Excitation gaps of finite-sized systems from Optimally-Tuned Range-Separated Hybrid Functionals:
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Transcript of Excitation gaps of finite-sized systems from Optimally-Tuned Range-Separated Hybrid Functionals:
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Leeor KronikDepartment of Materials & Interfaces,
Weizmann Institute of Science
Excitation gaps of finite-sized systems from Optimally-Tuned Range-Separated Hybrid Functionals:
5th Benasque TDDFT Workshop, January 2012
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The Group
Funding European Research CouncilIsrael Science Foundation
Germany-Israel FoundationUS-Israel Binational Science Foundation
Lise Meitner Center for Computational ChemistryAlternative Energy Research Initiative
Tami Zelovich
Ido Azuri Ariel BillerBaruch
FeldmanEli
Kraisler
Sivan Abramson
Andreas Karolewski(visiting)
Ofer Sinai
Anna Hirsch
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The people
Tamar Stein(Hebrew U)
Roi Baer
Sivan Refaely-Abramson
Natalia Kuritz
(Weizmann Inst.)
Kronik, Stein, Refaely-Abramson, Baer, J. Chem. Theo. Comp. (Perspectives Article), to be published
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Fundamental and optical gap – the quasi-particle picture
derivative discontinuity!IP
EA
Evac
(a) (b)
Eg Eopt
See, e.g., Onida, Reining, Rubio, RMP ‘02; Kümmel & Kronik, RMP ‘08
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Mind the gapThe Kohn-Sham gap underestimates the real gap
xcHOMOKS
LUMOKSg AIE
Perdew and Levy, PRL 1983;Sham and Schlüter, PRL 1983
derivative discontinuity!
Kohn-Sham eigenvalues do not mimic the quasi-particle picture
even in principle!
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H2TPPEn
ergy
[eV] -2.9
-4.7
-2.5
-5.2
-1.4
-6.2
-1.5
-6.2
-1.7
-6.4
2.11.92.12.2 4.71.8 2.7 4.8 4.7
GGA B3LYP OT-BNL GW-BSE EXP
2.0
-IP, -EA Eopt
TD TD TD
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Wiggle room: Generalized Kohn-Sham theory
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!!
- 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
)()()];([)(}][{ˆ rrrnvrVO iiiRionjS
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Hybrid functionals are a special case of Generalized Kohn-Sham theory!
)()()];([)];([)1(ˆ)];([)(21 2 rrrnvrnvaVarnVrV iii
slc
slxFHion
Does a conventional hybrid functional solve the gap problem?
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H2TPPEn
ergy
[eV] -2.9
-4.7
-2.5
-5.2
-1.4
-6.2
-1.5
-6.2
-1.7
-6.4
2.11.92.12.2 4.71.8 2.7 4.8 4.7
GGA B3LYP OT-BNL GW-BSE EXP
2.0
-IP, -EA Eopt
TD TD TD
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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?
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Range-separated hybrid functionalsCoulomb operator decomposition:
)(erf)(erfc 111 rrrrr
Short Range Long RangeEmphasize long-range exchange,
short-range exchange correlation!
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??
)()()];([)];([ˆ)];([)(21 ,,2 rrrnvrnvVrnVrV iii
slc
srx
lrFHion
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How to choose ?
);();1(HOMO NENE gsgs “Koopmans’ theorem”
Need both IP(D), EA(A) choose to best obey “Koopmans’ theorem” for both neutral donor and charged acceptor:
,0
2, ));();1(()(i
ii
iii
HOMO NENEJgsgs
Minimize
Tune, don’t fit, the range-separation parameter!
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Tuning the range-separation parameter
)1()1()()()( NIPNNIPNJ HH
)(min)( JJ opt
Neutral molecule (IP)
Anion (EA)
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H2TPPEn
ergy
[eV] -2.9
-4.7
-2.5
-5.2
-1.4
-6.2
-1.5
-6.2
-1.7
-6.4
2.11.92.12.2 4.71.8 2.7 4.8 4.7
GGA B3LYP OT-BNL GW-BSE EXP
2.0
-IP, -EA Eopt
TD TD TD
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Gaps of atoms
Stein, Eisenberg, Kronik, Baer, Phys. Rev. Lett., 105, 266802 (2010).
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Fundamental gaps of acenes
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7
Gap
(eV
)
n
Ref
BNL* orb. gap
0.31
0.28
0.25
0.22
0.20
0.19
Stein, Eisenberg, Kronik, Baer, Phys. Rev. Lett., 105, 266802 (2010).
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Fundamental gaps of hydrogenated Si
nanocrystals
GW: Tiago & Chelikowsky, Phys. Rev. B 73, 2006DFT: Stein, Eisenberg, Kronik, Baer,
PRL 105, 266802 (2010).s.
0
2
4
6
8
10
12
14
0 5 10 15
Ener
gy (e
V)
Diameter (Å)
-LumoGW EA-HOMOSeries4Exp IP
0.140.13
0.12
0.240.33
0.41
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6 6.5 7 7.5 8 8.5 9 9.5 10 10.55
6
7
8
9
10
11
Experimental ionization energy [eV]
-H
OM
O
Ionization Energy
[eV
]
EXPGWOT-BNLB3LYP
GW data: Blasé et al., PRB 83, 115103 (2011)
S. Refaely-Abramson, R. Baer, and L. Kronik, Phys.Rev. B 84 ,075144 (2011) [Editor’s choice].
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Optical gaps with Time-dependent DFTTDDFT: BNL results as accurate
as those of B3LYP
a – thiopheneb – thiadiazolec – benzothiadiazoled – benzothiazolee – flourenef – PTCDAg – C60
h – H2Pi – H2TPPj – H2Pc S. Refaely-Abramson, R. Baer, and L. Kronik, Phys.Rev. B 84 ,075144 (2011)
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The charge transfer excitation problem
Liao et al., J. Comp. Chem. 24, 623 (2003).
Time-dependent density functional theory (TDDFT), usingeither semi-local or standard hybrid functionals, can
seriously underestimate charge transfer excitation energies!Biphenylene – tetracyanoethylene:B3LYP: 0.77 eVExperiment: 2 eV
zincbacteriochlorin-phenylene-bacteriochlorin:
GGA (BLYP): 1.33 eV
CIS: 3.75 eV
Druew and Head-Gordon, J. Am. Chem. Soc. 126, 4007 (2004).
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The Mulliken limitIn the limit of well-separated donor and acceptor:
Neither the gap nor the ~1/r dependence obtained for standard functionals!
Both obtained with the optimally-tuned range-separated hybrid!
Coulomb attraction
RAEADIPh /1)()(CT
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Results – gas phase Ar-TCNE
Stein, Kronik, Baer, J. Am. Chem. Soc. (Comm.) 131, 2818 (2009).
Donor TD-PBE
TD-B3LYP
TD-BNL =0.5
TD-BNL
Best
Exp G0W0-BSE
GW-BSE(psc)
benzene 1.6 2.1 4.4 3.8 3.59 3.2 3.6toluene 1.4 1.8 4.0 3.4 3.36 2.8 3.3o-xylene 1.0 1.5 3.7 3.0 3.15 2.7 2.9Naphtha
lene 0.4 0.9 3.3 2.7 2.60 2.4 2.6MAE 2.1 1.7 0. 8 0.1 --- 0.4 0.1
Thygesen PRL ‘11
Blase APL ‘11
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Partial Charge Transfer: Coumarin dyes
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Sensitivity to the LR parameter
Wong, B. M.; Cordaro, J. G., J. Chem. Phys. 129, 214703 (2008).
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Instead of fitting: tuning
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
C343 NKX 2388 s-tran
s
NKX 2388 s-cis
NKX 2311
s-trans
NKX 2311 s-cis
NKX 2586 s-tran
s
NKX 2586 s-cis
NKX 2677
Mean MAD
Fit to CC2 -0.08 0.02 0.05 0.02 0.02 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
Dif
fere
nce
fro
m C
C2
(eV
)
Stein, T.; Kronik, L.; Baer, R., J. Chem. Phys. 131, 244119 (2009).
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Optical excitations: Fixing the La, Lb problem of oligoacenes
Kuritz, Stein, Baer, Kronik, J. Chem. Theo. Comp. 7, 2408 (2011).
2 3 4 5 61.10
1.60
2.10
2.60
3.10
3.60
4.10
4.60
5.10 LaCC2B3LYPBNL (Tuned)BP86*
N – number of benzene rings
exci
tatio
n en
ergy
[eV
]
2 3 4 5 61.10
1.60
2.10
2.60
3.10
3.60
4.10
4.60
5.10Lb CC2
B3LYPBNL (Tuned)BP86*
N-number of benzene rings
exci
tatio
n en
ergy
[eV
]
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HOMO-1
HOMO
LUMOLUMO +1
Energy LUMO
HOMO
1Lb excitation La excitation
Where’s the charge transfer?
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KEY: Mixing HOMO-LUMO“Charge-transfer-like” excitation
HOMO LUMO
LUMO-HOMO
LUMO+HOMO N. Kuritz, T. Stein, R. Baer, L. Kronik, JCTC 7, 2408
(2011).
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Conclusions
Kohn-Sham quasi-particle OpticalGW GW+BSERSH TD-RSH
Kronik, Stein, Refaely-Abramson, Baer, J. Chem. Theo. Comp. (Perspectives Article), to be published
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Two different paradigms for functional development and applications
Tuning is NOT fitting! Tuning is NOT semi-empirical!
From To
Choose the right tool (=range parameter) for the right reason (=Koopmans’ theorem)