The growing string method (GSM) has proven especially useful for locating chemical reaction paths at low computational cost. While many string methods use Cartesian coordinates, these methods can be substantially improved by changes in the coordinate system used for interpolation and optimization steps. The quality of the interpolation scheme is especially important because it determines how close the initial path is to the optimized reaction path, and this strongly affects the rate of convergence. In this article, a detailed description of the generation of internal coordinates (ICs) suitable for use in GSM as reactive tangents and in string optimization is given. Convergence of reaction paths is smooth because the IC tangent and orthogonal directions are better representations of chemical bonding compared to Cartesian coordinates. This is not only important quantitatively for reducing computational cost but also allows reaction paths to be described with smoothly varying chemically relevant coordinates. Benchmark computations with challenging reactions are compared to previous versions of GSM and show significant speedups. Finally, a climbing image scheme is included to improve the quality of the transition state approximation, ensuring high reliability of the method.

1.
A. T.
Ziegler
,
Chem. Rev.
91
(
5
),
651
(
1991
).
2.
W.
Kohn
,
A. D.
Becke
, and
R. G.
Parr
,
J. Phys. Chem.
100
(
31
),
12974
(
1996
).
3.
R. G.
Parr
and
W.
Yang
,
J. Am. Chem. Soc.
106
,
4049
(
1984
).
4.
Y.
Zhao
and
D. G.
Truhlar
,
J. Chem. Theory. Comput.
1
(
3
),
415
(
2005
).
5.
C. J.
Cramer
and
D. G.
Truhlar
,
Phys. Chem. Chem. Phys.
11
(
46
),
10757
(
2009
).
6.
A. T.
Bell
and
M.
Head-Gordon
,
Annu. Rev. Chem. Biomol. Eng.
2
,
453
(
2011
).
7.
F. J.
Keil
,
Top. Curr. Chem.
307
,
69
(
2012
).
8.
C. J.
Cerjan
and
W. H.
Miller
,
J. Chem. Phys.
75
,
2800
(
1981
).
9.
J.
Simons
,
P.
Jorgensen
,
H.
Taylor
, and
J.
Ozment
,
J. Phys. Chem.
87
,
2745
(
1983
).
10.
E.
Cancès
,
F.
Legoll
,
M. C.
Marinica
,
K.
Minoukadeh
, and
F.
Willaime
,
J. Chem. Phys.
130
,
114711
(
2009
).
11.
D.
Poppinger
,
Chem. Phys. Lett.
35
,
550
(
1975
).
12.
B.
Peters
,
W.-Z.
Liang
,
A. T.
Bell
, and
A.
Chakraborty
,
J. Chem. Phys.
118
(
21
),
9533
(
2003
).
13.
M.
Laio
and
M.
Parrinello
,
Proc. Natl. Acad. Sci. U.S.A.
99
(
20
),
12562
(
2002
).
14.
M.
Ianuzzi
,
A.
Laio
, and
M.
Parrinello
,
Phys. Rev. Lett.
90
(
23
),
238302
(
2003
).
15.
B.
Ensing
,
M.
de Vivo
,
Z.
Liu
,
P.
Moore
, and
M. L.
Klein
,
Acc. Chem. Res.
39
,
73
(
2006
).
16.
E. M.
Mueller
,
A.
de Meijere
, and
H.
Grubmueller
,
J. Chem. Phys.
116
(
2
),
897
(
2002
).
17.
A.
Samanta
and
W.
E
,
J. Chem. Phys.
136
,
124104
(
2012
).
18.
S. K.
Burger
and
P. W.
Ayers
,
J. Chem. Phys.
132
,
234110
(
2010
).
19.
M.
Chen
,
M. A.
Cuendet
, and
M. E.
Tuckerman
,
J. Chem. Phys.
137
,
024102
(
2012
).
20.
S.
Maeda
,
K.
Ohno
, and
K.
Morokuma
,
J. Chem. Theory Comput.
5
,
2734
2743
(
2009
).
21.
S.
Maeda
and
K.
Morokuma
,
J. Chem. Theory Comput.
7
,
2335
(
2011
).
22.
S.
Maeda
,
E.
Abe
,
M.
Hatanka
,
T.
Taketsugu
, and
K.
Morokuma
,
J. Chem. Theory Comput.
8
(
12
),
5058
5063
(
2012
).
23.
H. B.
Schlegel
,
J. Comput. Chem.
3
,
214
(
1982
).
24.
G.
Henkelman
and
H.
Jonsson
,
J. Chem. Phys.
111
,
7010
(
1999
).
25.
A.
Heyden
,
A. T.
Bell
, and
F. J.
Keil
,
J. Chem. Phys.
123
,
224101
(
2005
).
26.
J.
Baker
,
J. Comput. Chem.
7
,
385
(
1986
).
27.
D. J.
Wales
,
J. Chem. Soc., Faraday Trans.
88
,
653
657
(
1992
).
28.
J. M.
del Campo
and
A. M.
Koster
,
J. Chem. Phys.
129
,
024107
(
2008
).
29.
H. B.
Schlegel
,
WIREs Comput. Mol. Sci.
1
,
790
(
2011
).
30.
H. B.
Schlegel
,
J. Comput. Chem.
24
,
1514
(
2003
).
31.
J.
Baker
,
A.
Kessi
, and
B.
Delley
,
J. Chem. Phys.
105
,
192
(
1996
).
32.
V.
Bakken
and
T.
Helgaker
,
J. Chem. Phys.
117
(
2
),
9160
(
2002
).
33.
S. R.
Billeter
,
A. J.
Turner
, and
W.
Thiel
,
Phys. Chem. Chem. Phys.
2
,
2177
2186
(
2000
).
34.
J.
Baker
,
J. Comput. Chem.
14
(
9
),
1085
1100
(
1993
).
35.
P.
Pulay
and
G.
Fogarasi
,
J. Chem. Phys.
96
,
2856
2860
(
1992
).
36.
B.
Paizs
,
J.
Baker
,
S.
Suhai
, and
P.
Pulay
,
J. Chem. Phys.
113
,
6566
6572
(
2000
).
37.
R.
Granot
and
R. A.
Baer
,
J. Chem. Phys.
128
,
184111
(
2008
).
38.
S. A.
Ghasemi
and
S.
Goedecker
,
J. Chem. Phys.
135
,
014108
(
2011
).
39.
P. Y.
Ayala
and
H. B.
Schlegel
,
J. Chem. Phys.
107
,
375
(
1997
).
40.
C.
Peng
,
P. Y.
Ayala
,
H. B.
Schlegel
, and
M. J.
Frisch
,
J. Comput. Chem.
17
,
49
(
1996
).
41.
C.
Peng
and
H. B.
Schlegel
,
Isr. J. Chem.
33
,
449
(
1994
).
42.
G.
Mills
and
H.
Jónsson
,
Phys. Rev. Lett.
72
,
1124
(
1994
).
43.
G.
Henkelman
and
H.
Jonsson
,
J. Chem. Phys.
113
,
9978
(
2000
).
44.
G.
Henkelman
,
B. P.
Uberuaga
, and
H.
Jonsson
,
J. Chem. Phys.
113
,
9901
(
2000
).
45.
S. A.
Trygubenko
and
D. J.
Wales
,
J. Chem. Phys.
120
,
2082
(
2004
).
46.
J.
Chu
,
B.
Trout
, and
B. A.
Brooks
,
J. Chem. Phys.
119
,
12708
(
2003
).
47.
D.
Sheppard
,
R.
Terrell
, and
G.
Henkelman
,
J. Chem. Phys.
128
,
134106
(
2008
).
48.
W.
E
,
W.
Ren
, and
E.
Vanden-Eijnden
,
Phys. Rev. B
66
,
052301
(
2002
).
49.
W.
E
,
W.
Ren
, and
E.
Vanden-Eijnden
,
J. Phys. Chem. B
109
,
6688
(
2005
).
50.
W.
Ren
and
E.
Vanden-Eijnden
,
J. Chem. Phys.
126
,
164103
(
2007
).
51.
S. K.
Burger
and
W.
Yang
,
J. Chem. Phys.
124
,
054109
(
2006
).
52.
S. K.
Burger
and
W.
Yang
,
J. Chem. Phys.
127
,
164107
(
2007
).
53.
H.
Chaffey-Millar
,
A.
Nikodem
,
A. V.
Matveev
,
S.
Kruger
, and
N.
Rosch
,
J. Chem. Theory Comput.
8
,
777
(
2012
).
54.
J.
Jung
,
S.
Re
, and
S.
Ten-no
,
J. Chem. Phys.
138
,
044106
(
2013
).
55.
Y.
Liu
,
S. K.
Burger
, and
P. W.
Ayers
,
J. Math. Chem.
49
,
1915
1927
(
2011
).
56.
J. B.
Brokaw
,
K. R.
Haas
, and
J.-W.
Chu
,
J. Chem. Theory Comput.
5
(
8
),
2050
(
2009
).
57.
K. R.
Haas
and
J.-W.
Chu
,
J. Chem. Phys.
131
,
144105
(
2009
).
58.
B.
Peters
,
A.
Heyden
,
A. T.
Bell
, and
A.
Chakraborty
,
J. Chem. Phys.
120
,
7877
(
2004
).
59.
A.
Goodrow
,
A. T.
Bell
, and
M.
Head-Gordon
,
J. Chem. Phys.
129
,
174109
(
2008
).
60.
W. A.
Quapp
,
J. Chem. Phys.
122
,
174106
(
2005
).
61.
A.
Behn
,
P. M.
Zimmerman
,
A. T.
Bell
, and
M.
Head-Gordon
,
J. Chem. Theory Comput.
7
(
12
),
4019
(
2011
).
62.
S. M.
Sharada
,
P. M.
Zimmerman
,
A. T.
Bell
, and
M.
Head-Gordon
,
J. Chem. Theory Comput.
8
(
12
),
5166
(
2012
).
63.
A.
Behn
,
P. M.
Zimmerman
,
A. T.
Bell
, and
M.
Head-Gordon
,
J. Chem. Phys.
135
,
224108
(
2011
).
64.
T. A.
Halgren
and
W. N.
Lipscomb
,
Chem. Phys. Lett.
49
,
225
(
1977
).
65.
E. F.
Koslover
and
D. J.
Wales
,
J. Chem. Phys.
127
,
234105
(
2007
).
66.
J.
Baker
and
W. J.
Hehre
,
J. Comput. Chem.
12
,
606
(
1991
).
67.
W. J.
Hehre
,
L.
Radom
,
P. v. R.
Schleyer
, and
J. A.
Pople
,
Ab Initio Molecular Orbital Theory
(
John Wiley & Sons
,
New York
,
1986
).
68.
A. D.
Becke
,
Phys. Rev. A
38
,
3098
(
1988
).
69.
C.
Lee
,
W.
Yang
, and
R. G.
Parr
,
Phys. Rev. B
37
,
785
(
1988
).
70.
A. D.
Becke
,
J. Chem. Phys.
98
,
5648
(
1993
).
71.
Y.
Shao
,
L. F.
Molnar
 et al.,
Phys. Chem. Chem. Phys.
8
,
3172
(
2006
).
72.
A.
Banerjee
,
N.
Adams
,
J.
Simons
, and
R.
Shepard
,
J. Phys. Chem.
89
,
52
(
1985
).
73.
W.
Humphrey
,
A.
Dalke
, and
K.
Schulten
,
J. Mol. Graphics
14
,
33
38
(
1996
).
74.
A.
Perczel
,
O.
Farkas
,
I.
Jakli
,
I. A.
Topol
, and
I. G.
Csizmadia
,
J. Comput. Chem.
24
,
1026
(
2003
).
75.
Reference 61 optimized the string with a steepest descent algorithm and used LST for interpolation, requiring 213 gradients to reach convergence of F = 0.2. Using GSM-IC, this level of convergence required 69 gradients. For comparison with Ref. 59, where delocalized coordinates are used to optimize nodes along the string but the interpolation was not done in the same coordinate system, convergence of alanine dipeptide required 339 iterations for 11 nodes. Using the same convergence criteria as that reference but using GSM-IC, GSM-IC requires 213 iterations to converge.
76.
P. M.
Zimmerman
,
A.
Paul
,
Z.
Zhang
, and
C. B.
Musgrave
,
Inorg. Chem.
48
,
1069
1081
(
2009
).
77.
P. M.
Zimmerman
,
Z.
Zhang
, and
C. B.
Musgrave
,
Inorg. Chem.
49
(
19
),
8724
8728
(
2010
).
78.
L.
Roy
,
P. M.
Zimmerman
, and
A.
Paul
,
Chem.-Eur. J.
17
(
2
),
435
439
(
2011
).
79.
P. M.
Zimmerman
,
Z.
Zhang
, and
C. B.
Musgrave
,
J. Phys. Chem. Lett.
2
,
276
281
(
2011
).
80.
O.
Diels
and
L.
Alder
,
Liebigs Ann. Chem.
460
,
98
(
1928
).
81.
L. F.
Tietze
and
G.
Kettschau
, “
Stereoselective heterocyclic synthesis I
,”
Top. Curr. Chem.
189
,
1
(
1997
).
82.
Y. J.
Hong
and
D. J.
Tantillo
,
J. Am. Chem. Soc.
133
,
18249
18256
(
2011
).
83.
M.
Koeksal
,
Y.
Jin
,
R. M.
Coates
,
R.
Croteau
, and
D. W.
Christianson
,
Nature (London)
469
,
116
(
2011
).
You do not currently have access to this content.