the bonding-energy curve

The bonding-energy curve is a graph of a pair of atoms' potential energy against the distance between them, and it looks like a valley: high on both sides, dipping down to a low point in the middle. It answers, in one picture, everything about how strongly two atoms are bound. Imagine rolling a marble across the landscape — it will settle at the bottom of the valley, and that bottom is exactly where the atoms want to sit.

The shape comes from adding two competing energies. There is an attractive energy that becomes more negative (more favourable) as atoms approach, and a repulsive energy that shoots steeply positive when they get too close. A common textbook form is E = -A/r^n + B/r^m, where the first term is attraction and the second, with a larger exponent, is the sharp short-range repulsion. Their sum is the valley. The horizontal position of the lowest point is the equilibrium spacing (the bond length); the vertical depth of the valley below zero is the bond energy — how much energy you must supply to drag the atoms apart to infinity.

Two features of the valley predict a material's character. A deep valley means a strong bond: high melting point, because it takes a lot of thermal energy to shake atoms free. A narrow, steep-walled valley means a stiff bond: a high elastic modulus, because the atoms resist being pulled even slightly off the bottom. And a lopsided (asymmetric) valley is the hidden reason solids expand when heated, since a warmer atom rattles into the gentler outer wall and its average position shifts outward.