The Hamiltonian that describes any molecule or solid is,

\[ \begin{equation} \label{eq:htotal} H= -\sum\limits_i \frac{\hbar^2}{2m_e}\nabla^2_i -\sum\limits_a \frac{\hbar^2}{2m_a}\nabla^2_a -\sum\limits_{a,i} \frac{Z_ae^2}{4\pi\epsilon_0 |\vec{r}_i-\vec{r}_a|}+\sum\limits_{i< j} \frac{e^2}{4\pi\epsilon_0 |\vec{r}_i-\vec{r}_j|}+\sum\limits_{a< b} \frac{Z_aZ_be^2}{4\pi\epsilon_0 |\vec{r}_a-\vec{r}_b|} . \end{equation} \]

The first sum describes the kinetic energy of the electrons. The electrons are labeled with the subscript $i$. The second sum describes the kinetic energy of all of the atomic nuclei. The atoms are labeled with the subscript $a$. The third sum describes the attractive Coulomb interaction between the the positively charged nuclei and the negatively charge electrons. $Z_a$ is the atomic number (the number of protons) of nucleus $a$. The fourth sum describes the repulsive electron-electron interactions. Notice the plus sign before the sum for repulsive interactions. The fifth sum describes the repulsive nuclei-nuclei interactions.

This Hamiltonian neglects some small details like the spin-orbit interaction and relativistic effects. These effects will be ignored in this discussion. If they are relevant, they could be included as perturbations later. Any observable quantity of any solid can also be calculated from this Hamiltonian. It turns out, however, that solving the Schrödinger equation associated with this Hamiltonian is usually terribly difficult. Using Born-Oppenheimer approximation and making some assumptions about the electron-electron interactions, the many-electron wavefunction can often be written as an antisymmetrized product of single-electron wave functions that solve the Schrödinger equation,

\[ \begin{equation} \label{eq:schr} - \frac{\hbar^2}{2m_e}\nabla^2\psi(\vec{r}) +V(\vec{r})\psi(\vec{r}) = E\psi(\vec{r}), \end{equation} \]

where $V(\vec{r})$ is a periodic potential with the periodicity of the crystal.

There are many codes that can be used to calculate the electronic band structure of three-dimensional crystals. Some of them are found in the Wikipedia list of quantum chemistry and solid-state physics software. There are free software packages as well as commercial packages. Commonly used packages to calculate band structures are VASP, Quantum Espresso, FHI-AIMS, and ABINIT. These codes are used to calculate the electronic, magnetic, optical, mechanical, and thermodynamic properties of materials based on the arrangement of the atoms in the crystal. All of them have some limitations. It is advisable to follow a course on band structure calculations or to work with someone experienced with band structure calculations before you start using them.

Many band structure calculations have already been performed. A simple internet search of the form 'electronic band structure <crystal name>' will probably retrieve the band structure of the crystal you are interested in. The Materials Project is a database with many electron dispersion relations and densities of states.

A common strategy for calculating band structures is to guess a form for the wave function that contains some adjustable parameters. This form for the wavefunction is inserted into the Schrödinger equation, which results in some algebraic equations involving the adjustable parameters and the energy. The combination of parameters that minimizes the energy is the best solution for the wavefunction that is possible for the form of the initial guess. To illustrate how this works, two band structure methods, the plane wave method and tight binding, will be described.