In the hydrogen plus proton model of Eqn. (3) we have assumed that the electronic wave function/orbital rests on center a. What if we allow the orbital to move off center toward the bond midpoint, as described in Figure 1?
\(\psi=\phi_a\)
In this case, the \(-\zeta\) term in Eqn. (9) is replaced by a distance-dependent term, see Eqn. \eqref{eq:en_polmod}.
\(v_{aa}=-\zeta+\left\{\zeta e^{-2\zeta R_{ab}}+\frac{e^{-2\zeta R_{ab}}-1}{R_{ab}}\right\}\)
\begin{equation} {E_a} = \left\{ {\zeta {e^{ - 2\zeta {r_a}}} + \frac{{\left( {{e^{ - 2\zeta {r_a}}} - 1} \right)}}{{{r_a}}}} \right\} + \left\{ {\zeta {e^{ - 2\zeta {r_b}}} + \frac{{\left( {{e^{ - 2\zeta {r_b}}} - 1} \right)}}{{{r_b}}}} \right\} + \frac{1}{{{R_{ab}}}} \label{eq:en_polmod} \end{equation}
Since we are interested in bonding we'll again use hydrogen atom as our reference. The potential curve for this expression is given in Figure 2 for an internuclear distance of 3Å.
Click to rotate image.
This model demonstrates that build up of electron density between centers is not energetically attractive. Placing the electron symmetrically between the nuclei is nearly 300 kcal/mol uphill. In the region near R=0 we can see that the model of Eqn. \eqref{eq:en_polmod} and Figure 1 permits atomic electron density to be slightly polarized off-center due to charged particles or electric fields. This polarization effect should be included in the models of charged systems.6
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