 Hello everyone so welcome to the first basic notion seminar of this academic year. It's my pleasure to introduce our colleague Raph Kebauer if you haven't met him yet this is your chance to know what he does and he will tell us about using the computer as a microscope to understand materials. Can you hear me? Okay so hello good afternoon everyone so this is for me really first because I've never been I've given many talks here in this auditorium but never it's 5.30 in the afternoon and never to a mathematics audience so this is really first to me I never talked to a mathematics audience so I might be with what I've prepared completely out of what you what you expect or what you would like to see normally and so on so because of this I tried to to prepare something which is kind of a mixture there are some equations but they are mostly figures and some pictures and so on so it's all because I wasn't really sure where to place everything so I'm very happy if you interrupt me or if you ask questions or if you tell me to slow or we know this so just make it as informal as possible okay so what I would like to talk about is a subject which you might be interested in as mathematicians because as mathematicians you should like numbers I would imagine and if you think about numbers and computers one always hears about these big supercomputers which are some here in Europe mostly however now in China and in the UN in the US and then wonders what are these supercomputers in fact calculating okay so what these supercomputers are calculating is the first thing which comes to mind is probably the weather forecast and this is true and I remember some years ago when I did my PhD I was using one of the supercomputers at the time and there was always in the afternoon around 4 30 a time when the national it was in France at the time when the national weather forecast for the next day was calculated and all our jobs were stopped and then the weather forecast was calculated and then at a certain time when the weather forecast for the next day was ready we could continue doing it okay so the supercomputers are on one hand weather things but then there are today mainly two other main users of supercomputers one is also very much related to weather a climate prediction which today are very important and also we have not a supercomputer but we have kind of a smallish medium-sized cluster here in ICTP and also there about half of this cluster is used by our climate group calculating climate prediction so this is one thing and the other thing is what I will be talking about it's materials materials related computations in the more broadest sense so this is in fact one of the big number crunching activities is what I would like to show you here okay so I've written in the title that we use the computer as a microscope to look what happens in in a material I will try to explain you with some examples a little bit better what what I mean with this but let me start in a very informal way and let me start with a picture okay so here is a gentleman anyone of you has an idea of who this is one knows this gentleman this is beginning of the 20th century no so I show you in a different picture where he is younger and then you will yeah it is Edison in fact so on this younger picture here you probably recognize him not because his face is so different but because he holds in his hand the light bulb okay which he's invented so I'm not showing this because for a microscope which I've promised you one needs light to see something but because I would like to talk about the light bulb at his invention and mainly how he invented it okay so if you think about the light bulb you know in the interior of the light bulb there's a filament electrical current flows through the filament and it glows and it produces this light and now Edison had the challenge to find out which is a suitable material to use as a filament in the light bulb he wanted to produce and for some reason he has decided that the most probable good candidates are carbon-based materials so what he did is he was testing all kinds of carbon-based materials he could get a handle he used for example and wood from all kinds of different trees in the US he used filaments of other plants he used coconuts he even used hairs of the beard of a laboratory assistant which he had so he used kind of all kinds of materials he could put into a filament and try to make it glow in a light bulb it was a huge endeavor it took more than two years and it was also a huge logistic effort because he asked people all over the United States to send him all kinds of plants products where one could extract a filament from and by trying out many many many different things he finally settled on carbonized on a carbonized cotton thread so what we have today in our clothing carbonized and used that as a filament and it worked however so this was in my opinion or it's probably the first time that someone did a screening for materials used a huge array of different materials as for this intended usage and he finally settled on the one which he found best material screening something which goes on and on and on and has to until today still the most important source of inventing a new material for a given purpose in fact so Edison he was rather wrong because as I said he settled on the fact that it should be a carbon-based material for his filament but just few years later someone else proposed tungsten as a filament which works effect much much better and until today those light bulbs which are still around they are all based on filaments with tungsten so in the end in fact all his search for material was in a sense misguided by his first assumption that it should be some carbon-based filament so why am I talking about a search for materials well because this is one of the applications of this computational microscope which I would like to talk about okay so imagine you are in a situation you would like to develop something new perhaps not a light bulb but perhaps a drug for even illness or you would like to to produce a new photovoltaic panel or something so you know you look for a material with a given property and then you try out many many many things and this is very open very expensive and in fact industry spending huge amounts of money to do this an example might be finding a catalyst for a given reaction for example you want to produce fertilizers and then until today one uses iron-based catalysts for the Haber-Bosch reaction which produces fertilizers is a huge industry and also that was a catalyst which was found by Haber and Bosch by a search like this now one could save a lot of money if I didn't have to produce all these materials and send people to send you in from all corners plans or something like this if I could find out what is interesting just by doing some computations on a computer and I will show you one of the modern examples of how this can be done here so here people have been looking for photovoltaic panels I'm starting with this example because I'm working a lot in renewable energy and so on and this was at the time when it was published and got a famous case of a computational screening of materials so what did people try to do so they tried to find a material which is best suited to produce a photovoltaic panel other than the silicon panels we all know already so what I've shown here is on this axis you see energy and here on top I am is the solar spectrum so this indicates for each energy how much energy is irradiated on the earth's surface by the sun so at the same time so this obviously depends on the energy and there are certain frequencies there there's no light because of absorption in the atmosphere and you have this spectrum so you would like to transform this into electricity into energy and this is used normally done with the semiconductor materials and semiconductor materials are characterized by something which we physicists call an energy gap the energy gap in this context essentially means a minimum energy and you can collect the energy in the photons the solar light only for energies larger than the gap so in a sense therefore the get this energy gap for a good material should be as low as possible to capture as much sunlight as possible but the gap also must not be too low because the difference between the photon which comes onto your material and the gap this is a pure loss so the gap should not be too low because you have too many losses it should not be too large because you do not absorb the light okay and so these people have done a computational screening of many materials and they calculated something which they called spectroscopically limited maximum efficiency it doesn't matter it's a measure they calculate which indicates what will be the efficiency of a solar panel made out of it as a function of the gap which they also calculated and this is the result which they got so here you have a line this indicates as a function of the gap what is theoretically the possible maximum this is called the short kick wiser efficiency limit the theoretically possible efficiency limit you can have as I said too large a band gap you do not absorb anything too low a band gap you have too many losses and there's an optimal band gap you see between one and one point five an electron volt would be a typical bank and then they did in the computer they came with many many different materials you see here kappa gallium di teloride and so many materials and each one gives a dot on this line and amongst all these materials now one can find out ah these here probably which are close to the top are those materials which are worth synthesizing in the laboratory and examining more and in fact before this computational study some have already been known for example the kappa indian diselinate and kappa indian di teloride these have been known to be amongst the best materials for making these kind of solar cells but for example all these silver based materials they came out through this rather blind computational study so no this so at that point they were just theoretical materials but they also came it is not shown here the stability this is I will show more in the second graph of stability so the second example of which I want to show you when uses the computer to look into a material is this here so people also again would like to absorb sunlight in this case to produce hydrogen out of water and they have a screening materials which are of this type this type is called perovskite materials and perovskite materials are made out of two metals which here are called A and B which are in a cage like structure like this and then there is an octahedron of oxygen atoms like this so the inner atom here is a metal B here we have a metal A and we have oxygen here around so this is a structure AB03 an example a famous example is lead titanium would be here this year lead titanium oxygen 3 might be one of this class of materials and people have calculated with the computer with the methods which I will explain and first what Fernando just said how stable the material is and they came up for this the heat of formation so one would like this to be as low as possible to have as stable a material as possible so one would like it to be in the red region and they cagulate again the energy gap as before for absorbing solar light and as we've seen one would like here again to be in the region which is in the scale red so what have they done they have taken more than 3000 I think it's 3600 combination of these 60 something metals for the position A of the 60 the same kind of metals for the position B in the perovskite as I said before and for each combination of AB03 they cagulate these two quantities so you see for each combination here a little square and if I looks more closely into what this is then for each square they have they report two numbers the formation the heat of formation and the band gap so they have without first needing to produce those they had screened these 3800 or so materials and they came out that promising candidates are those where you see where they put kind of a green square around so out of 3600 they ended up with about 10 candidate perovskites which might be good for this so this is a second example of something where people have used a computer to look at a huge amount of materials to find out what is promising just two examples out of my own kind of field which is materials usable for renewable energy so this is an error as I've said we are using a lot of computer time not only for screening but say more generally for for doing this and it is in fact also if you look at the number of papers published in this area and so one discovers that this is an area which is growing very very fast it's one of the fastest growing errors in physical sciences and the question is why so in fact there are two main reasons one is of course when everyone would have guessed this computers today are much faster than computers some years ago so we can do things which were impossible for example the computer on which I did my PhD was probably all the the super computer together less powerful than my cellular phone today so it is really incredible how fast computer power has increased so one can do many things and also everything has become much cheaper in the field of computation and obviously this is a big enabler of this field the second is also even with the if there had been no progress in the computers today fortunately we are much cleverer I will show you in some minutes that in fact the task of cake living properties of material is a very very very difficult task I will show you why your mathematicians you will you will understand so it's a very difficult task and there has been a lot of progress over the last day 20 years of how we can tweak the equations and make them more easy to handle in the computer so the algorithms and the ways we approach the problem have become much better not the last I will say this in a minute also through many of improvements which have been invented here in Trieste so Trieste is kind of a very important city for these kind of things for new algorithms and approaches to this since I'm in ICTP and I'm speaking to this audience I would also like to say that this computational material science is ideally sweet for science in developing countries in my opinion because as it said here computers are cheap everyone today can have some computer and something one can do everywhere and even for science where you need kind of a supercomputer one only needs internet and even internet is these days more and more available everywhere so this is in my opinion some reason why also ICTP is and should invest in these kind of things because it is really an ideal way to do science and modern science also in in places where one cannot afford very expensive experimental equipment so this is why it is growing so fast but why is it useful well the first thing is what I've said the promise of new materials industry here there money this is of course one thing but the finding this computational screening which I said is just one point it is and it is in my opinion not even the most important point the other things is that it allows a more say a more profound understanding of what is happening for example imagines I'm experimentalist has some material and see some strange effect that it's unknown for this is due to simulating this material in the computer can either directly answer what it is due to or one can tweak things which in nature you cannot do for example you one thinks what is this effect you do is this purpose due to relativity is this a relativistic effect in an experiment you cannot switch off relativity or something but in a computer you can switch on and off relativity and your relativistic effect and in this way you can answer questions which you cannot easily answer if you do not do it mathematically on a computer okay and I would say that what is done here in ICTP is mainly of this second sort so when we are doing computation material science here we are mainly doing it to understand certain materials and certain properties of matter the third reason is also one can work with all kind of difficult substances try to do an experiment on uranium or plutonium and you will find out how difficult it is to do these kind of things while in the computer I have no problem of producing plutonium or uranium some calculations which in fact I've never done and my intention is not to work on radiative substances but so it's just to say although some other substance might be very toxic okay so you can on a computer you can do it and you you run the risk so now I've kind of shown you what how nice it is what one can do what people have done why it's ideal for everyone but how is this done how can we take a computer and calculate all this and this is in fact the main the main things I would like to tell you during this seminar so how can we do computation material science okay so let's start from the beginning and wonder what is it what is a material so we want to understand properties say of this chunk of matter so this here's a cartoon it's a number of of atoms in this case we have red atoms and blue atoms whatever it is it doesn't matter and matter generally always is a mixture of different kind of atoms like shown here with the different colors plus electrons around them okay so for all practical purposes we can imagine that atoms which we have are essentially point charges are charges which have really for us no structure they are really like a point with a given electrical charge sitting on top of them and a different atom will have a different charge so since they are all positively charged everything would fly apart what keeps matter and us together are obviously the electrons which are negatively charged and the electrons are in a sea around around the atoms they are creating chemical bonds as you know and the chemical bonds will make such that some atoms are bound more or less to the others and so on so this is what we would like to describe on a compute for doing this in fact the world of computational material scientists is separated into two classes one class of people to classical simulation with classical simulations one means that one takes this jungle no sorry this was the wrong button here we are so one takes this chunk of matter we've just seen before and one is not simulating electrons because the problem with electrons as you will see later is that they are quantum mechanical particles and then there are many identical particles and they have many properties which are in a sense difficult so in this kind of classical models one is throwing away electrons one is treating only the nuclei as a classical particle so each nucleus is well defined by the charge obviously it has and the position it has and perhaps its velocity if it is moving and the role of electrons in these classical models is taken by empirical interactions you can imagine them as if they were springs between the atoms and how strong the spring is might depend on obviously which kind of atoms are close together which other neighbors are there then there can be other interactions which depend on angles and so on so one calls this a force field so one replaces the electrons with some empirically fitted forces and force fields which give you an energy and which makes that everything is kept together so these are classical models it's pure Newton's equations which need to be solved and if your force field is very good you can do a lot of things and this part here of the world is what is mostly used so if you look would do some statistics of how many people are doing a or the other kind of simulation this is mostly used because this is extremely useful and works very well in biology and as you know there are many more biologists than there are physicists or chemists and therefore since this is such a huge community automatically there are many many more people doing this so this is very useful it's also in in other materials it's used very much but it is its main application lies in things which are somehow related to proteins to interactions of the drug for example with the protein with the red goes also to medicine and so say in the most general way I would say something somehow related to biology the other part is quantum models where one retains the electrons so one does not kind of empirically fit something but one tries to understand really what the electrons are doing want to mechanically so okay so this is in fact the the field where I am working in where I will talk a little bit more about it and the advantage here is of course you do not need a force field you do not need to have some a priori empirical fit of interactions and so on and the price to pace however that now you have to solve in the computer also for the electrons are doing okay so you can imagine that if you do not have to solve for the electrons but just solve some springs and some newton's equation you can be treat really very large systems here however the systems are relatively smaller what do I mean and I will give you some numbers in a second here however in a force field you will always have problems with chemistry so for example if an atom goes away from us if some chemical bond is broken the electrons do funny things and this is typically not really represented if you have some springs or some force field so here whenever chemistry happens you cannot do anything also if you are interested in something which depends on the electrons like I've shown you for the photovoltaics the properties of photoabsorption interaction with light you need electrons to do this these all is well described however in the guanto model so people here they can easily treat a million or several millions of atoms no problem and longer time scales here we treat typically 1000 atoms or so so this is just an order of magnitude to give you an idea so it's really systems which are much larger in one world here we have much smaller systems so now your mathematicians are finding what is the equation we have to solve okay so what do we need to do to do the quantum part the part where we retain also the electrons okay so in fact what we need to find is a so-called wave function now I'm not gonna give you a lesson about physics and what the wave function is let me just say your wave function is in principle a complex valued function which depends on the coordinates or of all your in this case electrons and we imagine with any electrons and the coordinates of all your nuclear very very N N nuclear in our system so we have to find a function which is defined by these coordinates it's complex value and it's everywhere space later I will show you that there are some more important things but in principle so for if you do quantum mechanics you need to find the way function of your system how is the way function determined where the way function must obey a very simple equation H psi is E psi H is an operator acting on this function here I will show it in a second and this is an eigenvalue equation what is typically important is only the lowest eigenvalue so we are interested only in the ground side in the lowest eigenvalue of this eigenvalue equation which gives you the ground state energy E and the corresponding eigenvector which is this wave function now H is the so-called Hamiltonian it's an operator which translates the energy of your system and in fact it is not difficult it's in fact a very easy operator as I said it represents the energy of your system and if you look closely so here in blue these two parts are very easy this is a kinetic energy p square is the momentum square divided by 2m this is just kinetic energy here for each nucleus this is kinetic energy for each electron so obviously this is part of the Hamiltonian then we have one part with a negative sign so this is attractive because it's the Coulomb interaction between an electron and a nucleus the nucleus of position r i and the electron position r i so this here gives you all the attractive interactions and then you have two repulsive interaction between the nuclei it's just Coulomb and here again between the electrons also this is just it really looks very easy this is just you see in fact this operator here the momentum operator is in fact just a second derivative in space okay so all you have here is a second order partial differential equation which we need to solve it really does not look difficult and why do it I say it's a form a double task the equations it looks extremely simple yeah but it is not why not because these these many body way functions they are really not nice so we have to try to simplify it and I should say all I'm saying here is for non relativistic physics one can also do relativistic things where the equations are a bit different but for us this is already difficult enough today okay because because electrons so you should know that these are not numbers here these are what we what mechanics call operators okay so and the wave function is never such that you have a very well-defined position in this thing and so when you integrate in the end in order to get your energy you integrate over all the positions and this and the measure of where where this here becomes infinite is in fact of measure zero so it does this does not really contribute to to the total energy this is not the problem the infinity here is not the problem the problem so let us try to make it easier here you have again the same Hamiltonian and if you look at some parts are perhaps larger than others first of all look kinetic energy in fact if there was not the blue part here then it would be very clear that the wave function looks like a delta peaks in space because here the an eigen function of a position operator is obviously something which is a well-defined position so the the wave function would be something which is like a delta peak or a very narrow function around some positions so everything would be very well defined if you didn't have kinetic energy the kinetic energy for being a second derivative it forces the wave function to be broader and having this characteristic nature which you know from your chemistry courses where you know how an orbital typically looks like okay so it is in fact what makes the system more quantum mechanical or more classical is in fact the strength of this part here in the Hamiltonian and if you look at this part here then you see that here we've always m the mass of the electron here we have the masses of the nuclei now electrons are much much lighter than nuclei in fact even the lightest nuclei which is hydrogen is about one thousand times heavier than an electron okay so because of this one over a big m and here one over a small m makes such that probably this term here is less important than that term here okay just by looking at the orders of magnitude in fact this is very often the very first approximation anyone is doing when he's saying okay we do as if there was no kinetic energy of the nuclei this is called also the Born-Oppenheimer approximation this means at the same time as I said all quantum effects a nuclei disappear if you do this because it's the quantum effects are introduced in fact by having a kinetic energy operator here you can also say you're doing the h bar going to zero limit some people call like this so but anyway so what does the classical approximation on the nuclei so with this Born-Oppenheimer approximation we do not take this into account in the Hamiltonian meaning we say all nuclei they are like a classical particle with a well-defined position in the acidic at one point and having just their Coulomb forces acting on the electrons so this is the first approximation we are doing and all which remains quantum mechanical are the electrons or so this is still difficult enough okay so now let's try to solve the problem for the electrons only by the nuclei we claim are pure the nuclei we claim are purely classical particles which are sitting at a given point in space so now I have to do a little bit of physics to show you where the difficulty comes from the difficulty comes from trying to build a valid wave function for electrons okay so how is a valid wave function built I will show you say a minute more about it so first of all if you have Ne electrons okay then let us choose Ne complex functions in space so these single particle orbitals are simply complex functions at every point in space and we require that of this function that it is square normalizable so it's normalized to one and also we to take Ne for them one for each electron and we also wonder they are orthogonal so their scalar product should be or must be zero so you take Ne orbitals for Ne electrons and you try to build a wave function from that this is done by building a so-called slatter determinant which looks like this okay so you have Ne orbitals and the determinant so this is what I call with the capital phi a determinant now for Ne electrons would be simply this so you all know your mathematicians you know very well what a determinant is just say this here is a function which looks like for example phi 1 of r1 phi 2 of r2 and so on times phi n of rn but then there are plus minus all kind of combinations why is this necessary it's necessary because the laws of quantum mechanics tell us that a valid n fermion wave function must be such that if you exchange two particles you must have a minus sign in front of it and a determinant perfectly satisfies so a determinant having a determinant character of your wave function is in fact a sufficient condition that your function wave function is valid but obviously I mean it depends on what you have chosen as single particles and being a determinant is not a necessary so sorry it is not necessary so the real wave function is not necessarily just one determinant in generally it is never so the real many body wave functions which we are looking at will be some linear combination of all possible determinants you can imagine for your n electrons okay so what you need to determine if you want to solve the electronic problem is you have to determine for all existing determinants what is the pre-factor this determines what is the many electron wave function okay so now why is this so complicated the reason is because not because it determines the complicated problem object no it's only complicated in latex until you have it there but it's not difficult to handle what is difficult is that there are so many many many possible determinants and the reason why there are so many possible determinants is the same reason as why we are not all millionaires because it's related to choosing a small number out of a big number so if you play in a lottery you have to choose I don't know six numbers out of 50 other numbers and you never win because you never get those numbers right here you have the same so you have to choose n e orbitals okay out of a huge number of orbitals and choosing this is something which grows factorially fast so you have a huge huge number of possibilities to choose these orbitals amongst all possible orbitals and for this there are so many many many possible determinants in your wave function now I would like to show you this with a stupid example it's really stupid but I like showing it because it really gives the a feeling for for the task attempt for doing this okay so imagine we would like to do what I've just explained we would like to understand the electronic properties of say a very easy molecule a benzene molecule benzene molecule is just 12 atoms six carbons oh no I really hate this thing here okay so we have six carbons six hydrogens 12 atoms and there are 42 electrons around it it's really something small it's really if you cannot do this that I mean why are we talking about this okay so let's try to do this benzene as we have said we have to solve the equation h psi we have seen what is what is the Hamiltonian is e psi for the electronic problem only we have taken away the nuclei this here will depend on positions of the 42 electrons which we have and the Hamiltonian we have seen it is kinetic energy interaction with the nuclei and interaction between the electrons this is all under control but now here we have the sum of our possible determinants let's try to build those determinants well to do this we first have to find out the single particle orbitals which ones can we take so we have to represent them in a computer the easiest thing is to put a grid around your molecule and to say we store the value of an orbital at each point on the grid okay so the you have to describe this lowercase phi of r and at each point of your grid you say it has that value you store this as an array in a computer so you have 10 times I mean this would be even one would certainly need a much finer grid but just for fixing some numbers imagine we fix a 10 times 10 times 10 grid okay so this means we have 1000 grid points on 1000 grid points you can create 1000 linearly independent linear combinations of functions there so we have in principle 1000 single particle orbitals in our hand okay 1000 because we have chosen this if you took a finer grid you will have more possible wave functions okay so hop next one so we have 1000 possible so-called basis functions then electrons also have kind of an angular momentum doesn't matter so something which we call a spin which can always be either up or down it doesn't matter very much what this is but so in fact since in each orbital you can put an electron up or down ways we have in fact 2000 so-called spin orbitals but we have 42 electrons so for each possible determinant we have to choose out of 2000 we have to choose 42 and now start choosing for 42 numbers out of 2000 and you are in the unfortunate situation that you have three 10 to the 87 determinants and now you want to do your computer on your physics on a computer so we'll have to store those numbers c alpha for the determinants to to characterize what is your electronic wave function so you will have to store on the computer this huge amount of eight byte numbers and what does this mean so it's already clear 10 to the 87 it's clear that it will not be possible but just to show you what this would mean so okay so these numbers here mean we have memory need of 2.5 10 to the 79 gigabyte however a computer today's computer memory is two gigabyte per square centimeter typically so we would need a computer memory which is one 10 to the 69 square kilometers and given that the earth surface is only five 10 to the nine square kilometers and this includes all oceans and everything this tells you that we need in fact 2.5 10 to the 59 planets earth of pure computer memory only to store a stupid electronic wave function for for a benzene molecule so i mean you see that this is what makes everything so complicated it's trying to get a handle computationally on on quantum effects is so difficult because the wave functions are so horrible objects okay so now i do not want you to think okay we forget about it and we cannot do wave functions in fact people can do wave functions so what quantum chemists have found ways around this for example one says who says that we need all these 10 to the 87 determinants in fact one of the most important approximations is we take only one and we find out which is the most important this is called Hartree Fock and then people find out have very clever tricks of finding determine important determinants with respect to unimportant determinants so there are ways around this but what i really would like to to keep in mind is quantum mechanical wave functions in materials they are a real problem okay so the question is could we do somehow this quantum mechanics without this horrible beast of the electronic wave function we would like to treat the electrons quantum mechanically because we want to be able to describe chemistry we want to know what happens when light shines on the electrons we would like to do all this but we would not like to be with the many body electronic wave function and the answer is yes this can be done and this is called the way this is done is called density function theory and this theory is in fact the workhorse for what what we people here are mainly doing and i will now try to explain you in fact the nature of it so it says theory and in fact the dft so density function theory or dft is an exact theory there's a theorem this theorem can be proven so all which you mathematicians like theorems proofs and so on and there's even the question of existence appears so things which mathematicians like okay so let's look again at the Hamiltonian in this case only the electronic one which we have seen before okay so with as said before we have kinetic energy we have interaction between the electrons and we the electrons interact with the nuclei so this part here i would like to call an external potential why this because it depends obviously on the positions of the nuclei and where they are but this is something which acts externally on the electrons in fact if you wonder so you take this Hamiltonian here and you wonder in which way is this Hamiltonian different in a benzene molecule then it is in a thin molecule or in a pair of sky which we've seen so you go to a different material in which way ways are the Hamiltonians different where they are different by what i've shown here in blue the number of electrons is different if you go to a different material okay but then apart from the material the number of electrons being different there's also these things which are in green it means the number the positions and the nuclear charges of the involved atoms okay and you see that in fact apart from the number of electrons all system specific properties are all in the second part they are all in the external potential all green parts are in fact here so it is the external potential which is characteristic for system A or system B or system C okay so now if we would like for a given number of electrons to calculate a property of the system what we can do is it is very difficult as i've shown but what we can do is when once we know this external potential so once we know the characteristic of the system we can solve this Schrodinger equation it will be very difficult on a computer as i've shown but in conceptually it's clear what we need to do we have to solve the eigenvalue equation it will give us a many body wave function and so this here comes here we solve this many body Schrodinger equation we get this here and once we have the wave function we have in fact access to all properties of the ground state of our system in particular we can calculate the charge density as a function of space okay so in fact the density is really simply to the square norm of psi if you're a mathematician this is why I've written you take this psi if you know it you simply have to integrate over all other positions you take the first the square of it and this defines an electronic density at each point in space so why i'm saying this but this theory which i would like to say what it is is called density function theory so this is where the density will somehow be important in what i want to say and so the important thing here is once you know the characteristic of your system due to an external potential then there is a unique definition there is a unique charge density which you can obtain by doing these steps so there is no doubt that one can go in this equation from left to right and obtain a charge density also note that the charge density is a very easy quantity it is something which is always non-negative because it's integral over some square so it's always non-negative and it's just a scalar at each point in space so this is something you can really store easily in a computer not like the many-body wave function okay so this is an easy quantity always positive always real and you can obtain it starting from the external potential now what is density function theory density function theory is this other error density function theory is that not only when you know the external potential you have clearly one density but density function theory means that also the other way around is true for a given density there can be only one external potential which is related to it via the Schrodinger equation so you can go this equation also the other way around it seems like something impossible how can such an easy thing here determine the wave function everything else but this is a theorem so this is proven it's called a Hohenberg theorem which one can prove and everything and in fact this theorem is worth a Nobel Prize so Walter Cohen here he won the Nobel Prize in chemistry for having proven this probably would not have wanted if he had only proven this and it would be useless but he has won because it's proven it and it proved so useful after all okay so he won a Nobel Prize for doing this I do not intend to to prove it to here even though it's really something which one can do in few minutes it's not a difficult proof but so the important thing is there is a one-to-one relationship between this external potential which contains the system characteristics and the density so as I say here again so since therefore from n you can get to v external but from v external you can go to the wave function there to all properties this means that all properties of the system are somehow or can be determined by the density alone n of r and as I say this is a very easy quantity which you can easily store on a computer so since all ground state properties are determined if you know the density one of the properties is for example the energy of the ground state so the energy which is what we are always looking for in fact we know that it is determined if you know the density so this this theorem this Hohenberg and Coen theorem tells us that there must exist a functional which given a density gives you the energy okay so n of r is a function and the energy therefore is a function of this function therefore a functional and so this is why it's called density functional theory because it proves or it says that there exists an energy function of the density which you have so we know that this exists and now the way forward is no longer solving the Schrodinger equation it is something much easier you take this functional here and you minimize it and that's it okay so you take the function what you do in the computer you have an easy quantity like the density you have some functional you minimize it and you get the ground state energy and this is in fact what we are doing so no no treating wave functions anymore we only work with the density um so we know that it exists because Hohenberg and Coen have proven it this functional but unfortunately they have not given us the exact form and now after many many years of work and so on it has been clear that it exists but this e of n is certainly not an easy function which we can write down or even if you knew it it will certainly not be something easy to evaluate in a computer so the big one of the big challenges in our field is finding approximation to e of n okay so I've decided later I have some slides if someone would like to know more about how this is approximated they are they are very clever ways but I think e of t is also so useful because one can get easy to use easy to implement approximation for this functional and once we have an approximation for it you just minimize and you get energy and so on so this is what people in this a oh we do quantum simulation density functional theory based this is it no more wave functions an approximation for an energy functional and you minimize it to get the density and the energy of your system so there are a lot of very nice things about this first of all it is what we call ab initio means we do not need empirical information no force field as I've shown no one needs to say ah two carbon atoms always interact like this no we solve the electronic problem and if the electrons want to make a bond they do a bond they want to make a triple bond they do a triple bond if you pull the two atoms the bond wants to break it all comes out of density function theory no empirical information is needed since you do not need empirical information it has a very strong predictive power this is what I've shown you in the beginning you can really predict of materials of which you only know what is the charge of the you only know which elements are there how many electrons both you have can do predictions okay so it is a good predictive power and very often it is very very accurate so many many properties we can calculate within very few percent one two percent we can calculate many many properties like lattice constants like heats or formation the kind of things I've shown you before can be calculated very accurately and many of these properties are also of technological interest so we can calculate so the energy obviously is what I've already shown you the density comes out because the density is the one which minimizes the function so we know the density since we have the energy it's also normally very easy to calculate the derivative of the energy with respect to the displacement of an atom this is the force on the nuclei so once you have the forces now you can move your atoms according to those forces and find where the forces become series we can find structures and equilibrium structures you can also put your atoms away from the minimum position get the forces and solve Newton's equation and have what we call molecular dynamics you can start making everything move around the temperatures and so on as you wish because at each moment you can calculate one to mechanically the forces which are there okay so then we obtain eigenmeste this is not important so we have also frequencies so once you are in a in an equilibrium configuration the frequency is the second derivative of which you obtained from the second derivative like the forces from it so you can what we call phonons and vibrational properties you can include the magnetic properties people use it a lot for ferroelectrics etc so there are many many exciting developments in this field and i can really calculate today many many properties based on density functional theory and this is in fact as i said for today many people are doing on on the fastest computers you can imagine we have some problems with it obviously so the problems are that it's still computational intensive it's not impossible you didn't need 10 to i don't know what planets earth for a wave function but still it is much more intensive than if you just do a classical simulation with the force field it's much more computationally intensive but it well you get something back for it and therefore our systems are much smaller say 1000 2000 atoms something like this and the time scales if you do a molecular dynamics necessarily is also much shorter so then we have a lot of other approximations may the main approximations rely on this function which i've shown you and these approximations often are very good and sometimes they are extremely bad so people like me who have been working for many years in this field have a feeling for when will your functional the density functional theory results be reliable and when they are not and unfortunately in those materials which are often the most interesting ones we get so-called strong correlation effects and we get troubled with our approximations okay also wonder was it's something special but i would like to show you i told you this is a very quick expanding field many people are working there and one measure how people in the science community measure the importance of a scientific work is by citations no okay so here you in fact this is what Nikola Mazzari a friend of mine has done he looked in fact in 2013 but it wouldn't have changed very much at the most cited papers in published by the american physical society so the physical society is the main body for where we will publish our results and the most cited papers of all times here number 1 to 22 are these and now what i always find incredible comes in red all the papers in red are related to what i've been talking to you so you see here so these are here all approximations here generous gradient approximation made simple this is one of the ways the density functional theory is approximated it's called gga generous gradient approximation then there are things where this is applied to different things and you see for example here all these kind of things at number 18 only is the atomic force microscopy or look here a barred and cooper schriefer noble price for who got the noble price for superconductivity is number 22 okay so i do not want to say that what we are doing is the most important and more important than superconductivity or whatever it is but what i want to say is what this clearly shows is that a lot of citation means there are a lot of people working in this field so this is really fitting rabbit expansion a lot of people there therefore a lot of citation and excitement so this is not some fringe area of physics this is really something which is in the heart of of physical sciences okay so i would also like to conclude by saying some things and that in fact trieste so here our city has for many many years been a cornerstone of this area of physical research why this well because since many years there are research groups well here in icdp in the condensed meta section where for example i am part of it also the applied physics section which is here without then we have in cisa since many years an important group doing a density functional theory based calculations at the university of trieste there are also people in the chemistry department doing quantum chemistry on these things the cna the italian national research council has people here they are mainly based at cisa who are working in this field so there are many research groups and we are kind of a very lively group of people doing doing things here in trieste also one of the most important discoveries in fact it was amongst the least which has just which i've just fleshed up is the so-called car parinello method for doing molecular dynamics combined with density function theory this has been invented here by car and parinello and in 1985 so in 1985 there was not yet so strictly a distinction between cisa and icdp and so on so i still until now i i think one of them was in cisa one was in icdp also but what is for sure they were in these buildings when in 1985 they invented their method and when in fact car or parinello in fact parinello is in the icdp scientific council roberto car and he is now in in switzerland roberto car is now in princeton and they are very close friends to us i'm a former student of roberto car for example and when they tell about this development of car parinello here in trieste their stories always start by saying oh it was a particularly cold winter and then they tell how strong the borough was blowing in that winter and they were working until you see they and then they developed this very very famous thing which brought the many many scientific prices and so on so this was developed here we are proud of it another very important thing one of the computer codes which actually do density function theory it's called quantum espresso it's an open source code everyone can download it for free and use it i'm also one of the developers of it and it is so it's used everywhere and it's mainly developed here in trieste by people just to show you again how important quantum espresso is i before coming down here i looked at the citations of our main paper where we have written quantum is what about this code and you see this paper has until now been inside 6,509 times and you see every year this year is not yet over so every year the number of citations is growing so this code and tft and so on is something which is very good health and being used very widely so you are in trieste so you can think you are really in one of the corners of a very important pillar of physical sciences okay so with this i think i can conclude and answer whatever questions you have well i cannot believe this this was not planned it is on the minute exactly i conclude