Density Functional Theory

The goal of density functional theory (DFT) is to approximate the ground state energy $E_g={\text{min}}_{\psi }⟨\psi |H|\psi ⟩$. We do this through an iterative algorithm

Minimize

$${\text{min}}_{\psi \to n(r)}⟨\psi |H|\psi ⟩={\text{min}}_{\psi \to n(r)}⟨\psi |T+V_{e-e}|\psi ⟩+\int d^{3}r{V}_{ext}(r)n(r)$$

Kohn-Sham

• Map many-body to independent-particle problem

$$V_{eff}(r)=V_{ext}(r)+\int \frac{n(r^{\prime })}{|r-r^{\prime }|}d{r}^{\prime }+V_{xc}$$

• Local density approximation (LDA) $n(r)={\sum }_{\nu =1}^N|{\psi }_{\nu }{|}^2$where ${\psi }_{\nu }$ are eigenstates of Hamiltonian with $V_{eff}$ as potential
• Successful for many s- and p-electron materials

Correlated materials

• Can DMFT capture metal-Mott insulator transition? Yes
• Mix DFT and Many Body approach
• Kondo singlet lowers energy.
• Spectral density evolves as $U/W$ in one band Hubbard model