Editing Electron density (section)

== Topology ==
The [[ground state]] electronic density of an [[atom]] is conjectured to be a [[Monotonic function|monotonically]] decaying function of the distance from the [[atomic nucleus|nucleus]].<ref>{{cite journal|last1=Ayers|first1=Paul W.|last2=Parr | first2= Robert G.|year=2003|title=Sufficient condition for monotonic electron density decay in many-electron systems|journal=International Journal of Quantum Chemistry|volume=95|issue=6|pages=877–881|doi=10.1002/qua.10622}}</ref>

=== 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, <math>\bar{\rho}</math>, about any given nucleus as<ref>{{cite journal|last=Kato|first=Tosio |year=1957|title=On the eigenfunctions of many-particle systems in quantum mechanics|journal=Communications on Pure and Applied Mathematics|volume=10|issue=2|pages=151–177|doi=10.1002/cpa.3160100201}}</ref>

:<math>\left.\frac{\partial}{\partial r_{\alpha}}\bar{\rho}(r_{\alpha})\right|_{r_{\alpha}=0} = -2Z_{\alpha}\bar{\rho}(0).</math>

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]] (<math>Z</math>).

=== Asymptotic behaviour ===
The nuclear cusp condition provides the near-nuclear (small <math>r</math>) density behaviour as

:<math>\rho(r) \sim e^{-2Z_{\alpha}r}\,.</math>

The long-range (large <math>r</math>) behaviour of the density is also known, taking the form<ref>{{cite journal|last1=Morrell|first1=Marilyn M.|last2=Parr|first2=Robert. G.|last3=Levy|first3=Mel|year=1975|title=Calculation of ionization potentials from density matrices and natural functions, and the long-range behavior of natural orbitals and electron density|journal=Journal of Chemical Physics|volume=62|issue=2|pages=549–554|doi=10.1063/1.430509|bibcode = 1975JChPh..62..549M |doi-access=free}}</ref>

:<math>\rho(r) \sim e^{-2\sqrt{2\mathrm{I}}r}\,.</math>

where I is the [[ionisation energy]] of the system.