1 / 33

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

Computational Solid State Physics 計算物性学特論　第７回. 7. Many-body effect I Hartree approximation, H artree-Fock approximation and Density functional method. Hartree approximation. N-electron Hamiltonian . ・ N-electron wave function . i-th spin-orbit. ortho-normal set.

doli
Télécharger la présentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.

More Related