1. 2/09/2015PHY 752 Spring 2015 -- Lecture 111 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 107 Plan for Lecture 11: Reading: Chapter 9 in MPM Approximations to the many electron problem 1.Hartree approximation 2.Hartree-Fock approximation 3.Density functional theory

2. 2/09/2015PHY 752 Spring 2015 -- Lecture 112 Note: Take-home exam scheduled for the week of March 2 nd.

3. 2/09/2015PHY 752 Spring 2015 -- Lecture 113 Born-Oppenheimer approximation Born & Huang, Dynamical Theory of Crystal Lattices, Oxford (1954) Electronic coordinates Atomic coordinates Quantum Theory of materials

4. 2/09/2015PHY 752 Spring 2015 -- Lecture 114 Quantum Theory of materials -- continued Effective nuclear interaction provided by electrons (Often treated classically)

5. 2/09/2015PHY 752 Spring 2015 -- Lecture 115 Consider electronic Hamiltonian Electron-electron interaction term prevents exactly separable electron wavefunction

6. 2/09/2015PHY 752 Spring 2015 -- Lecture 116 Hartree approximation to electronic wavefunction

7. 2/09/2015PHY 752 Spring 2015 -- Lecture 117 Variational equation for Hartree approximation -- continued Note: In principle, the self interaction term should be omitted from V ee (r), but often it is included.

8. 2/09/2015PHY 752 Spring 2015 -- Lecture 118 Hartree approximation -- continued In practice, the equations must be solved self-consistently

9. 2/09/2015PHY 752 Spring 2015 -- Lecture 119 Hartree approximation -- continued

10. 2/09/2015PHY 752 Spring 2015 -- Lecture 1110 Hartree-Fock approximation to electronic wavefunction

11. 2/09/2015PHY 752 Spring 2015 -- Lecture 1111 Hartree-Fock approximation to electronic wavefunction Second quantization formalism See Appendix C of MPM

12. 2/09/2015PHY 752 Spring 2015 -- Lecture 1112 Hartree-Fock approximation to electronic wavefunction -- continued

13. 2/09/2015PHY 752 Spring 2015 -- Lecture 1113 Variational equation for Hartree-Fock approximation -- continued Note that in the Hartree-Fock formalism, there is no spurious electron self-interaction.

14. 2/09/2015PHY 752 Spring 2015 -- Lecture 1114 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