Optimized effective potential method

Background via chain rule

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 ${\displaystyle \rho (r)}$

where the index ${\displaystyle s}$ 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

${\displaystyle v_{xc}(r)=\int d{r}^{\prime }\sum _s[\frac{\delta E_{xc}[\{{\varphi }_s\}]}{\delta {\varphi }_s(r^{\prime })}\frac{\delta {\varphi }_s(r^{\prime })}{\delta \rho (r)}+c.c.]}$

Unfortunately the functional derivative ${\displaystyle \delta {\varphi }_s/\delta \rho }$, 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 ${\displaystyle v_S(r)}$

${\displaystyle v_{xc}(r)=\iint d{r}^{\prime }d{r}^{″}\sum _s[\frac{\delta E_{xc}[\{{\varphi }_s\}]}{\delta {\varphi }_s(r^{\prime })}\frac{\delta {\varphi }_s(r^{\prime })}{\delta v_S(r^{″})}\underset{\equiv X_S^{-1}(r,r^{\prime })}{\underbrace{\frac{\delta v_S(r^{″})}{\delta \rho (r)}}}+c.c.]}$

where ${\displaystyle X_S^{-1}(r,r^{\prime })}$ is defined the inverse static Kohn-Sham (KS) response function.

Formalism

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

${\displaystyle E_x[\{{\varphi }_i\}]=-\frac{1}{2}\sum _i\sum _j(ij|ji)\equiv -\frac{1}{2}\sum _i\sum _j\iint drd{r}^{\prime }\frac{{\varphi }_i^{†}(r){\varphi }_j(r){\varphi }_j^{†}(r^{\prime }){\varphi }_i(r^{\prime })}{|r-r^{\prime }|}}$

where ${\displaystyle r,r^{\prime }}$ denote electronic coordinates, ${\displaystyle †}$ the hermitian conjugate.The static Kohn-Sham (KS) response function is given as

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

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

${\displaystyle \frac{\delta {\varphi }_s(r^{\prime })}{\delta v_S(r^{″})}={\varphi }_i(r^{\prime })\underset{G(r,r^{\prime })}{\underbrace{\sum _{t,t\ne i}\frac{{\varphi }_t^{†}(r^{\prime }){\varphi }_t(r)}{{\epsilon }_i-{\epsilon }_t}}}}$

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

${\displaystyle \int d{r}^{\prime }{v}_x(r^{\prime })X_S(r,r^{\prime })=t(r)}$

Implementation with a basis set

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

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

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