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Quantum Theory of Solids Mervyn Roy (S6) www2.le.ac.uk/departments/physics/people/mervynroy PA4311 Quantum Theory of Solids Course Outline 1. Introduction and background 2. The many-electron wavefunction - Introduction to quantum chemistry (Hartree, HF, and CI methods) 3. Introduction to density functional theory (DFT) - Periodic solids, plane waves and pseudopotentials 4. Linear combination of atomic orbitals 5. Effective mass theory 6. 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) PA4311 Quantum Theory of Solids Resources Lecture notes www2.le.ac.uk/departments/physics/people/mervynroy Abinit website - including comprehensive help and tutorials • www.abinit.org Books • Electronic Structure, RM Martin • Solid State Physics, Ashcroft and Mermin • Solid State Physics, Hook and Hall Plus many other relevant text books and online references – see the library Prerequisites • 3210 Quantum Mechanics • 2230 Condensed Matter Physics • Also – 214 Fourier Series, 372 Fourier Transforms, etc. PA4311 Quantum Theory of Solids The problem electron-ion interaction electron KE Ψ = Ψ −ℏ2 = 2 ℏ2 − ion 2 KE 2 − 1 2 + 2 ≠ 2 1 + 40 − 2 2 40 | − | electronelectron interaction 2 ≠ 40 − ion-ion interaction > 1023 electrons and ions Ψ is a function of electron co-ordinates, , (and spins), and ion co-ordinates, (and spins) But, mI ≫ so ion KE term is small PA4311 Quantum Theory of Solids Timescales From CA Ullrich, TimeDependent DensityFunction Theory, Oxford University Press (2012) PA4311 Quantum Theory of Solids The problem Born-Oppenheimer approximation - electrons react instantaneously to changes in nuclear positions −ℏ2 = 2 2 − 2 1 + 40 − 2 2 ≠ 40 − Or, in atomic units, 1 =− 2 2 − 1 + 2 1 ≠ − + But, still have N > 1023 electrons Ψ is a function of electron co-ordinates, , (and spins). Need to develop some approximations! PA4311 Quantum Theory of Solids + Constant depends on ion positions Why bother? The modern world is build upon our understanding of the electronic properties of solids Solid state (nano) physics, materials physics, space technology etc. etc. Spectroscopy e.g. astrophysics, Earth observation science – ExoMol line lists (TDDFT) Plasma physics… PA4311 Quantum Theory of Solids Highest cited papers in Physical Review suite of journals (2014) Citations Generalized Gradient Approximation Made Simple, JP Perdew, K Burke, and M Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996) 25 083 Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density, C Lee, W Yang, and RG Parr, Phys. Rev. B 37, 785 (1988) 24 292 Self-Consistent Equations Including Exchange and Correlation Effects, W Kohn and LJ Sham, Phys. Rev. 140, A1133 (1965) 18 399 Inhomogeneous Electron Gas, P Hohenberg and W Kohn, Phys. Rev. 136, B864 (1964) 15 629 Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, G Kresse and J Furthmüller, Phys. Rev. B 54, 11169 (1996) 14 495 Density-functional exchange-energy approximation with correct asymptotic behavior, AD Becke, Phys. Rev. A 38, 3098 (1988) 14 191 Special points for Brillouin-zone integrations, HJ Monkhorst, JD Pack, Phys. Rev. B 13, 5188 (1976) 12 938 From ultrasoft pseudopotentials to the projector augmented-wave method, G Kresse and D Joubert, Phys. Rev. B 59, 1758 (1999) 10 351 Helical microtubules of graphitic carbon, S. Ijima, Nature 354 , 56 (1991) (24 225) Electric field effect in atomically thin carbon films, K.S. Novoselov, A.K. Geim, et al. Science 306 (2004) (15 139) Highest cited paper on ADS astrophysics (2014) Maps of Dust Infrared Emission for Use in Estimation of Reddening and Cosmic Microwave Background Radiation Foregrounds, D.J. Schlegel et al, APJ 500, 525 (1998) 8920 Comparison with other areas - data from ADS Black hole 6765 Particle physics 4164 Metamaterials 2289 Aurora 1785 Gamma Ray Burst 1513 Density-Functional Thermochemistry 3. The Role of Exact Exchange, AD Becke, J. Chemical Physics 98, 5648 (1993) 46 280 Chemistry (data from WOK) PA4311 Quantum Theory of Solids 3210 revision (see Rae) Schrödinger equation - () = () Probability density - |()|2 ∗ Expectation values - = Variational principle - = ≥ 0 If = , then PA4311 Quantum Theory of Solids ∗ =0= − Question 1.1 If = is normalised and the are orthonormal, show that 2 = 1. ∗ subject If we wish to minimise = to the constraint that the are normalised, show that the appropriate Lagrange multiplier, = . PA4311 Quantum Theory of Solids Full variation and functionals is a functional of ∗ and ∗, = − − 1 ∗ , = ( − ) = ∫ ∗ − = 0 − =0 A functional maps a function onto a value, = 2 = . For example, 1 = + − = 2 1 See RM Martin App. A, or e.g. GC Evans, Functionals and their applications, Dover, New York, 1964 PA4311 Quantum Theory of Solids The N-electron wavefunction The -electron wavefunction depends on N spatial coordinates (and spins) Ψ(1 , 2 , … , N ) Electrons are indistinguishable: Ψ 1 , 2 , … , N 2 = Ψ 2 , 1 , … , N 2 Fermions are anti-symmetric: Ψ 1 , 2 , … , N = −Ψ 2 , 1 , … , N - they obey the Pauli exclusion principle See Tipler (4th Ed Sec. 36.6 on ‘The Schrödinger equation for 2 identical particles’) Expectation values = ΨΨ = Ψ ∗ 1 , 2 , … , Ψ(1 , 2 , … , ) 2 … PA4311 Quantum Theory of Solids