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13502 13503 13504 13505 13506 13507 13508 13509 13510 13511 13512 13513 13514 13515 13516 13517 13518 13519 13520 13521 13522 13523 13524 13525 13526 13527 13528 13529 13530 13531 13532 13533 13534 13535 13536 13537 13538 13539 13540 13541 13542 13543 13544 13545 13546 13547 13548 13549 13550 13551 13552 13553 13554 13555 13556 13557 13558 13559 13560 13561 13562 13563 13564 13565 13566 13567 13568 13569 13570 13571 13572 13573 13574 13575 13576 13577 13578 13579 13580 13581 13582 13583 13584 13585 13586 13587 13588 13589 13590 13591 13592 13593 13594 13595 13596 13597 13598 13599 13600 13601 13602 13603 13604 13605 13606 13607 13608 13609 13610 13611 | 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 |