1-3.The Kohn-Sham equations

08-17

來自專欄 A note on DFT

Rewrite the $H_{HK}$ in:

$egin{eqnarray} H_{HK}[ ho]&=&T_0+V_H+(V-V_H)+(+T-T_0) onumber\ &=&T_0+V_H+V_x+V_c onumber\ &=&T_0+V_H+V_{xc}\ H_{ext}[ ho]&=&T_0[ ho]+V_H[ ho]+V_{xc}[ ho]+V_{ext}[ ho] end{eqnarray}$

Using the Kohn-Sham Hamiltonian, we rewrite the $H_{ext}$ as $H_{KS}$

$egin{eqnarray} hat{H}_{KS}&equiv&hat{T}_0+hat{V}_H+hat{V}_{xc}+hat{V}_{ext}\ hat{T}_0&equiv&-frac{hbar^2}{2m} abla^2\ hat{V}_H&equiv&frac{e^2}{4piepsilon_0}int dvec{r}frac{ ho(vec{r})}{|vec{r}-vec{r}|}\ hat{V}_{xc}&equiv&frac{delta V_{xc}[ ho]}{delta ho} end{eqnarray}$

The exact ground-state density of an N-electron system is

$egin{eqnarray} ho(vec{r})&equiv&sum^N_{i=1}phi_i(vec{r})^*phi_i(vec{r}) end{eqnarray}$

Where the single-particle wave functions $phi_i$ are the N lowest-energy solutions of the Kohn-Sham equation

$egin{eqnarray} hat{H}_{KS}phi_i&=&epsilon_iphi_i end{eqnarray}$

Be aware that the single-particle wave functions $phi_i$ are not the wave functions of electrons! They describe mathematical quasi-particles, without a direct physical meaning. Only the ouver-all-density of these quasi-particles is guaranteed to be equal to the true electron density. Also the single-particle energies $epsilon_i$ are not single-electron energies.