Quantum Theory of Solids

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) • - Framework (Hohenberg-Kohn, Kohn-Sham) • - 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 • www.abinit.org

3. Last time… • Determinantal form for N-electron wavefunction(see Rae, Chapter 10) • electrons are indistinguishable and obey the PEP • Single determinant – Hartree-Fock method (exact inclusion of exchange) • If expand as a sum over determinants – • Full Configuration Interaction method describes both exchange and correlation • Full CI is, in principle, exact • - can converge systematically with number of single particle orbitals in the basis • But size of problem increases exponentially with N - limited to N~10

4. Kohn (1999) ‘In general the -electron wavefunction is not a legitimate scientific concept when • Nobel prize in Chemistry (1998) • e.g. storage required ~ bytes >> number of atoms in universe when Nobel Lecture: Electronic Structure of Matter: Wave Functions and Density Functionals. Nobelprize.org. Nobel Media AB, accessed 23 Oct 2014. www.nobelprize.org/nobel_prizes/chemistry/laureates/1998/kohn-lecture.html • Density functional theory • find all the useful properties of a system from without solving for • Electronic Structure, RM Martin (ch.s 4,6,7,12) • Time-Dependent Density Functional Theory, CA Ullrich (ch. 2) • Various review articles – e.g. U von Barth, Physica Scripta. T109, 9–39 (2004).

5. Hohenberg-Kohn HK 1. For any system of interacting particles in an external potential , the potential is uniquely determined by the ground state density - i.e. tells us everything about the system! 2. The energy can be defined as a functional of the density. The density that minimises is the exact ground state density,

6. 1. For any system of interacting particles in an external potential , the potential is uniquely determined by the ground state density Proof proceeds via 2 steps The potential uniquely determines the wavefunction No two different ground state wavefunctions can lead to the same ground state density • RM Martin: ‘…crucial for any practitioner in the field to understand’

7. 2. The energy can be defined as a functional of the density. The density that minimises is the exact ground state density, where is the universal Hohenberg-Kohn functional Find the exact ground state density by minimising But… the universal functional, is unknown!

8. Kohn-Sham Map the difficult many particle problem onto an auxiliary system that is easy (…easier…) to solve Assume the ground state density of the original interacting system is equal to that of some chosen non-interacting system Net result: independent particle equations with all the difficult many-body terms folded into the ‘exchange and correlation’ potential HK HK KS

9. Kohn-Sham Want to find by minimising Invent auxiliary non-interacting system with same total energy and density, where is a single determinant and Exchange and correlation energy makes up the difference Non-interacting system KE External potential Hartree interaction Trick is that is small

10. Total ground state energy, (Hartrees.), calculated within the LDA – CA Ullrich, Table 2.1

11. Kohn-Sham where Single particle equations are , where, Exchange and correlation potential