Dihydrogen cation

The hydrogen molecular ion, dihydrogen cation, or H2 , is the simplest molecular ion. It consists of two positively charged protons and a negatively charged electron, and may be formed by ionization of neutral hydrogen molecule.

It is of great historical and theoretical interest, since it contains only one electron and therefore no electron-electron interactions may occur. Therefore, can the electronic Schrödinger equation for this system article held internuclear distance solve (so-called Born- Oppenheimer approximation ) is analytic in a closed manner. The analytical solutions for the energy eigenvalues ​​are a generalization of Lambertian W function dar.

Because of its importance as the simplest molecular system, the hydrogen molecular ion is treated in most textbooks of quantum chemistry as an example. The first successful quantum mechanical treatment of the H2 was released by the Danish physicist Øyvind Burrau in 1927, just a year after the publication of the fundamental work on wave mechanics by Erwin Schrödinger. Previous attempts had been published in 1922 by Karel Niessen and Wolfgang Pauli, and in 1925 by Harold Urey. With a review article from 1928 Linus Pauling made ​​known to both the work of Burrau as well as by Walter Heitler and Fritz London on the hydrogen molecule to a wider audience.

The chemical bond in the H2 can be described as one-electron - covalent bond, which has a formal bond order of 1/2.

The hydrogen molecular ion is usually formed in molecular clouds in space and is of great importance for the chemistry in the interstellar medium.

Quantum mechanical treatment, Asymptotics and symmetries

The simplest electronic Schrodinger equation for the hydrogen molecular ion taken into account in addition to the electron one, the two cores, indicated by A and B at fixed locations in space. It can be written as

The electron-nucleus Coulomb potential function

, and the ( e ) of a given quantum energy state ( natural state ) with the electronic state function which depends on the location coordinates of the electron. An additive term which is a constant for given internuclear distance, was omitted in the potential function, since it only shifts the eigenvalue. The distances between the electron and the nuclei were labeled and. In atomic units, the Schrödinger equation becomes

The midpoint between the positions of the cores can be chosen as the origin of coordinates. Of general symmetry principles follows that the state functions can be characterized according to their symmetry space inversion behavior with respect to (R- r). There are state functions

That are " symmetric " with respect to space inversion, and state functions

Which are anti- symmetric under this symmetry operation:

We note that the permutation ( exchange ) of the cores having the same effect on the electronic state functions. For a many-electron system must, in addition to those just named symmetries, and the symmetry proper conduct of the state function with respect to permutations of the electrons ( Pauli exclusion principle cal ) be guaranteed. The Schrödinger equations for the symmetry- adapted wave functions are now

The ground state ( the lowest energy of discrete state) of the state, the corresponding state function is usually denoted by. The state function of the first excited state, is indicated with. The suffixes occurring here g and u (of even and odd ) characterize precisely the symmetry behavior under space inversion. Their use is standard for the identification of electronic states of diatomic molecules, while the labels e and u ( from English "even" and "odd" ) are used for states of atoms.

For large internuclear distances, the ( total ) energy eigenvalues ​​for these two lowest states have the same asymptotic expansion in inverse powers of the internuclear distance R:

The actual difference between the two energies is called the exchange splitting energy and is given by:

This expression vanishes exponentially with increase in the core -core distance. The leading term is obtained correctly only with the Holstein - Herring method. In a very similar manner asymptotic expansions in powers of 1 / R up to high order of Čížek et al were. for the lowest ten discrete states of the hydrogen molecular ion obtained ( in the case of detained cores). For any diatomic or polyatomic molecular systems can exchange energy splitting at large internuclear distance is very difficult to calculate. For the treatment of long-range interactions, including studies related to magnetism and charge exchange effects, but their knowledge is necessary. These effects are particularly important for the physical understanding of stars and atmospheres (terrestrial and extra- terrestrial).

The energies of the lowest discrete states are shown in the figure above. The values ​​can be obtained with any desired accuracy using a computer algebra program from the " generalized " Lambertian W function ( see Eq. Therein and the reference to the work of Scott, Aubert - Frécon, and Grotendorst ) but they were first obtained numerically, ODKIL called in double precision, using the most accurate available computer program. The red solid lines are states. The green dashed lines are states. Blue dotted line is a state, and the dotted line is a pink state. Although the " generalized " using the Lambertian W function obtained eigenvalue solutions replace these asymptotic expansions, they are very useful in practice, especially in the vicinity of the equilibrium distance. Such solutions are possible, because the partial differential equation representing the Schrodinger equation, coupled using prolate spheroidal coordinates in two ordinary differential equations is separable.

Education

The hydrogen molecular ion is formed in nature by the action of cosmic rays on hydrogen molecules. An electron is knocked out here and leaves the cation.

The cosmic ray particles have sufficient energy to ionize many molecules before they are stopped.

The maximum cross section is for very fast protons (70 keV) 2.5 × 10-16 cm2.

In nature, the ion further reacts with another hydrogen molecules:

Properties

The ionization energy of the hydrogen molecule is 15.603 eV, the dissociation energy of 1.8 eV.