Kohn–Sham equations

<h2 id="description">Description</h2>
In <a href="/facts/Physics/a7aH2y08">physics</a> and <a href="/facts/Quantum_chemistry/H9AXV0lx">quantum chemistry</a>, specifically <a href="/facts/Density_functional_theory/wwz4Gqdg">density functional theory</a>, the Kohn–Sham equation is the non-interacting <a href="/facts/Schr%25C3%25B6dinger_equation/3csgSIWF">Schrödinger equation</a> (more clearly, Schrödinger-like equation) of a fictitious system (the "Kohn–Sham system") of non-interacting particles (typically electrons) that generate the same <a href="/facts/Electronic_density/L5yDsk6j">density</a> as any given system of interacting particles.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a><a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> 
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 <a href="/facts/Density_Functional_Theory/wwz4Gqdg">Density Functional Theory</a> has been researched<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a><a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>), the Kohn–Sham wavefunction is a single <a href="/facts/Slater_determinant/oKfv2Av0">Slater determinant</a> constructed from a set of <a href="/facts/Atomic_orbital/tKhgl6mL">orbitals</a> that are the lowest-energy solutions to

(
          
            −
            
              
                
                  ℏ
                  
                    2
                  
                
                
                  2
                  m
                
              
            
            
              ∇
              
                2
              
            
            +
            
              v
              
                eff
              
            
            (
            
              r
            
            )
          
          )
        
        
          φ
          
            i
          
        
        (
        
          r
        
        )
        =
        
          ε
          
            i
          
        
        
          φ
          
            i
          
        
        (
        
          r
        
        )
        .
      
    
    {\displaystyle \left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+v_{\text{eff}}(\mathbf {r} )\right)\varphi _{i}(\mathbf {r} )=\varepsilon _{i}\varphi _{i}(\mathbf {r} ).}

This <a href="/facts/Eigenvalue%2c_eigenvector_and_eigenspace/8TjEoT8u">eigenvalue equation</a> is the typical representation of the Kohn–Sham equations. Here εi is the orbital energy of the corresponding Kohn–Sham orbital 
 
 
 
 
 φ
 
 i
 
 
 
 
 {\displaystyle \varphi _{i}}
 
, and the density for an N-particle system is

 
 
 
 ρ
 (
 
 r
 
 )
 =
 
 ∑
 
 i
 
 
 N
 
 
 
 |
 
 
 φ
 
 i
 
 
 (
 
 r
 
 )
 
 
 |
 
 
 2
 
 
 .
 
 
 {\displaystyle \rho (\mathbf {r} )=\sum _{i}^{N}|\varphi _{i}(\mathbf {r} )|^{2}.}

<h2 id="history">History</h2>
The Kohn–Sham equations are named after <a href="/facts/Walter_Kohn/WCQ3DRlI">Walter Kohn</a> and <a href="/facts/Lu_Jeu_Sham/9nr2lC49">Lu Jeu Sham</a>, who introduced the concept at the <a href="/facts/University_of_California%2c_San_Diego/srMClpNv">University of California, San Diego</a>, 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). <a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

<h2 id="kohnsham-potential">Kohn–Sham potential</h2>
In Kohn–Sham density functional theory, the total energy of a system is expressed as a <a href="/facts/Functional_(mathematics)/3Cp3PHBR">functional</a> of the charge density as

 
 
 
 E
 [
 ρ
 ]
 =
 
 T
 
 s
 
 
 [
 ρ
 ]
 +
 ∫
 d
 
 r
 
 
 
 v
 
 ext
 
 
 (
 
 r
 
 )
 ρ
 (
 
 r
 
 )
 +
 
 E
 
 H
 
 
 [
 ρ
 ]
 +
 
 E
 
 xc
 
 
 [
 ρ
 ]
 ,
 
 
 {\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 ],}

where Ts is the Kohn–Sham <a href="/facts/Kinetic_energy/aJRxUd4p">kinetic energy</a>, which is expressed in terms of the Kohn–Sham orbitals as

 
 
 
 
 T
 
 s
 
 
 [
 ρ
 ]
 =
 
 ∑
 
 i
 =
 1
 
 
 N
 
 
 ∫
 d
 
 r
 
 
 
 φ
 
 i
 
 
 ∗
 
 
 (
 
 r
 
 )
 
 (
 
 −
 
 
 
 ℏ
 
 2
 
 
 
 2
 m
 
 
 
 
 ∇
 
 2
 
 
 
 )
 
 
 φ
 
 i
 
 
 (
 
 r
 
 )
 ,
 
 
 {\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} ),}

vext is the external <a href="/facts/Potential_energy/I03KLbmn">potential</a> acting on the interacting system (at minimum, for a molecular system, the electron–nuclei interaction), EH is the Hartree (or Coulomb) energy

 
 
 
 
 E
 
 H
 
 
 [
 ρ
 ]
 =
 
 
 
 e
 
 2
 
 
 2
 
 
 ∫
 d
 
 r
 
 ∫
 d
 
 
 r
 
 ′
 
 
 
 
 
 ρ
 (
 
 r
 
 )
 ρ
 (
 
 
 r
 
 ′
 
 )
 
 
 
 |
 
 
 r
 
 −
 
 
 r
 
 ′
 
 
 |
 
 
 
 
 ,
 
 
 {\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} '|}},}

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 Kohn-Sham orbitals subject to the constraint that they are orthogonal,<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> this yields a time-independent Schrödinger equation with a scalar potential equal to the Kohn–Sham potential 

 
 
 
 
 v
 
 eff
 
 
 (
 
 r
 
 )
 =
 
 v
 
 ext
 
 
 (
 
 r
 
 )
 +
 
 e
 
 2
 
 
 ∫
 
 
 
 ρ
 (
 
 
 r
 
 ′
 
 )
 
 
 
 |
 
 
 r
 
 −
 
 
 r
 
 ′
 
 
 |
 
 
 
 
 
 d
 
 
 r
 
 ′
 
 +
 
 
 
 δ
 
 E
 
 xc
 
 
 [
 ρ
 ]
 
 
 δ
 ρ
 (
 
 r
 
 )
 
 
 
 ,
 
 
 {\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} )}},}

where the last term

 
 
 
 
 v
 
 xc
 
 
 (
 
 r
 
 )
 ≡
 
 
 
 δ
 
 E
 
 xc
 
 
 [
 ρ
 ]
 
 
 δ
 ρ
 (
 
 r
 
 )
 
 
 
 ,
 
 
 {\displaystyle v_{\text{xc}}(\mathbf {r} )\equiv {\frac {\delta E_{\text{xc}}[\rho ]}{\delta \rho (\mathbf {r} )}},}

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 <a href="/facts/Harris_functional/XHjjCwWT">Harris functional</a> theory.
The Kohn–Sham orbital energies εi, in general, have little physical meaning (see <a href="/facts/Koopmans%2527_theorem/H1W3vFAQ">Koopmans' theorem</a>). The sum of the orbital energies is related to the total energy as

 
 
 
 E
 =
 
 ∑
 
 i
 
 
 N
 
 
 
 ε
 
 i
 
 
 −
 
 E
 
 H
 
 
 [
 ρ
 ]
 +
 
 E
 
 xc
 
 
 [
 ρ
 ]
 −
 ∫
 
 
 
 δ
 
 E
 
 xc
 
 
 [
 ρ
 ]
 
 
 δ
 ρ
 (
 
 r
 
 )
 
 
 
 ρ
 (
 
 r
 
 )
 
 d
 
 r
 
 .
 
 
 {\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} .}

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 <a href="/facts/Koopmans%2527_theorem/H1W3vFAQ">Koopmans' theorem</a>).

<h2 id="references">References</h2>

<ol>
<li id="fn:1">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. <a href="https://doi.org/10.1103%2FPhysRev.140.A1133" target="_blank">https://doi.org/10.1103%2FPhysRev.140.A1133</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">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. <a href="978-0-19-509276-9" target="_blank">978-0-19-509276-9</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">"The Nobel Prize in Chemistry 1998". NobelPrize.org. Retrieved 2023-09-15. <a href="https://www.nobelprize.org/prizes/chemistry/1998/summary/" target="_blank">https://www.nobelprize.org/prizes/chemistry/1998/summary/</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Tomas Arias (2004). "Kohn–Sham Equations". P480 notes. Cornell University. Archived from the original on 2020-02-18. Retrieved 2021-01-14. <a href="https://web.archive.org/web/20200218032643/http://muchomas.lassp.cornell.edu/P480/Notes/dft/node11.html" target="_blank">https://web.archive.org/web/20200218032643/http://muchomas.lassp.cornell.edu/P480/Notes/dft/node11.html</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
</ol>

Kohn–Sham equations open-in-new

Kohn–Sham equations