1. Lecture 17: Excitations: TDDFT Successes and Failures of approximate functionals Build up to many-body methods Electronic 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: 1.Define excitations in a rigorous way 2.Analyze the meaning of excitations in independent-particle theories 3.Consider the Kohn-Sham approach Response functions play an important role and point the way toward the need for explicit many-body theoretical methods. We will consider the most important response functions – response to electric and magnetic fileds, in particular optical response

2. Lecture 17: Excitations: TDDFT Successes and Failures of approximate functionals Build up to many-body methods Electronic 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 response TDDFT -- 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 definitions 1. Electron addition/removal N → N+1 or N → N-1  E = E(N+1) – E(N) –  or  E = E(N) – E(N-1) –  Minimum gap E gap = [E(N+1) – E(N) –  ] – [E(N) – E(N-1) –  ] = E(N+1) + E(N-1) – 2 E(N) Note sign 2. Electron excitation at fixed number N  E* = E*(N) – E 0 (N) In General  E* < E gap

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

5. Electronic Addition/Removal – Independent-Particle Theories 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  Empty states with  are the possible states for adding electrons Filled states with  are the possible states for removing In a crystal (Chapter 4) these are the conduction bands and valence bands  i (k) that obey the Bloch theorem Electron addition/removal N → N+1 or N → N-1  E = E(N+1) – E(N) –  or  E = E(N) – E(N-1) –  Minimum gap E gap = [E(N+1) – E(N) –  ] – [E(N) – E(N-1) –  ] = E(N+1) + E(N-1) – 2 E(N) Note sign

6. Electronic Addition/Removal – Kohn-Sham Approach Empty states with  are the possible states for adding electrons Filled states with  are the possible states for removing In a crystal (Chapter 4) these are the conduction bands and valence bands  i (k) that obey the Bloch theorem 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 states Energy is NOT the sum of eigenvalues since there are corrections in the Hartree and Exc terms Kohn-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

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 systems Present approximations (like LDA and GGAs) lead to large quantitative errors What about strongly interacting systems? Present approximations (like LDA and GGAs) lead to qualitative errors in many cases What can be done?..... Later...

8. Powerful Experiment Angle Resolved Photoemission (Inverse Photoemission) Reveals Electronic Removal (Addition) Spectra Recent ARPES experiment on the superconductor MgB 2 Intensity plots show bands very close to those calculated Fig. 2.30 Domasicelli, et al.

9. Comparison of experiment and two different approximations for E xc 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 In independent particle approaches, there is no electron-hole interaction ---- thus excitations are simply combination of addition and removal Electron excitation at fixed number N  E* = E*(N) – E 0 (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 interaction Since electron-hole interaction is attractive  E* < E gap

11. Electron excitation at fixed number N Example of optical excitation Optical excitation is the spectra for absorption of photons with the energy going into electron excitations  E* = E*(N) – E 0 (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 N Example of optical excitation Example of CaF 2 – complete change of spectra due to electron-hole interaction Spectra for non-interacting electrons and holes Observed spectra changed by electron-hole interaction Excitons are the elementary excitations

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

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

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

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 – f xc (t-t’) ~  (t-t’) That is V xc (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 research Leads methods that explicitly treat inteacting particles! Lectures by Lucia Reining – Nov. 8, 10