# The Krieger-Li-Iafrate Approximation

I am having trouble understanding what KLI approximation is. Could someone explain it to me in an intuitive/simple way?

• Could you tell a bit more about it? – Mithoron Oct 13 '17 at 16:00
• Maybe they can – ParaH2 Oct 15 '17 at 12:01
• @Hexacoordinate-C Fell free to summarize that poster in an answer:) – andselisk Oct 17 '17 at 14:30

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.

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

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

Source: Maybe they can see comments above.

• Hope this helps, I didn't want to go deeper because you asked for a simple and intuitive explanation. But I am aware that the formula isn't very simple :P – Matthew Oct 17 '17 at 15:54
• Seriously? Direct copy-paste from a poster found by another user without referencing neither the user, nor the poster, nor the original paper? And formula as an image when there is MathJax up and running? And what is what in this equation? How it can be used and when? – andselisk Oct 17 '17 at 15:55
• @andselisk better than nothing. I am just surprised that no one used and extracted the information from that poster, which I ended up reading all and having interest for it. There are not very good sources on the internet about this topic. So I tried to do my best without changing a lot the wording because it should stay as it is. But it would always be great to have an expert talk about this (which I highly doubt that there is). – Matthew Oct 17 '17 at 16:02
• So, you don't even want to edit this answer to improve it in a slightest? – andselisk Oct 17 '17 at 16:03
• I'm on it, don't worry. – Matthew Oct 17 '17 at 16:07