Dissociation of the Hydrogen Molecule–Ion from the Viewpoint of the Integral Hellmann–Feynman Formula

Abstract

The dissociation energy of the hydrogen molecule–ion is computed and analyzed by means of the integral Hellmann–Feynman theorem. The molecule is considered to dissociate either unsymmetrically by the removal of one of the nuclei to infinity or symmetrically by the simultaneous removal of both nuclei to infinity. The energy change is computed for each path and compared with the difference of the energy expectation values for the initial and final states. The Hellmann–Feynman results are in general different for the two paths, while the differences of expectation values are not. The final separated-atom state is represented by the exact wavefunction. The initial state is represented by a variety of wavefunctions including variational functions in both AO bases and elliptic MO bases, wavefunctions with correct cusp behavior forced, wavefunctions which approximate correct long-range behavior, and wavefunctions formed from a linear combination of initial-state functions and final-state functions. A dissociation process wherein the nuclear charge increases upon dissociation is also examined. The integral Hellmann–Feynman method produces a 0 0 indeterminacy for symmetric dissociation, which is resolved by a limiting procedure. The integral Hellmann–Feynman approach involves the construct of transition density, which contains information about reorganization of electronic distribution from the initial to the final state, and which permits the dissociation energy to be expressed as the sum of the nuclear–nuclear repulsion energy and the classical electrostatic attraction energy of the nuclei for the transition density. Contour plots of the transition densities and plots of the transition densities along the internuclear axis are given.