Atoms in a bonded solid experience both attraction and short-range repulsion. Their competition can be summarized by an interatomic potential energy $U(r)$, where $r$ is separation.

A stable spacing $r_0$ occurs at a minimum of $U$:

$$\left.\frac{dU}{dr}\right|{r_0}=0,\qquad \left.\frac{d^2U}{dr^2}\right|{r_0}>0.$$

The force is

$$F(r)=-\frac{dU}{dr}.$$

For sufficiently small displacement from a smooth minimum, the potential can be approximated by a parabola:

$$U(r)\approx U(r_0)+\frac12 k(r-r_0)^2,$$

where

$$k=\left.\frac{d^2U}{dr^2}\right|_{r_0}.$$

The curvature $k$ measures how strongly the bond resists small changes in spacing. Steeper wells therefore tend to correspond to higher elastic stiffness.

The well depth measures the energy required to separate bonded atoms far apart, so it is related to cohesive strength and often to melting or sublimation temperature.

Real potentials are asymmetric: compression usually becomes costly more rapidly than equal-magnitude extension. As temperature raises the amplitude of atomic vibration, this asymmetry shifts the average spacing outward, providing an atomic explanation for thermal expansion.

This potential-energy picture does not specify whether the underlying cohesion is ionic, covalent, metallic or intermolecular. It is a common mechanical description that connects bonding to stiffness, cohesion and thermal expansion.