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In quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
, and in particular quantum chemistry
Quantum chemistry

Quantum chemistry is a branch of theoretical chemistry, which applies quantum mechanics and quantum field theory to address issues and problems in chemistry....
, the electronic density is a measure of the probability of an electron
Electron

The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle....
 occupying 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 ?(r) or n(r). The density is determined, through definition, by the normalized N-electron wavefunction
Wavefunction

A wave function or wavefunction is a mathematical tool used in quantum mechanics to describe any physical system. It is a function from a mathematical space that maps the possible states of the system into the complex numbers....
 which itself depends upon 4N variables (3N spatial and N spin
Spin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
 coordinates).






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In quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
, and in particular quantum chemistry
Quantum chemistry

Quantum chemistry is a branch of theoretical chemistry, which applies quantum mechanics and quantum field theory to address issues and problems in chemistry....
, the electronic density is a measure of the probability of an electron
Electron

The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle....
 occupying 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 ?(r) or n(r). The density is determined, through definition, by the normalized N-electron wavefunction
Wavefunction

A wave function or wavefunction is a mathematical tool used in quantum mechanics to describe any physical system. It is a function from a mathematical space that maps the possible states of the system into the complex numbers....
 which itself depends upon 4N variables (3N spatial and N spin
Spin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
 coordinates). Conversely, the density determines the wave function modulo a phase factor, providing the formal foundation of density functional theory
Density functional theory

Density functional theory is a quantum mechanics theory used in physics and chemistry to investigate the electronic structure of Many-body problem, in particular atoms, molecules, and the condensed phases....
.

Definition

The electronic density corresponding to a normalized N-electron wavefunction
Wavefunction

A wave function or wavefunction is a mathematical tool used in quantum mechanics to describe any physical system. It is a function from a mathematical space that maps the possible states of the system into the complex numbers....
 (with r and s denoting spatial and spin variables respectively) is defined as

where the operator corresponding to the density observable is

In Hartree-Fock
Hartree-Fock

In computational physics and computational chemistry, the Hartree-Fock method is an approximate method for the determination of the Stationary state wavefunction and Stationary state energy of a Many-body problem....
 and density functional
Density functional theory

Density functional theory is a quantum mechanics theory used in physics and chemistry to investigate the electronic structure of Many-body problem, in particular atoms, molecules, and the condensed phases....
 theories the wave function is typically represented as a single Slater determinant
Slater determinant

In quantum mechanics, a Slater determinant is an expression which describes the wavefunction of a multi-fermionic system that satisfies Skew-symmetric matrix requirements and subsequently the Pauli exclusion principle by changing Plus and minus signs upon exchange of fermions....
 constructed from N orbitals, fk, with corresponding occupations nk. In these situations the density simplifies to

General Properties


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
Sobolev space

In mathematics, a Sobolev space is a vector space of functions equipped with a normed space that is a combination of Lp norm of the function itself as well as its derivatives up to a given order....
 H1(R3). Together with the normalization and non-negativity this defines a space containing physically acceptable densities as

The second inequality places the density in the L3 space
Lp space

In mathematics, the Lp and lp spaces are spaces of p-integrable function, and corresponding sequence spaces....
. Together with the normalization property places acceptable densities within the intersection of L1 and L3 – a superset of .

Topology


The ground state electronic density of an atom
Atom

|-! bgcolor=gray | Properties|-||}The atom is a basic unit of matter consisting of a dense, central atomic nucleus surrounded by a electron cloud of electric charge electrons....
 is conjectured to be a monotonically
Monotonic function

In mathematics, a monotonic function is a function which preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....
 decaying function of the distance from the nucleus
Nucleus

Nucleus may refer to:...
.

Nuclear cusp condition


The electronic density displays cusps at each nucleus in a molecule as a result of the unbounded electron-nucleus Coulomb potential. This behavior is quantified by the Kato cusp condition formulated in terms of the spherically averaged density, , about any given nucleus as

That is, the radial derivative of the spherically averaged density, evaluated at any nucleus, is equal to twice the density at that nucleus multiplied by the negative of the atomic number
Atomic number

In chemistry and physics, the atomic number is the number of protons found in the atomic nucleus of an atom. It is conventionally represented by the symbol Z....
 (Z).

Asymptotic behavior


The nuclear cusp condition provides the near-nuclear (small r) density behavior as

The long-range (large r) behavior of the density is also known, taking the form

where I is the ionization energy of the system.

See also

  • Current density
    Probability current

    In quantum mechanics, the probability current is a concept describing the flow of probability density. In particular, if one pictures the probability density as an inhomogeneous fluid, then the probability current is the rate of flow of this fluid ....