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 and Density 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
Hartree　interaction

4. Charge density Hartree interaction :charge density operator

5. Hartree calculation for N>>1
Energy minimization with condition
Self-consistent Schröedinger equation 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 determinantor 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 Ｘ Pauli principle
no transfer
transfer
suppression of electron-electron Coulomb energy
No 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 EGS is the lowest limit of E[n].
Representability of the ground state energy.
:charge density

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

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, εi　means an ionization energy.

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

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