Optimized effective potential method

The OEP method was developed more than 10 years prior to the work of Pierre Hohenberg, Walter Kohn and Lu Jeu Sham in 1953 by R. T. Sharp and G. K. Horton {{cite journal|doi=10.1103/PhysRev.90.317|title=A Variational Approach to the Unipotential Many-Electron Problem|year=1953|last1=Sharp|first1=R. T.|last2=Horton|first2=G. K.|journal=Physical Review|volume=90|issue=2 |pages=317|bibcode=1953PhRv...90..317S }} in order to investigate, what happens to Hartree-Fock (HF) theory{{cite journal |first1=D. R. |last1=Hartree |author-link1=Douglas Hartree |title=The Wave Mechanics of an Atom with a Non-Coulomb Central Field |journal=Mathematical Proceedings of the Cambridge Philosophical Society |volume=24 |issue=1 |pages=111 |year=1928 |doi=10.1017/S0305004100011920 |bibcode=1928PCPS...24..111H |s2cid=121520012 }}{{cite journal |first=J. C. |last=Slater |title=The Self Consistent Field and the Structure of Atoms |journal=Physical Review |volume=32 |issue=3 |pages=339–348 |year=1928 |doi=10.1103/PhysRev.32.339 |bibcode=1928PhRv...32..339S }}{{cite journal |first=J. A. |last=Gaunt |title=A Theory of Hartree's Atomic Fields |journal=Mathematical Proceedings of the Cambridge Philosophical Society |volume=24 |issue=2 |pages=328–342 |year=1928 |doi=10.1017/S0305004100015851 |bibcode=1928PCPS...24..328G |s2cid=119685329 }}{{cite journal |first=J. C. |last=Slater |title=Note on Hartree's Method |journal=Physical Review |volume=35 |issue=2 |pages=210–211 |year=1930 |doi=10.1103/PhysRev.35.210.2 |bibcode=1930PhRv...35..210S }}{{cite journal |first=V. A. |last=Fock |title=Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems |language=de |journal=Zeitschrift für Physik |volume=61 |issue=1 |pages=126–148 |year=1930 |doi=10.1007/BF01340294 |bibcode=1930ZPhy...61..126F |s2cid=125419115 }} {{cite journal |first=V. A. |last=Fock |title="Selfconsistent field" mit Austausch für Natrium |language=de |journal=Zeitschrift für Physik |volume=62 |issue=11 |pages=795–805 |year=1930 |doi=10.1007/BF01330439 |bibcode=1930ZPhy...62..795F |s2cid=120921212 }} when, instead of the regular nonlocal exchange potential, a local exchange potential is demanded. Much later after 1990 it was found out that this ansatz is useful in density functional theory.

In density functional theory the exchange correlation (xc) potential is defined as the functional derivative of the exchange correlation (xc) energy with respect to the electron density $\backslash rho(r)${{Citation needed|date=March 2025}}

{{NumBlk|:|$v\_\{xc\}(r)\backslash equiv\backslash frac\{\backslash delta\; E\_\{xc\}[\backslash rho]\}\{\backslash delta\backslash rho(r)\}=\backslash frac\{\backslash delta\; E\_\{xc\}[\backslash \{\backslash phi\_\{s\}\backslash \}]\}\{\backslash delta\backslash rho(r)\}$|{{EquationRef|1}}}}

where the index $s$ denotes either occupied or unoccupied KS orbitals and eigenvalues. The problem is that, although the xc energy is in principle (due to the Hohenberg-Kohn (HK) theorem) a functional of the density, its explicit dependence on the density is unknown (only known in the simple Local density approximation (LDA) case), only its implicit dependence through the KS orbitals. That motivates the use of the chain rule

$v\_\{xc\}(r)=\backslash int\; dr\text{'}\backslash sum\_\{s\}\backslash bigg[\backslash frac\{\backslash delta\; E\_\{xc\}[\backslash \{\backslash phi\_\{s\}\backslash \}]\}\{\backslash delta\backslash phi\_\{s\}(r\text{'})\}\backslash frac\{\backslash delta\backslash phi\_\{s\}(r\text{'})\}\{\backslash delta\backslash rho(r)\}+c.c.\backslash bigg]$

Unfortunately the functional derivative $\backslash delta\backslash phi\_\{s\}/\backslash delta\backslash rho$, despite its existence, is also unknown. So one needs to invoke the chain rule once more, now with respect to the Kohn-Sham (KS) potential $v\_\{S\}(r)$

$v\_\{xc\}(r)=\backslash iint\; dr\text{'}dr$\sum_{s}\bigg[\frac{\delta E_{xc}[\{\phi_{s}\}]}{\delta\phi_{s}(r')}\frac{\delta\phi_{s}(r')}{\delta v_{S}(r)}\underbrace{\frac{\delta v_{S}(r'')}{\delta\rho(r)}}_{\equiv X_{S}^{-1}(r,r')}+c.c.\bigg]

where $X\_\{S\}^\{-1\}(r,r\text{'})$ is defined the inverse static Kohn-Sham (KS) response function.{{Citation needed|date=March 2025|reason=You need to include one of the standard sources}}

The KS orbital-dependent exact exchange energy (EXX) is given in Chemist's notation as

$E\_\{x\}[\backslash \{\backslash phi\_\{i\}\backslash \}]=-\backslash frac\{1\}\{2\}\backslash sum\_\{i\}\backslash sum\_\{j\}(ij|ji)\backslash equiv\; -\backslash frac\{1\}\{2\}\backslash sum\_\{i\}\backslash sum\_\{j\}\backslash iint\; drdr\text{'}\backslash frac\{\backslash phi\_\{i\}^\{\backslash dagger\}(r)\backslash phi\_\{j\}(r)\backslash phi\_\{j\}^\{\backslash dagger\}(r\text{'})\backslash phi\_\{i\}(r\text{'})\}$

where $r,r\text{'}$ denote electronic coordinates, $\backslash dagger$ the hermitian conjugate.The static Kohn-Sham (KS) response function is given as

{{NumBlk|:|$X\_\{S\}(r,r\text{'})\backslash equiv\backslash frac\{\backslash delta\; \backslash rho(r)\}\{\backslash delta\; v\_\{S\}(r\text{'})\}=\backslash sum\_\{i\}\backslash sum\_\{a\}\backslash bigg[\backslash frac\{\backslash phi\_\{i\}^\{\backslash dagger\}(r)\backslash phi\_\{a\}(r)\backslash phi\_\{a\}^\{\backslash dagger\}(r\text{'})\backslash phi\_\{i\}(r\text{'})\}\{\backslash varepsilon\_\{i\}-\backslash varepsilon\_\{a\}\}+c.c.\backslash bigg]$|{{EquationRef|2}}}}

where the indices $i$ denote occupied and $a$ unoccupied KS orbitals, $c.c.$ the complex conjugate. the right hand side (r.h.s.) of the OEP equation is

{{NumBlk|:|$t(r)=\backslash frac\{\backslash delta\; E\_\{x\}[\backslash \{\backslash phi\_\{i\}\backslash \}]\}\{\backslash delta\; v\_\{S\}(r)\}=\backslash sum\_\{i\}\backslash sum\_\{a\}\backslash bigg[\backslash frac\{\backslash phi\_\{i\}^\{\backslash dagger\}(r)\backslash phi\_\{a\}(r)\backslash langle\backslash phi\_\{a\}|\backslash hat\{v\}\_\{x\}^\{\backslash text\{NL\}\}|\backslash phi\_\{i\}\backslash rangle\}\{\backslash varepsilon\_\{i\}-\backslash varepsilon\_\{a\}\}+c.c.\backslash bigg]$|{{EquationRef|3}}}}

where $\backslash hat\{v\}\_\{x\}^\{\backslash text\{NL\}\}$ is the nonlocal exchange operator from Hartree-Fock (HF) theory but evaluated with KS orbitals stemming from the functional derivative $\backslash delta\; E\_\{xc\}[\backslash \{\backslash phi\_\{i\}\backslash \}]/\backslash delta\backslash phi\_\{i\}(r\text{'})$. Lastly note that the following functional derivative is given by first order static perturbation theory exactly

$\backslash frac\{\backslash delta\backslash phi\_\{s\}(r\text{'})\}\{\backslash delta\; v\_\{S\}(r\text{'}\text{'})\}=\backslash phi\_\{i\}(r\text{'})\backslash underbrace\{\backslash sum\_\{t,\; t\backslash neq\; i\}\backslash frac\{\backslash phi\_\{t\}^\{\backslash dagger\}(r\text{'})\backslash phi\_\{t\}(r)\}\{\backslash varepsilon\_\{i\}-\backslash varepsilon\_\{t\}\}\}\_\{G(r,r\text{'})\}$

which is a Green's function. Combining eqs. (1), (2) and (3) leads to the Optimized Effective Potential (OEP) Integral equation

$\backslash int\; dr\text{'}v\_\{x\}(r\text{'})X\_\{S\}(r,r\text{'})=t(r)$

Usually the exchange potential is expanded in an auxiliary basis set (RI basis) $\backslash \{f\_\{\backslash mu\}\backslash \}$ as $v\_\{x\}(r)=\backslash sum\_\{\backslash nu\}v\_\{x,\backslash nu\}f\_\{\backslash nu\}(r)$ together with the regular orbital basis $\backslash \{\backslash chi\_\{\backslash lambda\}\backslash \}$ requiring the so-called 3-index integrals of the form $(f\_\{\backslash nu\}|\backslash chi\_\{\backslash lambda\}\backslash chi\_\{\backslash kappa\})$ as the linear algebra problem

$\backslash textbf\; X\_\{\backslash text\{S\}\}\backslash textbf\; v\_\{\backslash text\{x\}\}=\backslash textbf\; t$

It shall be noted, that many OEP codes suffer from numerical issues.{{cite journal | author= Trushin, E. and Görling, A.|journal= The Journal of Chemical Physics| title=Numerically stable optimized effective potential method with standard Gaussian basis sets| volume=155| pages=054109| year=2021| doi=10.1063/5.0056431}} There are two main causes. The first is, that the Hohenberg-Kohn theorem is violated since for practical reasons a finite basis set is used, the second being that different spatial regions of potentials have different influence on the optimized energy leading e.g. to oscillations in the convergence from poor conditioning.

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Category:Density functional theory

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