ISSN:
1089-7690
Source:
AIP Digital Archive
Topics:
Physics
,
Chemistry and Pharmacology
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1063/1.458303
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