Submission declined on 23 May 2026 by Devonian Wombat (talk). This draft provides insufficient context for those unfamiliar with the subject. Please see the guide to writing better articles for how to improve your writing. If you would like to continue working on the submission, click on the "Edit" tab at the top of the window. If you have not resolved the issues listed above, your draft will be declined again and potentially deleted. If you need extra help, please ask us a question at the AfC Help Desk or get live help from experienced editors. Please do not remove reviewer comments or this notice until the submission is accepted. Where to get help If you need help editing or submitting your draft, please ask us a question at the AfC Help Desk or get live help from experienced editors. These venues are only for help with editing and the submission process, not to get reviews. If you need feedback on your draft, or if the review is taking a lot of time, you can try asking for help on the talk page of a relevant WikiProject. Some WikiProjects are more active than others so a speedy reply is not guaranteed. How to improve a draft Wikipedia:Contributing to Wikipedia – a basic overview on how to edit Wikipedia. Help:Wikitext – how to use the markup Help:Referencing for beginners – how to include references Wikipedia:Article development – how to develop your article Wikipedia:Writing better articles – how to improve your article Wikipedia:Verifiability – make sure your article includes reliable third-party sources You can also browse Wikipedia:Featured articles and Wikipedia:Good articles to find examples of Wikipedia's best writing on topics similar to your proposed article. Improving your odds of a speedy review To improve your odds of a faster review, tag your draft with relevant WikiProject tags using the button below. This will let reviewers know a new draft has been submitted in their area of interest. For instance, if you wrote about a female astronomer, you would want to add the Biography, Astronomy, and Women scientists tags. Add tags to your draft Editor resources Easy tools: Citation bot (help) | Advanced: Fix bare URLs Declined by Devonian Wombat 15 days ago. Last edited by Devonian Wombat 15 days ago. Reviewer: Inform author. Resubmit Please note that if the issues are not fixed, the draft will be declined again.

• Comment: This topic is clearly notable. Some of the references listed have >1000 citations.
  The current draft is unsuitable for an encyclopedia. The draft strength lies solely in some long equations which are themselves improperly sourced.
  The section "Formalism of GKS theory" is unsourced and starts by assuming the reader is familiar with calculational details of DFT. The early sections of content should target an undergraduate science major.
  The section "Kohn-Sham" is improperly sourced. The claim is "...one obtains the KS equations" but the citations are from 1964, long before GKS. These references could be used in a History section but not elsewhere in the article.
  "Hartree-Fock-Kohn-Sham (HF-KS)" is unsourced.
  "Hybrid formalism: GKS along the adiabatic connection (AC)" The GKS should not appear in section titles. The Gorling and Levy source is primary for the section, but could be used as a secondary source for an overview of GKS. According to a review like
  * Teale, A. M., Helgaker, T., Savin, A., Adamo, C., Aradi, B., Arbuznikov, A. V., ... & Yang, W. (2022). DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science. Physical chemistry chemical physics, 24(47), 28700-28781.
  this wing of GKS is not that important, but the source does provide a useful perspective.
  The intro is the closest to encyclopedic content but it is too short and does not summarize the article. It also has sourcing problems. For example Becke is mentioned but not sourced. The Seidl 1996 paper is cited but that does not verify that "GKS was introduced", we need a secondary source that cites Seidl for that. The Gorling-Levy "Hybrid schemes..." for example or the review cited above. Johnjbarton (talk) 02:01, 23 May 2026 (UTC)

This is a draft article. It is a work in progress open to editing by anyone. Please ensure core content policies are met before publishing it as a live Wikipedia article. Find sources: Google (books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL Easy tools: Citation bot (help) | Advanced: Fix bare URLs · Article logs · Draft logs. Last edited by Devonian Wombat (talk | contribs) 15 days ago. (Update) Finished drafting? Submit for review

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Generalized Kohn-Sham theory (GKS) is an extension of Kohn–Sham (KS) density functional theory (DFT) and Hartree–Fock (HF) theory. It is used to give a rigorous basis for Hybrid functionals, which mix some nonlocal exchange interaction from HF into the local exchange from DFT introduced by Becke. It also sees use in analyzing the band-gap structure of materials^[1]. GKS theory was introduced by Seidl, Görling, Vogl, Majewski and Levy in 1996^[2] and 1997^[3]. Roughly speaking it might be seen as a toy model for creating different formalisms in electronic structure theory.

Instead of the regular Levy constrained search formulation of DFT, where ${\displaystyle v(r)}$ is the external potential, the electron-nuclei Coulomb potential, ${\displaystyle \rho (r)}$ the electron density to obtain the groundstate energy ${\displaystyle E[v]}$ it is minimized ${\displaystyle \min _{\rho (r)\rightarrow N}}$ over all densities that yield the particle number ${\displaystyle N}$

${\displaystyle E[v]=\min _{\rho (r)\rightarrow N}{\bigg \{}F[\rho ]+\int drv(r)\rho (r){\bigg \}}}$

where the Hohenberg-Kohn (HK) functional is

${\displaystyle F[\rho ]=\min _{\Psi \rightarrow \rho (r)}\langle \Psi |{\hat {T}}+{\hat {V}}_{ee}|\Psi \rangle }$

here ${\textstyle {\hat {T}}=\sum _{i}-{\tfrac {1}{2}}\Delta _{i}}$ is the kinetic energy operator with the Laplacian ${\displaystyle \Delta \equiv \nabla ^{2}}$ and ${\textstyle {\hat {V}}_{ee}=\sum _{i<j}|r_{i}-r_{j}|^{-1}}$ the electron-electron interaction, it is minimized ${\displaystyle \min _{\Psi \rightarrow \rho (r)}}$ over all wave functions ${\displaystyle \Psi }$ that yield the appropriate density, in the formalism of GKS theory ^[2][4] one decomposes, partitions the electronic energy further into a HK-type functional ${\textstyle F^{S}[\rho ]}$ of a given subsystem denoted ${\textstyle S}$ and a remainder functional ${\textstyle R^{S}[\rho ]}$ as

${\displaystyle F[\rho ]=F^{S}[\rho ]+R^{S}[\rho ]}$

This leads to the GKS variational principle

${\displaystyle {\begin{aligned}E[v]=\min _{\rho (r)\rightarrow N}{\bigg \{}F^{S}[\rho ]+R^{S}[\rho ]+\int drv(r)\rho (r){\bigg \}}=\min _{\rho (r)\rightarrow N}{\bigg \{}\min _{\Phi \rightarrow \rho (r)}S[\Phi ]+R^{S}[\rho ]+\int drv(r)\rho (r){\bigg \}}=\\=\min _{\Phi \rightarrow N}{\bigg \{}S[\Phi ]+R^{S}[\rho [\Phi ]]+\int drv(r)\rho [\Phi ](r){\bigg \}}=\min _{\{\phi _{i}\}\rightarrow N}{\bigg \{}S[\{\phi _{i}\}]+R^{S}[\rho [\{\phi _{i}\}]]+\int drv(r)\rho [\{\phi _{i}\}](r){\bigg \}}\end{aligned}}}$

where the important step is that the two minima, in which both the density occur, ${\displaystyle \min _{\rho (r)\rightarrow N}\min _{\Phi \rightarrow \rho (r)}=\min _{\Phi \rightarrow N}}$ can be collapsed to one minimum independent of the density. It is searched over all Slater determinants ${\displaystyle \Phi =(N!)^{-1/2}\det(\phi _{i}(j))_{i,j\in \mathbb {N} _{\leq N}}}$ that yield the respective density or particle number. Here the HK-type functional of the subsystem is defined as

${\displaystyle F^{S}[\rho ]\equiv \min _{\Phi \rightarrow \rho (r)}S[\Phi ]=\min _{\{\phi _{i}\}\rightarrow N}S[\{\phi _{i}\}]}$

and the energy of the subsystem is given as

${\displaystyle E^{S}[v_{\text{eff}};\{\phi _{i}\}]=\min _{\rho (r)\rightarrow N}{\bigg \{}F^{S}[\rho ]+\int drv_{\text{eff}}(r)\rho (r){\bigg \}}=S[\{\phi _{i}\}]+\int drv_{\text{eff}}(r)\rho (r)}$

Here the effective potential is the sum of the external potential and remainder potential

${\displaystyle v_{\text{eff}}(r)=v(r)+\underbrace {\frac {\delta R^{S}[\rho ]}{\delta \rho (r)}} _{v_{R}(r)}}$

Minimizing the energy of the subsystem under constraint, that the set of orbitals ${\displaystyle \{\phi _{i}\}_{i\in \mathbb {N} _{\leq N}}}$ is orthonormalized ${\displaystyle \langle \phi _{i},\phi _{j}\rangle =\delta _{ij}}$ with the Kronecker delta ${\displaystyle \delta _{ij}}$ w.l.o.g. set diagonally, i.e. ${\textstyle {\frac {\delta {\mathcal {L}}[\{\phi _{i}\}]}{\delta \phi _{j}(r)}}={\frac {\delta E^{S}[\{\phi _{i}\}]}{\delta \phi _{j}(r)}}+{\frac {\delta }{\delta \phi _{j}(r)}}\varepsilon _{j}(\langle \phi _{j},\phi _{j}\rangle -1)=0}$ one obtains the GKS equations

${\displaystyle {\hat {O}}^{S}[\{\phi _{i}\}]\phi _{j}+v_{\text{eff}}\phi _{j}={\hat {O}}^{S}[\{\phi _{i}\}]\phi _{j}+v\phi _{j}+v_{R}\phi _{j}=\varepsilon \phi _{j}}$

The GKS operator ${\textstyle {\hat {O}}^{S}[\{\phi _{i}\}]\equiv {\frac {\delta S[\{\phi _{i}\}]}{\delta \phi _{j}(r)}}}$ as functional derivative of the functional ${\displaystyle S}$ is invariant under unitary orbital transformations. The choice of the Slater determinant functional ${\textstyle S[\Phi ]=S[\{\phi _{i}\}]}$ distinguishes the different GKS shemes.

Choosing the Slater determinant functional as the kinetic energy of a Slater determinant

${\displaystyle {\begin{aligned}S[\Phi ]\equiv T[\Phi ]\Rightarrow {\begin{cases}R^{s}[\rho ]=E_{\text{Hxc}}[\rho ]\\v_{R}=v_{\text{Hxc}}\end{cases}}\Rightarrow v_{\text{eff}}=v+v_{\text{Hxc}}=v_{S},\\O^{S}=-{\tfrac {1}{2}}\Delta \Rightarrow (O^{S}+v_{\text{eff}})\phi _{i}=(-{\tfrac {1}{2}}\Delta +v_{S})\phi _{i}=h_{\text{KS}}\phi =\varepsilon \phi _{i}\end{aligned}}}$

one obtains the Kohn–Sham (KS) equations^[5][6]. Here Hxc stands for the Hartree-exchange-correlation energy ${\displaystyle E_{\text{Hxc}}}$ or potential ${\displaystyle v_{\text{Hxc}}}$ respectively. ${\displaystyle v_{S}}$ is the KS potential and ${\displaystyle h_{\text{KS}}}$ the KS Hamiltonian.

Choosing the Slater determinant functional as the kinetic energy and electron-electron interaction of a Slater determinant

${\displaystyle {\begin{aligned}S[\Phi ]\equiv T[\Phi ]+E_{\text{Hx}}[\Phi ]\Rightarrow {\begin{cases}R^{s}[\rho ]=E_{\text{c}}^{\text{HF}}[\rho ]\\v_{R}=v_{\text{c}}^{\text{HF}}\end{cases}}\Rightarrow v_{\text{eff}}=v+v_{\text{c}}^{\text{HF}},\\O^{S}=-{\tfrac {1}{2}}\Delta +v_{\text{H}}+v_{\text{x}}^{\text{NL}}\Rightarrow (O^{S}+v_{\text{eff}})\phi _{i}=(-{\tfrac {1}{2}}\Delta +v+v_{\text{H}}+v_{\text{x}}^{\text{NL}}+v_{\text{c}}^{\text{HF}})\phi _{i}=h_{\text{HF-KS}}\phi _{i}=\varepsilon _{i}\phi _{i}\end{aligned}}}$

one obtains the Hartree-Fock-Kohn-Sham (HF-KS) equations with its Hamiltonian ${\displaystyle h_{\text{HF-KS}}}$. Here ${\displaystyle v_{\text{x}}^{\text{NL}}}$ is the nonlocal exchange operator from HF theory and ${\displaystyle v_{\text{c}}^{\text{HF}}}$ a HF correlation potential e.g. from Møller–Plesset perturbation theory.

Choosing the Slater determinant functional as the kinetic energy and electron-electron interaction of a Slater determinant scaled along the adiabatic connection (AC)^[3] with an interpolation parameter ${\displaystyle \alpha \in [0,1]}$

${\displaystyle {\begin{aligned}S[\Phi ]\equiv T[\Phi ]+\alpha E_{\text{Hx}}[\Phi ]\Rightarrow {\begin{cases}R_{\alpha }^{s}[\rho ]=\alpha E_{\text{c}}^{\text{HF}}+(1-\alpha )E_{\text{Hxc}}[\rho ]\\v_{R}^{\alpha }=\alpha v_{\text{c}}^{\text{HF}}+(1-\alpha )v_{\text{Hxc}}\end{cases}}\\\Rightarrow v_{\text{eff}}^{\alpha }=v+\alpha v_{\text{c}}^{\text{HF}}+(1-\alpha )v_{\text{Hxc}},O_{\alpha }^{S}=-{\tfrac {1}{2}}\Delta +\alpha v_{\text{H}}+\alpha v_{\text{x}}^{\text{NL}}\\\Rightarrow (O_{\alpha }^{S}+v_{\text{eff}}^{\alpha })\phi _{i}^{\alpha }=h_{\text{hyb}}^{\alpha }\phi _{i}^{\alpha }=\\(-{\tfrac {1}{2}}\Delta +v+\underbrace {\alpha v_{\text{H}}+(1-\alpha )v_{\text{H}}} _{v_{\text{H}}}+\alpha v_{\text{x}}^{\text{NL}}+\alpha v_{\text{c}}^{\text{HF}}+(1-\alpha )v_{\text{x}}+(1-\alpha )v_{\text{c}})\phi _{i}^{\alpha }=\varepsilon _{i}^{\alpha }\phi _{i}^{\alpha }\end{aligned}}}$

one obtains the global (double) hybrid equations interpolating between the KS and HF-KS equations ${\displaystyle h_{\text{hyb}}^{\alpha }\in [h_{\text{KS}},h_{\text{HF-KS}}]}$.