Paul R. C. KentMurilo L. Tiago

Fernando A. Reboredo Oak Ridge National Laboratory

Randolph Q. HoodLawrence Livermore National Laboratory

Excitations in Carbon Fullerenes calculated by GW Bethe-Salpeter and Quantum Monte Carlo

Kent / APS March Meeting New Orleans / March 2008

Acknowledgements

• Support− DOE

• Codes− PARSEC & RGWBS DFT, W-BSE, TDLDA− CASINO QMC

• Computational support− NCCS at ORNL− NERSC− LLNL− TACC

Kent / APS March Meeting New Orleans / March 2008

Questions that I will address

1. Do the GW-BSE and QMC correctly predict neutral and charged excitations of carbon fullerenes?− C60 is best characterized experimentally− We study 7 fullerenes C20-C80, including isomers

2. Where are improvements in computational methods required?

Kent / APS March Meeting New Orleans / March 2008

Methodology: QMC• Diffusion Monte Carlo is in principle exact, but fixed node

approximation introduces a variational error. First excitation energies of each symmetry are also exact, but non-variational in practice

• Trial wavefunction is single determinant of LDA orbitals− Costly &/or difficult to apply multi-determinants/orbital

optimization/backflow approaches in large systems

• More challenging calculation than for e.g. cohesive energy

LUMO

HOMO

Triplet energy = Etr ! Egs

Kent / APS March Meeting New Orleans / March 2008

• Full absorption spectrum, excitons. Bethe-Salpeter equation for e-h interactions

• We have applied two different levels of approximation

• GW0 and GWf often predict similar gaps, but differ in absolute energy levels compared to vacuum

GWf approximation

GW0 approximation

Methodology: GW-BSE

algorithm: Hybertsen & Louie (1985)

Tiago & Chelikowsky, PRB 73, 205334 (2006)

Del Sole et al., PRB 49, 8024 (1994)

Σ = i GWΓLDA

W = Vcoul + VcoulΠLDAVcoul

Σ = i GW

W = Vcoul + VcoulΠRPAVcoul

Kent / APS March Meeting New Orleans / March 2008

Fullerene geometries• DFT PBE geometries obtained from real-space and

plane-wave ground state calculations•

Kent / APS March Meeting New Orleans / March 2008

Focus on C50, C60, C70 for brevity

Kent / APS March Meeting New Orleans / March 2008

Results: Ionization Potentials• Good agreement for all methods

6

6.5

7

7.5

8

8.5

9

9.5

10

50 60 70

Ioni

zatio

n po

tent

ial (

eV)

Size (atoms)

ExperimentDelta SCF

GWfDMC

Kent / APS March Meeting New Orleans / March 2008

Results: Electron affinities• Systematic overestimation by GW, trends in DMC

unclear

2

2.5

3

3.5

4

4.5

5

50 60 70

Elec

tron

affin

ity (e

V)

Size (atoms)

ExperimentDelta SCF

GWfDMC

Kent / APS March Meeting New Orleans / March 2008

Results: First spin-triplet

• Stoke’s shifts estimated as max 0.2 eV from DFT not included in above data

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

50 60 70 80 90 100

Firs

t spi

n tri

plet

ene

rgy

(eV)

Size (atoms)

ExperimentTDLDA

GWBSEDMC

Kent / APS March Meeting New Orleans / March 2008

Results: First spin-triplet

• Stoke’s shifts estimated as max 0.2 eV from DFT not included in above data

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

50 60 70 80 90 100

Firs

t spi

n tri

plet

ene

rgy

(eV)

Size (atoms)

ExperimentTDLDA

GWBSEDMC

GW-BSE systematically low

Kent / APS March Meeting New Orleans / March 2008

Results: First spin-triplet

• Stoke’s shifts estimated as max 0.2 eV from DFT not included in above data

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

50 60 70 80 90 100

Firs

t spi

n tri

plet

ene

rgy

(eV)

Size (atoms)

ExperimentTDLDA

GWBSEDMC

DMC systematically high

Kent / APS March Meeting New Orleans / March 2008

Challenges to theory

• GW-BSE− Need to improve electron affinities without worsening other

quantities− Currently investigating self-consistent approaches

• QMC− Need improved trial wavefunctions with more optimal nodes− Non-systematic cancellation of nodal error is the primary error− Need compact multiconfigurational expansions &/or orbital

optimization for large systems− Pseudopotential evaluation related errors are small

Kent / APS March Meeting New Orleans / March 2008

Summary

• IP: well reproduced by all methods• EA: systematically overestimated by GW• Triplet: GW-BSE systematically low ~0.5eV (excl. Stokes)• Triplet: QMC systematically high ~0.8eV (excl. Stokes)•Delta SCF and TDLDA are surprisingly good, despite

being poor choices in nanotubes.

http://arxiv.org/abs/0803.0560

Kent / APS March Meeting New Orleans / March 2008

Kent / APS March Meeting New Orleans / March 2008

Results: Ionization Potentials

5

6

7

8

9

10

0 10 20 30 40 50 60 70 80 90 100 110 120

Ioni

zatio

n po

tent

ial (

eV)

Size (atoms)

Exp.dSCF

GWfDMC

Kent / APS March Meeting New Orleans / March 2008

Results: Electron affinities

0

1

2

3

4

5

6

0 10 20 30 40 50 60 70 80 90 100 110 120

Elec

tron

affin

ity (e

V)

Size (atoms)

Exp.dSCF

GWfDMC

Kent / APS March Meeting New Orleans / March 2008

Results: First spin-triplet

• Stoke’s shifts estimated as max 0.2 eV from DFT not included in above data

-1

-0.5

0

0.5

1

1.5

2

2.5

3

0 10 20 30 40 50 60 70 80 90 100 110 120

Firs

t spi

n tri

plet

ene

rgy

(eV)

Size (atoms)

Exp.TDLDA

GWBSEDMC