Monday, February 08, 2016

Brief news items

As I go on some travel, here are some news items that looked interesting to me:

  • Rumors are really heating up that LIGO has spotted gravity waves.  The details are similar to some things I'd heard, for what that's worth, though that may just mean that everyone is hearing the same rumors.  update:  Press conference coming (though they may just say that the expt is running well....)
  • The starship Enterprise is undergoing a refit.
  • This paper reports a photocatalytic approach involving asymmetric, oblong, core-shell semiconductor nanoparticles, plus a single Pt nanoparticle catalyst, that (under the right solution conditions) can give essentially 100% efficient hydrogen reduction - every photon goes toward producing hydrogen gas.   If the insights here can be combined with improved solution stability of appropriate nanoparticles, maybe there are ways forward for highly efficient water splitting or photo production of liquid fuels.
  • Quantum materials are like obscenity - hard to define, but you know it when you see it.

Sunday, February 07, 2016

What is density functional theory? part 3 - pitfalls and perils

As I've said, DFT proves that the electron density as a function of position contains basically all the information about the ground state (very cool and very non-obvious).  DFT has become of enormous practical use because one can use simple noninteracting electronic states plus the right functional (which unfortunately we can't write down in simple, easy-to-compute closed form, but we can choose various approximations) to find (a very good approximation to) the true, interacting density.

So, what's the problem, beyond the obvious issues of computing efficiency and the fact that we don't know how to write down an exact form for the exchange-correlation part of the functional (basically where all the bodies are buried)?  

Well, the noninteracting states that people like to use, the so-called Kohn-Sham orbitals, are seductive.  It's easy to think of them as if they are "real", meaning that it's very tempting to start using them to think about excited states and where the electrons "really" live in those states, even though technically there is no a priori reason that they should be valid except as a tool to find the ground state density.  This is discussed a bit in the comments here.  This isn't a completely crazy idea, in the sense that the Kohn-Sham states usually have the right symmetries and in molecules tend to agree well with chemistry ideas about where reactions tend to occur, etc.  However, there are no guarantees.

There are many approaches to do better (e.g., some statements that can be made about the lowest unoccupied orbital that let you determine not just the ground state energy but get a quantitative estimate of the gap to the lowest electronic excited state, and that has enabled very good computations of energy gaps in molecules and solids; time-dependent DFT, which looks at the general time-dependent electron density).  However, you have to be very careful.  Perhaps commenters will have some insights here.  

The bottom line:  DFT is intellectually deep, a boon to many practical calculations when implemented correctly, and so good at many things that the temptation is to treat it like a black box (especially as there are more and more simple-to-use commercial implementations) and assume it's good at everything.  It remains an impressive achievement with huge scientific impact, and unless there are major advances in other computational approaches, DFT and its relatives are likely the best bet for achieving the long-desired ability to do "materials by design".  

Thursday, February 04, 2016

What is density functional theory? part 2 - approximations

So, DFT contains a deep truth:   Somehow just the electronic density as a function of position within a system in its lowest energy state contains, latent within it, basically all of the information about that ground state.  This is the case even though you usually think that you should need to know the actual complex electronic wavefunction \(\Psi(\mathbf{r})\), and the density (\(\Psi^{*}\Psi\)) seems to throw away a bunch of information.

Moreover, thanks to Kohn and Sham, there is actually a procedure that lets you calculate things using a formalism where you can ignore electron-electron interactions and, in principle, get arbitrarily close to the real (including interaction corrections) density.  In practice, life is not so easy.  We don't actually know how to write down a readily computable form of the complete Kohn-Sham functional.  Some people have very clever ideas about trying to finesse this, but it's hard, especially since the true functional is actually nonlocal - it somehow depends on correlations between the density (and its spatial derivatives) at different positions.   In our seating chart analogy, we know that there's a procedure for finding the true optimal seating even without worrying about the interactions between people, but we don't know how to write it down nicely.  The correct procedure involves looking at whether each seat is empty or full, whether its neighboring seats are occupied, and even potentially the coincident occupation of groups of seats - this is what I mean by nonlocal.

Fig. from here.
We could try a simplifying local approximation, where we only care about whether a given chair is empty or full.  (If you try to approximate using a functional that depends only on the local density, you are doing LDA (the local density approximation)).  We could try to be a bit more sophisticated, and worry about whether a chair is occupied and how much the occupancy varies in different directions.  (If you try to incorporate the local density and its gradient, you are doing GGA (the generalized gradient approximation)).   There are other, more complicated procedures that add in additional nonlocal bits - if done properly, this is rigorous.  The real art in this business is understanding which approximations are best in which regimes, and how to compute things efficiently.

So how good can this be?  An example is shown in the figure (from a summer school talk by my friend Leeor Kronik).  The yellow points indicate (on both axes) the experimental values of the ionization energies for the various organic molecules shown.  The other symbols show different calculated ionization energies plotted vs. the experimental values.  A particular mathematical procedure with a clear theoretical justification (read the talk for details) that mixes in long-range and short-range contributions gives the points labeled with asterisks, which show very good agreement with the experiments.

Next time:  The conclusion, with pitfalls, perils, and general abuses of DFT.

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....

Wednesday, January 27, 2016

CalTech wins the whole internet - public outreach for quantum.

This makes my public outreach efforts look lame by comparison.  Well done!

Friday, January 15, 2016

What is a functional? Ex: the Action Principle

Working our way toward the biggest theory most people have never heard of, let's talk about functionals, using the non-rigorous language that physicists like and which annoys mathematicians.

Here's an analogy.   You want to drive from your house to the store.  There are many possible routes, and for each route we could come up with a single number that depends on the route - it could be the total distance traveled, or the total time it took to get from the house to the store, or it could be the total fuel consumed, or it could be the number of times you turned left minus the number of times you turned right.  We could take all your possible routes, and we could somehow process each possible route into a number.  The operation that chews on your route information and converts it to a number is a functional of your path from the house to the store.  (Why would you want to do this?  Well, perhaps you value your time, and you want to pick the route that has the least accumulated time.  Perhaps you value fuel costs, and you want to pick the route that has the least fuel consumption.  The point is, depending on what you care about, a functional can let you pick between alternatives, here the routes, that are described by a huge, effectively infinite number of variables.)

In the spirit of MTW, a function of a single variable is a machine that takes a number, chews on it, and spits out a number.   This could be \(y(x) = x^{2}\), for example.  A function of multiple variables is a machine that takes more than one number, chews on them, and spits out a number -- like \(y(x_{1}, x_{2}, x_{3}) = x_{1}^{2} + 3x_{2} - x_{3}\).  For this example, for any set of three numbers \( \{x_{1}, x_{2}, x_{3}\} \), you can compute a value of \(y\).  

A functional is the "continuum limit" of a function of multiple variables - it's a machine that takes an infinite number of numbers (!), chews on it, and spits out a single number.  We can cast our example of Fermat's principle of least time this way.  Suppose light starts out at point P, and we let it take some wild path like the one shown in the figure.  We're eventually going to have the light wind up at point Q.  How long does it take the light to get from P to the interface?  Well, that depends on how you think it goes.  If you knew all the intervening points \((x_{i},y_{i})\), you could compute the distance between successive points, and add up all the times.  The transit time \(t_{\mathrm{tot}}\) depends on the whole trajectory that the light takes from P to Q.  Instead of writing \(t_{\mathrm{tot}}(x_{1}, y_{1}, x_{2}, y_{2}, .....)\), we write \(t_{\mathrm{tot}}[x,y]\), where the square brackets indicate that this is a functional.  For any goofy trajectory we could draw from P to Q, we could compute \(t_{\mathrm{tot}}\).  Fermat's principle of least time says that the one actually taken by light is the one that gives the smallest value of \(t_{\mathrm{tot}}\).  Why does this work?  That's actually a very deep question, and I won't try to answer it now.

The Action Principle is the most famous example of showing that functionals can be incredibly useful in physics.  I'm going to do a simple 1d example involving mechanical motion of a particle, but everything I will say generalizes to much more complicated cases.  Suppose we have a particle that starts at some initial position position \(x_{\mathrm{i}}\) at some initial time \(t_{\mathrm{i}}\), and ends up at some final position \(x_{\mathrm{f}}\) at some final time \(t_{\mathrm{f}}\).  We want to know, how does the particle get there?  Which of the essentially infinite number of possible trajectories \(x(t)\) did the particle take?  (Note that by allowing any arbitrary path \(x(t)\), we're also basically permitting any arbitrary velocity as a function of time in there.)

The local way to answer this problem is to start with the particle at the initial location and time, and apply Newton's laws.  From its position find the force acting on the particle, use that force to find the acceleration, and take a little timestep forward, updating the particle's position and velocity.  Now repeat this.

The Action Principle is a global approach.  It says that there is some functional called the action, \(S[x(t)]\).  For any trajectory \(x(t)\), you can compute a number \(S\).  The trajectory that a classical particle takes is the one that starts and ends in the right places and times, and produces the minimum* value of \(S\).  The form of \(S\) contains all the physics.  (For a 1d particle obeying Newton's laws, the correct form for \(S\) is the integral as a function of time over the whole trajectory of (the kinetic energy minus the potential energy).)  This is one of the stranger things to learn when studying physics - with the right procedure for writing down and expression for \(S\), and the right procedure for minimizing it (techniques called variational calculus), it seems like the (global) Action Principle is nearly magical, giving you ways to solve problems that would seem hopelessly complex in traditional (local) approaches.   Why does this actually work?   Again, this is a deep question, and I'll revisit it some other time.  The fact that you can actually come up with a functional-based formalism does indicate that there is "hidden" structure to nature beyond what you might guess just from, e.g., Newton's laws.

To revisit the analogy:  If I told you that there was a way to predict how you would drive from home to the store based on a single number related to each possible route, you would realize:  (1) you don't necessarily have to know all the detailed rules of driving to find the preferred route, just how to calculate that number; and (2) there clearly is some deeper principle at work than just the rules of driving that picks out the route you take.

Next time, I'll finally get to the point about density functional theory.

*Technically, a maximum could also work here, but for many many cases, there is no maximum possible value of \(S\).

Sunday, January 10, 2016

"Local" vs "global" ways to solve physics problems

Inspired by a recent post of Ross McKenzie, I thought it would be fun to try to write a popularly accessible piece about the enormously successful, wholly remarkable  theory that most people have never heard of, density functional theory.

To get there will require a couple of steps.  First, it's important to appreciate that sometimes, thanks to the mathematical structure of the universe, it is possible to think about and solve physics problems with two seemingly very different approaches - call them "local" and "global".  In the local approach, we write down equations that describe the underlying problem in great detail, and by carefully working out their solution, we arrive at an answer.  In the global approach, we come at the problem from an overview perspective of considering possible solutions and figuring out which one is correct.

For example, let's think about a light ray propagating from point P (in air) to point Q (in water), as shown in the figure (courtesy wikipedia).  It turns out that light travels at a speed \(c/n\) in a medium, where \(c\) is the speed of light in vacuum, and \(n\) is the "index of refraction" that depends on the material and the frequency of the light.  (This is already short-hand for solving the complicated problem of electromagnetic radiation and its interactions with a material containing charges, something that Feynman wrote about elegantly in this book, based on these lectures.)  The "local" approach would be to write down the equations describing the electromagnetic light waves, and solve these, including the description of the air, the water, and their interface.  The result we would find is so simple and compact that we teach it to freshmen, Snell's Law:  \(n_{1}\sin(\theta_{1}) = n_{2}\sin(\theta_{2})\), where the angles are defined in the figure.

The "global" way to solve this problem (and again arrive at Snell's Law) was found by Fermat (yes, the one with the "last" theorem).  He didn't have the option of solving the microscopic equations governing the radiation, since he died two hundred years before Maxwell published them.  Instead, Fermat knew that light seems to travel in straight lines within a given medium.  Therefore, he considered all the possible paths that a light ray could take from P to Q (such as the blue and green alternatives shown in the modified figure), trying to figure out which combination of straight segments (and hence which angles) were picked out by nature.  The answer he posited was that the correct path for the light is the one that minimizes the overall time taken by the light in going from P to Q.   This does give Snell's Law as a consequence, and seems to hint at a deeper organizing principle or structure at work than just "we solved complex equations with tricky boundary conditions, and Snell's Law fell out".  (These days, if a student is asked to derive the Snell's Law from Fermat's Principle of Least Time, they would use calculus to do so, since that plus coordinate geometry provides a clear way to right down an expression for the transit time and a way to minimize that function.  Fermat couldn't do that, as modern calculus didn't exist at the time, though he was among the people thinking along those lines.  He was pretty sharp.)

Next up:  another example of a "global" approach, the Action Principle.