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PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 12:

PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 12: Reading: Chap. 4 in GGGPP; One-electron approximations to the many electron problem Hartree-Fock approximation Density functional theory. Quantum Theory of materials.

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PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 12:

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  1. PHY 752 Solid State Physics • 11-11:50 AM MWF Olin 103 • Plan for Lecture 12: • Reading: Chap. 4 in GGGPP; • One-electron approximations to the many electron problem • Hartree-Fock approximation • Density functional theory PHY 752 Fall 2015 -- Lecture 12

  2. PHY 752 Fall 2015 -- Lecture 12

  3. Quantum Theory of materials PHY 752 Fall 2015 -- Lecture 12

  4. Hartree approximation to electronic wavefunction PHY 752 Fall 2015 -- Lecture 12

  5. Variational equation for Hartree approximation -- continued Note: In principle, the self interaction term should be omitted from Vee(r), but often it is included. PHY 752 Fall 2015 -- Lecture 12

  6. Hartree approximation -- continued In practice, the equations must be solved self-consistently PHY 752 Fall 2015 -- Lecture 12

  7. Hartree approximation -- continued PHY 752 Fall 2015 -- Lecture 12

  8. Hartree-Fock approximation to electronic wavefunction PHY 752 Fall 2015 -- Lecture 12

  9. Hartree-Fock approximation to electronic wavefunction -- continued PHY 752 Fall 2015 -- Lecture 12

  10. Variational equation for Hartree-Fock approximation -- continued Note that in the Hartree-Fock formalism, there is no spurious electron self-interaction. PHY 752 Fall 2015 -- Lecture 12

  11. Hartree-Fock approximation – continued As for the Hartree formulation, the Hartree-Fock equations must be solved iteratively. At convergence, the Hartree-Fock electronic energy can be calculated from the one-electron orbitals and the charge density PHY 752 Fall 2015 -- Lecture 12

  12. Note: Hartree-Fock theory is generally the starting approximation for “quantum chemical” treatments of the electronic structure of atoms and molecules. More accurate calculations are based on multi-determinant expansions: PHY 752 Fall 2015 -- Lecture 12

  13. Evaluation of the Hartree-Fock equations for the jellium model – homogeneous electron gas PHY 752 Fall 2015 -- Lecture 12

  14. Hartree-Fock Equations PHY 752 Fall 2015 -- Lecture 12

  15. Some details -- PHY 752 Fall 2015 -- Lecture 12

  16. More details PHY 752 Fall 2015 -- Lecture 12

  17. Eigenvalues of the plane wave orbitals: PHY 752 Fall 2015 -- Lecture 12

  18. Total electronic energy of homogeneous electron gas in Hartree-Fock approximation Some details: PHY 752 Fall 2015 -- Lecture 12

  19. Some ideas – • John Slater suggested that the average exchange potential of the homogeneous electron gas could be used to estimate the exchange interaction of a material PHY 752 Fall 2015 -- Lecture 12

  20. Kohn-Sham’s approximate exchange Kohn & Sham argued that the effective exchange potential should be determined from the density derivative: PHY 752 Fall 2015 -- Lecture 12

  21. Comment on the spatial dependence of these approximations PHY 752 Fall 2015 -- Lecture 12

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