Resolution of a paradox in the calculation of exchange forces for H_2^+

What is it about?

Tang, Toennies and Yiu have shown that despite the inherent symmetry of H+2, wavefunctions obtained from a combination of the unsymmetrized polarization expansion and the 1/R expansion can be used in the Holstein—Herring formula to calculate for large internuclear distances R the leading (e−R) terms in the exchange energy between the lowest pair of states. However, the associated claim by Tang and Toennies that the polarization expansion of the wavefunction converges not to the gerade molecular wavefunction, but to an asymmetric function localized about a single nucleus, conflicts with other numerical and analytical results. We show by a limiting procedure that use of the infinite polarization expansion for the wavefunction in the Holstein—Herring formula provides a result that is not equal to the exact exchange energy, although it has the correct leading (e−R) behavior and is impressively close to the exact exchange energy for large R.

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Photo by Susann Schuster on Unsplash

Photo by Susann Schuster on Unsplash

Why is it important?

We resolve a mystery: why the approach used by Tang and Toennies which plugs the polarization wave function (from perturbation theory) into the Holstein-Herring surface integral gave good results for the exchange energy splittings of the lowest discrete states of the Hydrogen Molecular Ion yet their conclusions about the convergence of these wave functions was nonetheless incorrect.

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