Density functional theory

In the last chapter, we developed methods by which to treat systems of single particles, i.e. systems that don't have any particle-particle interaction. As we saw, this is rather straightforward.

Luckily for us, it is possible to treat interactions between electrons with high precision, while at the same time only solving single-particle equations. This is due to powerful results by Hohenberg, Kohn and Sham; of these, Walter Kohn won the Nobel prize in 1998 "for his development of the density-functional theory"\footnote{The Nobel Prize in Chemistry 1998. NobelPrize.org. Nobel Media AB 2021. Fri. 5 Feb 2021. }. The most relevant academics publications are in footnotes\footnote{Hohenberg, P.; Kohn, W. (1964). "Inhomogeneous Electron Gas". Physical Review. 136 (3B): B864}\footnote{Kohn, W.; Sham, L. J. (1965). "Self-Consistent Equations Including Exchange and Correlation Effects".

The Hohenberg-Kohn theorem

The Hohenberg-Kohn theorem says the following:

What does it mean? It means that we don't have to solve for the very complicated many-body wave function, if we can come up with some way to calculate the density instead. Also, we know that we have found the correct density if that density minimizes our energy functional \(E[\rho]\), where \(\rho (x)\) is the electron density.

Now, let's quickly recap what a functional is. It is not, in fact, much more complicated than a function. A function is a rule \(f(x)\) which takes in some number \(x\) and gives back another number, \(f(x)\). For example, the function \(f(x)=x^2\) associates to every number \(x\) its square.

A functional is just a rule which takes a function and gives back some number. Here's an example of a functional:

\begin{align} L[f] = \int _{0}^1f(x)\mathrm{d}x. \end{align}

As we see, given a function f, this functional gives back a number associated with it. The energy functional we're concerned with in this book is a functional of electron density: that is, it's a rule that gives back the energy of the system for a given electron density. Best of all, if we find the density (that satisfies some obvious constraints, like having the correct number of electrons and being positive) which gets the minimum energy out of the functional, we know that's the correct density!

First, some preliminary remarks. The Hamiltonian for an interacting system is

\begin{align} \hat{H} = \hat{T}+\hat{V}_{ee} + \hat{V}_{ext}, \end{align}

where \(\hat{T}\) is the kinetic energy operator, \(\hat{V}_{ee}\) the electron-electron interaction and \(\hat{V}_{ext}\) the external potential (like the ones we considered in Chapter 1; these could be, for example, caused by the nuclei of the atoms).

Now, two different systems of \(N\) electrons can't differ by anything more than the external potential. That is because if we take a system of \(N\) electrons, in all possible configurations the form of \(\hat{T}\) and \(\hat{V}_{ee}\) remains the same, since they are sums over the same number of electrons. The actual amount of energy could of course be different, but the operators themselves are exactly the same. Hence, it is sufficient that our electron density \(\rho (x)\) determine the external potential \(\hat{V}_{ext}\) uniquely.

We'll call \(\hat{T}+\hat{V}_{ee} = F[\rho (x)]\) and calculate the energy functional as follows (according to the rules of quantum mechanics): \(\langle \psi |\hat{F} + \hat{V}_{ext}|\psi \rangle \). Now, according to the variational principle, the following is true:

\begin{align} \langle \psi _2 | \hat{H}_ 1 | \psi _2 \rangle > \langle \psi _1 | \hat{H}_ 1 | \psi _1 \rangle \end{align}

meaning this: If we take the correct wave function, \(\psi _1\), and calculate the energy of the Hamiltonian \(\hat{H}_1\), then that energy will be lower than the energy calculated with an incorrect wave function \(\psi _2\). Finally, we can write the external

potential \(\hat{V}_{ext}\) with the following notation: \begin{align} \hat{V}_{ext} = \int _V \rho (x) v_{ext}(x)\mathrm{d}x. \end{align}

So we write the energy with the help of a local potential \(v_{ext}\). This gives us enough to prove the Hohenberg-Kohn theorem, which turns out to be easy.

The proof

We suppose the contrary of the first part of the theorem: there are two ground state densities that give the same energy, so the density doesn't determine the energy uniquely after all.

For \( H_1 = \hat{F} + \hat{V}_{ext1}\), \(H_2 = \hat{F} + \hat{V}_{ext2}\), we get by the variational principle \begin{align} \langle \psi _1 | \hat{H}_1 | \psi _1 \rangle < \langle \psi _2 | \hat{H}_1 | \psi _2\rangle = \langle \psi _2 | \hat{H}_2|\psi _2 \rangle + \langle \psi _2 | \hat{H}_1 - \hat{H}_2|\psi _2 \rangle \\ = \langle \psi _2 | \hat{H}_2|\psi _2 \rangle + \int \rho _0(x) (v_{ext1}(x)-v_{ext2}(x))\mathrm{d}x \end{align}

In the second line we used the fact that the only difference between the two Hamiltonians was the external potential, and the density is the same, so we can collect these terms under the same integral. But now we can do the same procedure by interchanging the numbers 1 and 2 everywhere; that is, \(H_1\) becomes \(H_2\) and vice versa, and we get a similar equation. Combining these inequalities leads to the nonsensical

\begin{align} \langle \psi _1 | \hat{H}_1 | \psi _1 \rangle +\langle \psi _2 | \hat{H}_2|\psi _2 \rangle < \langle \psi _1 | \hat{H}_1 | \psi _1 \rangle + \langle \psi _2 | \hat{H}_2|\psi _2 \rangle \end{align}

which is an obvious contradiction. Hence, our assumption that there are two electron densities determining the ground state energy leads to a logical contradiction; the reverse, then, must be true.

The second part of the theorem is equally easy to prove. If the electron density determines the energy, then it also determines the correct wave function. Hence, the variational principle applies also to the ground state energy functional.

If you don't entirely understand this proof, there is no reason to worry: you can take the Hohenberg-Kohn theorems as given and then peruse a more detailed proof later. It is the logical basis for what we are doing, but won't really figure in to the calculations.

Next, we get to the Kohn-Sham equation, which we will also immediately implement on the computer.

Chapter 1.6Index Chapter 2.2