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The dihydrogen cation or hydrogen molecular ion is a
cation An ion () is an atom or molecule with a net electrical charge. The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conven ...
(positive
ion An ion () is an atom or molecule with a net electrical charge. The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conv ...
) with formula . It consists of two hydrogen nuclei ( protons) sharing a single electron. It is the simplest
molecular ion Mass spectral interpretation is the method employed to identify the chemical formula, characteristic fragment patterns and possible fragment ions from the mass spectra. Mass spectra is a plot of relative abundance against mass-to-charge ratio. It i ...
. The ion can be formed from the
ionization Ionization, or Ionisation is the process by which an atom or a molecule acquires a negative or positive charge by gaining or losing electrons, often in conjunction with other chemical changes. The resulting electrically charged atom or molecule ...
of a neutral hydrogen molecule . It is commonly formed in molecular clouds in space, by the action of cosmic rays. The dihydrogen cation is of great historical and theoretical interest because, having only one electron, the equations of quantum mechanics that describe its structure can be solved in a relatively straightforward way. The first such solution was derived by Ø. Burrau in 1927, just one year after the wave theory of quantum mechanics was published.


Physical properties

Bonding in can be described as a covalent one-electron bond, which has a formal
bond order In chemistry, bond order, as introduced by Linus Pauling, is defined as the difference between the number of bonds and anti-bonds. The bond order itself is the number of electron pairs (covalent bonds) between two atoms. For example, in diato ...
of one half. The ground state energy of the ion is -0.597 
Hartree The hartree (symbol: ''E''h or Ha), also known as the Hartree energy, is the unit of energy in the Hartree atomic units system, named after the British physicist Douglas Hartree. Its CODATA recommended value is = The hartree energy is approxima ...
.


Isotopologues

The dihydrogen cation has six
isotopologues In chemistry, isotopologues are molecules that differ only in their isotopic composition. They have the same chemical formula and bonding arrangement of atoms, but at least one atom has a different number of neutrons than the parent. An exampl ...
, that result from replacement of one or more protons by nuclei of the other hydrogen
isotopes Isotopes are two or more types of atoms that have the same atomic number (number of protons in their nuclei) and position in the periodic table (and hence belong to the same chemical element), and that differ in nucleon numbers ( mass numbers ...
; namely, deuterium nuclei (
deuteron Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two stable isotopes of hydrogen (the other being protium, or hydrogen-1). The nucleus of a deuterium atom, called a deuteron, contains one proton and one n ...
s, ) or tritium nuclei (tritons, ). * = (the common one). * = (deuterium hydrogen cation). * = (dideuterium cation). * = (tritium hydrogen cation). * = (tritium deuterium cation). * = (ditritium cation).


Quantum mechanical analysis

The Schrödinger equation (in the clamped-nuclei approximation) for this cation can be solved in a relatively straightforward way due to the lack of electron–electron repulsion (
electron correlation Electronic correlation is the interaction between electrons in the electronic structure of a quantum system. The correlation energy is a measure of how much the movement of one electron is influenced by the presence of all other electrons. Atom ...
). The analytical solutions for the electronic energy eigenvalues are a ''generalization'' of the
Lambert W function In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse relation of the function , where is any complex number and is the exponential functio ...
which can be obtained using a
computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The d ...
within an
experimental mathematics Experimental mathematics is an approach to mathematics in which computation is used to investigate mathematical objects and identify properties and patterns. It has been defined as "that branch of mathematics that concerns itself ultimately with t ...
approach. Consequently, it is included as an example in most
quantum chemistry Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
textbooks. The first successful quantum mechanical treatment of was published by the Danish physicist Øyvind Burrau in 1927,
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 Karel Frederik Niessen (1895 in Velsen – 1967) was a Dutch theoretical physicist who made contributions to quantum mechanics and is known for the Pauli–Niessen model. Education Niessen began his studies in physics at the University of Utrech ...
and Wolfgang Pauli, and in 1925 by Harold Urey. In 1928, Linus Pauling published a review putting together the work of Burrau with the work of
Walter Heitler Walter Heinrich Heitler (; 2 January 1904 – 15 November 1981) was a German physicist who made contributions to quantum electrodynamics and quantum field theory. He brought chemistry under quantum mechanics through his theory of valence bo ...
and
Fritz London Fritz Wolfgang London (March 7, 1900 – March 30, 1954) was a German physicist and professor at Duke University. His fundamental contributions to the theories of chemical bonding and of intermolecular forces (London dispersion forces) are today c ...
on the hydrogen molecule.


Clamped-nuclei (Born–Oppenheimer) approximation

The electronic Schrödinger wave equation for the hydrogen molecular ion with two fixed nuclear centers, labeled ''A'' and ''B'', and one electron can be written as : \left( -\frac \nabla^2 + V \right) \psi = E \psi ~, where ''V'' is the electron-nuclear Coulomb potential energy function: : V = - \frac \left( \frac + \frac \right) and ''E'' is the (electronic) energy of a given quantum mechanical state (eigenstate), with the electronic state function ''ψ'' = ''ψ''(r) depending on the spatial coordinates of the electron. An additive term , which is constant for fixed internuclear distance ''R'', has been omitted from the potential ''V'', since it merely shifts the eigenvalue. The distances between the electron and the nuclei are denoted ''ra'' and ''rb''. In atomic units (''ħ'' = ''m'' = ''e'' = 4''ε''0 = 1) the wave equation is :\left( -\tfrac12 \nabla^2 + V \right) \psi = E \psi \qquad \mbox \qquad V = -\frac - \frac \; . We 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 the point group inversion operation ''i'' (r ↦ −r). There are wave functions ''ψ''g(r), which are ''symmetric'' with respect to ''i'', and there are wave functions ''ψ''u(r), which are ''antisymmetric'' under this symmetry operation: : \psi_(-) = \pm \psi_() \; . The suffixes ''g'' and ''u'' are from the German ''gerade'' and ''ungerade'') occurring here denote the symmetry behavior under the point group inversion operation ''i''. 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. The ground state (the lowest state) of is denoted X2Σ or 1sσg and it is gerade. There is also the first excited state A2Σ (2pσ''u''), which is ungerade. Asymptotically, the (total) eigenenergies ''E''''g''/''u'' for these two lowest lying states have the same asymptotic expansion in inverse powers of the internuclear distance ''R'': : E_ = - \frac12 - \frac + O\left(R^\right) + \cdots The actual difference between these two energies is called the exchange energy splitting and is given by: : \Delta E = E_ - E_ = \frac \, R \, e^ \left \, 1 + \frac + O\left(R^\right) \, \right which exponentially vanishes as the internuclear distance ''R'' gets greater. The lead term was first obtained by the Holstein–Herring method. Similarly, asymptotic expansions in powers of 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 internuclear 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 In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressio ...
from the generalized
Lambert W function In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse relation of the function , where is any complex number and is the exponential functio ...
(see eq. (3) 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. The red solid lines are 2Σ states. The green dashed lines are 2Σ states. The blue dashed line is a 2Π''u'' state and the pink dotted line is a 2Π''g'' state. Note that although the generalized
Lambert W function In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse relation of the function , where is any complex number and is the exponential functio ...
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 In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
of the wave equation here separates into two coupled
ordinary differential equations In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contras ...
using prolate spheroidal coordinates. The complete Hamiltonian of (as for all centrosymmetric molecules) does not commute with the point group inversion operation ''i'' because of the effect of the nuclear hyperfine Hamiltonian. The nuclear hyperfine Hamiltonian can mix the rotational levels of ''g'' and ''u'' electronic states (called ''ortho''-''para'' mixing) and give rise to ''ortho''-''para'' transitions


Occurrence in space


Formation

The dihydrogen ion is formed in nature by the interaction of cosmic rays and the hydrogen molecule. An electron is knocked off leaving the cation behind. :H2 + cosmic ray → + e + cosmic ray. Cosmic ray particles have enough energy to ionize many molecules before coming to a stop. The ionization energy of the hydrogen molecule is 15.603 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 with a cross section of . 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, () with a peak cross section at around of . An artificial plasma discharge cell can also produce the ion.


Destruction

In nature the ion is destroyed by reacting with other hydrogen molecules: : + H2 + H.


See also

* Symmetry of diatomic molecules * Dirac Delta function model (one-dimensional version of ) *
Di-positronium Di-positronium, or dipositronium, is an exotic molecule consisting of two atoms of positronium. It was predicted to exist in 1946 by John Archibald Wheeler, and subsequently studied theoretically, but was not observed until 2007 in an experiment ...
* Euler's three-body problem (classical counterpart) * Few-body systems *
Helium atom A helium atom is an atom of the chemical element helium. Helium is composed of two electrons bound by the electromagnetic force to a nucleus containing two protons along with either one or two neutrons, depending on the isotope, held together by ...
*
Helium hydride ion The helium hydride ion or hydridohelium(1+) ion or helonium is a cation ( positively charged ion) with chemical formula HeH+. It consists of a helium atom bonded to a hydrogen atom, with one electron removed. It can also be viewed as protonated ...
*
Trihydrogen cation The trihydrogen cation or protonated molecular hydrogen is a cation (positive ion) with formula , consisting of three hydrogen nuclei (protons) sharing two electrons. The trihydrogen cation is one of the most abundant ions in the universe. It ...
*
Triatomic hydrogen Triatomic hydrogen or H3 is an unstable triatomic molecule containing only hydrogen. Since this molecule contains only three atoms of hydrogen it is the simplest triatomic molecule and it is relatively simple to numerically solve the quantum mechan ...
*
Lambert W function In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse relation of the function , where is any complex number and is the exponential functio ...
* Molecular astrophysics * Holstein–Herring method * Three-body problem *
List of quantum-mechanical systems with analytical solutions Much insight in quantum mechanics can be gained from understanding the closed-form solutions to the time-dependent non-relativistic Schrödinger equation. It takes the form : \hat \psi\left(\mathbf, t\right) = \left - \frac \nabla^2 + V\left(\mat ...


References

{{DEFAULTSORT:Dihydrogen Cation Hydrogen physics Cations Quantum chemistry Quantum models