EXC!TING 1

Excited States:Manybody Perturbation Theory

& Time-dependent DFT

CAD, Montanuniversität Leoben & Universität Graz

Contents

Open Problems

ConceptsManybody perturbation theory (MBPT): GW & BSETime-dependent density functional theory (TDDFT)

Interpretation of Kohn-Sham statesFunctionals

ImplementationTechnicalities

Band gap problemExcited states

LAPW - specific problems & advantages

Results

Local Density Approximation (LDA)Generalized Gradient Approximation (GGA)

Ground state:

Excited state:Interpretation within one-particle pictureInterpretation of excited states in terms of ground state properties

Excited States Based on DFT?

Sources of discrepancies

Response function:Random phase approximation ignores electron-hole interactionManybody treatment needed

EXC!TING 2

DFT Basics

Hohenberg-Kohn theorem:

Kohn-Sham equation:

Interpretation of KS States

Hartree-Fock:

ionization energies

DFT:

Lagrange parameters

auxiliary functions

Janak's theorem

Koopman's theorem

manybody perturbation theory: GW approachshift of conduction bands: scissors operator

Electro-affinityIonization energy

Band gap

The Band Gap Problem

Even the exact KS solutions don't have to provide good band gaps!

EXC!TING 3

The GW Approach

Quasiparticle band structure:

Shift of conduction bands nearly independent of k.M. S. Hybertson and S. G. Louie, Phys. Rev. Lett. 55, 1418 (1985).M. S. Hybertsen and S. G. Louie, Phys. Rev. B 34, 5390 (1986).

Light Scattering

ES

hω

intraband transitioninterband transition

Ener

gy

wave vector

EFhω

Band Structure

kc

kv

RPA

The Selfenergy Correction

0 1 2 3 4 5 6-20-10

01020304050607080

Reε

Imε

Γ=0.05eV

Si

Die

lect

ric fu

nctio

n

Energy [eV]

Spectra: shift Imεnot band energies!Exact within RPA & GWMatrix elements rescaledSumrule violated

R. del Sole and R. Girlanda, Phys. Rev. B 48, 11789 (1993).

EXC!TING 4

BSE

Two-particle wave function:

Effective two-particle Schrödinger equation:

KS states from GS calculation

BSE

BSE: Implementation

EXC!TING 5

Implementation: the LAPW Method

Atomic spheres: atomic-like basis functions

Interstitial: planewave basis

Beyond RPA: Examples

Peter Puschnig, PhD Thesis, 2002.

Si

1 2 3 4 5 6

10

20

30

40

50

60

10 12 14 16 18 200

2

4

6

8

10

12 RPA BSE Experiment

Im ε

(ω)

ω [eV]

ω [eV]

LiF

BSE RPA

Solving BSE: LiF

EXC!TING 6

M. Rohlfing and S. G. Louie,Phys. Rev. Lett. 81, 2313 (1998).

FLi

hole

LAPW (all-electron) results:P. Puschnig and C. Ambrosch-Draxl,Phys. Rev. B 66, 165105 (2002).

Solving BSE: LiF

Pseudopotential results:

High precision neededmany k-points in small region of BZ

Huge BSE matricese.g. for GaN: 55000 x 55000

Inorganic semiconductors

Computational Effort

a few tens meVSmall binding energies

GaN

-10 -8 -6 -4 -2 0 2 4ΔV [%]

0

0.01

0.02

0.03

0.04

0.05

bind

ing

ener

gy [

eV]

-10 -8 -6 -4 -2 0 2 4ΔV [%]

-0.02

-0.01

0

0.01

0.02

bind

ing

ener

gy [

eV]

E⊥c E||c

EB= 38 meVEB= 10 meV

hexagonal, wurziteEg = 3.4 eV

R. Laskowski, N. E. Christensen, G. Santi, and CAD , Phys. Rev. B 72, 035204 (2005).

EXC!TING 7

Higher exciton binding energiesvalues up to 1 eV

Huge BSE matrices due to big unit cells order of magnitude: 100 atoms

Organic semiconductors

Localized states in real space

Computational Effort

S1

T1

hole

P. Puschnig, PhD Thesis, University Graz, 2002.

0 1 2 3 4 50

50

100

150

Im ε

z(ω)

ω [eV]

T1 S1RPA

Solving BSE: 1D Polyacetylene

P21/a

0.5 1.0 1.5 2.0 2.5 3.00

50

100

150

200

Im ε

(ω)

ω [eV]

S4T4

P. Puschnig and C. Ambrosch-Draxl, Phys. Rev. Lett. 89, 056405 (2002).

Solving BSE: 3D Polyacetylene

EXC!TING 8

Exciton Wavefunction: (CH)x

hole

MBPT versus TDDFT

Mixing of conceptsGW & BSE determined by 4 point functionsComputationally very demanding

MBPT: TDDFT:

Keep the spirit of DFTTDDFT involves 2 point functions onlyComputationally less costlyTDDFT more generally applicable than GW / BSE: applications in the linear-response regime & beyond (e.g. strong laser fields)

G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys. 74, 601 (2002).

(TD)-DFT

Fundamentals of DFT:

The density determines the potential up to a constant.The potential determines the Hamiltonian.The Hamiltonian determines the wavefunction.Every observable is a functional of the density.

Replace the system of interacting electrons by a fictitious system of non-interacting electrons with the same density

The electron density is the fundamental quantity.

EXC!TING 9

TDDFT Basics

Runge-Gross theorem:

Consider N electrons in a time-dependent external potential.Densities ρ and ρ' evolving from a common initial state under the influence of two potentials V and V' (both Taylor expandable about the initial time t0) are always different provided that the potentials differ by more than a purely time-dependent function:

E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984).

Thus there is a one-to-one mapping between densities and potentials.

TDDFT Basics

TD Kohn-Sham equation:

(TD)-DFT

DFT:

Hohenberg-Kohn theoremKohn-Sham systemKohn-Sham equationDensityApproximate xc-potential

TDDFT:

Runge-Gross theoremTD-Kohn-Sham systemTD Kohn-Sham equationResponse functionApproximate xc-kernel

Use same level of approximation!

EXC!TING 10

TDDFT in Linear-Response Regime

First-order density response:

Response function:

TDDFT in Linear-Response Regime

Kohn-Sham response function:

TDDFT: xc Kernels

TDDFT: DFT:LDAEXX…..

TD-LDATD-EXX…..

Universal!Contains all manybody effectsReplaces GW & BSE

EXC!TING 11

Thank you for your attention!