## Kohn-Sham approach To really solve this functional problem, a special approach is required, especially under the condition that we don't even know the $V_{ext}(\vec{r})$. Kohn-Sham method is suggested to solve this problem in a very clear way. They define the KS one electron operator as $\hat{h}_{i}^{KS} = -\frac{1}{2} \nabla^{2}_{i} - \sum_{k}^{nu} \frac{Z_{k}}{|r_{i}-r_{k}|} + \int \frac{\rho(r')}{|r_{i}-r'|}\mathrm{d}r +V_{xc}$ This $V_{xc}$ is the exchange-correlation energy. The first three terms are typically known interacting potentials, namely kinetic energy, nuclei-electron and electron-electron Columbic interactions. But the forth one, $V_{xc}$, [[Evaluating exchange-correlation energy|exchange-correlation energy]], is the bridge between single-electron problem and many-electron one. First, we create a non-interacting system, then we assume the ground state $\rho$ being the same for both the real and the noninteracting system. So theoretically, the properties of these two systems are fully determined and being the **same**! Therefore, although we do not know the real $V_{ext}$. But as long as $V_{xc}$ is accurate enough, we would solve exactly the same results as dealing with many-electron problems. **So one may see, the ultimate difficulty becomes finding a proper exchange-correlation energy $V_{xc}$ approximation with respect to $\rho$.** >[!Note] >Sometimes, the Hamiltonian for Kohn-Sham method can be written as >$\hat{H}_{KS} = -\frac{1}{2} \nabla^{2} + V_{KS} = -\frac{1}{2} \nabla^{2} +V_{ext} +V_{Hartree} +V_{xc}$ >Using $V_{Hartree}$ is just to identify the exchange E and e-e Coulombic interaction. To hear we can make a short summary The general procedure for solving DFT is shown in the following diagram. ![[Drawing 2023-10-10 20.01.00.excalidraw.svg]] >[!Info] >See more on wiki: https://en.wikipedia.org/wiki/Kohn%E2%80%93Sham_equations