= Pauling bond-energy excess
{c}
{title2=$\Delta_D$}
A <Pauling bond-energy excess> subtracts a chosen like-atom baseline from the unlike-atom <bond-dissociation energy>. For an unequal-input arithmetic example, stipulate $D_{AA}=200$, $D_{BB}=450$ and $D_{AB}=421.485\,\mathrm{kJ\,mol^{-1}}$, all with one dissociation convention. The <arithmetic mean> baseline and excess are
$$
D_{\rm base}=\frac{200+450}{2}=325\,\mathrm{kJ\,mol^{-1}},\qquad
\Delta_D=421.485-325=96.485\,\mathrm{kJ\,mol^{-1}}.
$$
The arithmetic-baseline <Pauling scale> model therefore gives $|\chi_{\rm P}(A)-\chi_{\rm P}(B)|=\sqrt{96.485/96.485}=1$. The <bond-dissociation energy> $421.485$ is not itself the excess: subtracting the baseline is essential. All three numbers are stipulated, not experimental values for named <chemical elements>.
\Image[/pauling-bond-energy-excess-construction.png]
{title=Constructing a <Pauling bond-energy excess> from stipulated <bond-dissociation energies> and converting it to a <dimensionless> <electronegativity> difference}
{id=image-pauling-bond-energy-excess-construction}
Changing the baseline changes the model. For the same stipulated inputs, a <geometric mean> baseline would be $\sqrt{200\times450}=300\,\mathrm{kJ\,mol^{-1}}$, yielding an excess of $121.485\,\mathrm{kJ\,mol^{-1}}$ and a difference of approximately $1.122$. It is a different construction, even though both start from the same <bond-dissociation energies>. State which baseline is used before comparing <electronegativity> differences.
The arithmetic-baseline <Pauling scale> also has a useful inverse: a difference of $0.5$ corresponds to an excess of $24.12125\,\mathrm{kJ\,mol^{-1}}$, and a difference of $2$ corresponds to $385.94\,\mathrm{kJ\,mol^{-1}}$. Doubling the <electronegativity> difference quadruples the modeled excess. Interchanging the endpoint labels changes neither quantity; the <square root> alone cannot choose the direction of <bond polarization>.
In <The Nature of the Chemical Bond. IV. The Energy of Single Bonds and the Relative Electronegativity of Atoms> (printed pp. 3576–3578, Table II and equation (1)), <Linus Pauling> organized excess <energies> for <chemical bonds> into an approximately additive map of their <square roots>. He also identified a limitation: his H–F map separation of $2$ predicted an excess of $4.00\,\mathrm{eV}$, exceeding his table's value by $1.23\,\mathrm{eV}$. Those are historical values and a historical model discrepancy, not modern recommended inputs.<The Nature of the Chemical Bond. IV. The Energy of Single Bonds and the Relative Electronegativity of Atoms by Linus Pauling>[printed pp. 3576–3578, Table II, equation (1), and paragraph beginning “These coördinates”]{ref}
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