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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)”