Electron density
It is a scalar quantity depending upon three spatial variables and is typically denoted as either
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.
In molecules, regions of large electron density are usually found around the atom, and its bonds.
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.
[1] Conjugated systems can sometimes represent regions where electromagnetic radiation is absorbed at different wavelengths resulting in compounds appearing coloured.
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,
can be written in terms of a sum of products of basis functions, φ: 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.
Graphical models, including electron density are a commonly employed tool in chemistry education.
[2] 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.
denoting spatial and spin variables respectively) is defined as[6] where the operator corresponding to the density observable is Computing
Further, for a system with kinetic energy T, the density satisfies the inequalities[7] For finite kinetic energies, the first (stronger) inequality places the square root of the density in the Sobolev space
Together with the normalization property places acceptable densities within the intersection of L1 and L3 – a superset of
The ground state electronic density of an atom is conjectured to be a monotonically decaying function of the distance from the nucleus.
[8] 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,
) behaviour of the density is also known, taking the form[10] where I is the ionisation energy of the system.
[11][12] 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.
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.
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.
