Reddit reviews Density Functional Theory: A Practical Introduction

This is probably one of the best DFT books I've ever read. It's such a good introduction.

If you heavily constrain the system, you can get an analytical solution (usually Particle Innabox is the first one students see), or if you have an extremely simple system like hydrogen or a highly ionized atom (read: only one nucleus and one electron).

But we've known about those for years and they're all fairly trivial compared to what theoretical chemists are looking for. Ultimately, what we want to calculate is where the electrons are in a system, because that gives us a lot of information about the system (whether it's a small molecule, a big molecule, a chunk of metal, etc) and its properties.

You don't directly solve the schrodinger equation for any system of practical interest. Instead, one of the more popular methods is to use a set of methods lumped under "Density-Functional Theory," which is more or less trying to solve for a representation of the electron density rather than individual electrons. There's also a few other, older methods out there like Hartree-Fock where the assumption is that there is some single wavefunction that can represent all electrons in a system, but as a result the method can't account for electron-electron interactions. There's also newer methods out there called Post-Hartree-Fock where they try to take into account some electron-electron interactions (called electron correlation). I'm not as familiar with them as I am with DFT, but I know they tend to be more expensive to run, but also tend to be more accurate than DFT.

If you're interested in DFT, here's a really good book to get started on it. It's intended more as an introduction for newcomers, and those who want a working knowledge of it, but it also has a bunch of book and paper recommendations in it, as well as a bunch of analogies to describe how it all works.