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)