Quantum Theory of Solids

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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)
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
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