The Krieger-Li-Iafrate Approximation
The optimized effective potential (OEP) is a natural connection between local density-functional theory and many-body perturbation theory, which is derived from the Sham-Schlüter equation. Converging the full set of OEP equations is quite a challenge. Therefore an approximation was made in order to reduce the integral equations to an analytically solvable one via a dominant orbital approximation, called The Krieger-Li-Iafrate (KLI) approximation.

$$\sum_{i,j\neq i}\frac{\phi_j^*(\mathbf{r})\phi_i^*(\mathbf{r})}{\epsilon_i-\epsilon_r}\left(f_i \langle \phi_i|v_x|\phi_j\rangle - \sum_{k, \alpha}\frac{d_{ik}^{\alpha}d_{kj}^{\alpha}}{2}(\epsilon_i-\epsilon_k)\left[\frac{f_i(1-f_k)}{\epsilon_i-\epsilon_k+\epsilon_{\alpha}}+\frac{f_k(1-f_i)}{\epsilon_i-\epsilon_k+\epsilon_{\alpha}}\right]\right) + c.c.$$

$$=\frac{1}{2}\sum_{\alpha,i,k}\omega_a\frac{\left|d_{ik}^{\alpha}\right|^2 f_i(1-f_k)}{(\epsilon_i-\epsilon_k-\epsilon_{\alpha})^2}\left(\left|\phi_k(\mathbf{r})\right|^2-\left|\phi_i(\mathbf{r})\right|^2\right)$$

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