XIII International Conference on the
applications of DFT in Chemistry and Physics
Lyon 2nd September 2009
GW renormalization of DFT molecular
electronic levels at the vicinity of a surface:
The image charge effect
Juan María García Lastra
Kristian Sommer Thygesen
Ángel Rubio
Outline
1. Introduction
2. Motivation
3. Our work
4. A simple model to explain the results
5. Outlook
Image charge
1.Introduction
Metal
q2
Vimg ( z ) 
4( z  z0 )
z
q
-q
Semiconductor
C60 on Ag(111)
q2
(  1)
Vimg ( z ) 
 r
4( z  z0 ) ( r  1)
Is it possible to
reproduce this effect
within DFT?
R. Hesper, L.H. Tjeng and G.A.
Sawatzky, Europhys. Lett. 40, 177 (1997)
Some definitions and equivalences in DFT
1.Introduction
Ionization Potential (IP)
Electron affinity (EA)
X  IP  X   e 
X   EA  X  e
D  IP  EA
Gap (D)
DFT
Vacuum
LUMO
 LUMO
Exact Vxc
IP   HOMO
EA   LUMO
D   LUMO   HOMO  C
HOMO
 HOMO
C is the derivative discontinuity
J.P. Perdew and M. Levy Phys. Rev. Lett. 51, 1884 (1983)
DSCF
1.Introduction
Alternative : DSCF
LUMO
IP
HOMO
EA
D=IP-EA
+
-2
Problem: EXTENDED SYSTEMS
1.Introduction
Many Body Perturbation Theory
The combination of a particle and its influence on the local environment
Propagators
S
Self-energy
R.D. Mattuck, A guide to Feynman Diagrams in the Many-Body Problem
1.Introduction
GW approximation
  , 
DFT
i
Good enough
DFT
i
L. Hedin, Phys. Rev. 139,
A796 (1965)
G0  i 
Initial guess
B. I. Lundqvist, Phys.
Kondens. Mater. 6, 193
(1967)
P  i 
EGS G
  i 
G G 0 , S 
G  G 0  G 0 SG
S  i 
W  i 
  ,  
QP
i
QP
i
F. Aryasetiawan and O.
Gunnarsson, Rep. Prog.
Phys. 61, 237 (1998)
G. Onida, L. Reining and A.
Rubio, Rev.Mod.Phys. 74,
601 (2002)
DFT vs. GW
1. Introduction
DFT + local xc-functionals underestimate
HOMO-LUMO gaps
Hartree-Fock is good for small molecules
(SI-free), but overestimates the gap for
extended systems
GW includes screening in the exchange
and this solves the gap problem.
Schilfgaarde, Kotani, and Faleev, PRL 96, 226402 (2006)
Hartree-Fock exchange
Screening correction
2.Motivation
Theoretical interest
2.Motivation
D. G. de Oteyza, J.M. García-Lastra et al., Adv. Func. Mater., accepted
STM
2.Motivation
Molecules and layers on surfaces
DIP and F16CuPc on
Cu(111)
D. G. de Oteyza, J.M. García-Lastra et al., Adv. Func. Mater., in press
Aromatic molecules
on Cu(110)
N. Atodiresei, V. Caciuc et al., PRL 102, 136809 (2009)
Conductance at molecular junctions
2.Motivation
Amine-Gold Linked Single-Molecule Circuits
SY Quek et al., Nano Lett 7, 3477 (2007)
Image Charge by dielectrics
S
D
SiO2
K. Kaasbjerg and K.
Flensberget, Nano Lett 8,
3809 (2008)
2.Motivation
Conductance at molecular junctions
SY Quek et al., Nano Lett 7, 3477 (2007)
3.Our work First-principles GW calculations: Physisorbed benzene
9 Å >Z>4 Å
DFT calculations performed
with PWSCF code (#)
G0W0 calculations performed
with the Yambo code(*).
Yambo:
G0W0  LDA, Plane wave
basis, norm-conserving
pseusopotentials, plasmon
pole approximation.
(#) S. Baroni et al. (2009), QUANTUM
ESPRESSO package, www.quantumespresso.org/
(*) A. Marini, C. Hogan, M. Grüning, D.
Varsano, Comp. Phys. Comm. 180, 1392
(2009).
See also: J. B. Neaton et al. Phys. Rev. Lett. 97, 216405 (2006)
Benzene Molecule
3.Our work
•Previously obtained by Neaton et
al.
•LDA underestimates the gap by a
factor of 2 (mainly due to Selfinteraction)
5.2 eV
10.5 eV
•GW HOMO-LUMO gap agrees
with experiment (IP-EA)
•LUMO predicted to be above the
vacuum level in GW, in agreement
with experiment
Experiment:
IP = 9.25 eV
EA = -1.15 eV
L. Klasinc et al., Pure Appl. Chem. 55, 289 (1983)
B.T. Hill, J. Chem. Soc. Perkin Trans. II 1027 (1998)
D = 10.4 eV
Substrates
3.Our work
Insulator and semiconductor
NaCl(001)
MgO(001)
8.9 eV
7.7 eV
•Same structure (fcc)
CaO(001)
BaO(001)
6.3 eV
4.0 eV
BaO(111)
•Varying the gap
•Varying the surface
Metallic surface!
Substrates
3.Our work
Metals
Pt(111)
Rh(111)
Ti(001)
Al(111)
Li(001)
sd
sd
sd
sp
s
•Different DOS at Fermi Level
•Similar interatomic distances
•Except Li: Electrons outer of the surface
Substrates
3.Our work
Semimetallic
•Benzene on Graphite(0001)
•Previously studied by
Neaton, Hybertsen and Louie,
PRL 97, 216405 (2006)
•Neaton et al. z = 3.25 Å
•Our work 4 Å < z < 9 Å
3.Our work
GW and LDA benzene HOMO-LUMO gaps
4.5 Å
J.M.G-L, A. R. and K.S.T.,
submitted
LDA gaps are independent of substrate and distance
Same result with other functionals (GGA, hybrid or exact exchange)
GW gaps show large variation across different surfaces
GW gap sensitive to atomistic details, e.g. surface plane (BaO)
3.Our work
Classical image charge model
Electrostatic energy of point
charge above a polarizable
medium:
q2
(  1)
Vimg ( z ) 
 r
4( z  z0 ) ( r  1)
Classical model describes the
physics of the gap reduction
qualitatively.
Best-fit values for  and z0:
Fitted for the gap: Different
values if HOMO or LUMO
are fitted independently
Dynamic interaction between
benzene orbitals and
surfaces: Bulk Dielectric
Constant is not a good
descriptor
3.Our work
Variation of HOMO and LUMO levels
Vacuum
Vacuum
GW: Symmetric effect on HOMO and LUMO. Exceptions Li and BaO(111)
LDA: HOMO level agrees better with GW than does LUMO
Very good agreement between LDA and GW for HOMO at metallic surfaces
(error cancellation in LDA between self-interaction and image charge)
3.Our work
General trends in level shifts
Insulator and semiconductor
Gap reduction increases with decreasing substrate band gap
General trends in level shifts
3. Our work
Metals
Gap reduction increases with increasing substrate DOS at EF
Li and BaO(111) deviate from general trend!
4. A simple model to explain the results GW S to second order in V
Renormalization of single electronic level,
interactions with substrate electrons:
Hartree-Fock exchange
 , by non-local
Screening correction
We truncate the expansion in the second order term
4. A simple model to explain the results
Semiconductors
L
Effective interaction strength
L
L
L
Substrate joint density of
states weighted by
particle-hole transitions
Metals
A simple model to explain the results
L
L proportional to JDOS
Slope of JDOS at
=0 proportional to
DOS at EF
The correction
increases if DOS at
EF increases
5.Outlook
•DFT is not able to reproduce image charge effect
•GW includes dynamic correlation (polarization) and solves the
problem
•Classic image potential describes the effect phenomenologically
•However microscopic description is required
•Renormalization of the gap in molecules follows the band gap in
semiconductors
•Renormalization of the gap in molecules follows the DOS at Fermi
level in metals
•It is possible to understand the results truncating at second order
the self energy.
A simple model to explain the results
Metals