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Leeor Kronik
Department of Materials and Interfaces,
Weizmann Institute of Science, Rehovoth 76100, Israel
Fundamental and optical gaps
from density functional theory
CSC 2013, Paphos, Cyprus
The people
Funding: European Research Council, Israel Science Foundation,
US-Israel Bi-National Foundation, Germany-Israel Foundation,
Lise Meitner Center for Computational Chemistry, Helmsley Foundation,
Wolfson Foundation, US Dept. of Energy (computer time)
Spectroscopy from DFT : The people
Tamar Stein
(Hebrew U)
Roi Baer
Sivan Refaely-
Abramson
Natalia
Kuritz
(Weizmann Inst.)
Shira
Weismann Piyush
Agrawal
Eli
Kraisler
Sahar
Sharifzadeh
(Berkeley Lab)
Jeff Neaton
Kronik, Stein, Refaely-Abramson, Baer,
J. Chem. Theo. Comp. (Perspectives Article) 8, 1515 (2012).
The Kohn-Sham equation
Exact mapping to a single particle problem!
(Kohn & Sham, Phys. Rev., 1965)
statesoccupied
i rrn 2|)(|)(
kinetic ion-e “many body” e-e
)()(2
1 2 rErVVV iiixcHartreeion
Exc[n(r)]
Exc is a unique, but complicated and unknown
functional of the charge density.
In principle: Exact ; In practice: approximate
)(
][)];([
rn
nErnV xc
xc
DFT: Functionals
LDA - Exc is approximated by its value per particle in a uniform electron gas weighted by the local density- local functional
GGA - Exc includes a dependence on the density gradient in order to account for density variations- semi-local functional
Hybrid Functionals - A fraction of exact exchange is mixed with GGA exchange and correlation
There are various approximations for the exchange-
correlation functional, for example:
The Challenge
B. Kippelen and J. L. Brédas, Energy Environ. Sci. 2, 251 (2009)
Photovoltaics
J. C. Cuevas, IAS Jerusalem, 2012
Molecular Junctions
L R
But Can we take DFT eigenvalues seriously?
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
Mind the gap
The Kohn-Sham gap underestimates the real gap
xc
HOMO
KS
LUMO
KSg 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!
H2TPP En
ergy
[eV
] -2.9
-4.7
-2.5
-5.2
-1.4
-6.2
-1.5
-6.2
-1.7
-6.4
2.1 1.9 2.1 2.2 4.7 1.8 2.7 4.8 4.7
GGA B3LYP OT-BNL GW-BSE EXP
2.0
-IP, -EA Eopt
TD TD TD
Gap information can still be extracted
by fractional charge considerations
From curvature corrections:
From an ensemble treatment:
Stein, Autschbach, Govind, Kronik, Baer,
J. Phys. Chem. Lett. 3, 3740 (2012).
Kraisler & Kronik,
Phys. Rev. Lett. 110, 126403 (2013).
but is not a direct assignment of
both HOMO and LUMO
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, Livshits, Salzner, 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
Hybrid functionals are a special case of
Generalized Kohn-Sham theory!
)()()];([)];([)1(ˆ)];([)(2
1 2 rrrnvrnvaVarnVrV iii
sl
c
sl
xFHion
Does a conventional hybrid
functional solve the gap problem?
H2TPP En
ergy
[eV
] -2.9
-4.7
-2.5
-5.2
-1.4
-6.2
-1.5
-6.2
-1.7
-6.4
2.1 1.9 2.1 2.2 4.7 1.8 2.7 4.8 4.7
GGA B3LYP OT-BNL GW-BSE EXP
2.0
-IP, -EA Eopt
TD TD TD
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:
)(erf)(erfc 111 rrrrr
Short Range Long Range
Emphasize 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)];([)(2
1 ,,2 rrrnvrnvVrnVrV iii
sl
c
sr
x
lr
FHion
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
i
i
i
ii
HOMO NENEJgsgs
Minimize
Tune, don’t fit, the range-separation parameter!
Tuning the range-separation parameter
)1()1()()()( NIPNNIPNJ HH
)(min)( JJ opt
Neutral molecule (IP) Anion (EA)
H2TPP En
ergy
[eV
] -2.9
-4.7
-2.5
-5.2
-1.4
-6.2
-1.5
-6.2
-1.7
-6.4
2.1 1.9 2.1 2.2 4.7 1.8 2.7 4.8 4.7
GGA B3LYP OT-BNL GW-BSE EXP
2.0
-IP, -EA Eopt
TD TD TD
Kronik, Stein, Refaely-Abramson, Baer, J. Chem. Theo. Comp. 8, 1515 (2012).
Gaps of atoms
Stein, Eisenberg, Kronik, Baer, Phys. Rev. Lett. 105, 266802 (2010).
Fundamental gaps of
hydrogenated Si nanocrystals
GW: Tiago & Chelikowsky, Phys. Rev. B 73, 2006
DFT: 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
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
]
EXP
GW
OT-BNL
B3LYP
GW data: Blase et al.,
PRB 83, 115103 (2011)
S. Refaely-Abramson, R. Baer, and L. Kronik, Phys.Rev. B 84 ,075144 (2011) [Editor’s choice].
Mulliken’s limit: 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
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!
)/()()(CT RaADh LH TDDFT
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 3.6
toluene 1.4 1.8 4.0 3.4 3.36 2.8 3.3
o-xylene 1.0 1.5 3.7 3.0 3.15 2.7 2.9
Naphtha
lene 0.4 0.9 3.3 2.7 2.60 2.4 2.6
MAE 2.1 1.7 0. 8 0.1 --- 0.4 0.1
Thygesen
PRL ‘11 Blase
APL ‘11
RSH for spectroscopy
Stein et al., J. Am. Chem. Soc. (Comm.) 131, 2818 (2009);
Stein et al., J. Chem. Phys. 131, 244119 (2009);
Refaely-Abramson et al., Phys. Rev. B 84 ,075144 (2011);
Kuritz et al., J. Chem. Theo. Comp. 7, 2408 (2011).
Optical excitations, including charge transfer
scenarios
Photoelectron Spectroscopy
Refaely-Abramson et al., Phys. Rev. Lett. 109, 226405 (2012);
Egger et al., to be published
Gap renormalization in organic crystals
Refaely-Abramson et al., Phys. Rev. B (RC) 88, 081204 (2013) .
Combination with van-der-Waals corrections
Agrawal et al., J. Chem. Theo. Comp. 9, 3473 (2013) .
Conclusions
Kohn-Sham quasi-particle Optical
GW GW+BSE
RSH TD-RSH
Kronik, Stein, Refaely-Abramson, Baer,
J. Chem. Theo. Comp. (Perspectives Article) 8, 1515 (2012).
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)