If we consider the interaction of a hydrogen atom (a proton plus and electron) with a proton we obtain the purely repulsive green curve in Figure 1. A physical interpretation of this repulsion arises from classical electrostatics.⁵ We know from classical electrostatics (Gauss' law) if a charged particle is outside a spherical charged surface, the interaction is as though the sphere were a point charge located at the center of the sphere. If the charged particle is inside the spherical surface, it sees a constant distance-independent potential. As given in Eqn. \eqref{eq:h2+wfxn} the hydrogenic electron density is an exponentially decaying spherical distribution.

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\begin{equation} \phi_a=\sqrt{\frac{\zeta^3}{\pi}}e^{-\zeta r_a} \label{eq:h2+wfxn} \end{equation}

Thus, as the proton penetrates shell upon shell of electron density it progressively becomes less and less distance-dependent approaching \(-\zeta\) at the nucleus (the red curve in Figure 1). The proton-proton repulsive interaction remains a full 1/R term (the blue curve). The repulsion swamps the attraction at all distances. The rapid increase in energy at roughly 3 Å corresponds to the initial proton penetration of the electron density.

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