What problem are we trying to solve?

We want to calculate the electronic structure of real materials and their physical properties by ab-initio method. A material is composed of atoms, while atoms themselves consist of electrons, protons, and neutrons. The dynamics of macroscopic particles such as electrons are governed by the laws of quantum mechanics. Quantum particles are described by the wave function.

$$\lambda \cdot p = h$$

where $h$ is the Plank constant. There are four fundamental forces of nature: electromagnetic, strong nuclear, weak nuclear, and gravitational influencing the influencing the dynamics of any particle. However, when it comes to understanding the behavior of materials, we only need to consider the electromagnetic force. The Wavefunction of an electron in a potential filed $(V)$ is calculated by solving the Schrödinger equation:

$$-\frac{\hbar^2}{2m} \nabla^2 \Psi(\textbf{r}, t) + V(\textbf{r}, t) = i\hbar \frac{\partial\Psi(\textbf{r}, t)}{\partial t}$$

Fortunately, in most practical purposes, the potential field is not a function of time $(t)$, or even if it is a function of time, they changes relatively slowly compared to the dynamics we are interested in. For example, the electrons inside a material are subjected to the Coulomb field of the nucleus. The nucleus is heavy and their motion is much slower than the motion of the electrons. In such situation, we can separate out the spatial and temporal parts of the wave function:

$$\Psi(\textbf{r}, t) = \psi(\textbf{r}) f(t)$$

That reduces our task to solving only time independent Schrödinger equation:

$$\left[-\frac{\hbar^2 \nabla^2}{2m} + v(\textbf{r})\right] \psi(\textbf{r}) = \epsilon \psi(\textbf{r})$$

Once we have the wavefunction, we can calculate the observables by taking the expectation values.

$$\braket{\psi_i | \psi_j} = \delta_{ij}$$ $$\braket{\psi_i | \hat{\mathcal{H}} | \psi_i} = \epsilon_i$$

However, the challenge is to solve the Schrödinger equation as a real physical system is consists of a large number of atoms. The Schrödinger equation becomes coupled many-body equation.

$$\left[-\frac{\hbar}{2m} \sum_{i=1}^N \nabla_i^2 + \sum_{i=1}^NV(\textbf{r}_i) + \sum_{i=1}^N \sum_{j<i}U(\textbf{r}_i, \textbf{r}_j)\right]\psi(\textbf{r}_1, \textbf{r}_2, ..., \textbf{r}_N) = E\psi(\textbf{r}_1, \textbf{r}_2, ..., \textbf{r}_N)$$

With today's available computing power, it is far from feasible to solve the actual electronic wavefunction of a condensed matter system, where $N$ is of the order of $10^{23}$.