Get Hydrogen Molecule-ion essential facts below. View Videos or join the Hydrogen Molecule-ion discussion. Add Hydrogen Molecule-ion to your PopFlock.com topic list for future reference or share this resource on social media.
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.
Hydrogen molecular ion with clamped nuclei A and B, internuclear distance R and plane of symmetry M.
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
where V is the electron-nuclear Coulomb potential energy function:
and E is the (electronic) energy of a given quantum mechanical state (eigenstate), with the electronic state function ? = ?(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 (? = m = e = 4??0 = 1) the wave equation is
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 ?g(r), which are symmetric with respect to i, and there are wave functions ?u(r), which are antisymmetric under this symmetry operation:
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 X2?+ g or 1s?g and it is gerade. There is also the first excited state A2?+ u (2p?u), which is ungerade.
Energies (E) of the lowest states of the hydrogen molecular ion as a function of internuclear distance (R) in atomic units. See text for details.
Asymptotically, the (total) eigenenergies Eg/u for these two lowest lying states have the same asymptotic expansion in inverse powers of the internuclear distance R:
The actual difference between these two energies is called the exchange energy splitting and is given by:
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 from the generalized Lambert W function (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 2?+ g states. The green dashed lines are 2?+ u states. The blue dashed line is a 2?u state and the pink dotted line is a 2?g 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.
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
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 .
^Clark R. Landis; Frank Weinhold (2005). Valency and bonding: a natural bond orbital donor-acceptor perspective. Cambridge, UK: Cambridge University Press. pp. 91-92. ISBN978-0-521-83128-4.CS1 maint: multiple names: authors list (link)
^Bressanini, Dario; Mella, Massimo; Morosi, Gabriele (1997). "Nonadiabatic wavefunctions as linear expansions of correlated exponentials. A quantum Monte Carlo application to H2+ and Ps2". Chemical Physics Letters. 272 (5-6): 370-375. Bibcode:1997CPL...272..370B. doi:10.1016/S0009-2614(97)00571-X.
^ abcdFábri, Csaba; Czakó, Gábor; Tasi, Gyula; Császár, Attila G. (2009). "Adiabatic Jacobi corrections on the vibrational energy levels of isotopologues". Journal of Chemical Physics. 130 (13): 134314. doi:10.1063/1.3097327. PMID19355739.
^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.
^Pauli W (1922). "Über das Modell des Wasserstoffmolekülions". Annalen der Physik. 373 (11): 177-240. doi:10.1002/andp.19223731101. Extended doctoral dissertation; received 4 March 1922, published in issue No. 11 of 3 August 1922.
^Pauling, L. (1928). "The Application of the Quantum Mechanics to the Structure of the Hydrogen Molecule and Hydrogen Molecule-Ion and to Related Problems". Chemical Reviews. 5 (2): 173-213. doi:10.1021/cr60018a003.