Why atoms hold together and how particles attract

An atom stays up because squeezing the electron costs energy

Intuition

Think of pushing a spring that is also being pulled in. The pull (the nucleus attracting the electron) wants everything small. The push-back (squeezing a wave makes it carry more energy) wants it big. They settle at one size. That size is the atom.

Lecture 2 asks why an atom does not collapse. Matter is held together by Coulomb forces, so why does the electron not fall into the nucleus? The answer is a balance between kinetic and potential energy.

The lecture uses the de Broglie relation $p = h/\lambda$ with $p = mu$. This gives the kinetic energy $\text{KE} = \tfrac{1}{2}mu^2 = p^2/2m = h^2/(2m\lambda^2)$. Squeeze the wave to a small $\lambda$ and KE goes up as $1/\lambda^2$. The potential energy of the electron (charge $-e$) at distance $r$ from a nucleus of charge $+Ze$ is $\text{PE} = -Ze^2/(4\pi\epsilon_0 r)$. Taking $r$ as roughly equal to $\lambda$, PE goes as $-1/\lambda$. The total energy is $E = \text{KE} + \text{PE} \propto 1/\lambda^2 - 1/\lambda$.

Small $\lambda$ makes the first term huge and positive, so the energy climbs. Large $\lambda$ makes both terms small, so the energy flattens out. In between there is a minimum. After accounting for the shape of the electron wave, the lecture finds $E_{min}$ at $\langle r\rangle = 1.5$ in units of $a_0$. Below that radius, KE grows so large that it stops the wave getting more compact.

Charges attract, and the pull dies off faster for neutral shapes

Intuition

Two bar magnets feel each other from a long way off if you think of single charges. A pair of tiny balanced pairs of charges (dipoles) only feel each other when they are close. The more balanced the charge shape, the faster the pull fades with distance.

Bond energy $V$ comes from the electronic properties of atoms and molecules. The lecture groups charge arrangements as a monopole (an ion), a dipole and a quadrupole, and gives one rule: $V \propto 1/r^{n+m-1}$, where $r$ is the separation and $n, m = 1$ for a monopole, 2 for a dipole and 4 for a quadrupole.

Exam detail

Falloff with distance from the slide: ion-ion (ionic bond) $1/r$. Ion-dipole $1/r^2$. Dipole-dipole $1/r^3$. Quadrupole-quadrupole $1/r^7$. Induced dipole-induced dipole $1/r^6$ (the $-C_6/r^6$ form).

The exact forms from the lecture are listed in the formulas for this page. The Coulomb form is $V(r) = -q_1q_2/(4\pi\epsilon_0 r)$. The dipole-dipole form is $V(r) = -\mu_1\mu_2/(2\pi\epsilon_0 r^3)$. For two induced dipoles, $V(r) = -C_6/r^6$, which the lecture writes in terms of the polarisabilities $\alpha$ and ionisation energies $I$.

Bonding balances attraction against repulsion, and electron counts decide structure

For a diatomic molecule, the lecture describes the energy curve in three steps. At large $r$ there is no interaction. As the atoms approach, the attraction between each atom’s electrons and the other’s nucleus lowers the energy, and this outweighs the electron-electron repulsion. At close distance the very strong repulsion between the positive nuclei dominates. The overall curve has a minimum, and that minimum is the bonding interaction.

The Pauli exclusion principle also matters. No two electrons in an atom may be in the same quantum state, so no two electrons share all four quantum numbers $n, l, m, s$, and no orbital can hold more than two electrons.

Electron counting then predicts what kind of bonding is possible. If the number of valence electrons equals the number of valence atomic orbitals, an inert gas configuration is possible through 2-centre, 2-electron covalent bonds and lone pairs. Carbon has 4 valence electrons and 4 valence orbitals ($1s^2, 2s^2, 2p^2$). If valence electrons are fewer than valence orbitals, covalent bonding alone cannot do it. Boron has 3 valence electrons and 4 valence orbitals ($1s^2, 2s^2, 2p^1$).

Isoelectronic elements have the same number of valence electrons, so they sit in the same group. Elements in the same group near the dividing line of the periodic table can still bond differently. The lecture compares glyoxylic acid, C$_2$H$_2$O$_3$, with Sn$_2$K$_2$O$_3$.

Pitfall

The lecture uses isoelectronic for valence electrons, meaning same group. Do not read it as the same total electron count.

To place electron groups so they repel each other least, the lecture reminds you of VSEPR. It gives the shape for the arrangement with the minimum potential energy.

Electron groups	0 lone pairs	1 lone pair	2 lone pairs	3 lone pairs
3	trigonal planar	bent
4	tetrahedral	trigonal pyramidal	bent
5	trigonal bipyramidal	see-saw	T-shaped	linear
6	octahedral	square pyramidal	square planar	T-shaped

Forces act in different, sometimes opposite, directions in molecules and in crystals. The lecture’s examples are NaCl (ionic lattice) and ice (hydrogen-bonded water molecules). For close-packed metals there are two choices for the third layer, both with 12 nearest neighbours. Hexagonal close-packed (hcp) repeats ABAB. Cubic close-packed (ccp, also called face-centred cubic, fcc) repeats ABCABC.

Recall

Why does the energy of the atom go up if the electron wave is squeezed very small?

The kinetic energy is $h^2/(2m\lambda^2)$, which grows as the wavelength shrinks.

Optional aside

The slide on E against r is a figure. The lecture gives E in units of 13.6 eV and the minimum near $\langle r\rangle \approx 1.5\,a_0$ with a value of about $-1$ on its axis. The page does not claim any other value for the minimum energy.