Kohn–Sham equations
The Kohn-Sham equations are a set of mathematical equations used in quantum mechanics to simplify the complex problem of understanding how electrons behave in atoms and molecules. They introduce fictitious non-interacting electrons and use them to find the most stable arrangement of electrons, which helps scientists understand and predict the properties of matter at the atomic and molecular scale.
Description
In
In the Kohn–Sham theory the introduction of the noninteracting kinetic energy functional Ts into the energy expression leads, upon functional differentiation, to a collection of one-particle equations whose solutions are the Kohn–Sham orbitals.
The Kohn–Sham equation is defined by a local effective (fictitious) external potential in which the non-interacting particles move, typically denoted as vs(r) or veff(r), called the Kohn–Sham potential. If the particles in the Kohn–Sham system are non-interacting fermions (non-fermion
This
History
The Kohn–Sham equations are named after Walter Kohn and Lu Jeu Sham, who introduced the concept at the University of California, San Diego, in 1965.
Kohn received a Nobel Prize in Chemistry in 1998 for the Kohn–Sham equations and other work related to density functional theory (DFT). [5]
Kohn–Sham potential
In Kohn–Sham density functional theory, the total energy of a system is expressed as a functional of the charge density as
where Ts is the Kohn–Sham kinetic energy, which is expressed in terms of the Kohn–Sham orbitals as
vext is the external potential acting on the interacting system (at minimum, for a molecular system, the electron–nuclei interaction), EH is the Hartree (or Coulomb) energy
and Exc is the exchange–correlation energy. The Kohn–Sham equations are found by varying the total energy expression with respect to a set of orbitals, subject to constraints on those orbitals,[6] to yield the Kohn–Sham potential as
The Kohn–Sham orbital energies εi, in general, have little physical meaning (see Koopmans' theorem). The sum of the orbital energies is related to the total energy as
Because the orbital energies are non-unique in the more general restricted open-shell case, this equation only holds true for specific choices of orbital energies (see Koopmans' theorem).
References
- ^ Kohn, Walter; Sham, Lu Jeu (1965). "Self-Consistent Equations Including Exchange and Correlation Effects". .
- ^
Parr, Robert G.; Yang, Weitao (1994). Density-Functional Theory of Atoms and Molecules. OL 7387548M.
- S2CID 119280339.
- S2CID 224802789.
- ^ "The Nobel Prize in Chemistry 1998". NobelPrize.org. Retrieved 2023-09-15.
- ^ Tomas Arias (2004). "Kohn–Sham Equations". P480 notes. Cornell University. Archived from the original on 2020-02-18. Retrieved 2021-01-14.