Editing Diatomic molecule

{{short description|Molecule composed of any two atoms}}
{{Use dmy dates|date=December 2020}}
[[File:Dinitrogen-3D-vdW.png|thumb|upright|A [[space-filling model]] of the diatomic molecule dinitrogen, N<sub>2</sub>]]

'''Diatomic molecules''' ({{ety|el|di-|two}}) are [[molecule]]s composed of only two [[atom]]s, of the same or different [[chemical element]]s. If a diatomic molecule consists of two atoms of the same element, such as [[hydrogen]] ({{chem2|H2}}) or [[oxygen]] ({{chem2|O2}}), then it is said to be [[homonuclear molecule|homonuclear]]. Otherwise, if a diatomic molecule consists of two different atoms, such as [[carbon monoxide]] ({{chem2|CO}}) or [[nitric oxide]] ({{chem2|NO}}), the molecule is said to be [[heteronuclear molecule|heteronuclear]]. The bond in a homonuclear diatomic molecule is [[non-polar]].

[[File:Diatomic molecules periodic table.svg|thumb|upright=2.5|A [[periodic table]] showing the elements that exist as [[homonuclear molecule|homonuclear]] diatomic molecules under typical laboratory conditions.]]

The only [[chemical element]]s that form stable homonuclear diatomic molecules at [[standard temperature and pressure]] (STP) (or at typical laboratory conditions of 1 [[bar (pressure)|bar]] and 25&nbsp;°C) are the [[gas]]es hydrogen ({{chem2|H2}}), [[nitrogen]] ({{chem2|N2}}), oxygen ({{chem2|O2}}), [[fluorine]] ({{chem2|F2}}), and [[chlorine]] ({{chem2|Cl2}}), and the liquid [[bromine]] ({{chem2|Br2}}).<ref>{{cite book|last=Hammond|first=C.R.|title=Handbook of Chemistry and Physics|year=2012|chapter-url=http://www.hbcpnetbase.com//articles/04_01_91.pdf |archive-url=https://web.archive.org/web/20111111050116/http://www.hbcpnetbase.com//articles/04_01_91.pdf |archive-date=2011-11-11 |url-status=live|chapter=Section 4: Properties of the Elements and Inorganic Compounds}}</ref>

The [[noble gas]]es ([[helium]], [[neon]], [[argon]], [[krypton]], [[xenon]], and [[radon]]) are also gases at STP, but they are [[monatomic]]. The homonuclear diatomic gases and noble gases together are called "elemental gases" or "molecular gases", to distinguish them from other gases that are [[chemical compound]]s.<ref>{{cite book |last= Emsley|first= J. |title= The Elements |location= Oxford|isbn = 9780198555681|publisher= Clarendon Press |year= 1989 |pages= 22–23 }}</ref>

At slightly elevated temperatures, the halogens [[bromine]] ({{chem2|Br2}}) and [[iodine]] ({{chem2|I2}}) also form diatomic gases.<ref name=Chemistry>{{cite book|title=Chemistry|author=Whitten, Kenneth W.|author2=Davis, Raymond E.|author3=Peck, M. Larry|author4=Stanley, George G.|year=2010|publisher=Brooks/Cole, Cengage Learning|pages=337–338|url=https://books.google.com/books?id=6Zwu9-qT0qQC&pg=PA337|edition=9th|isbn=9780495391630}}</ref> All halogens have been observed as diatomic molecules, except for [[astatine]] and [[tennessine]], which are uncertain.

Other elements form diatomic molecules when evaporated, but these diatomic species repolymerize when cooled. Heating ("cracking") elemental phosphorus gives [[diphosphorus]] ({{chem2|P2}}). Sulfur vapor is mostly [[disulfur]] ({{chem2|S2}}). [[Dilithium]] ({{chem2|Li2}}) and [[disodium]] ({{chem2|Na2}})<ref>{{cite journal |last1=Lu |first1=Z.W. |last2=Wang |first2=Q. |last3=He |first3=W.M. |last4=Ma |first4=Z.G. |title=New parametric emissions in diatomic sodium molecules |journal=Applied Physics B |date=July 1996 |volume=63 |issue=1 |pages=43–46 |doi=10.1007/BF01112836 |bibcode=1996ApPhB..63...43L |s2cid=120378643 }}</ref> are known in the gas phase. Di[[tungsten]] ({{chem2|W2}}) and di[[molybdenum]] ({{chem2| Mo2}}) form with [[sextuple bond]]s in the gas phase. [[Dirubidium]] ({{chem2|Rb2}}) is diatomic.

== Heteronuclear molecules ==
All other diatomic molecules are [[chemical compound]]s of two different elements. Many elements can combine to form [[heteronuclear molecule|heteronuclear]] diatomic molecules, depending on temperature and pressure.

Examples are gases [[carbon monoxide]] (CO), [[nitric oxide]] (NO), and [[hydrogen chloride]] (HCl).

Many 1:1 [[binary compound]]s are not normally considered diatomic because they are [[polymer]]ic at room temperature, but they form diatomic molecules when evaporated, for example gaseous MgO, SiO, and many others.

== Occurrence ==
Hundreds of diatomic molecules have been identified<ref>{{cite book |author1=Huber, K. P. |author2=Herzberg, G. |title= Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules |publisher= New York: Van Nostrand: Reinhold |year= 1979 |isbn=978-0-442-23394-5}}</ref> in the environment of the Earth, in the laboratory, and in [[list of molecules in interstellar space|interstellar space]]. About 99% of the [[Earth's atmosphere]] is composed of two species of diatomic molecules: nitrogen (78%) and oxygen (21%). The natural abundance of [[hydrogen|hydrogen (H<sub>2</sub>)]] in the Earth's atmosphere is only of the order of parts per million, but H<sub>2</sub> is the most abundant diatomic molecule in the universe. The interstellar medium is dominated by hydrogen atoms.

== Molecular geometry ==
All diatomic molecules are linear and characterized by a single parameter which is the [[bond length]] or distance between the two atoms. Diatomic nitrogen has a triple bond, diatomic oxygen has a double bond, and diatomic hydrogen, fluorine, chlorine, iodine, and bromine all have single bonds.<ref>{{cite book |last1=Brown |first1=Catrin |last2=Ford |first2=Mike |title=Standard Level Chemistry |date=2014 |publisher=Prentice Hall |isbn=9781447959069 |pages=123–125 |edition=2nd}}</ref>

== Historical significance ==
Diatomic elements played an important role in the elucidation of the concepts of element, atom, and molecule in the 19th century, because some of the most common elements, such as hydrogen, oxygen, and nitrogen, occur as diatomic molecules. [[John Dalton]]'s original atomic hypothesis assumed that all elements were monatomic and that the atoms in compounds would normally have the simplest atomic ratios with respect to one another. For example, Dalton assumed water's formula to be HO, giving the atomic weight of oxygen as eight times that of hydrogen,<ref>{{cite book|url=https://books.google.com/books?id=Bq8aDsluyggC&q=Dalton+assumed+water%27s+formula+to+be+HO%2C+giving+the+atomic+weight+of+oxygen+as+eight+times+that+of+hydrogen&pg=PA19|title=The Development of Chemical Principles|author-link1=Cooper H. Langford|last1=Langford|first1=Cooper Harold|last2=Beebe|first2=Ralph Alonzo|date=1995-01-01|publisher=Courier Corporation|isbn=9780486683591|language=en}}</ref> instead of the modern value of about 16. As a consequence, confusion existed regarding atomic weights and molecular formulas for about half a century.

As early as 1805, [[Joseph Louis Gay-Lussac|Gay-Lussac]] and [[Alexander von Humboldt|von Humboldt]] showed that water is formed of two volumes of hydrogen and one volume of oxygen, and by 1811 [[Amedeo Avogadro]] had arrived at the correct interpretation of water's composition, based on what is now called [[Avogadro's law]] and the assumption of diatomic elemental molecules. However, these results were mostly ignored until 1860, partly due to the belief that atoms of one element would have no [[chemical affinity]] toward atoms of the same element, and also partly due to apparent exceptions to Avogadro's law that were not explained until later in terms of dissociating molecules.

At the 1860 [[Karlsruhe Congress]] on atomic weights, [[Stanislao Cannizzaro|Cannizzaro]] resurrected Avogadro's ideas and used them to produce a consistent table of atomic weights, which mostly agree with modern values. These weights were an important prerequisite for the discovery of the [[periodic law]] by [[Dmitri Mendeleev]] and [[Lothar Meyer]].<ref>{{cite journal |author= Ihde, Aaron J. |title= The Karlsruhe Congress: A centennial retrospective |journal= Journal of Chemical Education |year= 1961 |volume= 38 |pages= 83–86 |url= http://search.jce.divched.org:8081/JCEIndex/FMPro?-db=jceindex.fp5&-lay=wwwform&combo=karlsruhe&-find=&-format=detail.html&-skip=0&-max=1&-token.2=0&-token.3=10 |access-date= 2007-08-24 |doi= 10.1021/ed038p83 |issue= 2 |bibcode= 1961JChEd..38...83I |archive-url= https://web.archive.org/web/20070928044137/http://search.jce.divched.org:8081/JCEIndex/FMPro?-db=jceindex.fp5&-lay=wwwform&combo=karlsruhe&-find=&-format=detail.html&-skip=0&-max=1&-token.2=0&-token.3=10 |archive-date= 28 September 2007 |url-status= dead }}</ref>

== Excited electronic states ==
Diatomic molecules are normally in their lowest or ground state, which conventionally is also known as the <math>X</math> state. When a gas of diatomic molecules is bombarded by energetic electrons, some of the molecules may be excited to higher electronic states, as occurs, for example, in the natural aurora; high-altitude nuclear explosions; and rocket-borne electron gun experiments.<ref name=gilmore1992/> Such excitation can also occur when the gas absorbs light or other electromagnetic radiation. The excited states are unstable and naturally relax back to the ground state. Over various short time scales after the excitation (typically a fraction of a second, or sometimes longer than a second if the excited state is [[metastability|metastable]]), transitions occur from higher to lower electronic states and ultimately to the ground state, and in each transition results a [[photon]] is emitted. This emission is known as [[fluorescence]]. Successively higher electronic states are conventionally named <math>A</math>, <math>B</math>, <math>C</math>, etc. (but this convention is not always followed, and sometimes lower case letters and alphabetically out-of-sequence letters are used, as in the example given below). The excitation energy must be greater than or equal to the energy of the electronic state in order for the excitation to occur.

In quantum theory, an electronic state of a diatomic molecule is represented by the [[molecular term symbol]]
<math display="block">^{2S+1} \Lambda (v)^{+/-}_{(g/u)}</math>
where <math>S</math> is the total electronic spin quantum number, <math>\Lambda</math> is the total electronic angular momentum quantum number along the internuclear axis, and <math>v</math> is the vibrational quantum number. <math>\Lambda</math> takes on values 0, 1, 2, ..., which are represented by the electronic state symbols <math>\Sigma</math>, <math>\Pi</math>, <math>\Delta</math>, ...
For example, the following table lists the common electronic states (without vibrational quantum numbers) along with the energy of the lowest vibrational level (<math>v=0</math>) of diatomic nitrogen (N<sub>2</sub>), the most abundant gas in the Earth's atmosphere.<ref name=laher1991/> 

The subscripts and superscripts after <math>\Lambda</math> give additional quantum mechanical details about the electronic state. The superscript <math>+</math> or <math>-</math> determines whether reflection in a plane containing the internuclear axis introduces a sign change in the wavefunction. The sub-script <math>g</math> or <math>u</math> applies to molecules of identical atoms, and when reflecting the state along a plane perpendicular to the molecular axis, states that does not change are labelled <math>g</math> (gerade), and states that change sign are labelled <math>u</math> (ungerade).

{| class="wikitable" style="margin-left:1em;"
|-
! State !! Energy{{efn|The "energy" units here are actually the reciprocal of the wavelength of a photon emitted in a transition to the lowest energy state. The actual energy can be found by multiplying the given statistic by the product of ''c'' (the speed of light) and ''h'' (the Planck constant); i.e., about 1.99 × 10<sup>−25</sup> joule-metres, and then multiplying by a further factor of 100 to convert from cm<sup>−1</sup> to m<sup>−1</sup>.}} (<math>T_0</math>, cm<sup>−1</sup>)
|-
| <math>X ^1\Sigma_g^+</math> || 0.0
|-
| <math>A ^3\Sigma_u^+</math> || 49754.8
|-
| <math>B ^3\Pi_g</math> || 59306.8
|-
| <math>W ^3\Delta_u</math> || 59380.2
|-
| <math>B' ^3\Sigma_u^-</math> || 65851.3
|-
| <math>a' ^1\Sigma_u^-</math> || 67739.3
|-
| <math>a ^1\Pi_g</math> || 68951.2
|-
| <math>w ^1\Delta_u</math> || 71698.4
|}
{{noteslist}}

The aforementioned [[fluorescence]] occurs in distinct regions of the [[electromagnetic spectrum]], called "[[emission spectrum|emission bands]]": each band corresponds to a particular transition from a higher electronic state and vibrational level to a lower electronic state and vibrational level (typically, many vibrational levels are involved in an excited gas of diatomic molecules). For example, N<sub>2</sub> <math>A</math>-<math>X</math> emission bands (a.k.a. Vegard-Kaplan bands) are present in the spectral range from 0.14 to 1.45 μm (micrometres).<ref name=gilmore1992/> A given band can be spread out over several nanometers in electromagnetic wavelength space, owing to the various transitions that occur in the molecule's rotational quantum number, <math>J</math>. These are classified into distinct sub-band branches, depending on the change in <math>J</math>.<ref name=levine1975/> The <math>R</math> branch corresponds to <math>\Delta J = +1</math>, the <math>P</math> branch to <math>\Delta J = -1</math>, and the <math>Q</math> branch to <math>\Delta J = 0</math>. Bands are spread out even further by the limited [[spectral resolution]] of the [[spectrometer]] that is used to measure the [[spectrum]]. The spectral resolution depends on the instrument's [[point spread function]].

== Energy levels ==
The [[molecular term symbol]] is a shorthand expression of the angular momenta that characterize the electronic quantum states of a diatomic molecule, which are also [[eigenstate]]s of the electronic molecular [[Hamiltonian (quantum mechanics)|Hamiltonian]]. It is also convenient, and common, to represent a diatomic molecule as two-point masses connected by a massless spring. The energies involved in the various motions of the molecule can then be broken down into three categories: the translational, rotational, and vibrational energies. The theoretical study of the rotational energy levels of the diatomic molecules can be described using the below description of the rotational energy levels. While the study of vibrational energy level of the diatomic molecules can be described using the harmonic oscillator approximation or using the quantum vibrational interaction potentials.<ref name="Swati">{{Cite journal|doi = 10.1016/j.ctta.2022.100073|title = Temperature guided behavioral transitions in confined helium: Gas-wall interaction effects on dynamics and transport in the cryogenic limit |year = 2022|last =  Mishra|first = Swati|journal = Chemical Thermodynamics and Thermal Analysis |volume = 7|issue = August |pages = 100073|doi-access = free}}</ref><ref name="Al-Raeei">{{Cite journal|doi = 10.1088/1361-648X/ac6a9b|title = Morse potential specific bond volume: a simple formula with applications to dimers and soft–hard slab slider |year = 2022|last = Al-Raeei|first = Marwan |journal = Journal of Physics: Condensed Matter |volume = 34|issue = 28|pages = 284001|pmid = 35544352 |doi-access = free|bibcode = 2022JPCM...34B4001A }}</ref> These potentials give more accurate energy levels because they take multiple vibrational effects into account.

Concerning history, the first treatment of diatomic molecules with quantum mechanics was made by [[Lucy Mensing]] in 1926.<ref>{{cite journal |last=Mensing |first=Lucy |title=Die Rotations-Schwingungsbanden nach der Quantenmechanik |journal=Zeitschrift für Physik |volume=36 |issue=11 |date=1926-11-01 |issn=0044-3328 |pages=814–823 |doi=10.1007/BF01400216 |bibcode=1926ZPhy...36..814M |s2cid=123240532 |language=German }}</ref>

=== Translational energies ===
The translational energy of the molecule is given by the [[kinetic energy]] expression:
<math display="block">E_\text{trans} = \frac{1}{2}mv^2</math>
where <math>m</math> is the mass of the molecule and <math>v</math> is its velocity.

=== Rotational energies ===
Classically, the kinetic energy of rotation is
<math display="block">E_\text{rot} = \frac{L^2}{2 I} </math>
where
* <math>L \,</math> is the [[angular momentum]]
* <math>I \,</math> is the [[moment of inertia]] of the molecule

For microscopic, atomic-level systems like a molecule, angular momentum can only have specific discrete values given by
<math display="block">L^2 = \ell(\ell+1) \hbar^2 </math>
where <math>\ell</math> is a non-negative integer and <math>\hbar</math> is the [[reduced Planck constant]].

Also, for a diatomic molecule the moment of inertia is
<math display="block">I = \mu r_0^2 </math>
where
* <math>\mu \,</math> is the [[reduced mass]] of the molecule and
* <math>r_0 \,</math> is the average distance between the centers of the two atoms in the molecule.

So, substituting the angular momentum and moment of inertia into {{math|''E''<sub>rot</sub>}}, the rotational energy levels of a diatomic molecule are given by:
<math style="" display="block">E_\text{rot} = \frac{\ell (\ell + 1) \hbar^2}{2 \mu r_0^2}, \quad \ell = 0, 1, 2, \dots</math>

=== Vibrational energies ===
Another type of motion of a diatomic molecule is for each atom to oscillate—or [[vibration|vibrate]]—along the line connecting the two atoms. The vibrational energy is approximately that of a [[quantum harmonic oscillator]]:
<math display="block">E_\text{vib} = \left(n + \tfrac{1}{2} \right)\hbar \omega, \quad n = 0, 1, 2, \dots,</math>
where
* <math>n</math> is an integer
* <math>\hbar</math> is the [[reduced Planck constant]] and
* <math>\omega</math> is the [[angular frequency]] of the vibration.

=== Comparison between rotational and vibrational energy spacings ===
The spacing, and the energy of a typical spectroscopic transition, between vibrational energy levels is about 100 times greater than that of a typical transition between [[rotational energy]] levels.

== Hund's cases==
{{Main|Hund's cases}}
The [[good quantum number]]s for a diatomic molecule, as well as good approximations of rotational energy levels, can be obtained by modeling the molecule using [[Hund's cases]].

== Mnemonics ==
The mnemonics ''BrINClHOF'', pronounced "Brinklehof",<ref>{{Cite web|url=https://www.mnemonic-device.com/chemistry/brinclhof-pronounced-brinklehoff/|title=Mnemonic BrINClHOF (pronounced Brinklehoff) in Chemistry|language=en|access-date=2019-06-01}}</ref> ''HONClBrIF'', pronounced "Honkelbrif",<ref name="sherman">{{cite book|last1=Sherman|first1=Alan|title=Chemistry and Our Changing World|date=1992|publisher=Prentice Hall|isbn=9780131315419|page=82|url=https://books.google.com/books?id=Ai8YAQAAMAAJ&q=honclbrif}}</ref> “HOBrFINCl”, pronounced “Hoberfinkel”, and ''HOFBrINCl'', pronounced "Hofbrinkle", have been coined to aid recall of the list of diatomic elements.  Another method, for English-speakers, is the sentence:  "''Never Have Fear of Ice Cold Beer''" as a representation of Nitrogen, Hydrogen, Fluorine, Oxygen, Iodine, Chlorine, Bromine.

== See also ==
* [[Symmetry of diatomic molecules]]
* [[AXE method]]
* [[Octatomic element]]
* [[Covalent bond]]
* [[Industrial gas]]

== References ==
{{reflist|2|refs=

<ref name=levine1975>{{citation |first1=Ira N. |last1=Levine |title=Molecular Spectroscopy |publisher=John Wiley & Sons |year=1975 |isbn=0-471-53128-6 |pages=508–9}}</ref>

<ref name=gilmore1992>
{{cite journal
| last1 = Gilmore
| first1 = Forrest R.
| last2 = Laher
| first2 = Russ R.
| last3 = Espy
| first3 = Patrick J.
| year = 1992
| title = Franck-Condon Factors, r-Centroids, Electronic Transition Moments, and Einstein Coefficients for Many Nitrogen and Oxygen Band Systems
| journal = [[Journal of Physical and Chemical Reference Data]]
| volume = 21
| issue = 5
| pages = 1005–1107
| doi=10.1063/1.555910
|bibcode = 1992JPCRD..21.1005G | url = https://apps.dtic.mil/sti/citations/ADA246065
| archive-url = https://web.archive.org/web/20170709141516/http://www.dtic.mil/docs/citations/ADA246065
| url-status = live
| archive-date = 9 July 2017
}}
</ref>

<ref name=laher1991>
{{cite journal
| last1 = Laher
| first1 = Russ R.
| last2 = Gilmore
| first2 = Forrest R.
| year = 1991
| title = Improved Fits for the Vibrational and Rotational Constants of Many States of Nitrogen and Oxygen
| journal = [[Journal of Physical and Chemical Reference Data]]
| volume = 20
| issue = 4
| pages = 685–712
| doi=10.1063/1.555892
|bibcode = 1991JPCRD..20..685L | url = https://apps.dtic.mil/sti/citations/ADA229996
| archive-url = https://web.archive.org/web/20180602001343/http://www.dtic.mil/docs/citations/ADA229996
| url-status = live
| archive-date = 2 June 2018
}}
</ref>

}}

== Further reading ==
* {{cite book |author1=Huber, K. P. |author2=Herzberg, G. |title = Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules |publisher = New York: Van Nostrand: Reinhold |year = 1979 |title-link=Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules }}
* {{cite book |author=Tipler, Paul |title=Physics For Scientists and Engineers: Vol. 1 |edition=4th |publisher=W. H. Freeman |year=1998 |isbn=1-57259-491-8}}

== External links ==
* [http://hyperphysics.phy-astr.gsu.edu/hbase/molecule/rotrig.html#c3 Hyperphysics] – Rotational Spectra of Rigid Rotor Molecules
* [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html Hyperphysics] – Quantum Harmonic Oscillator
* [http://www.3dchem.com/ 3D Chem] – Chemistry, Structures, and 3D Molecules
* [http://arquivo.pt/wayback/20160523113736/http://www.iumsc.indiana.edu/ IUMSC] – Indiana University Molecular Structure Center

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[[Category:Diatomic molecules| ]]
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