Introduction

Atoms, ions, and  molecules  emit or absorb characteristically; only certain energies of these species are possible; the energy of the photon (quantum of radiant energy)  emitted  or  absorbed corresponds to the difference between two permitted values of the energy of the species, or energy levels. (If the flux of photons incident upon the species is huge enough, simultaneous absorption of 2 or more photons might take place.)

Thus the energy levels might be studied via observing the differences between them. The absorption of radiant energy is accompanied through the promotion of the species from a lower to a higher energy level; the emission of radiant energy is accompanied via falling from a higher to a lower state; and if both procedures happen mutually, the condition is said resonance.

Definition of Rotational Spectroscopy

Rotational spectroscopy or microwave spectroscopy studies the absorption and emission of electromagnetic radiation (classically in the microwave region of the electromagnetic spectrum) via molecules connected through a analogous transform in the rotational  quantum number of the molecule.

Principle of Rotational Spectroscopy

Utilize of microwaves in spectroscopy essentially became possible due to the development of microwave technology for RADAR during World War II. Rotational spectroscopy is only actually practical in the gas phase where the rotational motion is quantized. In solids or liquids the rotational activity is generally quenched due to collisions.

Rotational spectrum from a molecule (to first order) requires that the molecule have a dipole moment and that there is dissimilarity between its centre of charge and its centre of mass, or consistently a division between 2 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 revolve  more quickly (in excitation)or gradually (in de-excitation). Diatomic molecules these as dioxygen (O₂), dihydrogen (H₂), and so on. Don't have a dipole moment and hence no purely rotational spectrum. However, electronic excitations can lead to asymmetric charge distributions and therefore give a net dipole moment to the molecule. Under such circumstances, such molecules will show a rotational spectrum.

Amongst the diatomic molecules, carbon monoxide (CO) has one of the simplest rotational   spectra. As for tri-atomic molecules, hydrogen cyanide (HC≡N) has a easy rotational spectrum for a linear molecule and hydrogen isocyanide (HN=C:) for a non-linear molecule. As the number of atoms rises the spectrum becomes more complex as lines due to dissimilar evolutions start overlapping.

Understanding the Rotational Spectrum

In quantum mechanics the free rotation of a molecule is quantized, that is the rotational energy and the angular momentum can only take certain fixed values; what these values  are  is simply related to the moment of inertia,  I, of the molecule. In general for any molecule, there are three moments of inertia: IA, IB and IC about three mutually orthogonal axes A, B, and C with the origin at the centre of mass of the system. A linear molecule is a particular case in this regard.  Such 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.

Classification of Molecules Based on Rotational Behaviour

The common convention is to describe the axes these that the axis A has the smallest moment of inertia (and hence the highest rotational frequency) and other axes such that. Sometimes the axis A may be associated with the symmetric axis of the molecule, if any. If these are the case, then IA require not be the smallest moment of inertia. To avoid confusion, we will stick through the former convention for the rest of the article. The scrupulous pattern of energy levels (and therefore of evolutions in the rotational spectrum) for a molecule, is concluded via its symmetry. A convenient way to look at the molecules is to divide them into 4 different classes (depend 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

Dealing through each in turn:

1.  Linear molecules:

As revealed earlier, for a linear molecule. For most of the functions, IA is taken to be zero. For a linear molecule, the division of lines in the rotational spectrum can be related straight to the moment of inertia of the molecule, and for a molecule of known atomic masses, can be utilized to find out the bond lengths (structure) directly. For diatomic molecules this procedure is trivial, and can be made from a single dimension of the rotational spectrum. For linear molecules through more atoms, rather more work is needed, and it is essential to compute molecules in that more than one isotope of each atom have been substituted (effectively this provides increase to a set of simultaneous equations that can be resolved for the bond lengths).

Instances or 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)

2.  Symmetric tops:

A symmetric top is a molecule in that 2 moments of inertia are the same.  As a matter of historical convenience, spectroscopists divide molecules into 2 classes of symmetric tops, Oblate symmetric tops (saucer or disc shaped) through IA = IB < IC and Prolate symmetric tops (rugby football, or cigar shaped) with IA < IB = IC. The spectra look rather different, and are instantly recognizable. As for linear molecules, the structure of symmetric tops (bond lengths and bond angles) can be construed from their spectra.

Instances of symmetric tops:

- Oblate: benzene (C₆H₆), cyclobutadiene (C₄H₄), ammonia (NH₃)

- Prolate: chloromethane (CH₃Cl), propyne (CH₃C≡CH)

3.  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).

- Instances of spherical tops: phosphorus tetramer (P₄), carbon tetrachloride (CCl₄), nitrogen tetrahydride (NH₄), ammonium ion (NH₄⁺), sulfur hexafluoride (SF₆)

4.  Asymmetric tops:

- As we would have guessed a molecule is termed an asymmetric top if all 3 moments of inertia are dissimilar. Most of the larger molecules are asymmetric tops, even when they contain an elevated degree of symmetry. Usually for such molecules an easy interpretation of the spectrum isn't usually probable.

Sometimes asymmetric tops have spectra that are alike to those of a linear molecule or a symmetric top; in that case the molecular structure must as well be alike to that of a linear molecule or a symmetric top. For the most common case, though, all that can be complete is to fit the spectra to 3 different moments of inertia. If the molecular formula is recognized, then educated guesses can be made of the possible structure, and from this guessed structure, the instants of inertia can be computed. If the computed moments of inertia agree well through the calculated moments of inertia, then the structure can be said to have been verified. For this approach to determining molecular structure, isotopic substitution is invaluable.

Instances of asymmetric tops: anthracene (C₁₄H₁₀), water (H₂O), nitrogen dioxide (NO₂).

Structure of Rotational Spectrum

Linear molecules

Fig: An Energy Level Diagram Showing some of the Transitions involved in the IR Vibration-rotation Spectrum of a Linear Molecule: P branch (where ΔJ = - 1), Q branch (not always allowed, ΔJ = 0) and R branch (ΔJ = 1)

These molecules have 2 degenerate modes of rotation (I_B = I_C, I_A= 0). Since we cannot distinguish between the two modes, we require only one rotational quantum number (J) to describe the rotational motion of the molecule.

The rotational energy levels (F(J)) of the molecule depend on rigid rotor model can be expressed as,

F(j) = B_eJ(J+1)   J=0,1,2,....

Where B_e is the rotational steady of the molecule and is related to the moment of inertia of the molecule I_B = I_C via

B_e = h/8 π² CI_B

Selection rules dictate that during release or absorption the rotational quantum number has to vary via unity i.e.

ΔJ = J' - J'' = +-1

Therefore the locations of the lines in a rotational spectrum will be specified via,

V_J'↔J'' = F(J') - F(J'') = 2Be(J''+1)    J'' = 0,1,2,.....

Where J''indicates the lower energy level and J' signifies higher energy level included in the transition. The height of the lines is computed via the distribution of the molecules in the different levels and the probability of evolution between 2 energy levels.

We examine that, for a stiff rotor, the conversion lines are uniformly spaced in the wave number space. Though, this isn't always the case, except for the stiff rotor model. For non-rigid rotor model, we require to consider modifies in the instant of inertia of the molecule. 2 primary causes for this are,

Centrifugal distortion

When a molecule rotates, the centrifugal force drags the atoms apart. As a result, the moment of inertia of the molecule rises, therefore decreasing. B_e To account for this a centrifugal distortion correction expression is added to the rotational energy levels of the molecule.

F(J) = B_e J(J +1) - D_e J² (J+1)²      J = 0,1,2....

Where D_e is the centrifugal distortion constant.

Accordingly the line spacing for the rotational mode transforms to,

V_{J'↔ J''}  = 2B_e (J''+1) - 4D_e (J''+1)³   J''=0,1,2..

Effect of vibration on rotation

A molecule is always in vibration. As the molecule vibrates its moment of inertia varies. Further, there is a fictitious  force, Coriolis coupling, between the vibrational motions of the nuclei in the rotating (non- inertial) frame. However, as long as the vibrational  quantum number doesn't 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, as well; if one is noticed in low vibrational and rotational quantum numbers only.

Symmetric top

The rotational motion of a symmetric top molecule can be explained through 2 independent rotational quantum numbers (because 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 2 independent axes, we connect one of the quantum number (J) by the total angular momentum of the molecule and the additional quantum number (K) with the angular momentum of the axis that has dissimilar moment of inertia (for instance axis C for oblate symmetric top and axis A for prolate symmetric tops). The rotational energy F(J,K) of these a molecule, depend on stiff rotor assumptions can be expressed in terms of the 2 earlier described rotational quantum numbers as follows,

F(J,K) = B˜J(J+1)+(A˜ - B˜)K²   J= 0,1,2,... and K = -J,-J+1, ...., -1,0,1,...,J-1,J

Where B˜ = h/8π²cI_B and A˜= h/8π²Ci_A for a prolate symmetric top molecule or A˜= h/8π²CI_B for an oblate molecule.

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

ΔJ = +-1 and ΔK =0

This is so because K is related through the axis about which the molecule is symmetric and hence has no net dipole moment in that direction. Therefore there is no interaction of this mode with the light particles (photon).

This provides the transition wave numbers of,

V_{J'↔ J''} , _K = F(J',K) - F(J'',K) = 2B˜ (J''+1)                 J'' = 0,1,2,...

Which is the same as in the case of a linear molecule.

In case of non-rigid rotors, the 1^st order centrifugal distortion correction is specified via,

F(J,K) = B˜J(J+1)+(A˜-B˜)K² -D˜_J J²(J+1)² - D˜_JK J(J+1)K² - D_KK⁴      J = 0,1,2, ... and K = -J,...,0,...,J

J = 0,1,2...

The suffixes on the centrifugal distortion constant D indicate the rotational mode involved and aren't a function of the rotational quantum number. The location of the transition lines on a spectrum is specified via,

V_{J'↔ J''} , _K   = F(J',K) - F(J'',K) = 2B˜(J''+1)-4D_J(J''+1)³-2D_JK (J''+1)K²       J''= 0,1,2,....

Spherical tops

Unlike other molecules, spherical top molecules contain no net dipole moment, and therefore they don't exhibit a pure rotational spectrum.

Asymmetric tops

The spectrum for such molecules generally includes many lines due to three different rotational modes and their combinations. The subsequent analysis is valid for the general case and collapses to the diverse special cases described above in the appropriate limit. From the moments of inertia one can describe an asymmetry parameter κ as

K = 2B-A-C /A-C

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

One can describe a scaled rotational Hamiltonian dependent on J and κ. The (symmetric) matrix representation of this Hamiltonian is banded, zero everywhere but the major diagonal and the 2^nd sub diagonal. The Hamiltonian can be formulated in 6 different settings, dependent on the mapping of the principal axes to lab axes and handedness. For the most asymmetric, right-handed illustration the diagonal components are, for  

H_k,k(κ) = κk²

and the 2^nd off-diagonal elements (independent of κ) are

H_k,k+2(κ) = √ [J(J+1)-(K+1)(K+2)][J(J+1)-K(K+1)]/2

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 specified via

A+C/2*J(J+1)+A-C/2*E_K(K)

Hyperfine interactions

In addition to the major structure that is examined 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 an extremely deep understanding of molecular quantum mechanics. The major interactions dependable for small changes in the spectra (additional splitting and shifts of lines) are due to magnetic and electrostatic interactions in the molecule. The particular strength of such interactions varies in dissimilar molecules, but in common, the order of such consequences (in decreasing importance) is:

1.  Electron spin - electron spin contact (this happens in molecules through 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 through 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 through one another).

Such interactions provide increase to the trait energy levels that are probed in 'magnetic resonance' spectroscopy these as NMR and ESR, where they symbolize the "zero field splitting" that are always present.

Experimental Determination of the Spectrum

Fourier transform infrared (FTIR) spectroscopy can be utilized to experimentally study rotational spectra. Typically, spectra at such wavelengths include rovibrational excitation, for example, excitation of together a vibrational and a rotational mode of a molecule.

Conventionally, microwave spectra were computed using an easy arrangement in that low pressure gas was introduced to a section of waveguide between a microwave source (of variable frequency) and a microwave detector. The spectrum was attained via sweeping the frequency of the source while detecting the intensity of the transmitted radiation. This experimental arrangement has a major difficulty related to the propagation of microwave radiation through waveguides. The physical size of the waveguide restricts the frequency of the radiation that can be transmitted through it. For a specified waveguide size (these as X-band) there is a cutoff frequency, and microwave radiation through smaller frequencies (longer wavelengths) can't be propagated through the waveguide. Additionally, as the frequency is raised, additional modes of propagation become possible, that correspond to different velocities of the radiation propagating down the waveguide (this can be envisaged as the radiation bouncing down the guide, at different angles of  reflection). The net consequence of these considerations is that each size of waveguide is only useful over a rather narrow range of frequencies and must be physically swapped out for a different size of waveguide once this frequency range is exceeded.

Within the last two years, a further development of Fourier Transform Microwave Spectroscopy has occurred, which may well introduce a new renaissance into microwave spectroscopy. This is the use of "Chirped Pulses" to provide an electromagnetic wave that has as its Fourier Transform a very wide range of microwave frequencies. (See University of Virginia, Chemistry Department website).

Applications

Microwave spectroscopy is commonly used in physical chemistry to determine the structure of small molecules (such as ozone, methanol, or water) with high precision. Other common techniques for determining molecular structure, such as X-ray crystallography don't work very well for several of such molecules (particularly the gases) and aren't as precise. However, microwave spectroscopy isn't helpful for determining the structures of huge molecules these as proteins.

Microwave spectroscopy is one of the principal means via that the elements of the universe are determined from the earth. It is particularly useful for detecting molecules 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 trying to research these molecules in the laboratory that Harry Kroto was led to the laboratory of Rick Smalley and Robert Curl, where it was probable to vaporize carbon under enormous energy conditions. This collaborative experiment led to the discovery of C₆₀, buckminsterfullerene, which led to the award of the year 1996 Nobel Prize in chemistry to Kroto, Smalley and Curl

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