Charge density

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https://en.wikipedia.org/wiki/Charge_density

In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m⁻³), at any point in a volume. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m⁻²), at any point on a surface charge distribution on a two dimensional surface. Linear charge density (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m⁻¹), at any point on a line charge distribution. Charge density can be either positive or negative, since electric charge can be either positive or negative.

Like mass density, charge density can vary with position. In classical electromagnetic theory charge density is idealized as a continuous scalar function of position ${\displaystyle \mathit{x}}$, like a fluid, and ${\displaystyle \rho (\mathit{x})}$, ${\displaystyle \sigma (\mathit{x})}$, and ${\displaystyle \lambda (\mathit{x})}$ are usually regarded as continuous charge distributions, even though all real charge distributions are made up of discrete charged particles. Due to the conservation of electric charge, the charge density in any volume can only change if an electric current of charge flows into or out of the volume. This is expressed by a continuity equation which links the rate of change of charge density ${\displaystyle \rho (\mathit{x})}$ and the current density ${\displaystyle \mathit{J}(\mathit{x})}$.

Since all charge is carried by subatomic particles, which can be idealized as points, the concept of a continuous charge distribution is an approximation, which becomes inaccurate at small length scales. A charge distribution is ultimately composed of individual charged particles separated by regions containing no charge. For example, the charge in an electrically charged metal object is made up of conduction electrons moving randomly in the metal's crystal lattice. Static electricity is caused by surface charges consisting of ions on the surface of objects, and the space charge in a vacuum tube is composed of a cloud of free electrons moving randomly in space. The charge carrier density in a conductor is equal to the number of mobile charge carriers (electrons, ions, etc.) per unit volume. The charge density at any point is equal to the charge carrier density multiplied by the elementary charge on the particles. However, because the elementary charge on an electron is so small (1.6⋅10⁻¹⁹ C) and there are so many of them in a macroscopic volume (there are about 10²² conduction electrons in a cubic centimeter of copper) the continuous approximation is very accurate when applied to macroscopic volumes, and even microscopic volumes above the nanometer level.

At even smaller scales, of atoms and molecules, due to the uncertainty principle of quantum mechanics, a charged particle does not have a precise position but is represented by a probability distribution, so the charge of an individual particle is not concentrated at a point but is 'smeared out' in space and acts like a true continuous charge distribution. This is the meaning of 'charge distribution' and 'charge density' used in chemistry and chemical bonding. An electron is represented by a wavefunction ${\displaystyle \psi (\mathit{x})}$ whose square is proportional to the probability of finding the electron at any point ${\displaystyle \mathit{x}}$ in space, so ${\displaystyle |\psi (\mathit{x}){|}^2}$ is proportional to the charge density of the electron at any point. In atoms and molecules the charge of the electrons is distributed in clouds called orbitals which surround the atom or molecule, and are responsible for chemical bonds.

Definitions

Continuous charges

Continuous charge distribution. The volume charge density ρ is the amount of charge per unit volume (three dimensional), surface charge density σ is amount per unit surface area (circle) with outward unit normal n̂, d is the dipole moment between two point charges, the volume density of these is the polarization density P. Position vector r is a point to calculate the electric field; r′ is a point in the charged object.

Following are the definitions for continuous charge distributions.

The linear charge density is the ratio of an infinitesimal electric charge dQ (SI unit: C) to an infinitesimal line element,

$${\displaystyle {\lambda }_q=\frac{dQ}{d\ell },}$$

similarly the surface charge density uses a surface area element dS

$${\displaystyle {\sigma }_q=\frac{dQ}{dS},}$$

and the volume charge density uses a volume element dV

$${\displaystyle {\rho }_q=\frac{dQ}{dV},}$$

Integrating the definitions gives the total charge Q of a region according to line integral of the linear charge density λ_q(r) over a line or 1d curve C,

$${\displaystyle Q={\int }_L{\lambda }_q(\mathbf{r})d\ell }$$

similarly a surface integral of the surface charge density σ_q(r) over a surface S,

$${\displaystyle Q={\int }_S{\sigma }_q(\mathbf{r})dS}$$

and a volume integral of the volume charge density ρ_q(r) over a volume V,

$${\displaystyle Q={\int }_V{\rho }_q(\mathbf{r})dV}$$

where the subscript q is to clarify that the density is for electric charge, not other densities like mass density, number density, probability density, and prevent conflict with the many other uses of λ, σ, ρ in electromagnetism for wavelength, electrical resistivity and conductivity.

Within the context of electromagnetism, the subscripts are usually dropped for simplicity: λ, σ, ρ. Other notations may include: ρ_ℓ, ρ_s, ρ_v, ρ_L, ρ_S, ρ_V etc.

The total charge divided by the length, surface area, or volume will be the average charge densities:

$${\displaystyle ⟨{\lambda }_q⟩=\frac{Q}{\ell },⟨{\sigma }_q⟩=\frac{Q}{S},⟨{\rho }_q⟩=\frac{Q}{V}.}$$

Free, bound and total charge

In dielectric materials, the total charge of an object can be separated into "free" and "bound" charges.

Bound charges set up electric dipoles in response to an applied electric field E, and polarize other nearby dipoles tending to line them up, the net accumulation of charge from the orientation of the dipoles is the bound charge. They are called bound because they cannot be removed: in the dielectric material the charges are the electrons bound to the nuclei.

Free charges are the excess charges which can move into electrostatic equilibrium, i.e. when the charges are not moving and the resultant electric field is independent of time, or constitute electric currents.

Total charge densities

In terms of volume charge densities, the total charge density is:

$${\displaystyle \rho ={\rho }_{\text{f}}+{\rho }_{\text{b}}.}$$

as for surface charge densities:

$${\displaystyle \sigma ={\sigma }_{\text{f}}+{\sigma }_{\text{b}}.}$$

where subscripts "f" and "b" denote "free" and "bound" respectively.

Bound charge

The bound surface charge is the charge piled up at the surface of the dielectric, given by the dipole moment perpendicular to the surface:

$${\displaystyle q_b=\frac{\mathbf{d}\cdot \hat{\mathbf{n}}}{|\mathbf{s}|}}$$

where s is the separation between the point charges constituting the dipole, ${\displaystyle \mathbf{d}}$ is the electric dipole moment, ${\displaystyle \hat{\mathbf{n}}}$ is the unit normal vector to the surface.

Taking infinitesimals:

$${\displaystyle d{q}_b=\frac{d\mathbf{d}}{|\mathbf{s}|}\cdot \hat{\mathbf{n}}}$$

and dividing by the differential surface element dS gives the bound surface charge density:

$${\displaystyle {\sigma }_b=\frac{d{q}_b}{dS}=\frac{d\mathbf{d}}{|\mathbf{s}|dS}\cdot \hat{\mathbf{n}}=\frac{d\mathbf{d}}{dV}\cdot \hat{\mathbf{n}}=\mathbf{P}\cdot \hat{\mathbf{n}}.}$$

where P is the polarization density, i.e. density of electric dipole moments within the material, and dV is the differential volume element.

Using the divergence theorem, the bound volume charge density within the material is

$${\displaystyle q_b=\int {\rho }_{b}dV=-{\oint }_S\mathbf{P}\cdot \hat{\mathbf{n}}dS=-\int \mathrm{\nabla }\cdot \mathbf{P}dV}$$

hence:

$${\displaystyle {\rho }_b=-\mathrm{\nabla }\cdot \mathbf{P}.}$$

The negative sign arises due to the opposite signs on the charges in the dipoles, one end is within the volume of the object, the other at the surface.

A more rigorous derivation is given below.

Derivation of bound surface and volume charge densities from internal dipole moments (bound charges)

The electric potential due to a dipole moment d is:

$${\displaystyle \phi =\frac{1}{4\pi {\epsilon }_0}\frac{(\mathbf{r}-{\mathbf{r}}^{\prime })\cdot \mathbf{d}}{|\mathbf{r}-{\mathbf{r}}^{\prime }{|}^3}}$$

For a continuous distribution, the material can be divided up into infinitely many infinitesimal dipoles

$${\displaystyle d\mathbf{d}=\mathbf{P}dV=\mathbf{P}{d}^3\mathbf{r}}$$

where dV = d³r′ is the volume element, so the potential is the volume integral over the object:

$${\displaystyle \phi =\frac{1}{4\pi {\epsilon }_0}\iiint \frac{(\mathbf{r}-{\mathbf{r}}^{\prime })\cdot \mathbf{P}}{|\mathbf{r}-{\mathbf{r}}^{\prime }{|}^3}{d}^3{\mathbf{r}}^{\prime }}$$

Since

$${\displaystyle {\mathrm{\nabla }}^{\prime }\left(\frac{1}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}\right)\equiv \left({\mathbf{e}}_x\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{\prime }}+{\mathbf{e}}_y\frac{\mathrm{\partial }}{\mathrm{\partial }{y}^{\prime }}+{\mathbf{e}}_z\frac{\mathrm{\partial }}{\mathrm{\partial }{z}^{\prime }}\right)\left(\frac{1}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}\right)=\frac{\mathbf{r}-{\mathbf{r}}^{\prime }}{|\mathbf{r}-{\mathbf{r}}^{\prime }{|}^3}}$$

where ∇′ is the gradient in the r′ coordinates,

$${\displaystyle \phi =\frac{1}{4\pi {\epsilon }_0}\iiint \mathbf{P}\cdot {\mathrm{\nabla }}^{\prime }\left(\frac{1}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}\right)d^3{\mathbf{r}}^{\prime }}$$

Integrating by parts

$${\displaystyle \phi =\frac{1}{4\pi {\epsilon }_0}\iiint \left[{\mathrm{\nabla }}^{\prime }\cdot \left(\frac{\mathbf{P}}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}\right)-\frac{1}{\mathbf{r}-{\mathbf{r}}^{\prime }}({\mathrm{\nabla }}^{\prime }\cdot \mathbf{P})\right]d^3{\mathbf{r}}^{\prime }}$$

using the divergence theorem:

${\displaystyle \phi =\frac{1}{4\pi {\epsilon }_0}}$ ${\displaystyle S}$ ${\displaystyle \frac{\mathbf{P}\cdot \hat{\mathbf{n}}d{S}^{\prime }}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}-\frac{1}{4\pi {\epsilon }_0}\iiint \frac{{\mathrm{\nabla }}^{\prime }\cdot \mathbf{P}}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}{d}^3{\mathbf{r}}^{\prime }}$

which separates into the potential of the surface charge (surface integral) and the potential due to the volume charge (volume integral):

${\displaystyle \phi =\frac{1}{4\pi {\epsilon }_0}}$ ${\displaystyle S}$ ${\displaystyle \frac{{\sigma }_{b}d{S}^{\prime }}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}+\frac{1}{4\pi {\epsilon }_0}\iiint \frac{{\rho }_b}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}{d}^3{\mathbf{r}}^{\prime }}$

that is

$${\displaystyle {\sigma }_b=\mathbf{P}\cdot \hat{\mathbf{n}},{\rho }_b=-\mathrm{\nabla }\cdot \mathbf{P}}$$

Free charge density

The free charge density serves as a useful simplification in Gauss's law for electricity; the volume integral of it is the free charge enclosed in a charged object - equal to the net flux of the electric displacement field D emerging from the object:

${\displaystyle {\mathrm{\Phi }}_D=}$ ${\displaystyle S}$ ${\displaystyle \mathbf{D}\cdot \hat{\mathbf{n}}dS=\iiint {\rho }_{f}dV}$

See Maxwell's equations and constitutive relation for more details.

Homogeneous charge density

For the special case of a homogeneous charge density ρ₀, independent of position i.e. constant throughout the region of the material, the equation simplifies to:

$${\displaystyle Q=V{\rho }_0.}$$

Proof

Start with the definition of a continuous volume charge density:

$${\displaystyle Q={\int }_V{\rho }_q(\mathbf{r})dV.}$$

Then, by definition of homogeneity, ρ_q(r) is a constant denoted by ρ_{q, 0} (to differ between the constant and non-constant densities), and so by the properties of an integral can be pulled outside of the integral resulting in:

$${\displaystyle Q={\rho }_{q,0}{\int }_{V}dV={\rho }_{0}V}$$

so,

$${\displaystyle Q=V{\rho }_{q,0}.}$$

The equivalent proofs for linear charge density and surface charge density follow the same arguments as above.

Discrete charges

For a single point charge q at position r₀ inside a region of 3d space R, like an electron, the volume charge density can be expressed by the Dirac delta function:

$${\displaystyle {\rho }_q(\mathbf{r})=q\delta (\mathbf{r}-{\mathbf{r}}_0)}$$

where r is the position to calculate the charge.

As always, the integral of the charge density over a region of space is the charge contained in that region. The delta function has the sifting property for any function f:

$${\displaystyle {\int }_R{d}^3\mathbf{r}f(\mathbf{r})\delta (\mathbf{r}-{\mathbf{r}}_0)=f({\mathbf{r}}_0)}$$

so the delta function ensures that when the charge density is integrated over R, the total charge in R is q:

$${\displaystyle Q={\int }_R{d}^3\mathbf{r}{\rho }_q={\int }_R{d}^3\mathbf{r}q\delta (\mathbf{r}-{\mathbf{r}}_0)=q{\int }_R{d}^3\mathbf{r}\delta (\mathbf{r}-{\mathbf{r}}_0)=q}$$

This can be extended to N discrete point-like charge carriers. The charge density of the system at a point r is a sum of the charge densities for each charge q_i at position r_i, where i = 1, 2, ..., N:

$${\displaystyle {\rho }_q(\mathbf{r})=\sum _{i=1}^N{q}_i\delta (\mathbf{r}-{\mathbf{r}}_i)}$$

The delta function for each charge q_i in the sum, δ(r − r_i), ensures the integral of charge density over R returns the total charge in R:

$${\displaystyle Q={\int }_R{d}^3\mathbf{r}\sum _{i=1}^N{q}_i\delta (\mathbf{r}-{\mathbf{r}}_i)=\sum _{i=1}^N{q}_i{\int }_R{d}^3\mathbf{r}\delta (\mathbf{r}-{\mathbf{r}}_i)=\sum _{i=1}^N{q}_i}$$

If all charge carriers have the same charge q (for electrons q = −e, the electron charge) the charge density can be expressed through the number of charge carriers per unit volume, n(r), by

$${\displaystyle {\rho }_q(\mathbf{r})=qn(\mathbf{r}).}$$

Similar equations are used for the linear and surface charge densities.

Charge density in special relativity

Further information: classical electromagnetism and special relativity and relativistic electromagnetism

In special relativity, the length of a segment of wire depends on velocity of observer because of length contraction, so charge density will also depend on velocity. Anthony French has described how the magnetic field force of a current-bearing wire arises from this relative charge density. He used (p 260) a Minkowski diagram to show "how a neutral current-bearing wire appears to carry a net charge density as observed in a moving frame." When a charge density is measured in a moving frame of reference it is called proper charge density.

It turns out the charge density ρ and current density J transform together as a four-current vector under Lorentz transformations.

Charge density in quantum mechanics

Main article: quantum mechanics

In quantum mechanics, charge density ρ_q is related to wavefunction ψ(r) by the equation

$${\displaystyle {\rho }_q(\mathbf{r})=q|\psi (\mathbf{r}){|}^2}$$

where q is the charge of the particle and |ψ(r)|² = ψ*(r)ψ(r) is the probability density function i.e. probability per unit volume of a particle located at r.

When the wavefunction is normalized - the average charge in the region r ∈ R is

$${\displaystyle Q={\int }_{R}q|\psi (\mathbf{r}){|}^2{d}^3\mathbf{r}}$$

where d³r is the integration measure over 3d position space.

Application

The charge density appears in the continuity equation for electric current, and also in Maxwell's Equations. It is the principal source term of the electromagnetic field; when the charge distribution moves, this corresponds to a current density. The charge density of molecules impacts chemical and separation processes. For example, charge density influences metal-metal bonding and hydrogen bonding. For separation processes such as nanofiltration, the charge density of ions influences their rejection by the membrane.

Electron density

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Electron_density

This article is about the quantum mechanical probability density of an electron. For the number density of electrons in a plasma, also called "electron density", see Plasma (physics).

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 ${\displaystyle \rho (\mathbf{\text{r}})}$ or ${\displaystyle n(\mathbf{\text{r}})}$. The density is determined, through definition, by the normalised ${\displaystyle N}$-electron wavefunction which itself depends upon ${\displaystyle 4N}$ variables ($3N$ 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.

Definition

The electronic density corresponding to a normalised ${\displaystyle N}$-electron wavefunction ${\displaystyle \mathrm{\Psi }}$ (with ${\displaystyle \mathbf{\text{r}}}$ and ${\displaystyle s}$ denoting spatial and spin variables respectively) is defined as

${\displaystyle \rho (\mathbf{r})=⟨\mathrm{\Psi }|\hat{\rho }(\mathbf{r})|\mathrm{\Psi }⟩,}$

where the operator corresponding to the density observable is

${\displaystyle \hat{\rho }(\mathbf{r})=\sum _{i=1}^N\delta (\mathbf{r}-{\mathbf{r}}_i).}$

Computing ${\displaystyle \rho (\mathbf{r})}$ as defined above we can simplify the expression as follows.

${\displaystyle \begin{array}{rl}\rho (\mathbf{r})& =\sum _{s_1}\cdots \sum _{s_N}\int \mathrm{d}{\mathbf{r}}_1\cdots \int \mathrm{d}{\mathbf{r}}_N\left(\sum _{i=1}^N\delta (\mathbf{r}-{\mathbf{r}}_i)\right)|\mathrm{\Psi }({\mathbf{r}}_1,s_1,{\mathbf{r}}_2,s_2,...,{\mathbf{r}}_N,s_N){|}^2\\ & =N\sum _{s_1}\cdots \sum _{s_N}\int \mathrm{d}{\mathbf{r}}_2\cdots \int \mathrm{d}{\mathbf{r}}_N|\mathrm{\Psi }(\mathbf{r},s_1,{\mathbf{r}}_2,s_2,...,{\mathbf{r}}_N,s_N){|}^2\end{array}}$

In words: holding a single electron still in position ${\displaystyle \mathbf{\text{r}}}$ 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 ${\displaystyle N}$ orbitals, ${\displaystyle {\phi }_k}$, with corresponding occupations ${\displaystyle n_k}$. In these situations, the density simplifies to

${\displaystyle \rho (\mathbf{r})=\sum _{k=1}^N{n}_k|{\phi }_k(\mathbf{r}){|}^2.}$

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

${\displaystyle \frac{1}{2}\int \mathrm{d}\mathbf{r}(\mathrm{\nabla }\sqrt{\rho (\mathbf{r})}{)}^2\le T.}$

${\displaystyle \frac{3}{2}{\left(\frac{\pi }{2}\right)}^{4/3}{\left(\int \mathrm{d}\mathbf{r}{\rho }^3(\mathbf{r})\right)}^{1/3}\le T.}$

For finite kinetic energies, the first (stronger) inequality places the square root of the density in the Sobolev space ${\displaystyle H^1({\mathbb{R}}^3)}$. Together with the normalization and non-negativity this defines a space containing physically acceptable densities as

${\displaystyle {\mathcal{J}}_N=\left\{\rho |\rho (\mathbf{r})\ge 0,{\rho }^{1/2}(\mathbf{r})\in H^1({\mathbf{R}}^3),\int \mathrm{d}\mathbf{r}\rho (\mathbf{r})=N\right\}.}$

The second inequality places the density in the L³ space. Together with the normalization property places acceptable densities within the intersection of L¹ and L³ – a superset of ${\displaystyle {\mathcal{J}}_N}$.

Topology

The ground state electronic density of an atom is conjectured to be a monotonically decaying function of the distance from the nucleus.

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 behaviour is quantified by the Kato cusp condition formulated in terms of the spherically averaged density, ${\displaystyle \overline{\rho }}$, about any given nucleus as

${\displaystyle {\frac{\mathrm{\partial }}{\mathrm{\partial }{r}_{\alpha }}\overline{\rho }(r_{\alpha })|}_{r_{\alpha }=0}=-2Z_{\alpha }\overline{\rho }(0).}$

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 (${\displaystyle Z}$).

Asymptotic behaviour

The nuclear cusp condition provides the near-nuclear (small ${\displaystyle r}$) density behaviour as

${\displaystyle \rho (r)\sim e^{-2Z_{\alpha }r}.}$

The long-range (large ${\displaystyle r}$) behaviour of the density is also known, taking the form

${\displaystyle \rho (r)\sim e^{-2\sqrt{2\mathrm{I}}r}.}$

where I is the ionisation energy of the system.

Response density

Another more-general definition of a density is the "linear-response density". This is the density that when contracted with any spin-free, one-electron operator yields the associated property defined as the derivative of the energy. For example, a dipole moment is the derivative of the energy with respect to an external magnetic field and is not the expectation value of the operator over the wavefunction. For some theories they are the same when the wavefunction is converged. The occupation numbers are not limited to the range of zero to two, and therefore sometimes even the response density can be negative in certain regions of space.

Overview

In molecules, regions of large electron density are usually found around the atom, and its bonds. In de-localised or conjugated systems, such as phenol, benzene and compounds such as hemoglobin and chlorophyll, the electron density is significant in an entire region, i.e., in benzene they are found above and below the planar ring. This is sometimes shown diagrammatically as a series of alternating single and double bonds. In the case of phenol and benzene, a circle inside a hexagon shows the delocalised nature of the compound. This is shown below:

In compounds with multiple ring systems which are interconnected, this is no longer accurate, so alternating single and double bonds are used. In compounds such as chlorophyll and phenol, some diagrams show a dotted or dashed line to represent the delocalization of areas where the electron density is higher next to the single bonds. Conjugated systems can sometimes represent regions where electromagnetic radiation is absorbed at different wavelengths resulting in compounds appearing coloured. In polymers, these areas are known as chromophores.

In quantum chemical calculations, the electron density, ρ(r), is a function of the coordinates r, defined so ρ(r)dr is the number of electrons in a small volume dr. For closed-shell molecules, ${\displaystyle \rho (\mathbf{r})}$ can be written in terms of a sum of products of basis functions, φ:

${\displaystyle \rho (\mathbf{r})=\sum _{\mu }\sum _{\nu }{P}_{\mu \nu }{\varphi }_{\mu }(\mathbf{r}){\varphi }_{\nu }(\mathbf{r})}$

Electron density calculated for aniline, high density values indicate atom positions, intermediate density values emphasize bonding, low values provide information on a molecule's shape and size.

where P is the density matrix. Electron densities are often rendered in terms of an isosurface (an isodensity surface) with the size and shape of the surface determined by the value of the density chosen, or in terms of a percentage of total electrons enclosed.

Molecular modeling software often provides graphical images of electron density. For example, in aniline (see image at right). Graphical models, including electron density are a commonly employed tool in chemistry education. Note in the left-most image of aniline, high electron densities are associated with the carbons and nitrogen, but the hydrogens with only one proton in their nuclei, are not visible. This is the reason that X-ray diffraction has a difficult time locating hydrogen positions.

Most molecular modeling software packages allow the user to choose a value for the electron density, often called the isovalue. Some software also allows for specification of the electron density in terms of percentage of total electrons enclosed. Depending on the isovalue (typical units are electrons per cubic bohr), or the percentage of total electrons enclosed, the electron density surface can be used to locate atoms, emphasize electron densities associated with chemical bonds , or to indicate overall molecular size and shape.

Graphically, the electron density surface also serves as a canvas upon which other electronic properties can be displayed. The electrostatic potential map (the property of electrostatic potential mapped upon the electron density) provides an indicator for charge distribution in a molecule. The local ionisation potential map (the property of local ionisation potential mapped upon the electron density) provides an indicator of electrophilicity. And the LUMO map (lowest unoccupied molecular orbital mapped upon the electron density) can provide an indicatory for nucleophilicity.

Experiments

Many experimental techniques can measure electron density. For example, quantum crystallography through X-ray diffraction scanning, where X-rays of a suitable wavelength are targeted towards a sample and measurements are made over time, gives a probabilistic representation of the locations of electrons. From these positions, molecular structures, as well as accurate charge density distributions, can often be determined for crystallised systems. Quantum electrodynamics and some branches of quantum field theory also study and analyse electron superposition and other related phenomena, such as the NCI index which permits the study of non-covalent interactions using electron density. Mulliken population analysis is based on electron densities in molecules and is a way of dividing the density between atoms to give an estimate of atomic charges.

In transmission electron microscopy (TEM) and deep inelastic scattering, as well as other high energy particle experiments, high energy electrons interacts with the electron cloud to give a direct representation of the electron density. TEM, scanning tunneling microscopy (STM) and atomic force microscopy (AFM) can be used to probe the electron density of specific individual atoms.

Spin density

Spin density is electron density applied to free radicals. It is defined as the total electron density of electrons of one spin minus the total electron density of the electrons of the other spin. One of the ways to measure it experimentally is by electron spin resonance, neutron diffraction allows direct mapping of the spin density in 3D-space.