DFT METHODS IN GAUSSIAN
Ivan Rostov,
Australian National University,
Canberra
E-mail: Ivan.Rostov@anu.edu.au
Variety Of Methods In Computational Chemistry
Cost
CCSD(T)
MP2 energy correction
Ab initio
Hartree-Fock QM
DFT methods
Semiempirical QM
MM Force Fields
Accuracy
2
Variety of Methods in Computational Chemistry
Quality


Ab initio MO Methods

CCSD(T)
quantitative (1~2 kcal/mol) but expensive
~N6

MP2
semi-quantitative and doable
~N4

HF
qualitative
~N2-3
Density Functional Theory


DFT
semi-quantitative and cheap
~N2-3
Semi-empirical MO Methods


Size dependence
AM1, PM3, MNDO
semi-qualitative
~N2-3
Molecular Mechanics Force Field

MM3, Amber, Charmm semi-qualitative (no bond-breaking)
~N1-2
3
Quantum Chemistry Basics
ˆ   E, where r , r r ,  ,   
H
1 2
N
1
2
N
Variational principle:
ˆ  EE
ˆ
H
exact  exact H exact
Born-Oppenheimer (clamped-nuclei) approximation
electrons are fast and moves in the field of fixed nuclei
N
M N
Zm
ZmZn
1
1
2
ˆ  
H




 
2 i 1 i m1 i 1 rim i  j rim mn Rmn
Hartree-Fock Approximation:
EHF  min E SD , where SD 
 SD  N



1
det1 ( x1 )  2 ( x2 )   N ( xN )
N!
4
Density Functional Theory Basics

 

 

 r1     * r1 , r2  rN ,  1 ,  2  N    r1 , r2  rN ,  1 ,  2  N d 3r2 d 3r3  d 3rN d 1d 2  d N
Hohenberg-Kohn Theorems (1964)







V
r

V

r
1. ext
ext
0  N , Z A , RA   Hˆ  0  E0 and other properties 
Therefore, instead of  dependent on 4N coordinates we would need just 0 dependent on just 3
coordinates
2. The variational principle for DFT
E   E 0 
E   Tk    E Ne    Eee   or
E   Tk    E Ne    J    Exc  
If we would know how to express each of those four terms

 
M

r

1  r  r2   
J     1
dr1dr2 ; E Ne     Z m    dr
2
r12
r  Rm
m 1
 
What about Tk[] and Exc[]?
Thomas, Fermi (1927)
2/3


3
Tk    3 2    r  5 / 3dr poor accuracy, as was formulated for the uniform electron gas
10
5
Density Functional Theory Basics
Kohn-Sham formalism resolves the problem with the kinetic energy term


1 N
Tk     i* r  2i r dr
2
N

 2
 r    i r 
The big unknown left is
EXC    EX    EC  
The Hartree-Fock case:












r

r

r

r
1
i 1
j 1
i 2
j 2  
EXHF    
dr1dr2 ;
2 i, j
r12
EC    0
6
Exchange functionals Ex[]
•
•
•
•
•
•
•
•
•
1/ 3
Slater:
with theoretical coefficient a =2/3. E     9  3  a  r 4 / 3 d 3 r
 
X

8  
Keyword: Used Alone: HFS, Comb. Form: S
Xαρ4/3 with the empirical coefficient of 0.7, usually used when this exchange functional
is used without a correlation functional
Keyword: Used Alone: XAlpha, Comb. Form: XA.
Becke 88: Becke's 1988 functional, which includes the Slater exchange along with
corrections involving the gradient of the density. E    f  r ,  r d 3r
X

Keyword: Used Alone: HFB, Comb.Form: B.
Perdew-Wang 91: The exchange component of Perdew and Wang's 1991 functional.
Keyword: Used Alone: N/A, Comb. Form: PW91.
Modified PW91: as modified by Adamo and Barone.
Keyword: Used Alone: N/A, Comb. Form: MPW.
Gill 96: The 1996 exchange functional of Gill.
Keyword: Used Alone: N/A, Comb. Form: G96.
PBE: The 1996 functional of Perdew, Burke and Ernzerhof.
Keyword: Used Alone: N/A, Comb. Form: PBE.
OPTX: Handy's OPTX modification of Becke's exchange functional.
Keyword: Used Alone: N/A, Comb. Form: O.
TPSS: The exchange functional of Tao, Perdew, Staroverov, and Scuseria.
Keyword: Used Alone: N/A, Comb. Form: TPSS. E    f  r ,  r ,  2  r d 3r
ρ4/3
X

7
Correlation functionals Ec[]
•
•
•
•
•
•
•
•
•
VWN: Vosko, Wilk, and Nusair 1980 correlation functional fitting the RPA solution to
the uniform electron gas, often referred to as Local Spin Density (LSD) correlation.
VWN V(VWN5): Functional which fits the Ceperly-Alder solution to the uniform
electron gas.
LYP: The correlation functional of Lee, Yang, and Parr which includes both local and
non-local terms.
PL (Perdew Local): The local (non-gradient corrected) functional of Perdew (1981).
P86 (Perdew 86): The gradient corrections of Perdew, along with his 1981 local
correlation functional.
PW91 (Perdew/Wang 91): Perdew and Wang's 1991 gradient-corrected correlation
functional.
B95 (Becke 95): Becke's τ-dependent gradient-corrected correlation functional
(defined as part of his one parameter hybrid functional.
PBE: The 1996 gradient-corrected correlation functional of Perdew, Burke and
Ernzerhof.
TPSS: The τ-dependent gradient-corrected functional of Tao, Perdew, Staroverov, and
Scuseria.
8
Popular combinations of Ex[] and Ec[]
•
•
•
SVWN=LSDA
SVWN5
BLYP
•
B3LYP
Hybrid functionals
Exchybr  a0 ExHF  1  a0 ExLDA  ax ExB88  EcLDA  ac EcGGA
a0  0.2; ax  0.72; ac  0.81
•
B3P86, B3PW91, B1B95 (1 parameter), B1LYP, MPW1PW91, B98, B971, B972,
PBE1PBE etc.
•
You can even construct your own. Gaussian provides such a functionality:
Exc = P2EXHF + P1(P4EXSlater + P3ΔExnon-local) + P6EClocal + P5ΔECnon-local
IOP(3/76),IOP(3/77) and IOP(3/78) setup P1 - P6
B3LYP =
BLYP IOp(3/76=1000002000) IOp(3/77=0720008000) IOp(3/78=0810010000)
9
New functionals in revision E01 of G03
Hybrid M05, MO5-2X with the same parameterization scheme but different set of
parameters (25!)
As reported by Donald Truhlar and Yan Zhao, M05 and MO5-2X outperform other
parameterized hybrid functionals in nonmetallic thermochemical kinetics,
thermochemistry and noncovalent interactions. MO5-2X is especially good for
calculation of the bond dissociation energies, stacking and hydrogen-bonding
interactions in nucleobase pairs
10
Time-Dependent DFT
Runge-Gross theorem


Vext r , t 
 r, t 
ˆ r , t 
H


r 1,,r n , t 
Runge-Gross equations:

 1
M

  

Z


r
', t  3 
2
m
i  j r , t             d r '   xc r , t  j r , t 
t
r  r'
m 1 r  R
 2


 2
 r , t     j r , t  ;  j t  0   0j
j


Exc r , t 
and  xc r , t  

r , t 
Linear response of the KS approximation
steff     K st ,uv  Puv  
uv
Pst   
nst
steff  
 s   t   
11
Time-Dependent DFT
A  B 1/ 2 A  B A  B 1/ 2 FI
  I2 FI
where
F  A  B 
1 / 2
XY
X ai  Pai  ; Yai  Pia  
Aia , jb   ij ab   a   i   K ia , jb
Bia , jb  K ia ,bj
K ia , jb
ia


 ia jb     r1  a r1 
*
i
 *  3 3
 2 E xc
HF

  j r2 b r2 d r1d r2  K ia , jb
 r1  r1 

 1


jb    i* r1  a r1   j r2 b* r2 d 3r1d 3r2
r12
K iaHF , jb  cx   ja ib  cx  0 for pure functional s
2

2
2 

1/ 2
f I  EI  E0  0 r 0   d ia  a   i  FiaI 
3
3  ia


 

where d ia   i r r  a* r d 3r
12
 Time Dependent DFT (TD-DFT) is widely used to calculate
molecular electronic excitation energies.
 Sufficiently accurate to be useful
 Sufficiently economical to apply to large molecules
 Not as accurate as highly correlated methods such as
CASPT2 or CC3
 Problems with Rydberg and Charge Transfer States, double
excitations, intensities
Pyrrole Rydberg States
CASPT2
B3LYP
PBE0
A2
5.22
4.64
4.93
B1
5.87
5.32
5.61
A2
5.97
5.31
5.61
B1
5.97
5.54
5.85
B2
6.09
6.06
6.28
B1
6.40
5.87
6.16
A2
6.42
5.78
6.04
A2
6.51
6.00
6.14
B2
6.53
6.32
6.53
A1
6.54
6.11
6.41
.46
.23
MAE
Basis set: aug-ccpvtz+R
14
14
CASPT2
B3LYP
W1 A’’
5.62
5.49
W2 A’’
5.79
5.73
NV1 A’
6.39
7.19
NV2 A’
6.49
7.23
CT1 A’
7.18
6.06
CT2 A’’
8.07
6.24
15
•Problems with Rydberg and Charge Transfer States are due to
incorrect long range potentials (also problems with the
response kernel, self-interaction)
•Standard DFT functionals are too short range
•Modifications to the long range part of the exchange potential
are needed
•One approach is to use range-separated functionals constructed
from different short range (high density) and long range (low
density) forms. Short and long range components evaluated
using different techniques
•Other approaches are possible e.g. orbital dependent potentials
16
Asymptotic Behavior of VXC(R).
Must be:

1
lim Vxc r     I   max
r 
r
D.J. Tozer, N.C. Handy (1998)
This is not observed for all model potentials listed earlier (exponential asymptotic)
Solution (T. Yanai and K. Hirao group, 2004)
1 1  erf ( r12 ) erf ( r12 )


r12
r12
r12
First term goes to Ex, while second calculated together
with J
x
2
2
erf ( x) 
exp

t
dt

 0
(T. Yanai, D.P.Tew, N.C.Handy, 2004)
1 1  a  erf ( r12 ) a  erf ( r12 )


r12
r12
r12
 
CAM-B3LYP: a = 0.19; =0.46,  = 0.33
17
Pyrrole Rydberg States
CASPT2
B3LYP
PBE0
CAMB3LYP
A2
5.22
4.64
4.93
5.10
B1
5.87
5.32
5.61
5.82
A2
5.97
5.31
5.61
5.83
B1
5.97
5.54
5.85
6.08
B2
6.09
6.06
6.28
6.38
B1
6.40
5.87
6.16
6.42
A2
6.42
5.78
6.04
6.37
A2
6.51
6.00
6.14
6.45
B2
6.53
6.32
6.53
6.72
A1
6.54
6.11
6.41
6.75
.46
.23
.13
MAE
Basis set: aug-ccpvtz+R
18
CASPT2
B3LYP
HCTH-AC
CAMB3LYP
W1 A’’
5.62
5.49
5.43
5.61
W2 A’’
5.79
5.73
5.70
5.84
NV1 A’
6.39
7.19
6.87
7.06
NV2 A’
6.49
7.23
6.98
7.36
CT1 A’
7.18
6.06
5.16
6.74
CT2 A’’
8.07
6.24
4.67
7.85
19
Retinal Proteins Chromophores
Table 1. Excitation energies (eV) and oscillator strengths for 6-cis-11-cis PSB11.1
•CASPT2 calculations performed on geometry optimized with the state averaged CAS(12,12)/6-31G(d). All TD-DFT calculations employed geometry
optimized with B3LYP/6-31G(d).
Method
Experiment (in methanol)
Experiment – solvent blue shift
CASPT2(12,12)1
TD-BP86
TD-B3LYP
TD-B3LYP
TD-B3LYP
TD-CAMB3LYP
TD-CAMB3LYP
TD-CAMB3LYP
TD-CAMB3LYP
Basis
6-31G(d)
6-31G+(d)
6-31G(d)
6-31G+(d)
SV(P)
6-31G(d)
6-31G+(d)
SV(P)
TZV(P)
S1
E
2.79, 2.82
2.41,2.43
2.41
2.14
2.35
2.31
2.33
2.50
2.46
2.48
2.45
S2
f
E
f
0.79
1.14
1.23
1.13
1.51
1.51
1.50
1.50
3.52
2.70
3.14
3.09
3.13
3.69
3.65
3.67
3.63
0.73
0.57
0.43
0.58
0.33
0.33
0.33
0.33
20
Retinal Proteins Chromophores
21
Retinal Proteins Chromophores
22
Retinal Proteins Chromophores
23
DFT references
1.
2.
3.
4.
W. Koch, M.C. Holthausen, A Chemist’s Guide to Density Functional Theory (WileyVCH Verlag GmbH, 2001)
M.E. Casida in Recent Advances in Density Functional Methods, Part 1 (World
Scientific, Singapore, 1995)
M.E. Casida in Recent Developments and Applications of Modern Density Functional
Theory, Theoretical and Computational Chemistry, vol 4., ed. by J.M. Seminario
(Elsevier, Amsterdam, 1996).
Marques M.A.L. and Gross E.K.U. Annu. Rev. Phys. Chem 55, 427 (2004).
24
Acknowledgements




Fujitsu Company for financial support and giving us opportunity
to visit the beautiful country of Japan
Hiro Hotta for constant help
Professor Shinkoh Nanbu for invitation to the Kyushu University
You, audience, for your attention
25