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The Duru–Kleinert transformation, named after İsmail Hakkı Duru and Hagen Kleinert, is a mathematical method for solving path integrals of physical systems with singular potentials, which is necessary for the solution of all atomic path integrals due to the presence of Coulomb potentials (singular like ).
For example, the three quantum numbers associated to an electron in a coulomb potential, like the hydrogen atom, form a complete set (ignoring spin).
The solution of the Schrödinger equation (wave equations) for the hydrogen atom uses the fact that the Coulomb potential produced by the nucleus is isotropic (it is radially symmetric in space and only depends on the distance to the nucleus).
The basic functional form is the Coulomb potential, which only falls off as r−1.
In solids, especially in metals and semiconductors, the electrostatic screening or screening effect reduces the electrostatic field and Coulomb potential of an ion inside the solid.
For the electron-nucleus potential, Thomas and Fermi employed the Coulomb potential energy functional
For the classical part of the electron-electron interaction, Thomas and Fermi employed the Coulomb potential energy functional