Electron density

Electron density or electronic density is the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial variables and is typically denoted as either ${\displaystyle \rho ({\textbf {r}})}$ or ${\displaystyle n({\textbf {r}})}$. The density is determined, through definition, by the normalised ${\displaystyle N}$-electron wavefunction which itself depends upon ${\displaystyle 4N}$ variables (${\textstyle 3N}$ spatial and ${\displaystyle N}$ spin coordinates). Conversely, the density determines the wave function modulo up to a phase factor, providing the formal foundation of density functional theory.

According to quantum mechanics, due to the uncertainty principle on an atomic scale the exact location of an electron cannot be predicted, only the probability of its being at a given position; therefore electrons in atoms and molecules act as if they are "smeared out" in space. For one-electron systems, the electron density at any point is proportional to the square magnitude of the wavefunction.