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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 or . The density is determined, through definition, by the normalised -electron wavefunction which itself depends upon variables ( spatial and 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. The electronic density corresponding to a normalised -electron wavefunction (with and denoting spatial and spin variables respectively) is defined as where the operator corresponding to the density observable is Computing as defined above we can simplify the expression as follows. In words: holding a single electron still in position we sum over all possible arrangements of the other electrons. The factor N arises since all electrons are indistinguishable, and hence all the integrals evaluate to the same value. In Hartree–Fock and density functional theories, the wave function is typically represented as a single Slater determinant constructed from orbitals, , with corresponding occupations . In these situations, the density simplifies to From its definition, the electron density is a non-negative function integrating to the total number of electrons. Further, for a system with kinetic energy T, the density satisfies the inequalities For finite kinetic energies, the first (stronger) inequality places the square root of the density in the Sobolev space .

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