Interpretation of electronic band structure diagram

Question

I want to understand the electronic band structure diagram of the following image, corresponding to $\text{MoS}_2$ (TMD):

I read about DFT (density functional theory).

DFT is based on solving the Schrodinger equation for a set of atoms. Through the Born Oppenheimer approach, it is possible to decouple the nuclear and electronic wave functions. We just then need to solve the Schrodinger equation for electrons. Using the two Hohenberg and Kohn theorems, we obtain the Kohn Scham equations: $$\bigg[-\frac{\hbar}{2m}\nabla^2+V(r)+V_H(r)+V_{XC}(r)\bigg]\psi_i(r)=\epsilon_i(r)\psi_i$$

Computationally the problem is solved as follows:

-Define n[r] (random)

-Solve Kohn Scham equation and find $\psi(r)$

-Calculate new n(r): $n(r)=2\sum\psi^{*}(r)\psi(r)$

-If n(r) calculated is equal to the original, the program will stop. If they're different, the program recalculate the Kohn Scham equation with n(r) calculated previously.

According to Hohenberg and Kohn's theorem 1, each electronic density uniquely corresponds to an energy in the ground state, therefore, once the program ends, knowing the electronic density, we know the energy in the ground state.

I don't understand how DFT relates to the various lines obtained from the graph and I don't understand what DFT is related to orbitals. How was the graph obtained based on the DFT?

Answer 1

$\epsilon_i$ in the Kohn-Sham equation actually has two indexes: $\epsilon_{nk}$, where $n$ and $k$ respectively, represent, the band and $k$ point indexes. So, for each point in a given line in that band-structure diagram you actually solve a Kohn-Sham equation for $\epsilon_{nk}$.

So, the vertical (y) axis represent $\epsilon$, whereas the horizontal (x) axis represents $k$ and different lines represent the bands $n$.