Computational Solid State Physics 計算物性学特論 第７回

1. Computational Solid State Physics 計算物性学特論　第７回 7.Many-body effect I Hartree approximation, Hartree-Fock approximation andDensity functional method

2. Hartree approximation N-electron Hamiltonian ・N-electron wave function i-th spin-orbit ortho-normal set

3. Expectation value of the energy single electron energy Hartreeinteraction

4. Charge density :charge density operator : charge density Hartree interaction

5. Hartree calculation for N>>1 Energy minimization with condition Self-consistent Schröedingerequation for the i-th state Electrostatic potential energy caused by electron-electron Coulomb interaction charge density

6. Hartree-Fock approximation • Pauli principle • Identical particles • Slater determinant • Exchange interaction • Hartree-Fock-Roothaan’s equation

7. Many electron Hamiltonian single electron Hamiltonian electron-electron Coulomb interaction

8. Slater determinant or N-electron wave function John Slater spin orbit Permutation of N numbers

9. Properties of Slater determinant or If Pauli principle Identical Fermi particles The Slater determinant satisfies both requirements of Pauli principle and identical Fermi particles on N-electron wave function.

10. Ground state energy Permutation of N numbers Orthonormal set

11. Expectation value of Hamiltonian

12. Expectation value of Hamiltonian

13. Expectation value of many-electron Hamiltonian Coulomb integral Exchange integral Hartree term: between like spin electrons and between unlike spin electrons Fock term: between like spin electrons

14. Exchange interaction Pauli principle Ｘ no transfer transfer No suppression of electron-electron Coulomb energy suppression of electron-electron Coulomb energy gain of exchange energy No exchange energy

15. Hartree-Fock calculation (1) Expansion by base functions

16. Hartree-Fock calculation (2) Calculation of the expectation value

17. Hartree-Fock calculation (3) Expectation value of N-electron Hamiltonian

18. Hartree-Fock calculation (4) Minimization of E with condition Hartree-Fock-Roothaan’s equation Exchange interaction is also considered in addition to electrostatic interaction.

19. Hartree-Fock calculation (5) Schröedinger equation for k-th state m: number of base functions N: number of electrons Self-consistent solution on C and P

20. Density functional theory • Density functional method to calculate the ground state of many electrons • Kohn-Sham equations to calculate the single particle state • Flow chart of solving Kohn-Sham equation

21. Many-electron Hamiltonian T: kinetic energy operator Vee: electron-electron Coulomb interaction vext: external potential

22. Variational principles • Variational principle on the ground state energy functional E[n]: The ground state energy EGSis the lowest limit of E[n]. • Representability of the ground state energy. :charge density

23. Hartree term Exchange correlation term Density-functional theory • Kohn-Sham total-energy functional for a set of doubly occupied electronic states

24. Kohn-Sham equations : Hartree potential of the electron charge density : exchange-correlation potential : excahnge-correlation functional

25. Kohn-Sham eigenvalues : Kinetic energy functional Janak’s theorem: If f dependence of εi is small, εimeans an ionization energy.

26. Local density approximation nX(r12) : Exchange-correlation energy per electron in homogeneous electron gas exchange hole distribution for like spin Local-density approximation satisfies the sum rule. Ｓｕｍ Ｒｕｌｅ: : exchange-correlation hole

27. Bloch’s theorem for periodic system G : Reciprocal lattice vectora :Lattice vector

28. Plane wave representation of Kohn-Sham equations

29. Supercell geometry Point defect Surface Molecule

30. Flow chart describing the computational procedure for the total energy calculation Conjugate gradient method Molecular-dynamics method

31. Hellman-Feynman force on ions (1) : for eigenfunctions

32. Hellman-Feynman force on ions (2) Electrostatic force between ions Electrostatic force between an ion and electron charge density

33. Problems 7 • Derive the single-electron Schröedinger equations in Hartree approximation. • Derive the single-electron Schröedinger equations in Hartree-Fock approximation. • Derive the Kohn-Sham equation in density functional method. • Solve the sub-band structure at the interface of the GaAs active channel in a HEMT structure in Hartree approximation.