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2013-11-19T10:11:07Z

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The '''hydrogen molecular ion''', '''dihydrogen cation''', or H2+, is the simplest [[molecular ion]]. It is composed of two positively charged [[proton]]s and one negatively charged [[electron]], and can be formed from [[ionization]] of a neutral [[hydrogen molecule]]. It is of great historical and theoretical interest because, having only one electron, the [[Schrödinger equation]] for the system can be solved in a relatively straightforward way due to the lack of electron–electron repulsion ([[electron correlation]]). The analytical solutions for the energy eigenvalues are a ''generalization'' of the [[Lambert W function]].<ref>{{cite journal |last=Scott |first=T. C. |last2=Aubert-Frécon |first2=M. |last3=Grotendorst |first3=J. |year=2006 |title=New Approach for the Electronic Energies of the Hydrogen Molecular Ion |journal=Chem. Phys. |volume=324 |issue=2–3 |pages=323–338 |doi=10.1016/j.chemphys.2005.10.031 |arxiv=physics/0607081 }}</ref> Thus, the case of clamped nuclei can be completely done analytically using a [[computer algebra system]] within an [[experimental mathematics]] approach. Consequently, it is included as an example in most [[quantum chemistry]] textbooks.

The first successful quantum mechanical treatment of H2+ was published by the Danish physicist Øyvind Burrau in 1927,<ref>{{Cite journal|author=Burrau Ø |title=Berechnung des Energiewertes des Wasserstoffmolekel-Ions (H2+) im Normalzustand. |journal=Danske Vidensk. Selskab. Math.-fys. Meddel. |volume=M 7:14 |issue= |pages=1–18 |year=1927| language=German| url=http://www.royalacademy.dk/CatalogEntry.asp?id=862}} {{Cite journal|author=Burrau Ø |title=The calculation of the Energy value of Hydrogen molecule ions (H2+) in their normal position |journal=Naturwissenschaften |volume=15 |issue=1 |pages=16–7 |year=1927| language=German| url=http://www.springerlink.com/content/h60148l4717uv805/fulltext.pdf|format=PDF | doi=10.1007/BF01504875}}</ref> just one year after the publication of wave mechanics by [[Erwin Schrödinger]]. Earlier attempts using the [[old quantum theory]] had been published in 1922 by [[Karel Niessen]]<ref>Karel F. Niessen ''Zur Quantentheorie des Wasserstoffmolekülions'', doctoral dissertation, University of Utrecht, Utrecht: I. Van Druten (1922) as cited in Mehra, Volume 5, Part 2, 2001, p. 932.</ref> and [[Wolfgang Pauli]],<ref>{{Cite journal|author=Pauli W |title=Über das Modell des Wasserstoffmolekülions |journal=Ann. D. Phys. |volume=373 |issue=11 |pages=177–240 |year=1922 |doi=10.1002/andp.19223731101}} Extended doctoral dissertation; received 4 March 1922, published in issue No. 11 of 3 August 1922.</ref> and in 1925 by [[Harold Urey]].<ref>{{Cite journal|author=Urey HC |title=The Structure of the Hydrogen Molecule Ion |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=11 |issue=10 |pages=618–21 |date=October 1925 |pmid=16587051 |pmc=1086173 |doi= 10.1073/pnas.11.10.618|url=}}</ref> In 1928, [[Linus Pauling]] published a review putting together the work of Burrau with the work of [[Walter Heitler]] and [[Fritz London]] on the hydrogen molecule.<ref>{{Cite journal|journal=Chemical Reviews |author=Pauling, L. |title=The Application of the Quantum Mechanics to the Structure of the Hydrogen Molecule and Hydrogen Molecule-Ion and to Related Problems |year=1928 |volume=5 |pages=173–213 |doi=10.1021/cr60018a003|issue=2}}</ref>

Bonding in H2+ can be described as a covalent [[one-electron bond]], which has a formal [[bond order]] of one half.<ref>{{Cite book|author=Clark R. Landis; Frank Weinhold |title=Valency and bonding: a natural bond orbital donor-acceptor perspective |publisher=Cambridge University Press |location=Cambridge, UK |year=2005 |pages=96–100 |isbn=0-521-83128-8 |oclc= |doi= |accessdate=}}</ref>

The ion is commonly formed in [[molecular cloud]]s in space, and is important in the chemistry of the [[interstellar medium]].

==Quantum mechanical treatment, symmetries, and asymptotics==
[[Image:hydrogen molecular ion.png|thumb|300px|right| Hydrogen molecular ion H2+ with clamped nuclei A and B, internuclear distance R and plane of symmetry M.]]The simplest electronic Schrödinger wave equation for the hydrogen molecular ion <math> H_2^{+}</math> is modeled with two fixed nuclear centers, labeled ''A'' and ''B'', and one electron. It can be written as
:<math>
\left( -\frac{\hbar^2}{2m} \nabla^2 + V \right) \psi = E \psi ~,
</math>
where <math> V </math> is the electron-nuclear Coulomb potential energy function:
:<math>
V = - \frac{e^{2}}{4 \pi \varepsilon_0 } \left( \frac{1}{r_a} + \frac{1}{r_b} \right)
</math>
and ''E'' is the (electronic) energy of a given quantum mechanical state (eigenstate), with the electronic state function <math> \psi=\psi(\mathbf{r}) </math> depending on the spatial coordinates of the electron. An additive term <math> 1/R </math>, which is constant for fixed inter-nuclear distance <math> R </math>, has been omitted from the potential <math> V</math>, since it merely shifts the eigenvalue. The distances between the electron and the nuclei are denoted <math>r_a^{}</math> and <math>r_b^{}</math>. In atomic units <math>(\hbar=m=e=4 \pi\varepsilon_0 =1)</math> the wave equation is
:<math>\left( {} - \frac{1}{2} \nabla^2 + V \right) \psi = E \psi \qquad \mbox{with} \qquad V = {} - \frac{1}{r_a^{}} - \frac{1}{r_b^{}} \; .
</math>
We can choose the midpoint between the nuclei as the origin of coordinates. It follows from general symmetry principles that the wave functions can be characterized by their symmetry behavior with respect to space inversion ('''r''' <math> \to </math> -'''r'''). There are wave functions :<math>\psi_{+}(\mathbf{r})</math>, which are ''symmetric'' with respect to space inversion, and there are wave functions :<math>\psi_{-}(\mathbf{r})</math>, which are ''anti-symmetric'' under this symmetry operation: <math> \psi_{\pm}(-{\mathbf{r}}) = {} \pm \psi_{\pm}({\mathbf r}) \; . </math>
The symmetry-adapted wave functions satisfy the same Schrödinger equation.

The ground state (the lowest discrete state) of <math> H_{2}^{+}</math> is denoted <math> {\rm X}_{}^{2}\Sigma_{\rm g}^{+}</math> <ref>{{cite book |last=Huber |first=K.-P. |authorlink2=Gerhard Herzberg |last2=Herzberg |first2=G. |year=1979 |title=Molecular Spectra and Molecular Structure. IV. Constants of Diatomic Molecules |location=New York |publisher=Van Nostrand Reinhold }}</ref> or <math>1s \sigma_{\rm g}^{}</math> and it is symmetric. There is also the first excited state <math> {\rm A}_{}^{2}\Sigma_{\rm u}^{+}</math> (<math> {\rm 2p}\sigma_{\rm u}^{}</math>), which is antisymmetric. (The suffixes [[Molecular term symbol|g and u]] are from the German ''gerade'' and ''ungerade'') occurring here denote just the symmetry behavior under space inversion. Their use is standard practice for the designation of electronic states of diatomic molecules, whereas for atomic states the terms ''even'' and ''odd'' are used. [[Image:h2plus figure 2.png|thumb|500px|right| Energies (E) of the lowest discrete states of the hydrogen molecular ion <math>H_2^{+}</math> as a function of inter-nuclear distance (R) in atomic units. See text for details.]] Asymptotically, the (total) eigenenergies <math>E_{\pm}</math> for these two lowest lying states have the same asymptotic expansion in inverse powers of the inter-nuclear distance ''R'':<ref>{{cite journal |last=Čížek |first=J. |last2=Damburg |first2=R. J. |last3=Graffi |first3=S. |last4=Grecchi |first4=V. |last5=Harrel II |first5=E. M. |last6=Harris |first6=J. G. |last7=Nakai |first7=S. |authorlink8=Josef Paldus |last8=Paldus |first8=J. |last9=Propin |first9=R. Kh. |last10=Silverstone |first10=H. J. |year=1986 |title=''1/R'' expansion for ''H2+'': Calculation of exponentially small terms and asymptotics |journal=[[Physical Review|Phys. Rev. A]] |volume=33 |issue= |pages=12–54 |doi=10.1103/PhysRevA.33.12 }}</ref>
:<math>
E_{\pm} = {} - \frac{1}{2} - \frac{9}{4 R^4} + O(R^{-6}) + \cdots
</math>
The actual difference between these two energies is called the [[exchange interaction|exchange energy]] splitting and is given by:<ref>{{cite journal |last=Scott |first=T. C. |authorlink2=Alexander Dalgarno |last2=Dalgarno |first2=A. |last3=Morgan III |first3=J. D. |year=1991 |title=Exchange Energy of ''H2+'' Calculated from Polarization Perturbation Theory and the [[Holstein–Herring method|Holstein-Herring Method]] |journal=[[Physical Review Letters|Phys. Rev. Lett.]] |volume=67 |issue=11 |pages=1419–1422 |doi=10.1103/PhysRevLett.67.1419 }}</ref>
:<math>
\Delta E = E_{-} - E_{+} = \frac{4}{e} \, R \, e^{-R} \left[ \, 1 + \frac{1}{2R} + O(R^{-2}) \, \right]
</math>
which exponentially vanishes as the inter-nuclear distance ''R'' gets greater. The lead term <math> {\textstyle \frac{4}{e}} R e^{-R} </math> was first obtained by the [[Holstein–Herring method]]. Similarly, asympotic expansions in powers of ''1/R'' have been obtained to high order by Cizek ''et al.'' for the lowest ten discrete states of the hydrogen molecular ion (clamped nuclei case). For general diatomic and polyatomic molecular systems, the exchange energy is thus very elusive to calculate at large inter-nuclear distances but is nonetheless needed for long-range interactions including studies related to magnetism and charge exchange effects. These are of particular importance in stellar and atmospheric physics.

The energies for the lowest discrete states are shown in the graph above. These can be obtained to within arbitrary accuracy using [[computer algebra]] from the generalized [[Lambert W function]] (see eq. <math>(3)</math> in that site and the reference of Scott, Aubert-Frécon, and Grotendorst) but were obtained initially by numerical means to within double precision by the most precise program available, namely ODKIL.<ref>{{cite journal |last=Hadinger |first=G. |last2=Aubert-Frécon |first2=M. |last3=Hadinger |first3=G. |year=1989 |title=The Killingbeck method for the one-electron two-centre problem |journal=[[Journal of Physics B|J. Phys. B]] |volume=22 |issue=5 |pages=697–712 |doi=10.1088/0953-4075/22/5/003 }}</ref> The red full lines are <math> {\rm {}}_{}^{2}\Sigma_{\rm g}^{+}</math> states. The green dashed lines are <math> {\rm {}}_{}^{2}\Sigma_{\rm u}^{+}</math> states. The blue dashed line is a <math> {\rm {}}_{}^{2}\Pi_{\rm u}</math> state and the pink dotted line is a <math> {\rm {}}_{}^{2}\Pi_{\rm g}</math> state. Note that although the generalized [[Lambert W function]] eigenvalue solutions supersede these asymptotic expansions, in practice, they are most useful near the [[bond length]]. These solutions are possible because the [[partial differential equation]] of the wave equation here separates into two coupled [[ordinary differential equations]] using [[prolate spheroidal coordinates]].

==Formation==
The dihydrogen ion is formed in nature by the interaction of [[cosmic ray]]s and the hydrogen molecule. An electron is knocked off leaving the cation behind.<ref name="eherbstastro">{{Cite journal|last=Herbst|first=E.|authorlink=|coauthors=|year=2000|month=|title=The Astrochemistry of H3+|journal=[[Philosophical Transactions of the Royal Society A]] |volume=358|issue=1774|pages=2523–2534|doi=10.1098/rsta.2000.0665|url=|accessdate=|quote=}}</ref>
:H2 + cosmic ray → H2+ + e- + cosmic ray.
Cosmic ray particles have enough energy to ionize many molecules before coming to a stop.

In nature the ion is destroyed by reacting with other hydrogen molecules:
:H2+ + H2 → [[Trihydrogen cation|H3+]] + H.

The ionization energy of the hydrogen molecule is 15.603 eV. The dissociation energy of the ion is 1.8 eV. High speed electrons also cause ionization of hydrogen molecules with a peak cross section around 50 eV. The peak cross section for ionization for high speed protons is 70000 eV with a cross section of 2.5x10−16 cm2. A cosmic ray proton at lower energy can also strip an electron off a neutral hydrogen molecule to form a neutral hydrogen atom and the dihydrogen cation, (p+ + H2 <big>→</big> H + H2+) with a peak cross section at around 8000 eV of 8x10−16 cm2.<ref>{{cite journal |first=Marco |last=Padovani |first2=Daniele |last2=Galli |first3=Alfred E. |last3=Glassgold |arxiv=0904.4149 |title=Cosmic-ray ionization of molecular clouds |year=2009 |work=Preprint }}</ref>

An artificial [[plasma discharge]] cell can also produce the ion.{{Citation needed|date=May 2013}}

==See also==
* [[Delta_potential#Double-well_Dirac_delta_function_model|Dirac Delta function model]] ( 1-D version of H2+)
* [[Di-positronium]]
* [[Euler's three-body problem]] (classical counterpart)
* [[Few-body systems]]
* [[Helium atom]]
* [[Helium hydride ion]]
* [[Trihydrogen cation]]
* [[Triatomic hydrogen]]
* [[Lambert W function]]
* [[Atomic and molecular astrophysics|Molecular astrophysics]]
* [[Holstein–Herring method]]
* [[Three-body problem]]

==References==
{{Reflist}}

{{DEFAULTSORT:Dihydrogen Cation}}
[[Category:Hydrogen physics]]
[[Category:Cations]]
[[Category:Quantum chemistry]]

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en>MrBill3: clean up using AWB