Orbital energies in DFTOrbital energies in DFTOrbital energies in DFTOrbital energies in DFT

Takao TsunedaTakao Tsuneda(F l C ll(F l C ll N t i lN t i l C tC t(Fuel Cell (Fuel Cell NanomaterialsNanomaterials Center, Center, 

Univ Yamanashi Japan)Univ Yamanashi Japan)Univ. Yamanashi, Japan)Univ. Yamanashi, Japan)Sep. 26, 2011Sep. 26, 2011

DCEN2011@Kyoto Univ.DCEN2011@Kyoto Univ.

ContentsContents1. DFT in Quantum Chemistry

Present status of DFT in Quantum ChemistryKohn‐Sham equation & its problems

2. Long‐range corrected (LC) DFTProblems that LC‐DFT has solvedLC‐DFT & its derivativesChemical reaction calculations

d l l lVan der Waals calculationsNonlinear optical property calculationsExcited state calculations

3. Orbital energiesMolecular orbitalsOrbital energiesOrbital energiesOrbital energy calculations by LC‐DFTWhy LC‐DFT reproduces orbital energies?Chemical reactions vs orbital energiesChemical reactions vs orbital energies

1. DFT in Quantum Chemistry1. DFT in Quantum Chemistry

DFT in Quantum ChemistryDFT in Quantum Chemistry

5000ers

In the field of QC, DFT studies are hardly found in the beginning of 1990’s but rapidly increases to become the main stream at present.

3000 3500 4000 4500 5000

emistry pa

pe TotalDFTHF and Post‐HF

Number of related papers published in electric journals of chemistry

1000 1500 2000 2500

uantum

 che

DFT‐related papers pull up the number of QC papers.

0 500

9192939495969798990001020304050607080910

# of q

Year 100 

The percentage of DFT‐related papers in QC papers

60 

80 

DFT stud

ies

At 2010, DFT‐related papers are more than 80% of QC papers, and 

20 

40 % of D

WorldJapan

are supposed to be more than 90% as far as applications go.

0 9192939495969798990001020304050607080910

Year

DFT & its problemsDFT & its problemsKohn‐Sham equation

ˆˆ ˆ2 [ ]xcE

F h J

Orbital (band) energy2 [ ]

2 [ ]

xc

xc

i i i iF h J

E h J E

Orbital (band)

Walter KohnNovel prize winner

The exchange‐correlation energy is expressed as a functional of density Exc[].This simple method reproduces accurate properties with Novel prize winner

for chemistry in 1998s s p e et od ep oduces accu ate p ope t es t

much less computational time than conventional ones.

Significant problems that DFTs have faced in QC calculations

Large systems Photochemical reactions High‐spin systems

VdW bonds Charge transfersT iti t

Magnetic propertiesOptical properties 

of long‐chain molecules

f

Transition momentsNonadiabatic &Spin‐forbiddentransitions

propertiesSpin‐orbitsplittingsDiradicalsBand gaps of semi‐

conductors &insulators

transitionsState crossingsBond dissociations

Diradicals

2. Long2. Long‐‐range corrected (LC) DFTrange corrected (LC) DFT

DFT problems which LC has solvedDFT problems which LC has solved

Hydrogen bond complexes

Dipole‐dipole complexes

Excited state calculations

Charge transfersRydberg

excitations

Excited state calculationsVan der Waals calculations

Long‐range correctioncomplexes complexes excitations

H

H

C C

H

H

F

F

C C

F

F

correction (LC) scheme

Stacking complexes Photochemistries of large systemsPhotocatalysisPhotosynthesis

LC scheme has solved or improved 

serious DFTDispersion complexes

Dipole‐induced dipole complexes

serious DFT problems in various chemical property 

calculations.

Optical property calculations

Orbital energy calculations

Solid state calculations

Cluster calculations

Chemical reaction calculations

Long‐chain polyenes

Water cluster anions

Band gaps of insulators

Reaction barrier heightsOrbital energies

The status of LCThe status of LC‐‐DFT in ScienceDFT in Science

Citations of LC‐related papers (TT)Brain & Heart & blood vessel diseases

Core paper density

>60

1270 citations/21 papers (ISI Web of Science, Sep. 10, 2011)

p p ( )neurological 

scienceInfection & Immunity

i

Obesity 0

Scientific impact of LC‐related papersNanosciencePost‐genome

Regenerative medicine

According to “ScienceMap 2008” (Ministry of Education, Culture, Sports, Science and Technology, Japan, 2010), LC l d f hChemistry

Plant science

LC‐related papers form a new research area of highest 1% citations in science. 

LC DFT is a ailable on the official ersion

Chemistry

Solid physics

LC‐DFT is available on the official version of Gaussian09 & GAMESS programs.

Environmental Science

Elementary particle & cosmological physics

LongLong‐‐range correction (LC) schemerange correction (LC) schemeThe correction for the long‐range part of general exchange functionals.

[H. Iikura, T. Tsuneda, T. Yanai and K. Hirao, J. Chem. Phys. 115, 3540, 2001; A. Savin, in ‘Recent Developments and Applications of Modern Density Functional Theory’ (Elsevier, 1996)]Developments and Applications of Modern Density Functional Theory  (Elsevier, 1996)]

1 f( ) f( )1

General exchange functionals for the short‐range part

i h l

Hartree‐Fock exchange intergral 

12 12

12 12 12

1 erf( ) erf( )1

r rr r r

is the only parameter

for the long‐range part

CAM i [T Y i D P T & N C H d Ch Ph L 91 551 2004 ]Functionals base on LC scheme

CAM‐series [T. Yanai, D.P. Tew, & N.C. Handy, Chem. Phys. Lett. 91, 551, 2004.]LC‐PBE [O. A. Vydrov, J. Heyd, A. V. Krukau, & G. E. Scuseria, J. Chem. Phys. 125, 074206, 2006.]LCgau‐series [J.‐W. Song et al., J. Chem. Phys. 127, 154109, 2007.]MCY series [A J Cohen P Mori Sanchez & W Yang J Chem Phys 126 191109 2007 ]MCY‐series [A. J. Cohen, P. Mori‐Sanchez, & W. Yang, J. Chem. Phys. 126, 191109, 2007.]BNL [E. Livshits & R. Baer, Phys. Chem. Chem. Phys., 9, 2937, 2007.]B97‐series [J.‐D. Chai & M. Head‐Gordon, J. Chem. Phys. 128, 084106, 2008.]

Chemical reactions of small moleculesChemical reactions of small molecules

Figs. Mean absolute deviations in calculated reaction barriers & reaction enthalpies for Reaction barrier

Transition state

10

rrie

r ol

) Hydrogen transfer reactionsN h d t f ti

reaction barriers & reaction enthalpies for Truhlar’s benchmark set (78 chemical reactions).

Reaction barrier

Reaction enthalpy

Reactant Product

6

8

eact

ion

bar

s（kc

al/m

o Nonhydrogen-transfer reactions

d

Reaction barrier energies

Reactant Product

2

4

MA

E of

reen

ergi

e [J. Song, T. Hirosawa, T. Tsuneda, and K. Hirao, J. Chem. Phys. 126, 154105, 2007; O. A. Vydrov and G. E. Scuseria, J. Chem. Phys.

0BOP PBE PBE0 B3LYP M05-2X LC-BOPLC-ωPBE

G. E. Scuseria, J. Chem. Phys. 125, 234109, 2006.]

8on Hydrogen transfer reactions

4

6

8

E of

reac

tioen

thal

pies

(k

cal/m

ol) Nonhydrogen transfer reactions

LC scheme drastically improves ti b i ith l

Reaction enthalpies

0

2

BOP B3LYP LC-BOP

MA

E e (reaction barriers with also modifying reaction enthalpies.

Van Van derder Waals bondsWaals bonds

Combining LC‐DFT with a dispersion functional ALL was applied to the calculations of van der Waals bonds.

[M. Kamiya, T. Tsuneda and K. Hirao, J. Chem. Phys. 117, 6010, 2002.]

0.75 SOPBOP

0.75 LC-BOP+ALL

0 30

0.45

0.60

mol

)

BOPPBEOPLC-SOPLC-BOPLC-PBEOPLC BLYP 0 30

0.45

0.60

ol)

LC BOP ALLmPWPW91mPW1PW91B3LYP+vdW

0.00

0.15

0.30

rgy

(kca

l/m LC-BLYPExact

0.00

0.15

0.30

gy (k

cal/m

B3LYP vdWMP2Exact.

0 45

-0.30

-0.15

Bond

Ene

-0.30

-0.15

Bon

d En

erg

ArAr

-0.75

-0.60

-0.45

-0 75

-0.60

-0.45B Ar2Ar2

3.0 4.0 5.0 6.0

Bond distance (Å)

0.753.0 4.0 5.0 6.0

Bond distance (Å)

[T. Sato, T. Tsuneda, and K. Hirao, JCP,

126, 234114, 2007.][T Sato T Tsuneda and K Hirao J Chem Phys 126

Van Van derder Waals and weak hydrogen bondsWaals and weak hydrogen bonds[T. Sato, T. Tsuneda and K. Hirao, J. Chem. Phys. 126, 234114, 2007.]

By combining with van der Waals correlation, LC DFT i t

Dispersioncomplexes

LC-DFT gives accurate bonding energies equivalently for weak bonds.

Stackingcomplexes

bonds.

Dipole-induceddipole complexes

Dipole-dipolecomplexes

MP2 considerably overestimates stacking energies due to the

d

p p

complexes near-degeneracy.

Hydrogen-bonded

Fig. Percent errors in binding energies.

complexes

Optical response properties of longOptical response properties of long‐‐chain chain polyenespolyenes

H[M. Kamiya, H. Sekino, T. Tsuneda, and K. Hirao, 

Longitudinal polarizabilityzz / unit cell

O2NNH2

H n

, ,J. Chem. Phys. 122, 234111, 2005.]

‐nitro,amino‐polyacetylene

zzn Longitudinal 

hyperpolarizability zzz/unit cell

zzn

n

# of unit cells n

z

zzz/n

LC scheme solve the overestimations of 

Longitudinal dipole moment /unit cell

optical properties in conventional DFTs.

# of unit cells n# of unit cells n

Second Second hyperpolarizabilitieshyperpolarizabilities of of diradicalsdiradicals[R. Kishi, S. Bonness, K. Yoneda, H. Takahashi, M. Nakano, E. Botek, B. Champagne, T. Kubo, K. Kamada, K. Ohta, and T. Tsuneda, J. Chem. Phys. 132, 094107, 2010.]

Fig. Diradical character dependence of second hyperpolarizability

LC‐UBLYP quantitatively reproduces second hyperpolarizabilities of diradicals.

hyperpolarizability

Table. Second hyperpolarizabilities of 1,4‐bis‐imidazol‐2 ylidenecyclohexa 2 5 diene (BI2Y) (x102 a u )

Method 6‐31G 6‐31G*+p

UHF 1736 2002

2‐ylidenecyclohexa‐2,5‐diene (BI2Y) (x102 a.u.).

UHF 1736 2002

UMP2 9387 9962

UCCSD 4474 ‐

21 HOMO LUMOn nTy T

UCCSD(T) 5244 ‐

UBLYP ‐129 ‐29821 , 21

y TT

Closed shell→y=0, Diradical→y=1

UB3LYP ‐377 ‐472

LC‐UBLYP 4310 6019

Electronic spectra in TDDFTElectronic spectra in TDDFTRydberg excitation energies Valence & Rydberg excitation energies

Oscillator strengths

100

150 LC-TDBOPBOPB3LYP

SAC‐CI results are set as 100%. LC scheme drastically improves Rydberg excitation energies & oscillator strengths

Oscillator strengths

50

100%

[Y Tawada T Tsuneda S Yanagisawa T Yanai and

excitation energies & oscillator strengths underestimated in time‐dependent DFT.

0

1Pu 1P 1A1 1B3u 1E1uN2 CO H2CO C2H4 C6H6

1u 1u+ 1 1B2 1A1 1B21B3u 1B1u 1E1u

[Y. Tawada, T. Tsuneda, S. Yanagisawa, T. Yanai and K. Hirao, J. Chem. Phys. 120, 8425, 2004.] 

Charge transfer excitations in TDDFTCharge transfer excitations in TDDFT2.0

LC-BOP

BOP

H

C C

HF

C C

F

1.5

AC-BOP

B3LYP

HF

SAC CI

CT(

5.0Å

) (eV

) H HF F

4

4.5

(eV

)CT energies

1.0

SAC-CI

-1/R

C

T(R

) -

C

2.5

3

3.5

atio

n E

nerg

y (

CT (ZnBC→BC)Q (ZnBC)Q (BC)Q (ZnBC)Q (BC)

LC‐TDBOPFig. CT energies of ethylene‐tetrafluoroethylene.

0.5

1.5

2

5.5 6 6.5 7 7.5 8

Exc

ita

Fig. The lowest excitation energies of Zn‐bacteriochlorin‐

0.0

5 6 7 8 9 10

Intramolecule distance R (Å)

3 5

4

4.5

gy (

eV) Q (BC)

Q (BC)Q (ZnBC)

Distance (Å)

TDB3LYP

Zn bacteriochlorinbacteriochlorin.

Dissociation limit

Exp. LC‐TDBOP TDB3LYP SAC‐CI

ECT (R=∞) 12.5 12.43 7.42 14.43 2

2.5

3

3.5

cita

tion

Ene

rg

( )CT (BC→ZnBC)CT (ZnBC→BC)

(R=∞)1.5

5.5 6 6.5 7 7.5 8

Dis tance (Å)

Exc

[Y. Tawada, T. Tsuneda, S. Yanagisawa, T. Yanai and K. Hirao, J. Chem. Phys. 120, 8425, 2004.] 

3. Orbital energies3. Orbital energies

Molecular Molecular orbitalsorbitals

MOs are expressed as a linear

Molecular orbitals

MOs are expressed as a linear combination of atomic orbitals.

MOs have been taken as unobservable (unphysical) due to the free orbital rotation (unitary transformation) in HF calculations.

Molecular orbital imagingBy experimental

MO imaging of N2 molecule.(a) experimental, (b) QC calculation. [Itatani et al., 

By experimental observations, MOs take increasing attentions in the field of QC. [ ,

Nature, 432, 867, 2004.]Q

Reaction analyses on molecular Reaction analyses on molecular orbitalsorbitals

Substitution reactions of aromatic compounds :

Electronic distribution of frontier orbitals of nitrobenzene.Frontier orbital theory (1952)

Substitution reactions of aromatic compounds :(1) Electrophilic reactions target HOMO density.(2) Nucleophilic reactions target LUMO density.(3) Radical reactions proceed as the sum of (1) & (2).(3) Radical reactions proceed as the sum of (1) & (2).

K. FukuiCyclization(Ring‐closing)

HOMO LUMO

Woodward‐Hoffmann rule (1965) ( g g)reactions

Con-rotatory

Dis-rotatory

R. B. Woodward R. Hoffmann

W d d & H ff d d t

rotatory rotatory

Woodward‐Hoffmann ruleWoodward & Hoffmann succeeded to explain experimental results by MO symmetry.

MOs are usually used in experiments to analyze and design chemical reactions.

Reaction analyses on orbital energiesReaction analyses on orbital energies

Koopmans’ theorem

Without SCF (orbital rotation), Hartree‐Fock calculations give occupied orbital energies as minus corresponding IP & unoccupied orbital energies as minus corresponding EA.g p g

No conventional theory including DFT satisfies this theorem.

Maximum hardness rule

Fukui function: HOMO & LUMO densities

Fukui functions of p‐Cl‐benzonitrile oxide.[A. Ponti & G. Molteni, Chem. Eur. J., 1156, 2006.]

A reaction index using orbital energies.As HOMO‐LUMO gaps increasing, molecules become stabilized.

[A. Ponti & G. Molteni, Chem. Eur. J., 1156, 2006.]

E 11 2 LUMO HOMO

⇒Reactions proceed to increase HOMO‐LUMO gaps.Global hardness:

NN 22 2 2

Conditions for reproducing orbital energiesConditions for reproducing orbital energies

Many studies have been reported to identify the cause why orbital energy gaps are usually underestimated especially on the context of band gapsusually underestimated especially on the context of band gaps.

Kohn‐Sham equation: ˆ ˆ xc i i ih J v Poor orbital energy gaps are originated from the discontinuity of exchange‐correlation potentials [Perdew, Parr, Levy, & Balduz, Phys. Rev. Lett. 49, 1691, 1982.]

xc i i i

p [ , , y, , y , , ]

0xc xc xc const.n n n nv v 1( ) ( )n nn n xc IP EA

⇒ For xc=0, HOMO & LUMO energies correspond to minus IP & EA, respectively.[Perdew & Levy, Phys. Rev. B 56, 1602, 1997; Casida, Phys. Rev. B 59, 4694, 1999.]

xc is attributed to the energy deviation of outermost orbital energies.[Sham & Schlüter, Phys. Rev. B 32, 3883, 1985.]

1 1( 1) ( )n n Orbital energies are not varied for outermost orbitals.

1 1( 1) ( )xc n nn n

Koopmans’ Ionization PotentialsKoopmans’ Ionization PotentialsFig. Deviations of HOMO energies from the minus vertical IPs (HOMO + ) i h Q b i

HF SiH4CH IP) with aug‐cc‐pVQZ basis set.

LC‐DFT  calculations give HOMO close t IP t f H H & N t

BOPB3LYPM05-2XLCgau-BOP

CH4PH3NH3HCOOHH2S to IP except for H, He & Ne atoms.

The HOMO of other functionals are significantly lower than IP

LCgau-BOPLC-BOP

H2SH2OCH2OC2H4C2H2 significantly lower than IP. 2 2CO2ClFCSCO

M th d MAE

LC‐DFT approximately satisfies Koopmans’ 

HClHFCl2P2F

Method MAE

LC‐BOP 0.266

LCgau 0.332

theorem.

The HOMO of HF are F2N2NeHeH

LCgau 0.332

M05‐2X 1.600

B3LYP 3.087

larger than IP.⇒ HF does not obey Koopmans’ theorem h th bit l

-10.0 -9.0 -8.0 -7.0 -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 HOMO IP (eV)

HBOP 4.580

HF 1.823

when the orbital relaxation is not neglected.

Koopmans’ Electron AffinitiesKoopmans’ Electron AffinitiesFig. Deviations of LUMO energies from the minus vertical EAs (LUMO + EA) with 

Q b i

HFBOP

SiH4CH4PH aug‐cc‐pVQZ basis set. 

The LUMO of LC‐DFT calculations are l t EA f ll t

B3LYPM05-2XLCgau-BOPLC BOP

PH3NH3HCOOHH2SH O very close to EA for all systems.

The LUMO of other functionals are considerably lower than EA

LC-BOP H2OCH2OC2H4C2H2CO2 considerably lower than EA.

LC functionals satisfyKoopmans’ theorem

CO2ClFCSCOHCl Method MAEKoopmans   theorem 

even for LUMO.HFCl2P2F2

Method MAE

LC‐BOP 0.142

LCgau 0.115N2NeHeH

M05‐2X 0.618

B3LYP 1.370

-6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0

LUMO + EA (eV)

BOP 2.116

HF 0.287

HOMOHOMO‐‐LUMOLUMO gapsgapsFig. Deviations of HOMO‐LUMO gaps from the IP – EA values given by CCSD(T)/CCSD/aug cc pVQZ

SiH4CH4PH3

HFBOP CCSD(T)/CCSD/aug‐cc‐pVQZ.

The gaps of LC functionals are very close to IP – EA of CCSD(T) except for

3NH3HCOOHH2SH2O

B3LYPM05-2XLCgau-BOP

close to IP EA of CCSD(T) except for H, He & Ne atoms.

For other functionals,

CH2OC2H4C2H2CO2ClF

LC-BOP

For other functionals,the HOMO‐LUMO gaps are significantly underestimated for most molecules.

ClFCSCOHClHF Method MAE

LC functionalsquantitatively 

HFCl2P2F2N2

Method MAE

LC‐BOP 0.282

LCgau 0.447reproduces orbital energies except for H, He & Ne atoms.

2NeHeH

M05‐2X 0.997

B3LYP 1.4398 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 0 0 1 0 2 0 BOP 2.514

HF 0.727

-8.0 -7.0 -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 (LUMO – HOMO) – (IP – EA)CCSD(T) (eV)

Fractional occupations to Fractional occupations to cationcation

10

12

ns fr

om

(eV)

HFBOPB3LYPM05 2X

Fig. Total energies of C2H4 calculated for varying occupation numbers of HOMO

Energies of fractionally‐occupied systems

6

8

gy v

aria

tion

ral e

nerg

y M05-2XLC-BOPLCgau-BOP

varying occupation numbers of HOMO to C2H4+ cation.

Only LC functionals give total energies HOMOLC-BOP

2

4

Tota

l ene

rgth

e ne

ut

Only LC functionals give total energies very close to a straight line, while other functionals give curved lines. 

IP=10.6eVHOMO=10.2eV

0-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

T

Shift of occupation numbers n

Table. Orbital energies of C2H4 & C2H4+ compared to the total energy gradients.

C2H4+ C2H4MethodC2H4 C2H4

LUMO ∂E/∂n(1) ∂E/∂n(0) HOMO

LC-BOP 10.22 10.17 10.71 10.68All functionals nearly satisfy Janak’s theorem:

i iE n

Janak’s theorem

B3LYP 13.48 13.46 7.59 7.60BOP 14.40 14.39 6.43 6.45HF 7 71 7 61 10 11 10 22

To make ∂E/∂n(0) identical to –IP, a straight line should be given for the energy.

f h l h ld b id i l HF 7.71 7.61 10.11 10.22M05-2X 11.92 11.67 9.17 9.16LCgau 10.37 10.33 10.53 10.56

HOMO of the neutrals should be identical to LUMO of the cations.= Sham‐Schlüter’s argument

Fractional occupations to anionFractional occupations to anion

3

3.5

om th

e )

HFBOPB3LYP

Fig. Total energies of C2H4 calculated for varying occupations of HOMO to C2H4 anion.

1

2

2.5

aria

tions

fren

ergy

(eV) M05-2X

LC-BOPLCgau-BOP

2 4

For anions, similar curves are given.

LC functionals always give ∂2E/∂n2

0 5

1

1.5

ene

rgy

vane

utra

l e

EA=2.3eV

LUMOLC-BOP=2.6eV

LC functionals always give ∂2E/∂n2close to zero.

Table. Orbital energies of C2H4 & C2H4

0 5

0

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Tota

l

C2H4 C2H4

2 4 2 4compared to the total energy gradients.

-0.5Shift of occupation numbers n Method

C2H4 C2H4LUMO ∂E/∂n(0) ∂E/∂n(1) HOMO

LC-BOP 2.60 2.62 1.97 1.94This also backs up that LC functionals satisfy Koopmans’ theorem approximately.

B3LYP 0.22 0.20 4.69 4.66BOP 0.78 0.72 5.74 5.68HF 3 56 3 95 2 23 2 11

y p pp y

Hybrid B3LYP & pure GGA BOP provide negative orbital energies for C2H4. HF 3.56 3.95 2.23 2.11

M05-2X 1.12 1.15 3.49 3.47LCgau 2.54 2.56 1.97 2.10

LUMO of neutrals should also be identical to HOMO of the anions.

Orbital energies for fractional occupationsOrbital energies for fractional occupations

Fig. Orbital energies of C2H4 calculated for varying occupation numbers of HOMO to C2H4 for -1 < n < 0 (left) and for 0 < n < 1 (right).

-8

-7

-6

/mol

)

4

5

6

l/mol

)

-11

-10

-9

rgie

s (k

cal

2

3

4

ergi

es (k

ca

-13

-12

-11

Orb

ital e

ner

HFBOPB3LYPM05-2xLC BOP

0

1

2

Orb

ital e

ne HFBOPB3LYPM05-2xLC BOP

-15

-14

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

O

Shift of occupation number n

LC-BOPLCgau-BOP -1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

OShift of occupation number n

LC-BOPLCgau-BOP

LC functionals give almost constant orbital energies for fractional occupations.Other functionals provide positive gradients raising  as increasing electrons.

Shift of occupation number n p

p p g g gHF gives negative gradients lowering  due to the lack of electron correlation.

Orbital energy derivativesOrbital energy derivatives

The derivative of the orbital energy in terms of the fractional occupation: 

Orbital energy derivative

* * 3 31( ) ( ') ( ) ( ') ''

xcii i i i

i

vd d

n

r r r r r rr r

Only the discrepancy in the self‐interaction through the sum of Coulomb and exchange‐correlation potential derivative dvxc/d directly affects the dependence of b l f lorbital energies on fractional occupations.

Coulomb self‐interaction potential

A normalized Gaussian‐type function:

The Coulomb self interaction potential approximated:

2( ) expii i r r

The Coulomb self‐interaction potential approximated: 2 212 12 3

1212

exp 2 2 cosˆSI ii

r r rrJ d

r

r

This form gives the accurate Coulomb energy for H atom.

1 2 1 212 12 12 120exp 2 exp 2 sinh 4i i ir r r r rr dr

SelfSelf‐‐interaction energiesinteraction energiesDistance from the nuclear

Exchange self‐interaction energy

Fig. The integral kernels of the exchange functional SI energies of HOMO of H atom.

The exchange functional SI energies are cancelled out with the Coulomb SI energy for all functionals but LDA

Distance from the nuclear

energy for all functionals but LDA.

Fi Th i t l k l f th h SI

Exchange self‐interaction energy through potential derivative

Fig. The integral kernels of the exchange SI energies through the  dvx/d of HOMO of H atom.

Only LC‐DFT gives a reasonable exchange SI energy even through dvx/d .

Why orbital energy Why orbital energy reproduciblitiesreproduciblities are different?are different?

Poorly‐given HOMO or He atomDistance from the nuclear

Fig. The integral kernels of the exchange SI energies through the dvx/d for the HOMO of He atom.

Even LC‐BOP slightly underestimates the exchange SI energy through dvx/d

Accurately‐given LUMO or He atomDistance from the nuclear

for the HOMO of He atom. 

Distance from the nuclearFig. The integral kernels of the exchange SI energies through the dvx/d of the LUMO of He atom.x

LUMO = ‐ EA= 2.653 eV

Only for LC‐BOP, the SI error is cancelled out even for the exchange SI through the potential derivative.

ConclusionsConclusions

L d (LC) DFTLong‐range corrected (LC) DFT

Applicabilities of LC‐DFT are reviewed. LC DFT has solved or improved various chemical properties that conventional DFTsLC‐DFT has solved or improved various chemical properties that conventional DFTs have poorly given: van der Waals bonds, charge transfers, optical properties, reaction barriers and enthalpies, and so on.

Orbital energies

LC‐DFT also enables us to calculate orbital energies quantitatively for the first timeLC DFT also enables us to calculate orbital energies quantitatively for the first time.To reproduce orbital energies quantitatively by DFT, self‐interaction energies through the potential derivatives should be excluded in exchange functionals. LC may be the best strategy to remove the self‐interaction energies.LC may be the best strategy to remove the self interaction energies.LC‐DFT suggests a new reaction analysis based on orbital energies.

AcknowledgmentsAcknowledgmentsgg

Prof. Kimihiko Hirao (RIKEN, Japan)

Long‐range corrected (LC) DFT

Hisayoshi IikuraDr. Muneaki Kamiya (Gifu Univ., Japan)Yoshihiro TawadaD T k hi S t (U i T k J )Dr. Takeshi Sato (Univ. Tokyo, Japan)

Orbital energies

Dr. Jong‐Won Song (RIKEN, Japan)Dr. Raman Kumar Singh (Yamanashi Univ.)Satoshi Suzuki (Kyushu Univ )Satoshi Suzuki (Kyushu Univ.)

Financial Support

Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT)