The method of dispersion correction as an add-on to standard Kohn–Sham density functional theory (DFT-D) has been refined regarding higher accuracy, broader range of applicability, and less empiricism. The main new ingredients are atom-pairwise specific dispersion coefficients and cutoff radii that are both computed from first principles. The coefficients for new eighth-order dispersion terms are computed using established recursion relations. System (geometry) dependent information is used for the first time in a DFT-D type approach by employing the new concept of fractional coordination numbers (CN). They are used to interpolate between dispersion coefficients of atoms in different chemical environments. The method only requires adjustment of two global parameters for each density functional, is asymptotically exact for a gas of weakly interacting neutral atoms, and easily allows the computation of atomic forces. Three-body nonadditivity terms are considered. The method has been assessed on standard benchmark sets for inter- and intramolecular noncovalent interactions with a particular emphasis on a consistent description of light and heavy element systems. The mean absolute deviations for the S22 benchmark set of noncovalent interactions for 11 standard density functionals decrease by 15%–40% compared to the previous (already accurate) DFT-D version. Spectacular improvements are found for a tripeptide-folding model and all tested metallic systems. The rectification of the long-range behavior and the use of more accurate C6 coefficients also lead to a much better description of large (infinite) systems as shown for graphene sheets and the adsorption of benzene on an Ag(111) surface. For graphene it is found that the inclusion of three-body terms substantially (by about 10%) weakens the interlayer binding. We propose the revised DFT-D method as a general tool for the computation of the dispersion energy in molecules and solids of any kind with DFT and related (low-cost) electronic structure methods for large systems.

1.
A. J.
Stone
,
The Theory of Intermolecular Forces
(
Oxford University Press
,
Oxford
,
1997
).
2.
I. G.
Kaplan
,
Intermolecular Interactions
(
Wiley
,
Chichester
,
2006
).
3.
S.
Grimme
,
J.
Antony
,
T.
Schwabe
, and
C.
Mück-Lichtenfeld
,
Org. Biomol. Chem.
5
,
741
(
2007
).
4.
J.
Gräfenstein
and
D.
Cremer
,
J. Chem. Phys.
130
,
124105
(
2009
).
5.
E. R.
Johnson
,
I. D.
Mackie
, and
G. A.
DiLabio
,
J. Phys. Org. Chem.
22
,
1127
(
2009
).
6.
T.
Sato
and
H.
Nakai
,
J. Chem. Phys.
131
,
224104
(
2009
).
7.
M.
Elstner
,
P.
Hobza
,
T.
Frauenheim
,
S.
Suhai
, and
E.
Kaxiras
,
J. Chem. Phys.
114
,
5149
(
2001
).
8.
S.
Grimme
,
J. Comput. Chem.
25
,
1463
(
2004
).
9.
P.
Jurečka
,
J.
Černý
,
P.
Hobza
, and
D. R.
Salahub
,
J. Comput. Chem.
28
,
555
(
2007
).
10.
Y.
Andersson
,
D. C.
Langreth
, and
B. I.
Lundqvist
,
Phys. Rev. Lett.
76
,
102
(
1996
).
11.
D. C.
Langreth
,
M.
Dion
,
H.
Rydberg
,
E.
Schröder
,
P.
Hyldgaard
, and
B. I.
Lundqvist
,
Int. J. Quantum Chem.
101
,
599
(
2005
).
12.
T.
Sato
,
T.
Tsuneda
, and
K.
Hirao
,
Mol. Phys.
103
,
1151
(
2005
).
13.
O. A.
von Lilienfeld
,
I.
Tavernelli
,
U.
Röthlisberger
, and
D.
Sebastiani
,
Phys. Rev. Lett.
93
,
153004
(
2004
).
14.
Y. Y.
Sun
,
Y. -H.
Kim
,
K.
Lee
, and
S. B.
Zhang
,
J. Chem. Phys.
129
,
154102
(
2008
).
15.
Y.
Zhao
and
D. G.
Truhlar
,
Acc. Chem. Res.
41
,
157
(
2008
).
16.
B.
Jeziorski
and
K.
Szalewicz
, in
Encyclopedia of Computational Chemisty
, edited by
P.
von Rague-Schleyer
(
Wiley
,
New York
,
1998
), Vol.
2
, p.
1376
.
17.
G.
Jansen
and
A.
Heßelmann
,
J. Phys. Chem. A
105
,
11156
(
2001
).
18.
M.
Pitoňák
and
A.
Heßelmann
,
J. Chem. Theory Comput.
6
,
168
(
2010
).
19.
C.
Tuma
and
J.
Sauer
,
Phys. Chem. Chem. Phys.
8
,
3955
(
2006
).
20.
K. E.
Yousaf
and
E. N.
Brothers
,
J. Chem. Theory Comput.
6
,
864
(
2010
).
21.
F.
Shimojo
,
Z.
Wu
,
A.
Nakano
,
R. K.
Kalia
, and
P.
Vashishta
,
J. Chem. Phys.
132
,
094106
(
2010
).
22.
O. A.
Vydrov
and
T.
Van Voorhis
,
Phys. Rev. Lett.
103
,
063004
(
2009
).
23.
J.
Klimeš
,
D. R.
Bowler
, and
A.
Michaelides
,
J. Phys.: Condens. Matter
22
,
022201
(
2010
).
24.
N.
Marom
,
A.
Tkatchenko
,
M.
Scheffler
, and
L.
Kronik
,
J. Chem. Theory Comput.
6
,
81
(
2010
).
25.
E. G.
Hohenstein
,
S. T.
Chill
, and
C. D.
Sherrill
,
J. Chem. Theory Comput.
4
,
1996
(
2008
).
26.
C. D.
Sherrill
, in
Reviews in Computational Chemistry
, edited by
K. B.
Lipkowitz
and
D. B.
Boyd
(
Wiley-VCH
,
New York
,
2009
), Vol.
26
, pp.
1
38
.
27.
G. S.
Tschumper
, in
Reviews in Computational Chemistry
, edited by
K. B.
Lipkowitz
and
D. B.
Boyd
(
Wiley-VCH
,
New York
,
2009
), Vol.
26
, pp.
39
90
.
28.
K.
Pernal
,
R.
Podeszwa
,
K.
Patkowski
, and
K.
Szalewicz
,
Phys. Rev. Lett.
103
,
263201
(
2009
).
29.
A. D.
Becke
,
Phys. Rev. A
38
,
3098
(
1988
).
30.
C.
Lee
,
W.
Yang
, and
R. G.
Parr
,
Phys. Rev. B
37
,
785
(
1988
).
31.
J. P.
Perdew
,
Phys. Rev. B
33
,
8822
(
1986
).
32.
J. P.
Perdew
,
Phys. Rev. B
34
,
7406
(
1986
).
33.
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
,
Phys. Rev. Lett.
77
,
3865
(
1996
).
34.
Y.
Zhang
and
W.
Yang
,
Phys. Rev. Lett.
80
,
890
(
1998
).
35.
S.
Grimme
,
J. Comput. Chem.
27
,
1787
(
2006
).
36.
J.
Tao
,
J. P.
Perdew
,
V. N.
Staroverov
, and
G. E.
Scuseria
,
Phys. Rev. Lett.
91
,
146401
(
2003
).
37.
A. D.
Becke
,
J. Chem. Phys.
98
,
5648
(
1993
).
38.
P. J.
Stephens
,
F. J.
Devlin
,
C. F.
Chabalowski
, and
M. J.
Frisch
,
J. Phys. Chem.
98
,
11623
(
1994
).
39.
C.
Adamo
and
V.
Barone
,
J. Chem. Phys.
110
,
6158
(
1999
).
40.
Y.
Zhao
and
D. G.
Truhlar
,
J. Phys. Chem. A
109
,
5656
(
2005
).
41.
S.
Grimme
,
J. Chem. Phys.
124
,
034108
(
2006
).
42.
S.
Grimme
,
J. Phys. Chem. A
109
,
3067
(
2005
).
43.
A. D.
Becke
and
E. R.
Johnson
,
J. Chem. Phys.
122
,
154104
(
2005
).
44.
A. D.
Becke
and
E. R.
Johnson
,
J. Chem. Phys.
123
,
154101
(
2005
).
45.
A.
Tkatchenko
and
M.
Scheffler
,
Phys. Rev. Lett.
102
,
073005
(
2009
).
46.
47.
A. D.
Becke
and
E. R.
Johnson
,
J. Chem. Phys.
124
,
014104
(
2006
).
48.
E. R.
Johnson
and
A. D.
Becke
,
J. Chem. Phys.
124
,
174104
(
2006
).
49.
E. R.
Johnson
and
A. D.
Becke
,
J. Chem. Phys.
123
,
024101
(
2005
).
50.
P.
Jurečka
,
J.
Šponer
,
J.
Černý
, and
P.
Hobza
,
Phys. Chem. Chem. Phys.
8
,
1985
(
2006
).
51.
J.
Kong
,
Z.
Gan
,
E.
Proynov
,
M.
Freindorf
, and
T. R.
Furlani
,
Phys. Rev. A
79
,
042510
(
2009
).
52.
F. A.
Gianturco
and
F.
Paesani
,
J. Chem. Phys.
113
,
3011
(
2000
).
53.
X.
Wu
,
M. C.
Vargas
,
S.
Nayak
,
V.
Lotrich
, and
G.
Scoles
,
J. Chem. Phys.
115
,
8748
(
2001
).
54.
Q.
Wu
and
W.
Yang
,
J. Chem. Phys.
116
,
515
(
2002
).
55.
U.
Zimmerli
,
M.
Parrinello
, and
P.
Koumoutsakos
,
J. Chem. Phys.
120
,
2693
(
2004
).
56.
G.
Murdachaew
,
S.
de Gironcoli
, and
G.
Scoles
,
J. Phys. Chem. A
112
,
9993
(
2008
).
57.
S. N.
Steinmann
,
G.
Csonka
, and
C.
Corminboeuf
,
J. Chem. Theory Comput.
5
,
2950
(
2009
).
58.
T.
Schwabe
and
S.
Grimme
,
Phys. Chem. Chem. Phys.
9
,
3397
(
2007
).
59.
Y.
Liu
and
W. A.
Goddard
 III
,
Mater. Trans.
50
,
1664
(
2009
).
60.
J. -D.
Chai
and
M.
Head-Gordon
,
Phys. Chem. Chem. Phys.
10
,
6615
(
2008
).
61.
S.
Grimme
,
Chem.-Eur. J.
10
,
3423
(
2004
).
62.
S.
Grimme
,
Angew. Chem., Int. Ed.
45
,
4460
(
2006
).
63.
H.
Casimir
and
D.
Polder
,
Phys. Rev.
73
,
360
(
1948
).
64.
M.
Lein
,
J. F.
Dobson
, and
E. K. U.
Gross
,
J. Comput. Chem.
20
,
12
(
1999
).
65.
C.
Kamal
,
T. K.
Ghanty
,
A.
Banerjee
, and
A.
Chakrabarti
,
J. Chem. Phys.
131
,
164708
(
2009
).
66.
G.
Starkschall
and
R.
Gordon
,
J. Chem. Phys.
56
,
2801
(
1972
).
67.
A.
Thakkar
,
H.
Hettema
, and
P.
Wormer
,
J. Chem. Phys.
97
,
3252
(
1992
).
68.
K. T.
Tang
and
J. P.
Toennies
,
J. Chem. Phys.
80
,
3726
(
1984
).
69.
N. A.
deLima
,
J. Chem. Phys.
132
,
014110
(
2010
).
70.
A.
Kumar
and
A. J.
Thakkar
,
J. Chem. Phys.
132
,
074301
(
2010
).
71.
M.
Dierksen
and
S.
Grimme
,
J. Phys. Chem. A
108
,
10225
(
2004
).
72.
B. M.
Axilrod
and
E.
Teller
,
J. Chem. Phys.
11
,
299
(
1943
).
73.
J.
Muto
,
Proc. Phys. Math. Soc. Jpn.
17
,
629
(
1943
).
74.
A.
Tkatchenko
and
O. A.
von Lilienfeld
,
Phys. Rev. B
78
,
045116
(
2008
).
75.
M.
Mantina
,
A. C.
Chamberlin
,
R.
Valero
,
C. J.
Cramer
, and
D. G.
Truhlar
,
J. Phys. Chem. A
113
,
5806
(
2009
).
76.
P.
Pyykkö
and
M.
Atsumi
,
Chem.-Eur. J.
15
,
186
(
2009
).
77.
See http://www.uni-muenster.de/Chemie.oc/grimme/ for a FORTRAN program implementing the DFT-D3 method and a file with available C6 coefficients.
78.
See supplementary material at http://dx.doi.org/10.1063/1.3382344 for optimized DFT-D3 parameter values for triple-zeta calculations, computational details, and details on the benchmark sets.
79.
C. J.
Cramer
and
D. G.
Truhlar
,
Phys. Chem. Chem. Phys.
11
,
10757
(
2009
).
80.
J. F.
Dobson
,
A.
White
, and
A.
Rubio
,
Phys. Rev. Lett.
96
,
073201
(
2006
).
81.
R.
Ahlrichs
,
F.
Furche
,
C.
Hättig
 et al, TURBOMOLE, version 6.0, Universität Karlsruhe
2009
. See http://www.turbomole.com.
82.
F.
Neese
,
ORCA—An Ab Initio, Density Functional and Semiempirical Program Package
(
University of Bonn
,
Germany
,
2007
).
83.
H. -J.
Werner
,
P. J.
Knowles
,
R.
Lindh
 et al, MOLPRO, version
2006
.1, a package of ab initio programs,” see http://www.molpro.net.
84.
F.
Weigend
,
F.
Furche
, and
R.
Ahlrichs
,
J. Chem. Phys.
119
,
12753
(
2003
).
85.
F.
Weigend
and
R.
Ahlrichs
,
Phys. Chem. Chem. Phys.
7
,
3297
(
2005
).
86.
A.
Schäfer
,
C.
Huber
, and
R.
Ahlrichs
,
J. Chem. Phys.
100
,
5829
(
1994
).
87.
B.
Metz
,
H.
Stoll
, and
M.
Dolg
,
J. Chem. Phys.
113
,
2563
(
2000
).
88.
K. A.
Peterson
,
D.
Figgen
,
E.
Goll
,
H.
Stoll
, and
M.
Dolg
,
J. Chem. Phys.
119
,
11113
(
2003
).
89.
O.
Vahtras
,
J.
Almlöf
, and
M. W.
Feyereisen
,
Chem. Phys. Lett.
213
,
514
(
1993
).
90.
F.
Weigend
,
Phys. Chem. Chem. Phys.
4
,
4285
(
2002
).
91.
F.
Weigend
and
M.
Häser
,
Theor. Chem. Acc.
97
,
331
(
1997
).
92.
F.
Weigend
,
M.
Häser
,
H.
Patzelt
, and
R.
Ahlrichs
,
Chem. Phys. Lett.
294
,
143
(
1998
).
93.
K.
Eichkorn
,
O.
Treutler
,
H.
Öhm
,
M.
Häser
, and
R.
Ahlrichs
,
Chem. Phys. Lett.
240
,
283
(
1995
).
94.
A.
Hellweg
,
C.
Hättig
,
S.
Höfener
, and
W.
Klopper
,
Theor. Chem. Acc.
117
,
587
(
2007
).
95.
F.
Furche
and
D.
Rappoport
, in
Theoretical and Computational Chemistry
, edited by
M.
Olivucci
(
Elsevier
,
Amsterdam
,
2005
), Vol.
16
.
96.
M.
Pitoňak
,
T.
Janowski
,
P.
Neogrády
,
P.
Pulay
, and
P.
Hobza
,
J. Chem. Theory Comput.
5
,
1761
(
2009
).
97.
W.
Klopper
and
H. P.
Lüthi
,
Mol. Phys.
96
,
559
(
1999
).
98.
L.
Goerigk
and
S.
Grimme
,
J. Chem. Theory Comput.
6
,
107
(
2010
).
99.
S.
Tsuzuki
,
K.
Honda
,
T.
Uchimaru
, and
M.
Mikami
,
J. Chem. Phys.
124
,
114304
(
2006
).
100.
E.
Goll
,
H. -J.
Werner
, and
H.
Stoll
,
Phys. Chem. Chem. Phys.
7
,
3917
(
2005
).
101.
N.
Runeberg
and
P.
Pyykkö
,
Int. J. Quantum Chem.
66
,
131
(
1998
).
102.
S.
Grimme
,
C.
Mück-Lichtenfeld
, and
J.
Antony
,
J. Phys. Chem. C
111
,
11199
(
2007
).
103.
C.
Mück-Lichtenfeld
and
S.
Grimme
,
Mol. Phys.
105
,
2793
(
2007
).
104.
A.
Sygula
,
F. R.
Fronczek
,
R.
Sygula
,
P. W.
Rabideau
, and
M. M.
Olmstead
,
J. Am. Chem. Soc.
129
,
3842
(
2007
).
105.
T.
Schwabe
and
S.
Grimme
,
Acc. Chem. Res.
41
,
569
(
2008
).
106.
J.
Antony
,
J.
Brüske
, and
S.
Grimme
,
Phys. Chem. Chem. Phys.
11
,
8440
(
2009
).
107.
J. -D.
Chai
and
M.
Head-Gordon
,
J. Chem. Phys.
131
,
174105
(
2009
).
108.
Y.
Zhao
and
D. G.
Truhlar
,
J. Phys. Chem. C
112
,
4061
(
2008
).
109.
M.
Korth
and
S.
Grimme
,
J. Chem. Theory Comput.
5
,
993
(
2009
).
110.
J.
Antony
and
S.
Grimme
,
J. Phys. Chem. A
111
,
4862
(
2007
).
111.
A.
Gulans
,
M. J.
Puska
, and
R. M.
Nieminen
,
Phys. Rev. B
79
,
201105
(R) (
2009
).
112.
D.
Řeha
,
H.
Valdés
,
J.
Vondrášek
,
P.
Hobza
,
A.
Abu-Riziq
,
B.
Crews
, and
M. S.
de Vries
,
Chem.-Eur. J.
11
,
6803
(
2005
).
113.
J.
Černý
,
P.
Jurečka
,
P.
Hobza
, and
H.
Valdés
,
J. Phys. Chem. A
111
,
1146
(
2007
).
114.
R.
Zacharia
,
H.
Ulbricht
, and
T.
Hertel
,
Phys. Rev. B
69
,
155406
(
2004
).
115.
E.
Ziambaras
,
J.
Kleis
,
E.
Schröder
, and
P.
Hyldgaard
,
Phys. Rev. B
76
,
155425
(
2007
).
116.
A. G.
Donchev
,
Phys. Rev. B
74
,
235401
(
2006
).
117.
B. M.
Wong
,
J. Comput. Chem.
30
,
51
(
2009
).
118.
Y.
Zhao
and
D. G.
Truhlar
,
Phys. Chem. Chem. Phys.
10
,
2813
(
2008
).
119.
E. R.
Johnson
,
P.
Mori-Sánchez
,
A. J.
Cohen
, and
W.
Yang
,
J. Chem. Phys.
129
,
204112
(
2008
).
120.
K.
Baldrigde
(private communication).
121.
M. S.
Marshall
,
R. P.
Steele
,
K. S.
Thanthiriwatte
, and
C. D.
Sherrill
,
J. Phys. Chem. A
113
,
13628
(
2009
).
122.
D.
Feller
,
D. A.
Dixon
, and
J. B.
Nicholas
,
J. Phys. Chem. A
104
,
11414
(
2000
).
123.
A.
Ruzsinszky
,
J. P.
Perdew
, and
G. I.
Csonka
,
J. Phys. Chem. A
109
,
11015
(
2005
).
124.
T. J.
Rockey
,
M.
Yang
, and
H. -L.
Dai
,
J. Phys. Chem. B
110
,
19973
(
2006
).
125.
S.
Grimme
,
C.
Mück-Lichtenfeld
, and
J.
Antony
,
Phys. Chem. Chem. Phys.
10
,
3327
(
2008
).

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