3 rotational spectroscopy1

Rotational Spectra 1

Chemistry Department University of Lagos

Rotational Spectroscopy

Rotational spectroscopy or microwave spectroscopy studies the absorption and emission of

electromagnetic radiation (typically in the microwave region of the electromagnetic spectrum) by

molecules associated with a corresponding change in the rotational quantum number of the

molecule. Rotational spectroscopy is only really practical in the gas phase where the rotational

motion is quantized. In solids or liquids the rotational motion is usually quenched due to

collisions.

Moment of force (often just moment) is a synonym for torque. The moment of inertia of a

molecule is a measure of how difficult it is to rotationally accelerate the molecule (SI units

kg·m2). It is a measure of an object's resistance to changes in its rotation rate - the larger the

moment of inertia, the smaller the increase in angular momentum for a given applied torque and

the slower the rate of rotation. Mathematically it is given as

∑

Where is the mass of the atom/particle and the distance from the rotation axis.

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: IA,

IB and IC about three mutually orthogonal axes A, B, and C with

the origin at the center of mass of the system. 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 .

Rotational spectrum from a molecule (to first order) requires that the molecule

have a dipole moment, i.e. µ ≠ 0, and that there be a difference between its

100GHz-1GHz

mm3cm30



center of charge and its 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 dioxygen (O2), dihydrogen (H2), etc. do not have a dipole moment

and hence no pure rotational spectrum. However, electronic excitations can lead to asymmetric

charge distributions and thus provide a net dipole moment to the molecule. Under such

circumstances, these molecules will exhibit a rotational spectrum.

The rotational energy levels

Because I depends on both the mass of the atoms and the geometry of the molecule, the

rotational spectroscopy will provide us with information about bond lengths and bond angles.

The classical expression for a body rotating about a given axis with angular velocity ω (in

radian) per second is

A body free to rotate about three mutually perpendicular axes has an energy given by:

This equation can be transformed into

The analogous quantum mechanical expressions can be obtained by substitution of the quantum

expressions for angular momentum,

J is the total angular momentum. It tells the relative orientation of the spin and orbital angular

momentum for several electrons. The angular momentum of an object is defined relative to a

fixed point

Classification of molecules based on rotational behavior

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,



1. Linear molecules (or linear rotors)

2. Symmetric tops (or symmetric rotors)

3. Spherical tops (or spherical rotors)

and

4. Asymmetric tops (asymmetric rotors)

Linear molecules:

A linear molecule is a special case. These molecules are

cylindrically symmetric and one of the moment of inertia

(IA, which is the moment of inertia for a rotation taking

place along the axis of the molecule) is negligible i.e.

. For most of the purposes, IA is taken to be

zero as all the atoms lie on the axis of rotation so are at zero

distance from it.

Examples of linear molecules: dioxygen (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)

Diatomic molecule: For diatomic molecules this process of

bond/structure determination is trivial, and can be made from

a single measurement of the rotational spectrum.

Rigid rotor

The bond distance: R=r = r1 + r2

The center of mass has a physical property: r1m1 = r2m2

The moment of inertia:

221

211 rmrmI

Figure 1: The rotation produces I

perpendicular to the plane of rotation.



The radial distances from mass center can be given by bond length

,

Substituting r1 and r2 into we get for I

Where,

is called the reduced mass

From quantum mechanics equation, the energy levels of a rigid rotor is

given

where J = 0, 1, 2, … and MJ = 0, ±1, ±2, …, ±J. the energy is

independent of MJ and there are 2J + 1 values of MJ for each J, making

each energy level to be (2J + 1) degenerate. MJ describes the orientation

of J in space much like the magnetic quantum number. The angular

momentum can only be oriented in the direction in which it takes

integral values. this is defined by MJ.

It follows that

The rotation energy is normally reported as the rotational term

Where B is called the rotational constant. It is given in wavenumber

terms, and a rotation constant B as

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

molecule IB = IC by,

Recall

Figure 2: Energy level spacing for linear

or spherical rotor



If the velocity of light is given in cm/s units the unit of B is cm-1

.

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

change by unity i.e. ∆J=J' - J'' = ±1. ∆MJ = 0, ±1, a rule which is important only if the molecule

is in an electric or magnetic field. The rotation constant of a molecule is characteristic to the

nature of molecule, and independent of J. In this notation J' and J'' mean the upper level and

lower level respectively with the relation J' - J'' =1

The transition → :

But and

(

) (

)

The wavenumber can be phrased either by lower

rotation level

or by upper rotation level.

Alternatively, one can say that the locations of the lines in a rotational spectrum will be given by

This wavenumber equals to the peak position of one band in the vibrational spectrum.

The rotation spectra contains peak series with peak

separation equal to each other.

Table 1. for rotation levels and energy spacing

J υ (cm-1

) ∆υ (cm-1

)

1 4B

2 6B 2B

3 8B 2B

4 10B 2B



The rotational levels are not equidistant, which can be seen from the second column of Table 1.

Example for CO:

the transition (J=0 J=1) for 12

C16

O is at 3.84235 cm-1

.

Given that C = 12.0000 ; O = 15.9994 amu;1 amu = 1 atomic mass unit = 1.6605402 x 10-27

kg;

h = 6.6260755 x10-34

Js; c = 2.99792458 x 1010

cm s-1

; Find r(CO)

246

2kgm

B

102.7992774

Bc8π

hI

= r2

B = 1.921175 cm-1

; = 1.1386378 x 10-26

kg

μ

Ir = 1.131 x 10

-10 m

0.1131 nm

Answer: C-O bondlength is 0.1131 nm.

Rotational absorption lines from 1H

35Cl gas were found at the following wavenumbers: 83.32

cm-1

, 104.13 cm -1

, 124.73 cm -1

, 145.37 cm -1

, 165.9 cm -1

, 186.23 cm -1

, 206.60 cm -1

, 226.86

cm -1

. Calculate the moment of inertia and the bond length of the molecule.

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Intensities of rotation spectral lines

Now we understand the locations (positions) of lines in the microwave spectrum, we can see

which lines are strongest. 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.

Apart from depending on the numerical value of the square of the transition moment varies

relatively little with J, intensities depend on the population of the lower state of a transition. The

population NJ of the Jth level relative to N0 is obtained from Boltzmann’s distribution law

where (2J + 1) is the degeneracy of the Jth

level. This degeneracy arises from the fact that, in the

absence of an electric or magnetic field, (2J + 1) levels, resulting from the number of values that

MJ can take, are all of the same energy: in other words they are degenerate. Values of (2J + 1),

exp(-Er/kT) and NJ /N0 are given for CO below, and illustrate the point that there are two



opposing factors in NJ /N0 . The (2J + 1) factor increases with J whereas the exp(-Er/kT) factor

decreases rapidly, so that increases at low J until, at higher J, the exponential factor wins and NJ

/N0 approaches zero.

The population therefore shows a maximum at a

value of J = Jmax corresponding to

Which gives

for B having dimensions of frequency.

Therefore intensity depends on population of state, the degeneracy and energy of the state.

Effect of Isotopes

Isotopes are atoms of the same element (that is, with the same number of protons in their atomic

nucleus), but having different numbers of neutrons. Consequently, changing a particular atom in

the molecule changes the atomic mass and therefore the total mass. This in turn changes the

moment of inertia and hence the rotational constant, B. There is usually no appreciable change in

the internuclear distance. For example consider carbonmonoxide, going from 12

C16

O to 13

C16

O,

there is a mass increase, therefore a reduction in B ( 1/I). This gives rise to lower energy levels.



Generally, the spectrum of heavier species gives rise to smaller separation between the lines than

that of the lighter species.

From this information one can determine: (i) isotopic masses accurately, to within 0.02% of

other methods for atoms in gaseous molecules; and (ii) isotopic abundances from the absorption

relative intensities.

Example:

Given : the below information

For 12

CO J=0 J=1is at 3.84235 cm-1

and at 3.67337 cm-1

for 13

CO (12

C = 12.0000 ; O =

15.9994 amu).What is isotopic mass of 13

C ?

B(12

CO) = 1.921175 cm-1

and B(13

CO) = 1.836685 cm-1

Now μ

1

I

1B

1.046001.836685

1.921175

CO)μ(

CO)μ(12

13

and 15.999412

15.999412

15.9994C)(

15.9994C)(1.046

13

13

(13

C) = 13.0006 amu

Is the bond length in 1HCl the same as in

2HCl? The wavenumbers of the J = rotational

transition for 1HCl and

2HCl are 20.8784 cm

-1 and 10.7840 cm

-1, respectively.

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Non rigid rotor

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

Equation for Erot(J) is only approximate. When a molecule rotates, the

centrifugal force pulls the atoms apart. (Centrifugal force (from Latin centrum "center" and

fugere "to flee") represents the effects of inertia that arise in connection with rotation and which

are experienced as an outward force away from the center of rotation. )As a result, the moment

of inertia of the molecule increases, thus decreasing the rotational constant B. In order to provide

a better description of the energy levels of a diatomic molecule, a centrifugal distortion term is

added to the energy.

DJ is called the centrifugal distortion constant, and is several orders of magnitude smaller than B.

It is a measure of stiffness of the molecule to rotation. The effect of rotation on a molecule. The

centrifugal force arising from rotation distorts the molecule, opening out bond angles and

stretching bonds slightly. The effect is to increase the moment of inertia of the molecule and

hence to decrease its rotational constant. It takes into account of the fact that as a real molecule

rotates faster and faster (i.e. with more energy), the bond stretches a little (if you swing a weight

attached to a piece of elastic in a circle, you can observe the same effect on a macroscopic scale).

It is large when the bond is easily stretched.

Accordingly the line spacing for the rotational mode changes to,

http://en.wikipedia.org/wiki/Latin

http://en.wikipedia.org/wiki/Inertia



The centrifugal distortion constant, DJ is related to the wavenumbers approximately by

1

224

3

cmkcrI32π

hD

Effect of vibration on rotation:

A molecule is always in vibration. As the molecule vibrates, its moment of inertia changes.

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. (More on this later)

Polyatomic linear molecules

For linear molecules with more atoms, Things get much more complicated, but the general

principles are the same. e.g. OCS HCCCl Ic = IB; IA = 0

The moment of inertia is greater than for diatomic molecule, B will be smaller and lines will

be more closely spaced. As usual, the molecule must have dipole moment for microwave

spectrum.

For a poly atomic molecule with N atoms, there are N-1 bond lengths to be determined and not

just 1 as in a diatomic molecule, so for OCS we must determine rCO, rCS i.e. two bondlengths are

unknown we need 2 values for IB - the second can come from an isotopically substituted

molecule, which has same bondlength (almost), but different mass.



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 lengths). e.g. consider 16

OC34

S, 18

OC34

S ….

O C S

Fro the center of mass moments moro + mCrC = mSrS

I = moro2 + mCrC

2 + mSrS

2

,

After several manipulations we obtain

Where m is the total mass of the molecule.

We may write

for the isotopic molecule, where is the isotopic mass and are not primed since it is

presumed that they are not affected by isotopic substitution.

In accurate work isotopic bond lengths differ, due to differences in zero point energy

J J+1 B(cm-1

)

0 1 … 2 0.4055 0.2027

1 2 0.8109 0.4054 0.2027

2 3 1.2163 0.4054 0.2027 Calculate

3 4 1.6217 0.4054 0.2027 IB

4 5 2.0271 0.4055 0.2027



Symmetric tops:

A symmetric top is a molecule in which two

moments of inertia are the same. They are divided

into two classes,

- Oblate symmetric tops (saucer or disc

shaped) with IA = IB < IC e.g benzene

(C6H6), cyclobutadiene (C4H4), ammonia

(NH3)

- Prolate symmetric tops (rugby football, or

cigar shaped) with IA < IB = IC e.g.

chloromethane (CH3Cl), propyne (CH3C≡CH)

The spectra look rather different, and are instantly recognizable. Like in the case of the linear

molecules, the structure of symmetric tops (bond lengths and bond angles) can be deduced from

their spectra.

The rotational motion of a symmetric top molecule can be described by two independent

rotational quantum numbers (since two axes have equal moments of inertia, the rotational motion

about these axes requires only one rotational quantum number for complete description). Instead

of defining the two rotational quantum numbers 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(J,K) of such a molecule, based on rigid rotor assumptions can be expressed in terms of the two

previously defined rotational quantum numbers as follows,

and

where

and

for a prolate symmetric top molecule or

for an

oblate molecule.

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

∆J=±1 and ∆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,

which is the same as in the

case of a linear molecule.

In case of non-rigid rotors, the

first order centrifugal distortion

correction is given by,

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 is given by,

1. Spherical tops:

A spherical top molecule, can be considered as a special case of symmetric tops with equal

moment of inertia about all three axes (IA = IB = IC).

Examples of spherical tops: phosphorus tetramer (P4), carbon tetrachloride (CCl4), nitrogen

tetrahydride (NH4), ammonium ion (NH4+), sulfur hexafluoride (SF6)

Unlike other molecules, spherical top molecules have no net dipole moment, and hence they do

not exhibit a pure rotational spectrum.

However, consider the methane molecule. Rotation about any of the C3 axes (i.e. any of the four

axes in methane containing a C–H bond) results in a centrifugal distortion in which the other

three hydrogen atoms are thrown outwards slightly from the axis. This converts the molecule

into a symmetric rotor and gives it a small dipole moment resulting in a very weak rotational

spectrum.



Part of the far-infrared

rotational spectrum of silane

(SiH4) is shown. It was

obtained under the following

conditions using a Michelson

interferometer: absorbing path

of 10.6 m and a pressure of 4.03

atm (4.086105 Pa), these

conditions indicating how very

weak the spectrum is. The

dipole moment has been

estimated from the intensities of the transitions to be 8.361076 D (2.7610735 C m).

The rotation term values for a spherical rotor are given by

This is an identical expression to that for a diatomic or linear polyatomic molecule and, as the

rotational selection rule is the same, namely, ∆J = ±1,the transition wavenumbers or frequencies

are given by

and adjacent transitions are separated by 2B.

All regular tetrahedral molecules, which belong to the Td point group, may show such a

rotational spectrum. However, those spherical rotors that are regular octahedral molecules and

that belong to the Oh point group do not show any such spectrum. The reason for this is that

when, for sample, SF6 rotates about aC4 axis (any of the F–S–F axes) no dipole moment is

produced when the other four fluorine atoms are thrown outwards.

2. Asymmetric tops:

a molecule is termed an asymmetric top if 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)

The spectrum for these molecules usually involves many lines due to three different rotational

modes and their combinations. The following analysis is valid for the general case and collapses

to the various special cases described above in the appropriate limit.

From the moments of inertia one can define an asymmetry parameter κ as

which varies from -1 for a prolate symmetric top to 1 for an oblate symmetric top.

One can define a scaled rotational Hamiltonian dependent on J and κ. The (symmetric) matrix

representation of this Hamiltonian is banded, zero everywhere but the main diagonal and the

second subdiagonal. The Hamiltonian can be formulated in six different settings, dependent on

the mapping of the principal axes to lab axes and handedness. For the most asymmetric, right-

handed representation the diagonal elements are, for | |

Hk,k(κ) = κk2

and the second off-diagonal elements (independent of κ) are

.

Diagonalising H yields a set of 2J + 1 scaled rotational energy levels Ek(κ). The rotational energy

levels of the asymmetric rotor for total angular momentum J are then given by

Degeneracy Stark effect

For each J there, are (2J+1) state with the same energy. This is called degeneracy. In the

presence of an external magnetic field this degenarcy is removed. This is the Stark effect. It is

the shifting and splitting of spectral lines of atoms and molecules due to the presence of an

external static electric field. It is the electric analogue of the Zeeman effect where a spectral line

is split into several components due to the presence of a magnetic field. The amount of splitting

and or shifting is called the Stark splitting or Stark shift.

In general one distinguishes first- and second-order Stark effects. The first-order effect is linear

in the applied electric field, while the second-order effect is quadratic in the field.



The Stark effect is responsible for the pressure broadening (Stark broadening) of spectral lines by

charged particles. When the split/shifted lines appear in absorption, the effect is called the

inverse Stark effect.

Hyperfine interactions:

In addition to the main structure that is observed in microwave spectra 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:

1. electron spin - electron spin interaction (this occurs in molecules with two or more

unpaired electrons, and is a magnetic-dipole / magnetic-dipole interaction)

2. electron spin - molecular rotation (the rotation of a molecule corresponds to a magnetic

dipole, which interacts with the magnetic dipole moment of the electron)

3. 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)).

4. 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)).

5. 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.

Applications

Microwave spectroscopy is commonly used in physical chemistry to determine the structure of

small molecules (such as ozone, methanol, or water; planarity and non-planarity of molecules;

accurate determination of geometric parameters such as bond lengths and bond angles) with high

precision. This is because it is highly sensitive, it gives high resolution and is non-destructive.

Other common techniques for determining molecular structure, such as X-ray crystallography

don't work very well for some of these molecules (especially the gases) and are not as precise.

http://en.wikipedia.org/wiki/NMR_spectroscopy

http://en.wikipedia.org/wiki/Electron_spin_resonance



However, microwave spectroscopy is not useful for determining the structures of large molecules

such as proteins.

Modern microwave spectrometers have very high resolution. When hyperfine structure can be

observed the technique can also provide information on the electronic structures of molecules.

provides number and energy difference of rotational isomers

determination of electric properties of the molecules (dipolar and quadrupolar moment)

Microwave spectroscopy is one of the principal means by which the constituents of the universe

are determined from the earth. It is particularly useful for analysis of the chemical composition in

the interstellar medium (ISM). One of the early surprises in interstellar chemistry was the

existence in the ISM of long chain carbon molecules. It was in attempting to research such

molecules in the laboratory that Harry Kroto was led to the laboratory of Rick Smalley and

Robert Curl, where it was possible to vaporize carbon under enormous energy conditions. This

collaborative experiment led to the discovery of C60, buckminsterfullerene, which led to the

award of the 1996 Nobel Prize in chemistry to Kroto, Smalley and Curl.

3 rotational spectroscopy1

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