Hello everyone. In this lecture, I'd like to make some remarks about the universal functional of density functional theory. Let's write down the formal definition. The total energy of a many-body system, E, is given as a functional of the density n. We express the total energy as a sum of universal functional F of n, which is the same for all systems with the same number of electrons and the same interaction, plus a part that depends on the external potential. Now all that Hohenberg and Kohn say is that the universal functional exists. But can we say more than that? In this lecture, I want to make two points. The first is that Kohn and Sham were smart when they introduced the exchange-correlation functional. The second will be to give you an illustration of what the exchange-correlation functional really is. So let's now come back to the total Hohenberg and Kohn energy functional. You have seen earlier that the universal functional F of n can be written as the sum of the non-interacting kinetic energy T_s plus the Hartree energy plus the exchange correlation energy. So this is now the universal energy functional written in the Kohn-Sham formalism. It defines the exchange correlation energy. Now these parts of the total energy functional are known. The only unknown part is the exchange correlation energy functional and it turns out that the exchange correlation energy is much smaller than the sum of non-interacting, kinetic and Hartree energies. Typically it's just a few percent. So here are two things to remember about the exchange-correlation energy functional. The exchange correlation energy is relatively small, so we may get away with approximating it. However, it turns out that the exchange-correlation contribution is very important for molecular bonding, which is why it is sometimes called nature's glue. So it is important that we do a good job approximating it. But what is the exchange-correlation functional? How can we really understand what it means? Well, here's what the functional does. It takes a function as input here, the density n of r, and it delivers a number as output. That number is the value of the exchange correlation energy. This recipe works for any mathematically reasonable input density. Now a good way of looking at this is to say that the exchange-correlation functional is like a library. For any given density n of r, we can look up the corresponding exact exchange correlation energy, which is given in one of the infinitely many books that are sitting on the shelves. Well, unfortunately, the door to the library is and remains locked. We don't have access to the exact exchange-correlation functional. So what can we do? Well, to construct the exact exchange-correlation functional, we would have to solve the full many-body problem for all conceivable external potentials and densities and this is clearly impossible. So we have no choice but to construct an approximation, doing the best we can to the exchange-correlation functional. Stay tuned to learn more about this in the following lectures.