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

2. 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) - Framework (Hohenberg-Kohn, Kohn-Sham) - 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 www.abinit.org Course Outline

3. PA4311 Quantum Theory of Solids Last time…

4. PA4311 Quantum Theory of Solids 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 www.nobelprize.org/nobel_prizes/chemistry/laureates/1998/kohn-lecture.html

5. PA4311 Quantum Theory of Solids Hohenberg-Kohn HK

6. PA4311 Quantum Theory of Solids Proof proceeds via 2 steps 1.The potential uniquely determines the wavefunction 2.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. PA4311 Quantum Theory of Solids

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 KS

9. PA4311 Quantum Theory of Solids Kohn-Sham Non-interacting system KE Hartree interaction Exchange and correlation energy makes up the difference External potential

10. PA4311 Quantum Theory of Solids Atom He-2.832.77-6.632.00-0.97 Ne-128.23127.74-309.9965.73-11.71 Ar-525.95524.97-1253.13231.46-29.24 Kr-2750.152747.81-6577.871171.72-91.82 Xe-7228.867225.10-17159.162880.92-175.71

11. PA4311 Quantum Theory of Solids Kohn-Sham Exchange and correlation potential