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

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

PA4311 Quantum Theory of Solids 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) Course Outline

PA4311 Quantum Theory of Solids Lecture notes www2.le.ac.uk/departments/physics/people/mervynroy www2.le.ac.uk/departments/physics/people/mervynroy Abinit website - including comprehensive help and tutorials Books Electronic Structure, RM Martin Solid State Physics, Ashcroft and Mermin Solid State Physics, Hook and Hall Prerequisites 3210 Quantum Mechanics 2230 Condensed Matter Physics Also – 214 Fourier Series, 372 Fourier Transforms, etc. Resources Plus many other relevant text books and online references – see the library

PA4311 Quantum Theory of Solids The problem electron KE electron-ion interaction electron- electron interaction ion-ion interaction ion KE

PA4311 Quantum Theory of Solids Timescales From CA Ullrich, Time- Dependent Density- Function Theory, Oxford University Press (2012)

PA4311 Quantum Theory of Solids The problem Born-Oppenheimer approximation - electrons react instantaneously to changes in nuclear positions Or, in atomic units, Need to develop some approximations! Constant depends on ion positions

PA4311 Quantum Theory of Solids The modern world is build upon our understanding of the electronic properties of solids Why bother? 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 Chemistry (data from WOK) Density-Functional Thermochemistry 3. The Role of Exact Exchange, AD Becke, J. Chemical Physics 98, 5648 (1993) 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) 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) Self-Consistent Equations Including Exchange and Correlation Effects, W Kohn and LJ Sham, Phys. Rev. 140, A1133 (1965) Inhomogeneous Electron Gas, P Hohenberg and W Kohn, Phys. Rev. 136, B864 (1964) Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, G Kresse and J Furthmüller, Phys. Rev. B 54, (1996) Density-functional exchange-energy approximation with correct asymptotic behavior, AD Becke, Phys. Rev. A 38, 3098 (1988) Special points for Brillouin-zone integrations, HJ Monkhorst, JD Pack, Phys. Rev. B 13, 5188 (1976) From ultrasoft pseudopotentials to the projector augmented-wave method, G Kresse and D Joubert, Phys. Rev. B 59, 1758 (1999) 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

PA4311 Quantum Theory of Solids 3210 revision (see Rae) Expectation values - Variational principle - thenIf

PA4311 Quantum Theory of Solids Question 1.1

PA4311 Quantum Theory of Solids Full variation and functionals

PA4311 Quantum Theory of Solids The N-electron wavefunction Expectation values See Tipler (4 th Ed Sec on ‘The Schrödinger equation for 2 identical particles’)