Electron density is a spatial quantity describing how electrons are distributed in an atom, molecule, or material. In microscopic electronic-structure theory, it represents the expected number of electrons per unit volume at a given position, rather than the trajectories of individual particles. It connects quantum mechanics with chemical bonding and experimental measurements of electronic structure. In plasma physics, the same term commonly denotes the local concentration of free electrons. Its interpretation therefore depends on which electrons and which spatial scale are being considered. (goldbook.iupac.org)
Definition and normalization
Electron number density is usually written (n(\mathbf r)), although (\rho(\mathbf r)) is common in chemistry and crystallography. For a system containing (N) electrons,
[ n(\mathbf r)\geq 0,\qquad \int n(\mathbf r),d^3r=N. ]
Its integral over a region gives the expected number of electrons in that region. Consequently, the density need not integrate to an integer over an arbitrary subvolume. For a one-electron system, the density is a normalized probability density; for an (N)-electron system, (n(\mathbf r)/N) gives the position probability density for a randomly selected electron. (goldbook.iupac.org)
Number density must be distinguished from charge density. The electronic charge density is
[ \rho_{\mathrm{charge}}(\mathbf r)=-e,n(\mathbf r), ]
where (e) is the positive elementary charge. Total charge density also includes the positive contribution of atomic nuclei. Thus a material can contain a large electron density while remaining electrically neutral. Number density has dimensions of inverse volume: its SI unit is (\mathrm{m}^{-3}), while molecular and crystallographic densities are often expressed as electrons per cubic ångström. (goldbook.iupac.org)
Quantum-mechanical description
For a normalized many-electron wave function (\Psi), electron density is obtained by integrating its squared magnitude over all electron coordinates except one, summing over spin coordinates, and multiplying by (N):
[ n(\mathbf r)=N\sum_{s_1,\ldots,s_N} \int |\Psi(\mathbf r,s_1,\mathbf r_2,s_2,\ldots,\mathbf r_N,s_N)|^2 ,d^3r_2\cdots d^3r_N. ]
This reduces a description involving many particle coordinates to a function of three spatial coordinates. It does not identify particular electrons or imply that they follow classical paths. (dft.uci.edu)
For a single-determinant orbital description, such as the Hartree–Fock method, density is the sum of squared occupied spin-orbitals, with spin summed out. In a closed-shell description using doubly occupied spatial orbitals,
[ n(\mathbf r)=2\sum_i|\phi_i(\mathbf r)|^2. ]
An orbital and an electron density are therefore different objects: an orbital is an amplitude, whereas density is nonnegative and incorporates contributions from occupied states. For correlated wave functions, a simple sum over a fixed set of doubly occupied orbitals generally does not suffice. (dft.uci.edu)
Density-functional theory
Electron density is the central variable of density-functional theory (DFT). The Hohenberg–Kohn results, published in 1964, established that, under their stated assumptions, the ground-state density determines the external scalar potential up to an additive constant. For fixed electron interactions, this determines the system’s Hamiltonian and its ground-state properties. Ground-state energy can consequently be expressed as a functional of the density. (nobelprize.org)
The Kohn–Sham formulation, introduced in 1965, uses an auxiliary noninteracting electron system whose orbitals reproduce the interacting system’s density. This makes orbital-based calculations possible without treating the full many-electron wave function directly. Practical calculations approximate the exchange-correlation contribution, so an exact theoretical foundation does not guarantee exact numerical densities. DFT is used to investigate molecules, crystals, surfaces, and their interactions. (nobelprize.org)
Chemical interpretation
Electron density provides a spatial description of chemical bonding. Comparisons with suitable reference densities reveal redistribution accompanying bond formation, polarization, and intermolecular interactions. Such comparisons are more informative than interpreting a high density value alone, because much of an atom’s density belongs to tightly bound core electrons. (journals.iucr.org)
Density analysis can also assign electron populations to atomic regions. These populations depend on how space or the electronic description is partitioned; an “atomic charge” is therefore not simply a directly observed local electron density. Density topology instead examines features such as stationary points and connecting paths. These methods characterize spatial organization, but their descriptors require interpretation and should not automatically be equated with a unique chemical bonding model. (journals.iucr.org)
Experimental reconstruction and density maps
In X-ray crystallography, X-rays are scattered by electrons. The resulting diffraction intensities supply information about structure-factor amplitudes. Electron density within a crystal’s unit cell is reconstructed by a Fourier synthesis using amplitudes and phases. Because ordinary intensity measurements do not directly provide the phases, reconstruction involves the phase problem. (journals.iucr.org)
Experimental maps are not exact, instantaneous pictures of electrons. Finite resolution limits spatial detail, while atomic motion and disorder smear the reconstructed distribution. In protein crystallography, maps support atomic-model building and assessment. Difference maps highlight features insufficiently explained or overrepresented by a model; weak density may reflect flexibility, multiple conformations, or incomplete occupancy rather than simple absence of an atom. (journals.iucr.org)
Maps from cryo-electron microscopy are related but physically distinct: electron scattering probes electrostatic potential, rather than electron number density alone. Calling every structural-imaging map an electron-density map can therefore obscure what was measured. (journals.iucr.org)
Free-electron density
In a plasma, electron density usually refers to free-electron number density rather than the complete bound-electron distribution of matter. For Earth’s ionosphere, a related quantity is total electron content: the free-electron density integrated along a propagation path, expressed as a column density. It is not a local volume density. These measurements help characterize ionospheric effects on radio signals, including propagation delays that vary with electron content and signal frequency. (swpc-drupal.woc.noaa.gov)