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[edit]
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 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 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
This eigenvalue equation is the typical representation of the Kohn–Sham equations. Here εi is the orbital energy of the corresponding Kohn–Sham orbital , and the density for an N-particle system is
History[edit]
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[edit]
In Kohn–Sham density functional theory, the total energy of a system is expressed as a functional of the charge density as
![{\displaystyle E[\rho ]=T_{s}[\rho ]+\int d\mathbf {r} \,v_{\text{ext}}(\mathbf {r} )\rho (\mathbf {r} )+E_{\text{H}}[\rho ]+E_{\text{xc}}[\rho ],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e22b4f091b030af2cb07280a0e70f54c39d9a98)
where Ts is the Kohn–Sham kinetic energy, which is expressed in terms of the Kohn–Sham orbitals as
![{\displaystyle T_{s}[\rho ]=\sum _{i=1}^{N}\int d\mathbf {r} \,\varphi _{i}^{*}(\mathbf {r} )\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\right)\varphi _{i}(\mathbf {r} ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51229c0eabea0945c109b6d5f4488b9ffeb04989)
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
![{\displaystyle E_{\text{H}}[\rho ]={\frac {e^{2}}{2}}\int d\mathbf {r} \int d\mathbf {r} '\,{\frac {\rho (\mathbf {r} )\rho (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f93f12919e1642e9ac240b6a121cf2de58b12b61)
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
![{\displaystyle v_{\text{eff}}(\mathbf {r} )=v_{\text{ext}}(\mathbf {r} )+e^{2}\int {\frac {\rho (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,d\mathbf {r} '+{\frac {\delta E_{\text{xc}}[\rho ]}{\delta \rho (\mathbf {r} )}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac289f865e3aa475741d4eab331a6bc6ba806d9)
![{\displaystyle v_{\text{xc}}(\mathbf {r} )\equiv {\frac {\delta E_{\text{xc}}[\rho ]}{\delta \rho (\mathbf {r} )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3970a0c00531c50b2e1d69060b6d7d3d0610613)
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
![{\displaystyle E=\sum _{i}^{N}\varepsilon _{i}-E_{\text{H}}[\rho ]+E_{\text{xc}}[\rho ]-\int {\frac {\delta E_{\text{xc}}[\rho ]}{\delta \rho (\mathbf {r} )}}\rho (\mathbf {r} )\,d\mathbf {r} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a6b2b8d3f9b70ee511198b9a54fc911a834d865)
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[edit]
- ^ Kohn, Walter; Sham, Lu Jeu (1965). "Self-Consistent Equations Including Exchange and Correlation Effects". Physical Review. 140 (4A): A1133–A1138. Bibcode:1965PhRv..140.1133K. doi:10.1103/PhysRev.140.A1133.
- ^ Parr, Robert G.; Yang, Weitao (1994). Density-Functional Theory of Atoms and Molecules. Oxford University Press. ISBN 978-0-19-509276-9. OCLC 476006840. OL 7387548M.
- ^ Wang, Hongmei; Zhang, Yunbo (2013). "Density-functional theory for the spin-1 bosons in a one-dimensional harmonic trap". Physical Review A. 88 (2): 023626. arXiv:1304.1328. Bibcode:2013PhRvA..88b3626W. doi:10.1103/PhysRevA.88.023626. S2CID 119280339.
- ^ Hu, Yayun; Murthy, G.; Rao, Sumathi; Jain, J. K. (2021). "Kohn-Sham density functional theory of Abelian anyons". Physical Review B. 103 (3): 035124. arXiv:2010.09872. Bibcode:2021PhRvB.103c5124H. doi:10.1103/PhysRevB.103.035124. 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.