Kohn–Sham equations explained

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 physics and quantum chemistry, specifically density functional theory, the Kohn–Sham equation is the non-interacting Schrödinger equation (more clearly, Schrödinger-like equation) of a fictitious system (the "Kohn–Sham system") of non-interacting particles (typically electrons) that generate the same density as any given system of interacting particles.^{[1] [2]}

In the Kohn–Sham theory the introduction of the noninteracting kinetic energy functional T_s 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 v_s(r) or v_eff(r), called the Kohn–Sham potential. If the particles in the Kohn–Sham system are non-interacting fermions (non-fermion Density Functional Theory has been researched^{[3] [4]}), the Kohn–Sham wavefunction is a single Slater determinant constructed from a set of orbitals that are the lowest-energy solutions to$$\backslash left(-\backslash frac\backslash nabla^2\; +\; v\_\backslash text(\backslash mathbf\; r)\backslash right)\; \backslash varphi\_i(\backslash mathbf\; r)\; =\; \backslash varepsilon\_i\; \backslash varphi\_i(\backslash mathbf\; r).$$

This eigenvalue equation is the typical representation of the Kohn–Sham equations. Here ε_i is the orbital energy of the corresponding Kohn–Sham orbital

\varphi _{i} , and the density for an N-particle system is$$\backslash rho(\backslash mathbf\; r)\; =\; \backslash sum\_i^N\; |\backslash varphi\_i(\backslash mathbf\; r)|^2.$$

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$$E[\backslash rho]\; =\; T\_s[\backslash rho]\; +\; \backslash int\; d\backslash mathbf\; r\backslash ,\; v\_\backslash text(\backslash mathbf\; r)\; \backslash rho(\backslash mathbf\; r)\; +\; E\_\backslash text[\backslash rho]\; +\; E\_\backslash text[\backslash rho],$$

where T_s is the Kohn–Sham kinetic energy, which is expressed in terms of the Kohn–Sham orbitals as$$T\_s[\backslash rho]\; =\; \backslash sum\_^N\; \backslash int\; d\backslash mathbf\; r\backslash ,\; \backslash varphi\_i^*(\backslash mathbf\; r)\; \backslash left(-\backslash frac\backslash nabla^2\backslash right)\; \backslash varphi\_i(\backslash mathbf\; r),$$

v_ext is the external potential acting on the interacting system (at minimum, for a molecular system, the electron–nuclei interaction), E_H is the Hartree (or Coulomb) energy$$E\_\backslash text[\backslash rho]\; =\; \backslash frac\; \backslash int\; d\backslash mathbf\; r\; \backslash int\; d\backslash mathbf\text{'}\backslash ,\; \backslash frac$$

,

and E_xc 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$$v\_\backslash text(\backslash mathbf\; r)\; =\; v\_\backslash text(\backslash mathbf)\; +\; e^2\; \backslash int\; \backslash frac$$

\,d\mathbf' + \frac,where the last term$$v\_\backslash text(\backslash mathbf\; r)\; \backslash equiv\; \backslash frac$$is the exchange–correlation potential. This term, and the corresponding energy expression, are the only unknowns in the Kohn–Sham approach to density functional theory. An approximation that does not vary the orbitals is Harris functional theory.

The Kohn–Sham orbital energies ε_i, in general, have little physical meaning (see Koopmans' theorem). The sum of the orbital energies is related to the total energy as$$E\; =\; \backslash sum\_i^N\; \backslash varepsilon\_i\; -\; E\_\backslash text[\backslash rho]\; +\; E\_\backslash text[\backslash rho]\; -\; \backslash int\; \backslash frac\; \backslash rho(\backslash mathbf)\; \backslash ,d\backslash mathbf.$$

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).

Notes and References

1. Kohn . Walter . Sham . Lu Jeu . 1965 . Self-Consistent Equations Including Exchange and Correlation Effects . . 140 . A1133–A1138 . 4A . 10.1103/PhysRev.140.A1133. 1965PhRv..140.1133K . free.
2. Book: Parr . Robert G. . Yang . Weitao . 1994 . Density-Functional Theory of Atoms and Molecules . . 978-0-19-509276-9 . 476006840 . 7387548M.
3. 10.1103/PhysRevA.88.023626. Density-functional theory for the spin-1 bosons in a one-dimensional harmonic trap. 2013. Wang. Hongmei. Zhang. Yunbo . Physical Review A. 88. 2. 023626 . 1304.1328. 2013PhRvA..88b3626W . 119280339.
4. 10.1103/PhysRevB.103.035124. Kohn-Sham density functional theory of Abelian anyons. 2021. Hu. Yayun. Murthy. G.. Rao. Sumathi. Jain. J. K. . Physical Review B. 103. 3. 035124 . 2010.09872. 2021PhRvB.103c5124H . 224802789.
5. Web site: The Nobel Prize in Chemistry 1998 . 2023-09-15 . NobelPrize.org . en-US.
6. Web site: Kohn–Sham Equations . P480 notes . Tomas Arias . 2004 . Cornell University . 2021-01-14 . 2020-02-18 . https://web.archive.org/web/20200218032643/http://muchomas.lassp.cornell.edu/P480/Notes/dft/node11.html . dead .