IPAM Excitations November 05helper.ipam.ucla.edu › publications › maws3 ›...
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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!