### 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 *E* _{ D }(*R*), and a separation constant *A* _{ D }(*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.

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