Interdimensional degeneracy and symmetry breaking in Ddimensional H^{ + }_{2}
Abstract
An interdimensional degeneracy linking the orbital angular momentum projection ‖m‖ and spatial dimension D gives Ddimensional eigenstates for H^{+}_{2} by simple correspondence with suitably scaled D=3 excited states. The wave equation for fixed nuclei is separable in Ddimensional spheroidal coordinates, giving generalized twocenter differential equations with parametric dependence on the internuclear distance R. By incorporating‖m‖ into D, the resulting eigenstates can be classified by the two dimensionindependent ``radial'' quantum numbers denoted in united atom notation by k and l‖m‖, corresponding, respectively, to the number of ellipsoidal and hyperboloidal nodal surfaces in the wave function. The two eigenparameters, the energy E_{D}(R), and a separation constant A_{D}(R) related to the total orbital angular momentum and the RungeLenz vector, have been determined numerically for the ground state and several low lying excited states for selected dimensions from D=2 to D=100.
The system simplifies greatly in the limit D→∞, where the electronic structure reduces to a classical electrostatic form with the electrons in a fixed geometrical configuration relative to the nuclei, akin to the traditional Lewis electrondot structure. For a given R, the energy E_{∞} reduces to the minimum of an effective potential surface and the separation constant A_{∞} reduces to a simple function of the energy. The surfaces are separable in spheroidal coordinates resulting in analytical expressions for the energy in terms of the coordinates. The surfaces exhibit a characteristic symmetry breaking as functions of R, changing from a single minimum surface in the united atom limit (R→0) to a double minimum surface in the separated atom limit (R→∞). Effects of this symmetry breaking are found at finite D as well. Analysis of excited state Ddimensional energies reveals that bonding in H^{+}_{2} is determined primarily by k, contrary to the standard scheme of bonding and antibonding molecular orbitals, which in the case of H^{+}_{2} correspond to even and odd l‖m‖, respectively. When the Ddimensional energies are examined as functions of 1/D, the resulting curves resemble typical perturbation diagrams with 1/D as the perturbation parameter.
 Publication:

Journal of Chemical Physics
 Pub Date:
 June 1990
 DOI:
 10.1063/1.458303
 Bibcode:
 1990JChPh..92.6668F
 Keywords:

 Hydrogen Ions;
 Molecular Excitation;
 Molecular Ions;
 Atomic Structure;
 Electron Energy;
 Ground State;
 Perturbation Theory;
 Wave Equations;
 Atomic and Molecular Physics