 Okay, does it work? It works. Okay, even if it is a school on phonons, my mission of phonons is quite small, so I need the microphone. Otherwise, you won't hear me. Okay, good morning to everybody, and thanks to the organizers for having invited me to open this very interesting workshop. And I have already lost the microphone. Here, you mean? No, better here. Okay, and I'm going from the University of Udine, if you don't know where it is, it's not far from here, 60, 70 kilometers. And I'm going to give a brief, actually not so brief introduction to density function to the very wasted potential method, which is what we are going to use during this workshop. I have a lot of slides, so I'm not going to try to fit everything into this hour. If something is left out, we'll see it later in the afternoon. So, first principle, where do we start from? We start from, at least ideally, from the Schrodinger equation for a system of nuclei and electrons in reciprocal interactions. And, well, we write this quite complex problem to solve. Then we, of course, we move to time-independent, the solution of time-independent equations. And then we have to disentangle in some way the motion of nuclei and of electrons. Well, sometimes you shouldn't, but in most cases, we start from the so-called adiabatic approximation. So, we start from the assumption that electrons follow adiabatically the motion of nuclei. So, mathematically, what you do is that you write this electron and nuclear wave function into a product of nuclear wave functions times electron wave function depending upon the non-clear position. Then you put inside the equation and by neglecting some terms, in particular the gradient of the electronic part times the gradient of the nuclear part with respect to nuclear positions, then you find that you can write the electronic term, the electronic equation like this in which the potential depends upon the position of nuclei. And then what you find here for the energy depends also upon the nuclei. And this is what defines the so-called potential energy surface. So, the energy as a function of the nuclei, energy that includes nuclear-nuclear refulsion energy and nuclear-electron attraction energy and also electronic energy. And so, the object of our desire is this potential energy surface, this energy as a function of nuclear position. And this is also what determines the potential energy of the nuclei. So, we can disentangle in this way the motion of electrons from the motion of nuclei. Once this is done, we have still to solve an electronic problem which is a complex many-body problem. And one of the typical ways to solve the electronic problem is to resort to the so-called density functional theory. It actually derives from an old idea which dates back to the 30s. So, the idea of considering the charge density instead of the wave function as the basic variable. So, try to solve for the charge density instead of solving for the whole electron. This one for the whole electron wave function. So, this is the correct many-body definition of the charge density. Of course, once you have a potential, so you have to solve the original problem depends upon this potential that is written here. So, once you have this potential, it's a function of a nuclear position that you assume to be fixed for the time being. And you have automatically the solution, but not automatically, but in principle you have the solution and a corresponding wave function and a corresponding charge density. So, you have a relation between potential on one side and the charge density of the ground state. I'm always talking about the ground state, the electronic ground state here on the other side. It's less obvious that you have also the opposite relation. So, given a charge density, a ground state charge density, you have a unique potential that has that charge density as ground state charge density. But this can be demonstrated and mathematically it's quite, it's even quite simple to show this and it goes under the name of Hohenberg and Concierge. So, but actually, so what one demonstrated is that there are no two different potential corresponding to the same, having the same charge density as ground state charge density. Or that if they are not the same, they differ by a constant. So, a constant does nothing, but the difference is just a constant. Now, this has some far-reaching consequences for instance, you may easily show that you can write the energy, the electronic energy as a functional. So, a functional of a charge density. So, a functional is a sort of generalization of the idea of a function. So, a function associates a value to another value. A functional associates a function to a value. So, a function of the charge density that can be written in this way as an interaction with the external potential, the potential of the nuclei of our electrons. Plus, a universal functional that doesn't contain the potential, the external potential, this one, and that this universal, so it's the same for whatever system. And another consequence is that the ground state charge density subject to the constraint of having the correct number of electrons, of course. So, if you have a capital N electrons, then you have the constraints that the integration or the integral of this charge density has to give N, capital N. Under this constraint, the charge density minimizes, the ground state charge density minimizes the energy. So, this functional is minimized by the ground state charge density. And this gives you a way, at least in principle, to solve the problem in terms of the charge density, which is a three-dimensional function, much simpler than the wave function, which is instead a complex many-body object. Of course, this would work great if we knew how this functional, this universal functional is done, how it is its properties. If we knew a way to write it explicitly, but this is, of course, not possible, we just know that it exists. Now, how can we manage, how can we use in practice this theorem? Now, the first answer came from Konnishan one year later, who introduced a set of objectives a set of auxiliary one-electron objects called con-sham orbitals. They find in such a way that they yield the same charge density as for the true system. So, you have the charge density of the true system. You introduce a number of one-electron con-sham orbitals such that for non-interacting electrons, so it's a fictitious system of non-interacting electrons, and the charge density, the true charge density is given as the sum of the square of the non-interacting electrons and this would be, actually the charge density for the system of non-interacting electrons is just the sum of the squares of the wave function function of the single electron. Under the constraint, of course, of orthonormality, these guarantees of orthonormality of the con-sham orbitals. This guarantees the correct normalization if you have the proper number of these con-sham orbitals. So you have assuming neglecting spin if you have capital N electrons and you need capital N one electron con-sham orbitals. Now we rewrite the energy functionals, which is still unknown, but we rewrite the energy functionals by extracting the most important parts from these functions. So we extract what we know, what we manage, what we can easily write. In particular, we extract the energy, the kinetic energy of the non-interacting electrons, which is not the kinetic energy, the true kinetic energy of the system of electrons. It's close, but it's not exactly the same. Then we extract the average electrostatic interaction. So the Hartree term is nothing but the electrostatic interaction between clouds of charges with charge density N of r. Then we introduce the interaction with the external potential. Here it's assumed that the external potential is a local potential. Then we leave everything else here inside. So this is the expression of the kinetic energy for non-interacting electrons. It's a usual form of the kinetic energy. This is the form of the Hartree energy. It's an electrostatic interaction. Of course, we don't know. Then at this point, we minimize the energy. So we say, okay, what are the concha morbidals that give the correct ground state, give the correct ground state always with the constant that they have to give the charge density of the system of real electrons, and with the constant that orbitals have to be orthogonal. Then in this way, by minimizing the energy, one finds the effective potential acting on electrons, on these non-interacting electrons, this auxiliary system of non-interacting electrons, and this is the form. So it's a traditional Hamiltonian with an effective potential and a kinetic part. So we have to solve these equations called the concha equations. This is clearly the external potential, the potential of the nuclei. This is the electrostatic potential, and it's also related to the functional derivative of the electrostatic energy. So it has this form, this potential of a cloud of charge. And we still don't know what we write formally as the functional derivative of the charge, density of the exchange correlation function with respect to the charge density. Notice that the charge density is here, it's somewhere here, and it depends upon the concha morbidals. So the problem is self-consistent. So the solution, the equation depends upon the solution. So you have to verify that the solutions are compatible with the potential you have assumed. And this is an alternative way to rewrite this energy as a sum of our concha magaine values. Those quantities here are called concha magaine energies or concha magaine values. You see that it's a sum of occupied orbitals. So the orbitals that contribute to the charge density minus double counting terms like this one for electrostatic energy. So it has to be subtracted out because it's present twice here. Minus this term that appears here plus the true, so to speak, exchange correlation energy. In R34, you have something similar, but this term and this term are the same. Well, they're not the same. This is two times this one. In density functional theory, this is not true. So this term and this term are not the same. They are different objects. Okay, so still we haven't solved the problem what is this exchange correlation energy. And so we have to figure out some approximations, a viable approximation for this exchange correlation functional. And the first historical proposal, well, actually it's maybe not the very first, but the first modern, so to speak, proposal for an approximation that's called local density approximation. So in the local density approximation, you write the exchange correlation energy this way by, let's say, as a density of the exchange correlation energy density times the charge density. And this energy correlation energy density is obtained from the results for the homogenous electron gas. So the homogenous electron gas is a model system in which you have a constant density. So you can compute for that system this quantity as a function of the homogenous charge density. Then locally, in each point, you assume that the exchange correlation energy density is the same as you would have in the homogenous electron gas for N uniform density equal to N of R, the local density. So that's the origin of this local density name. There are techniques that allow to solve practically exactly the homogenous electron gas, particularly quantum Monte Carlo techniques. So this function here, this function epsilon XC of N and zero, let's say, is known as a function of N zero. Actually, one uses the, more often than not, one uses not that zero, but the RS that is the parameter that is used in the theory of metals. So the volume of a sphere of radius RS is equal to the average volume per electron. Anyway, there are analytical formula quite simple that you can fit to Monte Carlo results for the electron gas. And so you can obtain this approximation. Apparently, one might say, okay, that's nice for systems in which the charge density varies slowly. So systems that are almost homogenous. So one might say, okay, metals have a relatively homogenous charge density, but what about all other cases like atoms, molecules, or cases in which the charge density is far from homogenous? Actually, this works better than one might expect. But still, one would like to take into account the lack of homogeneity of the charge density. So the fact that the charge density is a function that vaporizes with the position. So one might want to take into account the fact that there is a gradient of local gradient of the charge density, and it should be taken also into account. It's not that simple actually to write a functional of the charge density end of its gradient. So the first attempt failed miserably. You have to consider that there are a number of some rules and of results for the exact functional that have to be kept. Otherwise, the correct and the gradient corrective function are worse than the gradient uncorrected ones. But finally, some generalized gradient approximations so called have been proposed, and they work quite nicely. Of course, they depend upon not the gradient, but the modules of the gradient of the charge density. But in principle, you can write and you can generalize this form of the exchange correlation energy in this way in which this energy density depends upon the local density and the local gradient, the modules of the local gradients. Now, there are many possible functionals that one can define in these two classes. So these are the two, among the simple functionals, these are the two simplest classes, local density approximation, so function of the local density and local, generalize gradient approximation function of the local density and of the local gradients. There are many different functionals that can be written in this way. They all give, those of the same class, tend to give all very similar results because they are actually very similar. If you look at how they are defined and you discover that they are very similar in the region of charge density that is interesting for typical condensed matter systems. So they look all different, but they, in the end, they are almost equal. But still there are many different flavors and some flavors are slightly better for some characteristics and some are better for others. Yes, please? Some? Are conservative? There are some rules that have to be enforced that have to be respected. I don't know very well how these objects are derived, that there are some rules that have to be enforced, otherwise you get funny results, not so good results. Conservation law, you mean, no, but there are some rules that have to be enforced, otherwise you get funny results, not so good results. Conservation law, you mean, no, but actually those GGA's are always written in as function of the charge density and of not of the gradient, but of the gradient to the power four to four divided by three by divided by the charge density and there are functions of not direct of the gradient, but of a combinational gradient and of the charge density. Anyway, they are well defined functions. Yes? In the old year, you are saying that you are looking at the local density and why does it matter that one system is, you know, intelligent and that it is varied while the other one doesn't when in fact other people that I have is that although it is varying, at which point you are looking at the density that corresponds here. So you must then improve the variation. Yes and no, because the result would be exact only for homogeneous charge density. So the result for homogeneous charge density is exact. For non-homogeneous charge density, well, it's an approximation, it's an assumption. You are assuming that the exchange correlation energy is done like that, but the functional is an object that is more complex than a simple function of the local value. The functional is an object that takes a function and somehow produces a value. This is something. If you look at the exchange correlation potential in this approximation, then you see that it is a function of the local charge density, really a function, not a functional. So you are restricting the field of your possible solutions. Okay, this is the basic. So LDA and GGA are, let's say, the basic assumptions, the basic approximations, which work quite nicely for a variety of problems. They have some well-known shortcomings. So they may not be the best solution for an acceptable solution for all cases. You may have noticed that there is nothing like looking like a spin in this theory until now. In fact, the original density differential theory works for a strange object. So it works for spinless fermions, which are, of course, a contradiction. But the extension to fermions with spin has been derived, and what one finds is that the exchange correlation function is a function of the charge density, well, in the simplest case in which you choose a single quantization axis, and you assume that your states are either spin up or spin down. In that case, you have spin up and spin down charge density, and you can write an exchange correlation function as a function of spin up and spin down charge densities. Actually, more often than not, these exchange correlation energies are written in terms of the total charge density and the polarization, the local polarization. So the difference between spin up and spin down at a given point. Notice that this treatment of spin for historical reasons called local spin density, local spin density approximation, so one might think it works only for local density. So it's an extension of local density, but actually it works also for GGA, so it's an historical name. So what this LSD-A accolade means is that we assume a single quantization axis, and we assume to have all states, or all calcium states, with spin up or with spin down. There is also generalization to the general case in which you don't have a single quantization axis, and in which you write a functional while you consider your consium orbitals are not simple functions but spin also, so function with a spin up and spin down part. And you write the exchange correlation functional as a function of the total charge density plus a term that depends upon the local magnetization, which can be a vector, which can change in the system, while in local spin density you have magnetization up or down. In the general case, it's called non-colonial case, in non-colonial magnetism you have a magnetization that has a direction, a local direction that changes, that may change the function of the position. There are cases in which this is important, in which the local spin density approach is not sufficient because magnetization has different positions, different directions in different points. Notice that I have here introduced silently, not so silently because it's written explicitly here, another extension of DFT to fractional occupations. This is important for metals until now implicitly I have assumed that you have either occupied consium orbitals or empty consium orbitals that don't contribute to the charge density that you may also consider the case in which you have a fractional occupancy and this is important for metals. So you add in this case this occupancy which of course range from zero to one. Okay, all these actually works quite nicely, rather surprisingly also for atoms here. Those lights I borrowed from Stepanopoulos. So if you look at, if you can read numbers, not sure you can, but if you can, you notice that those numbers, both LDA and GGA numbers for the energy, this is the absolute energy of an atom. So something that you measure by taking an atom and removing one after the other electron. So you sum the ionization potential until the last one and then you have the total energy of atoms. And this quantity, this absolute quantity, this absolute energy that is quite well reproduced in LDA and GGA. LDA tend to underestimate the stability of atoms while GGA is better. Both have a funny problem which is quite well known. So negative ions are not stable. So negative ions in this framework don't exist. Of course, this is not exactly what one observes. For small molecules, one finds that binding energies and binding and bond lengths are also quite reasonable. LDA here shows a clear problem in that the binding energy tends to be overestimated. This is typical for LDA. It overestimates the binding while GGA is much better in this respect. If one looks at simple solids, one finds also a reasonable pattern of results and quite consistently LDA tends to overbind again. So the distance, the equilibrium lattice parameters are typically smaller than in LDA than the true ones. GGA tends to slightly overestimate those lattice parameters. And as a consequence, LDA is too strong, too high bulk modulate, so second derivative of the energy. While GGA tends to give somewhat softer materials. So materials in LDA are somewhat compressed and stiffer. Materials in GGA are somewhat slightly expanded and softer. And even for nastier objects like oxides, the results are not so bad. Here actually it's a little bit more... This is tricky, actually. So results of LDA may look reasonable, but if one looks more careful, then one sees a number of problems. What are these problems? These problems of LDA and GGA are invariably related to characteristics of the true exchange correlation function, which we still don't know, but we know a few things about it. In particular, we know two things that have to be true, necessarily. One is that if you have a single electron, if you look back at the formula, if you have a single electron, you have no exchange correlation potential. No, sorry, more exactly. You have an exchange correlation potential that must compensate the Hartree term because the electron cannot interact with itself. So this interaction of an electron with itself is called self-interaction. In Hartree-Fock, if you perform Hartree-Fock calculation for one electron, then you have exact removal of exact cancellation of the self-interaction by construction. This is not unfortunately true for all approximate density functionals. So invariably, you have that one electron interacts with itself. Now, the interaction is small, but it's sufficient to make up for most of the trouble of DFT. So in particular, what I was mentioning before about the negative ions that are not stable is a consequence of self-interaction. And many other cases, many other problems are a consequence of self-interaction. And another funny characteristic of the density function is that if you consider it as a function of the number of electrons, so if you consider it as, assume that the charge density is a continuous function, the total charge density is a continuous function of the number of electrons, so the number becomes a real variable. Now, you will have something that depends upon the number of electrons, but this something, this energy has to be, has to have a cusp corresponding to integer number of electrons. This is because if you add an electron or if you subtract an electron, the energy involved, the energies involved are different. Now, this is not something, both characteristics, both aspects are not obvious to enforce with a simple form of the Exxon's correlation function. Actually, it's impossible to have, to write a function, a simple function of the charge density of the gradient or something like that can enforce, because you can respect both of these aspects. So, what can you do with basic, simple LDA and UGA? Well, you can do calculations in condensed matter systems of all kinds because it's simple. So, it's a simple, these are simple approximations, so you can do quite complex cases with plain DFT, LDA, GGA, and the results are actually good, especially for weakly correlated semiconductors, but also for more difficult materials like the oxide I've shown before, structural properties come out quite well. We were quite satisfactory accuracy. But there are no problems, and one of the no problem is the infamous band gate problem. So, in principle, the difference between, you would like to have the difference between the highest occupied valence orbital and the lowest unoccupied conduction orbital, or homo-lumo in quantum chemistry dialect. You would have this difference of the eigenvalues to be your band gap. It is not, it is wildly underestimated. Even you may have cases in which your semiconductor turns into a metal. Germanium, for instance, is one of those borderline cases in which it's a semiconductor with a relatively small gap, but not as small. In fact, once upon a time, the first transistors were based on germanium. I'm old enough to remember when the germanium transistors were still existing and used them. But LDA germanium, the sort of semi-metal. I have mentioned before strongly correlated materials, so materials with strongly localized electronic states like 3D in transition metals. So these are the electronic states, atomic electronic states that tend to stay atomic-like even in condensed matter systems, and that are responsible for magnetism, for instance. For these materials, the incorrect oscillation of self-interaction is a problem because you have strongly localized electrons and so you have a significant self-interaction. Finally, if you look at how a GGLDA function is defined, it is defined as a function of the local potential gradient, of the local charge density gradient, sorry, which means that if you have two systems that do not overlap, the exchange correlation energy is simply the sum. No overlap, no interaction. But van der Waals' interactions don't depend upon overlap. They are interactions between fluctuating dipoles, which in principle are non-local, and so there is no way to account for van der Waals' interactions in using this so-called semi-local approximation. They are called semi-local because basically a local approximation of the charge density using the local value of the charge density and of the gradient. Yes? Yes? Local density or gradient? Yeah, gradient, yes. Well, yes, but... Yes, of course, the Coulomb interactions are accounted for exactly. So one of our interactions, yes, they are. It's a fluctuating dipole, fluctuating dipole interaction. Of course, if you have a fixed dipole, fixed dipole interaction like in water molecules, this is water, water has a dipole, water molecules have a dipole, and the interaction of this dipole is perfectly reproduced, is perfectly accounted for by the half return. Sure? But van der Waals is something more subtle. It's a perturbation theory. You can derive it from second-order perturbation theory. Okay, you may say, okay, I don't give a damn about van der Waals' interactions, but still there are a number of cases in which it's important instead. So what can we invent in order? Can we have people invented in order to deal with those shortcomings of density-function theory? Well, first approximation that somebody doesn't like, but that works quite nicely and it's quite cheap for... sorry, for strongly correlated cases. So materials with strongly localized atomic orbits are called DFT plus U. In DFT plus U, you add a correlation term of Hubbard time that has, in the simplified, in the most simple case of this form, where this N is a matrix of the occupancy, you project your conchium orbitals onto a selected manifold of atomic-like states. So for instance, 3D states for iron, you have this term, and this term has the important characteristics to distinguish between integer numbers of electrons and non-integer numbers. So it corrects in some way for the lack of discontinuity when you cross an integer number as a function of the charge density crosses, the total charge density crosses the integer number. So this cartoon is by Stephen of the Jirongli here, and it's meant to represent a localized orbital in a sea of delocalized bands as a function of the occupancy of the localized orbital. And so what you get from playing LDA or GGA is this continuous curve as a function of the occupancy of the state, while the correctest functional has this piecewise linear behavior, which is the correct behavior. As a consequence, here you tend to get integer occupancies, which is what you observe, while with LDA you tend to get mixed occupancies that are unphysical. So what happens sometimes is that you get good results with GGA or LDA because you happen to get the correct occupancy of localized orbitals, but sometimes it happens and sometimes it doesn't. So it's not something you can rely on it. A further advanced functional that is quite recent actually, well, not that recent actually, because this is a few years, but it has always been difficult to use. So called meta-GGA. Meta-GGA are what? They are the next step in the design of functional. So LDA charge density, local charge density only. GGA, local charge density, local gradient. Meta-GGA meta, this is beyond and then you add the dependency upon this quantity here. This quantity here that is sort... Well, no, it's not nabla square. It's tabla. Well, I'm not sure this is correctly defined. It's nabla, it's not nabla square, sorry. And this quantity you can write... You can use to write a functional that depends upon the charge density, the gradient of the charge density, and this so-called kinetic energy density. It's definitely more complex than GGA to use. Much more complex because when you do... When you compute the functional derivative of the energy of the exchange correlation energy, you have a term coming from here that changes your Hamiltonian. So it's not simply V over R, a potential for GGA, the exchange correlation potential, it's a local potential. So it's in a simple object. In beta-GGA, you have the simple object plus other objects that changes the Hamiltonian. So it's relatively complex to use and it's numerically not easy. These objects are quite nasty numerically. But recent meta-GGA functional seem to work very well. And in particular, this scan is the latest trend in the exchange correlation functional. This one, meta-GGA, not only this scan functional, it seems to give very accurate results, even if it's not very cheap, but it seems to work very nicely. Another approach that works quite nicely, but it's also a little bit computationally heavy, is called hybrid functionals. Hybrid functionals derive from quantum chemistry. So in the quantum chemistry world, the first reaction to DFT was, it doesn't work for some good reasons because it depends upon which kind of accuracy you are used to. But later it was realized that by mixing some exchange, some exact exchange, that is some Hartree-Fock exchange with DFT, so by mixing a little bit of Hartree-Fock with DFT, you could get some very accurate results. Now, what is the problem here? Well, the problem is that you have a lot of different density functionals available, and then you have even for the parameter. So problem number one is that you have an additional parameter. So they administer how much exact exchange you add to your DFT function. And the second problem is that if you work with plane waves, this is horribly slow. This part here, it has this form, is horribly slow. It scales as you have to make products of all states, of all occupied states, of all pairs of occupied states. So it's very slow. There are some recent developments that are very, very promising to speed up the calculation, the usage of hybrid functional with DFT. OK, coming to the problem of van der Waals, there are various approaches for this. One approach is from first principle, OK, I write down a functional that may account for non-local interactions like van der Waals. And in general, you have to write something that has this form. This is the most general, non-local form, not the most general, but a general form, a possible form for a non-local object. So you have the charge density as a function of r, the charge density as a function of r prime, and the kernel. And this kernel is also a function of the charge density, of the gradient of the charge density, and of the difference between r and r prime. So it's a rather complicated object that also exists in CRs, but it was basically unusable because too expensive. But there have been some algorithmic improvement, in particular this solace technique, that have allowed to write those non-local functionals in a way that is reasonably cheap for a computational point of view. So this is one possibility, non-local function. They work quite nicely. They tend to overestimate binding a little bit. Again, another problem is that you have a large choice. There is only one part. So you have the non-local term plus the local term, plus the axis correlation p, some gga. So you have an ample choice between different functionals. The alternative is to add Van der Waals' interaction manually, so to speak. So you have your gga that doesn't account for the Van der Waals' interaction, but you know how the Van der Waals' interaction works. It's something that more or less the main contribution, you may have also other term, but the main contribution is this one of r6. And this object here is related to the atomic polarizability. So one may try to fit for all atoms these coefficients. You also need this damping function because you have to turn the Van der Waals' interaction off when the charges start to overlap. Otherwise, you have divergence at zero. It's not really first principle of first and the half, but it works quite nicely. This Grime DFT plus D3 is the most accurate, and it works quite nicely. Otherwise, there are other further approaches that seems to work quite well, in particular, the Kachenko-Chefler approach and the exchange hold dipole model. And in these cases, instead of computing the coefficients here from a fixed table, and you compute those coefficients directly from the charge density with some method, so you do everything first principle, so to speak. Okay, I have been talking too much, so I'm way behind. Just a little quick introduction on how you actually solve those equations because I have, until now, I have been talking about DFT and results and never saying how you get those results. Of course, it's rather complex procedure that involves some numerical calculation. So I have mentioned that the Consham equations are self-consistent, so you have to solve for Consham orbitals under a potential that depends upon the solution themselves, so you have to do some self-consistency, and this is one possibility. So you solve those equations with the charge density that depends upon the Consham orbitals, ensuring also normality constraints, but this comes out almost automatically. There's also another approach, actually, which is sometimes more convenient, sometimes less, but it's basically equivalent. That is, you can say, okay, I have an energy as a function of Consham orbitals. Those Consham orbitals have to be orthogonal, orthonormal, actually, so this is a constraint I have to impose, but in the end, this is sort of a problem of minimization, so a problem of finding a minimum of a function, which is a quite well-known problem, and so the global minimization approach doesn't have Consham equations, it uses basically the same ingredients to minimize the energy directly. It's the same because this also comes out from the minimization of the energy. So here you are minimizing the energy by solving these equations, and here you are minimizing the energy by directly minimizing the energy, so it's quite the same. It's quite the same, and in the end, the minimization of the energy of a function, mathematically, it's much simpler if you know the gradients, gradients with respect to what? In this case, the gradients with respect to Consham orbitals. So the variables are the Consham orbitals, so you need this quantity, basically you need the functional derivative with respect to the Consham orbitals, and this is nothing but the Consham Hamiltonian times Consham orbital, minus a term that comes from the Tenshua orthonormality. This term, once you diagonalize it, yields a set consistency at the end, yields the Consham eigenvalues. I'm afraid that I have to stop here. Any questions? Yes? It's something that has been observed. I'm not aware of any of any case in which LDA doesn't overbind. I'm not sure if we can find a better justification than this. Okay, is it working? Yeah, so about LDA versus GGA, I don't know, one of the things that people say is that LDA is the correct principle for a uniform system, and then you expect it to be more wrong for inhomogeneous systems, so the atom is the most inhomogeneous system. One justification that is just a way to... I mean, it's a posteriori, so I mean, no one understood before.