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 Last time…

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

6. 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

7. PA4311 Quantum Theory of Solids Exchange correlation potential Exchange and correlation potential

8. PA4311 Quantum Theory of Solids Local density approximation

9. PA4311 Quantum Theory of Solids Local density approximation LDA works surprisingly well in a wide range of materials Atomic and molecular ground state energies within ~ 1-5% Molecular equilibrium distances and geometries within ~ 3% Fermi surfaces of bulk metals within ~ few % Lattice constants of solids within ~ 2% Vibrational frequencies and phonon energies within ~ 2%

10. PA4311 Quantum Theory of Solids Self consistent? No Yes Pre-condition or ‘seed’ SCF loop Self consistent field (SCF) Set maximum allowed number of steps

11. PA4311 Quantum Theory of Solids # Skeleton abinit input file (example for H 2 ) #Definition of the atoms ntypat 1 # There is only one type of atom znucl 1 # atomic number natom 2 # There are two atoms typat 1 1 # They both are of type 1, that is, Hydrogen xcart # cartesian co-ordinates for each atom -0.7 0.0 0.0 # Triplet giving the cartesian coordinates of atom 1, in Bohr 0.7 0.0 0.0 # Triplet giving the cartesian coordinates of atom 2, in Bohr #Definition of the SCF procedure nstep 10 # Maximal number of SCF cycles toldfe 1.0d-6 # Will stop when, twice in a row, the difference between two # consecutive total energies differ by less than toldfe (in Hartree) diemac 2.0 # precondition the SCF cycle (see documentation) iscf 7# default - use Pulay mixing of the density (update info from 7 steps)

12. PA4311 Quantum Theory of Solids Beyond the LDA – acronym zoo (ii) LDA GGA Meta-GGA Hyper-GGA hybrids Hartree Exact exchange Unoccupied orbitals Hundreds of GGA functionals: PBE, BLYP, PW91 etc. etc. PKZB, TPSS, VSXC etc. RPA exact CA Ullrich, Fig. 2.7

13. PA4311 Quantum Theory of Solids Kohn-Sham eigenvalues

14. PA4311 Quantum Theory of Solids DFT in crystals A crystal is a periodic structure Specified by types and positions of atoms in one repeat rules that describe the repetitions crystal = Bravais lattice + basis

15. PA4311 Quantum Theory of Solids a1a1 a2a2 graphene unit cell 2D crystal

16. PA4311 Quantum Theory of Solids wikipedia.org 3D crystal: zinc blende structure (diamond, Si, GaAs etc) www.seas.upenn.edu