http://debianws.lexgopc.com/wiki143/index.php?action=history&feed=atom&title=Rotational%E2%80%93vibrational_spectroscopyRotational–vibrational spectroscopy - Revision history2026-06-02T05:03:43ZRevision history for this page on the wikiMediaWiki 1.43.1http://debianws.lexgopc.com/wiki143/index.php?title=Rotational%E2%80%93vibrational_spectroscopy&diff=4907939&oldid=previmported>Jamesdp3: removed duplicate word in asymmetric top section2025-12-09T16:39:56Z<p>removed duplicate word in asymmetric top section</p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Previous revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:39, 9 December 2025</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l58">Line 58:</td>
<td colspan="2" class="diff-lineno">Line 58:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where positive ''m'' values refer to the R-branch and negative values refer to the P-branch. The term ω<sub>0</sub> gives the position of the (missing) Q-branch, the term <math>(B ^\prime+B^{\prime\prime})m</math> implies an progression of equally spaced lines in the P- and R- branches, but the third term, <math>(B^\prime-B^{\prime\prime})m^2</math>shows that the separation between adjacent lines changes with changing rotational quantum number. When <math>B^{\prime\prime}</math> is greater than <math>B^\prime</math>, as is usually the case, as ''J'' increases the separation between lines decreases in the R-branch and increases in the P-branch. Analysis of data from the infrared spectrum of [[carbon monoxide]], gives value of <math>B^{\prime\prime}</math> of 1.915&nbsp;cm<sup>−1</sup> and <math>B^{\prime}</math> of 1.898&nbsp;cm<sup>−1</sup>. The bond lengths are easily obtained from these constants as ''r''<sub>0</sub> = 113.3 pm, ''r''<sub>1</sub> = 113.6 pm.<ref>Banwell and McCash, p 70</ref> These bond lengths are slightly different from the equilibrium bond length. This is because there is [[zero-point energy]] in the vibrational ground state, whereas the equilibrium bond length is at the minimum in the potential energy curve. The relation between the rotational constants is given by</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where positive ''m'' values refer to the R-branch and negative values refer to the P-branch. The term ω<sub>0</sub> gives the position of the (missing) Q-branch, the term <math>(B ^\prime+B^{\prime\prime})m</math> implies an progression of equally spaced lines in the P- and R- branches, but the third term, <math>(B^\prime-B^{\prime\prime})m^2</math>shows that the separation between adjacent lines changes with changing rotational quantum number. When <math>B^{\prime\prime}</math> is greater than <math>B^\prime</math>, as is usually the case, as ''J'' increases the separation between lines decreases in the R-branch and increases in the P-branch. Analysis of data from the infrared spectrum of [[carbon monoxide]], gives value of <math>B^{\prime\prime}</math> of 1.915&nbsp;cm<sup>−1</sup> and <math>B^{\prime}</math> of 1.898&nbsp;cm<sup>−1</sup>. The bond lengths are easily obtained from these constants as ''r''<sub>0</sub> = 113.3 pm, ''r''<sub>1</sub> = 113.6 pm.<ref>Banwell and McCash, p 70</ref> These bond lengths are slightly different from the equilibrium bond length. This is because there is [[zero-point energy]] in the vibrational ground state, whereas the equilibrium bond length is at the minimum in the potential energy curve. The relation between the rotational constants is given by</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math>B_v=B_{eq}-\alpha \left(v+{1\over 2}\right)</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:<math>B_v=B_{eq}-\alpha \left(v+{1\over 2}\right)</math></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where ν is a vibrational quantum number and α is a vibration-rotation interaction constant which can be calculated when the B values for two different vibrational states can be found. For carbon monoxide ''r<sub>eq</sub> = 113.0 pm.<ref>Banwell and McCash, p69</ref></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>where ν is a vibrational quantum number and α is a vibration-rotation interaction constant which can be calculated when the B values for two different vibrational states can be found. For carbon monoxide ''r<sub>eq</sub><ins style="font-weight: bold; text-decoration: none;">'' </ins>= 113.0 pm.<ref>Banwell and McCash, p69</ref></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Nitric oxide]], NO, is a special case as the molecule is [[paramagnetic]], with one unpaired electron. Coupling of the electron spin angular momentum with the molecular vibration causes ''lambda-doubling''<ref group=note>Another example of lambda-doubling is found in the energy levels of the [[hydroxyl radical]].</ref> with calculated harmonic frequencies of 1904.03 and 1903.68 cm<sup>−1</sup>. Rotational levels are also split.<ref name=G>{{cite journal| last=Gillette|first=R. H.|author2=Eyster,Eugene H. |title=The Fundamental Rotation-Vibration Band of Nitric Oxide|journal=Phys. Rev.|date=1939|volume=56|issue=11|pages=1113–1119|doi=10.1103/PhysRev.56.1113|bibcode = 1939PhRv...56.1113G }}</ref></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Nitric oxide]], NO, is a special case as the molecule is [[paramagnetic]], with one unpaired electron. Coupling of the electron spin angular momentum with the molecular vibration causes ''lambda-doubling''<ref group=note>Another example of lambda-doubling is found in the energy levels of the [[hydroxyl radical]].</ref> with calculated harmonic frequencies of 1904.03 and 1903.68 cm<sup>−1</sup>. Rotational levels are also split.<ref name=G>{{cite journal| last=Gillette|first=R. H.|author2=Eyster,Eugene H. |title=The Fundamental Rotation-Vibration Band of Nitric Oxide|journal=Phys. Rev.|date=1939|volume=56|issue=11|pages=1113–1119|doi=10.1103/PhysRev.56.1113|bibcode = 1939PhRv...56.1113G }}</ref></div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l155">Line 155:</td>
<td colspan="2" class="diff-lineno">Line 155:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:Water infrared absorption coefficient.gif|thumb|350px|Absorption spectrum ([[attenuation coefficient]] vs. wavelength) of liquid water (red) <ref name="Bertie96">{{cite journal |author=Bertie J. E.|author2=Lan Z. |date=1996 |title=Infrared Intensities of Liquids XX: The Intensity of the OH Stretching Band of Liquid Water Revisited, and the Best Current Values of the Optical Constants of H2O(l) at 25°C between 15,000 and 1 cm<sup>−1</sup> |journal=Applied Spectroscopy |volume=50 |issue=8 | pages=1047–1057 |url=http://www.opticsinfobase.org/as/abstract.cfm?uri=as-50-8-1047 |access-date=2012-08-08 |bibcode = 1996ApSpe..50.1047B |doi = 10.1366/0003702963905385 |s2cid=97329854 |url-access=subscription }}</ref> atmospheric [[water vapor]] (green) <ref name=spectraiaoru>{{cite web |url=http://spectra.iao.ru/1024x563/en/mol/survey/1/ |title=Spectroscopy of Atmospheric Gases (spectral databases) |publisher=V.E. Zuev Institute of Atmospheric Optics SB RAS |access-date=August 8, 2012 |quote=... various data sources: HITRAN and GEISA spectral databanks, original data obtained by IAO researchers in collaboration with other scientists, H2O spectra simulated by Partridge and Schwenke etc... ... |url-status=dead |archive-url=https://archive.today/20130416222417/http://spectra.iao.ru/1024x563/en/mol/survey/1/ |archive-date=April 16, 2013 }}</ref><ref name="Aringer02">{{cite journal |author=Aringer B.|author2=Kerschbaum F.|author3=Jørgensen U. G. |date=2002 |title=H<sub>2</sub>O in stellar atmospheres |journal=Astronomy and Astrophysics |volume=395 |issue=3|pages=915–927 |publisher= EDP Sciences |doi=10.1051/0004-6361:20021313 |url=http://www.aanda.org/articles/aa/pdf/2002/45/aah3665.pdf |access-date=2012-08-08 |bibcode = 2002A&A...395..915A |doi-access=free }}</ref> and ice (blue line) <ref name="Warren84">{{cite journal |author=Warren S. G. |date=1984 |title=Optical constants of ice from the ultraviolet to the microwave |journal=Applied Optics |volume=23 |issue=8 |pages=1206 |url=http://www.atmos.washington.edu/~sgw/PAPERS/1984_Icemcx.pdf |access-date=2012-08-08 |bibcode = 1984ApOpt..23.1206W |doi = 10.1364/AO.23.001206 |pmid=18204705 }}</ref><ref name="AringerWarren02">{{cite journal |author=Warren S. G.|author2=Brandt R. E. |date=2008 |title=Optical constants of ice from the ultraviolet to the microwave: A revised compilation |journal=J. Geophys. Res. |volume=113 |issue=D14 |pages=D14220 |doi=10.1029/2007JD009744 |url=http://www.atmos.washington.edu/ice_optical_constants/Warren_and_Brandt_2008.pdf |access-date=2012-08-08 |bibcode = 2008JGRD..11314220W |doi-access=free }}</ref> between 667 nm and 200 μm.<ref name="WozniakDera07">{{cite book | author=Wozniak B.| author2=Dera J. | date = 2007 | title = Atmospheric and Oceanographic Sciences Library | publisher = Springer Science+Business Media. LLC | location=New York | url=https://www.springer.com/cda/content/document/cda_downloaddocument/9780387307534-c2.pdf | access-date=August 4, 2012 | isbn = 978-0-387-30753-4}}</ref> The plot for vapor is a transformation of data ''Synthetic spectrum for gas mixture "Pure H<sub>2</sub>O"'' (296K, 1 atm) retrieved from [[HITRAN|Hitran]] on the Web Information System.<ref name=hitraniaoru>{{cite web |url=http://hitran.iao.ru/ |title=Hitran on the Web Information System |publisher=Harvard-Smithsonian Center for Astrophysics (CFA), Cambridge, MA, USA; V.E. Zuev Institute of Atmosperic Optics (IAO), Tomsk, Russia |access-date=August 11, 2012 }}</ref>]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:Water infrared absorption coefficient.gif|thumb|350px|Absorption spectrum ([[attenuation coefficient]] vs. wavelength) of liquid water (red) <ref name="Bertie96">{{cite journal |author=Bertie J. E.|author2=Lan Z. |date=1996 |title=Infrared Intensities of Liquids XX: The Intensity of the OH Stretching Band of Liquid Water Revisited, and the Best Current Values of the Optical Constants of H2O(l) at 25°C between 15,000 and 1 cm<sup>−1</sup> |journal=Applied Spectroscopy |volume=50 |issue=8 | pages=1047–1057 |url=http://www.opticsinfobase.org/as/abstract.cfm?uri=as-50-8-1047 |access-date=2012-08-08 |bibcode = 1996ApSpe..50.1047B |doi = 10.1366/0003702963905385 |s2cid=97329854 |url-access=subscription }}</ref> atmospheric [[water vapor]] (green) <ref name=spectraiaoru>{{cite web |url=http://spectra.iao.ru/1024x563/en/mol/survey/1/ |title=Spectroscopy of Atmospheric Gases (spectral databases) |publisher=V.E. Zuev Institute of Atmospheric Optics SB RAS |access-date=August 8, 2012 |quote=... various data sources: HITRAN and GEISA spectral databanks, original data obtained by IAO researchers in collaboration with other scientists, H2O spectra simulated by Partridge and Schwenke etc... ... |url-status=dead |archive-url=https://archive.today/20130416222417/http://spectra.iao.ru/1024x563/en/mol/survey/1/ |archive-date=April 16, 2013 }}</ref><ref name="Aringer02">{{cite journal |author=Aringer B.|author2=Kerschbaum F.|author3=Jørgensen U. G. |date=2002 |title=H<sub>2</sub>O in stellar atmospheres |journal=Astronomy and Astrophysics |volume=395 |issue=3|pages=915–927 |publisher= EDP Sciences |doi=10.1051/0004-6361:20021313 |url=http://www.aanda.org/articles/aa/pdf/2002/45/aah3665.pdf |access-date=2012-08-08 |bibcode = 2002A&A...395..915A |doi-access=free }}</ref> and ice (blue line) <ref name="Warren84">{{cite journal |author=Warren S. G. |date=1984 |title=Optical constants of ice from the ultraviolet to the microwave |journal=Applied Optics |volume=23 |issue=8 |pages=1206 |url=http://www.atmos.washington.edu/~sgw/PAPERS/1984_Icemcx.pdf |access-date=2012-08-08 |bibcode = 1984ApOpt..23.1206W |doi = 10.1364/AO.23.001206 |pmid=18204705 }}</ref><ref name="AringerWarren02">{{cite journal |author=Warren S. G.|author2=Brandt R. E. |date=2008 |title=Optical constants of ice from the ultraviolet to the microwave: A revised compilation |journal=J. Geophys. Res. |volume=113 |issue=D14 |pages=D14220 |doi=10.1029/2007JD009744 |url=http://www.atmos.washington.edu/ice_optical_constants/Warren_and_Brandt_2008.pdf |access-date=2012-08-08 |bibcode = 2008JGRD..11314220W |doi-access=free }}</ref> between 667 nm and 200 μm.<ref name="WozniakDera07">{{cite book | author=Wozniak B.| author2=Dera J. | date = 2007 | title = Atmospheric and Oceanographic Sciences Library | publisher = Springer Science+Business Media. LLC | location=New York | url=https://www.springer.com/cda/content/document/cda_downloaddocument/9780387307534-c2.pdf | access-date=August 4, 2012 | isbn = 978-0-387-30753-4}}</ref> The plot for vapor is a transformation of data ''Synthetic spectrum for gas mixture "Pure H<sub>2</sub>O"'' (296K, 1 atm) retrieved from [[HITRAN|Hitran]] on the Web Information System.<ref name=hitraniaoru>{{cite web |url=http://hitran.iao.ru/ |title=Hitran on the Web Information System |publisher=Harvard-Smithsonian Center for Astrophysics (CFA), Cambridge, MA, USA; V.E. Zuev Institute of Atmosperic Optics (IAO), Tomsk, Russia |access-date=August 11, 2012 }}</ref>]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Asymmetric top molecules have at most one <del style="font-weight: bold; text-decoration: none;">or more </del>2-fold rotation <del style="font-weight: bold; text-decoration: none;">axes</del>. There are three unequal moments of inertia about three mutually perpendicular [[Moment of inertia#Principal axes|principal axes]]. The spectra are very complex. The transition wavenumbers cannot be expressed in terms of an analytical formula but can be calculated using numerical methods.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Asymmetric top molecules have at most one 2-fold rotation <ins style="font-weight: bold; text-decoration: none;">axis</ins>. There are three unequal moments of inertia about three mutually perpendicular [[Moment of inertia#Principal axes|principal axes]]. The spectra are very complex. The transition wavenumbers cannot be expressed in terms of an analytical formula but can be calculated using numerical methods.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The water molecule is an important example of this class of molecule, particularly because of the presence of water vapor in the atmosphere. The low-resolution spectrum shown in green illustrates the complexity of the spectrum. At wavelengths greater than 10 μm (or wavenumbers less than 1000&nbsp;cm<sup>−1</sup>) the absorption is due to pure rotation. The band around 6.3 μm (1590&nbsp;cm<sup>−1</sup>) is due to the HOH bending vibration; the considerable breadth of this band is due to the presence of extensive rotational fine structure. High-resolution spectra of this band are shown in Allen and Cross, p 221.<ref>{{cite journal|last=Dalby|first=F.W.|author2=Nielsen, H.H. |title=Infrared Spectrum of Water Vapor. Part I—The 6.26μ Region |journal=J. Chem. Phys.|date=1956|volume=25|issue=5|pages=934–940|doi=10.1063/1.1743146 |bibcode = 1956JChPh..25..934D }}</ref> The symmetric and asymmetric stretching vibrations are close to each other, so the rotational fine structures of these bands overlap. The bands at shorter wavelength are overtones and combination bands, all of which show rotational fine structure. Medium resolution spectra of the bands around 1600&nbsp;cm<sup>−1</sup> and 3700&nbsp;cm<sup>−1</sup> are shown in Banwell and McCash, p91.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The water molecule is an important example of this class of molecule, particularly because of the presence of water vapor in the atmosphere. The low-resolution spectrum shown in green illustrates the complexity of the spectrum. At wavelengths greater than 10 μm (or wavenumbers less than 1000&nbsp;cm<sup>−1</sup>) the absorption is due to pure rotation. The band around 6.3 μm (1590&nbsp;cm<sup>−1</sup>) is due to the HOH bending vibration; the considerable breadth of this band is due to the presence of extensive rotational fine structure. High-resolution spectra of this band are shown in Allen and Cross, p 221.<ref>{{cite journal|last=Dalby|first=F.W.|author2=Nielsen, H.H. |title=Infrared Spectrum of Water Vapor. Part I—The 6.26μ Region |journal=J. Chem. Phys.|date=1956|volume=25|issue=5|pages=934–940|doi=10.1063/1.1743146 |bibcode = 1956JChPh..25..934D }}</ref> The symmetric and asymmetric stretching vibrations are close to each other, so the rotational fine structures of these bands overlap. The bands at shorter wavelength are overtones and combination bands, all of which show rotational fine structure. Medium resolution spectra of the bands around 1600&nbsp;cm<sup>−1</sup> and 3700&nbsp;cm<sup>−1</sup> are shown in Banwell and McCash, p91.</div></td></tr>
<!-- diff cache key wiki143:diff:1.41:old-300090:rev-4907939:php=table -->
</table>imported>Jamesdp3http://debianws.lexgopc.com/wiki143/index.php?title=Rotational%E2%80%93vibrational_spectroscopy&diff=300090&oldid=previmported>OAbot: Open access bot: url-access updated in citation with #oabot.2025-05-24T19:45:53Z<p><a href="https://en.wikipedia.org/wiki/OABOT" class="extiw" title="wikipedia:OABOT">Open access bot</a>: url-access updated in citation with #oabot.</p>
<a href="http://debianws.lexgopc.com/wiki143/index.php?title=Rotational%E2%80%93vibrational_spectroscopy&diff=300090">Show changes</a>imported>OAbot