Lecture 17: Excitations: TDDFT Successes and Failures of approximate functionals Build up to many-body methods Electronic Structure of Condensed Matter,

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:1. Define excitations in a rigorous way2. Analyze the meaning of excitations in independent-particle theories3. 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

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 experimentsExcitation 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 DFTIn principle exact excitations from TD-Kohn-Sham!

Why are excitations harder than the ground state to approximate?Leads us to explicit many-body methods

Electronic ExcitationsFrom 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 Egap = [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) – E0(N)

In General E* < Egap

Powerful ExperimentAngle 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

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 Egap = [E(N+1) – E(N) – ] – [E(N) – E(N-1) – ]

= E(N+1) + E(N-1) – 2 E(N)

Note sign

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 statesEnergy 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

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 tolarge quantitative errors

What about strongly interacting systems?

Present approximations (like LDA and GGAs) lead toqualitative errors in many cases

What can be done?

. . . . . Later . . .

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.

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

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 NE* = 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 interaction

Since electron-hole interaction is attractive E* < Egap

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

E* = 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

Electron excitation at fixed number NExample of optical excitation

Example of CaF2 – complete change of spectra due to electron-hole interaction

Spectra for non-interacting

electrons and holes

Observed spectra changed by electron-holeinteraction

Excitons are the elementary excitations

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:

Dynamic Response Function Recall the static form: (Can be expressed in real

or reciprocal space)

At frequency this is simply:

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

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!

TDDFT - Kohn-Sham approachExact formulation of TDDTT – in principle

TDDFT - Adiabatic Approximation I

The simplest approximation – adiabatic assumption – fxc (t-t’) ~ (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, ….

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

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