= Solution
A <covalent bond> is stabilized by shared electronic states extending across neighboring atomic nuclei. In a simple two-centre description, combining comparable atomic orbitals produces a bonding state with enhanced <electron density> between the nuclei and an antibonding state with a node there. For equivalent normalized orbitals $\phi_A,\phi_B$ with real overlap $S$,
$$
\psi_b=\frac{\phi_A+\phi_B}{\sqrt{2(1+S)}},\qquad \psi_{ab}=\frac{\phi_A-\phi_B}{\sqrt{2(1-S)}}.
$$
Their <energy> ordering follows from the molecular <Hamiltonian>; constructive <interference> and reduced gradients help explain why the bonding combination can be lower. Opposite-spin <Electrons> can occupy the bonding orbital together. The familiar molecular-hydrogen bond is a basic example, while extended covalent networks form solids such as <diamond>. Orbital shape and hybridization make covalent bonding strongly directional and constrain bond angles. Occupying antibonding states reduces the net bond strength.
<Ionic bonding> instead emphasizes <electric charge> transfer and the electrostatic attraction of the resulting oppositely charged <ions>. Removing an <Electron> from one atom costs <ionization energy> and adding it to another gains <electron affinity>; these isolated-atom terms alone need not favor transfer. In a <crystal>, the electrostatic <lattice energy> can stabilize the ionic arrangement. For a schematic <lattice> of singly charged <ions> at nearest-neighbor distance $r$,
$$
U(r)=-\frac{A_Me^2}{4\pi\epsilon_0r}+\frac{B}{r^s},\qquad B>0,\quad s>1,
$$
where the geometrical <Madelung constant> $A_M$ sums attractions and repulsions throughout the <lattice>. Short-range overlap repulsion prevents collapse and gives an equilibrium separation. <Sodium chloride> is an example of an extended <ionic crystal>, rather than isolated NaCl molecules in the solid.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2014-bonding-models.png]
{title=Original schematic covalent electron sharing and the extended charge arrangement of an ionic crystal}
Ideal <ionic bonding> is comparatively nondirectional, so packing, <ion> sizes and <electric charge> balance strongly influence coordination. <ions> cannot freely migrate in the ordinary solid; melting or dissolution can make ionic <electric charge> transport possible. Covalent materials do not obey one universal <electrical conductivity> rule: their electronic bands and filling determine whether mobile electronic carriers are available.
The two descriptions are limiting models, not a sharp classification of every bond. Unequal electronegativities can produce polar covalent bonds with partial <electric charge> transfer, and a material may contain both covalent groups and ionic interactions between those groups. Chemical bonding reflects a balance of electronic <kinetic energy>, electron-nucleus attraction, electron-electron repulsion and nuclear repulsion; it cannot be reduced to a <force> between two assigned point charges in every material.
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