Tight-binding: aren't the electronic bands just the eigenvalues of the Hamiltonian? In the tight-binding model we choose a set of atomic orbitals $\{\phi_1(\textbf{r}), ..., \phi_N(\textbf{r})\}$ and estimate a transfer matrix and hopping matrix
$$
\begin{cases}
S_{ij}(\textbf{k}) = \langle \Phi_i(\textbf{k})\lvert\Phi_j(\textbf{k})\rangle\\
H_{ij}(\textbf{k}) = \langle \Phi_i(\textbf{k})\lvert \hat{H}\rvert\Phi_j(\textbf{k})\rangle
\end{cases}
$$
where
$$
\lvert \Phi_i(\textbf{r}, \textbf{k})\rangle= \sum_{\textbf{R}}e^{i\textbf{k}\textbf{R}}\lvert\phi_i(\textbf{r}- \textbf{R})\rangle
$$
We ask for eigenvectors of the Hamiltonian
$$\hat{H}\lvert \Psi_j(\textbf{r}, \textbf{k})\rangle = E_n(\textbf{k})\lvert \Psi_j(\textbf{r}, \textbf{k})\rangle $$
were we express them as linear combinations
$$
\lvert \Psi_j(\textbf{r}, \textbf{k})\rangle =  \sum_{j'}C_{jj'}\lvert \Phi_{j'}(\textbf{r}, \textbf{k})\rangle
$$
This leads to the secular system
$$
det\big[H-E_n(\textbf{k})S\big] = 0
$$
and by solving it we find the electronic bands and eigenvectors.
Now I am very confused because the problem seemed to be solvable by simply representing the Hamiltonian in the basis of the $\lvert \Phi_i(\textbf{r}, \textbf{k})\rangle$ vectors ($H_{ij}$) and diagonalising it.
So my question is: what's the point of doing the whole procedure (computing transfer and hopping matrix, solve secular system) if we can just calculate $H_{ij}$ and find its eigenvalues and eigenvectors? Aren't the electronic bands just the eigenvalues of the Hamiltonian?
 A: In solid state physics $H_{ij}$ is most often initially unknown thing. The goal of tight-binding model is to actually construct this hamiltonian in reasonable fashion. So theorists do something like that:

*

*Choose a set of basis functions (atomic orbitals in which electron wavefunctions are most expressed). For example, $p_z$ orbitals for carbon atoms in graphene and $d_{z^2}$, $d_{x^2-y^2}$, $d_{xz}$ orbitals for transition metal atoms in transition metal dichalcogenides.

*Then they introduce energy parameters responsible for overlapping orbitals. For atoms from one sublattice there can be one such a parameter (it can be named as exchange integral or hopping parameter) and for atoms from different sublattices there can be another set of hopping parameters.

The logic of step 2 can differ. You can see in some papers that energy of overlapping orbitals is computed for example using Slater-Koster approximation. Also symmetries of the system can say something about this hopping as a function of wavevector $k$, so there are approaches where using defined symmetries the number of tight-binding parameters is reduced. Then these parameters are often fitted to ab initio calculations (often DFT).
So finally yes, if you know $H_{ij}$ you can compute dispersion just solving eigenvalues problem. But tight-binding model is not about just that, it is about the method using which you can build your matrix which you will diagonalize.
