In
quantum chemistry
Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
, electron density or electronic density is the measure of the
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
of an
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
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
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
which itself depends upon
variables (
spatial and
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
coordinates). Conversely, the density determines the wave function modulo up to a phase factor, providing the formal foundation of
density functional theory
Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...
.
According to
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, due to the
uncertainty principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
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
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
.
Definition
The electronic density corresponding to a normalised
-electron
wavefunction
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
(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
In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two electro ...
constructed from
orbitals,
, with corresponding occupations
. 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
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense t ...
. Together with the normalization and non-negativity this defines a space containing physically acceptable densities as
:
The second inequality places the density in the
''L''3 space. Together with the normalization property places acceptable densities within the intersection of ''L''
1 and ''L''
3 – a superset of
.
Topology
The
ground state
The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...
electronic density of an
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...
is conjectured to be a
monotonically decaying function of the distance from the
nucleus
Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to:
*Atomic nucleus, the very dense central region of an atom
*Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA
Nucle ...
.
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,
, 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
The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei, this is equal to the proton number (''n''p) or the number of protons found in the nucleus of every ...
(
).
Asymptotic behaviour
The nuclear cusp condition provides the near-nuclear (small
) density behaviour as
:
The long-range (large
) behaviour of the density is also known, taking the form
:
where I is the
ionisation energy
Ionization, or Ionisation is the process by which an atom or a molecule acquires a negative or positive charge by gaining or losing electrons, often in conjunction with other chemical changes. The resulting electrically charged atom or molecule i ...
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
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
s, regions of large electron density are usually found around the
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...
, and its bonds. In de-localised or
conjugated system
In theoretical chemistry, a conjugated system is a system of connected p-orbitals with delocalized electrons in a molecule, which in general lowers the overall energy of the molecule and increases stability. It is conventionally represented as ...
s, such as
phenol
Phenol (also called carbolic acid) is an aromatic organic compound with the molecular formula . It is a white crystalline solid that is volatile. The molecule consists of a phenyl group () bonded to a hydroxy group (). Mildly acidic, it req ...
,
benzene
Benzene is an organic chemical compound with the molecular formula C6H6. The benzene molecule is composed of six carbon atoms joined in a planar ring with one hydrogen atom attached to each. Because it contains only carbon and hydrogen atoms, ...
and compounds such as
hemoglobin
Hemoglobin (haemoglobin BrE) (from the Greek word αἷμα, ''haîma'' 'blood' + Latin ''globus'' 'ball, sphere' + ''-in'') (), abbreviated Hb or Hgb, is the iron-containing oxygen-transport metalloprotein present in red blood cells (erythrocyte ...
and
chlorophyll
Chlorophyll (also chlorophyl) is any of several related green pigments found in cyanobacteria and in the chloroplasts of algae and plants. Its name is derived from the Greek words , ("pale green") and , ("leaf"). Chlorophyll allow plants to a ...
, 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
In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°.
Regular hexa ...
shows the delocalised nature of the compound. This is shown below:
![Phenol mesomeric structures](https://upload.wikimedia.org/wikipedia/commons/f/f1/Phenol_mesomeric_structures.png)
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
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...
is absorbed at different wavelengths resulting in compounds appearing coloured. In
polymer
A polymer (; Greek '' poly-'', "many" + ''-mer'', "part")
is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...
s, 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
In atomic physics and quantum chemistry, the electron configuration is the distribution of electrons of an atom or molecule (or other physical structure) in atomic or molecular orbitals. For example, the electron configuration of the neon atom ...
molecules,
can be written in terms of a sum of products of basis functions, φ:
:
![ElectronDensityAniline](https://upload.wikimedia.org/wikipedia/commons/2/29/ElectronDensityAniline.PNG)
where P is the
density matrix
In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using ...
. 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
Aniline is an organic compound with the formula C6 H5 NH2. Consisting of a phenyl group attached to an amino group, aniline is the simplest aromatic amine
In organic chemistry, an aromatic amine is an organic compound consisting of an aroma ...
(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
carbon
Carbon () is a chemical element with the symbol C and atomic number 6. It is nonmetallic and tetravalent
In chemistry, the valence (US spelling) or valency (British spelling) of an element is the measure of its combining capacity with o ...
s and
nitrogen
Nitrogen is the chemical element with the symbol N and atomic number 7. Nitrogen is a nonmetal and the lightest member of group 15 of the periodic table, often called the pnictogens. It is a common element in the universe, estimated at se ...
, but the
hydrogen
Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic, an ...
s with only one proton in their nuclei, are not visible. This is the reason that
X-ray diffraction
X-ray crystallography is the experimental science determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to diffract into many specific directions. By measuring the angles ...
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 bond
A chemical bond is a lasting attraction between atoms or ions that enables the formation of molecules and crystals. The bond may result from the electrostatic force between oppositely charged ions as in ionic bonds, or through the sharing of ...
s , 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
Electrostatics is a branch of physics that studies electric charges at rest (static electricity).
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber ...
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
In chemistry, HOMO and LUMO are types of molecular orbitals. The acronyms stand for ''highest occupied molecular orbital'' and ''lowest unoccupied molecular orbital'', respectively. HOMO and LUMO are sometimes collectively called the ''frontie ...
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
X-ray crystallography is the experimental science determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to diffract into many specific directions. By measuring the angles ...
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
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
and some branches of
quantum theory
Quantum theory may refer to:
Science
*Quantum mechanics, a major field of physics
*Old quantum theory, predating modern quantum mechanics
* Quantum field theory, an area of quantum mechanics that includes:
** Quantum electrodynamics
** Quantum ch ...
also study and analyse electron
superposition and other related phenomena, such as the
NCI index which permits the study of
non-covalent interactions
In chemistry, a non-covalent interaction differs from a covalent bond in that it does not involve the sharing of electrons, but rather involves more dispersed variations of electromagnetic interactions between molecules or within a molecule. The c ...
using electron density.
Mulliken population analysis Mulliken charges arise from the Mulliken population analysis and provide a means of estimating partial atomic charges from calculations carried out by the methods of computational chemistry, particularly those based on the linear combination of ato ...
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
Transmission electron microscopy (TEM) is a microscopy technique in which a beam of electrons is transmitted through a specimen to form an image. The specimen is most often an ultrathin section less than 100 nm thick or a suspension on a g ...
(TEM) and
deep inelastic scattering
Deep inelastic scattering is the name given to a process used to probe the insides of hadrons (particularly the baryons, such as protons and neutrons), using electrons, muons and neutrinos. It provided the first convincing evidence of the reality ...
, 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
A scanning tunneling microscope (STM) is a type of microscope used for imaging surfaces at the atomic level. Its development in 1981 earned its inventors, Gerd Binnig and Heinrich Rohrer, then at IBM Zürich, the Nobel Prize in Physics in 1986. ...
(STM) and
atomic force microscopy
Atomic force microscopy (AFM) or scanning force microscopy (SFM) is a very-high-resolution type of scanning probe microscopy (SPM), with demonstrated resolution on the order of fractions of a nanometer, more than 1000 times better than the op ...
(AFM) can be used to probe the electron density of specific individual atoms.
Spin density
Spin density is electron density applied to
free radical
A daughter category of ''Ageing'', this category deals only with the biological aspects of ageing.
Ageing
Ailments of unknown cause
Biogerontology
Biological processes
Causes of death
Cellular processes
Gerontology
Life extension
Metabo ...
s. 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
Electron paramagnetic resonance (EPR) or electron spin resonance (ESR) spectroscopy is a method for studying materials that have unpaired electrons. The basic concepts of EPR are analogous to those of nuclear magnetic resonance (NMR), but the spi ...
,
neutron diffraction allows direct mapping of the spin density in 3D-space.
See also
*
Difference density map In X-ray crystallography, a difference density map shows the spatial distribution of the difference between the measured electron density of the crystal and the electron density explained by the current model.
These coefficients are derived from th ...
*
Electron cloud
In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any spe ...
*
Electron configuration
In atomic physics and quantum chemistry, the electron configuration is the distribution of electrons of an atom or molecule (or other physical structure) in atomic or molecular orbitals. For example, the electron configuration of the neon atom ...
*
Resolution (electron density)
Resolution in terms of electron density is a measure of the resolvability in the electron density map of a molecule. In X-ray crystallography, resolution is the highest resolvable peak in the diffraction pattern, while resolution in cryo-electron m ...
*
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 co ...
*
Density functional theory
Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...
*
Probability current
In quantum mechanics, the probability current (sometimes called probability flux) is a mathematical quantity describing the flow of probability. Specifically, if one thinks of probability as a heterogeneous fluid, then the probability current is th ...
References
{{reflist
Electron
Atomic physics
Quantum chemistry
Density functional theory