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Quantum Theory of Solids

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

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

  2. 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)

  3. 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 • Prerequisites • 3210 Quantum Mechanics • 2230 Condensed Matter Physics • Also – 214 Fourier Series, 372 Fourier Transforms, etc. Plus many other relevant text books and online references – see the library

  4. The problem electron-ion interaction electron KE electron-electron interaction ion KE ion-ion interaction electrons and ions is a function of electron co-ordinates, (and spins), and ion co-ordinates, (and spins) But, so ion KE term is small

  5. Timescales From CA Ullrich, Time-Dependent Density-Function Theory, Oxford University Press (2012)

  6. The problem Born-Oppenheimer approximation - electrons react instantaneously to changes in nuclear positions Or, in atomic units, Constant depends on ion positions But, still have electrons is a function of electron co-ordinates, , (and spins). Need to develop some approximations!

  7. 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…

  8. 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)

  9. 3210 revision (see Rae) Schrödinger equation - Probability density - | Expectation values - Variational principle - If then

  10. Question 1.1 If is normalised and the are orthonormal, show that . If we wish to minimise subject to the constraint that the are normalised, show that the appropriate Lagrange multiplier,

  11. Full variation and functionals is afunctionalof and A functional maps a function onto a value, For example, See RM Martin App. A, or e.g. GC Evans, Functionals and their applications, Dover, New York, 1964

  12. 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

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