Interdimensional degeneracy and symmetry breaking in D-dimensional H + 2
Abstract
An interdimensional degeneracy linking the orbital angular momentum projection ‖m‖ and spatial dimension D gives D-dimensional eigenstates for H+2 by simple correspondence with suitably scaled D=3 excited states. The wave equation for fixed nuclei is separable in D-dimensional spheroidal coordinates, giving generalized two-center differential equations with parametric dependence on the internuclear distance R. By incorporating‖m‖ into D, the resulting eigenstates can be classified by the two dimension-independent ``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 ED(R), and a separation constant AD(R) related to the total orbital angular momentum and the Runge-Lenz 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 electron-dot 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 D-dimensional 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 D-dimensional 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:
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- Hydrogen Ions;
- Molecular Excitation;
- Molecular Ions;
- Atomic Structure;
- Electron Energy;
- Ground State;
- Perturbation Theory;
- Wave Equations;
- Atomic and Molecular Physics