en>The Deviant: Better calc spacing.

2013-04-19T21:15:56Z

Better calc spacing.

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The '''Kato theorem''', or '''Kato's cusp condition''', is used in computational [[quantum mechanics|quantum physics]].<ref>{{cite journal|last=Kato|first=Tosio|title=On the eigenfunctions of many-particle systems in quantum mechanics|journal=Communications on Pure and Applied Mathematics|year=1957|volume=10|issue=2|pages=151–177|doi=10.1002/cpa.3160100201}}</ref><ref>{{cite journal|last=March|first=N. H.|title=Spatially dependent generalization of Kato’s theorem for atomic closed shells in a bare Coulomb field|journal=Phys. Rev. A|year=1986|volume=33|issue=1|pages=88–89|doi=10.1103/PhysRevA.33.88|url=http://link.aps.org/doi/10.1103/PhysRevA.33.88|accessdate=16 June 2011|bibcode = 1986PhRvA..33...88M }}</ref> It states that for generalized Coulomb potentials, the [[electron density]] has a [[Cusp (singularity)|cusp]] at the position of the nuclei, where it satisfies
:<math> Z_k = - \frac{a_o}{2n(\mathbf{r})} \frac{dn(\mathbf{r})}{dr} |_{r \rightarrow \mathbf{R_k}} </math>

Here <math> \mathbf{R_k} </math> denotes the positions of the nuclei, <math> Z_k </math> their [[atomic number]] and <math> a_o = \left( \frac{h}{2 \pi m e} \right)^2 </math> is the [[Bohr radius]].

For a Coulombic system one can thus, in principle, read off all information necessary for completely specifying the Hamiltonian directly from examining the density distribution. This is also known as [[Edgar Bright Wilson|E. Bright Wilson's]] argument within the framework of [[density functional theory]] (DFT). The electron density of the ground state of a molecular system contains [[Cusp (singularity)|cusps]] at the location of the nuclei, and by identifying these from the total electron density of the system, the positions are thus established. From Kato's theorem, one also obtains the nuclear charge of the nuclei, and thus the external potential is fully defined. Finally, integrating the electron density over space gives the number of electrons, and the (electronic) [[Molecular Hamiltonian|Hamiltonian]] is defined. This is valid in a non-relativistic treatment within the [[Born-Oppenheimer approximation]], and assuming point-like nuclei.

==References==
<references />

{{DEFAULTSORT:Kato Theorem}}
[[Category:Quantum mechanics]]
[[Category:Theorems in quantum physics]]

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en>The Deviant: Better calc spacing.