Editing Rotational spectroscopy (section)

==Structure of rotational spectra==

===Spherical top===
Spherical top molecules have no net dipole moment. A pure rotational spectrum cannot be observed by absorption or emission spectroscopy 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 [[rotational–vibrational coupling|ro–vibrational spectroscopy]]. This occurs when a molecule is polar in the vibrationally excited state. For example, the molecule [[methane]] is a spherical top but the asymmetric C-H stretching band shows rotational fine structure in the infrared spectrum, illustrated in [[rotational–vibrational coupling|rovibrational coupling]]. This spectrum is also interesting because it shows clear evidence of [[Coriolis effect|Coriolis coupling]] in the asymmetric structure of the band.

===Linear molecules===
[[File:Rotational spectrum example.png|right|thumb|300px|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 particle|point masses]] connected by rigid bonds. A linear molecule lies 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 which describe 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,

:<math> F\left( J \right) = B J \left( J+1 \right) \qquad J = 0,1,2,...</math>
where <math> B </math> 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 molecular axis is unique, that is, <math> I_B = I_C, I_A=0 </math>, so 
:<math> B = {h \over{8\pi^2cI_B}}= {h \over{8\pi^2cI_C}}</math>
For a diatomic molecule
:<math> I=\frac{m_1m_2}{m_1 +m_2}d^2 </math> 
where ''m''<sub>1</sub> and ''m''<sub>2</sub> are the masses of the atoms and ''d'' is the distance between them.

[[Selection rules]] dictate that during emission or absorption the rotational quantum number has to change by unity; i.e., <math> \Delta J = J^{\prime} - J^{\prime\prime} = \pm 1 </math>. Thus, the locations of the lines in a rotational spectrum will be given by

:<math> \tilde \nu_{J^{\prime}\leftrightarrow J^{\prime\prime}} = F\left( J^{\prime} \right) - F\left( J^{\prime\prime} \right) = 2 B \left( J^{\prime\prime} + 1 \right) \qquad J^{\prime\prime} = 0,1,2,...</math>

where <math>J^{\prime\prime}</math> denotes the lower level and <math>J^{\prime}</math> denotes the upper level involved in the transition.

The diagram illustrates rotational transitions that obey the <math>\Delta J</math>=1 selection rule. The dashed lines show how these transitions map onto features that can be observed experimentally.  Adjacent <math>J^{\prime\prime}{\leftarrow}J^{\prime}</math> 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====
[[File:Populations of rotational states.png|thumb|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 the number of molecules in the ground state, ''N<sub>J</sub>''/''N''<sub>0</sub> is given by the [[Boltzmann distribution]] as
:<math>\frac{N_J}{N_0} = e^{-\frac{E_J}{kT}} = e^{-\frac {BhcJ(J+1)}{kT}}</math>, 
where ''k'' is the [[Boltzmann constant]] and ''T'' the [[absolute temperature]]. This factor decreases as ''J'' increases. The second factor is the [[Degenerate energy levels|degeneracy]] of the rotational state, which is equal to {{nowrap|2J + 1}}. This factor increases as ''J'' increases. Combining the two factors<ref>{{harvnb|Banwell|McCash|1994|p=40}}</ref>
:<math>\text{population} \propto (2J + 1)e^{-\frac{E_J}{kT}}</math> 

The maximum relative intensity occurs at<ref>{{harvnb|Atkins|de Paula|2006|p=449}}</ref><ref group=notes>This value of J corresponds to the maximum of the population considered as a continuous function of J. However, since only integer values of J are allowed, the maximum line intensity is observed for a neighboring integer J.</ref>
:<math>J = \sqrt{\frac{kT}{2hcB}} - \frac{1}{2}</math>

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

====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 <math> B </math>, 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.<ref name=Ban45>{{harvnb|Banwell|McCash|1994|p=45}}</ref>

:<math> F\left( J \right) = B J \left( J+1 \right) - D J^2 \left( J+1 \right)^2 \qquad J = 0,1,2,...</math>

where <math> D</math> is the centrifugal distortion constant.

Therefore, the line positions for the rotational mode change to

:<math> \tilde \nu_{J^{\prime}\leftrightarrow J^{\prime\prime}} = 2 B \left( J^{\prime\prime} + 1 \right) - 4 D \left( J^{\prime\prime} +1 \right)^3 \qquad J^{\prime\prime} = 0,1,2,...</math>
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 <math> D</math>  can be derived as
:<math> D = \frac{h^3}{32 \pi^4 I^2 r^2 k c}</math>
where ''k'' is the vibrational [[force constant]]. The relationship between <math> B</math> and <math> D</math> 
:<math> D=\frac{4 B^3}{\tilde \omega ^2}</math>
where <math> \tilde \omega </math> 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 and line positions.<ref name=Ban45/> A striking example concerns the rotational spectrum of [[hydrogen fluoride]] which was fitted to terms up to ''[J(J+1)]<sup>5</sup>''.<ref>{{cite journal|last=Jennings|first=D.A.|author2=Evenson, K.M |author3=Zink, L.R. |author4=Demuynck, C. |author5=Destombes, J.L. |author6=Lemoine, B |author7=Johns, J.W.C. |title=High-resolution spectroscopy of HF from 40 to 1100 cm<sup>−1</sup>: Highly accurate rotational constants|journal=Journal of Molecular Spectroscopy|date=April 1987|volume=122|issue=2|pages=477–480|doi=10.1016/0022-2852(87)90021-X|bibcode = 1987JMoSp.122..477J }}[http://tf.nist.gov/general/pdf/632.pdf pdf]</ref>

====Oxygen====
The electric dipole moment of the dioxygen molecule, {{chem|O|2}} is zero, but the molecule is [[paramagnetic]] with two unpaired electrons so that there are magnetic-dipole allowed transitions which 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 motion of the molecule. The energy difference between successive J terms in any of these triplets is about 2&nbsp;cm<sup>−1</sup> (60&nbsp;GHz), with the single exception of J = 1←0 difference which is about 4&nbsp;cm<sup>−1</sup>. 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 number K there are two allowed transitions. The <sup>16</sup>O nucleus has zero nuclear spin angular momentum, so that symmetry considerations demand that K have only odd values.<ref>{{cite journal|last=Strandberg|first=M. W. P.|author2=Meng, C. Y. |author3=Ingersoll, J. G. |title=The Microwave Absorption Spectrum of Oxygen|journal=Phys. Rev. |date=1949|volume=75|issue=10|pages=1524–8|doi=10.1103/PhysRev.75.1524|bibcode = 1949PhRv...75.1524S |url=http://dspace.mit.edu/bitstream/handle/1721.1/4963/RLE-TR-087-14236979.pdf%3Bjsessionid%3D90D2A8D5B14E50D8DE20B855FD5BAA97?sequence%3D1}}[http://dspace.mit.edu/bitstream/handle/1721.1/4963/RLE-TR-087-14236979.pdf pdf]</ref><ref>{{cite journal|last=Krupenie|first=Paul H.|title=The Spectrum of Molecular Oxygen|journal=Journal of Physical and Chemical Reference Data |date=1972|volume=1|issue=2|pages=423–534|doi=10.1063/1.3253101|url=https://www.nist.gov/data/PDFfiles/jpcrd8.pdf|bibcode = 1972JPCRD...1..423K }}</ref>

===Symmetric top===
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 [[molecular symmetry|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

:<math>
  F\left( J,K \right) = B J \left( J+1 \right) + \left( A - B \right) K^2 \qquad
  J = 0, 1, 2, \ldots \quad \mbox{and}\quad K = +J, \ldots, 0, \ldots, -J
</math>

where <math> B = {h\over{8\pi^2cI_B}} </math> and <math> A = {h\over{8\pi^2cI_A}} </math> for a ''prolate'' symmetric top molecule or <math> A = {h\over{8\pi^2cI_C}} </math> for an ''oblate'' molecule.

This gives the transition wavenumbers as

:<math> \tilde \nu_{J^{\prime}\leftrightarrow J^{\prime\prime},K} = F\left( J^{\prime},K \right) - F\left( J^{\prime\prime},K \right) 
= 2 B \left( J^{\prime\prime} + 1 \right) 
\qquad J^{\prime\prime} = 0,1,2,...</math>

which is the same as in the case of a linear molecule.<ref>{{harvnb|Hollas|1996|p=101}}</ref> With a first order correction for centrifugal distortion the transition wavenumbers become
:<math> \tilde \nu_{J^{\prime}\leftrightarrow J^{\prime\prime},K} = F\left( J^{\prime},K \right) - F\left( J^{\prime\prime},K \right) 
= 2 \left(B - 2D_{JK}K^2 \right)  
\left( J^{\prime\prime} + 1 \right)
-4D_J\left(J^{\prime\prime}+1\right)^3 \qquad J^{\prime\prime} = 0,1,2,...</math>

The term in ''D<sub>JK</sub>'' has the effect of removing degeneracy present in the rigid rotor approximation, with different ''K'' values.<ref>{{harvnb|Hollas|1996|p=102}} shows the effect on the microwave spectrum of {{chem|H|3|SiNCS}}.</ref>

===Asymmetric top===
[[File:Atmospheric terahertz transmittance at Mauna Kea (simulated).svg|300 px|thumb|Pure rotation spectrum of atmospheric water vapour measured at Mauna Kea (33 cm<sup>−1</sup> to 100 cm<sup>−1</sup>)]]

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 [[Diagonalizable matrix#Diagonalization|matrix diagonalization]] for each ''J'' value. Formulae are available for molecules whose shape approximates to that of a symmetric top.<ref>{{harvnb|Hollas|1996|p=103}}</ref>

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&nbsp;cm<sup>−1</sup>. 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.<ref>{{cite journal|last=Hall|first=Richard T.|author2=Dowling, Jerome M. |title=Pure Rotational Spectrum of Water Vapor|journal=J. Chem. Phys.|date=1967|volume=47|issue=7|pages=2454–61|doi=10.1063/1.1703330|bibcode = 1967JChPh..47.2454H }}
{{cite journal|last=Hall|first=Richard T.|author2=Dowling, Jerome M. |title=Erratum: Pure Rotational Spectrum of Water Vapor|journal=J. Chem. Phys.|date=1971|volume=54|issue=11|page=4968|doi=10.1063/1.1674785|bibcode = 1971JChPh..54.4968H |doi-access=free}}</ref>