Optimized effective potential method

The Optimized effective potential method (OEP)^{[1] [2]} in Kohn-Sham (KS) density functional theory (DFT) ^{[3] [4]} is a method to determine the potentials as functional derivatives of the corresponding KS orbital-dependent energy density functionals. This can be in principle done for any arbitrary orbital-dependent functional, but is most common for exchange energy as the so called Exact exchange method (EXX)^{[5] [6]} , which will be considered here.

Originally the OEP method was developed more than 10 years prior to the work of Pierre Hohenberg ^[3] , Walter Kohn and Lu Jeu Sham ^[4] in 1953 by R. T. Sharp and G. K. Horton ^[7] in order to investigate, what happens to Hartree-Fock (HF) theory ^{[8] [9] [10] [11] [12]} when instead of the regular nonlocal exchange potential a local exchange potential is demanded. Much later since 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

where the index ${\displaystyle s}${\displaystyle s} denotes either occupied or unoccupied KS orbitals. The problem is that, although the xc energy is in principle due to the Hohenberg-Kohn (HK) theorem ^[3] a functional of the density its explicit dependence of the density is unknown (only known in the simple Local density approximation (LDA) ^[3] case), but rather its implicit dependence through the KS orbitals. That motivates the use of the chain rule

${\displaystyle v_{xc}(r)=\int dr'\sum _{s}{\bigg [}{\frac {\delta E_{xc}[\{\phi _{s}\}]}{\delta \phi _{s}(r')}}{\frac {\delta \phi _{s}(r')}{\delta \rho (r)}}+c.c.{\bigg ]}}${\displaystyle v_{xc}(r)=\int dr'\sum _{s}{\bigg [}{\frac {\delta E_{xc}[\{\phi _{s}\}]}{\delta \phi _{s}(r')}}{\frac {\delta \phi _{s}(r')}{\delta \rho (r)}}+c.c.{\bigg ]}}

But unfortunately the functional derivative ${\displaystyle \delta \phi _{s}/\delta \rho }${\displaystyle \delta \phi _{s}/\delta \rho }, despite ist existence, is also unknown. So one needs to invoke the chain rule once more, now with respect to the Kohn-Sham (KS) potential

${\displaystyle v_{xc}(r)=\iint dr'dr''\sum _{s}{\bigg [}{\frac {\delta E_{xc}[\{\phi _{s}\}]}{\delta \phi _{s}(r')}}{\frac {\delta \phi _{s}(r')}{\delta v_{S}(r'')}}\underbrace {\frac {\delta v_{S}(r'')}{\delta \rho (r)}} _{\equiv X_{S}^{-1}(r,r')}+c.c.{\bigg ]}}${\displaystyle v_{xc}(r)=\iint dr'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 ${\displaystyle X_{S}^{-1}(r,r')}${\displaystyle X_{S}^{-1}(r,r')} is by definition the inverse static Kohn-Sham (KS) response function.

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

${\displaystyle E_{x}[\{\phi _{s}\}]=-{\frac {1}{2}}\sum _{i}\sum _{j}(ij|ji)}${\displaystyle E_{x}[\{\phi _{s}\}]=-{\frac {1}{2}}\sum _{i}\sum _{j}(ij|ji)}

The static Kohn-Sham (KS) response function is given as

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

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

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

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

${\displaystyle \int dr'v_{x}(r')X_{S}(r,r')=t(r)}${\displaystyle \int dr'v_{x}(r')X_{S}(r,r')=t(r)}

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

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

Lastly it shall be noted, that many OEP codes suffer from numerical issues.

• Density functional theory
• Computational chemistry
• Quantum chemistry
• Theoretical chemistry

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