# Quantum Theory of Solids

13 pages
7 views

View again

All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Share
Description
Quantum Theory of Solids. Mervyn Roy (S6 ) www2.le.ac.uk/departments/physics/people/mervynroy. Course Outline. Introduction and background The many-electron wavefunction - Introduction to quantum chemistry ( Hartree , HF, and CI methods) Introduction to density functional theory (DFT)
Transcript
Quantum Theory of SolidsMervyn Roy (S6)www2.le.ac.uk/departments/physics/people/mervynroyCourse Outline
• Introduction and background
• The many-electron wavefunction
• - Introduction to quantum chemistry (Hartree, HF, and CI methods)
• Introduction to density functional theory (DFT)
• - Periodic solids, plane waves and pseudopotentials
• Linear combination of atomic orbitals
• Effective mass theory
• ABINIT computer workshop (LDA DFT for periodic solids)
• Assessment: 70% final exam
• 30% coursework – mini ‘project’ report for ABINIT calculation
• (Set problems are purely formative)
• Last time…
• The modern world is build upon our understanding of the electronic properties of solids…
• Born-Oppenheimer approximation – electrons respond instantaneously to ion motion
• N-electron wavefunction contains all the information about the system
• is a function of spatial coordinates, and spins
• The variational principle is a useful starting point to find approximations to
• Rae 5th Ed. Sec. 7.3 – Variational principle & complete sets of states (q. 1.1)
• M. L. Boas, 2nd Ed. ,Ch. 4, Sec. 9 – Lagrange multipliers
• The N-electron wavefunctionThe -electron wavefunction depends on N spatial coordinates (and spins)Electrons are indistinguishable: Fermions are anti-symmetric: - they obey the Pauli exclusion principleSee Tipler (4th Ed Sec. 36.6 on ‘The Schrödinger equation for 2 identical particles’)Expectation valuesThe density operatorWe can calculate the electron density by finding the expectation value ofA hierarchy of methods
• Hartree
• ‘Independent’ particle approximation
• Hartree-Fock
• Exact inclusion of the exchange interaction
• Configuration Interaction
• Post Hartree-Fock methods attempt to include exchange and correlation
• The exponential wall
• Do we really need to know the full wavefunction?
• Hartree approximation
• ‘Independent’ electron picture – (electrons are distinguishable)
• Electrons interact via mean-field Coulomb potential - (respond to avg. charge density)
• Key points Replace interaction term with average potential, -electron wavefunction is separable, Must solve -single electron Schrödinger equations self-consistentlyTotal energy, , is the sum of single particle energies Single particle orbitalsHartree EquationsQuestion 2.1If the Schrödinger equation is separable so thatshow that the expectation value of the density operator , is Derivation of Hartree equationsAssume the independent electron form of the wavefunction,then minimise subject to the constraint that each is normalised.Question 2.2Assume that the full electron interaction can be replaced by a mean field term,Use the method of separation of variables to show that the -electron Schrödinger equation can be separated into single particle equations.Self consistent field approximationThe single particle equations must be solved self-consistentlyGuess Calculate Solve Eq.s - Use new Calculate new Self consistent?NoYesCalculation finishedHartree approximation
• Electrons are distinguishable & wavefunction is not antisymmetric
• - Pauli exclusion principle has to be put in by hand
• Electrons do not respond to the particular (as opposed to the average) configuration of the other N-1 electrons
• Self interaction problem
• Calculations are numerically complex
• But – Hartree-like calculations are important for modern DFT
• Hartree approximation
• Interaction effects (exchange and correlation) are important when the coulomb interaction energy is large compared to
• Hartree-like approximations better when
• Infinite square wellInteraction goes like goes like
Related Search
Similar documents

### online free Goldratt and the Theory of Constraints - The Quantum Leap in Management (Quistainable Business Solutions) [free] pdf

View more...

#### Urushiol 