|
The hydrogen molecular ion, dihydrogen cation, or H2+, is the simplest molecular ion. It is composed of two positively charged protons 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. 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,〔 〕 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 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.〕 and Wolfgang Pauli,〔 Extended doctoral dissertation; received 4 March 1922, published in issue No. 11 of 3 August 1922.〕 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 and Fritz London on the hydrogen molecule. Bonding in H2+ can be described as a covalent one-electron bond, which has a formal bond order of one half. The ion is commonly formed in molecular clouds in space, and is important in the chemistry of the interstellar medium. ==Quantum mechanical treatment, symmetries, and asymptotics== The simplest electronic Schrödinger wave equation for the hydrogen molecular ion is modeled with two fixed nuclear centers, labeled ''A'' and ''B'', and one electron. It can be written as : where 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 depending on the spatial coordinates of the electron. An additive term , which is constant for fixed inter-nuclear distance , has been omitted from the potential , since it merely shifts the eigenvalue. The distances between the electron and the nuclei are denoted , which are ''symmetric'' with respect to space inversion, and there are wave functions :, which are ''anti-symmetric'' under this symmetry operation: The symmetry-adapted wave functions satisfy the same Schrödinger equation. The ground state (the lowest discrete state) of is denoted or ( The actual difference between these two energies is called the exchange energy splitting and is given by: : which exponentially vanishes as the inter-nuclear distance ''R'' gets greater. The lead term was first obtained by the Holstein–Herring method. Similarly, asymptotic 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. 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 full lines are states. The green dashed lines are states. The blue dashed line is a state and the pink dotted line is a 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dihydrogen cation」の詳細全文を読む スポンサード リンク
|