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) • - 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. A hierarchy of methods • Hartree • ‘Independent’ particle approximation • Hartree-Fock • Exact inclusion of the exchange interaction • Configuration Interaction • Post Hartree-Fock methods attempt to include exchange and correlation • The exponential wall • Do we really need to know the full wavefunction?

4. Last time… Hartree approximation • ‘Independent’ electron picture – (electrons are distinguishable) • Electrons interact via mean-field Coulomb potential - (respond to avg. charge density) Hartree Equations Single particle orbitals Must solve -single electron Schrödinger equations self-consistently Total energy, , is the sum of single particle energies Problems - PEP not enforced, electrons do not respond to specific configuration of other electrons

5. Hartree-Fock • Electrons are indistinguishable and obey the Pauli exclusion principle • Exact inclusion of the exchangeinteraction • The N-electron wavefunction has the form of a Slater determinant

6. Hartree-Fock e.g. 2-electrons Wavefunction is antisymmetric - Pauli Exclusion Principle: if then

7. Question 2.3 Show that the 2-electron Slater determinant, is correctly normalised assuming the single particle orbitals and are orthonormal.

8. Electron spin Single particle orbital contains information on the electron spin, Wavefunctions with different spins are orthogonal Can write this formally in many different ways, e.g. notation of S Raimes, Many Electron Theory, North-Holland publishing company, 1972 spin coordinate space coordinates spin function, or where , wavefunction then ifand 0 if

9. Hartree-Fock Assume N-electron wavefunction has the form of a Slater determinant , then minimise subject to the constraint that each is normalised Hartree-Fock Equations exchange term (integral operator) direct term

10. Question 2.4 Starting from the 2-electron wavefunction, derive the Hartree-Fock equation, for two electrons by minimising subject to the constraint that each is normalised.

11. Hartree-Fock • Electrons are indistinguishable, and can lower their energy by exchanging (if spins are the same) • PEP is automatically enforced • But - still an approximation to the full wavefunction(electrons still don’t respond properly to theparticular configuration of the other N-1 electrons) • Calculations are much more difficult than in the Hartree approximation because of the integral operator • if can maybe use a Hartree-like approach. If might need to use post HF methods… • A number of variants of HF are popular in quantum chemistry • Hartree and HF methods are particularly poor for excited states

12. Post Hartree-Fock • HF Slater determinant is still an approximation to the full electron wavefunction because we still have an arbitrary constraint • If we solve S.E. exactly we will get a lower energy – the difference between and is often called the correlation energy • Physically, correlation describes the way that electrons tend to avoid each other Uniform electron gas is the pair distribution function Classically, average density Hartree Hartree-Fock – exchange hole Exact – ‘exchange and correlation’ hole radial distance

13. Configuration interaction Remove all constraints from the wavefunction, Expand as a sum over Slater determinants, each with a different configuration of single particle orbitals (full CI) • Expansion often split configurations by number of ‘excitations’ in each configuration Single excitations • electron ’excited’ from th occupied orbital to th unoccupied orbital double excitations Ground state determinant triple excitations

14. Configuration interaction • As usual, find the unknown coefficients by minimising the energy subject to the constraint that is normalised. • Then, as before, • Calculation of matrix elements, , is more complicated – but procedure is familiar.

15. Configuration interaction • Number of terms in expansion grows very rapidly • ‘Black art’ to truncate expansion for real systems - see e.g. J. Phys.: Conf. Ser. 242 (2010) • Must set maximum number of unoccupied orbitals • If must truncate number of excited state configurations (eg. CISD, CISDT etc.) • To be sure of the validity of an expansion must run calculation systematically • – check energy is minimised according to variationalprinciple • 5 electrons 2 electrons - from Phys. Rev. B 85 205432 (2012)

16. Quantum Chemistry: acronym zoo Method Exact solution Full CI Coupled cluster, CCSD, CCSDT, CISD, CISDT, Moller-Plesset, , CAS SCF, MC SCF etc. HF Limited basis Complete set of states Single particle basis STO-2G, STO-3G, STO-6G, STO-3G*, 3-21G, 3-21++G, 3-21G*, 3-21GSP, 4-31G, 4-22GSP, 6-31G, 6-31G-Blaudeau, 6-31++G, 6-31G*, 6-31G**, 6-31G*-Blaudeau, 6-31+G*, 6-31++G**, 6-31G(3df,3pd), 6-311G, 6-311G*, 6-311G**, 6-311+G*, 6-311++G**, 6-311++G(2d,2p), 6-311G(2df,2pd), 6-311++G(3df,3pd), MINI (Huzinaga), etc. etc. Full CI wavefunction tells us everything– but do we need it?

17. The exponential wall • Kohn (1999) “In general the -electron wavefunction is not a legitimate scientific concept when ” • it contains too much information… Usually we only need to know a few things – e.g. energy, polarizability, band structure, density etc. Imagine: store wavefunction on a grid, 10 points per dimension… atoms in the universe DFT - find all the useful properties of a system without solving for