The Schrödinger equation is undoubtedly the most important equation in molecular and solid state physics. Unfortunately, for many cases of interest, the Schrödinger equation is intractable. An intractable problem is one that is well defined and where a solution method is known but where the resources required to solve the problem are impractically large.

For instance, consider an atom with $N$ electrons and a nucleus with charge $Ze$ located at $\vec{r}=0$. The Hamiltonian is,

\begin{equation} \label{eq:many} H= - \sum \limits_i^N \frac{\hbar^2}{2m}\nabla_i^2 - \sum \limits_i^N \frac{Ze^2}{4\pi\epsilon_0 |\vec{r}_i|} + \sum \limits_{i\lt j}\frac{e^2}{4\pi\epsilon_0 |\vec{r}_i-\vec{r}_j| }. \end{equation}

The first sum represents the kinetic energies of the electrons, the second sum is the electron-nucleus interactions, and the third term is the electron-electron interactions.

In principle, you could search for the ground state of this system by trying different wavefunctions $\Psi$ and seeing which gives the lowest energy.

\begin{equation} E= \frac{\langle \Psi |H|\Psi\rangle}{\langle \Psi|\Psi\rangle}. \end{equation}

For the electrons of an atom, we expect the wavefunction to go to zero far from the nucleus so we can specify a wavefunction by providing its value on a finite grid of closely spaced points centered around the nucleus. We can interpolate the values of $\Psi$ between these point if we need to. For one, electron, if there are 100 grid points along the $x-$direction, 100 along the $y-$direction, and 100 along the $z-$direction, $\Psi$ would be specified by $100^3 = 10^6$ complex numbers. For $N$ electrons, the wavefunction depends on $N$ position vectors, $\Psi(\vec{r}_1,\vec{r}_2,\cdots\vec{r}_N)$. If we use 100 grid points along each coordinate axis $(x_1,\,y_1,\,z_1,\,x_2,\,y_2,\,z_2,\,\cdots\,x_N,\,y_N,\,z_N)$, we will need $100^{3N}=10^{6N}$ complex numbers to specify $\Psi$.

There are about $10^{80}$ atoms in the observable universe, so if there are more than 13 electrons in the atom, we will need to specify more complex numbers than there are atoms in the universe. Clearly, there are not enough computer resources to do this. Some approximation always has to be made when working with many-particle wavefunctions. These approximations always have limitations.