OUTLINE

1. Lecture 17: Excitations: TDDFT Successes and Failures of approximate functionals Build up to many-body methodsElectronic Structure of Condensed Matter, Physics 598SCM • Chapter 20, and parts of Ch. 2, 6, and 7 Also App. D and E). Today continues the turning point in the course that started with Lecture 15 on response functions. • Today we consider electronic excitations. The steps are: • Define excitations in a rigorous way • Analyze the meaning of excitations in independent-particle theories • Consider the Kohn-Sham approach • Response functions play an important role and point the way toward theneed for explicit many-body theoretical methods. We will consider themost important response functions – response to electric and magneticfileds, in particular optical response

2. Lecture 17: Excitations: TDDFT Successes and Failures of approximate functionals Build up to many-body methodsElectronic Structure of Condensed Matter, Physics 598SCM OUTLINE • Electronic Excitations • Electron addition/removal - ARPES experiments • Excitation with fixed number – Optical experiments • Non-interacting particles • Electron addition/removal - empty (filled) bands • Excitation with fixed number – spectra are just combination of addition/removal • Response functions • For fixed number – generalization of static response functions • The relation of  and 0 • Example of optical responseTDDFT -- Time Dependent DFT In principle exact excitations from TD-Kohn-Sham! • Why are excitations harder than the ground state to approximate? • Leads us to explicit many-body methods

3. Electronic Excitations From Chapter 2, sections 2.10 and 2.11 • Consider a system of N electrons - Rigorous definitions1. Electron addition/removal • N → N+1 or N → N-1 • DE = E(N+1) – E(N) – m or DE = E(N) – E(N-1) – m • Minimum gapEgap =[E(N+1) – E(N)– m ] – [E(N) – E(N-1) – m]=E(N+1) + E(N-1)– 2 E(N) Note sign • 2. Electron excitation at fixed number N • DE* = E*(N) – E0(N) • In General DE* < Egap

4. Powerful ExperimentAngle Resolved Photoemission (Inverse Photoemission)Reveals Electronic Removal (Addition) Spectra Comparison of theory (lines) and experiment (points) Germanium Silver A metal in “LDA” calculations! Improved many-Body Calculations Figs. 2.22, 2.23, 2.25

5. Electronic Addition/Removal – Independent-Particle Theories • Electron addition/removal • N → N+1 or N → N-1 • DE = E(N+1) – E(N) – m or DE = E(N) – E(N-1) – m • Minimum gapEgap =[E(N+1) – E(N)– m ] – [E(N) – E(N-1) – m]=E(N+1) + E(N-1)– 2 E(N) Note sign From the basic definitions (Section 3.5) the ground state at T=0 is constructed by filling the lowest N states up to the Fermi energy mEmpty states with e > m are the possible states for adding electrons Filled states with e < m are the possible states for removing In a crystal (Chapter 4) these are the conduction bands and valence bands ei(k) that obey the Bloch theorem

6. Electronic Addition/Removal – Kohn-Sham Approach • The Kohn-Sham theory replaces the original interacting-electron problem with a system of non-interacting “electrons” that move in an effective potential that depends upon all the electrons • Ground State – fill lowest N statesEnergy is NOT the sum of eigenvalues since there are corrections in the Hartree and Exc termsKohn-Sham eigenvalues - approximation for electron addition/removal energies -- the eigenvalues are simply the energies to add or remove non-interacting electrons - assuming that the potential remains constant Empty states with e > m are the possible states for adding electrons Filled states with e < m are the possible states for removing In a crystal (Chapter 4) these are the conduction bands and valence bands ei(k) that obey the Bloch theorem

7. Electronic Addition/Removal – Kohn-Sham Approach • When is it reasonable to approximate the addition/removal energies by the Kohn-Sham eigenvalues? • Reasonable – but NOT exact – for weakly interacting systemsPresent approximations (like LDA and GGAs) lead to largequantitative errors • What about strongly interacting systems?Present approximations (like LDA and GGAs) lead toqualitative errors in many casesWhat can be done? • . . . . . Later . . .

8. Powerful ExperimentAngle Resolved Photoemission (Inverse Photoemission)Reveals Electronic Removal (Addition) Spectra Recent ARPES experiment on the superconductor MgB2 Intensity plots show bands very close to those calculated Fig. 2.30 Domasicelli, et al.

9. Comparison of experiment and two different approximations for Exc in the Kohn-Sham approach The lowest gap in the set of covalent semiconductors LDA (also GGAs) give gaps that are too small – the “band gap problem” EXX (exact exchange) gives much better gaps Fig. 2.26

10. Electron excitation at fixed number N • Electron excitation at fixed number N • DE* = E*(N) – E0(N) • Can be considered as:1. Removing an electron leaving a hole (N-1 particles)2. Adding an electron (Total of N particles) • 3. Since both are present, there is an electron-hole interactionSince electron-hole interaction is attractiveDE* < Egap In independent particle approaches, there is no electron-hole interaction ---- thus excitations are simply combination of addition and removal

11. Electron excitation at fixed number NExample of optical excitation • Optical excitation is the spectra for absorption of photons with the energy going into electron excitations • DE* = E*(N) – E0(N) • Example of GaAs • Experiment shows electron-hole interaction directly – spectra is NOT the simple combination of non-interacting electrons and holes

12. Electron excitation at fixed number NExample of optical excitation • Example of CaF2 – complete change of spectra due to electron-hole interaction • Observed spectra changed by electron-holeinteraction • Excitons are the elementary excitations • Spectra for non-interacting electrons and holes

13. Dynamic Response Function 0 Recall the static response function for independent particles, for example the density response function: The dynamic response function for independent particlesat frequency w is: Matrix elements or Joint density of states multiplied by matrix elements

14. Dynamic Response Function  (Can be expressed in real or reciprocal space) Recall the static form: At frequency w this is simply:

15. Dynamic Response Function  What is the meaning of frequency dependence?Written as a function of time: Density response at different times Coulomb interaction – instantaneous in non-relativistic theory

16. Time Dependent DFT -- TDDFT Exact formulation of TDDTT – Gross and coworkers • Extends the Hohenberg-Kohn Theorems (section 6.4) • Exact theorems that time evolution of system is fully determined by the initial state (wave function) and the time dependent density! • But no hint of how to accomplish this! • Time Dependent Kohn-Sham (section 7.6) • Replace interacting-electron problem with a soluble non-interacting particle problem in a time dependent potential • Time evolution of the density of Kohn-Sham system is the same as the density of the interacting system!

17. TDDFT - Kohn-Sham approach Exact formulation of TDDTT – in principle

18. TDDFT - Adiabatic Approximation I The simplest approximation – adiabatic assumption – fxc (t-t’) ~ d(t-t’) That is Vxc (t) is assumed to be a function of the density at the same time When is this useful? Low frequencies, localized systems, … Now widely used in molecules, clusters, ….

19. TDDFT - Adiabatic Approximation II Calculations on semiconductor clusters – from small to hundreds of atoms Eigenvvalues Real-time method developed by us and others – can treat non-linear effects, etc. … . Not shown here -- See text

20. TDDFT - Beyond the Adiabatic Approximation Crucial in extended systems – much current researchLeads methods that explicitly treat inteacting particles! Lectures by Lucia Reining – Nov. 8, 10