Minimum mode following algorithms are widely used for saddle point searching in chemical and material systems. Common to these algorithms is a component to find the minimum curvature mode of the second derivative, or Hessian matrix. Several methods, including Lanczos, dimer, Rayleigh-Ritz minimization, shifted power iteration, and locally optimal block preconditioned conjugate gradient, have been proposed for this purpose. Each of these methods finds the lowest curvature mode iteratively without calculating the Hessian matrix, since the full matrix calculation is prohibitively expensive in the high dimensional spaces of interest. Here we unify these iterative methods in the same theoretical framework using the concept of the Krylov subspace. The Lanczos method finds the lowest eigenvalue in a Krylov subspace of increasing size, while the other methods search in a smaller subspace spanned by the set of previous search directions. We show that these smaller subspaces are contained within the Krylov space for which the Lanczos method explicitly finds the lowest curvature mode, and hence the theoretical efficiency of the minimum mode finding methods are bounded by the Lanczos method. Numerical tests demonstrate that the dimer method combined with second-order optimizers approaches but does not exceed the efficiency of the Lanczos method for minimum mode optimization.

1.
C.
Wert
and
C.
Zener
,
Phys. Rev.
76
,
1169
(
1949
).
2.
G. H.
Vineyard
,
J. Phys. Chem. Solids
3
,
121
(
1957
).
3.
C.
Lanczos
,
An Iteration Method for the Solution of the Eigenvalue Problem of Linear Differential and Integral Operators
(
United States Government Press Office
,
1950
).
4.
R.
Malek
and
N.
Mousseau
,
Phys. Rev. E
62
,
7723
(
2000
).
5.
G.
Henkelman
and
H.
Jónsson
,
J. Chem. Phys.
111
,
7010
(
1999
).
6.
A.
Heyden
,
A. T.
Bell
, and
F. J.
Keil
,
J. Chem. Phys.
123
,
224101
(
2005
).
7.
J.
Kästner
and
P.
Sherwood
,
J. Chem. Phys.
128
,
014106
(
2008
).
8.
R. A.
Horn
and
C. R.
Johnson
,
Matrix Analysis
(
Cambridge University Press
,
Cambridge
,
1985
).
9.
L. J.
Munro
and
D. J.
Wales
,
Phys. Rev. B
59
,
3969
(
1999
).
10.
W.
E
and
X.
Zhou
,
Nonlinearity
24
,
1831
(
2011
).
11.
J.
Leng
,
W.
Gao
,
C.
Shang
, and
Z.-P.
Liu
,
J. Chem. Phys.
138
,
094110
(
2013
).
12.
Y.
Saad
,
Numerical Methods for Large Eigenvalue Problems
(
SIAM
,
1992
), Vol.
158
.
13.
L. N.
Trefethen
and
D.
Bau
 III
,
Numerical Linear Algebra
(
SIAM
,
1997
), Vol.
50
.
14.
Y.
Saad
,
SIAM J. Numer. Anal.
17
,
687
(
1980
).
15.
J.
Kuczynski
and
H.
Wozniakowski
,
SIAM J. Matrix Anal. Appl.
13
,
1094
(
1992
).
16.
A.
Poddey
and
P. E.
Blöchl
,
J. Chem. Phys.
128
,
044107
(
2008
).
17.
R. A.
Olsen
,
G. J.
Kroes
,
G.
Henkelman
,
A.
Arnaldsson
, and
H.
Jónsson
,
J. Chem. Phys.
121
,
9776
(
2004
).
18.
A. F.
Voter
,
Phys. Rev. Lett.
78
,
3908
(
1997
).
20.
A.
Samanta
and
W.
E
,
J. Chem. Phys.
136
,
124104
(
2012
).
21.
S. T.
Chill
,
J.
Stevenson
,
V.
Ruehle
,
C.
Shang
,
P.
Xiao
,
D.
Wales
, and
G.
Henkelman
, “
Benchmarks for characterization of minima, transition states and pathways in atomic systems
,”
J. Chem. Phys.
(unpublished).
22.
See http://optbench.org/ for the optimization benchmarks.
23.
See https://wiki.fysik.dtu.dk/ase/ for information about the ASE project.
24.
See http://theory.cm.utexas.edu/henkelman/code/ to obtain the TSASE code.
You do not currently have access to this content.