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

Slides:

Advertisements

Similar presentations

Modelling of Defects DFT and complementary methods

Advertisements

Quantum Theory of Solids

Molecular Quantum Mechanics

Introduction to Molecular Orbitals

Chapter 3 Electronic Structures

Time-dependent density-functional theory University of Missouri

Density Functionals: Basic DFT Theory Sergio Aragon San Francisco State University CalTech PASI January 4-16, 2004.

Computational Solid State Physics 計算物性学特論第２回 2.Interaction between atoms and the lattice properties of crystals.

Density Functional Theory An introduction

Quantum Mechanics Discussion. Quantum Mechanics: The Schrödinger Equation (time independent)! Hψ = Eψ A differential (operator) eigenvalue equation H.

Fundamentals: the quantum-mechanical many-electron problem and the Density Functional Theory approach Javier Junquera.

The Hartree-Fock-Roothaan Method. Variational Principle.

Ground State of the He Atom – 1s State First order perturbation theory Neglecting nuclear motion 1 - electron electron 2 r 1 - distance of 1 to nucleus.

Molecular Modeling: Density Functional Theory C372 Introduction to Cheminformatics II Kelsey Forsythe.

Hartree-Fock Theory Patrick Tamukong North Dakota State University

Lecture 8: Introduction to Density Functional Theory Marie Curie Tutorial Series: Modeling Biomolecules December 6-11, 2004 Mark Tuckerman Dept. of Chemistry.

Physical Chemistry 2 nd Edition Thomas Engel, Philip Reid Chapter 21 Many-Electrons Atom.

Computational Solid State Physics 計算物性学特論第９回 9. Transport properties I: Diffusive transport.

6. Second Quantization and Quantum Field Theory

Instructor: Marco Buongiorno Nardelli 516C Cox - tel

Lectures Introduction to computational modelling and statistics1 Potential models2 Density Functional.

Computational Solid State Physics 計算物性学特論 5回

Atomic units The atomic units have been chosen such that the fundamental electron properties are all equal to one atomic unit. (me=1, e=1, = h/2 = 1,

Computational Solid State Physics 計算物性学特論第４回 4. Electronic structure of crystals.

Computational Solid State Physics 計算物性学特論第８回 8. Many-body effect II: Quantum Monte Carlo method.

CHEM 580 Week 1: From Schrodinger to Hartree-Fock

First principles electronic structure: density functional theory

Density Functional Theory (DFT) DFT is an alternative approach to the theory of electronic structure; electron density plays a central role in DFT. Why.

Statistical Mechanics and Multi- Scale Simulation Methods ChBE Prof. C. Heath Turner Lecture 02 Some materials adapted from Prof. Keith E. Gubbins:

Computational Solid State Physics 計算物性学特論第６回

Hartree-Fock Theory Erin Dahlke Department of Chemistry University of Minnesota VLab Tutorial May 25, 2006.

1 Physical Chemistry III ( ) Chapter 3: Atomic Structure Piti Treesukol Kasetsart University Kamphaeng Saen Campus.

Postulates Postulate 1: A physical state is represented by a wavefunction. The probablility to find the particle at within is. Postulate 2: Physical quantities.

Lecture 11. Quantum Mechanics

Theory From Quantum Mechanics to Density Functional Theory [based on Chapter 1, Sholl & Steckel (but at a more advanced level)] Quantum mechanics (~1920s)

PA4311 Quantum Theory of Solids Quantum Theory of Solids Mervyn Roy (S6) www2.le.ac.uk/departments/physics/people/mervynroy.

Background 1927: Introduction of the Thomas-Fermi model (statistics of electrons). 1964: Hohenberg-Kohn paper proving existence of exact Density Function.

Quantum Two 1. 2 Evolution of Many Particle Systems 3.

Lecture 17. Density Functional Theory (DFT)

Fundamentals of DFT R. Wentzcovitch U of Minnesota VLab Tutorial Hohemberg-Kohn and Kohn-Sham theorems Self-consistency cycle Extensions of DFT.

Density Functional Theory A long way in 80 years L. de Broglie – Nature 112, 540 (1923). E. Schrodinger – 1925, …. Pauli exclusion Principle.

Physics “Advanced Electronic Structure” Lecture 1. Theoretical Background Contents: 1. Historical Overview. 2. Basic Equations for Interacting Electrons.

Chemistry 700 Lectures. Resources Grant and Richards, Foresman and Frisch, Exploring Chemistry with Electronic Structure Methods (Gaussian Inc., 1996)

Last hour: Electron Spin Triplet electrons “avoid each other”, the WF of the system goes to zero if the two electrons approach each other. Consequence:

Ch 10. Many-Electron Atoms MS310 Quantum Physical Chemistry - Schrödinger equation cannot be solved analytically - Schrödinger equation cannot be solved.

Physics “Advanced Electronic Structure” Lecture 2. Density Functional Theory Contents: 1. Thomas-Fermi Theory. 2. Density Functional Theory. 3.

Lecture 5. Many-Electron Atoms. Pt

DENSITY FUNCTIONAL THEORY From the theory to its practical applications. - Matthew Lane; Professor J. Staunton.

Restricted and Unrestricted Hartree-Fock method Sudarshan Dhungana Phys790 Seminar (Feb15,2007)

2/09/2015PHY 752 Spring Lecture 111 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 107 Plan for Lecture 11: Reading: Chapter 9 in MPM Approximations.

Postulates Postulate 1: A physical state is represented by a wavefunction. The probablility to find the particle at within is. Postulate 2: Physical quantities.

PA4311 Quantum Theory of Solids Quantum Theory of Solids Mervyn Roy (S6) www2.le.ac.uk/departments/physics/people/mervynroy.

Time-Dependent Density Functional Theory (TDDFT) Takashi NAKATSUKASA Theoretical Nuclear Physics Laboratory RIKEN Nishina Center CNS-EFES Summer.

©2011, Jordan, Schmidt & Kable Lecture 13 Lecture 13 Self-consistent field theory This is how we do it.

Lecture 9. Many-Electron Atoms

Computational Physics (Lecture 22) PHY4061. In 1965, Mermin extended the Hohenberg-Kohn arguments to finite temperature canonical and grand canonical.

Electron density: Probability of finding one electron of arbitrary spin within a volume element dr 1 (other electrons may be anywhere). Properties of electron.

2/11/2015PHY 752 Spring Lecture 121 PHY 752 Solid State Physics 11-11:50 AM MWF Olin 107 Plan for Lecture 12: Reading: Chapter 9 in MPM Approximations.

Ch.1. Elementary Quantum Chemistry

Ground State of the He Atom – 1s State

PHY 752 Solid State Physics

PHY 752 Solid State Physics

Solid state physics Lecture 3: chemical bonding Prof. Dr. U. Pietsch.

Schrödinger Theory of the Electronic Structure of Matter from a ‘Newtonian’ Perspective Viraht Sahni.

Chemistry Department of Fudan University

Time-Dependent Density Functional Theory (TDDFT)

Quantum Chemistry / Quantum Mechanics

Hartree Self Consistent Field Method

PHY 752 Solid State Physics

Presentation transcript:

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 Hartree interaction :charge density operator

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 Pauli principle Identical particles Slater determinant Exchange interaction 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 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.

Ground state energy Permutation of N numbers Orthonormal set

Expectation value of Hamiltonian

Expectation value of Hamiltonian

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

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

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 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

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

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

Density-functional theory 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 nX(r12) : Exchange-correlation energy per electron in homogeneous electron gas exchange hole distribution for like spin ＳｕｍＲｕｌｅ: 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 Electrostatic force between an ion and electron charge density

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.