align:start position:0% for all our applications and for the lab align:start position:0% for all our applications and for the lab align:start position:0% for all our applications and for the lab sessions I guess they keep using Albert align:start position:0% sessions I guess they keep using Albert align:start position:0% sessions I guess they keep using Albert Einstein this is the 100th anniversary align:start position:0% Einstein this is the 100th anniversary align:start position:0% Einstein this is the 100th anniversary of is the famous year 1905 so just a align:start position:0% of is the famous year 1905 so just a align:start position:0% of is the famous year 1905 so just a little celebration one slide of a align:start position:0% little celebration one slide of a align:start position:0% little celebration one slide of a reminder of what we have seen in the align:start position:0% reminder of what we have seen in the align:start position:0% reminder of what we have seen in the previous previous lecture we had really align:start position:0% previous previous lecture we had really align:start position:0% previous previous lecture we had really developed the formalism and leading to align:start position:0% developed the formalism and leading to align:start position:0% developed the formalism and leading to the hartree-fock equations and the align:start position:0% the hartree-fock equations and the align:start position:0% the hartree-fock equations and the hartree-fock equation follow from a align:start position:0% hartree-fock equation follow from a align:start position:0% hartree-fock equation follow from a certif you know very simple and very align:start position:0% certif you know very simple and very align:start position:0% certif you know very simple and very beautiful path we have the Schrodinger align:start position:0% beautiful path we have the Schrodinger align:start position:0% beautiful path we have the Schrodinger equation and we have reformulated the align:start position:0% equation and we have reformulated the align:start position:0% equation and we have reformulated the Schrodinger equation in terms of the align:start position:0% Schrodinger equation in terms of the align:start position:0% Schrodinger equation in terms of the variational principle so we have a align:start position:0% variational principle so we have a align:start position:0% variational principle so we have a functional and we know that we can throw align:start position:0% functional and we know that we can throw align:start position:0% functional and we know that we can throw into that functional any arbitrary wave align:start position:0% into that functional any arbitrary wave align:start position:0% into that functional any arbitrary wave function and they'll give us an align:start position:0% function and they'll give us an align:start position:0% function and they'll give us an expectation value of the energy and set align:start position:0% expectation value of the energy and set align:start position:0% expectation value of the energy and set of the closer we get to the true ground align:start position:0% of the closer we get to the true ground align:start position:0% of the closer we get to the true ground state wave function the lower the align:start position:0% state wave function the lower the align:start position:0% state wave function the lower the Tenergy is going to be we are not we are align:start position:0% Tenergy is going to be we are not we are align:start position:0% Tenergy is going to be we are not we are never going to go below the ground state align:start position:0% never going to go below the ground state align:start position:0% never going to go below the ground state energy instead sort of a very powerful align:start position:0% energy instead sort of a very powerful align:start position:0% energy instead sort of a very powerful approach to try out a sort of align:start position:0% approach to try out a sort of align:start position:0% approach to try out a sort of possibilities and solution and in align:start position:0% possibilities and solution and in align:start position:0% possibilities and solution and in particular set of Hartree and Foca took align:start position:0% particular set of Hartree and Foca took align:start position:0% particular set of Hartree and Foca took this approach they wrote a set of the align:start position:0% this approach they wrote a set of the align:start position:0% this approach they wrote a set of the most general many-body wave function align:start position:0% most general many-body wave function align:start position:0% most general many-body wave function that can be written as a product of align:start position:0% that can be written as a product of align:start position:0% that can be written as a product of single particle orbitals that was align:start position:0% single particle orbitals that was align:start position:0% single particle orbitals that was actually the original heart resolution align:start position:0% actually the original heart resolution align:start position:0% actually the original heart resolution we functions written as data do not align:start position:0% we functions written as data do not align:start position:0% we functions written as data do not satisfy a fundamental symmetry of align:start position:0% satisfy a fundamental symmetry of align:start position:0% satisfy a fundamental symmetry of interacting fermions that is they are align:start position:0% interacting fermions that is they are align:start position:0% interacting fermions that is they are not antisymmetric and so what you do you align:start position:0% not antisymmetric and so what you do you align:start position:0% not antisymmetric and so what you do you take this product of single particle align:start position:0% take this product of single particle align:start position:0% take this product of single particle orbitals and you sum it with all the align:start position:0% orbitals and you sum it with all the align:start position:0% orbitals and you sum it with all the possible permutation with all the align:start position:0% possible permutation with all the align:start position:0% possible permutation with all the possible signs in front of so that the align:start position:0% possible signs in front of so that the align:start position:0% possible signs in front of so that the overall wave function is anti-symmetric align:start position:0% overall wave function is anti-symmetric align:start position:0% overall wave function is anti-symmetric and that can be sort of written align:start position:0% and that can be sort of written align:start position:0% and that can be sort of written compactly as what is called a Slater align:start position:0% compactly as what is called a Slater align:start position:0% compactly as what is called a Slater determinant here and basically now our align:start position:0% determinant here and basically now our align:start position:0% determinant here and basically now our unknowns align:start position:0% unknowns align:start position:0% unknowns are the N align:start position:0% are the N align:start position:0% are the N orbitals Phi and so we need to determine align:start position:0% orbitals Phi and so we need to determine align:start position:0% orbitals Phi and so we need to determine the shape of this n single particle align:start position:0% the shape of this n single particle align:start position:0% the shape of this n single particle orbitals and we want to determine them align:start position:0% orbitals and we want to determine them align:start position:0% orbitals and we want to determine them such that they minimize the expectation align:start position:0% such that they minimize the expectation align:start position:0% such that they minimize the expectation value of the variational principle and align:start position:0% value of the variational principle and align:start position:0% value of the variational principle and so that leads basically to a set of align:start position:0% so that leads basically to a set of align:start position:0% so that leads basically to a set of differential equation is just functional align:start position:0% differential equation is just functional align:start position:0% differential equation is just functional analysis and when you ask yourself what align:start position:0% analysis and when you ask yourself what align:start position:0% analysis and when you ask yourself what are the condition that those single align:start position:0% are the condition that those single align:start position:0% are the condition that those single particle orbitals need to satisfy in align:start position:0% particle orbitals need to satisfy in align:start position:0% particle orbitals need to satisfy in order to minimize the variational align:start position:0% order to minimize the variational align:start position:0% order to minimize the variational principle well this is it the align:start position:0% principle well this is it the align:start position:0% principle well this is it the hartree-fock equation so each a single align:start position:0% hartree-fock equation so each a single align:start position:0% hartree-fock equation so each a single particle orbital Phi of lambda need to align:start position:0% particle orbital Phi of lambda need to align:start position:0% particle orbital Phi of lambda need to satisfy basically a shredding ER like a align:start position:0% satisfy basically a shredding ER like a align:start position:0% satisfy basically a shredding ER like a question again as always there is a align:start position:0% question again as always there is a align:start position:0% question again as always there is a kinetic energy term here there is the align:start position:0% kinetic energy term here there is the align:start position:0% kinetic energy term here there is the interaction with the external potential align:start position:0% interaction with the external potential align:start position:0% interaction with the external potential that is just the potential of the nuclei align:start position:0% that is just the potential of the nuclei align:start position:0% that is just the potential of the nuclei and then come the so-called mean field align:start position:0% and then come the so-called mean field align:start position:0% and then come the so-called mean field terms so the electron lambda here will align:start position:0% terms so the electron lambda here will align:start position:0% terms so the electron lambda here will interact with each and every other align:start position:0% interact with each and every other align:start position:0% interact with each and every other electron move via an electrostatic align:start position:0% electron move via an electrostatic align:start position:0% electron move via an electrostatic interaction you see Phi star times Phi align:start position:0% interaction you see Phi star times Phi align:start position:0% interaction you see Phi star times Phi is the charge density coming from the align:start position:0% is the charge density coming from the align:start position:0% is the charge density coming from the orbital mu and they feel that they're align:start position:0% orbital mu and they feel that they're align:start position:0% orbital mu and they feel that they're the electron lambda feels is the align:start position:0% the electron lambda feels is the align:start position:0% the electron lambda feels is the electrostatic average density and in align:start position:0% electrostatic average density and in align:start position:0% electrostatic average density and in this app with sum over all the electrons align:start position:0% this app with sum over all the electrons align:start position:0% this app with sum over all the electrons including the electron lambda so up to align:start position:0% including the electron lambda so up to align:start position:0% including the electron lambda so up to now we have a system that is self align:start position:0% now we have a system that is self align:start position:0% now we have a system that is self interacting an electron lambda feels the align:start position:0% interacting an electron lambda feels the align:start position:0% interacting an electron lambda feels the electrostatic interaction with itself align:start position:0% electrostatic interaction with itself align:start position:0% electrostatic interaction with itself that in principle is not correct but align:start position:0% that in principle is not correct but align:start position:0% that in principle is not correct but luckily this next term that is called align:start position:0% luckily this next term that is called align:start position:0% luckily this next term that is called the exchange term cancels that exactly align:start position:0% the exchange term cancels that exactly align:start position:0% the exchange term cancels that exactly and the exchange term is that direct align:start position:0% and the exchange term is that direct align:start position:0% and the exchange term is that direct consequence of having written the trial align:start position:0% consequence of having written the trial align:start position:0% consequence of having written the trial wavefunction not just as a product of align:start position:0% wavefunction not just as a product of align:start position:0% wavefunction not just as a product of single particle orbital because up to align:start position:0% single particle orbital because up to align:start position:0% single particle orbital because up to now we would have sort of something align:start position:0% now we would have sort of something align:start position:0% now we would have sort of something closer to the heart equation but written align:start position:0% closer to the heart equation but written align:start position:0% closer to the heart equation but written as a proper antisymmetric wave function align:start position:0% as a proper antisymmetric wave function align:start position:0% as a proper antisymmetric wave function summing on all the permutation with them align:start position:0% summing on all the permutation with them align:start position:0% summing on all the permutation with them appropriate science' and so basically we align:start position:0% appropriate science' and so basically we align:start position:0% appropriate science' and so basically we are treating it like a question a great align:start position:0% are treating it like a question a great align:start position:0% are treating it like a question a great advantage with respect to the harsh align:start position:0% advantage with respect to the harsh align:start position:0% advantage with respect to the harsh equation is now the operator doesn't align:start position:0% equation is now the operator doesn't align:start position:0% equation is now the operator doesn't change depending on the index lambda align:start position:0% change depending on the index lambda align:start position:0% change depending on the index lambda because this sense if you want to go align:start position:0% because this sense if you want to go align:start position:0% because this sense if you want to go over all the electrons including lambda align:start position:0% over all the electrons including lambda align:start position:0% over all the electrons including lambda so our only constraint here is that we align:start position:0% so our only constraint here is that we align:start position:0% so our only constraint here is that we need to find the N lowest eigenstate of align:start position:0% need to find the N lowest eigenstate of align:start position:0% need to find the N lowest eigenstate of this single differential equation so if align:start position:0% this single differential equation so if align:start position:0% this single differential equation so if we have n electrons if you want it's not align:start position:0% we have n electrons if you want it's not align:start position:0% we have n electrons if you want it's not that we have n different align:start position:0% that we have n different align:start position:0% that we have n different differential equation like it was the align:start position:0% differential equation like it was the align:start position:0% differential equation like it was the case of the Hart equation but we have a align:start position:0% case of the Hart equation but we have a align:start position:0% case of the Hart equation but we have a identical differential equation that is align:start position:0% identical differential equation that is align:start position:0% identical differential equation that is written here and we need to find that align:start position:0% written here and we need to find that align:start position:0% written here and we need to find that the N lowest energy states and those align:start position:0% the N lowest energy states and those align:start position:0% the N lowest energy states and those will be our single particle or because align:start position:0% will be our single particle or because align:start position:0% will be our single particle or because in all of these that we have started align:start position:0% in all of these that we have started align:start position:0% in all of these that we have started from a variational principle so it's align:start position:0% from a variational principle so it's align:start position:0% from a variational principle so it's very easy to go beyond the hartree-fock align:start position:0% very easy to go beyond the hartree-fock align:start position:0% very easy to go beyond the hartree-fock we can say enlarge our variational align:start position:0% we can say enlarge our variational align:start position:0% we can say enlarge our variational classes we can add more Slater align:start position:0% classes we can add more Slater align:start position:0% classes we can add more Slater determinant with sort of different align:start position:0% determinant with sort of different align:start position:0% determinant with sort of different coefficients we can try to construct a align:start position:0% coefficients we can try to construct a align:start position:0% coefficients we can try to construct a more complex wave function and that align:start position:0% more complex wave function and that align:start position:0% more complex wave function and that solution will become better and better align:start position:0% solution will become better and better align:start position:0% solution will become better and better or we can sort of use a perturbation align:start position:0% or we can sort of use a perturbation align:start position:0% or we can sort of use a perturbation theory and so quantum chemistry has align:start position:0% theory and so quantum chemistry has align:start position:0% theory and so quantum chemistry has developed a number of techniques that align:start position:0% developed a number of techniques that align:start position:0% developed a number of techniques that are post r34 techniques that become align:start position:0% are post r34 techniques that become align:start position:0% are post r34 techniques that become systematically more and more accurate align:start position:0% systematically more and more accurate align:start position:0% systematically more and more accurate they are also more and more expensive align:start position:0% they are also more and more expensive align:start position:0% they are also more and more expensive and that's if you want the main align:start position:0% and that's if you want the main align:start position:0% and that's if you want the main limitation of that direction align:start position:0% limitation of that direction align:start position:0% limitation of that direction what we see today is something as they align:start position:0% what we see today is something as they align:start position:0% what we see today is something as they say Monty Python completely different align:start position:0% say Monty Python completely different align:start position:0% say Monty Python completely different and that will be set of density align:start position:0% and that will be set of density align:start position:0% and that will be set of density functional theory that if you want a align:start position:0% functional theory that if you want a align:start position:0% functional theory that if you want a theory that starts from a very different align:start position:0% theory that starts from a very different align:start position:0% theory that starts from a very different set of hypothesis the net result will be align:start position:0% set of hypothesis the net result will be align:start position:0% set of hypothesis the net result will be again a set of single particle equations align:start position:0% again a set of single particle equations align:start position:0% again a set of single particle equations the terrset are very similar actually align:start position:0% the terrset are very similar actually align:start position:0% the terrset are very similar actually formally to the hartree-fock equation align:start position:0% formally to the hartree-fock equation align:start position:0% formally to the hartree-fock equation but they have been derived in a align:start position:0% but they have been derived in a align:start position:0% but they have been derived in a completely different spirit density align:start position:0% completely different spirit density align:start position:0% completely different spirit density function theory tends to be less align:start position:0% function theory tends to be less align:start position:0% function theory tends to be less expensive than hartree-fock and overall align:start position:0% expensive than hartree-fock and overall align:start position:0% expensive than hartree-fock and overall tends to be more accurate especially for align:start position:0% tends to be more accurate especially for align:start position:0% tends to be more accurate especially for solid is much more accurate or you'll align:start position:0% solid is much more accurate or you'll align:start position:0% solid is much more accurate or you'll see when we discuss case studies the align:start position:0% see when we discuss case studies the align:start position:0% see when we discuss case studies the hartree-fock solution for the say align:start position:0% hartree-fock solution for the say align:start position:0% hartree-fock solution for the say interacting electron gas or in general align:start position:0% interacting electron gas or in general align:start position:0% interacting electron gas or in general for metals tends to make them align:start position:0% for metals tends to make them align:start position:0% for metals tends to make them semiconducting or insulating like so align:start position:0% semiconducting or insulating like so align:start position:0% semiconducting or insulating like so hard she folk tend towards very poorly align:start position:0% hard she folk tend towards very poorly align:start position:0% hard she folk tend towards very poorly for solids and that's why if you want align:start position:0% for solids and that's why if you want align:start position:0% for solids and that's why if you want density functional theory comes from the align:start position:0% density functional theory comes from the align:start position:0% density functional theory comes from the solid state community while hartree-fock align:start position:0% solid state community while hartree-fock align:start position:0% solid state community while hartree-fock that tends to work very well for atoms align:start position:0% that tends to work very well for atoms align:start position:0% that tends to work very well for atoms comes from the quantum chemistry align:start position:0% comes from the quantum chemistry align:start position:0% comes from the quantum chemistry community and all the theory was align:start position:0% community and all the theory was align:start position:0% community and all the theory was developed by world corner and coworkers align:start position:0% developed by world corner and coworkers align:start position:0% developed by world corner and coworkers you see the Homburg and con theorem the align:start position:0% you see the Homburg and con theorem the align:start position:0% you see the Homburg and con theorem the connection mapping during the six days align:start position:0% connection mapping during the six days align:start position:0% connection mapping during the six days but I would say it's only during the 70s align:start position:0% but I would say it's only during the 70s align:start position:0% but I would say it's only during the 70s that people started to be able to align:start position:0% that people started to be able to align:start position:0% that people started to be able to actually solve interesting cases using align:start position:0% actually solve interesting cases using align:start position:0% actually solve interesting cases using density functional theory and it's align:start position:0% density functional theory and it's align:start position:0% density functional theory and it's really the beginning of the 80s you'll align:start position:0% really the beginning of the 80s you'll align:start position:0% really the beginning of the 80s you'll see some cases here today in which align:start position:0% see some cases here today in which align:start position:0% see some cases here today in which people started calculating something align:start position:0% people started calculating something align:start position:0% people started calculating something that had sort of a direct application so align:start position:0% that had sort of a direct application so align:start position:0% that had sort of a direct application so we will see the phase diagram of silicon align:start position:0% we will see the phase diagram of silicon align:start position:0% we will see the phase diagram of silicon as a function of pressure or volume and align:start position:0% as a function of pressure or volume and align:start position:0% as a function of pressure or volume and sort of the first first principle align:start position:0% sort of the first first principle align:start position:0% sort of the first first principle prediction of properties of solids align:start position:0% prediction of properties of solids align:start position:0% prediction of properties of solids Walther corner for the development of align:start position:0% Walther corner for the development of align:start position:0% Walther corner for the development of the intervention theory got the Nobel align:start position:0% the intervention theory got the Nobel align:start position:0% the intervention theory got the Nobel Prize for chemistry in 1998 together align:start position:0% Prize for chemistry in 1998 together align:start position:0% Prize for chemistry in 1998 together with John popper that has been the align:start position:0% with John popper that has been the align:start position:0% with John popper that has been the person that has been sort of align:start position:0% person that has been sort of align:start position:0% person that has been sort of most that's been fundamental in the align:start position:0% most that's been fundamental in the align:start position:0% most that's been fundamental in the development of hartree-fock and poor align:start position:0% development of hartree-fock and poor align:start position:0% development of hartree-fock and poor posture terrific approaches in quantum align:start position:0% posture terrific approaches in quantum align:start position:0% posture terrific approaches in quantum chemistry okay so let's see sort of what align:start position:0% chemistry okay so let's see sort of what align:start position:0% chemistry okay so let's see sort of what is that the general idea behind the ends align:start position:0% is that the general idea behind the ends align:start position:0% is that the general idea behind the ends differential theory and in many ways align:start position:0% differential theory and in many ways align:start position:0% differential theory and in many ways will sort of start from idea that had align:start position:0% will sort of start from idea that had align:start position:0% will sort of start from idea that had been developed at the end of the 20s or align:start position:0% been developed at the end of the 20s or align:start position:0% been developed at the end of the 20s or at the beginning of the 30s what is align:start position:0% at the beginning of the 30s what is align:start position:0% at the beginning of the 30s what is nowadays calls the thomas fiering align:start position:0% nowadays calls the thomas fiering align:start position:0% nowadays calls the thomas fiering approach and again the basic idea here align:start position:0% approach and again the basic idea here align:start position:0% approach and again the basic idea here is that the wave function of a many body align:start position:0% is that the wave function of a many body align:start position:0% is that the wave function of a many body interacting problem is an object that is align:start position:0% interacting problem is an object that is align:start position:0% interacting problem is an object that is too complex to treta and it would be align:start position:0% too complex to treta and it would be align:start position:0% too complex to treta and it would be very very nice if we could instead try align:start position:0% very very nice if we could instead try align:start position:0% very very nice if we could instead try to deal with a simple object and sort of align:start position:0% to deal with a simple object and sort of align:start position:0% to deal with a simple object and sort of one of the choices could be the charge align:start position:0% one of the choices could be the charge align:start position:0% one of the choices could be the charge density so if you want a thomas and firm align:start position:0% density so if you want a thomas and firm align:start position:0% density so if you want a thomas and firm independently we're asking themselves align:start position:0% independently we're asking themselves align:start position:0% independently we're asking themselves well could we try to solve not really as align:start position:0% well could we try to solve not really as align:start position:0% well could we try to solve not really as trading an equation in the many-body align:start position:0% trading an equation in the many-body align:start position:0% trading an equation in the many-body wave function but solve something else align:start position:0% wave function but solve something else align:start position:0% wave function but solve something else in which our only unknown is the charge align:start position:0% in which our only unknown is the charge align:start position:0% in which our only unknown is the charge density if you think for a moment the align:start position:0% density if you think for a moment the align:start position:0% density if you think for a moment the charge density is one of the sort of align:start position:0% charge density is one of the sort of align:start position:0% charge density is one of the sort of fundamental variables in the description align:start position:0% fundamental variables in the description align:start position:0% fundamental variables in the description of an interacting electron problem and align:start position:0% of an interacting electron problem and align:start position:0% of an interacting electron problem and so this is this was the question can we align:start position:0% so this is this was the question can we align:start position:0% so this is this was the question can we do something just with the charge align:start position:0% do something just with the charge align:start position:0% do something just with the charge density and so what they did is writing align:start position:0% density and so what they did is writing align:start position:0% density and so what they did is writing out what we would call a heuristic align:start position:0% out what we would call a heuristic align:start position:0% out what we would call a heuristic functional that is trying to devise align:start position:0% functional that is trying to devise align:start position:0% functional that is trying to devise a set of terms that would give us the align:start position:0% a set of terms that would give us the align:start position:0% a set of terms that would give us the energy of a set of electrons in a align:start position:0% energy of a set of electrons in a align:start position:0% energy of a set of electrons in a potential just as a functional of their align:start position:0% potential just as a functional of their align:start position:0% potential just as a functional of their charge density and so you know sort of align:start position:0% charge density and so you know sort of align:start position:0% charge density and so you know sort of by now you could sort of think that some align:start position:0% by now you could sort of think that some align:start position:0% by now you could sort of think that some of you know the relevant terms will be align:start position:0% of you know the relevant terms will be align:start position:0% of you know the relevant terms will be electron-electron interactions electron align:start position:0% electron-electron interactions electron align:start position:0% electron-electron interactions electron interact and we could write a set of align:start position:0% interact and we could write a set of align:start position:0% interact and we could write a set of electrostatic term like the Hartree term align:start position:0% electrostatic term like the Hartree term align:start position:0% electrostatic term like the Hartree term in the heart or the hartree-fock align:start position:0% in the heart or the hartree-fock align:start position:0% in the heart or the hartree-fock equation that is just a functional of align:start position:0% equation that is just a functional of align:start position:0% equation that is just a functional of the charge density so this is sort of align:start position:0% the charge density so this is sort of align:start position:0% the charge density so this is sort of fairly easy it's also very easy to set align:start position:0% fairly easy it's also very easy to set align:start position:0% fairly easy it's also very easy to set up you know imagine what could be the align:start position:0% up you know imagine what could be the align:start position:0% up you know imagine what could be the interaction of the electrons with an align:start position:0% interaction of the electrons with an align:start position:0% interaction of the electrons with an external potential through the charge align:start position:0% external potential through the charge align:start position:0% external potential through the charge density will be just the integral of align:start position:0% density will be just the integral of align:start position:0% density will be just the integral of that external potential times the charge align:start position:0% that external potential times the charge align:start position:0% that external potential times the charge density what becomes really critical is align:start position:0% density what becomes really critical is align:start position:0% density what becomes really critical is you know finding a align:start position:0% you know finding a align:start position:0% you know finding a functional that will give us the quantum align:start position:0% functional that will give us the quantum align:start position:0% functional that will give us the quantum kinetic energy if you think in the align:start position:0% kinetic energy if you think in the align:start position:0% kinetic energy if you think in the Schrodinger equation the quantum kinetic align:start position:0% Schrodinger equation the quantum kinetic align:start position:0% Schrodinger equation the quantum kinetic energy is really the second derivative align:start position:0% energy is really the second derivative align:start position:0% energy is really the second derivative of the wave function and align:start position:0% of the wave function and align:start position:0% of the wave function and obtaining from a charge density only align:start position:0% obtaining from a charge density only align:start position:0% obtaining from a charge density only some insight into what could be the align:start position:0% some insight into what could be the align:start position:0% some insight into what could be the second Riv whatever the wavefunction is align:start position:0% second Riv whatever the wavefunction is align:start position:0% second Riv whatever the wavefunction is very complex if you think for a moment align:start position:0% very complex if you think for a moment align:start position:0% very complex if you think for a moment at the extreme case of a plane wave okay align:start position:0% at the extreme case of a plane wave okay align:start position:0% at the extreme case of a plane wave okay so a sine and cosine sort of in space align:start position:0% so a sine and cosine sort of in space align:start position:0% so a sine and cosine sort of in space remember the charge density given by a align:start position:0% remember the charge density given by a align:start position:0% remember the charge density given by a plane wave is a constant we just align:start position:0% plane wave is a constant we just align:start position:0% plane wave is a constant we just multiply the exponential times the align:start position:0% multiply the exponential times the align:start position:0% multiply the exponential times the company imaginary exponential times its align:start position:0% company imaginary exponential times its align:start position:0% company imaginary exponential times its complex conjugate that gives us a align:start position:0% complex conjugate that gives us a align:start position:0% complex conjugate that gives us a constant so all plane waves lead to a align:start position:0% constant so all plane waves lead to a align:start position:0% constant so all plane waves lead to a constant but obviously the quantum align:start position:0% constant but obviously the quantum align:start position:0% constant but obviously the quantum kinetic energy of a plane wave depends align:start position:0% kinetic energy of a plane wave depends align:start position:0% kinetic energy of a plane wave depends on the wave length of that plane wave align:start position:0% on the wave length of that plane wave align:start position:0% on the wave length of that plane wave because the second derivative is what align:start position:0% because the second derivative is what align:start position:0% because the second derivative is what counts up so what I'm trying to say is align:start position:0% counts up so what I'm trying to say is align:start position:0% counts up so what I'm trying to say is that when we look at sort of this as a align:start position:0% that when we look at sort of this as a align:start position:0% that when we look at sort of this as a possible wave function a align:start position:0% possible wave function a align:start position:0% possible wave function a function say of R and the charge density align:start position:0% function say of R and the charge density align:start position:0% function say of R and the charge density that comes from this is going to be a align:start position:0% that comes from this is going to be a align:start position:0% that comes from this is going to be a constant this wave function times this align:start position:0% constant this wave function times this align:start position:0% constant this wave function times this complex conjugate but the kinetic energy align:start position:0% complex conjugate but the kinetic energy align:start position:0% complex conjugate but the kinetic energy of this object align:start position:0% of this object align:start position:0% of this object is going to be minus 1/2 K square sorry align:start position:0% is going to be minus 1/2 K square sorry align:start position:0% is going to be minus 1/2 K square sorry plus 1/2 K square and so there is really align:start position:0% plus 1/2 K square and so there is really align:start position:0% plus 1/2 K square and so there is really not a good way for this extreme case align:start position:0% not a good way for this extreme case align:start position:0% not a good way for this extreme case that to correlate its charge density to align:start position:0% that to correlate its charge density to align:start position:0% that to correlate its charge density to the kinetic energy it's an ill-defined align:start position:0% the kinetic energy it's an ill-defined align:start position:0% the kinetic energy it's an ill-defined problem and this is really the align:start position:0% problem and this is really the align:start position:0% problem and this is really the difficulty ok so there isn't really a align:start position:0% difficulty ok so there isn't really a align:start position:0% difficulty ok so there isn't really a good way if you wanted to extract the align:start position:0% good way if you wanted to extract the align:start position:0% good way if you wanted to extract the information on the second derivative align:start position:0% information on the second derivative align:start position:0% information on the second derivative from just a charge density align:start position:0% from just a charge density align:start position:0% from just a charge density no matter sort of this objection they align:start position:0% no matter sort of this objection they align:start position:0% no matter sort of this objection they tried sort of to find a align:start position:0% tried sort of to find a align:start position:0% tried sort of to find a reasonable functional so without sort of align:start position:0% reasonable functional so without sort of align:start position:0% reasonable functional so without sort of trying to get the exact solution but try align:start position:0% trying to get the exact solution but try align:start position:0% trying to get the exact solution but try to find a reasonable functional that align:start position:0% to find a reasonable functional that align:start position:0% to find a reasonable functional that would give us a good estimate to the align:start position:0% would give us a good estimate to the align:start position:0% would give us a good estimate to the kinetic the quantum kinetic energy align:start position:0% kinetic the quantum kinetic energy align:start position:0% kinetic the quantum kinetic energy starting from the charge density and the align:start position:0% starting from the charge density and the align:start position:0% starting from the charge density and the solution to this problem that is align:start position:0% solution to this problem that is align:start position:0% solution to this problem that is something very important is what we align:start position:0% something very important is what we align:start position:0% something very important is what we could call a local density approximation align:start position:0% could call a local density approximation align:start position:0% could call a local density approximation so the problem here is that we ever align:start position:0% so the problem here is that we ever align:start position:0% so the problem here is that we ever known amo genius charge density align:start position:0% known amo genius charge density align:start position:0% known amo genius charge density everywhere in space and we try to figure align:start position:0% everywhere in space and we try to figure align:start position:0% everywhere in space and we try to figure out what could be the quantum kinetic align:start position:0% out what could be the quantum kinetic align:start position:0% out what could be the quantum kinetic energy of this non-homogeneous problem align:start position:0% energy of this non-homogeneous problem align:start position:0% energy of this non-homogeneous problem and align:start position:0% and align:start position:0% and set of the approximation that Thomason align:start position:0% set of the approximation that Thomason align:start position:0% set of the approximation that Thomason Fermi Dida was that are well dividing align:start position:0% Fermi Dida was that are well dividing align:start position:0% Fermi Dida was that are well dividing this non-homogeneous problem in a set of align:start position:0% this non-homogeneous problem in a set of align:start position:0% this non-homogeneous problem in a set of sort of infinitesimal volume in space align:start position:0% sort of infinitesimal volume in space align:start position:0% sort of infinitesimal volume in space and so it's a bit difficult to draw but align:start position:0% and so it's a bit difficult to draw but align:start position:0% and so it's a bit difficult to draw but suppose you have the density charge align:start position:0% suppose you have the density charge align:start position:0% suppose you have the density charge density coming from some atom or some align:start position:0% density coming from some atom or some align:start position:0% density coming from some atom or some molecule this is an align:start position:0% molecule this is an align:start position:0% molecule this is an non-homogeneous charge density align:start position:0% non-homogeneous charge density align:start position:0% non-homogeneous charge density distribution now what you do is you align:start position:0% distribution now what you do is you align:start position:0% distribution now what you do is you divide this in space in such a very align:start position:0% divide this in space in such a very align:start position:0% divide this in space in such a very small infinitesimal if you want volume align:start position:0% small infinitesimal if you want volume align:start position:0% small infinitesimal if you want volume and align:start position:0% and align:start position:0% and inside each volume the charge density align:start position:0% inside each volume the charge density align:start position:0% inside each volume the charge density can be approximated as a constant and align:start position:0% can be approximated as a constant and align:start position:0% can be approximated as a constant and what Thomas and Fermi said is well the align:start position:0% what Thomas and Fermi said is well the align:start position:0% what Thomas and Fermi said is well the contribution coming from this align:start position:0% contribution coming from this align:start position:0% contribution coming from this infinitesimal volume say the first one align:start position:0% infinitesimal volume say the first one align:start position:0% infinitesimal volume say the first one to the overall quantum kinetic energy align:start position:0% to the overall quantum kinetic energy align:start position:0% to the overall quantum kinetic energy will be given by that volume times the align:start position:0% will be given by that volume times the align:start position:0% will be given by that volume times the kinetic energy density of the align:start position:0% kinetic energy density of the align:start position:0% kinetic energy density of the homogeneous electron Gaza at that align:start position:0% homogeneous electron Gaza at that align:start position:0% homogeneous electron Gaza at that density so if again we partition all align:start position:0% density so if again we partition all align:start position:0% density so if again we partition all space we could have that you know the align:start position:0% space we could have that you know the align:start position:0% space we could have that you know the density in this little cube is point 5 align:start position:0% density in this little cube is point 5 align:start position:0% density in this little cube is point 5 here is point 6 here is point 7 outside align:start position:0% here is point 6 here is point 7 outside align:start position:0% here is point 6 here is point 7 outside it goes to 0 but we can actually align:start position:0% it goes to 0 but we can actually align:start position:0% it goes to 0 but we can actually calculate in some other way what would align:start position:0% calculate in some other way what would align:start position:0% calculate in some other way what would be the quantum kinetic energy of a align:start position:0% be the quantum kinetic energy of a align:start position:0% be the quantum kinetic energy of a homogeneous electron gas that's a align:start position:0% homogeneous electron gas that's a align:start position:0% homogeneous electron gas that's a problem that we can solve if the align:start position:0% problem that we can solve if the align:start position:0% problem that we can solve if the homogeneous electron gas is not align:start position:0% homogeneous electron gas is not align:start position:0% homogeneous electron gas is not interacting and we can solve it align:start position:0% interacting and we can solve it align:start position:0% interacting and we can solve it numerically even if it is interacting so align:start position:0% numerically even if it is interacting so align:start position:0% numerically even if it is interacting so we can know what is the quantum kinetic align:start position:0% we can know what is the quantum kinetic align:start position:0% we can know what is the quantum kinetic energy of a homogeneous gas with density align:start position:0% energy of a homogeneous gas with density align:start position:0% energy of a homogeneous gas with density point 5 density point 6 density point 7 align:start position:0% point 5 density point 6 density point 7 align:start position:0% point 5 density point 6 density point 7 and so we can also know what would be align:start position:0% and so we can also know what would be align:start position:0% and so we can also know what would be the quantum kinetic energy per unit of align:start position:0% the quantum kinetic energy per unit of align:start position:0% the quantum kinetic energy per unit of volume of data and so we'll say that align:start position:0% volume of data and so we'll say that align:start position:0% volume of data and so we'll say that this non-homogeneous system in blue will align:start position:0% this non-homogeneous system in blue will align:start position:0% this non-homogeneous system in blue will have an overall quantum kinetic energy align:start position:0% have an overall quantum kinetic energy align:start position:0% have an overall quantum kinetic energy that is given really by the integral align:start position:0% that is given really by the integral align:start position:0% that is given really by the integral across space and it's written here of align:start position:0% across space and it's written here of align:start position:0% across space and it's written here of the quantum kinetic energy of the align:start position:0% the quantum kinetic energy of the align:start position:0% the quantum kinetic energy of the homogeneous electron gas integrated over align:start position:0% homogeneous electron gas integrated over align:start position:0% homogeneous electron gas integrated over space and say for the non-interacting align:start position:0% space and say for the non-interacting align:start position:0% space and say for the non-interacting electrons ASSA is that really very easy align:start position:0% electrons ASSA is that really very easy align:start position:0% electrons ASSA is that really very easy to do so if you ever known interacting align:start position:0% to do so if you ever known interacting align:start position:0% to do so if you ever known interacting electron gas at a density Rho its align:start position:0% electron gas at a density Rho its align:start position:0% electron gas at a density Rho its quantum kinetic energy is just Rho to align:start position:0% quantum kinetic energy is just Rho to align:start position:0% quantum kinetic energy is just Rho to the 2/3 that then integrated time the align:start position:0% the 2/3 that then integrated time the align:start position:0% the 2/3 that then integrated time the unit volume gives as an Rho to the 5/3 align:start position:0% unit volume gives as an Rho to the 5/3 align:start position:0% unit volume gives as an Rho to the 5/3 so by integrating this quantity we would align:start position:0% so by integrating this quantity we would align:start position:0% so by integrating this quantity we would get an approximation this approximation align:start position:0% get an approximation this approximation align:start position:0% get an approximation this approximation is basically exact in the limit of a align:start position:0% is basically exact in the limit of a align:start position:0% is basically exact in the limit of a homogeneous system obviously and it will align:start position:0% homogeneous system obviously and it will align:start position:0% homogeneous system obviously and it will be sort of quite good in the limiter of align:start position:0% be sort of quite good in the limiter of align:start position:0% be sort of quite good in the limiter of a non homogeneous system the tears are align:start position:0% a non homogeneous system the tears are align:start position:0% a non homogeneous system the tears are very slowly changing charge density the align:start position:0% very slowly changing charge density the align:start position:0% very slowly changing charge density the more if you want a inhomogeneous your align:start position:0% more if you want a inhomogeneous your align:start position:0% more if you want a inhomogeneous your system becomes the less accurate this align:start position:0% system becomes the less accurate this align:start position:0% system becomes the less accurate this approximation is and of course something align:start position:0% approximation is and of course something align:start position:0% approximation is and of course something like an atom or a molecule is a very align:start position:0% like an atom or a molecule is a very align:start position:0% like an atom or a molecule is a very inhomogeneous system you go with the align:start position:0% inhomogeneous system you go with the align:start position:0% inhomogeneous system you go with the charge density align:start position:0% charge density align:start position:0% charge density that goes from zero to very high volumes align:start position:0% that goes from zero to very high volumes align:start position:0% that goes from zero to very high volumes close to the core of the nuclei align:start position:0% align:start position:0% so this is basically the overall on align:start position:0% so this is basically the overall on align:start position:0% so this is basically the overall on Saturday overall expression the Thomas align:start position:0% Saturday overall expression the Thomas align:start position:0% Saturday overall expression the Thomas and Fermi postulated for the energy of align:start position:0% and Fermi postulated for the energy of align:start position:0% and Fermi postulated for the energy of an inhomogeneous system they were saying align:start position:0% an inhomogeneous system they were saying align:start position:0% an inhomogeneous system they were saying well suppose that we have a system that align:start position:0% well suppose that we have a system that align:start position:0% well suppose that we have a system that there's a certain distribution of charge align:start position:0% there's a certain distribution of charge align:start position:0% there's a certain distribution of charge row without trying to solve the align:start position:0% row without trying to solve the align:start position:0% row without trying to solve the Schrodinger equation finding out the align:start position:0% Schrodinger equation finding out the align:start position:0% Schrodinger equation finding out the wavefunction and sort of go through that align:start position:0% wavefunction and sort of go through that align:start position:0% wavefunction and sort of go through that is a very complex many-body router we align:start position:0% is a very complex many-body router we align:start position:0% is a very complex many-body router we can actually set up postulate that the align:start position:0% can actually set up postulate that the align:start position:0% can actually set up postulate that the energy could be written again as an align:start position:0% energy could be written again as an align:start position:0% energy could be written again as an electrostatic energy you see set of each align:start position:0% electrostatic energy you see set of each align:start position:0% electrostatic energy you see set of each infinitesimal volume interacting with align:start position:0% infinitesimal volume interacting with align:start position:0% infinitesimal volume interacting with each other infinitesimal volume times align:start position:0% each other infinitesimal volume times align:start position:0% each other infinitesimal volume times via one over our electrostatic align:start position:0% via one over our electrostatic align:start position:0% via one over our electrostatic interaction then we have got an external align:start position:0% interaction then we have got an external align:start position:0% interaction then we have got an external potential again it's usually the align:start position:0% potential again it's usually the align:start position:0% potential again it's usually the columbic field of the nuclei and so the align:start position:0% columbic field of the nuclei and so the align:start position:0% columbic field of the nuclei and so the interaction between the electron and align:start position:0% interaction between the electron and align:start position:0% interaction between the electron and that external potential is just align:start position:0% that external potential is just align:start position:0% that external potential is just trivially given by Rho times V and the align:start position:0% trivially given by Rho times V and the align:start position:0% trivially given by Rho times V and the difficult term the quantum kinetic align:start position:0% difficult term the quantum kinetic align:start position:0% difficult term the quantum kinetic energy has been calculated with a local align:start position:0% energy has been calculated with a local align:start position:0% energy has been calculated with a local density approximation and this is the align:start position:0% density approximation and this is the align:start position:0% density approximation and this is the term that's not going to be very good align:start position:0% term that's not going to be very good align:start position:0% term that's not going to be very good again because it's very difficult to align:start position:0% again because it's very difficult to align:start position:0% again because it's very difficult to figure out what could be the curvature align:start position:0% figure out what could be the curvature align:start position:0% figure out what could be the curvature of our wave function align:start position:0% of our wave function align:start position:0% of our wave function just from the density that that wave align:start position:0% just from the density that that wave align:start position:0% just from the density that that wave function produces but anyhow this is a align:start position:0% function produces but anyhow this is a align:start position:0% function produces but anyhow this is a very simple expression to deal with so align:start position:0% very simple expression to deal with so align:start position:0% very simple expression to deal with so for any external potential V we can try align:start position:0% for any external potential V we can try align:start position:0% for any external potential V we can try to find out the row that minimizes this align:start position:0% to find out the row that minimizes this align:start position:0% to find out the row that minimizes this expression and this will be our thomas align:start position:0% expression and this will be our thomas align:start position:0% expression and this will be our thomas fermi solution align:start position:0% align:start position:0% there are obviously a number of problems align:start position:0% there are obviously a number of problems align:start position:0% there are obviously a number of problems are showing a moment and example of what align:start position:0% are showing a moment and example of what align:start position:0% are showing a moment and example of what the thomas fiering solution would give align:start position:0% the thomas fiering solution would give align:start position:0% the thomas fiering solution would give to an atom first of all i mean there is align:start position:0% to an atom first of all i mean there is align:start position:0% to an atom first of all i mean there is really no theoretical basis to this it's align:start position:0% really no theoretical basis to this it's align:start position:0% really no theoretical basis to this it's what we call a heuristic derivation align:start position:0% what we call a heuristic derivation align:start position:0% what we call a heuristic derivation Thomas an Fermi just wrote out what align:start position:0% Thomas an Fermi just wrote out what align:start position:0% Thomas an Fermi just wrote out what could be a regional energy functional align:start position:0% could be a regional energy functional align:start position:0% could be a regional energy functional and then try to sort of see what results align:start position:0% and then try to sort of see what results align:start position:0% and then try to sort of see what results it would give but there hasn't been any align:start position:0% it would give but there hasn't been any align:start position:0% it would give but there hasn't been any kind of you know formal derivation of align:start position:0% kind of you know formal derivation of align:start position:0% kind of you know formal derivation of that functional it's not like the align:start position:0% that functional it's not like the align:start position:0% that functional it's not like the hartree-fock equation that sort of align:start position:0% hartree-fock equation that sort of align:start position:0% hartree-fock equation that sort of derive just with some analysis from the align:start position:0% derive just with some analysis from the align:start position:0% derive just with some analysis from the variational principle align:start position:0% variational principle align:start position:0% variational principle another problem is that again it doesn't align:start position:0% another problem is that again it doesn't align:start position:0% another problem is that again it doesn't really sort of introduce the concept of align:start position:0% really sort of introduce the concept of align:start position:0% really sort of introduce the concept of anti symmetry that fermions need to have align:start position:0% anti symmetry that fermions need to have align:start position:0% anti symmetry that fermions need to have the fact that they many-body wave align:start position:0% the fact that they many-body wave align:start position:0% the fact that they many-body wave function needs to be antisymmetric upon align:start position:0% function needs to be antisymmetric upon align:start position:0% function needs to be antisymmetric upon exchange but you know there is no align:start position:0% exchange but you know there is no align:start position:0% exchange but you know there is no conceptual problem in adding and align:start position:0% conceptual problem in adding and align:start position:0% conceptual problem in adding and exchange energy to the previous align:start position:0% exchange energy to the previous align:start position:0% exchange energy to the previous functional using the same concept that align:start position:0% functional using the same concept that align:start position:0% functional using the same concept that the same idea of local density align:start position:0% the same idea of local density align:start position:0% the same idea of local density approximation suppose that we want to align:start position:0% approximation suppose that we want to align:start position:0% approximation suppose that we want to add an exchange term well we could look align:start position:0% add an exchange term well we could look align:start position:0% add an exchange term well we could look at what is the exchange energy coming align:start position:0% at what is the exchange energy coming align:start position:0% at what is the exchange energy coming from the hartree-fock equations say for align:start position:0% from the hartree-fock equations say for align:start position:0% from the hartree-fock equations say for a homogeneous electron gaza and that align:start position:0% a homogeneous electron gaza and that align:start position:0% a homogeneous electron gaza and that gives us a row to the one-third term and align:start position:0% gives us a row to the one-third term and align:start position:0% gives us a row to the one-third term and that's basically the exchange energy align:start position:0% that's basically the exchange energy align:start position:0% that's basically the exchange energy density and so for an inhomogeneous align:start position:0% density and so for an inhomogeneous align:start position:0% density and so for an inhomogeneous system we are going to sort of align:start position:0% system we are going to sort of align:start position:0% system we are going to sort of approximate its overall exchange energy align:start position:0% approximate its overall exchange energy align:start position:0% approximate its overall exchange energy just by taking the integral of that align:start position:0% just by taking the integral of that align:start position:0% just by taking the integral of that energy density that is one further times align:start position:0% energy density that is one further times align:start position:0% energy density that is one further times the sort of local value of the charge align:start position:0% the sort of local value of the charge align:start position:0% the sort of local value of the charge density and so we have a row to the 4/3 align:start position:0% density and so we have a row to the 4/3 align:start position:0% density and so we have a row to the 4/3 and so again it's a local density align:start position:0% and so again it's a local density align:start position:0% and so again it's a local density approximation align:start position:0% approximation align:start position:0% approximation they sort of great consequence of having align:start position:0% they sort of great consequence of having align:start position:0% they sort of great consequence of having this align:start position:0% this align:start position:0% this energy functional that depends only on R align:start position:0% energy functional that depends only on R align:start position:0% energy functional that depends only on R is that it is absolutely inexpensive align:start position:0% is that it is absolutely inexpensive align:start position:0% is that it is absolutely inexpensive from the computational point of view the align:start position:0% from the computational point of view the align:start position:0% from the computational point of view the only variable that we need to be align:start position:0% only variable that we need to be align:start position:0% only variable that we need to be concerned with is just align:start position:0% concerned with is just align:start position:0% concerned with is just escalara as a function of three align:start position:0% escalara as a function of three align:start position:0% escalara as a function of three coordinates that is the density as a align:start position:0% coordinates that is the density as a align:start position:0% coordinates that is the density as a function of Rho and it's what we call a align:start position:0% function of Rho and it's what we call a align:start position:0% function of Rho and it's what we call a linear scaling system if you double the align:start position:0% linear scaling system if you double the align:start position:0% linear scaling system if you double the size of your system the computational align:start position:0% size of your system the computational align:start position:0% size of your system the computational complexity just becomes double so it has align:start position:0% complexity just becomes double so it has align:start position:0% complexity just becomes double so it has a lot of very good things but it got a align:start position:0% a lot of very good things but it got a align:start position:0% a lot of very good things but it got a fundamental defect because of that align:start position:0% fundamental defect because of that align:start position:0% fundamental defect because of that approximation in the kinetic energy it align:start position:0% approximation in the kinetic energy it align:start position:0% approximation in the kinetic energy it actually does a very poor job in a align:start position:0% actually does a very poor job in a align:start position:0% actually does a very poor job in a describing a non homogeneous system so align:start position:0% describing a non homogeneous system so align:start position:0% describing a non homogeneous system so it would work reasonably well for align:start position:0% it would work reasonably well for align:start position:0% it would work reasonably well for something like a mental suppose that you align:start position:0% something like a mental suppose that you align:start position:0% something like a mental suppose that you want to describe a sodium or suppose you align:start position:0% want to describe a sodium or suppose you align:start position:0% want to describe a sodium or suppose you want to describe aluminum those are align:start position:0% want to describe aluminum those are align:start position:0% want to describe aluminum those are system in which the valence electron align:start position:0% system in which the valence electron align:start position:0% system in which the valence electron produce a charge density that is very align:start position:0% produce a charge density that is very align:start position:0% produce a charge density that is very homogeneous so a thomas fermi approach align:start position:0% homogeneous so a thomas fermi approach align:start position:0% homogeneous so a thomas fermi approach could actually work well and it's align:start position:0% could actually work well and it's align:start position:0% could actually work well and it's actually been used even very recently align:start position:0% actually been used even very recently align:start position:0% actually been used even very recently sort of quite successfully to describe align:start position:0% sort of quite successfully to describe align:start position:0% sort of quite successfully to describe problems like the surfaces of lithium align:start position:0% problems like the surfaces of lithium align:start position:0% problems like the surfaces of lithium the surfaces of aluminum what happens align:start position:0% the surfaces of aluminum what happens align:start position:0% the surfaces of aluminum what happens say what when these simple metals melt align:start position:0% say what when these simple metals melt align:start position:0% say what when these simple metals melt what happens to the sort of formation of align:start position:0% what happens to the sort of formation of align:start position:0% what happens to the sort of formation of defects in aluminum so there are a align:start position:0% defects in aluminum so there are a align:start position:0% defects in aluminum so there are a number of successes but sort of you know align:start position:0% number of successes but sort of you know align:start position:0% number of successes but sort of you know clear example of what goes wrong is if align:start position:0% clear example of what goes wrong is if align:start position:0% clear example of what goes wrong is if we study an inhomogeneous system like align:start position:0% we study an inhomogeneous system like align:start position:0% we study an inhomogeneous system like the argon atom and again if we think at align:start position:0% the argon atom and again if we think at align:start position:0% the argon atom and again if we think at the charge density of the argon atom as align:start position:0% the charge density of the argon atom as align:start position:0% the charge density of the argon atom as a function say of the radial distance align:start position:0% a function say of the radial distance align:start position:0% a function say of the radial distance from the centre from the nucleus well it align:start position:0% from the centre from the nucleus well it align:start position:0% from the centre from the nucleus well it will look something like this we have align:start position:0% will look something like this we have align:start position:0% will look something like this we have first a 1s and then we have the 2s and align:start position:0% first a 1s and then we have the 2s and align:start position:0% first a 1s and then we have the 2s and the 2p shells okay this is somewhat a align:start position:0% the 2p shells okay this is somewhat a align:start position:0% the 2p shells okay this is somewhat a poor depiction of that charge density if align:start position:0% poor depiction of that charge density if align:start position:0% poor depiction of that charge density if we try to solve the argon atom with a align:start position:0% we try to solve the argon atom with a align:start position:0% we try to solve the argon atom with a thomas fermi approach all these sort of align:start position:0% thomas fermi approach all these sort of align:start position:0% thomas fermi approach all these sort of you know fine structure of the core align:start position:0% you know fine structure of the core align:start position:0% you know fine structure of the core shells in the atoms is completely washed align:start position:0% shells in the atoms is completely washed align:start position:0% shells in the atoms is completely washed out okay so it gives you a reasonable align:start position:0% out okay so it gives you a reasonable align:start position:0% out okay so it gives you a reasonable approximation and a sort of an align:start position:0% approximation and a sort of an align:start position:0% approximation and a sort of an appropriate decay of the charge density align:start position:0% appropriate decay of the charge density align:start position:0% appropriate decay of the charge density as we move far away but a lot of those align:start position:0% as we move far away but a lot of those align:start position:0% as we move far away but a lot of those details have completely disappeared and align:start position:0% details have completely disappeared and align:start position:0% details have completely disappeared and for this reason really the Thomas film align:start position:0% for this reason really the Thomas film align:start position:0% for this reason really the Thomas film yeah align:start position:0% yeah align:start position:0% yeah wasn't developed beyond the firt is a align:start position:0% wasn't developed beyond the firt is a align:start position:0% wasn't developed beyond the firt is a bathroom sort of you know some of this align:start position:0% bathroom sort of you know some of this align:start position:0% bathroom sort of you know some of this recent application for the very specific align:start position:0% recent application for the very specific align:start position:0% recent application for the very specific case of solids that have a very ominous align:start position:0% case of solids that have a very ominous align:start position:0% case of solids that have a very ominous charge density the reason why we align:start position:0% charge density the reason why we align:start position:0% charge density the reason why we described it here is that because in align:start position:0% described it here is that because in align:start position:0% described it here is that because in many ways it's the grandfather of the align:start position:0% many ways it's the grandfather of the align:start position:0% many ways it's the grandfather of the ideas that were developed in the 60s in align:start position:0% ideas that were developed in the 60s in align:start position:0% ideas that were developed in the 60s in that's the functional theory and in align:start position:0% that's the functional theory and in align:start position:0% that's the functional theory and in particular the idea that for a moment we align:start position:0% particular the idea that for a moment we align:start position:0% particular the idea that for a moment we should focus not on the wavefunction but align:start position:0% should focus not on the wavefunction but align:start position:0% should focus not on the wavefunction but on the charge density of the system as align:start position:0% on the charge density of the system as align:start position:0% on the charge density of the system as the key ingredient align:start position:0% the key ingredient align:start position:0% the key ingredient the great difference between the Thomas align:start position:0% the great difference between the Thomas align:start position:0% the great difference between the Thomas Fermi approach and density functional align:start position:0% Fermi approach and density functional align:start position:0% Fermi approach and density functional theory is that density functional theory align:start position:0% theory is that density functional theory align:start position:0% theory is that density functional theory actually is a theory it starts with some align:start position:0% actually is a theory it starts with some align:start position:0% actually is a theory it starts with some theorems that are proven and then it align:start position:0% theorems that are proven and then it align:start position:0% theorems that are proven and then it shows what are the form of the equations align:start position:0% shows what are the form of the equations align:start position:0% shows what are the form of the equations that say a charge density need to align:start position:0% that say a charge density need to align:start position:0% that say a charge density need to satisfy in order to solve exactly the align:start position:0% satisfy in order to solve exactly the align:start position:0% satisfy in order to solve exactly the problem so in many ways the inste align:start position:0% problem so in many ways the inste align:start position:0% problem so in many ways the inste functional theory is an in principle at align:start position:0% functional theory is an in principle at align:start position:0% functional theory is an in principle at least an exact theory it's a top writes align:start position:0% least an exact theory it's a top writes align:start position:0% least an exact theory it's a top writes out what are the equation that the align:start position:0% out what are the equation that the align:start position:0% out what are the equation that the charge density needs to satisfy and align:start position:0% charge density needs to satisfy and align:start position:0% charge density needs to satisfy and those are absolutely equivalent to a align:start position:0% those are absolutely equivalent to a align:start position:0% those are absolutely equivalent to a Schrodinger equation for the wave align:start position:0% Schrodinger equation for the wave align:start position:0% Schrodinger equation for the wave function there are some difficulties and align:start position:0% function there are some difficulties and align:start position:0% function there are some difficulties and this is what we are going to sort of go align:start position:0% this is what we are going to sort of go align:start position:0% this is what we are going to sort of go into right now but sort of let me first align:start position:0% into right now but sort of let me first align:start position:0% into right now but sort of let me first give you the conceptual framework of align:start position:0% give you the conceptual framework of align:start position:0% give you the conceptual framework of density functional theory and sort of align:start position:0% density functional theory and sort of align:start position:0% density functional theory and sort of how it was derived and as usual we align:start position:0% how it was derived and as usual we align:start position:0% how it was derived and as usual we started from the Schrodinger equation align:start position:0% started from the Schrodinger equation align:start position:0% started from the Schrodinger equation okay so we start from the idea that in align:start position:0% okay so we start from the idea that in align:start position:0% okay so we start from the idea that in quantum mechanics for any given external align:start position:0% quantum mechanics for any given external align:start position:0% quantum mechanics for any given external potential you have a well-defined align:start position:0% potential you have a well-defined align:start position:0% potential you have a well-defined differential equation okay it's sort of align:start position:0% differential equation okay it's sort of align:start position:0% differential equation okay it's sort of very complex it describes a many-body align:start position:0% very complex it describes a many-body align:start position:0% very complex it describes a many-body wave function so in most practical cases align:start position:0% wave function so in most practical cases align:start position:0% wave function so in most practical cases we might not be able to solve it but align:start position:0% we might not be able to solve it but align:start position:0% we might not be able to solve it but everything is well-defined you have an align:start position:0% everything is well-defined you have an align:start position:0% everything is well-defined you have an external potential you have the align:start position:0% external potential you have the align:start position:0% external potential you have the differential equation that the many-body align:start position:0% differential equation that the many-body align:start position:0% differential equation that the many-body wave function needs to satisfy and so in align:start position:0% wave function needs to satisfy and so in align:start position:0% wave function needs to satisfy and so in principle you have the solution and so align:start position:0% principle you have the solution and so align:start position:0% principle you have the solution and so in that sense that sort of you know the align:start position:0% in that sense that sort of you know the align:start position:0% in that sense that sort of you know the first statement here is summarized for a align:start position:0% first statement here is summarized for a align:start position:0% first statement here is summarized for a given external potential and knowing how align:start position:0% given external potential and knowing how align:start position:0% given external potential and knowing how many electrons are going to fill this align:start position:0% many electrons are going to fill this align:start position:0% many electrons are going to fill this potential our quantum problem is align:start position:0% potential our quantum problem is align:start position:0% potential our quantum problem is formally completely defined in principle align:start position:0% formally completely defined in principle align:start position:0% formally completely defined in principle the solution exists unique we may not be align:start position:0% the solution exists unique we may not be align:start position:0% the solution exists unique we may not be able to calculate it but it exists and align:start position:0% able to calculate it but it exists and align:start position:0% able to calculate it but it exists and once we know the many-body wave function align:start position:0% once we know the many-body wave function align:start position:0% once we know the many-body wave function that solution we know everything about align:start position:0% that solution we know everything about align:start position:0% that solution we know everything about our quantum system okay so this is if align:start position:0% our quantum system okay so this is if align:start position:0% our quantum system okay so this is if you want the trivial part of the align:start position:0% you want the trivial part of the align:start position:0% you want the trivial part of the conclusion that is given an external align:start position:0% conclusion that is given an external align:start position:0% conclusion that is given an external potential we find by the shading align:start position:0% potential we find by the shading align:start position:0% potential we find by the shading equation the wave function the wave align:start position:0% equation the wave function the wave align:start position:0% equation the wave function the wave function determine all the properties of align:start position:0% function determine all the properties of align:start position:0% function determine all the properties of our system and in particular determine align:start position:0% our system and in particular determine align:start position:0% our system and in particular determine the ground state charge density so there align:start position:0% the ground state charge density so there align:start position:0% the ground state charge density so there is a unique pathway that starts from the align:start position:0% is a unique pathway that starts from the align:start position:0% is a unique pathway that starts from the external potential and leads us to the align:start position:0% external potential and leads us to the align:start position:0% external potential and leads us to the charge density the ground state charge align:start position:0% charge density the ground state charge align:start position:0% charge density the ground state charge density once you have defined a align:start position:0% density once you have defined a align:start position:0% density once you have defined a potential you in principle have uniquely align:start position:0% potential you in principle have uniquely align:start position:0% potential you in principle have uniquely defined what is the ground state charge align:start position:0% defined what is the ground state charge align:start position:0% defined what is the ground state charge density of your system and so in that align:start position:0% density of your system and so in that align:start position:0% density of your system and so in that sense we say that the ground state align:start position:0% sense we say that the ground state align:start position:0% sense we say that the ground state charge density the ground state energy align:start position:0% charge density the ground state energy align:start position:0% charge density the ground state energy and all the properties of our system are align:start position:0% and all the properties of our system are align:start position:0% and all the properties of our system are in some complex way a functional of our align:start position:0% in some complex way a functional of our align:start position:0% in some complex way a functional of our external potential and the number of align:start position:0% external potential and the number of align:start position:0% external potential and the number of electrons functional again you know can align:start position:0% electrons functional again you know can align:start position:0% electrons functional again you know can be anything and in this case it goes align:start position:0% be anything and in this case it goes align:start position:0% be anything and in this case it goes through the Schrodinger equation nothing align:start position:0% through the Schrodinger equation nothing align:start position:0% through the Schrodinger equation nothing sort of complex at this at this point align:start position:0% sort of complex at this at this point align:start position:0% sort of complex at this at this point the sort of remarkable result that no align:start position:0% the sort of remarkable result that no align:start position:0% the sort of remarkable result that no one had set of you know figured out align:start position:0% one had set of you know figured out align:start position:0% one had set of you know figured out between a 1964 and 1965 is that the align:start position:0% between a 1964 and 1965 is that the align:start position:0% between a 1964 and 1965 is that the opposite is also true and it's not align:start position:0% opposite is also true and it's not align:start position:0% opposite is also true and it's not trivial at all so what hohenberg and align:start position:0% trivial at all so what hohenberg and align:start position:0% trivial at all so what hohenberg and Cohn stated the first actually in 1964 align:start position:0% Cohn stated the first actually in 1964 align:start position:0% Cohn stated the first actually in 1964 was this that the ground state charge align:start position:0% was this that the ground state charge align:start position:0% was this that the ground state charge density is a align:start position:0% density is a align:start position:0% density is a fundamental quantity align:start position:0% fundamental quantity align:start position:0% fundamental quantity as fundamental as the external potential align:start position:0% as fundamental as the external potential align:start position:0% as fundamental as the external potential and in particular not only the external align:start position:0% and in particular not only the external align:start position:0% and in particular not only the external potential the terms uniquely the ground align:start position:0% potential the terms uniquely the ground align:start position:0% potential the terms uniquely the ground state charge density of yours system but align:start position:0% state charge density of yours system but align:start position:0% state charge density of yours system but also the vice versa is true that is align:start position:0% also the vice versa is true that is align:start position:0% also the vice versa is true that is given a ground state charge density in align:start position:0% given a ground state charge density in align:start position:0% given a ground state charge density in principle one can prove that there is a align:start position:0% principle one can prove that there is a align:start position:0% principle one can prove that there is a unique align:start position:0% unique align:start position:0% unique external potential for which that ground align:start position:0% external potential for which that ground align:start position:0% external potential for which that ground state charge density is the ground state align:start position:0% state charge density is the ground state align:start position:0% state charge density is the ground state solution for that external potential so align:start position:0% solution for that external potential so align:start position:0% solution for that external potential so if you have the external potential align:start position:0% if you have the external potential align:start position:0% if you have the external potential conceptually it's trivial to go through align:start position:0% conceptually it's trivial to go through align:start position:0% conceptually it's trivial to go through the Schrodinger equation and its align:start position:0% the Schrodinger equation and its align:start position:0% the Schrodinger equation and its solution to the charge density what align:start position:0% solution to the charge density what align:start position:0% solution to the charge density what hohenberg and corner are telling us and align:start position:0% hohenberg and corner are telling us and align:start position:0% hohenberg and corner are telling us and I'll just show you a sketch of the proof align:start position:0% I'll just show you a sketch of the proof align:start position:0% I'll just show you a sketch of the proof in a moment is that in principle if align:start position:0% in a moment is that in principle if align:start position:0% in a moment is that in principle if someone is giving you a charge density align:start position:0% someone is giving you a charge density align:start position:0% someone is giving you a charge density and is telling you this charge density align:start position:0% and is telling you this charge density align:start position:0% and is telling you this charge density is the ground state charge density of a align:start position:0% is the ground state charge density of a align:start position:0% is the ground state charge density of a number of electrons and electrons in an align:start position:0% number of electrons and electrons in an align:start position:0% number of electrons and electrons in an external potential in principle what is align:start position:0% external potential in principle what is align:start position:0% external potential in principle what is that external potential is an align:start position:0% that external potential is an align:start position:0% that external potential is an information that is completely contained align:start position:0% information that is completely contained align:start position:0% information that is completely contained into the charge density okay and it's align:start position:0% into the charge density okay and it's align:start position:0% into the charge density okay and it's not contained in a trivial way it's not align:start position:0% not contained in a trivial way it's not align:start position:0% not contained in a trivial way it's not that you can look at a ground state align:start position:0% that you can look at a ground state align:start position:0% that you can look at a ground state charge density and guess what the align:start position:0% charge density and guess what the align:start position:0% charge density and guess what the external potential is and that's where align:start position:0% external potential is and that's where align:start position:0% external potential is and that's where all the complexity of practical density align:start position:0% all the complexity of practical density align:start position:0% all the complexity of practical density functional Theory comes but from the align:start position:0% functional Theory comes but from the align:start position:0% functional Theory comes but from the conceptual and mathematical point of align:start position:0% conceptual and mathematical point of align:start position:0% conceptual and mathematical point of view these two quantities are absolutely align:start position:0% view these two quantities are absolutely align:start position:0% view these two quantities are absolutely equivalent from one you get the other align:start position:0% equivalent from one you get the other align:start position:0% equivalent from one you get the other and vice versa and align:start position:0% and vice versa and align:start position:0% and vice versa and the ascent of align:start position:0% the ascent of align:start position:0% the ascent of vice versa was not trivial and that is align:start position:0% vice versa was not trivial and that is align:start position:0% vice versa was not trivial and that is sort of you know what is contained in align:start position:0% sort of you know what is contained in align:start position:0% sort of you know what is contained in the so called first hohenberg and korn align:start position:0% the so called first hohenberg and korn align:start position:0% the so called first hohenberg and korn problem I I won't go through the align:start position:0% problem I I won't go through the align:start position:0% problem I I won't go through the derivation it's actually very simple align:start position:0% derivation it's actually very simple align:start position:0% derivation it's actually very simple I've printed it here in case you sort of align:start position:0% I've printed it here in case you sort of align:start position:0% I've printed it here in case you sort of want to read it but it's basically is a align:start position:0% want to read it but it's basically is a align:start position:0% want to read it but it's basically is a derivation and absurdum what they are align:start position:0% derivation and absurdum what they are align:start position:0% derivation and absurdum what they are saying is that if that external align:start position:0% saying is that if that external align:start position:0% saying is that if that external potential were not unique if there were align:start position:0% potential were not unique if there were align:start position:0% potential were not unique if there were two external potential that were align:start position:0% two external potential that were align:start position:0% two external potential that were different and would give the same ground align:start position:0% different and would give the same ground align:start position:0% different and would give the same ground state energy we would get to an absurdum align:start position:0% state energy we would get to an absurdum align:start position:0% state energy we would get to an absurdum okay so typical mathematical align:start position:0% okay so typical mathematical align:start position:0% okay so typical mathematical demonstration we suppose that there are align:start position:0% demonstration we suppose that there are align:start position:0% demonstration we suppose that there are two different external potential that align:start position:0% two different external potential that align:start position:0% two different external potential that give the same ground state as density align:start position:0% give the same ground state as density align:start position:0% give the same ground state as density and we show that we arrive to a align:start position:0% and we show that we arrive to a align:start position:0% and we show that we arrive to a conclusion that doesn't make sense so align:start position:0% conclusion that doesn't make sense so align:start position:0% conclusion that doesn't make sense so there can be only a single external align:start position:0% there can be only a single external align:start position:0% there can be only a single external potential and that's the proof and again align:start position:0% potential and that's the proof and again align:start position:0% potential and that's the proof and again it wasn't trivial I mean if you wanted a align:start position:0% it wasn't trivial I mean if you wanted a align:start position:0% it wasn't trivial I mean if you wanted a very basic statement but it took 40 align:start position:0% very basic statement but it took 40 align:start position:0% very basic statement but it took 40 years to be formulated and if actually align:start position:0% years to be formulated and if actually align:start position:0% years to be formulated and if actually not true in other cases that you know to align:start position:0% not true in other cases that you know to align:start position:0% not true in other cases that you know to first glance look very similar suppose align:start position:0% first glance look very similar suppose align:start position:0% first glance look very similar suppose that for a moment we want to discuss align:start position:0% that for a moment we want to discuss align:start position:0% that for a moment we want to discuss excited states you could say well if I align:start position:0% excited states you could say well if I align:start position:0% excited states you could say well if I have a charge density and I say this is align:start position:0% have a charge density and I say this is align:start position:0% have a charge density and I say this is an excited density of an excited align:start position:0% an excited density of an excited align:start position:0% an excited density of an excited electronic state maybe I could also align:start position:0% electronic state maybe I could also align:start position:0% electronic state maybe I could also recover the potential that has generated align:start position:0% recover the potential that has generated align:start position:0% recover the potential that has generated that and that's not true actually so align:start position:0% that and that's not true actually so align:start position:0% that and that's not true actually so there are sort of a number of cases in align:start position:0% there are sort of a number of cases in align:start position:0% there are sort of a number of cases in which this is not true but for this align:start position:0% which this is not true but for this align:start position:0% which this is not true but for this fundamental set of relation between the align:start position:0% fundamental set of relation between the align:start position:0% fundamental set of relation between the charge density of the ground state and align:start position:0% charge density of the ground state and align:start position:0% charge density of the ground state and external potential this is true so we align:start position:0% external potential this is true so we align:start position:0% external potential this is true so we have sort of moved away now our align:start position:0% have sort of moved away now our align:start position:0% have sort of moved away now our attention it's not any more than any align:start position:0% attention it's not any more than any align:start position:0% attention it's not any more than any body wave function that we want to focus align:start position:0% body wave function that we want to focus align:start position:0% body wave function that we want to focus but is the charge density the charge align:start position:0% but is the charge density the charge align:start position:0% but is the charge density the charge density is as much a fundamental align:start position:0% density is as much a fundamental align:start position:0% density is as much a fundamental variable of our problem is not a derived align:start position:0% variable of our problem is not a derived align:start position:0% variable of our problem is not a derived variable it's not something that comes align:start position:0% variable it's not something that comes align:start position:0% variable it's not something that comes from the wave function but is something align:start position:0% from the wave function but is something align:start position:0% from the wave function but is something that we can actually focus all our align:start position:0% that we can actually focus all our align:start position:0% that we can actually focus all our attention into and now align:start position:0% attention into and now align:start position:0% attention into and now we need to find the equivalent of the align:start position:0% we need to find the equivalent of the align:start position:0% we need to find the equivalent of the Schrodinger equation for the charge align:start position:0% Schrodinger equation for the charge align:start position:0% Schrodinger equation for the charge density this is what shredding had done align:start position:0% density this is what shredding had done align:start position:0% density this is what shredding had done in the 20s in 1925 he said this is the align:start position:0% in the 20s in 1925 he said this is the align:start position:0% in the 20s in 1925 he said this is the equation that quantum objects satisfy align:start position:0% equation that quantum objects satisfy align:start position:0% equation that quantum objects satisfy and I'll call it the Schrodinger align:start position:0% and I'll call it the Schrodinger align:start position:0% and I'll call it the Schrodinger equation now hohenberg ancona has shown align:start position:0% equation now hohenberg ancona has shown align:start position:0% equation now hohenberg ancona has shown that we don't need to think in terms of align:start position:0% that we don't need to think in terms of align:start position:0% that we don't need to think in terms of the wave function we can think in terms align:start position:0% the wave function we can think in terms align:start position:0% the wave function we can think in terms of the charge density as being the align:start position:0% of the charge density as being the align:start position:0% of the charge density as being the fundamental descriptor of our quantum align:start position:0% fundamental descriptor of our quantum align:start position:0% fundamental descriptor of our quantum system what is life that they need to align:start position:0% system what is life that they need to align:start position:0% system what is life that they need to show me that there is an equivalent of align:start position:0% show me that there is an equivalent of align:start position:0% show me that there is an equivalent of the shading equation that is we can align:start position:0% the shading equation that is we can align:start position:0% the shading equation that is we can write a align:start position:0% write a align:start position:0% write a density equation that is a sort of what align:start position:0% density equation that is a sort of what align:start position:0% density equation that is a sort of what will give me the ground state and sort align:start position:0% will give me the ground state and sort align:start position:0% will give me the ground state and sort of all the properties of the system and align:start position:0% of all the properties of the system and align:start position:0% of all the properties of the system and that's really the second hohenberg and align:start position:0% that's really the second hohenberg and align:start position:0% that's really the second hohenberg and corner theorem that is really writing align:start position:0% corner theorem that is really writing align:start position:0% corner theorem that is really writing out the aquiver the concept one of the align:start position:0% out the aquiver the concept one of the align:start position:0% out the aquiver the concept one of the shading an equation for the charge align:start position:0% shading an equation for the charge align:start position:0% shading an equation for the charge density and now sort of it becomes align:start position:0% density and now sort of it becomes align:start position:0% density and now sort of it becomes fairly conceptual okay so this is a the align:start position:0% fairly conceptual okay so this is a the align:start position:0% fairly conceptual okay so this is a the procedure align:start position:0% procedure align:start position:0% procedure we and all of this in the next few align:start position:0% we and all of this in the next few align:start position:0% we and all of this in the next few slides is still a conceptual procedure align:start position:0% slides is still a conceptual procedure align:start position:0% slides is still a conceptual procedure it will describe objects that are align:start position:0% it will describe objects that are align:start position:0% it will describe objects that are well-defined align:start position:0% well-defined align:start position:0% well-defined in principle that are conceptually align:start position:0% in principle that are conceptually align:start position:0% in principle that are conceptually well-defined but we still don't have a align:start position:0% well-defined but we still don't have a align:start position:0% well-defined but we still don't have a clue on you know what they look like in align:start position:0% clue on you know what they look like in align:start position:0% clue on you know what they look like in practice and all the sort of density align:start position:0% practice and all the sort of density align:start position:0% practice and all the sort of density functional application go through a align:start position:0% functional application go through a align:start position:0% functional application go through a procedure that will see later round that align:start position:0% procedure that will see later round that align:start position:0% procedure that will see later round that is the sort of connection mapping that align:start position:0% is the sort of connection mapping that align:start position:0% is the sort of connection mapping that gives a hint of what these objects look align:start position:0% gives a hint of what these objects look align:start position:0% gives a hint of what these objects look like but up to now we are going to align:start position:0% like but up to now we are going to align:start position:0% like but up to now we are going to introduce objects that are well defined align:start position:0% introduce objects that are well defined align:start position:0% introduce objects that are well defined in principle but we don't know how they align:start position:0% in principle but we don't know how they align:start position:0% in principle but we don't know how they look like and so that's right somehow align:start position:0% look like and so that's right somehow align:start position:0% look like and so that's right somehow density functional theory is a much less align:start position:0% density functional theory is a much less align:start position:0% density functional theory is a much less intuitive theory than something like align:start position:0% intuitive theory than something like align:start position:0% intuitive theory than something like hartree-fock ok so this is going to work align:start position:0% hartree-fock ok so this is going to work align:start position:0% hartree-fock ok so this is going to work the second hohenberg and confirm align:start position:0% the second hohenberg and confirm align:start position:0% the second hohenberg and confirm defining the fundamental equation for align:start position:0% defining the fundamental equation for align:start position:0% defining the fundamental equation for the charge density and this is the step align:start position:0% the charge density and this is the step align:start position:0% the charge density and this is the step for any charge density Rho so someone align:start position:0% for any charge density Rho so someone align:start position:0% for any charge density Rho so someone gives you someone draws you an arbitrary align:start position:0% gives you someone draws you an arbitrary align:start position:0% gives you someone draws you an arbitrary charge density well we know that there align:start position:0% charge density well we know that there align:start position:0% charge density well we know that there is an external potential of which that align:start position:0% is an external potential of which that align:start position:0% is an external potential of which that charge density is the ground state we align:start position:0% charge density is the ground state we align:start position:0% charge density is the ground state we don't know what it is honestly but we align:start position:0% don't know what it is honestly but we align:start position:0% don't know what it is honestly but we have proven that there is a unique align:start position:0% have proven that there is a unique align:start position:0% have proven that there is a unique external potential ok so because there align:start position:0% external potential ok so because there align:start position:0% external potential ok so because there is a unique external potential the align:start position:0% is a unique external potential the align:start position:0% is a unique external potential the reason am anybody's reading a question align:start position:0% reason am anybody's reading a question align:start position:0% reason am anybody's reading a question with that potential in there and there align:start position:0% with that potential in there and there align:start position:0% with that potential in there and there is a wave function that is going to be align:start position:0% is a wave function that is going to be align:start position:0% is a wave function that is going to be the ground state wave function of that align:start position:0% the ground state wave function of that align:start position:0% the ground state wave function of that many bodies Schrodinger equation so align:start position:0% many bodies Schrodinger equation so align:start position:0% many bodies Schrodinger equation so given a certain raw we know that an align:start position:0% given a certain raw we know that an align:start position:0% given a certain raw we know that an external potential exists and it's align:start position:0% external potential exists and it's align:start position:0% external potential exists and it's unique in the terms it determines a align:start position:0% unique in the terms it determines a align:start position:0% unique in the terms it determines a Schrodinger equation and that align:start position:0% Schrodinger equation and that align:start position:0% Schrodinger equation and that Schrodinger equation that Germans a align:start position:0% Schrodinger equation that Germans a align:start position:0% Schrodinger equation that Germans a ground state wave function that we call align:start position:0% ground state wave function that we call align:start position:0% ground state wave function that we call sign so what we are saying is that given align:start position:0% sign so what we are saying is that given align:start position:0% sign so what we are saying is that given a row in principle that sigh the ground align:start position:0% a row in principle that sigh the ground align:start position:0% a row in principle that sigh the ground state wave function of the Schrodinger align:start position:0% state wave function of the Schrodinger align:start position:0% state wave function of the Schrodinger equation in the external potential that align:start position:0% equation in the external potential that align:start position:0% equation in the external potential that is uniquely defined by the row is also a align:start position:0% is uniquely defined by the row is also a align:start position:0% is uniquely defined by the row is also a well-defined object again we don't know align:start position:0% well-defined object again we don't know align:start position:0% well-defined object again we don't know what it is but it is well-defined and align:start position:0% what it is but it is well-defined and align:start position:0% what it is but it is well-defined and because it's a well-defined object we align:start position:0% because it's a well-defined object we align:start position:0% because it's a well-defined object we can calculate the expectation value of align:start position:0% can calculate the expectation value of align:start position:0% can calculate the expectation value of that well define object of the quantum align:start position:0% that well define object of the quantum align:start position:0% that well define object of the quantum kinetic energy you know minus 1/2 sum align:start position:0% kinetic energy you know minus 1/2 sum align:start position:0% kinetic energy you know minus 1/2 sum over all I of the second derivatives and align:start position:0% over all I of the second derivatives and align:start position:0% over all I of the second derivatives and the electron-electron interaction just align:start position:0% the electron-electron interaction just align:start position:0% the electron-electron interaction just the 1 over RI minus RJ term so again align:start position:0% the 1 over RI minus RJ term so again align:start position:0% the 1 over RI minus RJ term so again this term is in principle align:start position:0% this term is in principle align:start position:0% this term is in principle well-defined and we call this term the align:start position:0% well-defined and we call this term the align:start position:0% well-defined and we call this term the universal density functional the T's for align:start position:0% universal density functional the T's for align:start position:0% universal density functional the T's for any given arbitrary Rho I align:start position:0% any given arbitrary Rho I align:start position:0% any given arbitrary Rho I in principle can define a number that is align:start position:0% in principle can define a number that is align:start position:0% in principle can define a number that is this number here is the Rho in principle align:start position:0% this number here is the Rho in principle align:start position:0% this number here is the Rho in principle from the row I have the external align:start position:0% from the row I have the external align:start position:0% from the row I have the external potential from the external potential I align:start position:0% potential from the external potential I align:start position:0% potential from the external potential I have the Schrodinger equation in align:start position:0% have the Schrodinger equation in align:start position:0% have the Schrodinger equation in principle I'm able to solve that align:start position:0% principle I'm able to solve that align:start position:0% principle I'm able to solve that shreddin equation found in principle the align:start position:0% shreddin equation found in principle the align:start position:0% shreddin equation found in principle the many-body ground state wave function align:start position:0% many-body ground state wave function align:start position:0% many-body ground state wave function that will be precise and I can calculate align:start position:0% that will be precise and I can calculate align:start position:0% that will be precise and I can calculate the expectation value of Phi of the align:start position:0% the expectation value of Phi of the align:start position:0% the expectation value of Phi of the quantum kinetic energy and of the align:start position:0% quantum kinetic energy and of the align:start position:0% quantum kinetic energy and of the electron-electron interaction term all align:start position:0% electron-electron interaction term all align:start position:0% electron-electron interaction term all well defined we have really no clue on align:start position:0% well defined we have really no clue on align:start position:0% well defined we have really no clue on how to calculate because we can't really align:start position:0% how to calculate because we can't really align:start position:0% how to calculate because we can't really do in practice any of the steps but this align:start position:0% do in practice any of the steps but this align:start position:0% do in practice any of the steps but this universal functional of the density is align:start position:0% universal functional of the density is align:start position:0% universal functional of the density is well-defined align:start position:0% well-defined align:start position:0% well-defined so with this universal functional that align:start position:0% so with this universal functional that align:start position:0% so with this universal functional that is now well defined align:start position:0% is now well defined align:start position:0% is now well defined we can write out something align:start position:0% we can write out something align:start position:0% we can write out something we can write an align:start position:0% we can write an align:start position:0% we can write an energy align:start position:0% energy align:start position:0% energy for any given external potential and for align:start position:0% for any given external potential and for align:start position:0% for any given external potential and for any given charge density and we write it align:start position:0% any given charge density and we write it align:start position:0% any given charge density and we write it as this so for any given charge density align:start position:0% as this so for any given charge density align:start position:0% as this so for any given charge density there will be a well-defined number that align:start position:0% there will be a well-defined number that align:start position:0% there will be a well-defined number that is this universal density function of align:start position:0% is this universal density function of align:start position:0% is this universal density function of data Rho Prime and then we add another align:start position:0% data Rho Prime and then we add another align:start position:0% data Rho Prime and then we add another term that is just trivially the integral align:start position:0% term that is just trivially the integral align:start position:0% term that is just trivially the integral of this V this external potential times align:start position:0% of this V this external potential times align:start position:0% of this V this external potential times the charge density Rho prime so again align:start position:0% the charge density Rho prime so again align:start position:0% the charge density Rho prime so again this new expression that we are written align:start position:0% this new expression that we are written align:start position:0% this new expression that we are written is well defined for any Rho prime and align:start position:0% is well defined for any Rho prime and align:start position:0% is well defined for any Rho prime and for any external potential we can align:start position:0% for any external potential we can align:start position:0% for any external potential we can calculate trivially disturber and in align:start position:0% calculate trivially disturber and in align:start position:0% calculate trivially disturber and in principle we know what this number is align:start position:0% principle we know what this number is align:start position:0% principle we know what this number is and this is if you want align:start position:0% and this is if you want align:start position:0% and this is if you want 1964-1965 align:start position:0% 1964-1965 align:start position:0% 1964-1965 the reformulation of quantum mechanics align:start position:0% the reformulation of quantum mechanics align:start position:0% the reformulation of quantum mechanics because now hohenberg and kona are able align:start position:0% because now hohenberg and kona are able align:start position:0% because now hohenberg and kona are able to prove that there is a variational align:start position:0% to prove that there is a variational align:start position:0% to prove that there is a variational principle that is for this expression align:start position:0% principle that is for this expression align:start position:0% principle that is for this expression written here for this function of Rho align:start position:0% written here for this function of Rho align:start position:0% written here for this function of Rho prime we can prove that for any Rho align:start position:0% prime we can prove that for any Rho align:start position:0% prime we can prove that for any Rho prime that we can throw in a the overall align:start position:0% prime that we can throw in a the overall align:start position:0% prime that we can throw in a the overall numerical value of this expression is align:start position:0% numerical value of this expression is align:start position:0% numerical value of this expression is always going to be either greater or align:start position:0% always going to be either greater or align:start position:0% always going to be either greater or equal to the ground state charge to the align:start position:0% equal to the ground state charge to the align:start position:0% equal to the ground state charge to the ground state energy that we would obtain align:start position:0% ground state energy that we would obtain align:start position:0% ground state energy that we would obtain from the shading equation in the align:start position:0% from the shading equation in the align:start position:0% from the shading equation in the presence of this external potential so align:start position:0% presence of this external potential so align:start position:0% presence of this external potential so now we have a well defined align:start position:0% now we have a well defined align:start position:0% now we have a well defined density functional so if you have an align:start position:0% density functional so if you have an align:start position:0% density functional so if you have an external potential the Z over R of your align:start position:0% external potential the Z over R of your align:start position:0% external potential the Z over R of your otama you can try out now not wave align:start position:0% otama you can try out now not wave align:start position:0% otama you can try out now not wave functions that are very difficult but align:start position:0% functions that are very difficult but align:start position:0% functions that are very difficult but you can try out charge density and the align:start position:0% you can try out charge density and the align:start position:0% you can try out charge density and the charge density that you the lowest align:start position:0% charge density that you the lowest align:start position:0% charge density that you the lowest expectation value the lowest value for align:start position:0% expectation value the lowest value for align:start position:0% expectation value the lowest value for this functional will be the ground state align:start position:0% this functional will be the ground state align:start position:0% this functional will be the ground state and charge density align:start position:0% align:start position:0% small problem we have no clue what this align:start position:0% small problem we have no clue what this align:start position:0% small problem we have no clue what this looks like as a function of Rho Prime align:start position:0% looks like as a function of Rho Prime align:start position:0% looks like as a function of Rho Prime but if we knew we would ever a align:start position:0% but if we knew we would ever a align:start position:0% but if we knew we would ever a wonderfully simple approach to quantum align:start position:0% wonderfully simple approach to quantum align:start position:0% wonderfully simple approach to quantum mechanics now we don't need to deal with align:start position:0% mechanics now we don't need to deal with align:start position:0% mechanics now we don't need to deal with the many body complexity we just align:start position:0% the many body complexity we just align:start position:0% the many body complexity we just minimize this expression as a function align:start position:0% minimize this expression as a function align:start position:0% minimize this expression as a function of Rho Prime and again it's sort of align:start position:0% of Rho Prime and again it's sort of align:start position:0% of Rho Prime and again it's sort of fairly easy to prove this variational align:start position:0% fairly easy to prove this variational align:start position:0% fairly easy to prove this variational principle but when it's ready to sit align:start position:0% principle but when it's ready to sit align:start position:0% principle but when it's ready to sit I've given you some readings so you are align:start position:0% I've given you some readings so you are align:start position:0% I've given you some readings so you are welcome if you are really interested in align:start position:0% welcome if you are really interested in align:start position:0% welcome if you are really interested in this to go back and read the first row align:start position:0% this to go back and read the first row align:start position:0% this to go back and read the first row invariant consider and read the second align:start position:0% invariant consider and read the second align:start position:0% invariant consider and read the second hohenberg and confit Rama but in many align:start position:0% hohenberg and confit Rama but in many align:start position:0% hohenberg and confit Rama but in many ways the the proof of this second align:start position:0% ways the the proof of this second align:start position:0% ways the the proof of this second hohenberg and Cohen theorem can be done align:start position:0% hohenberg and Cohen theorem can be done align:start position:0% hohenberg and Cohen theorem can be done again through the variational principle align:start position:0% again through the variational principle align:start position:0% again through the variational principle that is you know if we have a certain align:start position:0% that is you know if we have a certain align:start position:0% that is you know if we have a certain Rho prime well that again uniquely align:start position:0% Rho prime well that again uniquely align:start position:0% Rho prime well that again uniquely determine the ground state wave function align:start position:0% determine the ground state wave function align:start position:0% determine the ground state wave function Rho prime will determine an external align:start position:0% Rho prime will determine an external align:start position:0% Rho prime will determine an external potential that in principle is different align:start position:0% potential that in principle is different align:start position:0% potential that in principle is different from this but Rho prime will determine align:start position:0% from this but Rho prime will determine align:start position:0% from this but Rho prime will determine an external potential and will determine align:start position:0% an external potential and will determine align:start position:0% an external potential and will determine our wave function that is the solution align:start position:0% our wave function that is the solution align:start position:0% our wave function that is the solution of the many-body shading equation and if align:start position:0% of the many-body shading equation and if align:start position:0% of the many-body shading equation and if we take the expectation value of align:start position:0% we take the expectation value of align:start position:0% we take the expectation value of ara a meltonian with this external align:start position:0% ara a meltonian with this external align:start position:0% ara a meltonian with this external potential in this but evaluated on the align:start position:0% potential in this but evaluated on the align:start position:0% potential in this but evaluated on the wave function of C prime that comes from align:start position:0% wave function of C prime that comes from align:start position:0% wave function of C prime that comes from discharge density Rho prime well we can align:start position:0% discharge density Rho prime well we can align:start position:0% discharge density Rho prime well we can show that this expectation value here is align:start position:0% show that this expectation value here is align:start position:0% show that this expectation value here is just identical to this functional that I align:start position:0% just identical to this functional that I align:start position:0% just identical to this functional that I just written and for the variational align:start position:0% just written and for the variational align:start position:0% just written and for the variational principle then it needs to be greater or align:start position:0% principle then it needs to be greater or align:start position:0% principle then it needs to be greater or equal than zero align:start position:0% equal than zero align:start position:0% equal than zero I want sort of dwell into that and again align:start position:0% I want sort of dwell into that and again align:start position:0% I want sort of dwell into that and again you can look at the set of detail align:start position:0% you can look at the set of detail align:start position:0% you can look at the set of detail description in sort of some of the many align:start position:0% description in sort of some of the many align:start position:0% description in sort of some of the many references that I've given or that I've align:start position:0% references that I've given or that I've align:start position:0% references that I've given or that I've also posted on the website but what is align:start position:0% also posted on the website but what is align:start position:0% also posted on the website but what is conceptually important is that we have a align:start position:0% conceptually important is that we have a align:start position:0% conceptually important is that we have a new equation you okay so 1964-65 align:start position:0% new equation you okay so 1964-65 align:start position:0% new equation you okay so 1964-65 quantum mechanics turned around we don't align:start position:0% quantum mechanics turned around we don't align:start position:0% quantum mechanics turned around we don't have to think at many-body wave align:start position:0% have to think at many-body wave align:start position:0% have to think at many-body wave functions we can think just a charge align:start position:0% functions we can think just a charge align:start position:0% functions we can think just a charge density and align:start position:0% density and align:start position:0% density and all would be well align:start position:0% all would be well align:start position:0% all would be well apart from this detail that we don't align:start position:0% apart from this detail that we don't align:start position:0% apart from this detail that we don't know what that function of F of Rho is align:start position:0% know what that function of F of Rho is align:start position:0% know what that function of F of Rho is and so we have a conceptual approach but align:start position:0% and so we have a conceptual approach but align:start position:0% and so we have a conceptual approach but we don't have a practical approach to align:start position:0% we don't have a practical approach to align:start position:0% we don't have a practical approach to solve the density functional formulation align:start position:0% solve the density functional formulation align:start position:0% solve the density functional formulation of quantum mechanics and this is if you align:start position:0% of quantum mechanics and this is if you align:start position:0% of quantum mechanics and this is if you want a true to this day we don't know align:start position:0% want a true to this day we don't know align:start position:0% want a true to this day we don't know what is the exact form of f of Rho if we align:start position:0% what is the exact form of f of Rho if we align:start position:0% what is the exact form of f of Rho if we knew it sort of you know most of our align:start position:0% knew it sort of you know most of our align:start position:0% knew it sort of you know most of our sort of quantum mechanical computational align:start position:0% sort of quantum mechanical computational align:start position:0% sort of quantum mechanical computational problems would be trivially solved align:start position:0% problems would be trivially solved align:start position:0% problems would be trivially solved because solving that variational align:start position:0% because solving that variational align:start position:0% because solving that variational principle in the charge density would be align:start position:0% principle in the charge density would be align:start position:0% principle in the charge density would be most likely a trivial thing to do the align:start position:0% most likely a trivial thing to do the align:start position:0% most likely a trivial thing to do the issue is that not only we don't know but align:start position:0% issue is that not only we don't know but align:start position:0% issue is that not only we don't know but we have understood a lot of what that align:start position:0% we have understood a lot of what that align:start position:0% we have understood a lot of what that exchange correlation of what data align:start position:0% exchange correlation of what data align:start position:0% exchange correlation of what data Universal density functional is and align:start position:0% Universal density functional is and align:start position:0% Universal density functional is and it's very complex okay so it's unlikely align:start position:0% it's very complex okay so it's unlikely align:start position:0% it's very complex okay so it's unlikely that there is a sort of simple align:start position:0% that there is a sort of simple align:start position:0% that there is a sort of simple analytical expression of it as a align:start position:0% analytical expression of it as a align:start position:0% analytical expression of it as a function of the charge density only but align:start position:0% function of the charge density only but align:start position:0% function of the charge density only but you know the other sort of great piece align:start position:0% you know the other sort of great piece align:start position:0% you know the other sort of great piece of if you want quantum engineering by align:start position:0% of if you want quantum engineering by align:start position:0% of if you want quantum engineering by Walter Connor was finding out a very align:start position:0% Walter Connor was finding out a very align:start position:0% Walter Connor was finding out a very good approximation to that density align:start position:0% good approximation to that density align:start position:0% good approximation to that density functional okay we don't know what the align:start position:0% functional okay we don't know what the align:start position:0% functional okay we don't know what the exact one is but now what they are doing align:start position:0% exact one is but now what they are doing align:start position:0% exact one is but now what they are doing is well finding out one that is going to align:start position:0% is well finding out one that is going to align:start position:0% is well finding out one that is going to be very very closely similar to the align:start position:0% be very very closely similar to the align:start position:0% be very very closely similar to the exact one and so they are going to throw align:start position:0% exact one and so they are going to throw align:start position:0% exact one and so they are going to throw in some physical intuition to this align:start position:0% in some physical intuition to this align:start position:0% in some physical intuition to this problem that up to now if you want has align:start position:0% problem that up to now if you want has align:start position:0% problem that up to now if you want has been a mathematical problem align:start position:0% been a mathematical problem align:start position:0% been a mathematical problem it's another layer of complexity in this align:start position:0% it's another layer of complexity in this align:start position:0% it's another layer of complexity in this discussion so I hope I'm not losing you align:start position:0% discussion so I hope I'm not losing you align:start position:0% discussion so I hope I'm not losing you but sort of water align:start position:0% but sort of water align:start position:0% but sort of water Walter Condit I think here the young align:start position:0% Walter Condit I think here the young align:start position:0% Walter Condit I think here the young postdoc arriving from Cambridge Lucia align:start position:0% postdoc arriving from Cambridge Lucia align:start position:0% postdoc arriving from Cambridge Lucia had just done is a PhD in England and align:start position:0% had just done is a PhD in England and align:start position:0% had just done is a PhD in England and came there and certif you know I told align:start position:0% came there and certif you know I told align:start position:0% came there and certif you know I told him I have this new variational align:start position:0% him I have this new variational align:start position:0% him I have this new variational principle let's see what we can do to align:start position:0% principle let's see what we can do to align:start position:0% principle let's see what we can do to make it into a practical solution align:start position:0% make it into a practical solution align:start position:0% make it into a practical solution think they were in smithereens in San align:start position:0% think they were in smithereens in San align:start position:0% think they were in smithereens in San Diego probably at that time okay so this align:start position:0% Diego probably at that time okay so this align:start position:0% Diego probably at that time okay so this is what they are going to do remember align:start position:0% is what they are going to do remember align:start position:0% is what they are going to do remember sort of you know what is the problem we align:start position:0% sort of you know what is the problem we align:start position:0% sort of you know what is the problem we need to figure out what is a reasonable align:start position:0% need to figure out what is a reasonable align:start position:0% need to figure out what is a reasonable approximation to this functional here so align:start position:0% approximation to this functional here so align:start position:0% approximation to this functional here so what they say is well suppose that align:start position:0% what they say is well suppose that align:start position:0% what they say is well suppose that someone is given as a this charge align:start position:0% someone is given as a this charge align:start position:0% someone is given as a this charge density so we need in principle to find align:start position:0% density so we need in principle to find align:start position:0% density so we need in principle to find out what would be the many-body wave align:start position:0% out what would be the many-body wave align:start position:0% out what would be the many-body wave function that is solution of this align:start position:0% function that is solution of this align:start position:0% function that is solution of this external potential that corresponds to align:start position:0% external potential that corresponds to align:start position:0% external potential that corresponds to this charge density align:start position:0% this charge density align:start position:0% this charge density this is going to be very complex let's align:start position:0% this is going to be very complex let's align:start position:0% this is going to be very complex let's invent them a align:start position:0% invent them a align:start position:0% invent them a problem in which electrons do not align:start position:0% problem in which electrons do not align:start position:0% problem in which electrons do not interact between each other okay so align:start position:0% interact between each other okay so align:start position:0% interact between each other okay so electron so that's that's the sort of align:start position:0% electron so that's that's the sort of align:start position:0% electron so that's that's the sort of you know main problem in the Schrodinger align:start position:0% you know main problem in the Schrodinger align:start position:0% you know main problem in the Schrodinger equation that electrons interacting with align:start position:0% equation that electrons interacting with align:start position:0% equation that electrons interacting with each other introduce the two body align:start position:0% each other introduce the two body align:start position:0% each other introduce the two body electrostatic repulsion in the shading align:start position:0% electrostatic repulsion in the shading align:start position:0% electrostatic repulsion in the shading an equation and that what makes it align:start position:0% an equation and that what makes it align:start position:0% an equation and that what makes it difficult well what connection say is align:start position:0% difficult well what connection say is align:start position:0% difficult well what connection say is let's for a moment suppose that there is align:start position:0% let's for a moment suppose that there is align:start position:0% let's for a moment suppose that there is a system of electrons that don't align:start position:0% a system of electrons that don't align:start position:0% a system of electrons that don't interact the only thing that those align:start position:0% interact the only thing that those align:start position:0% interact the only thing that those so-called connection electrons fill is align:start position:0% so-called connection electrons fill is align:start position:0% so-called connection electrons fill is the external potential okay so those align:start position:0% the external potential okay so those align:start position:0% the external potential okay so those connection electron will solve will align:start position:0% connection electron will solve will align:start position:0% connection electron will solve will satisfy a Schrodinger equation that is align:start position:0% satisfy a Schrodinger equation that is align:start position:0% satisfy a Schrodinger equation that is much simpler because there is no align:start position:0% much simpler because there is no align:start position:0% much simpler because there is no electron electron interaction those align:start position:0% electron electron interaction those align:start position:0% electron electron interaction those connection electron the only thing that align:start position:0% connection electron the only thing that align:start position:0% connection electron the only thing that they feel is a new potential and they align:start position:0% they feel is a new potential and they align:start position:0% they feel is a new potential and they will have their own a quantum kinetic align:start position:0% will have their own a quantum kinetic align:start position:0% will have their own a quantum kinetic energy so what they are saying is for align:start position:0% energy so what they are saying is for align:start position:0% energy so what they are saying is for any given align:start position:0% any given align:start position:0% any given charge density Rho okay there is going align:start position:0% charge density Rho okay there is going align:start position:0% charge density Rho okay there is going to be align:start position:0% to be align:start position:0% to be non interacting set of electrons who's a align:start position:0% non interacting set of electrons who's a align:start position:0% non interacting set of electrons who's a ground Stata charge density is identical align:start position:0% ground Stata charge density is identical align:start position:0% ground Stata charge density is identical to row okay so we have said you know if align:start position:0% to row okay so we have said you know if align:start position:0% to row okay so we have said you know if we ever charge density Rho you can all align:start position:0% we ever charge density Rho you can all align:start position:0% we ever charge density Rho you can all go through you know find out the align:start position:0% go through you know find out the align:start position:0% go through you know find out the external potential that concerned align:start position:0% external potential that concerned align:start position:0% external potential that concerned they're all the shooting an equation align:start position:0% they're all the shooting an equation align:start position:0% they're all the shooting an equation them anybody interacting electrons align:start position:0% them anybody interacting electrons align:start position:0% them anybody interacting electrons solution but now what we are going to align:start position:0% solution but now what we are going to align:start position:0% solution but now what we are going to say is we can also think at a system of align:start position:0% say is we can also think at a system of align:start position:0% say is we can also think at a system of non-interacting electrons and we wanted align:start position:0% non-interacting electrons and we wanted align:start position:0% non-interacting electrons and we wanted those non-interacting electrons to fill align:start position:0% those non-interacting electrons to fill align:start position:0% those non-interacting electrons to fill a potential that is such that their align:start position:0% a potential that is such that their align:start position:0% a potential that is such that their ground state is going to give us a align:start position:0% ground state is going to give us a align:start position:0% ground state is going to give us a charge density that is identical to the align:start position:0% charge density that is identical to the align:start position:0% charge density that is identical to the charge density I am dealing with align:start position:0% charge density I am dealing with align:start position:0% charge density I am dealing with okay and we call that external potential align:start position:0% okay and we call that external potential align:start position:0% okay and we call that external potential the konchem potential okay so now for align:start position:0% the konchem potential okay so now for align:start position:0% the konchem potential okay so now for a charge dance if you don't only have to align:start position:0% a charge dance if you don't only have to align:start position:0% a charge dance if you don't only have to think at all the complexity that i've align:start position:0% think at all the complexity that i've align:start position:0% think at all the complexity that i've discussed up to now but you have also to align:start position:0% discussed up to now but you have also to align:start position:0% discussed up to now but you have also to think that for a charge density there is align:start position:0% think that for a charge density there is align:start position:0% think that for a charge density there is going to be the set of conan sham align:start position:0% going to be the set of conan sham align:start position:0% going to be the set of conan sham non-interacting electrons and there is align:start position:0% non-interacting electrons and there is align:start position:0% non-interacting electrons and there is going to be align:start position:0% going to be align:start position:0% going to be potential that is called the connection align:start position:0% potential that is called the connection align:start position:0% potential that is called the connection potential that is such that the ground align:start position:0% potential that is such that the ground align:start position:0% potential that is such that the ground state of the shredding an equation for align:start position:0% state of the shredding an equation for align:start position:0% state of the shredding an equation for non interacting electron that is without align:start position:0% non interacting electron that is without align:start position:0% non interacting electron that is without the electron-electron interaction in align:start position:0% the electron-electron interaction in align:start position:0% the electron-electron interaction in that connection potential will give us a align:start position:0% that connection potential will give us a align:start position:0% that connection potential will give us a wave function and a ground data that is align:start position:0% wave function and a ground data that is align:start position:0% wave function and a ground data that is that leads to a charge density identical align:start position:0% that leads to a charge density identical align:start position:0% that leads to a charge density identical to the charge density and sort of align:start position:0% to the charge density and sort of align:start position:0% to the charge density and sort of dealing with align:start position:0% dealing with align:start position:0% dealing with okay what do we do with this well at align:start position:0% okay what do we do with this well at align:start position:0% okay what do we do with this well at this stage align:start position:0% this stage align:start position:0% this stage there is a sort of you know great align:start position:0% there is a sort of you know great align:start position:0% there is a sort of you know great simplification that align:start position:0% simplification that align:start position:0% simplification that for the Schrodinger equation of non align:start position:0% for the Schrodinger equation of non align:start position:0% for the Schrodinger equation of non interacting electron we actually know align:start position:0% interacting electron we actually know align:start position:0% interacting electron we actually know what is the exact solution so it's align:start position:0% what is the exact solution so it's align:start position:0% what is the exact solution so it's actually very simple to solve a align:start position:0% actually very simple to solve a align:start position:0% actually very simple to solve a Schrodinger equation in which the align:start position:0% Schrodinger equation in which the align:start position:0% Schrodinger equation in which the electrons do not interact because now align:start position:0% electrons do not interact because now align:start position:0% electrons do not interact because now this later the term Ananta is actually align:start position:0% this later the term Ananta is actually align:start position:0% this later the term Ananta is actually the exact solution so if you have a set align:start position:0% the exact solution so if you have a set align:start position:0% the exact solution so if you have a set of non-interacting electrons you to have align:start position:0% of non-interacting electrons you to have align:start position:0% of non-interacting electrons you to have the electron-electron terming that's align:start position:0% the electron-electron terming that's align:start position:0% the electron-electron terming that's reading a question the Slater align:start position:0% reading a question the Slater align:start position:0% reading a question the Slater determinant is not only a good align:start position:0% determinant is not only a good align:start position:0% determinant is not only a good approximation but it's actually the align:start position:0% approximation but it's actually the align:start position:0% approximation but it's actually the exact solution okay so for this align:start position:0% exact solution okay so for this align:start position:0% exact solution okay so for this non-interacting set of lectrons we can align:start position:0% non-interacting set of lectrons we can align:start position:0% non-interacting set of lectrons we can solve everything exactly and in align:start position:0% solve everything exactly and in align:start position:0% solve everything exactly and in particular we can calculate say what is align:start position:0% particular we can calculate say what is align:start position:0% particular we can calculate say what is the kinetic energy of the set of align:start position:0% the kinetic energy of the set of align:start position:0% the kinetic energy of the set of non-interacting electrons align:start position:0% non-interacting electrons align:start position:0% non-interacting electrons okay so now we can set up you know F align:start position:0% okay so now we can set up you know F align:start position:0% okay so now we can set up you know F somehow through the physical way of the align:start position:0% somehow through the physical way of the align:start position:0% somehow through the physical way of the composer this mysterious dense align:start position:0% composer this mysterious dense align:start position:0% composer this mysterious dense difunctional in two different terms okay align:start position:0% difunctional in two different terms okay align:start position:0% difunctional in two different terms okay so what you're actually doing via the align:start position:0% so what you're actually doing via the align:start position:0% so what you're actually doing via the connection mapping is align:start position:0% connection mapping is align:start position:0% connection mapping is extracting from here terms that are very align:start position:0% extracting from here terms that are very align:start position:0% extracting from here terms that are very large and that we know how to write we align:start position:0% large and that we know how to write we align:start position:0% large and that we know how to write we know how to calculate and then sort of align:start position:0% know how to calculate and then sort of align:start position:0% know how to calculate and then sort of you know hopefully we are going to align:start position:0% you know hopefully we are going to align:start position:0% you know hopefully we are going to remain once we have extracted all these align:start position:0% remain once we have extracted all these align:start position:0% remain once we have extracted all these terms that we know how to define we align:start position:0% terms that we know how to define we align:start position:0% terms that we know how to define we remain with something that is very small align:start position:0% remain with something that is very small align:start position:0% remain with something that is very small okay and that will find another align:start position:0% okay and that will find another align:start position:0% okay and that will find another approximation numerical approximation align:start position:0% approximation numerical approximation align:start position:0% approximation numerical approximation for it so connection say well we have align:start position:0% for it so connection say well we have align:start position:0% for it so connection say well we have this well define density functional we align:start position:0% this well define density functional we align:start position:0% this well define density functional we extract two terms that are well defined align:start position:0% extract two terms that are well defined align:start position:0% extract two terms that are well defined and these two terms that sort of you align:start position:0% and these two terms that sort of you align:start position:0% and these two terms that sort of you know the great achievement actually align:start position:0% know the great achievement actually align:start position:0% know the great achievement actually contain most of the physics of our align:start position:0% contain most of the physics of our align:start position:0% contain most of the physics of our problem and the sort of small term that align:start position:0% problem and the sort of small term that align:start position:0% problem and the sort of small term that is left over we are going to approximate align:start position:0% is left over we are going to approximate align:start position:0% is left over we are going to approximate in some simple way and that really the align:start position:0% in some simple way and that really the align:start position:0% in some simple way and that really the approximation that they found worked align:start position:0% approximation that they found worked align:start position:0% approximation that they found worked very well and that's one sort of density align:start position:0% very well and that's one sort of density align:start position:0% very well and that's one sort of density functions really became a practical align:start position:0% functions really became a practical align:start position:0% functions really became a practical theory and so in this sort of align:start position:0% theory and so in this sort of align:start position:0% theory and so in this sort of density functional the first physical align:start position:0% density functional the first physical align:start position:0% density functional the first physical large term that they extract is the align:start position:0% large term that they extract is the align:start position:0% large term that they extract is the quantum kinetic energy that we call the align:start position:0% quantum kinetic energy that we call the align:start position:0% quantum kinetic energy that we call the s naught of the real system because align:start position:0% s naught of the real system because align:start position:0% s naught of the real system because again even if it's well defined we don't align:start position:0% again even if it's well defined we don't align:start position:0% again even if it's well defined we don't know how to do that but what they were align:start position:0% know how to do that but what they were align:start position:0% know how to do that but what they were able to write is the quantum kinetic align:start position:0% able to write is the quantum kinetic align:start position:0% able to write is the quantum kinetic energy of this align:start position:0% energy of this align:start position:0% energy of this knowning Dhingra problem so for a given align:start position:0% knowning Dhingra problem so for a given align:start position:0% knowning Dhingra problem so for a given charge density there is this set of align:start position:0% charge density there is this set of align:start position:0% charge density there is this set of connection non-interacting electrons align:start position:0% connection non-interacting electrons align:start position:0% connection non-interacting electrons that lives in a potential such that they align:start position:0% that lives in a potential such that they align:start position:0% that lives in a potential such that they have the same ground state charge align:start position:0% have the same ground state charge align:start position:0% have the same ground state charge density and their kinetic energy is align:start position:0% density and their kinetic energy is align:start position:0% density and their kinetic energy is trivial because it's going to be just align:start position:0% trivial because it's going to be just align:start position:0% trivial because it's going to be just the kinetic energy of the Slater align:start position:0% the kinetic energy of the Slater align:start position:0% the kinetic energy of the Slater determinant just a sum of single align:start position:0% determinant just a sum of single align:start position:0% determinant just a sum of single particle term so for a charge density align:start position:0% particle term so for a charge density align:start position:0% particle term so for a charge density now there is a well-defined quantum align:start position:0% now there is a well-defined quantum align:start position:0% now there is a well-defined quantum kinetic energy that is not the true align:start position:0% kinetic energy that is not the true align:start position:0% kinetic energy that is not the true point on kinetic energy of the system align:start position:0% point on kinetic energy of the system align:start position:0% point on kinetic energy of the system but is the quantum kinetic energy of align:start position:0% but is the quantum kinetic energy of align:start position:0% but is the quantum kinetic energy of this sort of align:start position:0% this sort of align:start position:0% this sort of associated system of non-interacting align:start position:0% associated system of non-interacting align:start position:0% associated system of non-interacting electrons but this term is now align:start position:0% electrons but this term is now align:start position:0% electrons but this term is now well-defined they say well let's extract align:start position:0% well-defined they say well let's extract align:start position:0% well-defined they say well let's extract another term that is well defined that align:start position:0% another term that is well defined that align:start position:0% another term that is well defined that is just a hearty electrostatic energy of align:start position:0% is just a hearty electrostatic energy of align:start position:0% is just a hearty electrostatic energy of a charge density distribution okay so if align:start position:0% a charge density distribution okay so if align:start position:0% a charge density distribution okay so if we look at a charge density distribution align:start position:0% we look at a charge density distribution align:start position:0% we look at a charge density distribution in which each infinitesimal volume align:start position:0% in which each infinitesimal volume align:start position:0% in which each infinitesimal volume interacts with each other infinitesimal align:start position:0% interacts with each other infinitesimal align:start position:0% interacts with each other infinitesimal volume with an electrostatic interaction align:start position:0% volume with an electrostatic interaction align:start position:0% volume with an electrostatic interaction that's going to be the term and you know align:start position:0% that's going to be the term and you know align:start position:0% that's going to be the term and you know what we are laughter is now something align:start position:0% what we are laughter is now something align:start position:0% what we are laughter is now something that they call them the exchange align:start position:0% that they call them the exchange align:start position:0% that they call them the exchange correlation term that is everything else align:start position:0% correlation term that is everything else align:start position:0% correlation term that is everything else okay so f in principle is an exact align:start position:0% okay so f in principle is an exact align:start position:0% okay so f in principle is an exact quantity we are now able to define a align:start position:0% quantity we are now able to define a align:start position:0% quantity we are now able to define a quantum kinetic energy term that is an align:start position:0% quantum kinetic energy term that is an align:start position:0% quantum kinetic energy term that is an exact quantity but it is not really the align:start position:0% exact quantity but it is not really the align:start position:0% exact quantity but it is not really the quantum kinetic energy of the true align:start position:0% quantum kinetic energy of the true align:start position:0% quantum kinetic energy of the true system but we sort of say you know this align:start position:0% system but we sort of say you know this align:start position:0% system but we sort of say you know this is going to be equal to a well-defined align:start position:0% is going to be equal to a well-defined align:start position:0% is going to be equal to a well-defined term plus another well-defined term plus align:start position:0% term plus another well-defined term plus align:start position:0% term plus another well-defined term plus a third term that we don't know so we align:start position:0% a third term that we don't know so we align:start position:0% a third term that we don't know so we have sort of decompose the quantity that align:start position:0% have sort of decompose the quantity that align:start position:0% have sort of decompose the quantity that we have no clue on what it is into three align:start position:0% we have no clue on what it is into three align:start position:0% we have no clue on what it is into three terms of which two terms are align:start position:0% terms of which two terms are align:start position:0% terms of which two terms are well-defined and all our cluelessness is align:start position:0% well-defined and all our cluelessness is align:start position:0% well-defined and all our cluelessness is contained in the third term and we call align:start position:0% contained in the third term and we call align:start position:0% contained in the third term and we call this third term the exchange correlation align:start position:0% this third term the exchange correlation align:start position:0% this third term the exchange correlation but the sort of physical advantage of align:start position:0% but the sort of physical advantage of align:start position:0% but the sort of physical advantage of having done this is that it turns out align:start position:0% having done this is that it turns out align:start position:0% having done this is that it turns out that these two terms capture a lot of align:start position:0% that these two terms capture a lot of align:start position:0% that these two terms capture a lot of the complexity of your problem and this align:start position:0% the complexity of your problem and this align:start position:0% the complexity of your problem and this term align:start position:0% term align:start position:0% term tends to be fairly small okay so that's align:start position:0% tends to be fairly small okay so that's align:start position:0% tends to be fairly small okay so that's a that that's all actually that's why it align:start position:0% a that that's all actually that's why it align:start position:0% a that that's all actually that's why it was very well because somehow they align:start position:0% was very well because somehow they align:start position:0% was very well because somehow they managed to capture the complexity of our align:start position:0% managed to capture the complexity of our align:start position:0% managed to capture the complexity of our system and so align:start position:0% align:start position:0% once that exchange correlation term is align:start position:0% once that exchange correlation term is align:start position:0% once that exchange correlation term is defined and it's approximated in some align:start position:0% defined and it's approximated in some align:start position:0% defined and it's approximated in some way that we'll see in a moment our align:start position:0% way that we'll see in a moment our align:start position:0% way that we'll see in a moment our problem is now well-defined because we align:start position:0% problem is now well-defined because we align:start position:0% problem is now well-defined because we really have a variational principle align:start position:0% really have a variational principle align:start position:0% really have a variational principle remember the universal density align:start position:0% remember the universal density align:start position:0% remember the universal density functional class the external potential align:start position:0% functional class the external potential align:start position:0% functional class the external potential plus the charge density in the field of align:start position:0% plus the charge density in the field of align:start position:0% plus the charge density in the field of the external potential align:start position:0% the external potential align:start position:0% the external potential minimizes the sort of new variational align:start position:0% minimizes the sort of new variational align:start position:0% minimizes the sort of new variational principle that comes from the hohenberg align:start position:0% principle that comes from the hohenberg align:start position:0% principle that comes from the hohenberg and confusion and so we can write it a align:start position:0% and confusion and so we can write it a align:start position:0% and confusion and so we can write it a variational principle that is this align:start position:0% variational principle that is this align:start position:0% variational principle that is this quantity align:start position:0% quantity align:start position:0% quantity with the constraint that the number of align:start position:0% with the constraint that the number of align:start position:0% with the constraint that the number of electrons should be equal to n ax should align:start position:0% electrons should be equal to n ax should align:start position:0% electrons should be equal to n ax should be minimum and as usual when you sort of align:start position:0% be minimum and as usual when you sort of align:start position:0% be minimum and as usual when you sort of you know you write a variational align:start position:0% you know you write a variational align:start position:0% you know you write a variational principle you are saying that sort of align:start position:0% principle you are saying that sort of align:start position:0% principle you are saying that sort of you know the differential of that align:start position:0% you know the differential of that align:start position:0% you know the differential of that quantity needs to be equal to zero or if align:start position:0% quantity needs to be equal to zero or if align:start position:0% quantity needs to be equal to zero or if you want I mean this is a generic term align:start position:0% you want I mean this is a generic term align:start position:0% you want I mean this is a generic term you have a set of what are called Euler align:start position:0% you have a set of what are called Euler align:start position:0% you have a set of what are called Euler Lagrange equation basically but it's align:start position:0% Lagrange equation basically but it's align:start position:0% Lagrange equation basically but it's nothing else than differential analysis align:start position:0% nothing else than differential analysis align:start position:0% nothing else than differential analysis that is you're asking yourself what are align:start position:0% that is you're asking yourself what are align:start position:0% that is you're asking yourself what are going to be the conditions that need to align:start position:0% going to be the conditions that need to align:start position:0% going to be the conditions that need to be satisfied by the charge density in align:start position:0% be satisfied by the charge density in align:start position:0% be satisfied by the charge density in order to satisfy the variational align:start position:0% order to satisfy the variational align:start position:0% order to satisfy the variational principle there is always this sort of align:start position:0% principle there is always this sort of align:start position:0% principle there is always this sort of one-to-one correspondence you have a align:start position:0% one-to-one correspondence you have a align:start position:0% one-to-one correspondence you have a variational principle it gives you align:start position:0% variational principle it gives you align:start position:0% variational principle it gives you differential equation or you have align:start position:0% differential equation or you have align:start position:0% differential equation or you have differential equation you can rewrite align:start position:0% differential equation you can rewrite align:start position:0% differential equation you can rewrite them in a variational principle we have align:start position:0% them in a variational principle we have align:start position:0% them in a variational principle we have seen that for the Schrodinger equation align:start position:0% seen that for the Schrodinger equation align:start position:0% seen that for the Schrodinger equation and we see this in particular now in align:start position:0% and we see this in particular now in align:start position:0% and we see this in particular now in explicitly for the align:start position:0% explicitly for the align:start position:0% explicitly for the connection orbitals so I'll actually go align:start position:0% connection orbitals so I'll actually go align:start position:0% connection orbitals so I'll actually go directly to the explicit expression of align:start position:0% directly to the explicit expression of align:start position:0% directly to the explicit expression of the connection align:start position:0% the connection align:start position:0% the connection orbitals again remember that what we align:start position:0% orbitals again remember that what we align:start position:0% orbitals again remember that what we have done is we have defined a align:start position:0% have done is we have defined a align:start position:0% have done is we have defined a variational principle that acts on a align:start position:0% variational principle that acts on a align:start position:0% variational principle that acts on a universal density functional f plus the align:start position:0% universal density functional f plus the align:start position:0% universal density functional f plus the charge density and external potential align:start position:0% charge density and external potential align:start position:0% charge density and external potential and we have decomposed that we have align:start position:0% and we have decomposed that we have align:start position:0% and we have decomposed that we have extracted from this Universal functional align:start position:0% extracted from this Universal functional align:start position:0% extracted from this Universal functional sort of terms that are large and align:start position:0% sort of terms that are large and align:start position:0% sort of terms that are large and physical and we have sort of pushed all align:start position:0% physical and we have sort of pushed all align:start position:0% physical and we have sort of pushed all the many body complexity of the problem align:start position:0% the many body complexity of the problem align:start position:0% the many body complexity of the problem in something that we call the exchange align:start position:0% in something that we call the exchange align:start position:0% in something that we call the exchange correlation functional that is again a align:start position:0% correlation functional that is again a align:start position:0% correlation functional that is again a functional of the charge density we align:start position:0% functional of the charge density we align:start position:0% functional of the charge density we don't know yet what that function of the align:start position:0% don't know yet what that function of the align:start position:0% don't know yet what that function of the charge density is but luckily it's going align:start position:0% charge density is but luckily it's going align:start position:0% charge density is but luckily it's going to be small so in a moment we'll align:start position:0% to be small so in a moment we'll align:start position:0% to be small so in a moment we'll approximate it and then we ask ourselves align:start position:0% approximate it and then we ask ourselves align:start position:0% approximate it and then we ask ourselves what are the align:start position:0% what are the align:start position:0% what are the variational what are the differential align:start position:0% variational what are the differential align:start position:0% variational what are the differential equation that derive from this align:start position:0% equation that derive from this align:start position:0% equation that derive from this variational principle well in principle align:start position:0% variational principle well in principle align:start position:0% variational principle well in principle I had written them here okay we just align:start position:0% I had written them here okay we just align:start position:0% I had written them here okay we just need to take the variation with respect align:start position:0% need to take the variation with respect align:start position:0% need to take the variation with respect to the charge density and imposing the align:start position:0% to the charge density and imposing the align:start position:0% to the charge density and imposing the lagrange multiplication constraint and align:start position:0% lagrange multiplication constraint and align:start position:0% lagrange multiplication constraint and so this this would be heat basically the align:start position:0% so this this would be heat basically the align:start position:0% so this this would be heat basically the case the charge density needs to satisfy align:start position:0% case the charge density needs to satisfy align:start position:0% case the charge density needs to satisfy this set of equation the sort of align:start position:0% this set of equation the sort of align:start position:0% this set of equation the sort of functional derivative of this non align:start position:0% functional derivative of this non align:start position:0% functional derivative of this non interacting quantum kinetic energy plus align:start position:0% interacting quantum kinetic energy plus align:start position:0% interacting quantum kinetic energy plus a number of terms that really contain align:start position:0% a number of terms that really contain align:start position:0% a number of terms that really contain the external potential the Hart align:start position:0% the external potential the Hart align:start position:0% the external potential the Hart interaction and exchange correlation align:start position:0% interaction and exchange correlation align:start position:0% interaction and exchange correlation need to be equal to the Lagrange align:start position:0% need to be equal to the Lagrange align:start position:0% need to be equal to the Lagrange multiplier that fixes the number the align:start position:0% multiplier that fixes the number the align:start position:0% multiplier that fixes the number the number of electrons align:start position:0% number of electrons align:start position:0% number of electrons we are not able to calculate this align:start position:0% we are not able to calculate this align:start position:0% we are not able to calculate this functional derivative because remember align:start position:0% functional derivative because remember align:start position:0% functional derivative because remember the quantum kinetic energy of the non align:start position:0% the quantum kinetic energy of the non align:start position:0% the quantum kinetic energy of the non interacting system is again written as a align:start position:0% interacting system is again written as a align:start position:0% interacting system is again written as a Slater determinant and so there is sort align:start position:0% Slater determinant and so there is sort align:start position:0% Slater determinant and so there is sort of you know this step back in which even align:start position:0% of you know this step back in which even align:start position:0% of you know this step back in which even if we had written everything in terms of align:start position:0% if we had written everything in terms of align:start position:0% if we had written everything in terms of a charge density we are not able to align:start position:0% a charge density we are not able to align:start position:0% a charge density we are not able to explicitly calculate even not only the align:start position:0% explicitly calculate even not only the align:start position:0% explicitly calculate even not only the derivative of the true interacting align:start position:0% derivative of the true interacting align:start position:0% derivative of the true interacting electrons kinetic energy with respect to align:start position:0% electrons kinetic energy with respect to align:start position:0% electrons kinetic energy with respect to Rho but we are not even able to align:start position:0% Rho but we are not even able to align:start position:0% Rho but we are not even able to calculate the functional derivative of align:start position:0% calculate the functional derivative of align:start position:0% calculate the functional derivative of the non interacting kinetic energy with align:start position:0% the non interacting kinetic energy with align:start position:0% the non interacting kinetic energy with respect to Rho but what we are able is align:start position:0% respect to Rho but what we are able is align:start position:0% respect to Rho but what we are able is actually to calculate the derivative of align:start position:0% actually to calculate the derivative of align:start position:0% actually to calculate the derivative of that non interacting kinetic energy with align:start position:0% that non interacting kinetic energy with align:start position:0% that non interacting kinetic energy with respect align:start position:0% respect align:start position:0% respect to the orbitals that describe the align:start position:0% to the orbitals that describe the align:start position:0% to the orbitals that describe the connection align:start position:0% connection align:start position:0% connection remember that you know this align:start position:0% remember that you know this align:start position:0% remember that you know this non-independent connection electrons align:start position:0% non-independent connection electrons align:start position:0% non-independent connection electrons have an exact solution that is a Slater align:start position:0% have an exact solution that is a Slater align:start position:0% have an exact solution that is a Slater determinant and so we know they're align:start position:0% determinant and so we know they're align:start position:0% determinant and so we know they're trivial many-body wave function is a align:start position:0% trivial many-body wave function is a align:start position:0% trivial many-body wave function is a Slater determinant composed by single align:start position:0% Slater determinant composed by single align:start position:0% Slater determinant composed by single particle orbitals and the functional align:start position:0% particle orbitals and the functional align:start position:0% particle orbitals and the functional derivative of data in the independent align:start position:0% derivative of data in the independent align:start position:0% derivative of data in the independent non-interacting electrons kinetic energy align:start position:0% non-interacting electrons kinetic energy align:start position:0% non-interacting electrons kinetic energy with respect to the single particle align:start position:0% with respect to the single particle align:start position:0% with respect to the single particle orbital is now trivial and is just minus align:start position:0% orbital is now trivial and is just minus align:start position:0% orbital is now trivial and is just minus 1/2 L square so at the end of all these align:start position:0% 1/2 L square so at the end of all these align:start position:0% 1/2 L square so at the end of all these sort of complex align:start position:0% sort of complex align:start position:0% sort of complex formulation what we are left with it's align:start position:0% formulation what we are left with it's align:start position:0% formulation what we are left with it's something very simple and probably align:start position:0% something very simple and probably align:start position:0% something very simple and probably something you should focus your align:start position:0% something you should focus your align:start position:0% something you should focus your attention from now on we have now a set align:start position:0% attention from now on we have now a set align:start position:0% attention from now on we have now a set of connection equation that are the align:start position:0% of connection equation that are the align:start position:0% of connection equation that are the differential equation that the electrons align:start position:0% differential equation that the electrons align:start position:0% differential equation that the electrons need to satisfy in order to satisfy the align:start position:0% need to satisfy in order to satisfy the align:start position:0% need to satisfy in order to satisfy the variational principle with the caveat align:start position:0% variational principle with the caveat align:start position:0% variational principle with the caveat a-- that in this connection equation align:start position:0% a-- that in this connection equation align:start position:0% a-- that in this connection equation there is a Therma an exchange align:start position:0% there is a Therma an exchange align:start position:0% there is a Therma an exchange correlation term that we still don't align:start position:0% correlation term that we still don't align:start position:0% correlation term that we still don't know what it is it's at the formally align:start position:0% know what it is it's at the formally align:start position:0% know what it is it's at the formally defined as the functional derivative of align:start position:0% defined as the functional derivative of align:start position:0% defined as the functional derivative of the exchange correlation energy with align:start position:0% the exchange correlation energy with align:start position:0% the exchange correlation energy with respect to the charge density but we align:start position:0% respect to the charge density but we align:start position:0% respect to the charge density but we need to approximate somewhere and what align:start position:0% need to approximate somewhere and what align:start position:0% need to approximate somewhere and what this equation described is not anymore align:start position:0% this equation described is not anymore align:start position:0% this equation described is not anymore the true electrons in your system but align:start position:0% the true electrons in your system but align:start position:0% the true electrons in your system but they describe these cousins of the true align:start position:0% they describe these cousins of the true align:start position:0% they describe these cousins of the true electrons they describe this connection align:start position:0% electrons they describe this connection align:start position:0% electrons they describe this connection non-interacting electrons that have align:start position:0% non-interacting electrons that have align:start position:0% non-interacting electrons that have their own orbital sy i and that will align:start position:0% their own orbital sy i and that will align:start position:0% their own orbital sy i and that will give us a ground state charge density align:start position:0% give us a ground state charge density align:start position:0% give us a ground state charge density that if the exchange correlation align:start position:0% that if the exchange correlation align:start position:0% that if the exchange correlation functional was exact would be not only align:start position:0% functional was exact would be not only align:start position:0% functional was exact would be not only this is obviously the same ground state align:start position:0% this is obviously the same ground state align:start position:0% this is obviously the same ground state energy of our interacting electron align:start position:0% energy of our interacting electron align:start position:0% energy of our interacting electron system but it would be set of the exact align:start position:0% system but it would be set of the exact align:start position:0% system but it would be set of the exact solution of the problem align:start position:0% solution of the problem align:start position:0% solution of the problem ok so this equation looked a lot like a align:start position:0% ok so this equation looked a lot like a align:start position:0% ok so this equation looked a lot like a Schrodinger equation they look a lot if align:start position:0% Schrodinger equation they look a lot if align:start position:0% Schrodinger equation they look a lot if you want like the hartree-fock equation align:start position:0% you want like the hartree-fock equation align:start position:0% you want like the hartree-fock equation that we had written before because what align:start position:0% that we had written before because what align:start position:0% that we had written before because what we are saying is that align:start position:0% align:start position:0% Kaneesha electron I feel a quantum align:start position:0% Kaneesha electron I feel a quantum align:start position:0% Kaneesha electron I feel a quantum kinetic energy operator feels a hearty align:start position:0% kinetic energy operator feels a hearty align:start position:0% kinetic energy operator feels a hearty operator feels the external potential align:start position:0% operator feels the external potential align:start position:0% operator feels the external potential and then fills this sort of you know align:start position:0% and then fills this sort of you know align:start position:0% and then fills this sort of you know remaining term that is the exchange align:start position:0% remaining term that is the exchange align:start position:0% remaining term that is the exchange correlation potential again if we knew align:start position:0% correlation potential again if we knew align:start position:0% correlation potential again if we knew what were this exact exchange align:start position:0% what were this exact exchange align:start position:0% what were this exact exchange correlation potential we would have an align:start position:0% correlation potential we would have an align:start position:0% correlation potential we would have an exact solution to the problem but we align:start position:0% exact solution to the problem but we align:start position:0% exact solution to the problem but we know very good approximation and then if align:start position:0% know very good approximation and then if align:start position:0% know very good approximation and then if you want find in the ground state is not align:start position:0% you want find in the ground state is not align:start position:0% you want find in the ground state is not very different now finding the ground align:start position:0% very different now finding the ground align:start position:0% very different now finding the ground state of the hartree-fock equation with align:start position:0% state of the hartree-fock equation with align:start position:0% state of the hartree-fock equation with the caveat that actually this term here align:start position:0% the caveat that actually this term here align:start position:0% the caveat that actually this term here is going to be much simpler than the align:start position:0% is going to be much simpler than the align:start position:0% is going to be much simpler than the exchange term of the hartree-fock align:start position:0% exchange term of the hartree-fock align:start position:0% exchange term of the hartree-fock equation if you go back to the first align:start position:0% equation if you go back to the first align:start position:0% equation if you go back to the first slide to the hartree-fock equation the align:start position:0% slide to the hartree-fock equation the align:start position:0% slide to the hartree-fock equation the last term is a align:start position:0% last term is a align:start position:0% last term is a numerically very complex expression in align:start position:0% numerically very complex expression in align:start position:0% numerically very complex expression in which we sort of take the orbital and we align:start position:0% which we sort of take the orbital and we align:start position:0% which we sort of take the orbital and we put it inside align:start position:0% put it inside align:start position:0% put it inside integral differential operator now it's align:start position:0% integral differential operator now it's align:start position:0% integral differential operator now it's become simpler and that's all if you align:start position:0% become simpler and that's all if you align:start position:0% become simpler and that's all if you want so the connection equation looks align:start position:0% want so the connection equation looks align:start position:0% want so the connection equation looks very similar in practice they are align:start position:0% very similar in practice they are align:start position:0% very similar in practice they are simpler to solver they tend to be more align:start position:0% simpler to solver they tend to be more align:start position:0% simpler to solver they tend to be more accurate in most cases and that's at the align:start position:0% accurate in most cases and that's at the align:start position:0% accurate in most cases and that's at the end what leads to the success but what align:start position:0% end what leads to the success but what align:start position:0% end what leads to the success but what is critical for all of this is having a align:start position:0% is critical for all of this is having a align:start position:0% is critical for all of this is having a reasonable approximation to the exchange align:start position:0% reasonable approximation to the exchange align:start position:0% reasonable approximation to the exchange correlation potential if we add the align:start position:0% correlation potential if we add the align:start position:0% correlation potential if we add the exact exchange correlation potential align:start position:0% exact exchange correlation potential align:start position:0% exact exchange correlation potential everything would be exact in this align:start position:0% everything would be exact in this align:start position:0% everything would be exact in this formulation we would find a connection align:start position:0% formulation we would find a connection align:start position:0% formulation we would find a connection independent electrons that we are sort align:start position:0% independent electrons that we are sort align:start position:0% independent electrons that we are sort of you know the ground state electrons align:start position:0% of you know the ground state electrons align:start position:0% of you know the ground state electrons for that charge density that is align:start position:0% for that charge density that is align:start position:0% for that charge density that is ultimately equal to the charge density align:start position:0% ultimately equal to the charge density align:start position:0% ultimately equal to the charge density of the interacting electrons in this align:start position:0% of the interacting electrons in this align:start position:0% of the interacting electrons in this external potential align:start position:0% align:start position:0% ok and align:start position:0% align:start position:0% we have the align:start position:0% we have the align:start position:0% we have the Euler Lagrangian or connection align:start position:0% Euler Lagrangian or connection align:start position:0% Euler Lagrangian or connection differential equation in the previous align:start position:0% differential equation in the previous align:start position:0% differential equation in the previous page I written here and sort of you know align:start position:0% page I written here and sort of you know align:start position:0% page I written here and sort of you know just for reference also what would be align:start position:0% just for reference also what would be align:start position:0% just for reference also what would be the total energy of the system and align:start position:0% the total energy of the system and align:start position:0% the total energy of the system and usually if you had align:start position:0% usually if you had align:start position:0% usually if you had independent electron the total energy of align:start position:0% independent electron the total energy of align:start position:0% independent electron the total energy of the system is trivially the sum of each align:start position:0% the system is trivially the sum of each align:start position:0% the system is trivially the sum of each of the single particle energies okay if align:start position:0% of the single particle energies okay if align:start position:0% of the single particle energies okay if you have ten electrons and they don't align:start position:0% you have ten electrons and they don't align:start position:0% you have ten electrons and they don't interact with each other you can align:start position:0% interact with each other you can align:start position:0% interact with each other you can calculate what is the energy of each of align:start position:0% calculate what is the energy of each of align:start position:0% calculate what is the energy of each of these ten electron sum all of them and align:start position:0% these ten electron sum all of them and align:start position:0% these ten electron sum all of them and that's the total energy of the system in align:start position:0% that's the total energy of the system in align:start position:0% that's the total energy of the system in this case it's it's it's more complex align:start position:0% this case it's it's it's more complex align:start position:0% this case it's it's it's more complex and the total energy of the system can't align:start position:0% and the total energy of the system can't align:start position:0% and the total energy of the system can't be really written as that but it's got align:start position:0% be really written as that but it's got align:start position:0% be really written as that but it's got other terms that depend on the charge align:start position:0% other terms that depend on the charge align:start position:0% other terms that depend on the charge density but sort of this is you know in align:start position:0% density but sort of this is you know in align:start position:0% density but sort of this is you know in summary what your total energy is and align:start position:0% summary what your total energy is and align:start position:0% summary what your total energy is and again is nothing else than kinetic align:start position:0% again is nothing else than kinetic align:start position:0% again is nothing else than kinetic energy term sort of a heart return align:start position:0% energy term sort of a heart return align:start position:0% energy term sort of a heart return function of the charge density this align:start position:0% function of the charge density this align:start position:0% function of the charge density this exchange correlation functional and the align:start position:0% exchange correlation functional and the align:start position:0% exchange correlation functional and the interaction interaction between external align:start position:0% interaction interaction between external align:start position:0% interaction interaction between external potential and the charge density but align:start position:0% potential and the charge density but align:start position:0% potential and the charge density but this is actually different align:start position:0% align:start position:0% from the sum of the eigenvalues that align:start position:0% from the sum of the eigenvalues that align:start position:0% from the sum of the eigenvalues that would be the sum of the expectation align:start position:0% would be the sum of the expectation align:start position:0% would be the sum of the expectation values of a I align:start position:0% values of a I align:start position:0% values of a I calculated align:start position:0% align:start position:0% on the single particle orbital where T align:start position:0% on the single particle orbital where T align:start position:0% on the single particle orbital where T is again just a simple quantum kinetic align:start position:0% is again just a simple quantum kinetic align:start position:0% is again just a simple quantum kinetic energy and bks is this connection align:start position:0% energy and bks is this connection align:start position:0% energy and bks is this connection potential so if you want to calculate align:start position:0% potential so if you want to calculate align:start position:0% potential so if you want to calculate the total energy of your system even if align:start position:0% the total energy of your system even if align:start position:0% the total energy of your system even if it made of independent electron you align:start position:0% it made of independent electron you align:start position:0% it made of independent electron you can't some just a single particle align:start position:0% can't some just a single particle align:start position:0% can't some just a single particle orbitals but you have to sort of deal align:start position:0% orbitals but you have to sort of deal align:start position:0% orbitals but you have to sort of deal with this expression nothing complex in align:start position:0% with this expression nothing complex in align:start position:0% with this expression nothing complex in this is that sort of a caveat that is align:start position:0% this is that sort of a caveat that is align:start position:0% this is that sort of a caveat that is relevant when you want to sort of you align:start position:0% relevant when you want to sort of you align:start position:0% relevant when you want to sort of you know this is the reason why we can't align:start position:0% know this is the reason why we can't align:start position:0% know this is the reason why we can't really find out the equivalent of the align:start position:0% really find out the equivalent of the align:start position:0% really find out the equivalent of the Koopman theorems for hartree-fock this align:start position:0% Koopman theorems for hartree-fock this align:start position:0% Koopman theorems for hartree-fock this is why at the end there's a single align:start position:0% is why at the end there's a single align:start position:0% is why at the end there's a single particle align:start position:0% particle align:start position:0% particle energies are ultimately not physically align:start position:0% energies are ultimately not physically align:start position:0% energies are ultimately not physically meaningful they're sort of you know done align:start position:0% meaningful they're sort of you know done align:start position:0% meaningful they're sort of you know done gives us the total energy of the system align:start position:0% gives us the total energy of the system align:start position:0% gives us the total energy of the system just by taking the sum over all of them align:start position:0% just by taking the sum over all of them align:start position:0% just by taking the sum over all of them okay so in order to make this into a align:start position:0% okay so in order to make this into a align:start position:0% okay so in order to make this into a practical algorithm the only part that align:start position:0% practical algorithm the only part that align:start position:0% practical algorithm the only part that remains is finding an approximation to align:start position:0% remains is finding an approximation to align:start position:0% remains is finding an approximation to that exchange correlation term to that align:start position:0% that exchange correlation term to that align:start position:0% that exchange correlation term to that last term remember we had sort of align:start position:0% last term remember we had sort of align:start position:0% last term remember we had sort of defined is the NC functional we have align:start position:0% defined is the NC functional we have align:start position:0% defined is the NC functional we have been able to extract two meaningful align:start position:0% been able to extract two meaningful align:start position:0% been able to extract two meaningful terms the Hartree align:start position:0% terms the Hartree align:start position:0% terms the Hartree electrostatic energy and the non align:start position:0% electrostatic energy and the non align:start position:0% electrostatic energy and the non interacting connection kinetic energy align:start position:0% interacting connection kinetic energy align:start position:0% interacting connection kinetic energy and we have said what is left is a align:start position:0% and we have said what is left is a align:start position:0% and we have said what is left is a function of the charge density that we align:start position:0% function of the charge density that we align:start position:0% function of the charge density that we call the exchange correlation functional align:start position:0% call the exchange correlation functional align:start position:0% call the exchange correlation functional how we are going to approximate data align:start position:0% how we are going to approximate data align:start position:0% how we are going to approximate data well we go back to the thomas fermi idea align:start position:0% well we go back to the thomas fermi idea align:start position:0% well we go back to the thomas fermi idea we are going to do a local density align:start position:0% we are going to do a local density align:start position:0% we are going to do a local density approximation to data exchange align:start position:0% approximation to data exchange align:start position:0% approximation to data exchange correlation functional so again what we align:start position:0% correlation functional so again what we align:start position:0% correlation functional so again what we want to calculate is the exchange align:start position:0% want to calculate is the exchange align:start position:0% want to calculate is the exchange correlation energy for any arbitrary align:start position:0% correlation energy for any arbitrary align:start position:0% correlation energy for any arbitrary charge density sometimes I call the align:start position:0% charge density sometimes I call the align:start position:0% charge density sometimes I call the charge density and sometimes I call the align:start position:0% charge density and sometimes I call the align:start position:0% charge density and sometimes I call the charge density Rho but they are always align:start position:0% charge density Rho but they are always align:start position:0% charge density Rho but they are always the same so how do we do this well we align:start position:0% the same so how do we do this well we align:start position:0% the same so how do we do this well we don't have the full solution but what we align:start position:0% don't have the full solution but what we align:start position:0% don't have the full solution but what we can say again is that for a you know mo align:start position:0% can say again is that for a you know mo align:start position:0% can say again is that for a you know mo genius charge density that is you know align:start position:0% genius charge density that is you know align:start position:0% genius charge density that is you know changes values and then drops to zero I align:start position:0% changes values and then drops to zero I align:start position:0% changes values and then drops to zero I can calculate the exchange correlation align:start position:0% can calculate the exchange correlation align:start position:0% can calculate the exchange correlation energy for this charge density align:start position:0% energy for this charge density align:start position:0% energy for this charge density distribution align:start position:0% distribution align:start position:0% distribution by sort of you know decomposing Gita in align:start position:0% by sort of you know decomposing Gita in align:start position:0% by sort of you know decomposing Gita in infinitesimal volume align:start position:0% infinitesimal volume align:start position:0% infinitesimal volume inside each infinitesimal volume I can align:start position:0% inside each infinitesimal volume I can align:start position:0% inside each infinitesimal volume I can say the charge density is constant and align:start position:0% say the charge density is constant and align:start position:0% say the charge density is constant and you see I make a local dense the align:start position:0% you see I make a local dense the align:start position:0% you see I make a local dense the approximation that is I say the align:start position:0% approximation that is I say the align:start position:0% approximation that is I say the contribution to the overall align:start position:0% contribution to the overall align:start position:0% contribution to the overall exchange correlation energy of this align:start position:0% exchange correlation energy of this align:start position:0% exchange correlation energy of this non-homogeneous system can be broken align:start position:0% non-homogeneous system can be broken align:start position:0% non-homogeneous system can be broken down and each infinitesimal volume will align:start position:0% down and each infinitesimal volume will align:start position:0% down and each infinitesimal volume will give its own contribution to the total align:start position:0% give its own contribution to the total align:start position:0% give its own contribution to the total exchange correlation density this is you align:start position:0% exchange correlation density this is you align:start position:0% exchange correlation density this is you know in principle it's not correct I align:start position:0% know in principle it's not correct I align:start position:0% know in principle it's not correct I mean our problem doesn't have to be align:start position:0% mean our problem doesn't have to be align:start position:0% mean our problem doesn't have to be local in any way actually as people say align:start position:0% local in any way actually as people say align:start position:0% local in any way actually as people say this exchange correlation functional the align:start position:0% this exchange correlation functional the align:start position:0% this exchange correlation functional the true one although we don't know what it align:start position:0% true one although we don't know what it align:start position:0% true one although we don't know what it is and we know that is ultra non-local align:start position:0% is and we know that is ultra non-local align:start position:0% is and we know that is ultra non-local so it can't be decomposed into terms align:start position:0% so it can't be decomposed into terms align:start position:0% so it can't be decomposed into terms that independently sum up so in align:start position:0% that independently sum up so in align:start position:0% that independently sum up so in principle we can do this but in practice align:start position:0% principle we can do this but in practice align:start position:0% principle we can do this but in practice it tends to be a good approximation for align:start position:0% it tends to be a good approximation for align:start position:0% it tends to be a good approximation for a lot of cases and so what is going to align:start position:0% a lot of cases and so what is going to align:start position:0% a lot of cases and so what is going to be the contribution to the exchange align:start position:0% be the contribution to the exchange align:start position:0% be the contribution to the exchange correlation energy from this align:start position:0% correlation energy from this align:start position:0% correlation energy from this infinitesimal volume where say the align:start position:0% infinitesimal volume where say the align:start position:0% infinitesimal volume where say the charge density there is a 0.5 well what align:start position:0% charge density there is a 0.5 well what align:start position:0% charge density there is a 0.5 well what we need to do is we need to find out align:start position:0% we need to do is we need to find out align:start position:0% we need to do is we need to find out what is the exchange correlation energy align:start position:0% what is the exchange correlation energy align:start position:0% what is the exchange correlation energy for the homogeneous electron gas that is align:start position:0% for the homogeneous electron gas that is align:start position:0% for the homogeneous electron gas that is at this density that's something that align:start position:0% at this density that's something that align:start position:0% at this density that's something that with some advanced computational align:start position:0% with some advanced computational align:start position:0% with some advanced computational techniques we can actually find out align:start position:0% techniques we can actually find out align:start position:0% techniques we can actually find out almost exactly so we would know if we align:start position:0% almost exactly so we would know if we align:start position:0% almost exactly so we would know if we add a homogeneous charge density point 5 align:start position:0% add a homogeneous charge density point 5 align:start position:0% add a homogeneous charge density point 5 everywhere what would be the charge align:start position:0% everywhere what would be the charge align:start position:0% everywhere what would be the charge density per unit volume and we can find align:start position:0% density per unit volume and we can find align:start position:0% density per unit volume and we can find out what is you know the exchange align:start position:0% out what is you know the exchange align:start position:0% out what is you know the exchange correlation charge density per unit align:start position:0% correlation charge density per unit align:start position:0% correlation charge density per unit volume not only four point five point align:start position:0% volume not only four point five point align:start position:0% volume not only four point five point six point seven any finger and what we align:start position:0% six point seven any finger and what we align:start position:0% six point seven any finger and what we are saying is that in this align:start position:0% are saying is that in this align:start position:0% are saying is that in this non-homogeneous problem we construct the align:start position:0% non-homogeneous problem we construct the align:start position:0% non-homogeneous problem we construct the overall exchange correlation energy by align:start position:0% overall exchange correlation energy by align:start position:0% overall exchange correlation energy by summing up these different pieces and so align:start position:0% summing up these different pieces and so align:start position:0% summing up these different pieces and so this is what separately and other did in align:start position:0% this is what separately and other did in align:start position:0% this is what separately and other did in 1980 they basically found out align:start position:0% 1980 they basically found out align:start position:0% 1980 they basically found out what was the align:start position:0% what was the align:start position:0% what was the almost exact sort of closely to align:start position:0% almost exact sort of closely to align:start position:0% almost exact sort of closely to numerical exact solution for the align:start position:0% numerical exact solution for the align:start position:0% numerical exact solution for the homogeneous electron gas the Tisza for a align:start position:0% homogeneous electron gas the Tisza for a align:start position:0% homogeneous electron gas the Tisza for a system in which you have only electrons align:start position:0% system in which you have only electrons align:start position:0% system in which you have only electrons homogeneously so the charge density is align:start position:0% homogeneously so the charge density is align:start position:0% homogeneously so the charge density is identical everywhere and those electron align:start position:0% identical everywhere and those electron align:start position:0% identical everywhere and those electron interact so you can calculate the energy align:start position:0% interact so you can calculate the energy align:start position:0% interact so you can calculate the energy of this interacting electron problem align:start position:0% of this interacting electron problem align:start position:0% of this interacting electron problem exactly as a function of the density align:start position:0% exactly as a function of the density align:start position:0% exactly as a function of the density okay so you change the density in your align:start position:0% okay so you change the density in your align:start position:0% okay so you change the density in your sort of volume and you find out what is align:start position:0% sort of volume and you find out what is align:start position:0% sort of volume and you find out what is this energy and then you can calculate align:start position:0% this energy and then you can calculate align:start position:0% this energy and then you can calculate what is for you know any of these align:start position:0% what is for you know any of these align:start position:0% what is for you know any of these density what is the align:start position:0% density what is the align:start position:0% density what is the connection quantum kinetic energy you align:start position:0% connection quantum kinetic energy you align:start position:0% connection quantum kinetic energy you can find out what is the Hartree align:start position:0% can find out what is the Hartree align:start position:0% can find out what is the Hartree electrostatic energy and so you can also align:start position:0% electrostatic energy and so you can also align:start position:0% electrostatic energy and so you can also find out for the specific case of the align:start position:0% find out for the specific case of the align:start position:0% find out for the specific case of the homogeneous gas you can find out align:start position:0% homogeneous gas you can find out align:start position:0% homogeneous gas you can find out numerically what would be the exchange align:start position:0% numerically what would be the exchange align:start position:0% numerically what would be the exchange correlation density and so that's align:start position:0% correlation density and so that's align:start position:0% correlation density and so that's basically a function so for the align:start position:0% basically a function so for the align:start position:0% basically a function so for the homogeneous gas that is for the casing align:start position:0% homogeneous gas that is for the casing align:start position:0% homogeneous gas that is for the casing which n doesn't depend on R people found align:start position:0% which n doesn't depend on R people found align:start position:0% which n doesn't depend on R people found out what was basically these align:start position:0% out what was basically these align:start position:0% out what was basically these exchange correlation energy it was align:start position:0% exchange correlation energy it was align:start position:0% exchange correlation energy it was calculated align:start position:0% calculated align:start position:0% calculated as a function this is a function align:start position:0% as a function this is a function align:start position:0% as a function this is a function of what people call RS Araiza is the align:start position:0% of what people call RS Araiza is the align:start position:0% of what people call RS Araiza is the radius of a sphere that contains one align:start position:0% radius of a sphere that contains one align:start position:0% radius of a sphere that contains one electron so it sort of you know inverse align:start position:0% electron so it sort of you know inverse align:start position:0% electron so it sort of you know inverse quantity with respect to the density so align:start position:0% quantity with respect to the density so align:start position:0% quantity with respect to the density so numerical calculation what are called align:start position:0% numerical calculation what are called align:start position:0% numerical calculation what are called quantum Monte Carlo calculation really align:start position:0% quantum Monte Carlo calculation really align:start position:0% quantum Monte Carlo calculation really solved the interacting ash reading an align:start position:0% solved the interacting ash reading an align:start position:0% solved the interacting ash reading an equation problem but for the specific align:start position:0% equation problem but for the specific align:start position:0% equation problem but for the specific case of an electron gas that there's a align:start position:0% case of an electron gas that there's a align:start position:0% case of an electron gas that there's a homogeneous density they were able to do align:start position:0% homogeneous density they were able to do align:start position:0% homogeneous density they were able to do that for various density and so now we align:start position:0% that for various density and so now we align:start position:0% that for various density and so now we have a function for the homogeneous align:start position:0% have a function for the homogeneous align:start position:0% have a function for the homogeneous problem for the non-homogeneous problem align:start position:0% problem for the non-homogeneous problem align:start position:0% problem for the non-homogeneous problem we take a local density approximation align:start position:0% we take a local density approximation align:start position:0% we take a local density approximation and we say that the overall exchange align:start position:0% and we say that the overall exchange align:start position:0% and we say that the overall exchange correlation energy is given by the align:start position:0% correlation energy is given by the align:start position:0% correlation energy is given by the integral over all the infinitesimal align:start position:0% integral over all the infinitesimal align:start position:0% integral over all the infinitesimal volume and each infinitesimal volume align:start position:0% volume and each infinitesimal volume align:start position:0% volume and each infinitesimal volume will have a certain density and will align:start position:0% will have a certain density and will align:start position:0% will have a certain density and will contribute with you know with its own align:start position:0% contribute with you know with its own align:start position:0% contribute with you know with its own density if the density is going to be align:start position:0% density if the density is going to be align:start position:0% density if the density is going to be equal to here this will be the value of align:start position:0% equal to here this will be the value of align:start position:0% equal to here this will be the value of the contribution of that infinitesimal align:start position:0% the contribution of that infinitesimal align:start position:0% the contribution of that infinitesimal volume if the density somewhere else align:start position:0% volume if the density somewhere else align:start position:0% volume if the density somewhere else corresponds to this this will be the align:start position:0% corresponds to this this will be the align:start position:0% corresponds to this this will be the correspondent so we really patch up this align:start position:0% correspondent so we really patch up this align:start position:0% correspondent so we really patch up this overall align:start position:0% overall align:start position:0% overall exchange correlation term from all the align:start position:0% exchange correlation term from all the align:start position:0% exchange correlation term from all the little infinitesimal volume exactly as align:start position:0% little infinitesimal volume exactly as align:start position:0% little infinitesimal volume exactly as Thomas Fermi had done but now we do it align:start position:0% Thomas Fermi had done but now we do it align:start position:0% Thomas Fermi had done but now we do it for a align:start position:0% for a align:start position:0% for a atoma that is a much smaller term in our align:start position:0% atoma that is a much smaller term in our align:start position:0% atoma that is a much smaller term in our problem Thomas and Fermi at Donita for align:start position:0% problem Thomas and Fermi at Donita for align:start position:0% problem Thomas and Fermi at Donita for the quantum kinetic energy instead what align:start position:0% the quantum kinetic energy instead what align:start position:0% the quantum kinetic energy instead what connection do it they do it for what is align:start position:0% connection do it they do it for what is align:start position:0% connection do it they do it for what is left from their Universal functional align:start position:0% left from their Universal functional align:start position:0% left from their Universal functional once you have taken out of the align:start position:0% once you have taken out of the align:start position:0% once you have taken out of the electrostatic and once you have taken align:start position:0% electrostatic and once you have taken align:start position:0% electrostatic and once you have taken out the quantum kinetic energy of the align:start position:0% out the quantum kinetic energy of the align:start position:0% out the quantum kinetic energy of the non-interacting electrons at this point align:start position:0% non-interacting electrons at this point align:start position:0% non-interacting electrons at this point in time align:start position:0% in time align:start position:0% in time if you want 1980 and even before without align:start position:0% if you want 1980 and even before without align:start position:0% if you want 1980 and even before without the computation with some sort of align:start position:0% the computation with some sort of align:start position:0% the computation with some sort of analytical approximations to this curve align:start position:0% analytical approximations to this curve align:start position:0% analytical approximations to this curve the inste functional theory becomes not align:start position:0% the inste functional theory becomes not align:start position:0% the inste functional theory becomes not only a theory but also a practical align:start position:0% only a theory but also a practical align:start position:0% only a theory but also a practical algorithm we have a sort of expression align:start position:0% algorithm we have a sort of expression align:start position:0% algorithm we have a sort of expression for the exchange correlation term and so align:start position:0% for the exchange correlation term and so align:start position:0% for the exchange correlation term and so now it's just a matter of trying to find align:start position:0% now it's just a matter of trying to find align:start position:0% now it's just a matter of trying to find out what the solution to these problems align:start position:0% out what the solution to these problems align:start position:0% out what the solution to these problems are and because somehow conceptually we align:start position:0% are and because somehow conceptually we align:start position:0% are and because somehow conceptually we start from the homogeneous electron gasa align:start position:0% start from the homogeneous electron gasa align:start position:0% start from the homogeneous electron gasa it turns out that you know this approach align:start position:0% it turns out that you know this approach align:start position:0% it turns out that you know this approach worked especially well for solids I mean align:start position:0% worked especially well for solids I mean align:start position:0% worked especially well for solids I mean the valence electrons in a solid align:start position:0% the valence electrons in a solid align:start position:0% the valence electrons in a solid are a much less structured than the align:start position:0% are a much less structured than the align:start position:0% are a much less structured than the electrons in a molecule that you know align:start position:0% electrons in a molecule that you know align:start position:0% electrons in a molecule that you know they need to drop to zero so the charge align:start position:0% they need to drop to zero so the charge align:start position:0% they need to drop to zero so the charge dance in a solid overall varies less align:start position:0% dance in a solid overall varies less align:start position:0% dance in a solid overall varies less dramatically as a function of space than align:start position:0% dramatically as a function of space than align:start position:0% dramatically as a function of space than the electron density in atoms and align:start position:0% the electron density in atoms and align:start position:0% the electron density in atoms and molecules and these are actually sort of align:start position:0% molecules and these are actually sort of align:start position:0% molecules and these are actually sort of you know what were summarized that the align:start position:0% you know what were summarized that the align:start position:0% you know what were summarized that the numerical result of separately and Dalda align:start position:0% numerical result of separately and Dalda align:start position:0% numerical result of separately and Dalda so they had calculated this exchange align:start position:0% so they had calculated this exchange align:start position:0% so they had calculated this exchange correlation energy as a function of the align:start position:0% correlation energy as a function of the align:start position:0% correlation energy as a function of the density and that was actually a align:start position:0% density and that was actually a align:start position:0% density and that was actually a computational curve a set of dots and align:start position:0% computational curve a set of dots and align:start position:0% computational curve a set of dots and this is often cited again Purdue and align:start position:0% this is often cited again Purdue and align:start position:0% this is often cited again Purdue and zoomer in a sort of paper of death among align:start position:0% zoomer in a sort of paper of death among align:start position:0% zoomer in a sort of paper of death among other things a sort of you know align:start position:0% other things a sort of you know align:start position:0% other things a sort of you know suggested align:start position:0% suggested align:start position:0% suggested analytical interpolation of all the align:start position:0% analytical interpolation of all the align:start position:0% analytical interpolation of all the numerical data and so you see it align:start position:0% numerical data and so you see it align:start position:0% numerical data and so you see it something somehow exotic but once it's align:start position:0% something somehow exotic but once it's align:start position:0% something somehow exotic but once it's defined this is just not even a align:start position:0% defined this is just not even a align:start position:0% defined this is just not even a functional is just a function of the align:start position:0% functional is just a function of the align:start position:0% functional is just a function of the charge density so it's something that is align:start position:0% charge density so it's something that is align:start position:0% charge density so it's something that is very simple to calculate align:start position:0% very simple to calculate align:start position:0% very simple to calculate in practice and so at this point density align:start position:0% in practice and so at this point density align:start position:0% in practice and so at this point density functional theory is a well-defined align:start position:0% functional theory is a well-defined align:start position:0% functional theory is a well-defined theory so you see 1980 Satterlee and align:start position:0% theory so you see 1980 Satterlee and align:start position:0% theory so you see 1980 Satterlee and alder do this quantum Monte Carlo align:start position:0% alder do this quantum Monte Carlo align:start position:0% alder do this quantum Monte Carlo calculation find out sort of what is align:start position:0% calculation find out sort of what is align:start position:0% calculation find out sort of what is this exchange correlation energy per align:start position:0% this exchange correlation energy per align:start position:0% this exchange correlation energy per doing zoom can write out a simple align:start position:0% doing zoom can write out a simple align:start position:0% doing zoom can write out a simple interpolation 1982 sort of the first align:start position:0% interpolation 1982 sort of the first align:start position:0% interpolation 1982 sort of the first time that I think we see sort of where align:start position:0% time that I think we see sort of where align:start position:0% time that I think we see sort of where all of this is going Marvin Cohen in align:start position:0% all of this is going Marvin Cohen in align:start position:0% all of this is going Marvin Cohen in Berkeley sort of you know has been align:start position:0% Berkeley sort of you know has been align:start position:0% Berkeley sort of you know has been working for two or three years align:start position:0% working for two or three years align:start position:0% working for two or three years aleksander was that reason him number of align:start position:0% aleksander was that reason him number of align:start position:0% aleksander was that reason him number of his students they have been able to align:start position:0% his students they have been able to align:start position:0% his students they have been able to actually write out all the electronic align:start position:0% actually write out all the electronic align:start position:0% actually write out all the electronic structure codes that are able to solve align:start position:0% structure codes that are able to solve align:start position:0% structure codes that are able to solve the density functional equation for the align:start position:0% the density functional equation for the align:start position:0% the density functional equation for the case of a periodic solid and so they align:start position:0% case of a periodic solid and so they align:start position:0% case of a periodic solid and so they address the case of silicon sort of the align:start position:0% address the case of silicon sort of the align:start position:0% address the case of silicon sort of the most important material in electronics align:start position:0% most important material in electronics align:start position:0% most important material in electronics and so what they do is they're able now align:start position:0% and so what they do is they're able now align:start position:0% and so what they do is they're able now to calculate you know the energy of that align:start position:0% to calculate you know the energy of that align:start position:0% to calculate you know the energy of that system as a function of the atomic align:start position:0% system as a function of the atomic align:start position:0% system as a function of the atomic position and in particular as a function align:start position:0% position and in particular as a function align:start position:0% position and in particular as a function of the lattice parameter so you know align:start position:0% of the lattice parameter so you know align:start position:0% of the lattice parameter so you know first thing that they do is they take align:start position:0% first thing that they do is they take align:start position:0% first thing that they do is they take silicon in it die among the structure so align:start position:0% silicon in it die among the structure so align:start position:0% silicon in it die among the structure so you know the FCC lattice with two atoms align:start position:0% you know the FCC lattice with two atoms align:start position:0% you know the FCC lattice with two atoms as a basis and they calculate that align:start position:0% as a basis and they calculate that align:start position:0% as a basis and they calculate that energy as a function of the lattice align:start position:0% energy as a function of the lattice align:start position:0% energy as a function of the lattice parameter and it looks something like align:start position:0% parameter and it looks something like align:start position:0% parameter and it looks something like this and then obviously you know as you align:start position:0% this and then obviously you know as you align:start position:0% this and then obviously you know as you have learned by now you look at what is align:start position:0% have learned by now you look at what is align:start position:0% have learned by now you look at what is the minimum of that energy and it is the align:start position:0% the minimum of that energy and it is the align:start position:0% the minimum of that energy and it is the theoretical prediction of the lattice align:start position:0% theoretical prediction of the lattice align:start position:0% theoretical prediction of the lattice parameter and this Mac on you know one align:start position:0% parameter and this Mac on you know one align:start position:0% parameter and this Mac on you know one percent error they look at the second align:start position:0% percent error they look at the second align:start position:0% percent error they look at the second derivative this curvature here is really align:start position:0% derivative this curvature here is really align:start position:0% derivative this curvature here is really the bulk models of your problem again align:start position:0% the bulk models of your problem again align:start position:0% the bulk models of your problem again you know five ten percent error and then align:start position:0% you know five ten percent error and then align:start position:0% you know five ten percent error and then they say well let's suppose that we have align:start position:0% they say well let's suppose that we have align:start position:0% they say well let's suppose that we have silicon not in the diamond phase but align:start position:0% silicon not in the diamond phase but align:start position:0% silicon not in the diamond phase but let's suppose that we have silicon in align:start position:0% let's suppose that we have silicon in align:start position:0% let's suppose that we have silicon in the beta T in phase and so you know this align:start position:0% the beta T in phase and so you know this align:start position:0% the beta T in phase and so you know this is also experimentally known and we know align:start position:0% is also experimentally known and we know align:start position:0% is also experimentally known and we know in the better teen what is the lattice align:start position:0% in the better teen what is the lattice align:start position:0% in the better teen what is the lattice parameter of silicon and we know from align:start position:0% parameter of silicon and we know from align:start position:0% parameter of silicon and we know from the Maxwell construction what is the align:start position:0% the Maxwell construction what is the align:start position:0% the Maxwell construction what is the pressure align:start position:0% pressure align:start position:0% pressure at which we would have a transition from align:start position:0% at which we would have a transition from align:start position:0% at which we would have a transition from say diamanda to beta Tina and again you align:start position:0% say diamanda to beta Tina and again you align:start position:0% say diamanda to beta Tina and again you know I can't remember what was the error align:start position:0% know I can't remember what was the error align:start position:0% know I can't remember what was the error but is substantially correct and you align:start position:0% but is substantially correct and you align:start position:0% but is substantially correct and you know they were able to actually sort of align:start position:0% know they were able to actually sort of align:start position:0% know they were able to actually sort of calculate the sort of complex zoology of align:start position:0% calculate the sort of complex zoology of align:start position:0% calculate the sort of complex zoology of all the high pressure phases of silicon align:start position:0% all the high pressure phases of silicon align:start position:0% all the high pressure phases of silicon and it was in remarkable agreement with align:start position:0% and it was in remarkable agreement with align:start position:0% and it was in remarkable agreement with experiment so 1982 this is the in align:start position:0% experiment so 1982 this is the in align:start position:0% experiment so 1982 this is the in enjoyment in particular Marvin Cohen in align:start position:0% enjoyment in particular Marvin Cohen in align:start position:0% enjoyment in particular Marvin Cohen in Berkeley shows that you know for a-- align:start position:0% align:start position:0% Marvin align:start position:0% align:start position:0% Cohen for a realistic case' density align:start position:0% Cohen for a realistic case' density align:start position:0% Cohen for a realistic case' density functional theory is able really to give align:start position:0% functional theory is able really to give align:start position:0% functional theory is able really to give us quantitative prediction Marvin Cohen align:start position:0% us quantitative prediction Marvin Cohen align:start position:0% us quantitative prediction Marvin Cohen has actually become this year the align:start position:0% has actually become this year the align:start position:0% has actually become this year the president of the American Physical align:start position:0% president of the American Physical align:start position:0% president of the American Physical Society okay so this is really the align:start position:0% Society okay so this is really the align:start position:0% Society okay so this is really the beginning of density functional theory align:start position:0% beginning of density functional theory align:start position:0% beginning of density functional theory as a practical approach and in many ways align:start position:0% as a practical approach and in many ways align:start position:0% as a practical approach and in many ways what has happened between align:start position:0% what has happened between align:start position:0% what has happened between 1982 and today is that we have become align:start position:0% 1982 and today is that we have become align:start position:0% 1982 and today is that we have become better and better at solving the align:start position:0% better and better at solving the align:start position:0% better and better at solving the algorithm for this overall still complex align:start position:0% algorithm for this overall still complex align:start position:0% algorithm for this overall still complex computational problem and you see a lot align:start position:0% computational problem and you see a lot align:start position:0% computational problem and you see a lot of this in the next two lectures that align:start position:0% of this in the next two lectures that align:start position:0% of this in the next two lectures that follows and we have become align:start position:0% follows and we have become align:start position:0% follows and we have become somewhat better not really dramatically align:start position:0% somewhat better not really dramatically align:start position:0% somewhat better not really dramatically better in calculating that exchange align:start position:0% better in calculating that exchange align:start position:0% better in calculating that exchange correlation energy in a way sort of you align:start position:0% correlation energy in a way sort of you align:start position:0% correlation energy in a way sort of you know the ideas of align:start position:0% know the ideas of align:start position:0% know the ideas of Coney Shama from 1965 of having a local align:start position:0% Coney Shama from 1965 of having a local align:start position:0% Coney Shama from 1965 of having a local density approximation is still very good align:start position:0% density approximation is still very good align:start position:0% density approximation is still very good I mean it's not used nowadays anymore align:start position:0% I mean it's not used nowadays anymore align:start position:0% I mean it's not used nowadays anymore that much but you know it's as close as align:start position:0% that much but you know it's as close as align:start position:0% that much but you know it's as close as you know what we can do now is not align:start position:0% you know what we can do now is not align:start position:0% you know what we can do now is not really that much better and you know as align:start position:0% really that much better and you know as align:start position:0% really that much better and you know as you can imagine sort of you know what align:start position:0% you can imagine sort of you know what align:start position:0% you can imagine sort of you know what people have done that was a bit better align:start position:0% people have done that was a bit better align:start position:0% people have done that was a bit better was introducing gradients in your align:start position:0% was introducing gradients in your align:start position:0% was introducing gradients in your problem so you have you're trying to align:start position:0% problem so you have you're trying to align:start position:0% problem so you have you're trying to guess what the energy of an align:start position:0% guess what the energy of an align:start position:0% guess what the energy of an inhomogeneous system comes align:start position:0% inhomogeneous system comes align:start position:0% inhomogeneous system comes starting from what you know about the align:start position:0% starting from what you know about the align:start position:0% starting from what you know about the homogeneous electron Gaza well maybe you align:start position:0% homogeneous electron Gaza well maybe you align:start position:0% homogeneous electron Gaza well maybe you should somehow throw in into your align:start position:0% should somehow throw in into your align:start position:0% should somehow throw in into your problem also the first derivative the align:start position:0% problem also the first derivative the align:start position:0% problem also the first derivative the gradient of the density and so people align:start position:0% gradient of the density and so people align:start position:0% gradient of the density and so people did that fairly sooner in the early 80s align:start position:0% did that fairly sooner in the early 80s align:start position:0% did that fairly sooner in the early 80s and sort of you using the gradients was align:start position:0% and sort of you using the gradients was align:start position:0% and sort of you using the gradients was actually much worse there is there was align:start position:0% actually much worse there is there was align:start position:0% actually much worse there is there was you know a miracle in the local density align:start position:0% you know a miracle in the local density align:start position:0% you know a miracle in the local density approximation in which the actual align:start position:0% approximation in which the actual align:start position:0% approximation in which the actual expression of the local density align:start position:0% expression of the local density align:start position:0% expression of the local density approximation satisfy satisfies a lot of align:start position:0% approximation satisfy satisfies a lot of align:start position:0% approximation satisfy satisfies a lot of symmetry properties and scaling align:start position:0% symmetry properties and scaling align:start position:0% symmetry properties and scaling properties of what would be the exact align:start position:0% properties of what would be the exact align:start position:0% properties of what would be the exact exchange correlation functional the time align:start position:0% exchange correlation functional the time align:start position:0% exchange correlation functional the time people patina gradients all these sort align:start position:0% people patina gradients all these sort align:start position:0% people patina gradients all these sort of you know symmetries and scaling align:start position:0% of you know symmetries and scaling align:start position:0% of you know symmetries and scaling properties were sort of thrown to the align:start position:0% properties were sort of thrown to the align:start position:0% properties were sort of thrown to the dogs and actually the GGAs sorry them align:start position:0% dogs and actually the GGAs sorry them align:start position:0% dogs and actually the GGAs sorry them the gradient approximation were working align:start position:0% the gradient approximation were working align:start position:0% the gradient approximation were working much much worse and so people needed to align:start position:0% much much worse and so people needed to align:start position:0% much much worse and so people needed to realize a sort of in the late align:start position:0% realize a sort of in the late align:start position:0% realize a sort of in the late 80s at the work of axle-back a of John align:start position:0% 80s at the work of axle-back a of John align:start position:0% 80s at the work of axle-back a of John Purdue especially a lotta that you sort align:start position:0% Purdue especially a lotta that you sort align:start position:0% Purdue especially a lotta that you sort of need to introduce gradients in ways align:start position:0% of need to introduce gradients in ways align:start position:0% of need to introduce gradients in ways that still satisfy a lot of the these align:start position:0% that still satisfy a lot of the these align:start position:0% that still satisfy a lot of the these analytical forms and in many ways by now align:start position:0% analytical forms and in many ways by now align:start position:0% analytical forms and in many ways by now there is a sort of generalized align:start position:0% there is a sort of generalized align:start position:0% there is a sort of generalized exchange correlation functional that align:start position:0% exchange correlation functional that align:start position:0% exchange correlation functional that being set of developed in the mid 90s by align:start position:0% being set of developed in the mid 90s by align:start position:0% being set of developed in the mid 90s by / - Kieran Burke now at Rutgers and align:start position:0% / - Kieran Burke now at Rutgers and align:start position:0% / - Kieran Burke now at Rutgers and Matthew but yes elder horf that is align:start position:0% Matthew but yes elder horf that is align:start position:0% Matthew but yes elder horf that is called the PBE functional that is align:start position:0% called the PBE functional that is align:start position:0% called the PBE functional that is becoming a sort of the workhorse so a align:start position:0% becoming a sort of the workhorse so a align:start position:0% becoming a sort of the workhorse so a lot of the time you see sort of density align:start position:0% lot of the time you see sort of density align:start position:0% lot of the time you see sort of density functional calculation than in the PBE align:start position:0% functional calculation than in the PBE align:start position:0% functional calculation than in the PBE GGA approximation but again you know align:start position:0% GGA approximation but again you know align:start position:0% GGA approximation but again you know these are important improvements but if align:start position:0% these are important improvements but if align:start position:0% these are important improvements but if you want just you know sort of very align:start position:0% you want just you know sort of very align:start position:0% you want just you know sort of very little on top of the local density align:start position:0% little on top of the local density align:start position:0% little on top of the local density approximation of the sixties align:start position:0% approximation of the sixties align:start position:0% approximation of the sixties the chemistry community is also sort of align:start position:0% the chemistry community is also sort of align:start position:0% the chemistry community is also sort of you know than a number of very align:start position:0% you know than a number of very align:start position:0% you know than a number of very intriguing align:start position:0% intriguing align:start position:0% intriguing developments in particular there are align:start position:0% developments in particular there are align:start position:0% developments in particular there are things that are Treefolk does very well align:start position:0% things that are Treefolk does very well align:start position:0% things that are Treefolk does very well in particular because you have the sort align:start position:0% in particular because you have the sort align:start position:0% in particular because you have the sort of exchange term in hartree-fock you align:start position:0% of exchange term in hartree-fock you align:start position:0% of exchange term in hartree-fock you cancel remember the self interaction say align:start position:0% cancel remember the self interaction say align:start position:0% cancel remember the self interaction say in the single electron problem coming align:start position:0% in the single electron problem coming align:start position:0% in the single electron problem coming from the heart rate electrostatic align:start position:0% from the heart rate electrostatic align:start position:0% from the heart rate electrostatic problem the instant functional theory in align:start position:0% problem the instant functional theory in align:start position:0% problem the instant functional theory in theory in its exact formulation would be align:start position:0% theory in its exact formulation would be align:start position:0% theory in its exact formulation would be self interaction corrected but in align:start position:0% self interaction corrected but in align:start position:0% self interaction corrected but in practice it is not if you solve the align:start position:0% practice it is not if you solve the align:start position:0% practice it is not if you solve the hydrogen atom with density functional align:start position:0% hydrogen atom with density functional align:start position:0% hydrogen atom with density functional theory you have that the electron align:start position:0% theory you have that the electron align:start position:0% theory you have that the electron interacts with the charge density align:start position:0% interacts with the charge density align:start position:0% interacts with the charge density created by this thing by the electron align:start position:0% created by this thing by the electron align:start position:0% created by this thing by the electron itself and so what sort of the quantum align:start position:0% itself and so what sort of the quantum align:start position:0% itself and so what sort of the quantum chemistry community is Danna is well align:start position:0% chemistry community is Danna is well align:start position:0% chemistry community is Danna is well they said let's take you know Lda is align:start position:0% they said let's take you know Lda is align:start position:0% they said let's take you know Lda is like that only take ggas that seemed to align:start position:0% like that only take ggas that seemed to align:start position:0% like that only take ggas that seemed to work very well but let's actually align:start position:0% work very well but let's actually align:start position:0% work very well but let's actually construct an exchange correlation align:start position:0% construct an exchange correlation align:start position:0% construct an exchange correlation functional that has a little bit of disa align:start position:0% functional that has a little bit of disa align:start position:0% functional that has a little bit of disa but got also a little bit of what we align:start position:0% but got also a little bit of what we align:start position:0% but got also a little bit of what we know worked well in the hartree-fock align:start position:0% know worked well in the hartree-fock align:start position:0% know worked well in the hartree-fock equation so they construct hybrid align:start position:0% equation so they construct hybrid align:start position:0% equation so they construct hybrid functional in which there are sort of align:start position:0% functional in which there are sort of align:start position:0% functional in which there are sort of pure density functional terms and sort align:start position:0% pure density functional terms and sort align:start position:0% pure density functional terms and sort of our three fork exchange term mixed in align:start position:0% of our three fork exchange term mixed in align:start position:0% of our three fork exchange term mixed in it makes the equation much more complex align:start position:0% it makes the equation much more complex align:start position:0% it makes the equation much more complex and if you want a is a set of less pure align:start position:0% and if you want a is a set of less pure align:start position:0% and if you want a is a set of less pure formulation of density functional theory align:start position:0% formulation of density functional theory align:start position:0% formulation of density functional theory but it can work reasonably well or very align:start position:0% but it can work reasonably well or very align:start position:0% but it can work reasonably well or very well especially again for atoms and align:start position:0% well especially again for atoms and align:start position:0% well especially again for atoms and molecules and this is a this is where we align:start position:0% molecules and this is a this is where we align:start position:0% molecules and this is a this is where we are basically with exchange correlation align:start position:0% are basically with exchange correlation align:start position:0% are basically with exchange correlation functional I think I'll stop here for align:start position:0% functional I think I'll stop here for align:start position:0% functional I think I'll stop here for today because it's a - a lot of work align:start position:0% today because it's a - a lot of work align:start position:0% today because it's a - a lot of work what we'll start seeing in the next align:start position:0% what we'll start seeing in the next align:start position:0% what we'll start seeing in the next class is a sort of you know how we align:start position:0% class is a sort of you know how we align:start position:0% class is a sort of you know how we actually solve this equation in practice align:start position:0% actually solve this equation in practice align:start position:0% actually solve this equation in practice on march 8th you will go into your align:start position:0% on march 8th you will go into your align:start position:0% on march 8th you will go into your second lab in which you'll actually align:start position:0% second lab in which you'll actually align:start position:0% second lab in which you'll actually study the energy of a solid using align:start position:0% study the energy of a solid using align:start position:0% study the energy of a solid using density functional theory what I said align:start position:0% density functional theory what I said align:start position:0% density functional theory what I said today is probably the last of the align:start position:0% today is probably the last of the align:start position:0% today is probably the last of the conceptual lectures and I understand align:start position:0% conceptual lectures and I understand align:start position:0% conceptual lectures and I understand that some of it is very complex there align:start position:0% that some of it is very complex there align:start position:0% that some of it is very complex there are a-- there is reading material posted align:start position:0% are a-- there is reading material posted align:start position:0% are a-- there is reading material posted on the stellar website there is the align:start position:0% on the stellar website there is the align:start position:0% on the stellar website there is the corner of get opals paper on the insta align:start position:0% corner of get opals paper on the insta align:start position:0% corner of get opals paper on the insta functional theory and some of the align:start position:0% functional theory and some of the align:start position:0% functional theory and some of the readings that I've given are very useful align:start position:0% readings that I've given are very useful align:start position:0% readings that I've given are very useful that the two best books that are also align:start position:0% that the two best books that are also align:start position:0% that the two best books that are also cited at the end of this lecture are align:start position:0% cited at the end of this lecture are align:start position:0% cited at the end of this lecture are probably the one by hawker or the one by align:start position:0% probably the one by hawker or the one by align:start position:0% probably the one by hawker or the one by power and young both called the density align:start position:0% power and young both called the density align:start position:0% power and young both called the density functional Theory or density functional align:start position:0% functional Theory or density functional align:start position:0% functional Theory or density functional theory in practice and they are cited on align:start position:0% theory in practice and they are cited on align:start position:0% theory in practice and they are cited on the last page otherwise er this is it align:start position:0% the last page otherwise er this is it align:start position:0% the last page otherwise er this is it for today and see you next week