Excitation gaps of finite-sized systems from OptimallyTuned Range-Separated Hybrid Functionals:
Leeor Kronik
Department of Materials & Interfaces,
Weizmann Institute of Science
5th Benasque TDDFT Workshop, January 2012
The Group
Tami
Zelovich
Ido
Azuri
Eli
Kraisler
Andreas
Baruch Karolewski Ariel
Feldman (visiting) Biller
Anna Ofer Sivan
Hirsch Sinai Abramson
Funding
European Research Council
Israel Science Foundation
Germany-Israel Foundation
US-Israel Binational Science Foundation
Lise Meitner Center for Computational Chemistry
Alternative Energy Research Initiative
The people
Roi Baer
Tamar Stein
(Hebrew U)
Sivan RefaelyAbramson
Natalia
Kuritz
(Weizmann Inst.)
Kronik, Stein, Refaely-Abramson, Baer,
J. Chem. Theo. Comp. (Perspectives Article), to be published
Fundamental and optical gap – the quasi-particle picture
Evac
EA
derivative
discontinuity!
IP
Eg
(a)
Eopt
(b)
See, e.g., Onida, Reining, Rubio, RMP ‘02; Kümmel & Kronik, RMP ‘08
Mind the gap
The Kohn-Sham gap underestimates the real gap
Eg  I  A  
LUMO
KS
Perdew and Levy, PRL 1983;
Sham and Schlüter, PRL 1983

HOMO
KS
  xc
derivative
discontinuity!
Kohn-Sham eigenvalues do not mimic
the quasi-particle picture
even in principle!
H2TPP
TD
TD
TD
Energy [eV]
-1.4
-1.7
-2.5
-2.9
1.8
-1.5
2.0
2.7
2.2
4.8
2.1
4.7
1.9
4.7
2.1
-4.7
-5.2
-6.2
-6.2
-6.4
-IP, -EA Eopt
GGA
B3LYP
OT-BNL
GW-BSE
EXP
Wiggle room: Generalized Kohn-Sham theory
- Map to a partially interacting electron gas that is
represented by a single Slater determinant.
- Seek Slater determinant that minimizes an energy
functional S[{φi}] while yielding the original density
- Type of mapping determines the functional form
Oˆ
S

[{ j }]  V ion ( r )  v R ([ n ]; r )  i ( r )   i i ( r )
Seidl, Goerling, Vogl, Majevski, Levy, Phys. Rev. B 53, 3764 (1996).
Kümmel & Kronik, Rev. Mod. Phys. 80, 3 (2008)
Baer et al., Ann. Rev. Phys. Chem. 61, 85 (2010).
- Derivative discontinuity problem possibly
mitigated by non-local operator!!
Hybrid functionals are a special case of
Generalized Kohn-Sham theory!
 1 2

sl
sl
ˆ



V
(
r
)

V
([
n
];
r
)

a
V

(
1

a
)
v
([
n
];
r
)

v
([
n
];
r
)

 i ( r )   i  i ( r )
ion
H
F
x
c
 2

Does a conventional hybrid
functional solve the gap problem?
H2TPP
TD
TD
TD
Energy [eV]
-1.4
-1.7
-2.5
-2.9
1.8
-1.5
2.0
2.7
2.2
4.8
2.1
4.7
1.9
4.7
2.1
-4.7
-5.2
-6.2
-6.2
-6.4
-IP, -EA Eopt
GGA
B3LYP
OT-BNL
GW-BSE
EXP
Need correct asymptotic potential!
Can’t work without full exact exchange!
But then, what about correlation?
How to have your cake and eat it too?
Range-separated hybrid functionals
Coulomb operator decomposition:
r
1
1
1
 r erfc ( r )  r erf ( r )
Short Range
Long Range
Emphasize long-range exchange,
short-range exchange correlation!
 1 2

lr , 
sr , 
sl
    V ion ( r )  V H ([ n ]; r )  VˆF  v x ([ n ]; r )  v c ([ n ]; r )  i ( r )   i i ( r )
 2

See, e.g.: Leininger et al., Chem. Phys. Lett. 275, 151 (1997)
Iikura et al., J. Chem. Phys. 115, 3540 (2001)
Yanai et al., Chem. Phys. Lett. 393, 51 (2004)
Kümmel & Kronik, Rev. Mod. Phys. 80, 3 (2008).
But how to balance ??
How to choose ?

“Koopmans’ theorem”   HOMO  E gs ( N  1;  )  E gs ( N ;  )
Need both IP(D), EA(A)  choose  to best obey
“Koopmans’ theorem” for both neutral donor and
charged acceptor:
Minimize
J ( ) 

i0,
 ,i
(  HOMO  E gs ( N i  1;  )  E gs ( N i ;  ))
i
i
2
Tune, don’t fit, the range-separation parameter!
Tuning the range-separation parameter




J (  )   H ( N )  IP ( N )   H ( N  1)  IP ( N  1)
Neutral molecule (IP)
J (  opt )  min J (  )
Anion (EA)
H2TPP
TD
TD
TD
Energy [eV]
-1.4
-1.7
-2.5
-2.9
1.8
-1.5
2.0
2.7
2.2
4.8
2.1
4.7
1.9
4.7
2.1
-4.7
-5.2
-6.2
-6.2
-6.4
-IP, -EA Eopt
GGA
B3LYP
OT-BNL
GW-BSE
EXP
Gaps of atoms
Stein, Eisenberg, Kronik, Baer, Phys. Rev. Lett., 105, 266802 (2010).
Fundamental gaps of acenes
10
Gap (eV)
9
Ref
0.31
BNL* orb. gap
0.28
8
7
0.25
6
0.22
5
0.20
0.19
4
0
1
2
3
4
5
6
7
n
Stein, Eisenberg, Kronik, Baer, Phys. Rev. Lett., 105, 266802 (2010).
Fundamental gaps of
hydrogenated Si nanocrystals
14
0.41
12
0.33
Energy (eV)
10
0.24
8
6
-Lumo
GW EA
-HOMO
Series4
Exp IP
4
2
0
0
0.13
0.14
0.12
5
10
15
Diameter (Å)
GW: Tiago & Chelikowsky, Phys. Rev. B 73, 2006
DFT: Stein, Eisenberg, Kronik, Baer,
PRL 105, 266802 (2010).
Ionization Energy
11
10
- HOMO[eV]
9
8
7
GW data: Blasé et al.,
PRB 83, 115103 (2011)
EXP
6
GW
OT-BNL
B3LYP
5
6
6.5
7
7.5
8
8.5
9
9.5
10
10.5
Experimental ionization energy [eV]
S. Refaely-Abramson, R. Baer, and L. Kronik, Phys.Rev. B 84 ,075144 (2011) [Editor’s choice].
Optical gaps with Time-dependent DFT
TDDFT: BNL results as accurate
as those of B3LYP
a – thiophene
b – thiadiazole
c – benzothiadiazole
d – benzothiazole
e – flourene
f – PTCDA
g – C60
h – H2P
i – H2TPP
j – H2Pc
S. Refaely-Abramson, R. Baer, and L. Kronik, Phys.Rev. B 84 ,075144 (2011)
The charge transfer excitation problem
Time-dependent density functional theory (TDDFT), using
either semi-local or standard hybrid functionals, can
seriously underestimate charge transfer excitation energies!
Biphenylene –
tetracyanoethylene:
B3LYP: 0.77 eV
Experiment: 2 eV
Liao et al., J. Comp. Chem. 24, 623 (2003).
zincbacteriochlorin-phenylenebacteriochlorin:
GGA (BLYP): 1.33 eV
CIS: 3.75 eV
Druew and Head-Gordon, J. Am. Chem. Soc. 126, 4007 (2004).
The Mulliken limit
In the limit of well-separated donor and acceptor:
h  CT  IP ( D )  EA ( A )  1 / R
Coulomb
attraction
Neither the gap nor the ~1/r dependence
obtained for standard functionals!
Both obtained with the optimally-tuned
range-separated hybrid!
Results – gas phase Ar-TCNE
Donor
benzene
toluene
o-xylene
Naphtha
lene
MAE
TDPBE
TDB3LYP
TDBNL
=0.5
TDBNL
Best 
Exp
G0W0BSE
GWBSE
(psc)
1.6
2.1
4.4
3.8
3.59
3.2
3.6
1.4
1.8
4.0
3.4
3.36
2.8
3.3
1.0
1.5
3.7
3.0
3.15
2.7
2.9
0.4
0.9
3.3
2.7
2.60
2.4
2.6
2.1
1.7
0. 8
0.1
---
0.4
0.1
Thygesen
Blase
PRL ‘11 APL ‘11
Stein, Kronik, Baer, J. Am. Chem. Soc. (Comm.) 131, 2818 (2009).
Wong, B. M.; Cordaro, J. G., J. Chem. Phys. 129, 214703 (2008).
0.25
0.20
Difference from CC2 (eV)
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
-0.20
-0.25
C343
Fit to CC2
-0.08
NKX
2388
s-tran
s
0.02
NKX
2388
s-cis
NKX
2311
s-cis
0.05
NKX
2311
strans
0.02
NKX
2586
s-cis
NKX
2677
Mean
MAD
0.02
NKX
2586
s-tran
s
0
0
-0.04
0.00
0.03
BNL* 1
0.01
0.09
0.11
0.03
0
-0.05
-0.05
-0.02
0.02
0.05
BNL* 2 or 3
-0.08
-0.06
-0.01
-0.1
-0.11
-0.17
-0.17
-0.23
-0.12
0.12
Stein, T.; Kronik, L.; Baer, R., J. Chem. Phys. 131, 244119 (2009).
Optical excitations:
Fixing the La, Lb problem of oligoacenes
4.60
B3LYP
5.10
BNL (Tuned)
4.60
BP86*
4.10
3.60
3.10
2.60
2.10
1.60
1.10
excitation energy [eV]
La
5.10
excitation energy [eV]
CC2
CC2
Lb
B3LYP
BNL (Tuned)
BP86*
4.10
3.60
3.10
2.60
2.10
1.60
1.10
2
3
4
5
6
N – number of benzene rings
2
3
4
5
N-number of benzene rings
Kuritz, Stein, Baer, Kronik, J. Chem. Theo. Comp. 7, 2408 (2011).
6
Where’s the charge transfer?
LUMO +1
LUMO
HOMO
Energy
LUMO
HOMO
HOMO-1
1L
b
excitation
La excitation
LUMO
HOMO
LUMO+HOMO
LUMO-HOMO
N. Kuritz, T. Stein, R. Baer, L. Kronik, JCTC 7, 2408 (2011).
Conclusions
Kohn-Sham
quasi-particle
Optical
GW
GW+BSE
RSH
TD-RSH
Kronik, Stein, Refaely-Abramson, Baer,
J. Chem. Theo. Comp. (Perspectives Article), to be published
Two different paradigms for functional
development and applications
From
To
Tuning is NOT fitting! Tuning is NOT semi-empirical!
Choose the right tool (=range parameter)
for the right reason (=Koopmans’ theorem)