Roi BaerInstitute of Chemistry,

The Fritz Haber Center for Molecular Dynamics

The Hebrew University of Jerusalem, 

Jerusalem, Israel

• Ms. Ester Livshits (PhD student, Jerusalem)• Dr. Helen Eisenberg (Postdoc, Jerusalem)• Tamar Gershon (PhD student Jerusalem)Collaborators • Tamar Gershon (PhD student, Jerusalem)• Professor Leeor Kronik (Weizmann)• Professor Daniel Neuhauser (UCLA)• Professor Anna Krylov (USC)

Collaborators and Students

• Professor Anna Krylov (USC)

• Israel Science Foundation (ISF)• US-Israel Binational Science Foundation

(BSF)Funding (BSF)• FAME network of excellence

Funding

• An electron is repelled not only by other electrons, but also by itselfSelf

• Good at the short range (emulates correlation)

• But much too much at long range 

Self repulsion

• Static polarizability• Stability of anions• Charge quantization in decoupled systems

Unphysical b h i

g q p y• Charge transfer excitations• Rydberg excitations

behavior

1 1 2 2Take a charge at and bring another charge from infinity to . q qr r

1 22 12 1 2

12

The energy for this i .s: q q

E rr

= = −r r

1 3 2 3Next bring another charge from infinity to :q q q q

q E = +r3 3 313 23

Next, bring another charge from infinity to :q Er r

= +r

1The total energy is a sum over pairs:

2i j

i j ij

q qE

r≠

= ∑2 i j ijr≠

( ) ( )( )3 31

For continuous charge dist: ( density of charge)2

n nE d rd r n

′′=

′−∫r r

rr r

But this includes self interaction - analogous to the termsi j=

( ) ( ) ( ) ( ) ( )In Hartree-Fock theory we have: , ; ,occN

i inρ φ φ ρ′ ′= =∑r r r r r r r( ) ( ) ( ) ( ) ( )

( ) ( ) ( )1

2

3 3 3 3r,r1 1

The 2-electron repulsion is: d rd r2 2 r-r

i

ee

n nE d rd r

ρ=

′′′ ′= −

′ ′−∫ ∫r r

r rDirect or Hatree Exchange

So in HF theory self-repulsion is automatically removed.

.5

V(x) .3

.4

High and widebarrier

V0 0

.1

.2

x-2.0 -1.5 -1.0 -.5 0.0 .5 1.0 1.5 2.0

0.0

Self repulsionNo self repulsion

.3

.4

.5

High and widebarrier

p

.3

.4

.5

High and widebarrier

V(x)

.1

.2

.3 ba e

V(x)

.1

.2

x-2.0 -1.5 -1.0 -.5 0.0 .5 1.0 1.5 2.0

0.0

x-2.0 -1.5 -1.0 -.5 0.0 .5 1.0 1.5 2.0

0.0

• LSDA/GGA: no barrier• Scheffler et al:

– Electron to O2 too early – O - is strongly attracted– O2 is strongly attracted– Introduced “spin rules”– Lowering the spin on the

triplet O2 was forbidden – This is a strange ad hoc

mechanism

J. Behler, B. Delley, S. Lorenz, K. Reuter, and M. Scheffler,, Phys. Rev. Lett. 94 (3), 036104 (2005).

E. 

PBE

Livshits, R

B3LYP

R. Baer andB3LYP d R. Koslof

BNL

ff, work in BNL progress ((2009)

1 1 121 1ˆ2

ˆ2

1j

j j j jj

Ur

T′≠ ′

∇ == − ∑ ∑

( )Unique gs WF of electrons having gs density nψ = r( )q g g g y gs

ψ

( )0Unique gs WF of non-int Fermions having gs density nψ = r

ˆ ˆˆgs gs

E T V Uψ ψ= + +

0 0ˆ ˆˆ

CT V U E nψ ψ ⎡ ⎤= + + + ⎢ ⎥⎣ ⎦

( ) ( ) 30 0

ˆ ˆgs gsV v n d r Vψ ψ ψ ψ= =∫ r r

ˆ ˆˆ ˆ⎡ ⎤0 0C gs gs

E n T U T Uψ ψ ψ ψ⎡ ⎤ = + − +⎢ ⎥⎣ ⎦

0 0ˆ ˆ

C gs gsT n T Tψ ψ ψ ψ⎡ ⎤ = −⎢ ⎥⎣ ⎦

⎡ ⎤ ⎡ ⎤0 0C CE n T n⎡ ⎤ ⎡ ⎤< >⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

•Theorem: A physical quantity cannot have self-repulsion•Corollary: Expectation values of physical operators are self-repulsion free•Corollary: Ec and Tc are self repulsion-free

U E n E nψ ψ ⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦( )

0 0

Self-repulsion freeH X

U E n E nψ ψ +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

( ) ( ) 3 31 n nE n d rd r

′⎡ ⎤ ′=⎢ ⎥⎣ ⎦ ∫∫

r r

( )2

Has self-repulsion

HE n d rd r=⎢ ⎥⎣ ⎦ ′−∫∫ r r

( )2

′

( )Has self-repulsion

( )3 3

,12X

E n d rd rρ ′

⎡ ⎤ ′= −⎢ ⎥⎣ ⎦ ′−∫∫r r

r r

( )Has self-attraction

0 0ˆ ˆˆ ˆ

C gs gsE n T U T Uψ ψ ψ ψ⎡ ⎤ = + − +⎢ ⎥⎣ ⎦( )self repulsion-free

( ) ( )( )⎡ ⎤ ∫ ( ) ( )( )( )

3

ˆˆ ˆ

,SLC XC X

T U T E

E n f n d r En

ψ ψ ψ ψ+ +

⎡ ⎤ = −⎢ ⎥⎣ ⎦ ∇∫ r r

( )0 0gs gs HT U T Eψ ψ ψ ψ+ − +

Long range self repulsion

0 0ˆ ˆˆ ˆ

C gs gsE n T U T Uψ ψ ψ ψ⎡ ⎤ = + − +⎢ ⎥⎣ ⎦ 0 0C gs gs

ψ ψ ψ ψ⎢ ⎥⎣ ⎦

( )0 0

1ˆ ˆ ˆC j jE n Y Y Y yψ ψ ψ ψ⎡ ⎤ = − = −⎢ ⎥⎣ ⎦ ∑ r r( )0 0 2C gs gs j j

jj

yγ γ γψ ψ ψ ψ′≠

⎢ ⎥⎣ ⎦ ∑

re γ− ( )erfc rγ1 1

0.25

0.30

ey

rγ =( )erfc r

yrγ

γ=

1 11

yrγ γ

=+

( ) ( )1

0 10

0.15

0.20

ˆ ˆ0 Y Y E T Eψ ψ ψ ψ

( ) ( )0

10y r y r

r ∞= =

2 3 4 5 6 7 8

0.05

0.10

0 00 :

gs gs C C CY Y E T Eγ γγ ψ ψ ψ ψ= − = − <

ˆ ˆ: 0Y Y Eγ ψ ψ ψ ψ= ∞ = >

Conclusion: There exists a for which there is equality γ0 0

: 0gs gs CY Y Eγ γγ ψ ψ ψ ψ= ∞ − = >

0 0ˆ ˆ

C gs gsE n Y Yγ γψ ψ ψ ψ⎡ ⎤ = −⎢ ⎥⎣ ⎦ 0 0

ˆC gs gs

y y

gs gs H XY E Eγ γ

γ γ

γψ ψ⎢ ⎥⎣ ⎦

⎡ ⎤= − −⎢ ⎥⎣ ⎦Fix γ and findGood approx for f(n,dn):

↓

( ) ( )( ) 3 yH b

↓

∫

•Becke (B3LYP)•…•Savin•Hirao( ) ( )( ) 3,

y

XC XHybC

f n nE d r E γγ ∇= −∫ r r

( ) ( )1∫∫

Hirao•Handy•Baer+Neuhauser•Yang

( )

( ) ( ) 3 31,

2y

XE y d rd rγ

γρ ′ ′ ′= − −∫∫ r r r r •Tozer•Truhlar•Head-Gordon•Gerber-Angyan

( )erfc1 1, ,

1

rr ey

r r r

γ

γ

γ

γ

−

=+

Gerber Angyan•Perdew-Scuseria•…

1 r r rγ+“B3LYP” typeLong-range self-repulsion

“RSH” typeNo long-range self-repulsion

1 Cε

11

C

C Ctγ ε

=+ −

1.0g

1 3

1sr

n∝

0 6

0.8

0.4

0.6

0.2

0 5 10 15 20rs

γ HOMOε− HOMOε−SCFΔγ 10a−wγ 10a−w

100

10

Unpolarized Polarized

1g

0.1

0.010 01 0 1 1 100.01 0.1 1 10

rs

E. Livshits and R. Baer, PCCP  9, 2932(2007)

Pair correlation function taken from:P. Gori-Giorgi and J. P. Perdew, Phys. Rev. B 66, 165118 (2002).

• In B3LYP – OK• In RSH – NOT OK• Must be: γ system dependent

30

25

HOMO B3LYP

Exp. Vertical Theorem:Theorem:

Th i i tiTh i i ti

15

20

(eV)

Basis: cc-pVTZThe ionization The ionization

potential of the potential of the

t i tlt i tl

10

IP system is exactly system is exactly

equal to the HOMO equal to the HOMO

5

energyenergy

0

• M. Levy and J. P. Perdew, in Density Functional Methods in Physics, edited by R. Dreizler and J. Perovidencia (Plenum, New York, 1985), pp. 11.

• C.‐O. Almbladh and U. von‐Barth, Phys. Rev. B 31, 3231 (1985).

γ HOMOε− HOMOε−SCFΔγ 10a−wγ 10a−w

Molecule

γ(a0

-1)

IP exp

IPw=0

IPw=0.1

IPB3LYP

( );HOMO Nε γ

Li2 0.3 5.1 -0.1 0.0 -0.2N2 0.6 15.6 -1.0 -0.6 0.0O2 0.7 12.1 -1.2 -0.9 -0.7F 0 7 15 7 -0 8 -0 4 0 1

( )( ) ( ); 1;E N E Nγ γ= − −

F2 0.7 15.7 -0.8 -0.4 0.1Na2 0.4 4.9 -0.2 -0.1 -0.3P2 0.8 10.6 -0.4 -0.3 0.0S2 0.5 9.4 -0.4 -0.2 -0.2Cl2 0.5 11.5 -0.3 0.0 0.1BeH 0.6 8.2 -0.2 -0.1 -0.2CO 0.6 14.0 -0.3 -0.1 0.0HF 0 7 16 0 -0 4 0 0 0 0HF 0.7 16.0 -0.4 0.0 0.0HCl 0.5 12.8 0.0 0.3 0.2NH3 0.5 10.2 -0.6 -0.4 -0.5PH3 0.4 9.9 -0.6 -0.4 -0.6

E. Livshits and R. Baer, PCCP 9, 2932 (2007)

Mean -0.5 -0.2 -0.2

RMS 0.6 0.4 0.3

‐0.48

‐0.52

‐0.50

HF

H2+

QCHEM 3.1

0 56

‐0.54

E(E h)

B3LYP

BLYP

cc-pVTZ

Same

‐0.58

‐0.56 Same problems appear in:

‐0.62

‐0.60 He2+, Ne2

+

etc0 1 2 3 4 5 6 7 8 9

H‐H distance (A)

R Merkle A Savin and H Preuss J Chem Phys 97 9216 (1992)R. Merkle, A. Savin, and H. Preuss, J. Chem. Phys. 97, 9216 (1992).Y. Zhang and W. Yang, J. Chem. Phys. 109, 2604 (1998).A. Ruzsinszky, J. P. Perdew, G. I. Csonka, O. A. Vydrov, and G. E. Scuseria, J. Chem. Phys. 126, 104102 (2007).

LR R+

r → ∞

L

( )( ) ( ) ( )

1

2R R

GS L R

E r E E +

= ±

→ +

RR+

( ) ( ) ( )

4

R R

...2

correctE r E E

r

α

→ +

− +R

0 5 0 5( )1

Density Picture:

n n n+R0.5 R0.5( )

( ) ( )0.5

214

2

L Rn n n

E r E Rr

= +

= +

‐0.50

‐0.48

By using RSH we get id f l i

0 54

‐0.52 HF

B3LYP

BLYP

rid of repulsion

But this is not enough

h l l‐0.56

‐0.54

E(E h) BLYP

γ=0.5There is also large error at r = r0Wrong as mptotic

‐0.60

‐0.58 Wrong asymptotic curve (≠ -α/2r4)

‐0.62

0 1 2 3 4 5 6 7 8 9

H‐H distance (A)

E. Livshits and R. Baer, J. Phys. Chem A 112, 12789 (2008).

( ) 0 1 0.5 0.5E E Eγ γγ − −Δ = −

1 4

1.6

Ne2+

( ) γ γ

E HF γ→ ∞

0 8

1.0

1.2

1.4

)

Ne2+

He2+

H2+ ΔE

ΔE

0 2

0.4

0.6

0.8

ΔE (eV)

LDA γ → 0

ΔE

0 4

‐0.2

0.0

0.2

( )Åeqr

‐0.4

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

γ (1/a0)

( )eq

2ωR+R+ R++RR0.5++R0.5+

E. Livshits and R. Baer, J. Phys. Chem A 112, 12789 (2008).

E. Livs

• Have γ so: localized-delocalized degenerate• Have γ so: localized-delocalized degenerate

hits and R.

• Energies in kcal/mole• Energies in kcal/moleProperty R BLYP B3LYP HF LC LC* Exp

Baer, J. P

h

Property R BLYP B3LYP HF BNL BNL* Exp.

Enthalpy by atomization

H 66 65 60.9 60.9 60.9 61He 82 75 43 74 59 55N 75 60 2 59 34 32

hys. Chem

ANe 75 60 2 59 34 32

Enthalpy by asymptote

H NA NA 60.9 50 60.9 61He NA NA 43 42 59 55Ne NA NA 2 27 34 32H 1 1 1 1 1 06 1 2 1 06 1 05

A 112, 1278

H 1.1 1.1 1.06 1.2 1.06 1.05He 1.2 1.1 1.075 1.2 1.078 1.080Ne 1.9 1.9 1.7 1.760 1.72 1.765H 2.7 2.9 3.3 2.9 3.3 3.32H 1 7 2 0 2 5 2 1 2 5 2 42

( )eqr Å

2ω

89 (2008).

2ω

He 1.7 2.0 2.5 2.1 2.5 2.42Ne 0.5 0.6 0.9 0.726 0.8 0.729H NA NA 1 0.6 1 1He NA NA 0.98 NA 0.98 1N NA NA 1 01 NA 1 02 1

2ω

R effα α ( ) 51

2effE r rα ′=

R refα α

Ne NA NA 1.01 NA 1.02 1H 5.3 5.6 4.51 5.8 4.51 4.50He 1.6 1.5 1.34 1.8 1.41 1.38Ne 3.1 2.9 2.4 3.2 2.70 2.66

ff

( )30

daugRa a

( )2ff

By using γ* several y g γproblems are solvedp

Repulsion instead of

i

No degeneracy

b kiProper well depth at r=r0

Fine bond lengths

“Good vibrations”

Leading term in asymptotic

form of i l

Excellent atomic l i biliattraction breaking depth at r r0 lengths vibrations potential -

correctpolarizability

E. Livshits and R. Baer, J .Phys. Chem, A 112, 12789 (2008).

Energies (kcal/mol) of (H2O)2+ relative to 

Method Vertical neutral

Hemi-bonded

2 2the proton‐transferred geometry

neutral bonded

BLYP 6.7 ‐8.3B3LYP -1.6

E. Livshits, A. I. Krylov and R. Baer, work in progress 

BNL*(0.56) 23.3 8.2EOM-IP-CCSD 20.0 5.3EOM-IP-CC(2.3) 21.8 7.4

p g(2009).

Neutral dimer Hemi‐bonded

Proton‐transferred

• We looked into the Anthracene-TCNE charge transfer complex in solution (CH2Cl2)

1• In ideal CT excitation: 1CT ArH A

h IP EAR

νε

= − −

J. M. Masnovi, E. A. Seddon, and J. K. Kochi, Can. J. of Chem . 62 (11), 2552 (1984).

• First stage is to find the appropriate γ• This requires finding a γ that gives proper q g γ g p p

HOMO energies for IP(ArH) and EA(TCNE)γ IP (ANT) (eV) IP (TCNE-) = EA (eV) Σrγ IP (ANT) (eV) IP (TCNE ) = EA (eV) Σr

ΔESCF ‐εHOMO r ΔESCF ‐εHOMO r0.01 6.09 4.03 0.34 0.59 -0.90 2.53 2.870 02 6 09 4 22 0 31 1 69 0 73 1 43 1 740.02 6.09 4.22 0.31 1.69 -0.73 1.43 1.74

0.033 6.14 4.44 0.28 1.72 -0.52 1.30 1.580.05 6.18 4.71 0.24 1.77 -0.24 1.14 1.38

0 3 6 91 7 02 0 02 2 44 2 34 0 04 0 060.3 6.91 7.02 -0.02 2.44 2.34 0.04 0.060.4 7.04 7.35 -0.04 2.69 2.88 -0.07 0.120.5 7.08 7.56 -0.07 2.89 3.27 -0.13 0.20

• One obtains an optimal value of 0.3

• Checking the Mulliken law in gas phase3.8

3 6

3.7TDDFT g=0.3

Mulliken

3.5

3.6

E (eV)

3 3

3.4

3.2

3.3

0 02 0 025 0 03 0 035 0 04 0 045 0 050.02 0.025 0.03 0.035 0.04 0.045 0.05

1/R(a0)1CT ArH A

h I EAR

ν = − −

• We took a B3LYP optimized geometry for the complex and performed TDDFT calculation

Ar B3LYP BNLγ=0.5

BNL γ* ExpΕ f γ* Ε f Ε f2 1 0 03 4 4 0 33 3 8 0 03 3 59 0 02Benzene 2.1 0.03 4.4 0.33 3.8 0.03 3.59 0.02

Toluene 1.8 0.04 4.0 0.32 3.4 0.03 3.36 0.03o‐Xylene 1.5 ~0 3.7 0.31 3.0 0.01 3.15 0.05Naphthalene 0.9 ~0 3.3 0.32 2.7 ~0 2.60 0.01

T. Stein, L. Kronik and R. Baer, JACS to be published (2008).

Substituent PBE B3‐LYP

BNL Exp26Ε(γ=0.5)  γ∗ Ε(γ=0.3)

None 0.9 1.0 2.3 0.31 1.82 1.739‐cyano Fail 0.5 2.6 0.30 2.03 2.019‐chloro 0.9 1.0 2.3 0.31 1.82 1.749‐carbo‐ 0 8 0 9 2 4 0 30 1.84 1.849 carbomethoxy

0.8 0.9 2.4 0.30 1.84 1.84

9‐methyl 1.0 1.1 2.1 0.30 1.71 1.559‐nitro 0 6 0 9 2 8 0 30 2 12 2 039‐nitro 0.6 0.9 2.8 0.30 2.12 2.039.10‐dimethyl

1.3 1.4 2.1 0.30 1.77 1.44

9 formyl 0 8 1 0 2 5 0 30 1 95 1 909‐formyl 0.8 1.0 2.5 0.30 1.95 1.909‐formyl 10‐chloro

0.8 0.9 2.5 0.30 1.96 1.96

T. Stein, L. Kronik and R. Baer, JACS to be published (2008).

• Kohn-Sham bad gaps (50-70% too small)• Because missing “derivative discontinuity”g y• Hartree-Fock theory terrible: 500% too large.

LDAE E

γ−

g g

Hartree Fock LDA

g g

E E

E E−

−

0.2

0.16

0.18 NaCl* C

Aγ

ε=

0 1

0.12

0.14

∗ MgO

optAε −

0.22C =

0 06

0.08

0.1γ∗

AlPC

0.8A=

0.02

0.04

0.06 AlP

AlN

SiC AlA

GaP

Si

0

0 2 4 6 8 10 12 14

SiC AlAsAlSb

εOpt

3.0

2.5

2.0

ap (e

V)

ExpGaP AlP

1.0

1.5

Ban

d G

a ExpgHSELSDA

AlSbSiC

0.5

1.0 LSDAAlSb

0.01 0 1 5 2 0 2 5

Si

1.0 1.5 2.0 2.5Experimental Band gap (eV)

H. Eisenberg, R. Baer, work in progress (2008).

8.0

7.0

)

6.0

Ban

d G

ap (e

V)

Exp

γ

HSEMgO

4.0

5.0

B HSE

LSDA

AlN

MgO

NaCl

3.0

4.0

4 0 5 0 6 0 7 0 8 0

CNaCl

AlN

4.0 5.0 6.0 7.0 8.0

Experimental Band gap (eV)

H. Eisenberg, R. Baer, work in progress (2008).

Dogmatic spirit Pragmatic spirit spirit

Hybrid (B3LYP/RSH) theory

γ system depende

nt

Ab initio ways to

determine γ

“Systems too difficult for DFT”

In B3LYP: In RSH: γγ

insensitive to density (factor 3)

γvery

sensitive (factor ~100)

Symmetric cation

radicals

Charge-transfer

excitationsBand gaps