1-2.The theorems of H-K

08-17

來自專欄 A note on DFT

First theorem: The ground-state expectation value of any observable $hat{O}$ is a unique(one-to-one) functional of the exact ground-state electron density:

$egin{eqnarray} O[ ho]equiv<Psi|hat{O}|Psi> end{eqnarray}$

Second theorem: $H_{V_{ext}}[ ho]$ reaches its minimal value (equal to the ground-state total energy) for the ground-state density corresponding to $V_{ext}$ .

$egin{eqnarray} H_{ext}[ ho]&equiv&<Psi|hat{T}+hat{V}|Psi>+<Psi|hat{V}_{ext}|Psi> onumber\ &=&H_{HK}[ ho]+int dvec{r} ho(vec{r})V_{ext}(vec{r}) end{eqnarray}$

Note that Theyre only suitable for the ground-state density.