical composition

of the interstellar medium using radio telescopes.

Contents

1 Applications

2 Overview

o 2.1 Classification of molecular rotors o 2.2 Selection rules o 2.3 Units o 2.4 Effect of vibration on rotation o 2.5 Effect of rotation on vibrational spectra

3 Structure of rotational spectra

o 3.1 Spherical top o 3.2 Linear molecules

3.2.1 Rotational line intensities 3.2.2 Centrifugal distortion 3.2.3 Oxygen

o 3.3 Symmetric top o 3.4 Asymmetric top

4 Quadrupole splitting

5 Stark and Zeeman effects

6 Rotational Raman spectroscopy 7 Instruments and Methods

o 7.1 Absorption cells and Stark modulation o 7.2 Fourier transform microwave (FTMW) spectroscopy

7.2.1 The Balle-Flygare FTMW spectrometer 7.2.2 The Chirped-Pulse FTMW spectrometer

8 Notes 9 References

10 Bibliography 11 External links

Applications

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/en.wikipedia.org/wiki/Rotational_spectroscopy#Bibliographyhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#External_linkshttp://en.wikipedia.org/wiki/Rotational_spectroscopy#External_linkshttp://en.wikipedia.org/wiki/Rotational_spectroscopy#External_linkshttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Bibliographyhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Referenceshttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Noteshttp://en.wikipedia.org/wiki/Rotational_spectroscopy#The_Chirped-Pulse_FTMW_spectrometerhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#The_Balle-Flygare_FTMW_spectrometerhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Fourier_transform_microwave_.28FTMW.29_spectroscopyhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Absorption_cells_and_Stark_modulationhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Instruments_and_Methodshttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Rotational_Raman_spectroscopyhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Stark_and_Zeeman_effectshttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Quadrupole_splittinghttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Asymmetric_tophttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Symmetric_tophttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Oxygenhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Centrifugal_distortionhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Rotational_line_intensitieshttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Linear_moleculeshttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Spherical_tophttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Structure_of_rotational_spectrahttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Effect_of_rotation_on_vibrational_spectrahttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Effect_of_vibration_on_rotationhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Unitshttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Selection_ruleshttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Classification_of_molecular_rotorshttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Overviewhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#Applicationshttp://en.wikipedia.org/wiki/Radio_telescopehttp://en.wikipedia.org/wiki/Interstellar_mediumhttp://en.wikipedia.org/wiki/Electric_dipole_momenthttp://en.wikipedia.org/wiki/Stark_effecthttp://en.wikipedia.org/wiki/Moment_of_inertiahttp://en.wikipedia.org/wiki/Rigid_rotorhttp://en.wikipedia.org/wiki/Vibronic_spectroscopyhttp://en.wikipedia.org/wiki/Rotational-vibrational_spectroscopyhttp://en.wikipedia.org/wiki/Rotational-vibrational_spectroscopy


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Rotational spectroscopy has primarily been used to investigate fundamental aspects of molecular

physics. It is a uniquely precise tool for the determination of molecular structure in gas phase

molecules. It can be used to establish barriers to internal rotation such as that associated with the

rotation of the CH3 group relative to the C6H4Cl group in chlorotoluene (C7H7Cl).[2]

When fine or

hyperfine structure can be observed, the technique also provides information on the electronic

structures of molecules. Much of current understanding of the nature of weak molecular

interactions such as van der Waals, hydrogen and halogen bonds has been established through

rotational spectroscopy. In connection with radio astronomy, the technique has a key role in

exploration of the chemical composition of the interstellar medium. Microwave transitions are

measured in the laboratory and matched to emissions from the interstellar medium using a radio

telescope. NH3 was the first stable polyatomic molecule to be identified in the interstellarmedium.[3] The measurement of chlorine monoxide[4] is important for atmospheric chemistry.

Current projects in astrochemistry involve both laboratory microwave spectroscopy andobservations made using modern radiotelescopes such as the Atacama Large Millimetre Array

(ALMA).[5]

Unlike NMR , Infrared and UV-Visible spectroscopies, microwave spectroscopy hasnot yet found widespread application in analytical chemistry.

Overview

A molecule in the gas phase is free to rotate relative to a set of mutually orthogonal axes of fixed

orientation in space, centered on the center of mass of the molecule. Free rotation is not possible

for molecules in liquid or solid phases due to the presence of intermolecular forces. Rotation

about each unique axis is associated with a set of quantized energy levels dependent on the

moment of inertia about that axis and a quantum number. Thus, for linear molecules the energy

levels are described by a single moment of inertia and a single quantum number, J. For symmetric

tops there are two moments of inertia and two rotational quantum numbers to consider. Analysis

of spectroscopic data with the expressions detailed below results in quantitative determination of

the value(s) of the moment(s) of inertia. From these precise values of the molecular structure and

dimensions may be obtained.

For a linear molecule, analysis of the rotational spectrum provides values for the rotational

constant and the moment of inertia of the molecule, and, knowing the atomic masses, can be used

to determine the bond length directly. For diatomic molecules this process is straightforward. For

linear molecules with more than two atoms it is necessary to measure the spectra of two or moreisotopologues, such as 16O12C32S and 16O12C34S. This allows a set of simultaneous equations to be

set up and solved for the bond lengths).[notes 2]

It should be noted that a bond length obtained inthis way is slightly different from the equilibrium bond length. This is because there is zero-point

energy in the vibrational ground state, to which the rotational states refer, whereas the equilibrium

bond length is at the minimum in the potential energy curve. The relation between the rotationalconstants is given by

where ν is a vibrational quantum number and α is a vibration-rotation interaction constant whichcan be calculated if the B values for two different vibrational states can be found.[6]

For other molecules, if the spectra can be resolved and individual transitions assigned both bond

lengths and bond angles can be deduced. When this is not possible, as with most asymmetric tops,

all that can be done is to fit the spectra to three moments of inertia calculated from an assumedmolecular structure. By varying the molecular structure the fit can be improved, giving a

http://en.wikipedia.org/wiki/Chlorotoluenehttp://en.wikipedia.org/wiki/Chlorotoluenehttp://en.wikipedia.org/wiki/Chlorotoluenehttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-3http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-3http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-3http://en.wikipedia.org/wiki/Van_der_Waalshttp://en.wikipedia.org/wiki/Van_der_Waalshttp://en.wikipedia.org/wiki/Van_der_Waalshttp://en.wikipedia.org/wiki/Hydrogen_bondhttp://en.wikipedia.org/wiki/Hydrogen_bondhttp://en.wikipedia.org/wiki/Hydrogen_bondhttp://en.wikipedia.org/wiki/Halogen_bondhttp://en.wikipedia.org/wiki/Halogen_bondhttp://en.wikipedia.org/wiki/Halogen_bondhttp://en.wikipedia.org/wiki/Radio_astronomyhttp://en.wikipedia.org/wiki/Radio_astronomyhttp://en.wikipedia.org/wiki/Radio_astronomyhttp://en.wikipedia.org/wiki/Interstellar_mediumhttp://en.wikipedia.org/wiki/Interstellar_mediumhttp://en.wikipedia.org/wiki/Interstellar_mediumhttp://en.wikipedia.org/wiki/Interstellar_mediumhttp://en.wikipedia.org/wiki/Interstellar_mediumhttp://en.wikipedia.org/wiki/Interstellar_mediumhttp://en.wikipedia.org/wiki/Radio_telescopehttp://en.wikipedia.org/wiki/Radio_telescopehttp://en.wikipedia.org/wiki/Radio_telescopehttp://en.wikipedia.org/wiki/Radio_telescopehttp://en.wikipedia.org/wiki/Ammoniahttp://en.wikipedia.org/wiki/Ammoniahttp://en.wikipedia.org/wiki/Ammoniahttp://en.wikipedia.org/wiki/Ammoniahttp://en.wikipedia.org/wiki/Polyatomichttp://en.wikipedia.org/wiki/Polyatomichttp://en.wikipedia.org/wiki/Polyatomichttp://en.wikipedia.org/wiki/Interstellar_mediumhttp://en.wikipedia.org/wiki/Interstellar_mediumhttp://en.wikipedia.org/wiki/Interstellar_mediumhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-4http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-4http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-4http://en.wikipedia.org/wiki/Chlorine_monoxidehttp://en.wikipedia.org/wiki/Chlorine_monoxidehttp://en.wikipedia.org/wiki/Chlorine_monoxidehttp://en.wikipedia.org/wiki/Chlorine_monoxidehttp://en.wikipedia.org/wiki/Atmospheric_chemistryhttp://en.wikipedia.org/wiki/Atmospheric_chemistryhttp://en.wikipedia.org/wiki/Atmospheric_chemistryhttp://en.wikipedia.org/wiki/Atacama_Large_Millimeter_Arrayhttp://en.wikipedia.org/wiki/Atacama_Large_Millimeter_Arrayhttp://en.wikipedia.org/wiki/Atacama_Large_Millimeter_Arrayhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-6http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-6http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-6http://en.wikipedia.org/wiki/Nuclear_magnetic_resonancehttp://en.wikipedia.org/wiki/Nuclear_magnetic_resonancehttp://en.wikipedia.org/wiki/Nuclear_magnetic_resonancehttp://en.wikipedia.org/wiki/Infrared_spectroscopyhttp://en.wikipedia.org/wiki/Infrared_spectroscopyhttp://en.wikipedia.org/wiki/Infrared_spectroscopyhttp://en.wikipedia.org/wiki/UV-visible_spectroscopyhttp://en.wikipedia.org/wiki/UV-visible_spectroscopyhttp://en.wikipedia.org/wiki/UV-visible_spectroscopyhttp://en.wikipedia.org/wiki/Analytical_chemistryhttp://en.wikipedia.org/wiki/Analytical_chemistryhttp://en.wikipedia.org/wiki/Analytical_chemistryhttp://en.wikipedia.org/wiki/Gas_phasehttp://en.wikipedia.org/wiki/Gas_phasehttp://en.wikipedia.org/wiki/Gas_phasehttp://en.wikipedia.org/wiki/Orthogonalhttp://en.wikipedia.org/wiki/Orthogonalhttp://en.wikipedia.org/wiki/Orthogonalhttp://en.wikipedia.org/wiki/Center_of_masshttp://en.wikipedia.org/wiki/Center_of_masshttp://en.wikipedia.org/wiki/Center_of_masshttp://en.wikipedia.org/wiki/Intermolecular_forcehttp://en.wikipedia.org/wiki/Intermolecular_forcehttp://en.wikipedia.org/wiki/Intermolecular_forcehttp://en.wikipedia.org/wiki/Rigid_rotor#Quantum_mechanical_linear_rigid_rotorhttp://en.wikipedia.org/wiki/Rigid_rotor#Quantum_mechanical_linear_rigid_rotorhttp://en.wikipedia.org/wiki/Rigid_rotor#Quantum_mechanical_linear_rigid_rotorhttp://en.wikipedia.org/wiki/Rigid_rotor#Quantum_mechanical_linear_rigid_rotorhttp://en.wikipedia.org/wiki/Bond_lengthhttp://en.wikipedia.org/wiki/Bond_lengthhttp://en.wikipedia.org/wiki/Bond_lengthhttp://en.wikipedia.org/wiki/Diatomichttp://en.wikipedia.org/wiki/Diatomichttp://en.wikipedia.org/wiki/Diatomichttp://en.wikipedia.org/wiki/Isotopologuehttp://en.wikipedia.org/wiki/Isotopologuehttp://en.wikipedia.org/wiki/Simultaneous_equationshttp://en.wikipedia.org/wiki/Simultaneous_equationshttp://en.wikipedia.org/wiki/Simultaneous_equationshttp://en.wikipedia.org/wiki/Bond_lengthhttp://en.wikipedia.org/wiki/Bond_lengthhttp://en.wikipedia.org/wiki/Bond_lengthhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-7http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-7http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-7http://en.wikipedia.org/wiki/Zero-point_energyhttp://en.wikipedia.org/wiki/Zero-point_energyhttp://en.wikipedia.org/wiki/Zero-point_energyhttp://en.wikipedia.org/wiki/Zero-point_energyhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-8http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-8http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-8http://en.wikipedia.org/wiki/Bond_lengthhttp://en.wikipedia.org/wiki/Bond_lengthhttp://en.wikipedia.org/wiki/Bond_lengthhttp://en.wikipedia.org/wiki/Bond_lengthhttp://en.wikipedia.org/wiki/Molecular_geometryhttp://en.wikipedia.org/wiki/Molecular_geometryhttp://en.wikipedia.org/wiki/Molecular_geometryhttp://en.wikipedia.org/wiki/Molecular_geometryhttp://en.wikipedia.org/wiki/Bond_lengthhttp://en.wikipedia.org/wiki/Bond_lengthhttp://en.wikipedia.org/wiki/Bond_lengthhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-8http://en.wikipedia.org/wiki/Zero-point_energyhttp://en.wikipedia.org/wiki/Zero-point_energyhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-7http://en.wikipedia.org/wiki/Bond_lengthhttp://en.wikipedia.org/wiki/Simultaneous_equationshttp://en.wikipedia.org/wiki/Isotopologuehttp://en.wikipedia.org/wiki/Diatomichttp://en.wikipedia.org/wiki/Bond_lengthhttp://en.wikipedia.org/wiki/Rigid_rotor#Quantum_mechanical_linear_rigid_rotorhttp://en.wikipedia.org/wiki/Rigid_rotor#Quantum_mechanical_linear_rigid_rotorhttp://en.wikipedia.org/wiki/Rigid_rotor#Quantum_mechanical_linear_rigid_rotorhttp://en.wikipedia.org/wiki/Intermolecular_forcehttp://en.wikipedia.org/wiki/Center_of_masshttp://en.wikipedia.org/wiki/Orthogonalhttp://en.wikipedia.org/wiki/Gas_phasehttp://en.wikipedia.org/wiki/Analytical_chemistryhttp://en.wikipedia.org/wiki/UV-visible_spectroscopyhttp://en.wikipedia.org/wiki/Infrared_spectroscopyhttp://en.wikipedia.org/wiki/Nuclear_magnetic_resonancehttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-6http://en.wikipedia.org/wiki/Atacama_Large_Millimeter_Arrayhttp://en.wikipedia.org/wiki/Atmospheric_chemistryhttp://en.wikipedia.org/wiki/Chlorine_monoxidehttp://en.wikipedia.org/wiki/Chlorine_monoxidehttp://en.wikipedia.org/wiki/Chlorine_monoxidehttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-4http://en.wikipedia.org/wiki/Interstellar_mediumhttp://en.wikipedia.org/wiki/Interstellar_mediumhttp://en.wikipedia.org/wiki/Interstellar_mediumhttp://en.wikipedia.org/wiki/Polyatomichttp://en.wikipedia.org/wiki/Ammoniahttp://en.wikipedia.org/wiki/Radio_telescopehttp://en.wikipedia.org/wiki/Radio_telescopehttp://en.wikipedia.org/wiki/Radio_telescopehttp://en.wikipedia.org/wiki/Interstellar_mediumhttp://en.wikipedia.org/wiki/Interstellar_mediumhttp://en.wikipedia.org/wiki/Radio_astronomyhttp://en.wikipedia.org/wiki/Halogen_bondhttp://en.wikipedia.org/wiki/Hydrogen_bondhttp://en.wikipedia.org/wiki/Van_der_Waalshttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-3http://en.wikipedia.org/wiki/Chlorotoluene


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qualitative estimate of the structure. Isotopic substitution is invaluable when using this approach

to the determination of molecular structure.

Classification of molecular rotors

In quantum mechanics the free rotation of a molecule is quantized, so that the rotational energy

and the angular momentum can take only certain fixed values, which are related simply to themoment of inertia, , of the molecule. For any molecule, there are three moments of inertia: ,

and about three mutually orthogonal axes A, B, and C with the origin at the center of mass

of the system. The general convention, used in this article, is to define the axes such that

, with axis corresponding to the smallest moment of inertia. Some authors,

however, define the axis as the molecular rotation axis of highest order.

The particular pattern of energy levels (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

Spherical tops (spherical rotors) All three moments of inertia are equal to each other:. Examples of spherical tops include phosphorus tetramer (P4), carbon

tetrachloride (CCl4) and other tetrahalides, methane (CH4), silane, (SiH4), sulfur

hexafluoride (SF6) and other hexahalides. The molecules all belong to the cubic point

groups Td or Oh.

Linear molecules. For a linear molecule the moments of inertia are related by

. For most purposes, can be taken to be zero. Examples of linear

molecules include dioxygen, O2, dinitrogen, N2, carbon monoxide, CO, hydroxy radical,

OH, carbon dioxide, CO2, hydrogen cyanide, HCN, carbonyl sulfide, OCS, acetylene

(ethyne, HC≡CH) and dihaloethynes. These molecules belong to the point groups C∞v orD∞h

Symmetric tops (symmetric rotors) A symmetric top is a molecule in which two moments

of inertia are the same, or . By definition a symmetric top must have

a 3-fold or higher order rotation axis. As a matter of convenience, spectroscopists divide

molecules into two classes of symmetric tops, Oblate symmetric tops (saucer or disc

shaped) with and Prolate symmetric tops (rugby football, or cigar

shaped) with . The spectra look rather different, and are instantly

recognizable. Examples of symmetric tops include

Oblate: benzene, C6H6, ammonia, NH3 Prolate: chloromethane, CH3Cl, propyne, CH3C≡CH

As a detailed example, ammonia has a moment of inertia IC = 4.4128 × 10−47 kg m2 about

the 3-fold rotation axis, and moments IA = IB = 2.8059 × 10−47 kg m2 about any axis

perpendicular to the C3 axis. Since the unique moment of inertia is larger than the other

two, the molecule is an oblate symmetric top.[7]

Asymmetric tops (asymmetric rotors) The three moments of inertia have different values.

Examples of small molecules that are asymmetric tops include water, H2O and nitrogen

dioxide, NO2 whose symmetry axis of highest order is a 2-fold rotation axis. Most large

molecules are asymmetric tops.

Selection rules

http://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Angular_momentum_quantizationhttp://en.wikipedia.org/wiki/Angular_momentum_quantizationhttp://en.wikipedia.org/wiki/Angular_momentum_quantizationhttp://en.wikipedia.org/wiki/Rotational_energyhttp://en.wikipedia.org/wiki/Rotational_energyhttp://en.wikipedia.org/wiki/Rotational_energyhttp://en.wikipedia.org/wiki/Angular_momentumhttp://en.wikipedia.org/wiki/Angular_momentumhttp://en.wikipedia.org/wiki/Angular_momentumhttp://en.wikipedia.org/wiki/Moment_of_inertiahttp://en.wikipedia.org/wiki/Moment_of_inertiahttp://en.wikipedia.org/wiki/Center_of_masshttp://en.wikipedia.org/wiki/Center_of_masshttp://en.wikipedia.org/wiki/Center_of_masshttp://en.wikipedia.org/wiki/Molecular_symmetry#Elementshttp://en.wikipedia.org/wiki/Molecular_symmetry#Elementshttp://en.wikipedia.org/wiki/Molecular_symmetry#Elementshttp://en.wikipedia.org/wiki/Energy_levelhttp://en.wikipedia.org/wiki/Energy_levelhttp://en.wikipedia.org/wiki/Energy_levelhttp://en.wikipedia.org/wiki/Allotropes_of_phosphorus#White_phosphorushttp://en.wikipedia.org/wiki/Allotropes_of_phosphorus#White_phosphorushttp://en.wikipedia.org/wiki/Allotropes_of_phosphorus#White_phosphorushttp://en.wikipedia.org/wiki/Allotropes_of_phosphorus#White_phosphorushttp://en.wikipedia.org/wiki/Allotropes_of_phosphorus#White_phosphorushttp://en.wikipedia.org/wiki/Carbon_tetrachloridehttp://en.wikipedia.org/wiki/Carbon_tetrachloridehttp://en.wikipedia.org/wiki/Carbon_tetrachloridehttp://en.wikipedia.org/wiki/Carbon_tetrachloridehttp://en.wikipedia.org/wiki/Carbon_tetrachloridehttp://en.wikipedia.org/wiki/Carbon_tetrachloridehttp://en.wikipedia.org/wiki/Methanehttp://en.wikipedia.org/wiki/Methanehttp://en.wikipedia.org/wiki/Methanehttp://en.wikipedia.org/wiki/Methanehttp://en.wikipedia.org/wiki/Methanehttp://en.wikipedia.org/wiki/Silanehttp://en.wikipedia.org/wiki/Silanehttp://en.wikipedia.org/wiki/Silanehttp://en.wikipedia.org/wiki/Silanehttp://en.wikipedia.org/wiki/Silanehttp://en.wikipedia.org/wiki/Sulfur_hexafluoridehttp://en.wikipedia.org/wiki/Sulfur_hexafluoridehttp://en.wikipedia.org/wiki/Sulfur_hexafluoridehttp://en.wikipedia.org/wiki/Sulfur_hexafluoridehttp://en.wikipedia.org/wiki/Sulfur_hexafluoridehttp://en.wikipedia.org/wiki/Sulfur_hexafluoridehttp://en.wikipedia.org/wiki/Molecular_point_grouphttp://en.wikipedia.org/wiki/Molecular_point_grouphttp://en.wikipedia.org/wiki/Molecular_point_grouphttp://en.wikipedia.org/wiki/Molecular_point_grouphttp://en.wikipedia.org/wiki/Oxygenhttp://en.wikipedia.org/wiki/Oxygenhttp://en.wikipedia.org/wiki/Oxygenhttp://en.wikipedia.org/wiki/Oxygenhttp://en.wikipedia.org/wiki/Nitrogenhttp://en.wikipedia.org/wiki/Nitrogenhttp://en.wikipedia.org/wiki/Nitrogenhttp://en.wikipedia.org/wiki/Nitrogenhttp://en.wikipedia.org/wiki/Carbon_monoxidehttp://en.wikipedia.org/wiki/Carbon_monoxidehttp://en.wikipedia.org/wiki/Carbon_monoxidehttp://en.wikipedia.org/wiki/Hydroxyl_radicalhttp://en.wikipedia.org/wiki/Hydroxyl_radicalhttp://en.wikipedia.org/wiki/Hydroxyl_radicalhttp://en.wikipedia.org/wiki/Hydroxyl_radicalhttp://en.wikipedia.org/wiki/Carbon_dioxidehttp://en.wikipedia.org/wiki/Carbon_dioxidehttp://en.wikipedia.org/wiki/Carbon_dioxidehttp://en.wikipedia.org/wiki/Carbon_dioxidehttp://en.wikipedia.org/wiki/Hydrogen_cyanidehttp://en.wikipedia.org/wiki/Hydrogen_cyanidehttp://en.wikipedia.org/wiki/Hydrogen_cyanidehttp://en.wikipedia.org/wiki/Carbonyl_sulfidehttp://en.wikipedia.org/wiki/Carbonyl_sulfidehttp://en.wikipedia.org/wiki/Carbonyl_sulfidehttp://en.wikipedia.org/wiki/Acetylenehttp://en.wikipedia.org/wiki/Acetylenehttp://en.wikipedia.org/wiki/Acetylenehttp://en.wikipedia.org/wiki/Acetylenehttp://en.wikipedia.org/wiki/Molecular_symmetry#Elementshttp://en.wikipedia.org/wiki/Molecular_symmetry#Elementshttp://en.wikipedia.org/wiki/Molecular_symmetry#Elementshttp://en.wikipedia.org/wiki/Oblate_spheroidhttp://en.wikipedia.org/wiki/Oblate_spheroidhttp://en.wikipedia.org/wiki/Oblate_spheroidhttp://en.wikipedia.org/wiki/Prolatehttp://en.wikipedia.org/wiki/Prolatehttp://en.wikipedia.org/wiki/Prolatehttp://en.wikipedia.org/wiki/Oblate_spheroidhttp://en.wikipedia.org/wiki/Oblate_spheroidhttp://en.wikipedia.org/wiki/Benzenehttp://en.wikipedia.org/wiki/Benzenehttp://en.wikipedia.org/wiki/Benzenehttp://en.wikipedia.org/wiki/Benzenehttp://en.wikipedia.org/wiki/Benzenehttp://en.wikipedia.org/wiki/Benzenehttp://en.wikipedia.org/wiki/Ammoniahttp://en.wikipedia.org/wiki/Ammoniahttp://en.wikipedia.org/wiki/Ammoniahttp://en.wikipedia.org/wiki/Ammoniahttp://en.wikipedia.org/wiki/Prolatehttp://en.wikipedia.org/wiki/Prolatehttp://en.wikipedia.org/wiki/Chloromethanehttp://en.wikipedia.org/wiki/Chloromethanehttp://en.wikipedia.org/wiki/Chloromethanehttp://en.wikipedia.org/wiki/Chloromethanehttp://en.wikipedia.org/wiki/Chloromethanehttp://en.wikipedia.org/wiki/Methylacetylenehttp://en.wikipedia.org/wiki/Methylacetylenehttp://en.wikipedia.org/wiki/Methylacetylenehttp://en.wikipedia.org/wiki/Methylacetylenehttp://en.wikipedia.org/wiki/Methylacetylenehttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-9http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-9http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-9http://en.wikipedia.org/wiki/Water_%28molecule%29http://en.wikipedia.org/wiki/Water_%28molecule%29http://en.wikipedia.org/wiki/Water_%28molecule%29http://en.wikipedia.org/wiki/Water_%28molecule%29http://en.wikipedia.org/wiki/Water_%28molecule%29http://en.wikipedia.org/wiki/Nitrogen_dioxidehttp://en.wikipedia.org/wiki/Nitrogen_dioxidehttp://en.wikipedia.org/wiki/Nitrogen_dioxidehttp://en.wikipedia.org/wiki/Nitrogen_dioxidehttp://en.wikipedia.org/wiki/Nitrogen_dioxidehttp://en.wikipedia.org/wiki/Nitrogen_dioxidehttp://en.wikipedia.org/wiki/Nitrogen_dioxidehttp://en.wikipedia.org/wiki/Nitrogen_dioxidehttp://en.wikipedia.org/wiki/Water_%28molecule%29http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-9http://en.wikipedia.org/wiki/Methylacetylenehttp://en.wikipedia.org/wiki/Chloromethanehttp://en.wikipedia.org/wiki/Prolatehttp://en.wikipedia.org/wiki/Ammoniahttp://en.wikipedia.org/wiki/Benzenehttp://en.wikipedia.org/wiki/Oblate_spheroidhttp://en.wikipedia.org/wiki/Prolatehttp://en.wikipedia.org/wiki/Oblate_spheroidhttp://en.wikipedia.org/wiki/Molecular_symmetry#Elementshttp://en.wikipedia.org/wiki/Acetylenehttp://en.wikipedia.org/wiki/Acetylenehttp://en.wikipedia.org/wiki/Acetylenehttp://en.wikipedia.org/wiki/Carbonyl_sulfidehttp://en.wikipedia.org/wiki/Hydrogen_cyanidehttp://en.wikipedia.org/wiki/Carbon_dioxidehttp://en.wikipedia.org/wiki/Hydroxyl_radicalhttp://en.wikipedia.org/wiki/Hydroxyl_radicalhttp://en.wikipedia.org/wiki/Hydroxyl_radicalhttp://en.wikipedia.org/wiki/Carbon_monoxidehttp://en.wikipedia.org/wiki/Nitrogenhttp://en.wikipedia.org/wiki/Oxygenhttp://en.wikipedia.org/wiki/Molecular_point_grouphttp://en.wikipedia.org/wiki/Molecular_point_grouphttp://en.wikipedia.org/wiki/Molecular_point_grouphttp://en.wikipedia.org/wiki/Sulfur_hexafluoridehttp://en.wikipedia.org/wiki/Sulfur_hexafluoridehttp://en.wikipedia.org/wiki/Sulfur_hexafluoridehttp://en.wikipedia.org/wiki/Silanehttp://en.wikipedia.org/wiki/Methanehttp://en.wikipedia.org/wiki/Carbon_tetrachloridehttp://en.wikipedia.org/wiki/Carbon_tetrachloridehttp://en.wikipedia.org/wiki/Carbon_tetrachloridehttp://en.wikipedia.org/wiki/Allotropes_of_phosphorus#White_phosphorushttp://en.wikipedia.org/wiki/Energy_levelhttp://en.wikipedia.org/wiki/Molecular_symmetry#Elementshttp://en.wikipedia.org/wiki/Center_of_masshttp://en.wikipedia.org/wiki/Moment_of_inertiahttp://en.wikipedia.org/wiki/Angular_momentumhttp://en.wikipedia.org/wiki/Rotational_energyhttp://en.wikipedia.org/wiki/Angular_momentum_quantizationhttp://en.wikipedia.org/wiki/Quantum_mechanics


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Main article: selection rules

Transitions between rotational states can be observed in molecules with a permanent electric

dipole moment.[8][notes 3]

A consequence of this rule is that no microwave spectrum can be

observed for centrosymmetric linear molecules such as N2 (dinitrogen) or HCCH (ethyne), which

are non-polar. Tetrahedral molecules such as CH4 (methane), which have both a zero dipole

moment and isotropic polarizability, would not have a pure rotation spectrum but for the effect of

centrifugal distortion; when the molecule rotates about a 3-fold symmetry axis a small dipole

moment is created, allowing a weak rotation spectrum to be observed by microwave

spectroscopy.[9]

With symmetric tops, the selection rule for electric-dipole-allowed pure rotation transitions is

Δ K =0, Δ J = ±1. Moreover the quantum number K is limited to have values between and including

+ J to - J .[10]

For Raman spectra the general rule is that the molecular polarizability must be anisotropic, whichmeans that it is not the same in all directions.[11] Polarizability is a 3-dimensional tensor that can

be represented as an ellipsoid. The polarizability ellipsoid of spherical top molecules is in factspherical so those molecules show no rotational Raman spectrum. For all other molecules both

Stokes and anti-Stokes lines[notes 4] can be observed and they have similar intensities due to the fact

than many rotational states are thermally populated. The selection rule for linear molecules is ΔJ

= 0, ±2. The reason for the value of 2 is that the ellipsoid must rotate twice during a transition.

The selection rule for symmetric top molecules is

Δ K = 0

If K =0, then Δ J = ±2

If K ≠ 0, then Δ J = 0, ±1, ±2

Transitions with Δ J = +1 are said to belong the an R series, whereas transitions with Δ J = +2 belong to an S series.[12]

Units

The units used for rotational constants depend on the type of measurement. With infrared spectra,

the unit of measurement is usually wavenumbers per cm, written as cm−1 and shown with the

symbol . Wavenumbers per cm is literally the number of waves in one centimeter, or the

reciprocal of wavelength in cm. On the other hand, microwave spectra are usually measured in

Gigahertz. The relationship between the two units is derived from the expression

where ν is a frequency, λ is a wavelength and c is the velocity of light. It follows that

As 1 GHz = 109 hz, the numerical conversion can be expressed as

Effect of vibration on rotation

http://en.wikipedia.org/wiki/Selection_ruleshttp://en.wikipedia.org/wiki/Selection_ruleshttp://en.wikipedia.org/wiki/Selection_ruleshttp://en.wikipedia.org/wiki/Electric_dipole_momenthttp://en.wikipedia.org/wiki/Electric_dipole_momenthttp://en.wikipedia.org/wiki/Electric_dipole_momenthttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-10http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-10http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-10http://en.wikipedia.org/wiki/Dinitrogenhttp://en.wikipedia.org/wiki/Dinitrogenhttp://en.wikipedia.org/wiki/Dinitrogenhttp://en.wikipedia.org/wiki/Ethynehttp://en.wikipedia.org/wiki/Ethynehttp://en.wikipedia.org/wiki/Ethynehttp://en.wikipedia.org/wiki/Methanehttp://en.wikipedia.org/wiki/Methanehttp://en.wikipedia.org/wiki/Methanehttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-12http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-12http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-12http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-13http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-13http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-13http://en.wikipedia.org/wiki/Raman_spectroscopyhttp://en.wikipedia.org/wiki/Raman_spectroscopyhttp://en.wikipedia.org/wiki/Raman_spectroscopyhttp://en.wikipedia.org/wiki/Polarizabilityhttp://en.wikipedia.org/wiki/Polarizabilityhttp://en.wikipedia.org/wiki/Polarizabilityhttp://en.wikipedia.org/wiki/Anisotropichttp://en.wikipedia.org/wiki/Anisotropichttp://en.wikipedia.org/wiki/Anisotropichttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-14http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-14http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-14http://en.wikipedia.org/wiki/Tensorhttp://en.wikipedia.org/wiki/Tensorhttp://en.wikipedia.org/wiki/Tensorhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-15http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-15http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-15http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-16http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-16http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-16http://en.wikipedia.org/wiki/Hzhttp://en.wikipedia.org/wiki/Hzhttp://en.wikipedia.org/wiki/Frequencyhttp://en.wikipedia.org/wiki/Frequencyhttp://en.wikipedia.org/wiki/Frequencyhttp://en.wikipedia.org/wiki/Wavelengthhttp://en.wikipedia.org/wiki/Wavelengthhttp://en.wikipedia.org/wiki/Wavelengthhttp://en.wikipedia.org/wiki/Velocity_of_lighthttp://en.wikipedia.org/wiki/Velocity_of_lighthttp://en.wikipedia.org/wiki/Velocity_of_lighthttp://en.wikipedia.org/wiki/Velocity_of_lighthttp://en.wikipedia.org/wiki/Wavelengthhttp://en.wikipedia.org/wiki/Frequencyhttp://en.wikipedia.org/wiki/Hzhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-16http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-15http://en.wikipedia.org/wiki/Tensorhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-14http://en.wikipedia.org/wiki/Anisotropichttp://en.wikipedia.org/wiki/Polarizabilityhttp://en.wikipedia.org/wiki/Raman_spectroscopyhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-13http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-12http://en.wikipedia.org/wiki/Methanehttp://en.wikipedia.org/wiki/Ethynehttp://en.wikipedia.org/wiki/Dinitrogenhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-10http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-10http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-10http://en.wikipedia.org/wiki/Electric_dipole_momenthttp://en.wikipedia.org/wiki/Electric_dipole_momenthttp://en.wikipedia.org/wiki/Electric_dipole_momenthttp://en.wikipedia.org/wiki/Selection_rules


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The population of vibrationally excited states follows a Boltzmann distribution, so low frequency

vibrational states are appreciably populated even at room temperatures. As the moment of inertia

is higher when a vibration is excited, the rotational constants, B decrease. Consequently, the

rotation frequencies in each vibration state are different from each other. This can give rise to

"satellite" lines in the rotational spectrum. An example is provided by cyanodiacetylene, H-

C≡C−C≡C-C≡N,[13]

Further, there is a fictitious force, Coriolis coupling, between the vibrational motion of the nuclei

in the rotating (non-inertial) frame. However, as long as the vibrational quantum number does not

change (i.e., the molecule is in only one state of vibration), the effect of vibration on rotation is

not important, because the time for vibration is much shorter than the time required for rotation.

The Coriolis coupling is often negligible, too, if one is interested in low vibrational and rotational

quantum numbers only.

Effect of rotation on vibrational spectra

Main article: Rotational-vibrational spectroscopy

Historically, the theory of rotational energy levels was developed to account for observations of

vibration-rotation spectra of gases in infrared spectroscopy, which was used before microwavespectroscopy had become practical. To a first approximation the energy of rotation is added to, or

subtracted from the energy of vibration. The vibration-rotation wavenumbers of transitions for aharmonic oscillator with rigid rotor are given by

In reality, this expression has to be modified for the effects of anharmonicity of the vibrations, for

centrifugal distortion and for Coriolis coupling.[14] The plus sign implies simultaneous excitation

of both vibration and rotation, giving the so-called R branch in the spectrum, whereas with theminus sign a quantum of rotational energy is lost while a quantum of vibrational energy is gained,

giving the P branch. The pure vibration, Δ J =0, gives rise to the Q branch of the spectrum.

Because of the thermal population of the rotational states the P branch is slightly less intense than

the R branch.

Rotational constants obtained from infrared measurements are in good accord with those obtained

by microwave spectroscopy while the latter usually offers greater precision.

Structure of rotational spectra

Spherical top

Spherical top molecules have no net dipole moment. A pure rotational spectrum cannot beobserved by absorption or emission spectrocopy because there is no permanent dipole moment

whose rotation can be accelerated by the electric field of an incident photon. Also the polarizability is isotropic, so that pure rotational transitions cannot be observed by Raman

spectroscopy either. Nevertheless, rotational constants can be obtained by ro-vibrational

spectroscopy. This occurs when a molecule is polar in the vibrationally excited state. For

example, the molecule methane is a symmetric top but the asymmetric C-H stretching band shows

rotational fine structure in the infrared spectrum, illustrated in rovibrational coupling. This

spectrum is also interesting because it shows clear evidence of Coriolis coupling in the

asymmetric structure of the band.

http://en.wikipedia.org/wiki/Cyanodiacetylenehttp://en.wikipedia.org/wiki/Cyanodiacetylenehttp://en.wikipedia.org/wiki/Cyanodiacetylenehttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-17http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-17http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-17http://en.wikipedia.org/wiki/Fictitious_forcehttp://en.wikipedia.org/wiki/Fictitious_forcehttp://en.wikipedia.org/wiki/Fictitious_forcehttp://en.wikipedia.org/wiki/Coriolis_effecthttp://en.wikipedia.org/wiki/Coriolis_effecthttp://en.wikipedia.org/wiki/Coriolis_effecthttp://en.wikipedia.org/wiki/Rotational-vibrational_spectroscopyhttp://en.wikipedia.org/wiki/Rotational-vibrational_spectroscopyhttp://en.wikipedia.org/wiki/Rotational-vibrational_spectroscopyhttp://en.wikipedia.org/wiki/Infrared_spectroscopyhttp://en.wikipedia.org/wiki/Infrared_spectroscopyhttp://en.wikipedia.org/wiki/Infrared_spectroscopyhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-18http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-18http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-18http://en.wikipedia.org/wiki/Rovibrational_couplinghttp://en.wikipedia.org/wiki/Rovibrational_couplinghttp://en.wikipedia.org/wiki/Rovibrational_couplinghttp://en.wikipedia.org/wiki/Rovibrational_couplinghttp://en.wikipedia.org/wiki/Methanehttp://en.wikipedia.org/wiki/Methanehttp://en.wikipedia.org/wiki/Methanehttp://en.wikipedia.org/wiki/Rovibrational_couplinghttp://en.wikipedia.org/wiki/Rovibrational_couplinghttp://en.wikipedia.org/wiki/Rovibrational_couplinghttp://en.wikipedia.org/wiki/Coriolis_effecthttp://en.wikipedia.org/wiki/Coriolis_effecthttp://en.wikipedia.org/wiki/Coriolis_effecthttp://en.wikipedia.org/wiki/Coriolis_effecthttp://en.wikipedia.org/wiki/Rovibrational_couplinghttp://en.wikipedia.org/wiki/Methanehttp://en.wikipedia.org/wiki/Rovibrational_couplinghttp://en.wikipedia.org/wiki/Rovibrational_couplinghttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-18http://en.wikipedia.org/wiki/Infrared_spectroscopyhttp://en.wikipedia.org/wiki/Rotational-vibrational_spectroscopyhttp://en.wikipedia.org/wiki/Coriolis_effecthttp://en.wikipedia.org/wiki/Fictitious_forcehttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-17http://en.wikipedia.org/wiki/Cyanodiacetylene


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Linear molecules

Energy levels and line positions calculated in the rigid rotor approximation

The rigid rotor is a good starting point from which to construct a model of a rotating molecule. It

is assumed that component atoms are point masses connected by rigid bonds. A linear moleculelies on a single axis and each atom moves on the surface of a sphere around the centre of mass.

The two degrees of rotational freedom correspond to the spherical coordinates θ and φ whichdescribe the direction of the molecular axis, and the quantum state is determined by two quantum

numbers J and M. J defines the magnitude of the rotational angular momentum, and M its

component about an axis fixed in space, such as an external electric or magnetic field. In the

absence of external fields, the energy depends only on J. Under the rigid rotor model, the

rotational energy levels, F (J), of the molecule can be expressed as,

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

molecule. In a linear molecule the moment of inertia about an axis perpendicular to the molecularaxis is unique, that is, , so

For a diatomic molecule

where m1 and m2 are the masses of the atoms and d is the distance between them.

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Selection rules dictate that during emission or absorption the rotational quantum number has to

change by unity; i.e., . Thus, the locations of the lines in a rotational

spectrum will be given by

where denotes the lower level and denotes the upper level involved in the transition.

The diagram illustrates rotational transitions that obey the =1 selection rule. The dashed lines

show how these transitions map onto features that can be observed experimentally. Adjacent

transitions are separated by 2 B in the observed spectrum. Frequency or wavenumber

units can also be used for the x axis of this plot.

Rotational line intensities

Rotational level populations with Bhc/kT = 0.05. J is the quantum number of the lower rotational

state

The probability of a transition taking place is the most important factor influencing the intensity

of an observed rotational line. This probability is proportional to the population of the initial state

involved in the transition. The population of a rotational state depends on two factors. The number

of molecules in an excited state with quantum number J, relative to t he number of molecules in

the ground state, N J /N 0 is given by the Boltzmann distribution as

,

where k is the Boltzmann constant and T the absolute temperature. This factor decreases as Jincreases. The second factor is the degeneracy of the rotational state, which is equal to 2J+1. This

factor increases as J increases. Combining the two factor s[15]

The maximum relative intensity occurs at

The diagram at the right shows an intensity pattern roughly corresponding to the spectrum above

it.

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Centrifugal distortion

When a molecule rotates, the centrifugal force pulls the atoms apart. As a result, the moment of

inertia of the molecule increases, thus decreasing the value of , when it is calculated using the

expression for the rigid rotor. To account for this a centrifugal distortion correction term is added

to the rotational energy levels of the diatomic molecule.

where is the centrifugal distortion constant.

Therefore, the line positions for the rotational mode change to

In consequence, the spacing between lines is not constant, as in the rigid rotor approximation, but

decreases with increasing rotational quantum number.

An assumption underlying these expressions is that the molecular vibration follows simple

harmonic motion. In the harmonic approximation the centrifugal constant D can be derived as

where k is the vibrational force constant. The relationship between B and D

where : is the harmonic vibration frequency, follows. If anharmonicity is to be taken into

account, terms in higher powers of J should be added to the expressions for the energy levels andline positions.[16] A striking example concerns the rotational spectrum of hydrogen fluoride which

was fitted to terms up to [J(J+1)]5.[17]

Oxygen

The electric dipole moment of the dioxygen molecule, O2 is zero, but the molecule is

paramagnetic with two unpaired electrons so that there are magnetic-dipole allowed transitionswhich can be observed by microwave spectroscopy. The unit electron spin has three spatial

orientations with respect to the given molecular rotational angular momentum vector, K, so that

each rotational level is split into three states, J = K + 1, K, and K - 1, each J state of this so-called

p-type triplet arising from a different orientation of the spin with respect to the rotational motionof the molecule. The energy difference between successive J terms in any of these triplets is about

2 cm−1

(60 GHz), with the single exception of J = 1←0 difference which is about 4 cm−1

.Selection rules for magnetic dipole transitions allow transitions between successive members of

the triplet (ΔJ = ±1) so that for each value of the rotational angular momentum quantum numberK there are two allowed transitions. The 16O nucleus has zero nuclear spin angular momentum, so

that symmetry considerations demand that K have only odd values.[18][19]

Symmetric top

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For symmetric rotors a quantum number J is associated with the total angular momentum of the

molecule. For a given value of J, there is a 2 J +1- fold degeneracy with the quantum number, M

taking the values + J ...0 ... - J . The third quantum number, K is associated with rotation about the

principal rotation axis of the molecule. In the absence of an external electrical field, the rotational

energy of a symmetric top is a function of only J and K and, in the rigid rotor approximation, the

energy of each rotational state is given by

where and for a prolate symmetric top molecule or

for an oblate molecule.

This gives the transition wavenumbers as

which is the same as in the case of a linear molecule.[20] With a first order correction for

centrifugal distortion the transition wavenumbers become

The term in D JK has the effect of removing degeneracy present in the rigid rotor approximation,

with different K values.[21]

Asymmetric top

Pure rotation spectrum of atmospheric water vapour measured at Mauna Kea (33 cm-1 to 100 cm-

1)

The quantum number J refers to the total angular momentum, as before. Since there are three

independent moments of inertia, there are two other independent quantum numbers to consider,

but the term values for an asymmetric rotor cannot be derived in closed form. They are obtained

by individual matrix diagonalization for each J value. Formulae are available for molecules whose

shape approximates to that of a symmetric top.[22]

http://en.wikipedia.org/wiki/Molecular_symmetryhttp://en.wikipedia.org/wiki/Molecular_symmetryhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-24http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-24http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-24http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-25http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-25http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-25http://en.wikipedia.org/wiki/Diagonalizable_matrix#Diagonalizationhttp://en.wikipedia.org/wiki/Diagonalizable_matrix#Diagonalizationhttp://en.wikipedia.org/wiki/Diagonalizable_matrix#Diagonalizationhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-26http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-26http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-26http://en.wikipedia.org/wiki/File:Atmospheric_terahertz_transmittance_at_Mauna_Kea_%28simulated%29.pnghttp://en.wikipedia.org/wiki/File:Atmospheric_terahertz_transmittance_at_Mauna_Kea_%28simulated%29.pnghttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-26http://en.wikipedia.org/wiki/Diagonalizable_matrix#Diagonalizationhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-25http://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-24http://en.wikipedia.org/wiki/Molecular_symmetry


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The water molecule is an important example of an asymmetric top. It has an intense pure rotation

spectrum in the far infrared region, below about 200 cm−1

. For this reason far infrared

spectrometers have to be freed of atmospheric water vapour either by purging with a dry gas or by

evacuation. The spectrum has been analyzed in detail.[23]

Quadrupole splitting

When a nucleus has a spin quantum number, I , greater than 1/2 it has a quadrupole moment. In

that case, coupling of nuclear spin angular momentum with rotational angular momentum causes

splitting of the rotational energy levels. If the quantum number J of a rotational level is greater

than I , 2 I +1 levels are produced; but if J is less than I , 2 J +1 levels result. The effect is known as

hyperfine splitting. For example, with14 N ( I = 1) in HCN, all levels with J > 0 are split into 3. The

energy of the sub-levels are proportional to the nuclear quadrupole moment and a function of F

and J . where F = J + I , J + I -1, ..., 0, ... | J - I |. Thus, observation of nuclear quadrupole splitting permits the magnitude of the nuclear quadrupole moment to be determined.[24] This is an

alternative method to the use of nuclear quadrupole resonance spectroscopy. The selection rule forrotational transitions becomes[25]

Stark and Zeeman effects

In the presence of a static external electric field the 2 J +1 degeneracy of each rotational state is

partly removed, an instance of a Stark effect. For example in linear molecules each energy level is

split into J +1 components. The extent of splitting depends on the square of the electric field

strength and the square of the dipole moment of the molecule.[26]

In principle this provides ameans to determine the value of the molecular dipole moment with high precision. Examples

include carbonyl sulfide, OCS, with μ = 0.71521 ± 0.00020 Debye. However, because thesplitting depends on μ2, the orientation of the dipole must be deduced from quantum mechanical

considerations.[27]

A similar removal of degeneracy will occur when a paramagnetic molecule is placed in amagnetic field, an instance of the Zeeman effect. Most species which can be observed in the

gaseous state are diamagnetic . Exceptions, known as odd molecules, include nitric oxide, NO,nitrogen dioxide, NO2, some chlorine oxides and the hydroxyl radical. The Zeeman effect has

been observed with dioxygen, O2[28]

Rotational Raman spectroscopyMolecular rotational transitions can also be observed by Raman spectroscopy. Rotational

transitions are Raman-allowed for any molecule with an anisotropic polarizability which includes

all molecules except for spherical tops. This means that rotational transitions of molecules with no

permanent dipole moment, which cannot be observed in absorption or emission, can be observed,

by scattering, in Raman spectroscopy. Very high resolution Raman spectra can be obtained by

adapting a Fourier Transform Infrared Spectrometer . An example is the spectrum of 15 N2. It

shows the effect of nuclear spin, resulting in intensities variation of 3:1 in adjacent lines. A bond

length of 109.9985 ± 0.0010 pm was deduced from the data.[29]

Instruments and Methods

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_effecthttp://en.wikipedia.org/wiki/Electric_fieldhttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-29http://en.wikipedia.org/wiki/Nuclear_quadrupole_resonancehttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-28http://en.wikipedia.org/wiki/Quadrupole_momenthttp://en.wikipedia.org/wiki/Hyperfine_splittinghttp://en.wikipedia.org/wiki/Quadrupolehttp://en.wikipedia.org/wiki/Rotational_spectroscopy#cite_note-27


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The great majority of contemporary spectrometers use a mixture of commercially available and

bespoke components which users integrate according to their particular needs. Instruments can be

broadly categorised according to their general operating principals. Although rotational transitions

can be found across a very broad region of the electromagnetic spectrum, fundamental physical

constraints exist on the operational bandwidth of instrument components. It is often impractical

and costly to switch to measurements within an entirely different frequency region. The

instruments and operating principals described below are generally appropriate to microwave

spectroscopy experiments conducted at frequencies between 6 and 24 GHz.

Absorption cells and Stark modulation

A microwave spectrometer can be most simply constructed using a source of microwave

radiation, an absorption cell into which sample gas can be introduced and a detector such as a

superheterodyne receiver . A spectrum can be obtained by sweeping the frequency of the source

while detecting the intensity of transmitted radiation. A simple section of waveguide can serve as

an absorption cell. An important variation of the technique in which an alternating current is

applied across electrodes within the absorption cell results in a modulation of the frequencies of

rotational transitions. This is referred to as Stark modulation and allows the use of phase-sensitivedetection methods offering improved sensitivity. Absorption spectroscopy allows the study of

samples that are thermodynamically stable at room temperature. The first study of the microwave

spectrum of a molecule (NH3) was performed by Cleeton & Williams in 1934.[30] Subsequent

experiments exploited powerful sources of microwaves such as the klystron, many of which were

developed for radio detection and ranging (RADAR ) during the Second World War . The numberof experiments in microwave spectroscopy surged immediately after the war. By 1948, Walter

Gordy was able to prepare a review of the results contained in approximately 100 research papers.[31] Commercial versions[32] of microwave absorption spectrometer were developed by

Hewlett Packard in the 1970s and were once widely used for fundamental research. Most researchlaboratories now exploit either Balle-Flygare or chirped-pulse Fourier transform microwave

(FTMW) spectrometers.

Fourier transform microwave (FTMW) spectroscopy

The theoretical framework [33] underpinning FTMW spectroscopy is analogous to that used to

describe FT-NMR spectroscopy. The behaviour of the evolving system is described by optical

Bloch equations. First, a short (typically 0-3 microsecond duration) microwave pulse is

introduced on resonance with a rotational transition. Those molecules that absorb the energy from

this pulse are induced to rotate coherently in phase with the incident radiation. De-activation ofthe polarisation pulse is followed by microwave emission that accompanies decoherence of the

molecular ensemble. This free induction decay occurs on a timescale of 1-100 microsecondsdepending on instrument settings. Following pioneering work by Dicke and co-workers in the

1950s,[34] the first FTMW spectrometer was constructed by Ekkers and Flygare in 1975.[35]

The Balle-Flygare FTMW spectrometer

Balle, Campbell, Keenan and Flygare demonstrated that the FTMW technique can be applied

within a "free space cell" comprising an evacuated chamber containing a Fabry-Perot cavity.[36]

This technique allows a sample to be probed only milliseconds after it undergoes rapid cooling to

only a few degrees Kelvin in the throat of an expanding gas jet. This was a revolutionary

development because (i) cooling molecules to low temperatures concentrates the available

population in the lowest rotational energy levels. Coupled with benefits conferred by the use of aFabry-Perot cavity, this brought a great enhancement in the sensitivity and resolution of

spectrometers along with a reduction in the complexity of observed spectra; (ii) it became possible to isolate and study molecules that are very weakly bound because there is insufficient

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energy available for them to undergo fragmentation or chemical reaction at such low

temperatures. William Klemperer was a pioneer in using this instrument for the exploration of

weakly bound interactions. While the Fabry-Perot cavity of a Balle-Flygare FTMW spectrometer

can typically be tuned into resonance at any frequency between 6 and 18 GHz, the bandwidth of

individual measurements is restricted to about 1 MHz. An animation illustrates the operation of

this instrument which is currently the most widely used tool for microwave spectroscopy.[37]

The Chirped-Pulse FTMW spectrometer

Noting that digitisers and related electronics technology had significantly progressed since the

inception of FTMW spectroscopy, B.H. Pate at the University of Virginia[38] designed a

spectrometer [39]

which retains many advantages of the Balle-Flygare FT-MW spectrometer while

innovating in (i) the use of a high speed (>4 GS/s) arbitrary waveform generator to generate a

"chirped" microwave polarisation pulse that sweeps up to 12 GHz in frequency in less than a

microsecond and (ii) the use of a high speed (>40 GS/s) oscilloscope to digitise and Fourier

transform the molecular free induction decay. The result is an instrument that allows the study of

weakly bound molecules but which is able to exploit a measurement bandwidth (12 GHz) that is

greatly enhanced compared with the Balle-Flygare FTMW spectrometer. Modified versions of theoriginal CP-FTMW spectrometer have been constructed by a number of groups in the UnitedStates, Canada and Europe.[40][41] The instrument offers a broadband capability that is highly

complementary to the high sensitivity and resolution offered by the Balle-Flygare design.

Notes

1. Jump up ^ The spectrum was measured over a couple of hours with the aid of a chirped- pulse Fourier transform microwave spectrometer at the University of Bristol.

2. Jump up ^ For a symmetric top, the values of the 2 moments of inertia can be used toderive 2 molecular parameters. Values from each additional isotopologue provide the

information for one more molecular parameter. For asymmetric tops a single isotopologue provides information for at most 3 molecular parameters.

3. Jump up ^ Such transitions are called electric dipole-allowed transitions. Othertransitions involving quadrupoles, octupoles, hexadecapoles etc. may also be allowed but

the spectral intensity is very much smaller, so these transitions are difficult to observe.

Magnetic-dipole-allowed transitions can occur in paramagnetic molecules such as

dioxygen, O2 and nitric oxide, NO

4. Jump up ^ In Raman spectroscopy the photon energies for Stokes and anti-Stokesscattering are respectively less than and greater than the incident photon energy. See the

energy-level diagram at Raman spectroscopy.

Coming soon, interpretation of spectra and solving molecular structures!
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