Dihydrogen cation

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 (see Lambert W function and references therein for more details on this function). Thus, the case of clamped nuclei can be completely done analytically using a Computer algebra system. 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 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 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.

Quantum mechanical treatment, Symmetries and Asymptotics

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 $H_{2}^{+}$ is modeled with two fixed nuclear centers, labeled A and B, and one electron. It can be written as:

$\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V\right)\psi =E~,$

where $V$ is the electron-nuclear Coulomb potential energy function:

$V=-{\frac {e^{2}}{4\pi \varepsilon _{0}}}\left({\frac {1}{r_{a}}}+{\frac {1}{r_{b}}}\right)$

and E is the (electronic) energy of a given quantum mechanical state (eigenstate), with the electronic state function $\psi =\psi (\mathbf {r} )$ depending on the spatial coordinates of the electron. An additive term $1/R$, 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 $r_{a}^{}$ and $r_{b}^{}$. In atomic units ($\hbar =m=e=4\pi \varepsilon _{0}=1$) the wave equation is:

$\left({}-{\frac {1}{2}}\nabla ^{2}+V\right)\psi =E\psi \qquad {\mbox{with}}\qquad V={}-{\frac {1}{r_{a}^{}}}-{\frac {1}{r_{b}^{}}}\;.$

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 $\to$ -r). There are wave functions :$\psi _{+}(\mathbf {r} )$, which are symmetric with respect to space inversion, and there are wave functions :$\psi _{-}(\mathbf {r} )$, which are anti-symmetric under this symmetry operation: $\psi _{\pm }(-{\mathbf {r} })={}\pm \psi _{\pm }({\mathbf {r} })\;.$ We note that the permutation (exchange) of the nuclei has a similar effect on the electronic wave function. We only mention that for a many-electron system proper behavior of $\psi$ with respect to the permutational symmetry of the electrons (Pauli exclusion principle) must be guaranteed, in addition to those symmetries just discussed above. Now the Schrödinger equations for these symmetry-adapted wave functions are:

${\begin{aligned}\left(-{\frac {1}{2}}\nabla ^{2}+V\right)\psi _{+}=E_{+}\psi _{+}\\\left(-{\frac {1}{2}}\nabla ^{2}+V\right)\psi _{-}=E_{-}\psi _{-}\end{aligned}}$

The ground state (the lowest discrete state) of $H_{2}^{+}$ is the ${\rm {X}}_{}^{2}\Sigma _{\rm {g}}^{+}$ state with the corresponding wave function $\psi _{+}$ denoted as $1s\sigma _{\rm {g}}^{}$. There is also the first excited ${\rm {A}}_{}^{2}\Sigma _{\rm {u}}^{+}$ state, with its $\psi _{-}$ labeled as ${\rm {2p}}\sigma _{\rm {u}}^{}$. (The suffixes 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.

Energies (E) of the lowest discrete states of the Hydrogen Molecular Ion $H_{2}^{+}$ as a function of internuclear distance (R) in atomic units. See text for explanation.

Asymptotically, the (total) eigenenergies $E_{\pm }$ for these two lowest lying states have the same asymptotic expansion in inverse powers of the internuclear distance R :

$E_{\pm }={}-{\frac {1}{2}}-{\frac {9}{4R^{4}}}+O(R^{-6})+\cdots$

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

$\Delta E=E_{-}-E_{+}={\frac {4}{e}}\,R\,e^{-R}\left[\,1+{\frac {1}{2R}}+O(R^{-2})\,\right]$

which exponentially vanishes as R gets larger. 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). 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).

The energies for the lowest discrete states are shown in the graph on the left. The red full lines are ${\rm {}}_{}^{2}\Sigma _{\rm {g}}^{+}$ states. The green dashed lines are ${\rm {}}_{}^{2}\Sigma _{\rm {u}}^{+}$ states. The blue dashed line is a ${\rm {}}_{}^{2}\pi _{\rm {u}}$ state and the pink dotted line is a ${\rm {}}_{}^{2}\pi _{\rm {g}}$ state.

Note that although, in principle, the general eigenvalue solutions in terms of the generalized Lambert W function - see eq. $(3)$ of "Generalizations" in that site - supersede these asymptotic expansions, in practice, they are most useful near the bond length. 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.

See also

• Trihydrogen cation