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-âTriplet spli ng of S states is about 1 eV. Jellium model: Hartree-âFock theory of free electrons. ⢠No periodic poten#al.