Computational Solid State Physics 7 Manybody effect I

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

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

Expectation value of the energy single electron energy Hartree　interaction

Charge density : charge density operator : charge density Hartree interaction

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

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

Many electron Hamiltonian single electron Hamiltonian electron-electron Coulomb interaction

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

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

Ground state energy Permutation of N numbers Orthonormal set

Expectation value of Hamiltonian

Expectation value of Hamiltonian

Expectation value of manyelectron Hamiltonian Coulomb integral Hartree term: between like spin electrons and between unlike spin electrons Exchange integral Fock term: between like spin electrons

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

Hartree-Fock calculation (1) Expansion by base functions

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

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

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

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

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

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

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

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

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

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

Local density approximation n. X(r 12) : 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

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

Plane wave representation of Kohn-Sham equations

Supercell geometry Point defect Surface Molecule

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

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

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

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