Hohenberg-Kohn theorems

I assume you know nothing about the Hohenberg-Kohn theorems. Here is the link for the paper that form the basis for the Kohn-Sham Density functional theory. Now, you download the paper and find skimming through the paper. You may not understand anything. Cool! It is not just you.

Here is a journey to understand Hohenberg-Kohn theorems.

The title is "Inhomogeneous electron gas". What we understand from this? We know electrons. What is electron gas? Collection of large number of electrons which behave like free electrons. Right?

Gas is defined as follow:
"a substance or matter in a state in which it will expand freely to fill the whole of a container, having no fixed shape (unlike a solid) and no fixed volume (unlike a liquid)."

Now, we may consider such gas which consist of electrons. However, materials have have nucleus which are positively charged and attract electrons and forms some weired kind of patterns/orbits/etc in the real materials. So, the electrons are not homogeneous inside a material and they must be in-homogeneous. Thus the title give some intuition on the subject of the paper.

Let us jump on the the formalism section.

First the authors start with the definition of the Hamiltonian and how it is actually written in different contributions added together.

H = T + U + V

Here, T, U and V stands for kinetic energy, electron-electron repulsion energy, and V is the electron-nucleus attractive potential energy.