VIVI 0£'ì#Õ è&É ÛFþ ö&O&O - Kobe University1985 - 1995 GXG\GlGTGxG0GyFø ì Û '¨1 N#.MD 2H DFT ×6õ XH ¨ g!· 2H MDG0G GCG GdGy GcG5GnGV&ÊFúFùGzG GDG &É ÛH 9× 2x

VIIVI

201919 11111 6620119 1111 6

Georg Wilhelm Friedrich Hegel

Georg Simon Ohm

→

Albert Einstein Richard Feynman

1926 – 1937 HF VB

1938 – 1949

1950 - 1960 MDDNA

1961 - 1968 CCGVB RNA

1969 - 1984 DFT MDGaussian

1985 - 1995 1 MD DFTMD C60

1996 – 2005 Order-N QM/MMDFT ES

2006– AIiPS

DFT

85% DFT

90 DFT

ISI Web of Science

DFT

0

200

400

600

800

1000

1200

94 96 98 00 02 04 06 08 10 12 14 16 18

# of

DFT

stud

ies i

n bi

olog

y

Year

0

1000

2000

3000

4000

5000

6000

91 93 95 97 99 01 03 05 07 09 11 13 15 17

# of

qua

ntum

chem

istry

pap

ers

Year

TotalDFTHF and Post-HF

DFT

SCF

21 1 1 1 eff 1 1 1 1 1 1

1

22 1 2 2 eff 2 2 2 2 2 2

2

1 2ˆ( ) ( ) ( ) ( ) ( )2

1 2ˆ( ) ( ) ( ) ( ) ( )2

A

A

h Vr

h Vr

r r r r r

r r r r r

1 1 2 2( ) ( )r r

D. R. Hartree

1 2ˆ ˆˆ ( ) ( )H h hr r

3

31

21

2

3

H2

2 21 2

1 1 2 2 12

1 1 1 1 1 1 1 1ˆ2 2 A B A B

Hr r r r r R

2 2 22

2 2 2ii i ix y z

( )

J. R. Oppenheimer

LCAO-MO

J. E. Lennard-Jones

σ :

σ* : 1 2 11 22

1212

,12 2h h

SS

1 2 11 22

1212

,12 2h h

SS

C. A. Coulson

δS=0

3 * 2 21 11 2 22 1 2 12

2 23 *1 2 1 2 12

3 *

3 *

ˆ( ) ( ) 22( ) ( )

ˆ( ) ( )

( ) ( )

ij i j

ij i j

d h C h C h C C hC C C C Sd

h d h

S d

r r r

r r r

r r r

r r r 1 2

0C C

11 12 12

12 12 220h h S

h S h

W. Pauli

2

1 2 1 2( , , , , , , ) ( , , , , , , )i j N j i Nr r r r r r r r r r1 21

W. Heisenberg

P. A. M. Dirac

[Heisenberg, Dirac, 1926]

1 1 1

1 2

1

1( , , , )!

N

N

N N NN

r rr r r

r r

1r1, 1

!NN NrN NN

1 1 1 1 1

1 2 1 2 21 2

1 2 1 2 2

1 1

( ) ( ) ( )1 ( ) ( ) ( )( , , , )

(2 )!( ) ( ) ( )

1 det(2 )!

n

nN

n n n n

n

n

n

r r rr r rr r rr r r

11n

22 1 2 2n

22

n

*

*1 , 1

ˆ2 2

n n

i ij iji i j

d HE h J K

d

3 * 2ne

3 3 * *1 2 1 1 2 2

12

3 3 * *1 2 1 1 2 2

12

1( ) ( ) ( )2

1( ) ( ) ( ) ( )

1( ) ( ) ( ) ( )

i i i

ij i i j j

ij i j j i

h d V

J d dr

K d dr

r r r r

r r r r r r

r r r r r r

J. C. Slater

1

ab initio

2

Ab initio

3 *1 1 2 2 2 1

12

3 *1 1 2 2 2 1

12

ˆˆ ˆ ˆ(2 )

1ˆ ( ) ( ) ( ) ( ) ( )

1ˆ ( ) ( ) ( ) ( ) ( )

j jj

j i j j i

j i j i j

F h J K

J dr

K dr

r r r r r r

r r r r r r

V. A. Focki E/ i=0

1 , 1

2 2n n

i ij iji i j

E h J K

ˆi i iF

SCF

i

Ab initio SCF

{ i} F

F

i(1) { (1)

i(1)} F(1)

i( )

{εi( )

i(n-1)

i(n) {εi

(n-1)} {εi(n)}

C. C. J. Roothaan

{χp}

S=EF i

F: S: Ci: 0||

ˆ

SFSCFC

i

iiiiiiF

basisN

i p pi ip

C χC

99.8

→

0.1 kcal/mol

[Löwdin, Phys. Rev. 97, 1509, 1955.]

P. -O. Löwdincorrelation exact HFE E E

rij=0

N

ji ijrhH 1ˆˆ EH

rij→0

r12

1 2

12

1 2

12

HF 0

HF 0

(1,2;1,2) 0

1(1,2;1,2) (1,1) (2,2)2

r

r P P

HFExact

2

r12→0 0 r12

0012 12122 rrr

HF

1

N

[Hohenberg & Kohn, Phys. Rev. B, 136, 864, 1964.]

W. KohnP. Hohenberg

{ i}

Veff

DFT

{ i}{ i}

DFT[Kohn & Sham, Phys. Rev. A, 140, 1133, 1965.]

21 ( )2 eff i i iV r

3 ( )( ) ( ) ( )eff xcV V d Vr'r r r' rr r'

occ

ii

2

3 * 2 31( ) ( ) ( ) ( ) [ ]2 xci i

iE d d V J Er r r r r r

W. Kohn L. J. Sham

SCF

DFT

DFT

CI

[Schipper et al., Phys. Rev. A 57, 1729,1998.]

LDA ρGGA LDA ρ

GGA meta-GGA GGA τGGA hybrid-GGA GGA

[Perdew et al., J. Chem. Phys. 123, 062201, 2006.]

DFT

LDA

XSVWN

GGA

BLYPPBE

Hybrid GGA

B3LYPPBE0

GGALC, CAMB3LYP, B97

GGAB97, HCTH, M06

1985 1995 2005

: vs

:vs

2010

GGA

GGA

:vs

:vs

20001990

←(MAE ~1 kcal/mol)

[Medvedev, Bushmarinov, Sun, Perdew, and Lyssenko, Science 355, 49, 2017; ibid. 356, 496c, 2017.]

21 ( )2 eff i i iV r

DFT→

:

(i.e., ) 80

( )

[Iikura, Tsuneda, Yanai and Hirao, J. Chem. Phys. 115, 3540, 2001.]

occ occlr 3 3 * * 12

x 1 2 1 2 1 212

erf( )1 ( ) ( ) ( ) ( )2 i j j i

i j

rE d dr

r r r r r r

1/ 2 1/3/(2 ), (9 )a k k K12 12

12 12 12

1 erf( ) erf( )1 r rr r r

3 4/3

3 32

1 812 3

1 1erf 2 4 exp 3 42 4

E d K a

a a a aa a

rsrx

:CAM-B3LYP [Yanai, Tew, & Handy, Chem. Phys. Lett. 91, 551, 2004.]LC- PBE [Vydrov, Heyd, Krukau, & Scuseria, J. Chem. Phys. 125, 074206, 2006.]BNL [Livshits & Baer, Phys. Chem. Chem. Phys. 9, 2937, 2007.]

B97-series [Chai & Head-Gordon, J. Chem. Phys. 128, 084106, 2008.]Tuned RSH [Baer, Livshits, & Salzner, Ann. Rev. Phys. Chem. 61, 85, 2010.]

-

-

Excited state calculations

(LC)

DFT

-1 0 1 2 3 4 5 6 7 8 9 10HOMO + IP (eV)

BOPB3LYPLC-BOP

SiH4CH4PH3NH3

HCOOHH2SH2O

CH2OC2H4C2H2CO2ClFCSCOHClHFCl2P2F2N2NeHe

H-1 0 1 2 3 4 5 6

LUMO + EA (eV)

BOPB3LYPLC-BOP

SiH4CH4PH3NH3

HCOOHH2SH2O

CH2OC2H4C2H2CO2ClFCSCOHClHFCl2P2F2N2NeHe

H

HOMO LUMO

ˆ ˆi i ih J Vxc

[Tsuneda, Song, Suzuki, & Hirao, J. Chem. Phys. 133, 174101, 2010.]

HOMO HOMO = IPLUMO LUMO = EA

LC

-200-180-160-140-120-100-80-60-40-20

0

0 5 10 15 20

Ener

gy (k

cal/m

ol)

Number of atoms

epsilo: LC-BLYPepsilo: LC-BLYP-VEA: LC-BLYP-VIP: LC-BLYPepsilo: PBEepsilo: PBE-VEA: PBE-VIP: PBE

HOMO

LUMO

HOMOLUMO

HOMO LUMOIP EA

. HOMO-LUMO

LC-DFT: PBE: HOMO-LUMO

HOMO-LUMO

[T. Tsuneda, J. Comput. Chem. (MorokumaFestschrift) 40, 206, 2019.]

( 1) ( )p p q pE n E n E nq q q

+10-1

E(n)

n

[Perdew et al., Phys. Rev. Lett. 49, 1691, 1982; Yang et al., Phys. Rev. Lett. 84, 5172, 2000.]

ii

En

HOMO HOMO = IPLUMO LUMO = EA

IP

EA( 0)E nn

HOMO

( 0)E nn

LUMO

[Sham and Schlüter, Phys. Rev. B 32, 3883, 1985.]

+ =

[Janak, Phys. Rev. B 103, 7165, 1978.]

[Itatani et al., Nature 432, 867, 2004]:

DFT

CO2 HOMO

[Vozzi et al., Nature Phys. 7, 822, 2011]

PTCDA[Puschnig et al., Phys. Rev. B 7, 235427, 2011]

nF nB nI nR

0 0 0 0

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

2 0 0 0

0 2 0 0

0 0 2 0

0 0 0 2

F B I R

F B I R

n n n n

n n n nE

F B I R

nF nB nI nR

1 0 0 1 (IR) 1 1 0 0

0 1 1 0

2 0 0 1

3 0 0 1

2 1 0 0

1 0 0 2

2 0 0 2

2 2 0 0

[Jensen, “Introduction to Computational Chemistry, 2nd Ed. (Wiley, Chichester, UK), 2007.]

Coupled

' 'AU F

, ,

3 3 * *,

12

3 3 * *xc 2

( )

1( ) ( ) ( ) ( )

( ) ( ) ( , ) ( ) ( )

ia jb ij ab a i ia jb

ia jb i a j b

i a j b

A K

K d dr

d d f

1 2 1 1 2 2

1 2 1 1 1 2 2

r r r r r r

r r r r r r r r

3 *' i aiadF r r r r

F’ 1

12

E q FμF Qr

= + + 12 + 16 ⋯1

2

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

E –

E reac

tant

(har

tree

) B3LYPLC-BOP

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

ƞ –ƞ r

eact

ant (

eV)

Diels-Alder

B3LYP:

LC-BOP: ( )

( )

→

1,3-butadiene+ethyleneHOMO-LUMO

[R. K. Singh & T. Tsuneda, J. Comput. Chem., 34, 379, 2013.]

Diels-Alder : C2H4 + C4H6 → C6H10

[Tsuneda & Singh, J. Comput. Chem. 35, 1093, 2014.]

&

-0.65-0.6

-0.55-0.5

-0.45-0.4

-0.35-0.3

0 0.2 0.4 0.6 0.8 1Normalized IRC

0.020.030.040.050.060.070.08

→

(< 0.25)

(> 0.25)

Orb

ital e

nerg

y (h

artr

ee)

Occ

upie

d

U

nocc

upie

d

[Tsuneda et al., Phys. Chem. Chem. Phys. 20, 14211, 2018.]

C2H4 + C4H6 → C6H10 Diels-Alder.

↔

C4H6 C2H4 :C4

24% 2

DFT

1.2.3.

→

DFT

1.2.3.

→

DFT

[Tsuneda, The Chemical Record, in press.]

DFT[Rupp, von Lilienfeld & Burke, J. Phys. Chem. 148, 241743, 2018.]

AI

DFT

DFT

1.2.3.4. DFT

1. →→

2. →→

3.

4.

[Tsuneda, The Chemical Record, in press.]

“ , ( , 2012)”

“T. Tsuneda, Density Functional Theory in Quantum Chemistry

(Springer, 2014)”