The Kato theorem, or Kato's cusp condition (after Japanese mathematician
Tosio Kato), is used in computational
quantum physics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
.
It states that for generalized
Coulomb potentials, the
electron density
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 typical ...
has a
cusp
A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth.
Cusp or CUSP may also refer to:
Mathematics
* Cusp (singularity), a singular point of a curve
* Cusp catastrophe, a branch of bifu ...
at the position of the nuclei, where it satisfies
:
Here
denotes the positions of the nuclei,
their
atomic number
The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of its atomic nucleus. For ordinary nuclei composed of protons and neutrons, this is equal to the proton number (''n''p) or the number of pro ...
and
is the
Bohr radius
The Bohr radius () is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an at ...
.
For a Coulombic system one can thus, in principle, read off all information necessary for completely specifying the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
directly from examining the density distribution. This is also known as
E. Bright Wilson's argument within the framework 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 ...
(DFT). The electron density of the ground state of a molecular system contains
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) Hamiltonian is defined. This is valid in a non-relativistic treatment within the
Born–Oppenheimer approximation
In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the assumption that the wave functions of atomic nuclei and electrons in a molecule can be treated separately, based on the fact that the nuclei are much h ...
, and assuming point-like nuclei.
References
Theorems in quantum mechanics
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