Asymptotic exchange energies for ${\mathbf{H}}_{\mathbf{2}}$

An analytical approximation of the asymptotic exchange energy of two interacting hydrogen atoms is obtained. This approximation depends only on functions of the internuclear distance $R$, which remain bounded as $R\to \infty$ and is derived using the Herring-Holstein surface-integral technique. It is found that, for large $R$, the exchange energy is $O\left(R^3\exp (-2R)\right)$ in contrast to earlier approximations of $O\left(R^{2.5}\exp (-2R)\right)$. Our result is similar to the classic Heitler-London expression without the unphysical term $O\left(R^3\ln \left(R\right)\exp (-2R)\right)$.

Figure 1

The exchange energy $\Delta E=\frac{1}{2}(E_{+}-E_{-})$ is plotted for the three formulas in the range of $90<R<100$. The upper curve is the Heitler-London formula, the middle curve is the Herring-Flicker formula, and the lower curve is obtained from the formula derived here [Eq. (44)] (energies and separations in a.u.).

Figure 2

The exchange energy $\Delta E=\frac{1}{2}(E_{+}-E_{-})$ is plotted for the three formulas in the range of $10<R<12$. The upper curve is the Heitler-London formula, the middle curve is the Herring-Flicker formula, and the lower curve is obtained from the formula derived here [Eq. (44)] (energies and separations in a.u.).