| 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 |
| |
|
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| align:start position:0% |
| of is the famous year 1905 so just a |
| little celebration one slide of a |
|
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| 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 |
|
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| align:start position:0% |
| previous previous lecture we had really |
| |
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| align:start position:0% |
| previous previous lecture we had really |
| developed the formalism and leading to |
|
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| align:start position:0% |
| developed the formalism and leading to |
| |
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| align:start position:0% |
| developed the formalism and leading to |
| the hartree-fock equations and the |
|
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| align:start position:0% |
| the hartree-fock equations and the |
| |
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| align:start position:0% |
| the hartree-fock equations and the |
| hartree-fock equation follow from a |
|
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| align:start position:0% |
| hartree-fock equation follow from a |
| |
|
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| align:start position:0% |
| hartree-fock equation follow from a |
| certif you know very simple and very |
|
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| align:start position:0% |
| certif you know very simple and very |
| |
|
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| 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 |
|
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| align:start position:0% |
| equation and we have reformulated the |
| |
|
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| align:start position:0% |
| equation and we have reformulated the |
| Schrodinger equation in terms of the |
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| align:start position:0% |
| Schrodinger equation in terms of the |
| |
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| align:start position:0% |
| Schrodinger equation in terms of the |
| variational principle so we have a |
|
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| 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 |
|
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| 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 |
|
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| 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 |
|
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| 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 |
|
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| 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 |
|
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| 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 |
|
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| 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 |
