We still do not know the functional, and we do not know the density. So stay tuned, goodbye. Okay. So instead of the many-body wave function, we use the density. But what do we do with the Hamiltonian, then? There's some kind of functional, but what does that look like? Did I miss something in the lecture? Hello there. Any problem? Hello? Hello. We feel you have doubts. Or questions. That's completely normal at this stage, maybe you want to explain a bit your doubts or questions? Sure. How should I start? So far I understand that we want to use the density as our prime object of investigation, rather than the many-body wave function. Since the many-body wave function is quite high dimensional, the density only depends on three spatial coordinates. It is not only the fact that the many-body wave function is complicated to use, it's also complicated to calculate. Actually it's impossible, except for some really simple systems. Yes, I understand that. So although we have a definition of the Hamiltonian and the many-body problem in terms of its ingredients. Like for instance, in this slide as we've seen before, we cannot solve the Schrodinger equation and find the wave function. But what bothers me about the Hohenberg-Kohn picture is that we are talking about the functional which connects everything. But what is the functional? Good point. What I mean is, can we express this function in the mathematical formula in an equation in terms of the density? Very good point. There is more to it than just your questions. So far, we have not indeed tackled all issues. What is the functional? How do we connect the ingredient with observable? Even other questions are pertinent at this stage. Yeah, in my lectures for instance, I have already told you this. To work with the density functional means to move from a simple functional of a complicated quantity, to an unknown and probably complicated functional of a simple quantity. So the complicated quantity is the many-body wave function and the simpler quantity, the density. But here's the good question. Why the density? Is that the obvious choice? What do you mean by the density not being an obvious choice? Are there alternatives beyond the description of a system via the density or a many-body wave function, or is it good to choose the density for a reason we have not discussed yet? Okay, let's see. In order to clarify these points, at least a little bit. Many things would be clearer going on with this course, this and next week. So to clarify better, let me mention some points that suggest that we are indeed in the good track, okay? Let me make the devil's advocate. You should not learn DFT just because someone told you to do so, and you should not use DFT just because many people use it. There should be some good some winning arguments, and to convince you, we must also consider the counter arguments. So let's raise doubts and questions. You want to start with a question? Yes, of course. In the previous lecture, we have always spoken about the total energy of the system. How for example, can I write the total energy as a functional of the density? Well, we do not know the total energy functional of the density. But we know approximations for some pieces of the total energy. We know approximations for the kinetic energy and for the electron-electron interaction term. Have you just seen in the historical part about the Thomas-Fermi approximation or Hartree. The Hartree term for the electron-electron interaction captures the essential of classical electrostatic. The rest beyond these simple approximations, can be considered smaller and prone to approximations. We will see this next week. Okay. So we have approximations for the total energy, but can't we hope to find something better or even something exact? Well, about this, we must live with the fact that in many-body physics, we almost always can only calculate approximately. Otherwise, we could simply solve the many-body Schrodinger equation, right? Okay, then let's say I accept these approximations to the total energy as given. But what about other observables? Well, this is more complicated at this stage. But we could make the guess that the electrostatic would be always a big ingredient in the recipe. Always? No, no. We know that electrostatics is not always the most important thing. I think that we have situations where that is quite clear. For example, you can measure the band structure of a ferromagnet like nickel, for each spin separately. The band structure is different for majority spin, which is aligned with the magnetization, and minority spin which is opposite. The difference at the Fermi level is called exchange splitting. You can measure it. So it's an observable. I think this will depend more on exchange, on the Fock term, than on the Hartree. So I think it's dangerous to claim that electrostatics is always the biggest contribution. It's only a starting point of course. In most cases, we have to go beyond Hartree. Okay. But you just said that everything beyond electrostatics is small enough to be approximated in some way? Well, yes, and we will see next week that lots of approximations are possible, but it is true they're not satisfying in all situations. You see, it is difficult to conceive approximations for something that depend on the density in a very implicit and delicate way. What do you mean by implicit or delicate? Implicit means the exact functional is maybe not an explicit expression, you can never write down a formula. Delicate means, sometimes small differences in the density leads to very important differences in some observables, and you can imagine that makes it very difficult to find a good approximation. Can you give me maybe an example for the second part? Well, of course. Let's consider two very different systems. Titanium oxide and vanadium oxide. The first one is used as pigment, food coloring, as photo catalysts, and it's an insulator. The second, used as sensor or in batteries is metal and so a conductor. So qualitatively different materials. Now, have a look at the electron densities. They are extremely similar. A functional, has to depend on the density to capture this tiny difference and convert it into a macroscopic discernible observable. One is a metal, and the other is an insulator after all. Okay, I understand the problem now, but then a stupid question. Why are we considering the density then? Because from what you said, it seems not very ideal. Not a stupid question at all. Let's see, what are in your opinion, the good points of the density? Well, it's a very simple quantity. It's much easier than the many-body wave function. I can visualize it or even look at it. Exactly. We can also measure the density, and let's not forget that it does not refer to a single particle picture. It is a true many-body quantity. Yeah, it's a true many-body quantity, and that's not good for us. It's a new problem because it means that most often we are not able to calculate it. I'm really glad that Georg shares his doubts with us. Because let me insist, density functional theory has two problems. One, we do not know the functional, and what I mean is we do not know how an observable depends on the density. But second point, we do not even know what we should put into this functional because we do not know the density. But that is the same if we used many-body Schrödinger equation, we are not able to calculate the many-body wave function, so we approximate it. Is an approximation to the many-body wave function better or worse than using an approximated density? Sure, sure. But at least in the case of the wave function, we would know the functional. It is an integral with the many-body wave function. In the case of the density, we know nothing. If at least we knew the density, that would eliminate one argument against DFT. But can't you think about the functional of something else, something that is both simple and known? Sure, at least in principle. At the beginning of the course in the module dedicated to observables, you remember we said that an observable can be seen as dependent on the external potential. You remember did you pay attention? The external potential is an ingredient we know. It defines the system, for example, the potential given by the nuclei. Yeah, and in the first part of the Hohenberg-Kohn theorem, was about the one-to-one correspondence between the wave function and the external potential. So we can certainly say that an observable is a unique functional of the external potential. Great. Why don't we do external potential functional theory then? We could in principle. I'm afraid I do not know how to proceed. How do we write, for instance, the kinetic energy in terms of the external potential? We know it in principle. Here it is. I admit of course that this is again too complicated to calculate. Otherwise we could calculate all observables like that. By the way, also the density. So again, we would have to approximate. Now we are into very difficult questions. For example, is it easier to find a good approximation for the kinetic energy in terms of the density or in terms of the external potential? There are people who search for potential functionals. It could be very promising. Yeah, I agree. Let me mention then this attempt do not concern only the external potential. For example, we could consider another fundamental quantity like the entropy density and formulate an entropy density functional theory. But can we determine the entropy density better than the electron density? Not at all. Unlike the external potential, and more like the electron density, we do not know the entropy density. But maybe we can find functionals of the entropy density, which is just an example better or more easily than of the electron density. Who knows, maybe we have not tried hard enough. Yeah, that's exactly my point. Density functional theory has a long history of lots of ideas for approximations, but is it because people have understood that this is the best way to go? Or did they just not try hard enough to find alternatives? We may be too conservative. I mean, the overall idea to use the electron density was suggested already in the 20s, but that is no guarantee that it must be the end of the story. Yeah, I completely agree. We shouldn't settle on what we are used to and look for alternatives with open mind. However, I think that the choice of the density was not driven only by the historical development of the theory of electronic structure. Other reasons can come into play, for example, for what concern the unknown density. I've already told you that you can use the variational theorem to find the density. You mean if we know the total energy as a functional of the density, we can find its minimum, and that gives us the density. That's true in principle, but again, we do not know the total energy functional. Yeah, but as you say it's true in principle. Okay. It's still a good point. Somehow reduces two problems into one problem. Your remark reminds me that this minimization idea actually also leads to a concept that I like very much, which is the concept that one can build an auxiliary system. That is something I will explain in the following modules. Okay, I'm looking forward to that as well. I've heard a lot about density functional theory being very popular in simulations everywhere. So I'm eager to learn more about this. Very good. I'm happy to see that you are still motivated about DFT, in spite of all the doubts and criticism we are raising. By the way, DFT was not so smoothly accepted by all communities, and anyway not immediately. For example, did you know that still in 1980, 16 years after the Hohenberg-Kohn theorem, and more than 50 years after Thomas-Fermi or Hartree, many chemists didn't fully believe in the possibility to express observables in terms of the density alone? Really? Yes. In a review article published in the International Journal of Quantum Chemistry, we can read. "There still seems to be a misguided belief that a one-particle density can determine the exact N-body ground-state." Levy and Perdew replied to that in 1982 in the same journal, saying that the belief is most definitely not misguided. So in the early '80s, the consensus about DFT was not so unanimous as we consider it today. Anyway, it's time to move on. I hope we were able to answer some of your doubts or at least to make you understand that these doubts are the good approach. It is important to remain critical and open. Well, thanks for showing up on my screen, and bye then. Thank you for bringing up this discussion. That was really fun. Bye. Thank you, bye. Bye.
