## Begin of DFT, Hohenberg–Kohn theorems Since the [[Hartree-Fork method#^ed2c7d|HF method]] does not take correlation into account, a new method would be necessary if one wanna get better/more accurate results. What if one considers the eigenfunction problem as, inherently, a many-electron one? And that is density functional theory. The fundamental of DFT is the Hohenberg-Kohn theorems. 1. H-K existence theorem For any system of interacting particles in the external potential $V_{ext}(\vec{r})$, The potential is determined uniquely (except for the difference of a constant) by the ground state particle density $\rho_{o}(r)$. >[!Note] > Since the Hamiltonian is fully determined, except for a constant shift of $E$, it follows many-body wave function for all states, and determined. > ALL properties of the system are DETERMINED given only the GROUND STATE ENERGY. 2. H-K variational theorem An universal *functional* for the energy $E(\rho)$ in terms of the density $\rho(r)$ can be defined, valid for any external field $V_{ext}(\vec{r})$. For any particular $V_{ext}(\vec{r})$, the exact growth state energy of the system is a global minimum value of this functional and the density $\rho(r)$ that minimize the functional is the exact ground state $\rho_{0}(r)$. >[!Note] >The [variational method](https://en.wikipedia.org/wiki/Variational_method_(quantum_mechanics)) still valid. This two theorems show: Once $\rho$ is determined/known, everything else is known. So if we take $\rho$ as a *variable* (functional), then the answer/solution itself is fixed. Even we don't know the question, i.e., $V_{ext}(\vec{r})$. >[!Info] >See more on >https://en.wikipedia.org/wiki/Density_functional_theory >https://people.chem.ucsb.edu/metiu/horia/OldFiles/115C/KH_Ch4.pdf >https://www.tcm.phy.cam.ac.uk/~pdh1001/thesis/node17.html >Next chapter: [[Kohn-Sham approach]].