Exact and Variational Energies for the Hydrogen Molecular Ion and the Helium Hydride Ion

For the lowest two states of the homonuclear or heteronuclear one-electron two-center system with nuclear charges and , exact (red curves) and approximate (blue curves) energies are shown. Depending on the choice made, total energies including the nuclear repulsion ("potential energy curves"), electronic energies, or differences between the exact and approximate energies are plotted as a function of the internuclear separation. At each nuclear center, one hydrogenic 1 orbital is employed for the LCAO (Linear Combination of Atomic Orbitals) approximation. By adjusting the parameters in the exponents and of the 1 orbitals, the quality of the variational approximation can be controlled, and by changing the charge parameter one can study the stability properties of the system.


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Assuming the Born–Oppenheimer approximation, the quantum energies of two-center systems depend parametrically on the internuclear distance. The behavior of the corresponding potential energy curves determines the binding property of the molecular system. By continuous variation of the nuclear charges, for growing the transition from stable to metastable bonding and eventually to instability can be followed and critical charges can be determined that separate the stable, metastable, and unstable regimes (see, e.g., H. Hogreve, "On the Stability of the One-Electron Bond," The Journal of Chemical Physics, 98(7), 1993 pp. 5579–5594, where the exact energies were also computed, and J. Ackermann and H. Hogreve, "The Magnetic Two-Centre Problem: Stability and Critical Bonding," Physics Letters A, 372(32), 2008 pp. 5314–5317 for a recent extension to systems in magnetic fields). This Demonstration corroborates, in particular, that the multiply charged one-electron systems and fail to be stable.
Among the various aspects illustrated by the LCAO upper bound energies, it is interesting to observe that—as long as the variational parameters are not re-optimized for each internuclear separation—one has to compromise between a globally optimal and a locally most accurate description of the exact energies.
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