Editing Rotational spectroscopy (section)

==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 force]]s. 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, <math>J</math>, which defines the magnitude of the rotational angular momentum.

For nonlinear molecules which are symmetric rotors (or symmetric tops - see next section), there are two moments of inertia and the energy also depends on a second rotational quantum number, <math>K</math>, which defines the vector component of rotational angular momentum along the [[molecular symmetry|principal symmetry axis]].<ref>{{harvnb|Atkins|de Paula|2006|p=444}}</ref> 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 [[Rigid rotor#Quantum mechanical linear rigid rotor|rotational constant]]<ref group=notes>This article uses the molecular spectroscopist's convention of expressing the rotational constant <math>B</math> in cm<sup>−1</sup>. Therefore <math>B</math> in this article corresponds to <math>\bar B = B/hc</math> in the Rigid rotor article.</ref> and the moment of inertia of the molecule, and, knowing the atomic masses, can be used to determine the [[bond length]] directly. For [[diatomic|diatomic molecules]] this process is straightforward. For linear molecules with more than two atoms it is necessary to measure the spectra of two or more [[isotopologue]]s, such as <sup>16</sup>O<sup>12</sup>C<sup>32</sup>S and <sup>16</sup>O<sup>12</sup>C<sup>34</sup>S. This allows a set of [[simultaneous equations]] to be set up and solved for the [[bond length]]s).<ref group=notes>For a symmetric top, the values of the 2 moments of inertia can be used to derive 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.</ref> A bond length obtained in this 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 rotational constants is given by
:<math>B_v = B - \alpha\left(v + \frac{1}{2}\right)</math>

where v is a vibrational quantum number and α is a vibration-rotation interaction constant which can be calculated if the B values for two different vibrational states can be found.<ref>{{harvnb|Banwell|McCash|1994|p=99}}</ref>

For other molecules, if the spectra can be resolved and individual transitions assigned both [[bond length]]s and [[Molecular geometry|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 assumed molecular structure. By varying the molecular structure the fit can be improved, giving a 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 [[angular momentum quantization|quantized]], so that the [[rotational energy]] and the [[angular momentum]] can take only certain fixed values, which are related simply to the [[moment of inertia]], <math> I </math>, of the molecule. For any molecule, there are three moments of inertia: <math>I_A</math>, <math>I_B</math> and <math>I_C</math> 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 <math>I_A \leq I_B \leq I_C</math>, with axis <math>A</math> corresponding to the smallest moment of inertia. Some authors, however, define the <math>A</math> axis as the molecular [[molecular symmetry#Elements|rotation axis]] of highest order.

The particular pattern of [[energy level]]s (and, hence, of transitions in the rotational spectrum) for a molecule is determined by its symmetry. A convenient way to look at the molecules is to divide them into four different classes, based on the symmetry of their structure. These are

{{glossary}}
{{term|Spherical tops (spherical rotors)}}{{defn|
All three moments of inertia are equal to each other: <math>I_A = I_B = I_C</math>. Examples of spherical tops include [[Allotropes of phosphorus#White phosphorus|phosphorus tetramer ({{chem|P|4}})]], [[Carbon tetrachloride|carbon tetrachloride ({{chem|CCl|4}})]] and other tetrahalides, [[methane|methane ({{chem|CH|4}})]], [[silane|silane, ({{chem|SiH|4}})]], [[Sulfur hexafluoride|sulfur hexafluoride ({{chem|SF|6}})]] and other hexahalides. The molecules all belong to the cubic [[molecular point group|point group]]s T<sub>d</sub> or O<sub>h</sub>.
}}
{{term|Linear molecules}}{{defn|
For a linear molecule the moments of inertia are related by <math>I_A \ll I_B = I_C </math>. For most purposes, <math>I_A</math> can be taken to be zero. Examples of linear molecules include [[Oxygen|dioxygen ({{chem|O|2}})]], [[nitrogen|dinitrogen ({{chem|N|2}})]], [[Carbon monoxide|carbon monoxide (CO)]], [[Hydroxyl radical|hydroxy radical (OH)]], [[Carbon dioxide|carbon dioxide (CO<sub>2</sub>)]], [[Hydrogen cyanide|hydrogen cyanide (HCN)]], [[Carbonyl sulfide|carbonyl sulfide (OCS)]], [[Acetylene|acetylene (ethyne (HC≡CH)]] and dihaloethynes. These molecules belong to the point groups C<sub>∞v</sub> or D<sub>∞h</sub>.
}}
{{term|Symmetric tops (symmetric rotors)}}{{defn|
A symmetric top is a molecule in which two moments of inertia are the same, <math>I_A = I_B</math> or <math>I_B = I_C</math>. By definition a symmetric top must have a 3-fold or higher order [[molecular symmetry#Elements|rotation axis]]. As a matter of convenience, spectroscopists divide molecules into two classes of symmetric tops, ''[[Oblate spheroid|Oblate]] symmetric tops'' (saucer or disc shaped) with <math>I_A = I_B < I_C</math> and ''[[Prolate]] symmetric tops'' (rugby football, or cigar shaped) with <math>I_A < I_B = I_C </math>. The spectra look rather different, and are instantly recognizable. Examples of symmetric tops include
; [[Oblate spheroid|Oblate]]: [[Benzene|Benzene, {{chem|C|6|H|6}}]]; [[ammonia|ammonia, {{chem|NH|3}}]]; [[xenon tetrafluoride|xenon tetrafluoride, {{chem|Xe|F|4}}]]
; [[Prolate]]: [[Chloromethane|Chloromethane, {{chem|CH|3|Cl}}]], [[methylacetylene|propyne, {{chem|CH|3|C≡CH}}]]

As a detailed example, ammonia has a moment of inertia {{nowrap|''I''<sub>C</sub> {{=}} 4.4128 × 10<sup>−47</sup> kg m<sup>2</sup>}} about the 3-fold rotation axis, and moments {{nowrap|''I''<sub>A</sub> {{=}} ''I''<sub>B</sub> {{=}} 2.8059 × 10<sup>−47</sup> kg m<sup>2</sup>}} about any axis perpendicular to the C<sub>3</sub> axis. Since the unique moment of inertia is larger than the other two, the molecule is an oblate symmetric top.<ref>Moment of inertia values from {{harvnb|Atkins|de Paula|2006|p=445}}</ref>
}}
{{term|Asymmetric tops (asymmetric rotors)}}{{defn|
The three moments of inertia have different values. Examples of small molecules that are asymmetric tops include [[Water (molecule)|water, {{chem|H|2|O}}]] and [[Nitrogen dioxide|nitrogen dioxide, {{chem|NO|2}}]] whose symmetry axis of highest order is a 2-fold rotation axis. Most large molecules are asymmetric tops.
}}
{{glossary end}}

===Selection rules===
{{Main|selection rules}}

====Microwave and far-infrared spectra====
Transitions between rotational states can be observed in molecules with a permanent [[electric dipole moment]].<ref>{{harvnb|Hollas|1996|p=95}}</ref><ref group=notes>Such transitions are called electric dipole-allowed transitions. Other transitions 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]], {{chem|O|2}} and [[nitric oxide]], NO</ref> A consequence of this rule is that no microwave spectrum can be observed for centrosymmetric linear molecules such as {{chem|N|2}} ([[dinitrogen]]) or HCCH ([[ethyne]]), which are non-polar. Tetrahedral molecules such as {{chem|CH|4}} ([[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.<ref>{{harvnb|Hollas|1996|p=104}} shows part of the observed rotational spectrum of [[silane]]</ref>

With symmetric tops, the selection rule for electric-dipole-allowed pure rotation transitions is {{nowrap|Δ''K'' {{=}} 0}}, {{nowrap|Δ''J'' {{=}} ±1}}. Since these transitions are due to absorption (or emission) of a single photon with a spin of one, [[conservation of angular momentum]] implies that the molecular angular momentum can change by at most one unit.<ref>{{harvnb|Atkins|de Paula|2006|p=447}}</ref> Moreover, the quantum number ''K'' is limited to have values between and including +''J'' to -''J''.<ref>{{harvnb|Banwell|McCash|1994|p=49}}</ref>

====Raman spectra====
For [[Raman spectroscopy|Raman spectra]] the molecules undergo transitions in which an ''incident'' photon is absorbed and another ''scattered'' photon is emitted. The general selection rule for such a transition to be allowed is that the molecular [[polarizability]] must be [[anisotropic]], which means that it is not the same in all directions.<ref>{{harvnb|Hollas|1996|p=111}}</ref> Polarizability is a 3-dimensional [[tensor]] that can be represented as an ellipsoid. The polarizability ellipsoid of spherical top molecules is in fact spherical so those molecules show no rotational Raman spectrum. For all other molecules both [[Stokes line|Stokes]] and anti-Stokes lines<ref group=notes>In Raman spectroscopy the photon energies for Stokes and anti-Stokes scattering are respectively less than and greater than the incident photon energy. See the energy-level diagram at [[Raman spectroscopy]].</ref> can be observed and they have similar intensities due to the fact that many rotational states are thermally populated. The selection rule for linear molecules is ΔJ = 0, ±2. The reason for the values ±2 is that the polarizability returns to the same value twice during a rotation.<ref>{{harvnb|Atkins|de Paula|2006|pp=474–5}}</ref> The value ΔJ = 0 does not correspond to a molecular transition but rather to [[Rayleigh scattering]] in which the incident photon merely changes direction.<ref name="Banwell 1994 loc=Section 4.2, p. 105, Pure Rotational Raman Spectra">{{harvnb|Banwell|McCash|1994|loc=Section 4.2, p. 105, ''Pure Rotational Raman Spectra''}}</ref>

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 to the ''R'' series, whereas transitions with {{nowrap|Δ''J'' {{=}} +2}} belong to an ''S'' series.<ref name="Banwell 1994 loc=Section 4.2, p. 105, Pure Rotational Raman Spectra"/> Since Raman transitions involve two photons, it is possible for the molecular angular momentum to change by two units.

===Units===
The units used for rotational constants depend on the type of measurement. With infrared spectra in the [[wavenumber]] scale (<math>\tilde \nu</math>), the unit is usually the [[inverse centimeter]], written as cm<sup>−1</sup>, which is literally the number of waves in one centimeter, or the reciprocal of the wavelength in centimeters (<math>\tilde\nu = 1 / \lambda</math>). On the other hand, for microwave spectra in the frequency scale (<math>\nu</math>), the unit is usually the [[gigahertz]]. The relationship between these two units is derived from the expression
:<math> \nu \cdot \lambda = c,</math>

where ν is a [[frequency]], λ is a [[wavelength]] and ''c'' is the [[velocity of light]]. It follows that 
:<math>
  \tilde \nu / \text{cm}^{-1} =
  \frac{1}{\lambda / \text{cm}} =
  \frac{\nu / \text{s}^{-1}}{c / \left(\text{cm} \cdot \mathrm{s}^{-1}\right)} =  
  \frac{\nu / \text{s}^{-1}}{2.99792458 \times 10^{10}}.
</math>

As 1&nbsp;GHz = 10<sup>9</sup> Hz, the numerical conversion can be expressed as 
:<math>\tilde\nu / \text{cm}^{-1} \approx \frac{\nu / \text{GHz}}{30}.</math>

===Effect of vibration on rotation===
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.<ref>{{cite journal|last=Alexander|first=A. J.|author2=Kroto, H. W. |author3=Walton, D. R. M. |title=The microwave spectrum, substitution structure and dipole moment of cyanobutadiyne|journal=J. Mol. Spectrosc.|date=1967|volume=62|issue=2|pages=175–180|doi=10.1016/0022-2852(76)90347-7|bibcode = 1976JMoSp..62..175A }} Illustrated in {{harvnb|Hollas|1996|p=97}}</ref>

Further, there is a [[fictitious force]], [[Coriolis effect|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|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 microwave spectroscopy had become practical. To a first approximation, the rotation and vibration can be treated as [[Separable partial differential equation|separable]], so the energy of rotation is added to the energy of vibration. For example, the rotational energy levels for linear molecules (in the rigid-rotor approximation) are

:<math>E_\text{rot} = hc BJ(J + 1).</math>

In this approximation, the vibration-rotation wavenumbers of transitions are

:<math>\tilde\nu = \tilde\nu_\text{vib} + B''J''(J'' + 1) - B'J'(J' + 1),</math>

where <math>B''</math> and <math>B'</math> are rotational constants for the upper and lower vibrational state respectively, while <math>J''</math> and  <math>J'</math> are the rotational quantum numbers of the upper and lower levels. In reality, this expression has to be modified for the effects of anharmonicity of the vibrations, for centrifugal distortion and for Coriolis coupling.<ref>{{harvnb|Banwell|McCash|1994|p=63}}.</ref>

For the so-called ''R'' branch of the spectrum, <math>J' = J'' + 1</math> so that there is simultaneous excitation of both vibration and rotation. For the ''P'' branch, <math>J' = J'' - 1</math> so that a quantum of rotational energy is lost while a quantum of vibrational energy is gained. The purely vibrational transition, <math>\Delta J=0</math>, 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.