Electron Density

In quantum mechanics, an electron is not a point-like ball on a fixed orbit — it is described by a wavefunction ψ(r) that spreads through space as a "probability cloud." The quantity ρ(r) = |ψ(r)|² is the probability density — the larger ρ(r) is at a point, the more likely a measurement will find the electron there. This does not mean the electron is "smeared out" like a fluid — a single measurement always finds the whole electron at one point, but the statistics of many measurements follow exactly the shape of ρ(r). This is a direct consequence of the uncertainty principle — the electron has no classical trajectory.

Bohr's 1913 model treated the electron as a particle on a circular orbit around the nucleus — this simple picture gave correct energy levels for hydrogen but was physically wrong. Schrödinger showed in 1926 that a complete description of the electron requires a wavefunction, not a trajectory. Max Born gave ψ its modern probabilistic interpretation that same year — |ψ|² is a probability density. In 1964, Hohenberg and Kohn proved that ρ(r) alone (without the wavefunction) is sufficient to describe all properties of a system — this became the foundation of DFT and earned Kohn the Nobel Prize in 1998.

The exact wavefunction of a hydrogen-like atom separates into radial and angular parts:

Definition of Density: The probability density is the squared modulus of the wavefunction:

Normalization: The integral over all space equals 1 — the electron is definitely somewhere:

Nodes: The number of radial nodes (spheres where ρ=0) equals n-l-1. The number of angular nodes (planes/cones) equals l. The larger n, the more nodes — this is the geometric analog of increasing energy (fitting more "wave" into the same space).

For the hydrogen atom, ρ(r) is calculated analytically directly from the Schrödinger equation — which is why this page computes it separately, with exact formulas. In multi-electron molecules no analytical solution exists — this is exactly where DFT comes in (see the "Molecular DFT" page): Kohn-Sham equations numerically find ρ(r), from which the total energy and other properties are derived. The second tab on this page ("Molecular Density") uses results obtained from that same DFT calculation — Mulliken population analysis distributes the total ρ(r) onto individual atoms in an approximate way.

The "Hydrogen Atom" page already computes R_nl(r) — there you see the radial cross-section in 1D. This page shows the same physics in full 3D, including the angular part — which is why the well-known shapes of s, p, d orbitals (sphere, dumbbell, four-leaf clover) only appear here. The "Molecular DFT" page applies an analogous principle to multi-atom systems.

Knowledge of electron density determines the nature of chemical bonds (where two atoms' clouds overlap), reactivity (high-density regions attract electrophiles), material conductivity (electron delocalization in metals), and spectroscopic properties. STM (scanning tunneling microscopy) directly measures surface electron density with atomic resolution.

Select n, l, m. Follow the rule l < n and |m| ≤ l — otherwise an error will appear.

l=0 (s) — spherical symmetry, always. l=1 (p) — dumbbell shape, 3 orientations (m=-1,0,1). l=2 (d) — more complex, 4-5 lobed shapes.

Each point is a possible location where the electron might be found — point density is higher in high-probability regions. Color indicates the relative magnitude of ρ.

In the second sub-tab you will see the electron distribution of a real molecule, coming from the "Molecular DFT" page.

Higher n → a larger, more "inflated" orbital with more nodes. Higher l (at the same n) → energy slightly increases in multi-electron atoms (s<p<d<f), though this is degenerate for hydrogen.