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Beyond  the  independent   electron  approxima8on   Ross  McKenzie   condensedconcepts.blogspot.com   Reading:  AshcroC  &  Mermin,  Chapter  17  

Key  ques8ons   •  What  periodic  poten8al  should  we   use?   •  Why  is  the  independent  electron   approxima8on  so  successful?   •  When  does  it  break  down?  

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Key  concepts   •  Self-­‐consistent  fields   •  Hartree-­‐Fock  theory:  exchange   energy   •  Jellium  and  the  local  density  approx.   •  Screening   •  Fermi  liquid  theory   •  Strongly  correlated  electron   materials  

Comparison  of  electron  kine8c  energy   and  Coulomb  repulsion  energy  

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The  theory  of  everything!   •  The  many-­‐body  Schrodinger  equa8on  with   Hamiltonian  

•  Finding  the  eigenstates  is  a  highly  non-­‐trivial   problem!  

Hartree  approxima8on   •  Electron  moves  in  the  average  poten8al  due   to  all  the  other  electrons.   •   Many-­‐par8cle  Schrodinger  equn.  is  reduced   to  a  one-­‐par8cle  Schrodinger  equn.   •  Self  consistent  field   •  17.7  

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Hartree-­‐Fock  approxima8on   •  Takes  into  account  an8-­‐symmetry  of  many-­‐ par8cle  wavefunc8on  (electrons  are   fermions!)   •  Slater  determinants   •  Exchange  interac8on   •  17.15  

Two-­‐electron  problem   •  A&M.  pp.  674-­‐679  

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Energy  levels  of  Helium  atom   •  Singlet-­‐Triplet  spli^ng   of  S  states  is  about  1  eV  

Jellium  model:  Hartree-­‐Fock  theory  of   free  electrons   •  No  periodic  poten8al   •  17.19   •  17.22   •  17.23   •  17.26  local  exchange  energy   •  Cf.  Density  func8onal  theory  (1998  Chemistry   Nobel  Prize  ,  Kohn    )  

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Screening   •  Intui8ve  picture  

•  17.36  

Thomas-­‐Fermi  screening   •  17.49-­‐17.55  

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Plasmons   •  Frequency  dependent  screening  

Conclusions   •  Going  beyond  the  independent  electron   approxima8on  is  a  highly  non-­‐trivial  quantum   many-­‐body  problem.   •  Hartree-­‐Fock  theory  treats  the  electrons  in  a   mean-­‐field  sense.     •  Fermi-­‐Dirac  sta8s8cs  leads  to  the  exchange   interac8on.   •  Screening  significantly  reduces  the  effect  of   electron-­‐electron  interac8ons.  

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RM8 electron-electron interactions.pptx

Energy levels of Helium atom. • Singlet-‐Triplet spli ng of S states is about 1 eV. Jellium model: Hartree-‐Fock theory of free electrons. • No periodic poten#al.

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