Tuesday, February 02, 2016

What is density functional theory? part 1.

In previous posts, I've tried to introduce the idea that there can be "holistic" approaches to solving physics problems, and I've attempted to give a lay explanation of what a functional is (short version: a functional is a function of a function - it chews on a whole function and spits out a number.).  Now I want to talk about density functional theory, an incredibly valuable and useful scientific advance ("easily the most heavily cited concept in the physical sciences"), yet one that is basically invisible to the general public.

Let me try an analogy.  You're trying to arrange the seating for a big banquet, and there are a bunch of constraints:  Alice wants very much to be close to the kitchen.  Bob also wants to be close to the kitchen.  However, Alice and Bob both want to be as far from all other people as possible.  Etc. Chairs can't be on top of each other, but you still need to accommodate the full guest list.  In the end you are going to care about the answers to certain questions:  How hard would it be to push two chairs closer to each other? If one person left, how much would all the chairs need to be rearranged to keep everyone maximally comfortable?    You could imagine solving this problem by brute force - write down all the constraints and try satisfying them one person at a time, though every person you add might mean rearranging all the previously seated people.  You could also imagine solving this by some trial-and-error method, where you guess an initial arrangement, and make adjustments to check and see if you've improved how well you satisfy everyone.  However, it doesn't look like there's any clear, immediate strategy for figuring this out and answering the relevant questions.

The analogy of DFT here would be three statements.  First, you'd probably be pretty surprised if I told you that if I gave you the final seating positions of the people in the room, that would completely specify and nail down the answer to any of those questions up there that you could ask about the room.1  Second, there is a math procedure (a functional that depends on the positions of all of the people in the room that can be minimized) to find that unique seating chart.2  Third, even more amazingly, there is some mock-up of the situation where we don't have to worry about the people-people interactions directly, yet (minimizing a functional of the positions of the non-interacting people) would still give us the full seating chart, and therefore let us answer all the questions.3

For a more physicsy example:  Suppose you want to figure out the electronic properties of some system.  In something like hydrogen gas, H2, maybe we want to know where the electrons are, how far apart the atoms like to sit, and how much energy it takes to kick out an electron - these are important things to know if you are a chemist and want to understand chemical reactions, for example.  Conceptually, this is easy:  In principle we know the mathematical rules that describe electrons, so we should be able to write down the relevant equations, solve them (perhaps with a computer if we can't find nice analytical solutions), and we're done.  In this case, the equation of interest is the time-independent form of the Schroedinger equation.  There are two electrons in there, one coming from each hydrogen atom.  One tricky wrinkle is that the two electrons don't just feel an attraction to the protons, but they also repel each other - that makes this an "interacting electron" problem.  A second tricky wrinkle is that the electrons are fermions.  If we imagine swapping (the quantum numbers associated with) two electrons, we have to pick up a minus sign in the math representation of their quantum state.  We do know how to solve this problem (two interacting electrons plus two much heavier protons) numerically to a high degree of accuracy.  Doing this kind of direct solution gets prohibitively difficult, however, as the number of electrons increases.

So what do we do?  DFT tells us:
1If you actually knew the total electron density as a function of position, \(n(\mathbf{r})\), that would completely determine the properties of the electronic ground state.  This is the first Hohenberg-Kohn theorem.

2There is a unique functional \(E[n(\mathbf{r})]\) for a given system that, when minimized, will give you the correct density \(n(\mathbf{r})\).  This is the second Hohenberg-Kohn theorem.

3You can set up a system where, with the right functional, you can solve a problem involving noninteracting electrons that will give you the true density \(n(\mathbf{r})\).  That's the Kohn-Sham approach, which has actually made this kind of problem solving practical.

The observations by Kohn and Hohenberg are very deep.  Somehow just the electronic density encodes a whole lot more information than you might think, especially if you've had homework experience trying to solve many-body quantum mechanics problems.  The electronic density somehow contains complete information about all the properties of the lowest energy many-electron state.  (In quantum language, knowing the density everywhere in principle specifies the expectation value of any operator you could apply to the ground state.)

The advance by Kohn and Sham is truly great - it describes an actual procedure that you can carry out to really calculate those ground state properties.  The Kohn-Sham approach and its refinements have created the modern field of "quantum chemistry".

More soon....


Anonymous said...

I admit I don't know anything about this but it seems the basic reasoning is as follows:

Reduce the many-electron *interacting* problem to a non-interacting 1-electron system.

I assume this is an abuse of mathematics because by definition correlation is everything that cannot by covered at the 1-electron level.

Anyways, at the practical level, "success" of DFT is made possible by incorporating empirical methods and benchmarking against wave function based calculations. So why is it so fantastic if it is not reliable for even the simplest systems and the alphabet zoo of functionals makes it so-called simplicity a farce?

Its as if condensed matter has its own scandal going similar to string theory. DFT, only game in town...the record citations prove it.

Douglas Natelson said...

Anon, the basic reasoning is: You can prove rigorously that the density as a fn of position alone tells you everything about the ground state (! very surprising, on its face !), and there is a procedure involving non-interacting electrons that can get you arbitrarily close to the true density a fn of position. In principle, if you knew how to write down the exchange-correlation piece of the functional and do the math in a computationally efficient way, you could get exact answers. That was Russ McKenzie's point in the post that I linked in my first writing about this.

I'm not arguing that DFT is flawless, but you are giving it very short shrift by implying that it's some epicycle-like farce or that it's widespread utility is a scandal. It's also not the only game in town, and it does actually make quantitative, testable predictions. (which I will discuss further shortly)