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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)
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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) • - 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 wavefunction The -electron wavefunction depends on N spatial coordinates (and spins) Electrons are indistinguishable: Fermions are anti-symmetric: - they obey the Pauli exclusion principle See Tipler (4th Ed Sec. 36.6 on ‘The Schrödinger equation for 2 identical particles’) Expectation values
The density operator We can calculate the electron density by finding the expectation value of
A 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-consistently Total energy, , is the sum of single particle energies Single particle orbitals Hartree Equations
Question 2.1 If the Schrödinger equation is separable so that show that the expectation value of the density operator , is
Derivation of Hartree equations Assume the independent electron form of the wavefunction, then minimise subject to the constraint that each is normalised.
Question 2.2 Assume 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 approximation The single particle equations must be solved self-consistently Guess Calculate Solve Eq.s - Use new Calculate new Self consistent? No Yes Calculation finished
Hartree 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 well Interaction goes like goes like
