時間依存密度汎関数法 - 理化学研究所®Ÿ際のTDDFT計算における定式 時間依存Kohn-Sham方程式 →エルミート性がないので実数固有値が保証されない

5.1. 5.1.

RungeRunge--GrossGross

v(r,t) v(r,t)vstatvpert

Hohenberg-KohnTaylor

v(r,t)(r,t)Schrdingerv

[](r,t) Hohenberg-Kohn

A[]

A[] 1i(r,t)Schrdinger

1veff

Axc

v(r,t) v(r,t)vstatvpert

Hohenberg-KohnTaylor

v(r,t)(r,t)Schrdingerv

[](r,t) Hohenberg-Kohn

A[]

A[] 1i(r,t)Schrdinger

1veff

Axc

Runge-Gross [Runge and Gross, Phys. Rev. Lett., 52, 997, 1984.]Runge-Gross [Runge and Gross, Phys. Rev. Lett., 52, 997, 1984.]

)()(1

0

tHt

itdtAt

t

=

=1

0

1

0

),(),()()(][ 3t

t

t

ttvtddttH

titdtA rrr

),()],(;,[),(21 2 tttvt

ti ieffi rrrr =

+

),(][

'),'(),()],(;,[ 3

tAtdtvttv xcseff rrr

rrrrr

+

+= ),(][

),(][

tE

tA xcxc

rr

KohnKohn--ShamSham

RGv(r,t)=vstat(r)+v(r,t) Fouriert

K

Kohn-Sham

RGv(r,t)=vstat(r)+v(r,t) Fouriert

K

Kohn-Sham

Runge-GrossRunge-Gross

= ji jiji P, * )()()(),( rrr = i ii tft

2),(),( rr

+

=

=

klklklijij

ji

ijij

ji

ijjjii

jsi

ij

PKvff

P

ffv

P

)()()(

)(

)()(

,0

',',',''

)'()'()'()(

)()(')'()'('

1)()(' *2

*33**33, rrrr

rrrrrrrr

rrrr

lkxcjilkjikl

ijklij

EddddPv

K +=

=

{ }[ ]{ }[ ]

=

=+===

=++

=+++

00

YX

*L*MML

r

r

1001

,)((0),(),(

),()(

),()(

,,,,,,,

0

0,,,,,

0,,,,,

lkijklijlkijijljkiklij

jijiijijij

jikllkjilklkjijiljki

ijlkklijlkklijjiljki

KMKLvPYPX

vPKPK

vPKPK

)

Kohn-Sham

TDDFTTDDFT

Kohn-Sham

iLM

Mp-Lp

Kohn-Sham

iLM

Mp-Lp

Kohn-Sham[R. Bauernschmitt and R. Ahlrichs, CPL, 256, 454, 1996.]

Kohn-Sham[R. Bauernschmitt and R. Ahlrichs, CPL, 256, 454, 1996.]

=

00

YX

*L*MML

1001

( )( )( ) ( )YXYXLMLM +=++ 2pppp

( ) ( )( ) ( ) ( )( ) ( ) ( )YXLMYX

YXYXLMLMLM

+=+

+=++2/1

22/12/1

'

''pp

pppppp

''3 ,),,,,( == rfdExc

TDDFTTDDFT

TDDFTTDDFT

5.2. 5.2.

ab initio

valence

1

crossingavoided crossing

Rydberg

TDDFTTDDFT

DreuwHead-Gordon

CT

Q

Dreuw & Head-Gordon,

JACS, 126, 4007, 2004.Dreuw, Weisman & Head-Gordon,

JCP, 119, 2943, 2003.

TDDFT

LCLC--TDDFTTDDFT

[Y. Tawada, T. Tsuneda, S. Yanagisawa, T. Yanai and K. Hirao,

JCP, 120, 8425, 2004.]

TDDFT

K:

longjbiabj

shortxc

ai

bjaijbia

KddEE

ddr

K

,33

2**

33

12

**,

)()()()()()()(

)()(1)()(

++

+

=

212221

11

212211

rrrrrr

rr

rrrrrr

( ) ( ) ( ) ( ) ( ) 23132112

122

*1

*, 2

1 rrrrrr ddr

rerfK biajlong

jbia =

=

+=

=

bjiajbia

jbiaiaabijjbia

KB

KA

,,

,, )(

YX

1001

YX

ABBA

TDDFTTDDFTRydbergRydberg

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

0

0.2

0.4

0.6

0.8

1

1.2

1.4

N2 CO H2CO C2H4 C6H6

Molecules

Mean a

bso

lute

devia

tions (

eV

)

LC-BOP

BOP

B3LYP

SAC-CI

0

0.5

1

1.5

2

2.5

N2 CO H2CO C2H4 C6H6Molecules

Mean a

bso

lute

devia

tions (

eV

)

LC-BOP

BOP

B3LYP

SAC-CI

Rydberg

120,88,86,90.0,95.370.3941.2949.7158.091E1uC6H6

29.037.3127.5512.8533.551B1u

4.08.075.653.497.421B3uC2H4

2.81,1.7,1.92.952.641.753.751B2

6.05,3.2,3.8,3.64.261.652.113.961A1

4.13,2.8,3.8,3.21.882.241.681.171B2H2CO

17.69.638.478.668.961CO

27.915.673.970.6915.991u+24.38.141.380.287.611u

N2

Exp.SAC-CIB3LYPBOPLC-BOPStateMolecule

0.0

0.5

1.0

1.5

2.0

5 6 7 8 9 10

LC-BOP

BOP

AC-BOP

B3LYP

HF

SAC-CI

-1/R

C

T(R

) -

CT(

5.0

) (eV

)

Intramolecule distance R ()

TDDFTTDDFT

H

H

C C

H

H

F

F

C C

F

F

12.5Exp.

14.437.4212.43ECT (R=)SAC-CIB3LYPLC-BOP

1.5

2

2.5

3

3.5

4

4.5

5.5 6 6.5 7 7.5 8

Distance ()

Exc

itatio

n E

nerg

y (e

V) Q (BC)

Q (BC)Q (ZnBC)CT (BCZnBC)CT (ZnBCBC)

1.5

2

2.5

3

3.5

4

4.5

5.5 6 6.5 7 7.5 8

Distance ()

Exc

itatio

n E

nerg

y (e

V)

CT (ZnBCBC)Q (ZnBC)Q (BC)Q (ZnBC)Q (BC)

LC-BOP

B3LYP

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

TDDFTTDDFT

TDDFTLC

MD.[M. Chiba, T. Tsuneda, and K. Hirao, JCP, in press.]

TDDFTLC

MD.[M. Chiba, T. Tsuneda, and K. Hirao, JCP, in press.]

TDDFT[Furche & Ahlrichs, JCP, 117, 7433, 2002.]

++=

PVWSPh xc )(

+++'

')(

''

' )()()|(

YXYXf xc

GAMESSLC-TDDFT

GAMESSLC-TDDFT

-0.50

0.51

1.52

2.53

3.54

4.55

5.56

0 30 60 90twisted angle (degree)

Exc

itatio

n E

nerg

ies

(eV

)

CTLEGround

-0.50

0.51

1.52

2.53

3.54

4.55

5.56

0 30 60 90

twisted angle /degree

Ex

cit

ati

on

En

erg

ies

/e

V

CT

LE

Ground

-0.50

0.51

1.52

2.53

3.54

4.55

5.56

0 30 60 90twisted angle (degree)

Exc

itati

on

En

erg

ies (

eV

)

CTLEGround

DMABNDMABN

4-N,N-dimethylaminobenzonitrile (DMABN)TICT

hCN N CN N +

:LE

LC-BOPLEPES

: TDDFT: cc-pVTZ: PCM

LC-TDBOP

TDB3LYPTDBOP

0

1

2

3

4

5

6

0 30 60 90twisted angle (degree)

Exc

ited

Ene

rgie

s (e

V)

LCBOPBOPB3LYPRef (MRMP)

LC-BOP MRMP CT.

CT LEPESCT

0

1

2

3

4

5

6

7

0 10 20 30 40 50 60 70 80 90

Twisted angle/degree

Energ

ies

(eV

)

LE

CT

Ground

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60 70 80 90

Twisted angle/degree

Osc

illat

or st

reng

th CT

LE

DMABNDMABN

DMABNCT

LC-TDBOPDMABNPES

LE (eV)CT (eV)ABS (eV)

3.862.204.49PCM-TDB3LYPExp.

PCM-LC-TDBOP

4.25

4.76

3.44?2.66

4.172.87

CTCTLE

[Zachariasse et al., CPL, 372, 1997.]

(PCM)DFTLE

CT

KohnKohn--ShamSham

Kohn-Sham

v0p

U

Kohn-Sham

v0p

U

)'()'()'()(

)()(')'()'('

1)()(' *2

*33**33, rrrr

rrrrrrrr

rrrr

lkxcjilkjiklij

EddddK +=

{ }[ ]{ }[ ] ),()(

),()(

0,,,,,

0,,,,,

r

r

jikllkjilklkjijiljki

ijlkklijlkklijjiljki

vPKPK

vPKPK

=++

=++

pHU =

[ ] UpTr4=

[ ] [ ])"()'()(

)"()'()()"()'()("'64

Tr2Tr2

,,

3333 rrr

rrrrrrrrr

GGG

kjickbjai

xccbackbjai

EdddUUU

++++++=

UUUU +=+= ,,ai

aiaipqpqpq UHpG

TDDFTTDDFT

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

JCP,122, 234111, 2005.]

/unit cell

,-nitro,amino-polyacetylene

z

zz/n

O2NNH2

H

H n

/unit cell

zz/n

zzz

/n

/unit cell