The latter region is often described by a trust radius, R. If the. P-RFO step is longer than the trust radius, the step is simply scaled down by a multiplicative factor.
Locating transition structures by mode following: A comparison of six methods on the Ar8 Lennard-Jones potential Frank Jensen Department of Chemistry, Odense University, DK-5230 Odense M., Denmark
~Received 10 November 1994; accepted 24 January 1995! Six different methods for walking from a minimum on a potential energy surface to a transition structure are tested on the Lennard-Jones surface for a cluster of eight argon atoms. The six methods consist of two Newton–Raphson-type algorithms using augmented Hessians, two methods for following gradient extremals, one following the intrinsic reaction coordinate on the image potential, and a constrained optimization technique. Only if the lowest mode of a given symmetry is followed can these methods locate transition structures in a stable manner. Optimizations along the higher modes display erratic or no convergence. The analysis shows that this is due to two factors: Hessian eigenvectors in general provide a poor direction for the uphill walk, and the presence of bifurcations along the path. © 1995 American Institute of Physics.
I. INTRODUCTION
Locating stationary points on potential energy surfaces ~PES! are requisite for many applications in computational chemistry. Locating minima is relatively easy; the gradient always gives a direction along which the energy can be lowered. Locating transition structures ~TSs!, which are first order maxima on the PES, is much more difficult. Several strategies have been proposed for locating saddle points on multidimensional surfaces.1 These may be divided into three catagories: ~i! methods based on some form of interpolation between reactant and product; ~ii! optimizations based on local information ~energy derivatives!; and ~iii! dynamical or statistical methods. Methods ~i! assume that the reactant and product are known and that a single TS connects these minima. Included in this category are linear and quadratic synchronous reaction paths,2 coordinate driving,3 constrained optimization techniques,4,5 and ridge methods.6 The methods belonging to category ~ii! are minimization of the gradient norm,7 different variations of Newton–Raphson approaches,8 –14 and gradient extremal algorithms.15–17 Only local information ~i.e., energy derivatives at a single point! are needed for advancing the algorithm, and they can be started at any point on the PES, which may be close to a minimum. The dynamical and statistical methods ~iii! generate approximations to TSs by molecular dynamics or Monte Carlo techniques.18 When the surface of interest contains only a few minima, and only the lowest TSs connecting these minima are of interest, many of the above methods perform well. The potential number of TSs grows factorial with the number of minima, however, many of these potential TSs may not exist. Methods relying on interpolation between the reactant and product lose some of their attraction as the number of minima increases. Furthermore, these methods can only locate one TS between two minima. For some systems, like rearrangement of atomic or molecular clusters or conformational changes in flexible molecules, it may be difficult to even find all the minima. In some cases it may be desirable to locate all TSs leading to a given minimum. If that is possible, then the intrinsic 6706
J. Chem. Phys. 102 (17), 1 May 1995
reaction coordinate ~IRC! can be followed downhill to establish connecting minima. By a series of such TS/IRC cycles, possible pathways on the complete surface may, in principle, be mapped out. The present paper will analyze the performance of six different methods for ‘‘walking’’ from a minimum to a TS. In the 2-dimensional case this corresponds to walking ‘‘uphill’’ through a ‘‘valley’’ to a ‘‘pass.’’ We note that there is no guarantee that all valleys will lead to a TS; a valley may be ‘‘blind,’’ with the TS located sideways.19 The above-mentioned methods almost all focus on following the lowest mode from a given minimum towards a TS. A few also claim the ability to walk along higher modes to locate other TSs, but very few actual examples exist. The advantages and disadvantages of optimization algorithms, including the present six methods have usually been discussed based on theoretical considerations and performances on 2-dimensional model surfaces or simple molecular reactions like the HNC to HCN isomerization. However, many applications of computational chemistry involve complex reactions on multidimensional surfaces. At present there are no examples of a direct comparison between different methods on more complicated surfaces. We have chosen a model system consisting of a cluster of eight argon atoms with the energy described by a pairwise Lennard-Jones potential for evaluating the performance. The resulting 18dimensional PES is a moderately complicated system with 8 minima and 42 TSs.13 The present investigation is focused on the feasibility of using uphill walking for locating TSs, and less on which strategy is the more efficient in terms of computer time. Analytical gradients and Hessians have thus been calculated whenever they were required. It is possible that Hessians generated by updating schemes may be sufficiently accurate for some applications. Furthermore, a fixed step size has been used for advancing the geometry. More efficient schemes would allow this step size to change dynamically during the optimization. The paper is organized as follows: In Sec. II, a brief description of the theory for each of the six methods are given. Section III describes computational details, and Sec. IV contains the results for walking from the lowest minimum
0021-9606/95/102(17)/6706/13/$6.00
© 1995 American Institute of Physics
Frank Jensen: A comparison on Ar8 Lennard-Jones potential
on the Ar8 surface. The performance of the methods is analysed in detail in Sec. V. II. DESCRIPTION OF THE ALGORITHMS
Expanding the energy to second order around the current geometry x 0 % ~ x2x ! , E ~ x! 5E ~ x ! 1gt ~ x2x ! 1 1 ~ x2x ! t H 0
0
0
2
0
and requiring the gradient to be zero, gives the standard Newton–Raphson ~NR! formula % 21 g. ~ x2x ! 52H 0
In the space of the Hessian eigenvectors this may be written as Dy5 ~ d y 1 , d y 2 ,... d y 3N-6 ! ,
d y i 52
fi , ei
where y i and f i are the projection of x and g along the Hessian eigenvector with eigenvalue e i . The 6 ~5! translational and rotational degrees of freedom are assumed to have been projected out of the Hessian prior to diagonalization,20 and the gradient contribution along these modes removed. If the Hessian is mass weighted before diagonalization, the eigenvectors at a stationary point correspond to the vibrational normal modes, and the term mode is used interchangeably with the eigenvector. As the real surface contains terms of higher order, the NR formula may be used as an iterate for stepping towards a stationary point. The NR approach attempts to converge on the ‘‘nearest’’ stationary point, and a ~usually very! good estimate of the TS geometry thus becomes mandatory. The small convergence radius of the straight NR iterate makes this impractical for routine use. If the starting geometry is outside the region where the Hessian has the desired nature ~i.e., one negative eigenvalue for a TS search!, the Hessian may be modified by introducing suitable shift parameter~s! such that the augmented Hessian has the desired structure. The resulting step can be written as Dy5 ~ d y 1 , d y 2 ,••• d y 3N-6 ! ,
d y i 52
fi . e i 2l
We will use the term augmented Hessian to describe methods where the step can be parameterized as above. Different strategies have been proposed for choosing a suitable l.8 –13 One popular method was suggested by Banerjee et al. and Baker,10 often denoted ‘‘eigenvector following.’’ Instead of the straight second order Taylor expansion, the energy is written as a rational function expansion. The resulting partitioned rational function optimization ~P-RFO! treats the TS mode separately from the minimization modes, which give the following formulas for determining l
(
iÞTS
f i2 l2 e i
5l,
1 1 l TS5 eTS6 2 2
Ae
2 2 TS14 f TS.
The sign in the lTS equation is chosen such that lTS is larger than eTS , and l is chosen such that it is smaller than the lowest non-TS Hessian eigenvalue. This ensures that the energy is maximized along the TS mode and minimized
6707
along the perpendicular directions. The user initially selects an eigenvector as the TS mode, at subsequent iterations this is chosen as the eigenvector which has the largest overlap with the previous TS mode. As the optimization converges on a stationary point, the shift parameters go toward zero and the predicted step approaches the standard NR step. One disadvantage of this method is that the P-RFO step length is often so large that the resulting step will take the geometry outside the region where the second order expansion is valid. The latter region is often described by a trust radius, R. If the P-RFO step is longer than the trust radius, the step is simply scaled down by a multiplicative factor. If this factor is much smaller than 1.0, it follows that the resulting step direction may not be best for the given trust radius. The best step with a step size equal to the trust radius can be determined by choosing a single l such that
(
iÞTS
f i2 ~ e i 2l !
21
2 f TS
~eTS1l) 2
5R 2 .
l is chosen such that it is smaller than the lowest non-TS eigenvalue and smaller than 2eTS . This procedure has been labeled the quadratic approximation ~QA! by Culot et al.12 If the pure NR step is within the trust radius, a straight NR iterate is done. The QA method is exactly equivalent to the trust region image method ~TRIM! by Helgaker.21 In the latter language, the image potential22,23 is constructed by inverting the sign of one of the Hessian eigenvalues and the component of the gradient along this eigenvector ~changing the sign of eTS and f TS!. Optimization of this image potential is then carried out by choosing a single l such that the step length is equal to the trust radius. As seen from the above equation this is equivalent to choosing lTS52l for the minimization modes. Sun and Ruedenberg have shown that a NR-type minimization often follows a different path than what is obtained from a steepest descent run when starting from the same geometry. They may even converge to different minima.24 The QA/TRIM algorithm can be considered as a NR-type minimization of the image potential. Sun and Ruedenberg have analyzed the concept of an image potential in detail and suggested that a method of walking from a minimum to a TS would be to follow the IRC on the image potential ~IPIRC!.23 It is important to realize that there is, in general, no ‘‘potential’’ associated with the image gradient field. The image gradient cannot be integrated to yield an image energy function, nor is the image Hessian the derivative of the image gradient. To conform with other work we will use the term image potential although there is no potential associated with the image gradient field. Sun and Ruedenberg considered only the image potential formed by inverting the lowest Hessian eigenvalue. If starting from a minimum, this can only be used to follow the lowest mode towards a TS. For following higher modes, one must consider all the possible 3N-6 different image potentials, corresponding to inverting the sign of any one of the Hessian eigenvalues. At a given geometry, there are thus 3N-6 different image potentials. After the first point, where the user selects which eigenvalue to invert, the image mode
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Frank Jensen: A comparison on Ar8 Lennard-Jones potential
is chosen based on the largest overlap of the Hessian eigenvectors with the previous image mode. At a given geometry the resulting image potential is thus determined by the history of the walk, i.e., the algorithm by which the geometry is advanced from one point to the next. It is conceivable that two different paths from the same starting geometry to the same final geometry will end up on two different image potentials. The values of the image gradient and Hessian are therefore not uniquely determined by the geometry and the starting conditions; they also depend on the algorithm used for changing the geometry. This is especially critical if the Hessian eigenvalue of the image mode becomes degenerate with one of the other modes at a given point along the path. The lack of a potential function which corresponds to the image gradient field makes the termination of the IPIRC problematic. On the normal surface the IRC run is terminated once the energy starts to rise. This criterion cannot be used on the image potential. Using the energy of the regular potential and terminating the run when the energy starts to decrease will only work if the IRC path is followed exactly. One may instead look for a minimum in the gradient norm, if this is sufficiently close to zero, it is likely that a stationary point is nearby. In the present case we calculate the pure NR step if the image Hessian is positive definite. If this steplength is less than 0.10 a.u. or the present step size, we switch to the straight NR iterate which usually converge to the stationary point in three or four steps. The Gonzales– Schlegel method25 has been used for following the IRC. Sun and Ruedenberg have used their quadratic IRC method on a 2-dimensional model surface.24 While both the augmented Hessian methods maximize the energy along the TS mode and minimize the energy along the perpendicular modes, they make no particular effort to stay close to the valley floor. If the reaction path is curved, they will, in general, walk on one of the valley sides, and may cross over a ridge. Furthermore, they may converge on stationary points which do not have the required one negative eigenvalue in the Hessian. To follow the gentlest ascent from a minimum to a TS, one may consider gradient extremals ~GEs!.15–17 A GE is the path along which the gradient norm changes least. It may be shown that the gradient is an eigenvector of the Hessian, Hg5 e g, on the GE, and that the GE is a line connecting points on steepest descent paths where the curvature is zero.17 Contrary to the IRC, which is the steepest descent path from the TS to the minimum, the GE is defined locally, i.e., it may be determined without prior knowledge of the TS. We have considered two algorithms for following GEs. The first, due to Jørgensen, Jensen, and Helgaker ~GE-JJH!, takes advantage of the fact that the gradient is an eigenvector of the Hessian on the GE.16 At a given point away from the GE a step may be calculated which reduces the gradient components along the perpendicular modes to zero. Employing a second order expansion of the energy and limiting the steplength to a given trust radius, this may be formulated as % 21 g1 a v, ~ x2x0 ! 52P% H
P% 512vvt .
The vector v is the Hessian eigenvector of the mode being followed. The projection matrix P removes the TS
mode from the Hessian and the first term in the predicted step is a pure NR step in the perpendicular directions. If the length of this step is smaller than the allowed step size, an additional step is taken along the TS mode with a chosen such that the resulting step has length R. If the step length determined from the projected NR formula is longer than the allowed step size, the step is scaled down and no step is taken along the TS mode ~a50!. In the eigenvector space the step may be written as Dy5 ~ d y 1 , d y 2 ,... d y 3N-6 ! ,
d y iÞTS52
fi , ei
d y TS5 a .
The GE-JJH algorithm may also be considered as a variation of the augmented Hessian methods, where the optimization ~which is not necessarily a minimization, see below! along the perpendicular modes always is performed before any further advance along the TS mode. The second GE algorithm considered is due to Sun and Ruedenberg ~GE-SR!.17 In this case the tangent to the GE path is constructed and a step is taken along the tangent to produce a predictor geometry. At the predictor geometry a correction step is calculated which brings the geometry back to the GE path. Both the construction of the tangent and the correction step require knowledge of one component of the third derivative of the energy, which may be calculated from two Hessians at slightly displaced geometries. In the coordinate system where the first coordinate is along the gradient and the rest makes the Hessian diagonal, denoted y 8 ,26 the GE tangent, e, can be obtained by solving the following set of equations: 3N-6
(
A ni e i 5B n e 1 ,
n52,3,...,3N-6,
i52
A ni 5
] H 8ni ] y 81
u gu 1 d ni e n ~ e n 2 e 1 ! ,
B n 52A n1 52
] H 8n1 ] y 81
u gu .
Note the minus sign in the last equation, there is a mis8 and e i are Hessian elements in print in the original paper. H ni the primed coordinate system ~off-diagonal and diagonal elements, respectively!. The value of e 1 is determined by normalization. The corrector step is calculated as follows: 3N-6
8 , d y i8 52 ( $ A% 8 † ~ A% 8 A% 8 † ! 21 % in u gu H n1 n52
i51,2,...,3N-6, A 8ni 5
8 ] H ni ] y 18
u gu 1 ~ H8 2 ! ni 2H 811 H 8ni ,
n52,3,...,3N-6.
The A 8ni elements reduce to A ni for points on the GE. For more details the reader is referred to the original paper.17 In the above implementation, the GE-JJH method only accidentally determines points actually on the GE path. The step calculated from the projected NR formula brings the
J. Chem. Phys., Vol. 102, No. 17, 1 May 1995
Frank Jensen: A comparison on Ar8 Lennard-Jones potential
geometry towards the GE path, but either the allowed step size is not large enough to actually reach the path, or if it is, the remaining step size taken along the TS eigenvector results in a geometry off the path. A more serious problem is that there is no guarantee that the direction of the Hessian TS eigenvector is along the GE path. The GE-SR algorithm has the potential of following a GE path as closely as desired. In the original formulation, a step is taken along the GE tangent to a predictor geometry. At this point a correction step is calculated from the partial third derivative of the energy. At the predictor geometry an approximation to the GE tangent at the corrector geometry is also calculated. This approximate GE tangent is calculated 8 from the same set of equation as above, but using the A ni matrix elements instead of A ni . Provided that the correction step is not too large, this approximate tangent may be good enough for producing the next predictor step. This strategy was chosen such that only one partial third derivative of the energy is required for each combined predictor–corrector step. To allow very accurate determinations of the GE, we have modified the original algorithm by adding a microiterate on the correction step, i.e., as long as the correction step is longer than a specified threshold, only the correction step is taken and a new set of predictor–corrector directions are calculated. The present data have been generated using a criteria of 1025 a.u. for the corrector threshold. This ensures that all predictor steps are taken from points actually on the GE. A potential problem with the GE-SR algorithm is that the corrector step takes the geometry towards a point on the nearest GE ~nearest does not necessarily mean the shortest in terms of geometric distance!. If the path is very curved, or two GEs come close together, the predictor step may take the geometry so far off the exact path that the corrector step point towards a different GE. This is especially troublesome near stationary points where 3N-6 different GEs cross. Abashkin and Russo have proposed a different approach for determining approximate reaction paths from a minimum to a TS by a series of constrained optimizations.27 A point on the reaction path is determined as the geometry which minimizes the energy, subject to the constraint that the geometry must be on a hypersphere with a given radius and the minimum as the expansion point. By gradually increasing the radius of the hypersphere, a geometry close to the actual TS may eventually be located. This algorithm is in spirit very closely related to the bracketing scheme proposed by Dewar et al.4 The success of this method depends on the validity of two assumptions. First, the algorithm will only be useful for walking up modes which do not ‘‘turn back’’ on themselves, i.e., it is conceivable that some TSs are closer to the minimum in terms of distance than the point on the path which is farthest away from the minimum. In such cases the algorithm will suddenly produce a very large change in geometry for a small increase in the hypersphere radius and the energy will start to decrease. This discontinuity is similar to what is often observed when generating an approximate reaction path by driving a selected internal coordinate. Second, the constrained optimization for following modes other than the lowest may be numerically quite tricky. In this case the geometry of interest is not the one with the lowest energy on
6709
the hypersphere, but rather a specific stationary point on the hypersphere. For following the lowest mode ~to either side!, the stationary point is a ~local! minimum, but for following the nth lowest mode, the desired stationary point on the hypersphere has (n21) negative eigenvalues, at least near the minimum. The algorithm formulated by Abashkin and Russo solve the constrained optimization ~CONOPT! problem by explicitly eliminating one of the variable by the constraint condition.27 Internal coordinates were used as the variables, and the unique variable to eliminate was chosen based on the nature of the problem. Only the gradient of the energy was used in the constrained optimization. We have chosen to formulate the problem directly in Cartesian coordinates, and making use of both first and second derivatives. The 3N-6 unique coordinates have been chosen as those which make the Hessian diagonal, and the coordinate eliminated is the one corresponding to the Hessian mode being followed uphill. Following Abashkin and Russo, the constraint condition is given by
F(
3N-6
R5
~ y i 2a i ! 2
i51
G
1/2
,
where a is the coordinates for the minimum and R is the hypersphere radius. Solving for a specific coordinate gives
F
3N-7
y TS5a TS6 R 2 2
(
~ y i 2a i !
2
i51
G
1/2
.
Choosing either the 1 or 2 sign corresponds to following the mode in each of the two possible directions. The derivatives of the reduced energy function, E 8 (y 1 ,y 2 ,•••y 3N-7 ,R) may be written as
]E8 ]E ] E ] y TS 5 1 , ] y j ] y j ] y TS ] y j where the derivative of y TS is given by
F
] y TS 57 ~ y j 2a j ! R 2 2 ]y j
3N-7
(
~ y i 2a i !
i51
2
G
21/2
.
The corresponding formulas for the second derivatives of the reduced energy function are
] 2E 8 ] 2E ] 2 E ] y TS ] 2 E ] y TS 5 1 1 ] y j ] y k ] y j ] y k ] y j ] y TS ] y k ] y k ] y TS ] y j 1
] 2 E ] y TS ] y TS ] E ] 2 y TS 1 , 2 ] y TS ] y j ] y k ] y TS ] y j ] y k
F
] 2 y TS 57 ~ y j 2a j !~ y k 2a k ! R 2 2 ] y j] y k
F
3N-7
6 d jk R 2 2
J. Chem. Phys., Vol. 102, No. 17, 1 May 1995
(
i51
~ y i 2a i !
2
3N-7
(
~ y i 2a i !
i51
G
21/2
.
2
G
23/2
J. Chem. Phys., Vol. 102, No. 17, 1 May 1995
182
E
E
E
E
7
E
E
10
E
E
10
181
1
5
E
172
E
36
9
E
37
E
E
152 34
1
10
E
1
28
151
1
171
25
16
28
141
142 37
1
1
/1
37
1
5
13
7
12
112
111
2
34
E
E
5
E
1
16
1
5
5
1
10
24
1
37
4
7
5
10
5
28
5
36
92
D
4
9
1
7
24
82
1
91
8
81
D
10
72
9
7
71
36
6
36
52
6
1
51
7
9
7
1
30
E
E
4
E
5
28
7
4
D
1
1
5
9
7
1
50
E
E
E
10/1
1
5
28
34/4
/38
9
1/9
4
2
1
1
16
5
5
7
1
1
E
E
E
2
1
1
1
7
1
4
1
1
5
10
7
6
1
7
5
7
1
2
E
E
E
1
34
28
16
7
4
24
4
2
3
30
9
1
4
7
5
7
1
5
E
E
E
1
1
9
2
1
2
1
5
2
5
1
D
5
3
1
7
9
7
1
10
E
E
E
28
1
1
7
4
26
26
29
3
7
9
7
1
20
QA/TRIM
E
E
4
E
2
5
3
4
10
6
16
1
1
5
9
7
1
30
E
E
4
E
18
23
2
7
16
D
3
16
5
24
7
1
50
E
E
1
E
1/
1
/1
16
1
7
4/
1
6
2
2
30
7
1
36
1
5
5
7
1
1
E
E
1
E
1
9
1
16/
1
6
6
1/7
1
2
16
9
1
2
36
1
23
9
5
7
1
2
E
E
1
E
5
16
5
16
9
7
10
1
5
1
4
6
1
7
1
5
7
1
5
E
E
E
4
1
1
7
1
10
2
2
1
1
5
1
16
28
2
1
1
5
1
5
7
1
E
E
E
E
25
6
7
7
1
6
36
5
1
5
7
1
20
IPIRC 10
E
E
E
E
10
16
7
1
1
4
28
5
1
5
7
5
1
5
7
1
30
5
5
5
1
7
1
1
6
5
9
7
1
50
5
7
5
7
1
1
7
7
5
7
1
2
E
E
7
7
7
7
5
7
1
5
E
7
7
7
7
1
7
5
7
1
E
E
E
7
7
7
7
7
7
7
5
7
1
20
GE-JJH 10
E
E
E
7
1
7
7
7
7
7
1
5
5
7
1
30
E
E
E
E
E
1
7
9
1
9
7
5
9
7
1
50
E
E
D
E
5
7
1
1
E
E
D
E
5
7
1
2
E
E
D
E
5
7
1
5
E
E
D
E
5
7
1
E
E
D
E
7
1
20
GE-SR 10
E
E
D
E
/D
1
30
E
E
E
D
D
D
D
D/
/D
50
E
E
D
5
7
1
1
E
E
D
5
7
1
2
E
E
D
5
7
1
5
E
E
D
5
7
1
10
E
E
E
E
D
D
7
1
20
CONOPT
E
E
E
E
D
5
7
1
30
E
E
E
E
D
D
D
1
50
D: Dissociation. E: Energy raises above 0 kcal/mol. Notation 1/9 indicates that two different TS’s are found, even though the two directions should be equivalent by symmetry. Blanks indicate no convergence on location of a stationary point which is not a TS. Header shows stepsize in units of 0.01 a.u. Total pathlength is 10 a.u.
a
9
1
9
7
1
1
18
9
7
1
1
1
9
7
1
4
9
22
7
1
20
P-RFO
10
32
7
21
5
7
1
1
2
31
1
Mode
TABLE I. Transition structures located by mode following starting from the lowest energy minimum.a
6710 Frank Jensen: A comparison on Ar8 Lennard-Jones potential
Frank Jensen: A comparison on Ar8 Lennard-Jones potential
The resulting optimization of the reduced energy function in the 3N-7 dimensional space is done by a straight NR iterate. The start geometry for the optimization is initially generated by taking a step along the chosen eigenvector of the Hessian at the minimum, and subsequent points at larger constraint radii are generated by linear extrapolation along the two previous optimized points. The length of the extrapolation step is defined as the step size, the increment between subsequent hypersphere radii on which the optimization is carried out is thus always smaller than or equal to the step size. For sufficiently small step sizes, the guess geometry is close enough to the desired stationary point that the straight NR iterate converges. The main objective of the different walking algorithms is to get sufficiently near a TS that a standard NR procedure will converge on the exact TS geometry. In the present investigation we switch to a pure NR iterate if the Hessian has the correct structure ~one negative eigenvalue which is the mode being followed! and the corresponding NR step is smaller than 0.10 a.u. or the present step size.
TABLE II. Transition structure energies ~kcal/mol! and minima connected by IRC.a
III. COMPUTATIONAL DETAILS
The energy surface used is a Lennard-Jones potential with e5121 K and s53.4 Å.13 Gradients and Hessians have been calculated analytically. The overall translational and rotational modes are projected out of the Hessian.20 All algorithms used ~non-mass weighted! Cartesian coordinates as variables. The initial geometry off the minimum for the P-RFO, QA/TRIM, GE-JJH, and GE-SR methods has been generated by taking a step along the chosen eigenvector with a step size of 0.01 a.u. The initial step for the IPIRC and CONOPT methods depends on the allowed step size. Two sets of runs have been made. In one series the step size was varied ~0.01, 0.02, 0.05, 0.10, 0.20, 0.30, 0.50 a.u.! and the maximum number of steps allowed was taken so the total path length was 10 a.u. The 10 a.u. distance is for all the TSs longer than the corresponding pathlength for following the IRC from the TS to the minimum, in the majority of the cases by a factor of 3 to 4. The other series employed a fixed step size of 0.10 a.u. but allowed a total of 1000 steps, i.e., a total pathlength of 100 a.u. The image potential was constructed by diagonalizing the Hessian, projecting the gradient along each eigenvector, inverting the sign of the chosen eigenvalue and corresponding gradient component, and backtransforming the derivatives to the original coordinate system. Optimizations were stopped if all atoms belonging to one fragment of the cluster was found to be more than 6 Å away from all atoms belonging to another fragment, such terminations are labeled as dissociative in the tables. In principle, only the totally symmetric modes need to be followed in both directions as the two paths generated by following nonsymmetric modes should be equivalent by symmetry. In all cases, however, have all the normal modes been followed in both directions. Tsai and Jordan have studied the same Ar8 system, they found a total of 41 TSs.13 In the present work, we located one additional TS, thus there have so far been located 42 TSs on the Ar8 Lennard-Jones surface ~geometries are available upon request!. The 42 TSs have been labeled according to
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a
TS
Energy
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
24.636 177 24.606 850 24.550 832 24.522 010 24.517 781 24.512 561 24.498 953 24.498 228 24.495 144 24.492 717 24.483 392 24.481 675 24.471 735 24.471 564 24.467 193 24.463 679 24.461 648 24.454 668 24.454 657 24.451 956 24.447 965 24.446 716 24.434 980 24.432 849 24.429 132 24.417 717 24.408 043 24.400 136 24.398 884 24.393 170 24.390 486 24.382 656 24.344 885 24.333 271 24.329 736 24.319 228 24.304 258 24.300 088 24.294 894 24.288 413 24.276 910 24.031 338
Min. A ~mode! S tot 1 1 2 1 1 1 1 2 1 1 2 2 2 2 2 1 3 1 3 1 4 1 1 1 1 2 4 1 1 1 1 2 7 1 8 1 1 1 1 1 7 1
~2! ~1! ~2! ~1! ~2! ~1! ~2! ~1! ~2! ~1! ~1! ~1! ~1! ~1! ~1! ~1! ~1! ~1! ~3! ~1! ~1! ~1! ~2! ~2! ~1! ~1! ~1! ~1! ~2! ~2! ~1! ~1! ~1! ~1! ~1! ~2! ~1! ~1! ~1! ~2! ~1! ~2!
2.5 3.0 4.8 3.3 5.1 3.5 3.2 5.0 3.5 3.3 5.0 4.9 5.2 4.9 6.0 3.9 2.3 4.1 2.4 4.1 2.5 4.1 3.8 4.5 4.1 6.7 4.0 4.3 4.3 4.6 4.3 7.9 3.6 5.2 3.6 5.8 6.3 6.2 6.1 6.6 5.8 8.1
Min. B ~mode! S tot 2 1 2 2 1 2 3 5 4 3 6 6 8 7 2 1 6 1 7 1 8 1 1 1 1 3 3 6 7 8 6 2 8 1 8 1 6 7 6 8 8 1
~1! ~1! ~2! ~1! ~2! ~1! ~1! ~1! ~1! ~1! ~1! ~1! ~1! ~1! ~1! ~1! ~1! ~1! ~1! ~1! ~1! ~1! ~2! ~2! ~1! ~1! ~1! ~1! ~1! ~1! ~1! ~1! ~1! ~1! ~1! ~2! ~1! ~1! ~1! ~1! ~1! ~2!
2.2 3.0 4.8 4.8 5.1 4.9 2.1 2.3 2.3 2.1 2.0 2.0 2.0 1.9 6.0 3.9 2.2 4.1 2.2 4.1 2.1 4.1 3.8 4.5 4.1 3.7 4.0 3.0 3.0 2.9 3.2 7.9 3.4 5.2 3.6 5.8 4.9 4.8 4.8 5.0 5.4 8.1
Value in parenthesis is the mode along which the IRC enter the minimum. S tot is the path length in a.u.
energy. IRC paths for establishing reaction paths from the TSs to the minima have been done by the Gonzales–Schlegel method with a step size of 0.01 Å.26 IV. RESULTS
We have run the six algorithms with seven different stepsizes ~0.01, 0.02, 0.05, 0.10, 0.20, 0.30, and 0.50 a.u.! along each of the normal modes in both directions, for all eight nonequivalent minima on the PES. The results for walking up the normal modes from the lowest minimum are given in Table I. The numbers in the table refer to TSs which has been labeled according to their energies. A list of all the TSs, their energies, and the minima they connect by means of IRCs is given in Table II. In the runs in Table I the total allowed pathlength was 10 a.u. Blank entries indicate no convergence
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within the allowed number of steps, or convergence to a stationary point which is not a TS. Walks which lead to geometries indicative of dissociation are labeled with a D, those which lead to high energies are labeled with E. Results from starting from other minima give qualitatively similar features and have been omitted. Of the 42 TSs given in Table II, 26 have IRCs which lead to minimum 1. Out of these 26 TSs, 12 connect minimum 1 with an equivalent minimum 1, 3 connect it to minimum 2, 2 connect it to minimum 3, 1 connects it to minimum 4, 4 connect it to minimum 6, 2 connect it to minimum 7 and 2 connect it to minimum 8. There are no TS which connect minimum 1 with minimum 5, in fact there is only one TS associated with minimum 5, this is TS8 which connects minima 2 and 5. There are several other pairs of minima which are not connected by a single TS, e.g., minimum 4 has no direct path to minima 6 or 7, and minimum 6 has no direct path to minima 7 or 8. The lowest minimum has C s symmetry, and modes number 1, 4, 6, 10, 12, 13, and 16 belong to the a 9 representation. These modes need, in principle, only be followed in one direction as the other direction is equivalent by symmetry. However, to model real applications, the starting minimum geometry was generated by optimization of a nonsymmetric structure to a gradient norm of 1029 a.u. This results in a starting geometry which deviates from having exact C s symmetry by errors of ;1029 a.u. All modes were then followed in both directions. Looking across the methods in Table I, it is seen that all locate TS1 by following the lowest mode, independent of the employed step size except for the GE-SR method at step size 0.50. The second lowest mode is totally symmetric and two different TSs are located by walking in the two different directions. Along the positive direction, all methods find TS7, again with the exception of the GE-SR and CONOPT methods at large step sizes. Along the negative direction the P-RFO method finds TS9, the QA/TRIM finds TS5 or TS9 dependent on step size, and the other methods converge on TS5, again with some instability for the larger step sizes. The walks along the remaining higher modes show a completely different picture. The augmented Hessian and IPIRC methods essentially find TSs at random, the GE-JJH method only locates TSs for large step sizes, and the GE-SR and CONOPT methods do not converge at all. Note that the failure to locate TSs is not due to following dissociative pathways, very few of the walks result in geometries indicative of dissociation. Neither is the failure due to following modes which have atoms colliding; only three of the walks along the two highest modes lead consistently to high energy structures. The number of runs which do not converge depends on the total length of the walk. If the maximum length is increased by a factor of 10 ~1000 steps of 0.10!, the results given in Table III are obtained. Of the 36 walks, 27 converge on a TS for the P-RFO, 31 for the QA/TRIM, and 32 for the IPIRC method. The longer GE walks do not result in location of more TSs. The longer allowed pathlength has no influence on the CONOPT walks as they terminate long before the 10 a.u. path length in Table I.
TABLE III. Transition structures located by mode following starting from the lowest energy minimum. Step size50.10 a.u., total pathlength5100 a.u.a Mode
P-RFO
QA/TRIM
IPIRC
GE-JJH
GE-SR
1 21 22 31 32 4 51 52 6 71 72 81 82 91 92 10 111 112 12 13 141 142 151 152 16 171 172 181 182
1 7 9 37 1 1 S2 S2 1 9 4 S D 13 28 1
1 7 9 7 15 1 S2 3 11 26 6 5 D 1 5 5 2 5 1 2 1 2 9 1 1 E 13 E E
1 7 5 1 5 6 1 2 28 16 1 5 6 1 1 2 2 10
1 7 5 7 1 7 7
1 7 5 S3 S5 S0 S5
7/ 7
S6
7
S0
a
5 4 1 14 1 S 37 38 E 10 E E
D
1 7 25 1 4 E
E
E E
E E
E D E E
D: Dissociation. E: Energy raises above 0 kcal/mol. Sn: Stationary point with n negative Hessian eigenvalues. Blanks indicate no convergence. Notation 7/ indicate that only one of the two direction lead to convergence, although they should be equivalent by symmetry.
The augmented Hessian walks which fail to converge in the limit of very long path lengths fall into three categories: ~i! convergence on a stationary point which is not a TS, typically a point with two negative eigenvalues in the Hessian; ~ii! the energy and gradient increase to large values, indicative of a mode which moves the atoms toward each other; ~iii! the algorithm stalls in the sense that the optimization keep moving in the same area of the PES which apparently does not contain a stationary point. Category ~i! is typically observed when following totally symmetric modes, where one of the nonsymmetric modes may develop a negative eigenvalue, but there is no gradient component along this direction to break the symmetry. Category ~ii! is normally seen when following the highest two or three modes. The behavior in category ~iii! has only been observed for the P-RFO method, following mode 111~Table III! is one such example. The random aspect, or more precisely, the extreme sensitivity of convergence characteristics to changes in optimization parameters for the augmented Hessian and IPIRC methods may be illustrated by the seven runs with different step sizes along mode 10 for the QA/TRIM method. As mentioned above, the minimum deviates by ;1029 a.u. from being exactly C s symmetric. The two starting geometries formed by displacing the geometry along the two directions
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of the eigenvector thus also differ from being symmetry equivalent by ;1029 a.u. Nevertheless, after a number of steps this error is magnified sufficiently that different convergence is observed, for example with a step size of 0.01 a.u. the walk locates either TS1 or TS9. The other six step sizes locate four different TSs and two fail to converge! This clearly illustrates that the convergence behavior of the augmented Hessian and IPIRC methods for walking along the higher modes is very sensitive to details in the starting geometry and the optimization parameters. The reason for this sensitivity will be discussed in Sec. V. The CONOPT method converges smoothly for the lowest two modes, with some exceptions for the larger step sizes. None of the other runs converge on a TS, the reason being either ~i! the optimized geometry for two successive hypersphere radii suddenly change significantly and the energy starts to decrease, or ~ii! the constrained optimization for a given hypersphere has not converged after 100 steps. The first scenario indicates passage of a maximum, without the path passing sufficiently close to a TS that the NR iterate is activated. The second type of failure may be due either to a poor estimate of the starting point for the optimization ~large step size! or the straight NR iterate oscillates between two geometries without achieving convergence. The ~i! reason is by far the most common type of failure, but the type ~ii! failure ~oscillation! is, for example, observed in the 22 walk with a step size of 0.20 ~Table I!. V. ANALYSIS AND DISCUSSION
From Tables I and III it is clear that the behavior of the optimizations along the lowest two modes are significantly different than walks along higher modes. The low-lying modes generally display stable convergence characteristics, i.e., convergence to the same TS for different step sizes. Along the higher modes, the augmented Hessian and IPIRC methods essentially find TSs more or less at random. For the GE methods is it even more clear that there is a fundamental difference between following the lowest mode of a given symmetry28 and following the higher modes. The GE-SR method, which for small step sizes follows the GE path very closely, converges on a TS in only three cases. For all 8 minima and 42 TSs, there are only 10 such pairs which are connected directly by GEs, these are given in Table IV. For minimum 5 there is no direct GE path leading to a TS. These results are not artifacts or difficulties in following the GE uphill. The GE paths which can be followed uphill can also be followed downhill by starting GE walks from the TS along the negative Hessian eigenvector. Starting GE walks from TSs other than those given in Table IV do not lead to minima. Why does the GE-SR method locate so few TSs? A closer analysis of the GE walks along the higher modes show that the reason for this is the presence of bifurcations along the path. At some point along the walk, one of the perpendicular Hessian eigenvalues change from positive to negative. This corresponds to a valley splitting into two side valleys.29 The GE path which hugs the valley floor before the bifurcation point, continues on the ridge between the two side valleys after the bifurcation. This GE path may in some
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cases lead to stationary points which have more than one negative eigenvalue in the Hessian. As we are not interested in such points, these walks just continue and appear in Table I as ‘‘not converging.’’ Near such stationary points the GE switching problem mentioned in Sec. II is sometimes observed. In some cases the long GE walks shown in Table III return to the same minimum they started from without passing through other stationary points. Most of the GE-SR walks in Table III terminate because the microiterate for the corrector step fail to converge. For the lowest two modes and sufficiently small stepsizes the results from the GE-JJH method mirror those from the GE-SR method. Along the higher modes, the GE-JJH method has trouble converging the optimization along the perpendicular modes, as shown below. This may be due to the fixed step size employed, incorporation of a dynamical trust radius may alleviate this problem. However, even if this problem is solved, the GE-JJH method will display the same problem as the GE-SR, it will follow the GE continuing on the ridge. This is due to the fact that the projected NR step taken along the perpendicular modes is an optimization step, and not a minimization step. Before the bifurcation point the perpendicular eigenvalues are positive, and the projected NR step is a minimization step. However, after the bifurcation point one of the perpendicular eigenvalues is negative, and the projected NR step is towards a first order maxima in the perpendicular directions, i.e., a point on the ridge. For the larger stepsizes the GE-JJH may occasionally come close enough to a TS that the pure NR step converges on it. The augmented Hessian methods always take a minimization step along the perpendicular modes, this is ensured by the l shift parameter. Why is it then that these methods perform so poorly when walking along the higher modes? The analysis below shows that the fundamental problem is that the Hessian TS eigenvector in general does not give the correct uphill direction, except for the lowest mode of a given symmetry. Before analyzing some of the walks in detail, we need to discuss the proper reference. The usual definition of the reaction path is the steepest descent curve traced from the TS to the minimum. At each step along such a curve the Hessian eigenvectors can be calculated, and by tracing the mode which has the largest overlap with the TS mode in the previous step, the entry mode to the minimum may be estab-
TABLE IV. Transition structures and minima which are connected directly by gradient extremals. Minimum
Mode
TS
1 1 1 2 3 3 4 6 7 8
1 21 22 1 1 3 1 11 11 11
1 7 5 8 7 19 9 11 19 21
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FIG. 1. Distance to gradient extremal for the GE-JJH, P-RFO, QA/TRIM, IPIRC, and CONOPT paths with a step size of 0.01 a.u. for walks along mode 1.
FIG. 3. Overlap between the GE tangent and the Hessian TS eigenvector along the mode 1 GE path.
lished. Of the 26 reaction paths leading to minimum 1, 15 enter along the lowest mode and 11 along the second lowest mode, as shown in Table II. These two modes have different symmetry; in general, an IRC path enters the minimum along the lowest mode of the given symmetry.30 This clearly illustrates the intrinsic problem of establishing methods which can follow the IRC in the uphill direction. For example, for the present system, such methods would have to leave the minimum along the same mode but locate 15 different TSs! For the TSs which cannot be reached by uphill walking along the lowest mode of a given symmetry, the question is thus whether there exists a path, different from the IRC, which leads from the minimum to the TS. Such a path leaves the minimum along one of the higher modes and it can enter the TS either ~i! along the negative Hessian eigenvector, or ~ii! along one of the positive modes. In case ~i! the alternate path will approach the IRC asymptotically near the TS. Case ~ii! corresponds to a path which encounters a bifurcation, as one of the perpendicular modes must change from positive to negative curvature. Furthermore, the energy must decrease in the final stage of the optimization as the approach to the TS is along a mode with positive curvature. Such a path would
be very different from the usual definition of a reaction path, but it may of course be relevant for locating TSs. We will consider a GE of type ~i! to be the logical choice for defining such alternative reaction paths. For the lowest mode of a given symmetry, we will consider the GE as the reference path, and not the IRC.31 Displaying the behavior of different optimization methods on two-dimensional surfaces is possible by means of, for example, contour diagrams, but impossible for the present 18-dimensional system. Furthermore, there are 36 different walks to analyze for each of the six method and each of the seven step sizes. We have chosen to show some details of three of these runs: ~1! the walk along mode 1 to TS1, ~2! the walk along mode 22 which converges on TS5 or TS9 depending on method and stepsize, and ~3! the walk along mode 142 as a representative for the situation which occurs along the higher modes. To measure the difference between the paths generated by the six methods, we use the shortest distance to the reference curve, and plot this distance as a function of the pathlength of the reference curve.32 Figure 1 shows the distance to the GE curve for the GE-JJH, P-RFO, QA/TRIM, IPIRC, and CONOPT methods with a stepsize of 0.01 a.u. The GE-
FIG. 2. Distance to gradient extremal for the GE-JJH, P-RFO, QA/TRIM, IPIRC, and CONOPT paths with a step size of 0.10 a.u. for walks along mode 1.
FIG. 4. Distance to gradient extremal for the GE-JJH, P-RFO, QA/TRIM, IPIRC, and CONOPT paths with a step size of 0.01 a.u. for walks along mode 22.
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FIG. 5. Distance to gradient extremal for the GE-JJH, P-RFO, QA/TRIM, IPIRC, and CONOPT paths with a step size of 0.10 a.u. for walks along mode 22.
FIG. 7. Energy as a function of pathlength for the GE-SR, GE-JJH, P-RFO, QA/TRIM, IPIRC, and CONOPT methods with a step size of 0.01 a.u. for walks along mode 142.
JJH method follows the exact GE closely. The QA/TRIM and IPIRC runs show very similar characteristics. The P-RFO method is the poorest with a maximum deviation of ;0.15 a.u. Figure 2 shows the same plot but with a stepsize of 0.10 a.u. The deviations for the P-RFO, QA/TRIM, and GE-JJH methods increase while the IPIRC is very similar to that in Fig. 1. Figure 3 shows the overlap between the GE tangent and the Hessian TS eigenvector along the GE path. If the GE tangent is taken as the correct uphill direction, this measures how well the Hessian TS eigenvector approximate the correct path direction. The dip in this curve to 0.97 at a path distance of 0.95 a.u. coincides with the region where the deviation of the augmented Hessian and IPIRC paths from the GE increases most, Figs. 1 and 2. The same three types of plots for the walk along mode 2are given as Figs. 4, 5, and 6. The difference between the GE and the P-RFO, QA/TRIM, and IPIRC increases rapidly at a path distance of ;0.60 a.u. The P-RFO, and QA/TRIM at the larger step size, dissipates into a valley which leads to TS9. The CONOPT method here performs better than the P-RFO, QA/TRIM, and IPIRC methods. Figure 6 shows that the overlap between the GE tangent and the Hessian TS eigenvector drops to 0.5 around a path distance of ;0.60 a.u., i.e.,
the Hessian TS eigenvector provides a poor uphill direction. In this region the P-RFO and QA/TRIM methods thus climb the valley side at an angle of 60° with the correct uphill direction. For the smaller step size the QA/TRIM method recovers sufficiently to eventually converge to the correct TS5, while the P-RFO, and QA/TRIM at larger step size, walk over the ridge to the valley leading to TS9. For a stepsize of 0.01 a.u. the IPIRC and QA/TRIM paths are very similar, but the IPIRC results appears to be more robust toward an increase in step size. Figure 7 shows the energy as a function of path length for walking along mode 142 to the bifurcation point ~path length 1.35 a.u.! with a step size of 0.01 a.u. The GE-JJH method follows the GE closely until a pathlength of 0.25 where it stalls. The algorithm becomes trapped in a region of the PES where the optimization along the perpendicular modes become cyclic, i.e., two subsequent steps effectively cancel each other. As the perpendicular optimization does not converge, the path is never further advanced. The P-RFO, QA/TRIM, and IPIRC methods start to deviate from the GE around a path length of 0.30. The QA/TRIM and IPIRC methods are close until a path length of 0.60 where they split apart. Figure 8, showing the overlap between the GE tangent
FIG. 6. Overlap between the GE tangent and the Hessian TS eigenvector along the mode 22 GE path.
FIG. 8. Overlap between the GE tangent and the Hessian TS eigenvector along the mode 142 GE path to the bifurcation point.
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TABLE V. Overlap (S) between GE tangent and Hessian TS eigenvector along the GE path to the bifurcation point. Mode
S min
% path (S,0.9)
1 21 22 31 32 4 51 52 6 71 72 81 82 91 92 10 111 112 12 13 141 142 151 152 16 171 172 181 182
0.97 0.74 0.52 0.37 0.69 0.48 0.52 0.63 0.51 0.01 0.46 0.02 0.67 0.77 0.63 0.78 0.85 0.55 0.65 0.66 0.89 0.83 0.87 0.70 0.85 0.56 0.83 0.54 0.60
0 10 5 55 27 73 87 36 26 78 49 40 94 76 18 52 85 91 96 39 11 64 19 98 58 72 66 80 86
and the Hessian TS mode, again indicates that it is the poor direction of the eigenvector that causes the P-RFO, QA/ TRIM, and IPIRC methods to diverge from the GE. The overlap rapidly drops to a value of 0.83, and in contrast to the cases shown in Figs. 3 and 6, remains below 0.9. The CONOPT method follows the GE quite closely, and continues on after the bifurcation point. The similarity of the GE and the CONOPT paths may be understood as follows. The GE is determined by the condition that the derivative of the gradient norm is zero, subject to the constraint that the energy is constant. Points on the CONOPT path are determined by the condition that the derivative of the energy is zero, subject to the constraint that the distance to the minimum is constant. While these conditions clearly are distinct, it appears not unreasonable that the geometries they generate are somewhat similar, given that there is some general connection between the energy and the distance from the minimum. Both the GE-SR and CONOPT methods encounter bifurcations, which explains why none of the CONOPT walks along higher modes converge on a TS. In the vicinity of the bifurcation point, the CONOPT should switch direction. However, the starting geometry for the next constrained optimization is generated by linear extrapolation along the two previous points, and the path thus continues on the ‘‘ridge,’’ analogous to the GE-SR method. As illustrated above, the main problem of the augmented Hessian and IPIRC methods is the poor direction of the Hessian TS mode. Table V shows the smallest overlap between
the GE tangent and the TS mode along the GE before the bifurcation point. Also shown is the fraction of the path where the overlap is less than 0.9. Except for the lowest mode, the minimum overlaps are typically ;0.7. An overlap of 0.7 correspond to an angle of 45°, i.e., maximizing the energy along this direction and minimizing along the other modes corresponds to climbing the valley side in a direction which forms an angle of 45° with the valley floor. Note also that some modes have regions where the overlap is essentially zero! The table also shows that the walks along the higher modes have low overlaps at a much larger fraction of the path than the lowest mode of each symmetry. It is clear that steps which constantly maximize the energy along a direction which deviates significantly from the correct one, quickly makes the optimization essentially a random walk on the PES. The analysis indicates that all optimization methods which assume that a Hessian eigenvector gives a correct uphill direction will be unsuitable for locating TSs by walking up valleys, except for the lowest mode of each symmetry. This encompasses all variations of ~pseudo! Newton– Raphson and augmented Hessian methods,8 –14 as well as methods based on the image potential.21–23 The latter methods attempt to transform a minimax problem to a minimization, but the direction along which the maximization is turned into a minimization is taken as one of the Hessian eigenvectors. If the maximizing direction on the regular surface points in the wrong direction, the minimization step on the image potential is also wrong. The formulas for the GE tangent allow some insight into the difference between walking along the lowest and the higher modes. Consider the simple 2-dimensional case. The GE tangent, e, is given by
H J
e5e 1 1, A 22 5
B2 , A 22
] H 22 u gu 1 e 2 ~ e 2 2 e 1 ! , ]y1
B 2 52
] H 21 u gu . ]y1
e 1 is a normalization constant. Direction ‘‘1’’ is along the gradient and ‘‘2’’ is the perpendicular direction. On the GE path, these directions are identical to the Hessian eigenvectors and 1 is the eigenvector being followed. The angle between the GE tangent and the Hessian eigenvector being followed is given by e 1 , i.e., cosine to the angle is e 1 . The TS eigenvector will thus deviate significantly from the GE tangent when B 2 is large and/or A 22 is small. A 22 will be small when ~i! ] H 22 / ] y 1 is small and either e2>0 or e2>e1 , or ~ii! e 2 ( e 2 2 e 1 )>2 u g u ] H 22 / ] y 1 . If the lowest mode is being followed from the minimum to the TS, e1 will change from positive to negative and will be below e2 along the whole path. Only condition ~ii! can then make A 22 small and in the general case there is no reason that the two terms should cancel. However, if the highest mode is being followed, e1 will start off by being larger than e2 and must at some point be degenerate with e2 in order to become nega-
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tive at the TS. At this point the last term in A 22 disappears. Assuming that the ] H i j / ] y 1 elements are roughly of the same magnitude, e 1 thus become >0.7. This is a typical value for the minimum overlap value observed along the GE path, Table V. Only if the coupling term between the mode being followed and the perpendicular mode, B 2 , is zero, is the Hessian eigenvector parallel to the GE tangent. This will be the case if the two modes have different symmetry. The fundamental difference between following the lowest and the higher modes is thus that the latter encounter points where Hessian eigenvalues cross. In the vicinity of such crossings, or when the coupling is large, the Hessian eigenvector gives a poor uphill direction. It is important to realize that the deviation of the GE tangent and the TS mode has nothing to do with the indeterminate nature of the Hessian eigenvector at points where the Hessian has degenerate eigenvalues. In actual calculations, the exact point where the Hessian eigenvalues cross will essentially never be sampled. However, near such crossing points, the coupling between the two modes plays a strong role in determining the direction of the eigenvectors. Thus when the TS mode couples strongly to another mode, the direction of the TS eigenvector changes rapidly and it becomes a poor indicator for the uphill direction. In the general N-dimensional case, one has to consider coupling from the mode being followed to all other perpendicular modes. This allows us to formulate the condition for when a Hessian eigenvector can be used for followed a reaction path from a minimum to a TS in a stable fashion. The coupling between the mode being followed and close-lying modes should be small. Near mode crossing points, the coupling should be essentially zero. This will be the case if the modes have different symmetry, but can also be fulfilled if the modes are localized in different parts of the system. An example of the latter would be rotation of methyl groups in opposite ends of a molecule. Note that there is no claim that it will be impossible to locate TSs by using Hessian eigenvectors to walk along higher modes which couple strongly with lower modes. It just implies that such walks locate TSs in an essentially random fashion, as clearly illustrated by the data in Table I. The present Ar8 system is a ‘‘worst case’’ example of the problems associated with following higher modes. Not only is the interaction between all pairs of atoms of the same strength, but the clusters are also geometrically tight, implying that all Hessian modes of the same symmetry couple strongly. As a consequence, even the lowest totally symmetric mode is quite difficult to follow as it couples with the next higher mode in a certain region of the path. Of the two augmented Hessian methods, the QA/TRIM performs better than the P-RFO method. At small stepsizes, the IPIRC and QA/TRIM methods give very similar results. However, the IPIRC results appear less sensitive to increases in the step size. This may be seen from the walks along mode 22 ~Table I!, the QA/TRIM method switches from TS5 to TS9 for a step size of 0.10 a.u. while the IPIRC method can tolerate step size up to 0.30 a.u. before the switch occurs. The better stability may also be seen from the Figs. 1, 2 and 4, 5. If the paths generated by the present methods are com-
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pared to the IRCs leading from TS1, TS5, and TS7 to minimum 1, it is generally found that the GE is the one which is closest to the IRC. The maximum deviation is 0.05 a.u. for the walk along mode 1, 0.12 a.u. for the walk along mode 21, and 0.20 a.u. for the walk along mode 22. The GE pathlength is longer than the IRC by 0.02, 0.13, and 0.11 a.u. for the walks along modes 1, 21, and 22, respectively. The maximum deviation is generally found in the region where the GE tangent deviates most from the TS eigenvector. This may qualitatively be understood by noting that the gradient is the tangent to the IRC path, and that the gradient is a Hessian eigenvector on the GE. If the GE tangent coincide with the Hessian TS eigenvector along the whole curve, the GE thus is identical to the IRC. The remaining 23 IRCs which lead to minimum 1 also enter along the two lowest modes. They enter the two valleys from the side. If an alternative path, such as a GE, exists which connects the minimum with one of these 23 TSs, it will thus be a very different path. The IRC connects the minimum with the TS along the lowest mode while the alternative path connects it by one of the higher modes. The difference between such reaction paths may be important for theories which treat the dynamical aspect of the reaction by an expansion around the reaction path. VI. CONCLUSION
The present investigation has shown that methods for walking towards TSs using Hessian eigenvectors as the step direction only can display stable optimization characteristics when walking along modes which are virtually uncoupled from all lower modes, and only weakly coupled to higher modes. Other methods relying on constrained optimizations or following gradient extremals encounter bifurcation points along the path, which they, in the current implementations, are unable to handle in a satisfactory manner. ACKNOWLEDGMENTS
The author would like to thank Professor K. D. Jordan and Professor K. Ruedenberg for helpful information, and Professor H. J. Aa. Jensen for constructive discussions. ~a! K. Mu¨ller, Angew. Chem. Int. Ed. Engl. 19, 1 ~1980!; ~b! H. B. Schlegel, Adv. Chem. Phys. 67, 249 ~1987!. 2 ~a! T. A. Halgren and W. N. Lipscomb, Chem. Phys. Lett. 49, 225 ~1977!; ~b! S. Bell, J. S. Crighton, and R. Fletcher, ibid. 82, 122 ~1981!; ~c! S. Bell and J. S. Crighton, J. Chem. Phys. 80, 2464 ~1984!; ~d! A. Jensen, Theor. Chim. Acta 63, 269 ~1983!; ~e! O. Tapia and J. Andre´s, Chem. Phys. Lett. 109, 471 ~1984!. 3 ~a! M. J. Rothman and L. L. Lohr, Jr., Chem. Phys. Lett. 70, 405 ~1980!; ~b! U. Burkert and N. L. Allinger, J. Comput. Chem. 3, 40 ~1982!; ~c! I. H. Williams and G. M. Maggiora, J. Mol. Struct. 89, 365 ~1982!. 4 M. J. S. Dewar, E. F. Healy, and J. J. P. Stewart, J. Chem. Soc. Faraday Trans. 2 80, 227 ~1984!. 5 ~a! P. G. Mezey, M. R. Peterson, and I. G. Csizmadia, Can. J. Chem. 55, 2941 ~1977!; ~b! K. Mu¨ller and L. D. Brown, Theor. Chim. Acta 53, 75 ~1979!; ~c! R. Elber and M. Karplus, Chem. Phys. Lett. 139, 375 ~1987!; ~d! A. Uitsky and R. Elber, J. Chem. Phys. 92, 1510 ~1990!; ~e! R. Czerminsky and R. Elber, Int. J. Quantum Chem. Quantum Chem. Symp. 24, 167 ~1990!; ~f! R. Czerminsky and R. Elber, J. Chem. Phys. 92, 5580 ~1990!; ~g! C. Choi and R. Elber, ibid. 94, 751 ~1991!; ~h! S. Fischer and M. Karplus, Chem. Phys. Lett. 194, 252 ~1992!; ~i! S. Fischer, S. Mich1
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Frank Jensen: A comparison on Ar8 Lennard-Jones potential
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Beck, Chem. Phys. Lett. 147, 13 ~1988!; ~d! D. J. Wales and R. S. Berry, J. Chem. Phys. 92, 4283 ~1990!; ~e! R. J. Hinde and R. S. Berry, ibid. 99, 2942 ~1993!; ~f! J. Schlitter, M. Engels, P. Kru¨ger, E. Jacoby, and A. Wollmer, Mol. Simul. 10, 291 ~1993!; ~g! H. A. Gabb and S. C. Harvey, J. Am. Chem. Soc. 115, 4218 ~1993!. 19 M. J. S. Dewar and S. Kirschner, J. Am. Chem. Soc. 93, 4290, 4291, 4292 ~1971!. 20 W. H. Miller, N. C. Handy, and J. E. Adams, J. Chem. Phys. 72, 99 ~1980!. 21 T. Helgaker, Chem. Phys. Lett. 182, 503 ~1991!. 22 ~a! C. M. Smith, Theor. Chim. Acta 74, 85 ~1988!; ~b! C. M. Smith, Int. J. Quantum Chem. 37, 773 ~1990!. 23 J. Sun and K. Ruedenberg, J. Chem. Phys. 101, 2157 ~1994!. 24 ~a! J. Sun and K. Ruedenberg, J. Chem. Phys. 99, 5257, 5269, 5276 ~1993!; ~b! 100, 6101 ~1994!. 25 ~a! C. Gonzales and H. B. Schlegel, J. Chem. Phys. 90, 2154 ~1989!; ~b! J. Phys. Chem. 94, 5523 ~1990!. The Gonzales–Schlegel method for determining points on the IRC consists of a series of constrained optimizations. An analytical image Hessian cannot be used in this optimization as it is not the derivative of the image gradient. We have instead used Hessians updated by the Powell formula. 26 For points on the GE, the y and y 8 coordinate systems are identical since the gradient here is an eigenvector of the Hessian. Only exception may be that the 1 direction is not the one corresponding to the lowest Hessian eigenvalue, but rather along the gradient. 27 ~a! Y. Abashkin and N. Russo, J. Chem. Phys. 100, 4477 ~1994!; ~b! Y. Abashkin, N. Russo, and M. Toscano, Int. J. Quantum Chem. 52, 695 ~1994!. 28 The lowest mode of a given symmetry is one which is symmetry distinct from lower modes along the whole path. 29 It is possible that a second GE crosses the one being followed before the bifurcation point, and it is the crossing GE that should be followed at this point. The problem of GE crossings will be considered in a separate publication. 30 Tachibana and K. Fukui, Theor. Chim. Acta 51, 189 ~1979!. 31 Note that there may not be a GE linking a minimum with a TS directly via the lowest mode of a given symmetry, see Table IV. 32 The reference curve is given by a pointwise description ~generated by the GE-SR method with a step size of 0.001 a.u.!, and the shortest distance is calculated by linear interpolation between the two nearest points. The distance between two sets of coordinates is defined as
F( 3N
Dist5
~ x1 i 2x2 i !
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