Foundations — Electronegativity — Pauling, Mulliken, Allred-Rochow scales

Before you can read the parent note on Electronegativity, you must own every letter and symbol it fires at you. This page builds them one at a time, from nothing, in the order they depend on each other.

0. The picture behind everything: a shared electron pair

Look at the figure. The two circles are atomic nuclei (the dense positive centres of atoms). The two dots between them are the shared bonding electrons. If both atoms pull equally, the dots sit dead-centre. If the right atom pulls harder, the dots drift right — that drift is electronegativity in action.

1. Symbols that describe a single atom

Before we talk about pulling, we need to name the parts of one atom.

The picture: a small blob at the atom's centre carrying $+Z$ units of charge, with electrons circling outside it.

2. Why the inner electrons hide the nucleus: $Z_{\text{eff}}$

A valence electron does not feel the full pull of all $Z$ protons, because inner electrons sit between it and the nucleus and cancel part of that pull.

The picture (see figure): the full nucleus glows $+Z$, but a shell of inner electrons throws a "shadow," so the outer electron only sees the dimmer core charge $Z_{\text{eff}}$.

Why the topic needs this: the Allred–Rochow scale computes the force on a bonding electron, and that force depends on the charge the electron truly feels — which is $Z_{\text{eff}}$, not $Z$. We get $Z_{\text{eff}}$ from Slater's Rules (a bookkeeping recipe for $S$). See Effective Nuclear Charge for the full story.

3. The force that does the pulling: Coulomb's law

Read it as a sentence: the pull $F$ gets bigger when the felt charge $Z_{\text{eff}}$ is bigger, and gets smaller when the distance $r$ is bigger — and $r$ appears squared, so distance matters a lot.

The figure shows two atoms: a small one with high $Z_{\text{eff}}$ (short, strong amber arrow = big force) and a big one with low $Z_{\text{eff}}$ (long reach, weak arrow). This single picture is the trend: small + high $Z_{\text{eff}}$ → high electronegativity.

Why the topic needs this: this force ratio is the beating heart of Allred–Rochow. Everything after $0.359\times (\text{Coulomb ratio})$ is just rescaling into Pauling-like numbers.

4. Energies we can actually measure

Two atomic energies feed the Mulliken scale. Both are real, lab-measurable numbers for isolated atoms.

5. The energy stored in a bond: bond energy $E$

Why the topic needs this: Pauling noticed that a bond between two different atoms is often stronger than expected. That "extra glue" comes from charge separation (Ionic Character of Bonds), and its size reveals the electronegativity gap.

6. The star of the show: $\chi$ and $\Delta \chi$

Pauling's formula produces only this gap, never absolute values — which is exactly why the scale must be anchored at hydrogen (${\chi }_{\text{H}}=2.20$). The bigger $\Delta \chi$, the more the shared pair drifts, feeding Dipole Moment and Fajans Rules.

How the foundations feed the topic

Read the map below top-to-bottom. Every box is a symbol or quantity defined above, using its full name so nothing is shorthand: the atomic number and screening constant combine into effective nuclear charge; that plus the covalent radius gives the Coulomb force, which powers Allred–Rochow. Ionization energy and electron affinity power Mulliken. Bond energy and the geometric mean give the excess energy, which powers Pauling. All three scales land on the same electronegativity $\chi$.

Equipment checklist

Cover the right side and test yourself. If any answer is fuzzy, reread that section before opening the parent note.