Development and application of a hybrid method involving interpolationand ab initio calculations for the determination of transition states

Anthony Goodrow,1 Alexis T. Bell,2,a� and Martin Head-Gordon3,a�

1University of California—Berkeley, Berkeley, California 94720-1462, USA2Department of Chemical Engineering, University of California, Berkeley, Berkeley,California 94720-1462, USA3Department of Chemistry, University of California, Berkeley, Berkeley, California 94720-1462, USA

�Received 18 July 2008; accepted 11 September 2008; published online 6 November 2008�

Transition state search algorithms, such as the nudged elastic band can fail, if a good initial guessof the transition state structure cannot be provided. The growing string method �GSM� �J. Chem.Phys. 120, 7877 �2004�� eliminates the need for an initial guess of the transition state. While thismethod only requires knowledge of the reactant and product geometries, it is computationallyintensive. To alleviate the bottlenecks in the GSM, several modifications were implemented:Cartesian coordinates were replaced by internal coordinates, the steepest descent method forminimization of orthogonal forces to locate the reaction path was replaced by the conjugate gradientmethod, and an interpolation scheme was used to estimate the energy and gradient, thereby reducingthe calls to the quantum mechanical �QM� code. These modifications were tested to measure thereduction in computational time for four cases of increasing complexity: the Müller–Brownpotential energy surface, alanine dipeptide isomerization, H abstraction in methanol oxidation, andC–H bond activation in oxidative carbonylation of toluene to p-toluic acid. These examples showthat the modified GSM can achieve two- to threefold speedups �measured in terms of the reductionin actual QM gradients computed� over the original version of the method without compromisingaccuracy of the geometry and energy of the final transition state. Additional savings incomputational effort can be achieved by carrying out the initial search for the minimum energypathway �MEP� using a lower level of theory �e.g., HF/STO-3G� and then refining the MEP usingdensity functional theory at the B3LYP level with larger basis sets �e.g., 6-31G�, LANL2DZ�. Thus,a general strategy for determining transition state structures is to initiate the modified GSM using alow level of theory with minimal basis sets and then refining the calculation at a higher level oftheory with larger basis sets. © 2008 American Institute of Physics. �DOI: 10.1063/1.2992618�

I. INTRODUCTION

Efficient methods for finding transition states �TSs� arerequired for studies of the mechanism and kinetics of chemi-cal reactions.1–5 During the past several decades, two classesof TS-finding methods have been developed, often referredto as surface-walking methods and interpolation methods.6–8

Surface-walking methods require calculation of the potentialgradient and Hessian in addition to the reactant configura-tion. The most common of these algorithms are those basedon restricted-step Newton–Raphson methods,9 such asthat developed by Cerjan and Miller10 and later modified byBanerjee et al.11 and Baker12 as the partitioned rational func-tion optimization �P-RFO� method. Such methods tend towork well if information about the potential energy surface isknown or can be calculated easily, but they are slow to con-verge, particularly for large systems with a large number oflow frequency modes. For complex systems, the computa-tional burden can become prohibitive for determining theHessian matrix analytically or numerically. Although meth-ods exist for updating the Hessian matrix, there are conver-

gence problems associated with them and they may not leadto the correct TS.5 Interpolation methods require knowledgeof the reactant and product configurations in order to gener-ate a sequence of structures �nodes� between them. The sumof the orthogonal forces acting on each node is then mini-mized, resulting in the placement of the nodes on the mini-mum energy pathway �MEP�.

A common interpolation method is the nudged elasticband �NEB�.13,14 The NEB connects the reactant and productstates by a series of intermediate structures �typically 10–20�with springlike interactions between each node. Upon con-vergence, the band joining all the structures is the MEP. It isnoted, though, that for this method to be successful, a goodinitial guess of the TS is needed, which can be a challengefor complex reaction systems. When the guess for the initialreaction path is far from the MEP, electronic structure calcu-lations along the path can fail to converge. Finally, ifHooke’s constants used to keep the nodes equally spacedalong the path are not chosen appropriately, convergence canbe slow or the resulting path can deviate significantly fromthe MEP.7

Similar to the NEB, E et al.15 introduced the zero tem-perature string method �SM� for finding the MEP. A stringjoining the reactant and product configurations in the SM

a�Authors to whom correspondence should be addressed. Electronic ad-dresses: alexbell@berkeley.edu and mhg@bastille.cchem.berkeley.edu.

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moves in the direction of the orthogonal force at each nodeon the string. Rather than using Hooke’s constants to keepthe nodes equally spaced along the pathway in the NEB, theSM redistributes the nodes along the string after each move.A class of methods based on the SM has emergedrecently.7,16,17 One such method is the growing SM �GSM�developed by Peters et al.7 In contrast to the NEB and SM,the GSM does not require a guess of the TS geometry orMEP,7 and consequently, electronic structure calculations arefar less likely to fail. Several improvements have been maderecently to the SM and GSM by E et al.18 and Aguilar-Mogaset al.,19 respectively. Rather than equally spacing the nodesin terms of arclength, the spacing is determined by usingenergy-weighted arclength, enabling the achievement of afiner resolution of nodes near the TS.18,19 In addition, amethod has been proposed for avoiding numerical instabilityin the calculation of the tangent vector for nodes near a re-actant or product minimum or near the TS.19 However, evenwith these improvements, the GSM remains quite computa-tionally intensive and, hence, there is incentive to find addi-tional methods for decreasing the computational burden.

This paper presents a modified version of the GSM de-veloped to alleviate the computational bottleneck associatedwith the original version of this algorithm. Three separateimprovements are introduced: �1� the use of internal coordi-nates instead of Cartesian coordinates, �2� the use of theconjugate gradient method instead of the steepest descentmethod for the minimization of orthogonal forces during theevolution step, and �3� the use of an interpolation scheme forpoints on the potential energy surface in order to minimizecalls to the quantum mechanics �QM� code. These modifica-tions are then implemented within the original GSM. Themethodology and strategy for the use of a hybrid low level/high level of theory to determine TSs using the modifiedGSM is also described. The performance of the modifiedGSM is then assessed for four reactions of increasing com-plexity. These examples demonstrate that significant reduc-tion in computational effort can be achieved without a loss inaccuracy using the modified GSM. The computational sav-ings associated with the use of a hybrid low level/high levelof theory approach are discussed with each example.

II. THEORY

Prior to discussing the improvements made to the GSM,it is useful to summarize the method briefly. There are fourdistinct steps involved in growing a string of nodes and thenplacing these nodes along the MEP. The creation of a stringis initiated by locating the geometries and energies of thereactant and product states. Then two additional nodes arelocated along the linear synchronous transit path joining thereactant and product configurations. One of these nodes isadded close to the reactant state and the other is added closeto the products. The string is then grown independently fromthe reactant and product ends and the string segments arejoined together once the fraction of vacant arclength betweenthe two string segments reaches zero.

The second step is evolution of the two string segments.During this step, the QM code is used to calculate the energy

and forces at each node. The orthogonal force, f�, on eachnode is then minimized using the steepest descent method.This method generates a new geometry for the node by mov-ing it a prescribed distance along the direction of steepestdescent. The QM code is then used again to calculate theenergy and forces for the new node geometry. This sequenceof steps is repeated until the norm of the orthogonal force oneach node, �f��, falls below a user-specified tolerance, f�

tol.The resulting string is parallel to the gradient at each pointalong the path. The evolution of the string is the most com-putationally demanding part of the GSM method because ofrepeated calls to the QM code to calculate the energy andforces.

In the third step, the string is reparametrized by redis-tributing the nodes uniformly along a string of normalizedarclength after each evolution step. The arclength of thestring, s, is calculated by integrating along a cubic splinejoining the nodes in each string segment. If the two stringsegments have been joined, then a single cubic splinethrough all nodes is integrated. The arclength is then normal-ized such that the reactant and product are located at s=0 ands=1, respectively. The remaining nodes in the normalizedstring are equally spaced in terms of arclength. For example,in a fully joined string containing 11 nodes, the nodes arelocated at s=0, 1

10 , 210 , . . ., and 1. The computational cost of

the reparametrization is negligible compared to a single QMforce calculation.

Once the string has been evolved and reparametrized,the fourth step is the growth of the string. As the string growsinward from both the reactant and product ends indepen-dently, a new node can be added to each advancing stringsegment. A new node is added to the string if the number ofnodes, n, is less than the maximum number, Nnode. The loca-tion of the node on the string is determined by extrapolatinga fitted cubic spline through the existing nodes. Once thedistance between the two advancing nodes reaches zero, thestring grown from the reactant end merges with the stringgrown from the product end. The number of nodes remainsconstant and the evolution step proceeds until the sum of thenorm of the orthogonal forces on all nodes, �n=1

Nnode�f�,n�, isless than a user-specified value, F�

tol. The geometry corre-sponding to the highest energy on the converged string isthen used as the TS estimate in a conventional TS searchcalculation, using the P-RFO method.11,12

A. Coordinate system

The first modification to the GSM is the replacement ofthe Cartesian coordinates by nonredundant delocalized inter-nal coordinates �henceforth referred to simply as internal co-ordinates� for the description of molecular geometries. Pre-vious work has shown that the choice of coordinate systemcan have a major influence on the rate of convergence ingeometry optimizations and that a coordinate system involv-ing minimal coupling between individual coordinates ispreferred.20–23 Internal coordinates, i.e., bond lengths, bondangles, and dihedral �or torsion� angels, are weakly coupledin contrast to the Cartesian coordinates, which are heavilycoupled. Baker et al.23 described a general procedure for

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generating a complete, nonredundant set of delocalized inter-nal coordinates from the Cartesian coordinates for any mo-lecular configuration. This procedure involves two steps: �1�relating displacements in the Cartesian coordinates, �X, todisplacements in primitive internal coordinates, �q �whichmay be redundant�, using the Wilson B matrix24 �i.e., �q=B�X� and �2� diagonalizing the matrix G=BBT to giveeigenvalues that are positive or zero. For a species of Natoms, there are 3N-6 eigenvectors with positive eigenval-ues, and the remaining eigenvalues are zero. The eigenvec-tors corresponding to the 3N-6 positive eigenvalues of Grepresent the set of nonredundant delocalized internal coor-dinates, �.

B. Conjugate gradient

The second modification to the GSM is to use the con-jugate gradient method, instead of the previously used steep-est descent method, to minimize the orthogonal forces on thenodes during the evolution of the string. The steepest descentmethod moves a node from point xi to xi+1 on the potentialenergy surface by moving in the direction of the local down-hill gradient, −�E�xi�. While this minimization scheme in-corporates gradient information, it has the disadvantage oftaking many small steps.25 The conjugate gradient removesthese difficulties. While this method involves more compu-tations to compute the search direction, it requires feweroverall steps to achieve convergence. Additionally, the con-vergence characteristics of the conjugate gradient method aremuch better than that of the steepest descent method becausethe current gradient is the Hessian-conjugate with respect tothe previous gradient search directions.25 As a consequence,the convergence of the conjugate gradient is superlinear,whereas the convergence of the steepest descent method islinear. The difference in the rates of convergence betweenthese two methods can lead to large savings in computationaltime when the conjugate gradient method is used, especiallyin cases where the potential energy surface is relativelyflat.25,26 It should also be noted that still more efficient mini-mization procedures exist, such as global-BFGS,27 where allavailable gradient information �from each node� is used ineach individual local optimization. In our present context, itis not clear whether the reduction in the number of forcecalculations would offset the advantage of using an interpo-lation method to reduce the number of calls to the QM code�see below�.

C. Potential energy surface interpolation

The third modification to the GSM is the use of a poten-tial energy surface �PES� interpolation method. Currently,the GSM makes frequent calls to the QM code to performforce calculations. Significant savings in computational timecan be made by estimating the energy of a given geometrythrough an interpolation method, rather than computing itdirectly. An efficient method for estimating the energy ofpoints along the potential energy surface has been developedby Collins and co-workers.28–33 This method uses amodified-Shepard interpolation scheme.34 Data points on thePES calculated previously by the QM code are used to esti-

mate the energy of new points by means of a weighted sumof Taylor series. In their initial formulation, Collins andco-workers28–33 used reciprocal atom-atom distances to de-scribe the geometry of a molecule. Since internal coordinateswere used for the present study, the interpolation approach ofCollins and co-workers was rewritten in these coordinates.

A second-order Taylor series expansion as a function ofthe internal coordinates, �, was used to approximate the localpotential energy in the vicinity of a calculated point, ��i�, asshown in Eq. �1�. The term E���i�� is the QM calculatedenergy for configuration ��i� and �E /��n is the QM calcu-lated analytical gradient. The second derivative term,�2E /��n��m, is approximated by using a unit matrix for theHessian �the default for many QM codes�. An analytical Hes-sian is not used because its calculation is prohibitively ex-pensive and does not provide a significant improvement overusing an approximate Hessian in the accuracy of the energyfor interpolated points. Thus, Ti��� represents the Taylor se-ries estimate of the energy of geometry � based on data pointi, where the indices n and m in Eq. �1� refer to one of the3N-6 components of �.

Ti��� = E���i�� + �n=1

3N−6

��n − �n�i��� �E

��n�

��i�

+1

2 �n=1

3N−6

�m=1

3N−6

��n − �n�i����m − �m�i��

�� �2E

��n � �m�

��i�. �1�

A weighted sum of Taylor series expansions is used to esti-mate the energy of a known geometry, �, on the potentialenergy surface, Eq. �2�. The summation is over all datapoints Ndata calculated previously by the QM code.

E��� � �i=1

Ndata

wi���Ti��� . �2�

Data points closer to the configuration that is to be cal-culated by interpolation are weighted more heavily than datapoints that are further away. The weighting function wi��� isbased on confidence lengths determined from a Bayesiananalysis of the errors in the gradient.31,35 Each Taylor seriesexpansion has an associated confidence length for each of the3N-6 directions in space. The normalized weighting functionwi��� is given by Eq. �3� and involves determination of theprimitive weights vi���. The un-normalized primitiveweights vi��� and the confidence lengths dn�i� are determinedfrom Eqs. �4� and �5�, respectively. Previous work byThompson et al.30 and Bettens and Collins35 showed empiri-cally that the best values for the parameters in Eq. �4� areq=2 and p=12 for a wide range of PES. These values for qand p are consistent with recent findings by Dawes et al.36,37

where the range q=2–4 and p=8–12 were found to workbest for fits that included the energy, gradient, and Hessian,as in Eq. �1�. In the two-part weighting function in Eq. �4�, if�n−�n�i��dn�i�, then the first term on the right-hand sidedominates and vi decreases slowly as �n−�n�i� increases.However, if �n−�n�i��dn�i�, then the second term dominates

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and vi decreases rapidly to zero. In other words, pointswithin a confidence length dn�i� contribute heavily to theestimate of E���, while points that lie outside contribute verylittle. The accuracy of the interpolated energy in Eq. �1� wasshown to not be strongly dependent on the number of pointsM �Ref. 37� or the value of the error tolerance Etol �Ref. 35�used in calculating the confidence length dn�i�. Details con-cerning Eqs. �4� and �5� have been discussed by Collins andco-workers.31,35

wi��� =vi���

�j=1

Ndata

v j���

, �3�

vi��� = �n=1

3N−6 � �n − �n�i�dn���

�2 q/2

+ �n=1

3N−6 � �n − �n�i�dn���

�2 p/2�−1

, �4�

dn�i�−6 =1

M�j=1

M � �E

��n�

��j�−� �Ti

��n�

��j� ��n�j� − �n�i���2

Etol2 ���j� − ��i��6 .

�5�

D. Modified-GSM algorithm

The three modifications described above were imple-mented in the evolution step of the GSM, as shown in Fig. 1.The reactant and product geometries are specified in the Car-tesian coordinates, which is the default output for many mo-lecular visualization software packages. The center of massand list of coordinates are adjusted to match in the reactantand product states, so as to ensure that the geometries of bothstates are defined consistently in the standard orientation. For

instance, the ith atom in the reactant matches the ith atom inthe product, and the center of mass of both species is at theorigin. Using the standard orientation for each geometry,QCHEM 3.0 �Ref. 38� was used to determine the set of internalcoordinates following the procedure of Baker et al.,23

described previously.A library of information about the PES surface is built

by saving the delocalized internal coordinates, energy, ana-lytical gradient, and approximate Hessian from each forcecalculation for a given molecular geometry. This informationis used to determine the search direction for the conjugategradient method and for the potential energy surface interpo-lation scheme.

If the norm of the orthogonal force, �f��, on a node isless than a user-specified tolerance, f�

tol, then the evolutionstep for that node is complete and the calculation proceeds tothe next node. Once the evolution step is complete for allnodes, a new node is added �if the number of nodes is lessthan Nnode�, and the string is reparametrized. However, if�f��� f�

tol then the conjugate gradient method is used to de-termine the direction to move on the potential energy sur-face. The Polak-Ribiére method is used within the conjugategradient method to calculate the scalar �, a multiplicativefactor needed to determine successive conjugate directionvectors. The Polak–Ribière method is preferred over theFletcher–Reeves method because the it converges muchmore rapidly.25,26 Whereas the Polak–Ribière and Fletcher–Reeves methods are identical for a quadratic surface, for anonquadratic surface where the conjugate property betweensuccessive search directions does not hold rigorously, thePolak–Ribière method periodically restarts the conjugategradient method by setting �=0. Previous work has shownthat as the number of iterations in the conjugate gradientmethod increases, the Polak–Ribière method proceeds moresmoothly toward a minimum than the Fletcher–Reevesmethod.25

Moving in the conjugate gradient direction, the energyand forces of the new configuration can be calculated eitherusing the QM code or by interpolation. If the criterion forinterpolation is satisfied, then the library of previous calcu-lated structures is used as the basis for estimation. The crite-rion that was used for using the interpolation scheme is thatthere must be at least M points within a confidence length ofdtol. Choosing a large value for M and dtol would result ininterpolating often, but with less accuracy; however, choos-ing a low value of M and dtol would result in interpolatingrarely, but with high accuracy. In this work, a value of M=5 and dtol=1.0dnode, where dnode is the curvilinear distancebetween adjacent nodes, was found to balance the frequencyof interpolating with the accuracy of the interpolated results.The energy tolerance, Etol in Eq. �5� was set to 0.062 kcal/mol �0.1 mH�, as in previous work.35

If any of the points in the final string were determined byinterpolation, then a single-point energy calculation was per-formed using the QM code. This ensures that interpolatedpoints with an erroneous energy estimate are corrected.

The geometry corresponding to the highest energy in theconverged string is used as the starting geometry in a TS

QChemConverts X� �

Determines E, E, E

Conjugate GradientComputes Step Size

Is || || < ?

Yes

No

Interpolation SchemeEstimates E, E

Start with n NodesGeometry in

Cartesian Coordinates

For each node

Advance to Next Node

InterpolationCriterionSatisfied

Yes

No

∇

∇ 2∇

Library of Points{�, E, E, E}∇ 2∇

tolf⊥⊥f

FIG. 1. A flow sheet for the evolution step of the modified GSM. Theorthogonal force f�, refers to the negative of the component of the gradientthat is orthogonal to the reaction path. The criterion for using the interpola-tion scheme is if there are M points within a confidence length dtol.

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search calculation. The eigenvector following �EF� algo-rithm, which uses the P-RFO method described by Baker,12

is used in QCHEM with internal coordinates to optimize theTS estimate from the GSM to the final converged TS geom-etry. The EF scheme in QCHEM works by maximizing alongthe lowest Hessian mode while at the same time minimizingalong all other modes. Often, the starting TS estimate hasmany negative Hessian eigenvalues that are eliminated dur-ing the calculation, with the final converged TS geometryhaving one negative Hessian eigenvalue.

E. Hybrid low-level/high-level strategy

A further aim of this work was to find a means for re-ducing the computational time required to find TSs. To thisend, we investigated a strategy in which the search for a TSwas initiated using a low level of theory/small basis sets �i.e.,Hartree–Fock, force field, and semiempirical methods� andthen refined using a higher level of theory/larger basis sets,

which is more computationally demanding. For instance, thecomputational time for a single force calculation can be re-duced by two orders of magnitude by switching from usinglarge basis sets �e.g., 6-31G�� to smaller basis sets �e.g.,STO-3G� because the calculation time is O�b3�–O�b4�,where b is number of basis functions. Hence, a small reduc-tion in the number of basis functions used has a large impacton reducing the computational time. This strategy, shown inFig. 2, is conceptually similar to that used for geometry op-timizations, in which an initial geometry is sought using alow level of theory and then refined using a higher level oftheory.5 In the context of the GSM, a string connecting thereactant and product is grown using a low level of theory anda minimal basis set �e.g., HF/STO-3G�. Next, the TS esti-mate from this calculation is used as a starting geometry fora TS search at a higher level of theory carried out with largerbasis sets �e.g., density functional theory using

No

TS Found

Yes

Grow String atLow-Level of Theory

Did TS SearchConverge?

Improve TS Estimatefrom Low-Level String

using High-Level Theory

Refine String atHigh-Level of Theory

Optimize TS Estimatefrom High-Level String

using QChem TS Search

No

Yes

Did TS SearchConverge?

Add More Nodes toString & Restart

Calculation

FIG. 2. A flow sheet for a TS-finding strategy that uses a combined lowlevel/high level of theory approach. A TS search using a high level of theorycan be initiated directly from the converged low-level string. Otherwise, ifthe topology of the PES is correct, the low-level string can be further refinedat a high level of theory.

-1.5 -1 -0.5 0 0.5 1-0.5

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5 1-0.5

0

0.5

1

1.5

2

Saddle Point (Transition State)Minimumx

y

A

B

CTS1

TS2

FIG. 3. �Color online� Contour plot of the Müller–Brown potential energysurface showing the three minima �A, B, C� and two saddle points, or TSs�TS1, TS2�.

1 10 100 1000

0.01

0.1

|xsp

-xes

t|

Gradient Evaluations

GSMmod-GSMInterpolated Pt

FIG. 4. �Color online� Error in the estimate of the highest saddle point fromthe true saddle point �TS1� on the MB PES. Results are shown at 18 nodesfor the GSM and modified GSM. The open circles represent points where aninterpolation step was performed in the modified GSM.

0.0 0.2 0.4 0.6 0.8 1.0

-140

-120

-100

-80

-60

-40

GSMmod-GSM

Ene

rgy

(arb

.uni

ts)

Normalized Arclength

FIG. 5. �Color online� Final energy profile of the MEP for the MB PESusing the GSM and the modified GSM for 18 nodes. The points in eachcurve are joined with a spline.

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B3LYP /6-31G��. However, if the TS search calculation failsto converge, then the converged string at the low level oftheory can be used as the initial pathway for a subsequentstring calculation at a high level of theory. The TS estimatefrom this refined calculation can then be used to start a TSsearch.

It is noted that for each of the systems investigated, a TSsearch performed using the TS estimate attained from a low-level string calculation could not be converged. This findingindicates that it is essential to refine the string calculationusing a high level of theory prior to completing the TSsearch. Without refining the string at a high level of theory,the energetics of the overall reaction tend to be erroneousand the resulting TS estimate is often a poor initial guesscontaining many negative Hessian eigenvalues. However,this strategy is valuable in providing an initial reaction path-way with a small computational cost that can then be refinedat a higher level of theory.

III. RESULTS AND DISCUSSION

The following four examples compare the efficiency ofthe modified GSM with the original GSM for finding TSs fora set of reactions of increasing complexity. Unless noted, 11nodes were used in the final string. The GSM and modifiedGSM were interfaced with QCHEM 3.0 �Ref. 38� to performthe QM energy and force calculations. The same conver-gence criteria were used for both forms of the GSM. A per-pendicular gradient �f���0.040 hartree /bohr �f�

tol� was re-quired before a new node was added to the string, theextrapolated distance for placement of a new node was0.08���1 /Nnode��, and a perpendicular gradient �f���0.002 hartree /bohr �F�

tol� was required on each node forconvergence of the string. The highest energy structure ineach string calculation was used as the initial geometry for aTS search. Each refined TS structure was found to have onlyone imaginary frequency.

The GSM and modified GSM were compared on thebasis of the number of QM gradient calculations required to

TABLE I. Comparison of TS geometries from final string calculations foralanine dipeptide isomerization. The energies are relative to the reactant, C5.

Calculation method Species�

�deg��

�deg�E

�kcal/mol�

GSM �B3LYP /6-31G��

C5 −161.2 165.1 0.0TS 107.3 −139.7 7.6

C7AX 71.7 −58.2 −1.40

Modified GSM �B3LYP /6-31G��

C5 −161.2 165.1 0.0TS 105.8 −150.9 7.2

C7AX 71.7 −58.2 −1.40

Perczel et al.a �B3LYP /6-31G��

C5 −156.8 162.6 0.0TS 114.4 −147.9 7.33

C7AX 72.3 −54.7 −1.35

aReference 44.

FIG. 6. �Color online� Alanine dipeptide isomerization from species C5

�reactant� to C7AX �product�.

10 100 10000.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

GSMmod-GSMInterpolated Pt

RM

SE

rror

QM Gradients

FIG. 7. �Color online� Error in the estimate of the highest saddle point fromthe true saddle point for alanine dipeptide isomerization. Results are shownat 11 nodes for the GSM and modified-GSM. The open circles representpoints where an interpolation step was performed in the modified GSM.

0.0 0.2 0.4 0.6 0.8 1.0-2

0

2

4

6

8

GSMmod-GSM

Ene

rgy

(kca

l/mol

)

Normalized Arclength

FIG. 8. �Color online� Final energy profile of the MEP for alanine dipeptideisomerization using the GSM and the modified GSM for 11 nodes. Thereaction path shown for the modified GSM is after single-point QM calcu-lations have been performed on interpolated points that exist in the finalstring. The points in each curve are joined with a spline.

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reach the estimated TS. For both methods, gradient calcula-tions represent nearly all the computational time. The timerequired for calculating the search direction in the conjugategradient method and for estimating the energy of interpolatedgeometries is negligible compared to the time required for asingle gradient calculation.

A. Müller–Brown potential energy surface

The Müller–Brown PES �Ref. 39� is commonly used tobenchmark the performance of new TS-findingmethods.7,17,39–41 The MEP on this PES differs significantlyfrom the linearly interpolated path and, as seen in Fig. 3,there are two saddle points �TS1, TS2� located between threelocal minima �A, B, C� on the MEP. Since the Müller–BrownPES is analytic, the energy and gradient at any point can beevaluated rapidly. Figure 4 shows the number of gradientevaluations versus the root-mean-square error between thehighest saddle point �TS1� and the TS estimate in going frompoint A �reactant� to point B �product�, using 18 nodes.

In the original paper by Müller and Brown,39 124-155function evaluations were needed to determine the TS1, and73-74 to determine TS2, using the constrained simplex opti-mization procedure. �The exact number of evaluations re-quired varied depending on the starting point.� Using theGSM, both TSs were determined from one calculation, re-quiring 152 gradient evaluations.7 �The time needed to per-form a gradient evaluation is equivalent to that for a singlefunction evaluation on the Müller–Brown PES.� A clear ad-vantage of the GSM over other methods, such as the NEB,can be seen from this example: the GSM does not requireinitial information about the reaction path and it is able tofind multiple TSs �when they exist� by connecting the reac-tant and product. The capability of the GSM to determinemultiple TSs has also been noted recently by Chempath andBell42 in their study of methane oxidation to formaldehydeon isolated molybdate species supported on silica.

By implementing the modifications to the GSM, only 75gradient evaluations were required to find both TSs. Thepoints in Fig. 4 identify geometries for which the energy wasestimated using the interpolation scheme. In total, 62 pointswere interpolated on the Müller–Brown PES in the course ofidentifying the MEP. Figure 5 compares the final strings ob-tained from the GSM and the modified GSM. As can be seen,the results of both methods agree quite well, and the valuesof the energy for both TS1 and TS2 agree with the actualvalue. Thus, the results of the modified GSM on the Müller–

Brown PES give a hint about the speedup in cases wherecalculating the energy and gradient are nontrivial.

B. Alanine dipeptide isomerization

Alanine dipeptide isomerization is another well-studiedexample that has been used to evaluate the performance ofTS-finding methods.43 The system consists of 22 atoms and66 Cartesian coordinates. Using internal coordinates, theisomerization of alanine dipeptide can be described by therotation of two dihedral angles � and �, shown in Fig. 6.44

The GSM and modified GSM calculations were performed atthe B3LYP /6-31G� level of theory. A single gradient �i.e.,force� calculation for the geometries of alanine dipeptide re-quired �8 min. Figure 7 compares the number of gradientscomputed by the QM code required to achieve convergenceusing the GSM and the modified GSM. The GSM requires980 QM gradient calculations,7 whereas the modified GSMrequires only 339. Several factors contribute to the markedimprovement. First, internal coordinates are much better atdescribing this reaction than the Cartesian coordinates.Rather than keeping track of all the 66 Cartesian coordinates,the isomerization reaction can be described just by examin-ing the two dihedral angles. Second, the conjugate gradientmethod results in very efficient convergence of the finalstring containing 11 nodes �requiring 87 QM gradient calcu-lations�. Third, 45 points are interpolated by using the PESinterpolation method. These results are similar to those re-ported recently by Quapp45 using a modified conjugate gra-dient method and the Newton projections,46,47 which are usedto determine where to move each node on the string, for thedetermination of the TS.

An analysis of the relative contribution of various im-provements to the GSM was carried out. As noted above, theoriginal version of the GSM algorithm, which uses Cartesiancoordinates and the steepest decent method, required 980QM gradient calculations. Replacement of the Cartesian co-ordinates by internal coordinates reduced the number of QMgradient calculations to 651, and the replacement of thesteepest decent algorithm by the conjugate gradient algo-rithm reduced the number of QM gradient calculations to435. Finally, use of the PES interpolation scheme, furtherreduced the number of QM gradient calculations to 339. Theuse of internal coordinates is very helpful because the reac-tion involves the movement of 16 atoms around two dihedralangles.

TABLE II. Time required to determine the TS for alanine dipeptide isomerization.

Calculation methodFunctional/basis

setNo. of

QM gradient CPU time/gradientCPU time

for TS optimizationTotal

CPU timea

GSM string grown at B3LYP /6-31G�

TS search at B3LYP /6-31G�

B3LYP /6-31G� 980 �8 min 35 min 130 h

Modified-GSM string grown at B3LYP /6-31G�

TS search at B3LYP /6-31G�

B3LYP /6-31G� 339 �8 min 36 min 46 h

Modified-GSM string grown at HF/STO-3G,refined at B3LYP /6-31G� TS searchat B3LYP /6-31G�

HF/STO-3G 324 �10–15 s 35 min 32 h

B3LYP /6-31G� 216 �8 min

aTotal CPU time=CPU time for growing string+CPU time for optimizing TS estimate.

174109-7 Modified growing string method J. Chem. Phys. 129, 174109 �2008�

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The large deviation from the TS geometry occurringfrom 20 to 28 QM gradients in the case of the modified GSMcan be explained by looking at the PES calculated by Perczelet al.44 The string starts moving uphill to a TS that is higherin energy than the actual TS for this reaction. The problem israpidly corrected by calculating QM gradients instead of us-ing the interpolation scheme. This same problem is encoun-tered in the GSM from 102 to 108 QM gradients.

Neither the GSM nor the modified GSM approaches theTS monotonically, as shown in Fig. 7. The reason why thereare periodic increases in the error of the TS estimate as moreQM gradients are computed is due to the addition of a newnode �or nodes� along the reaction pathway. When a newnode is added along the reaction pathway, there is initiallylittle information about the local PES, and this can result indeviations away from the TS. These deviations are quicklycorrected in the GSM and modified GSM.

Typically, when points are interpolated successfully onthe PES, the estimate of the TS becomes better without hav-ing to make additional QM gradient calculations. However,at points where the interpolation criterion is barely satisfied,the next step often produces a point for which the criterion isno longer satisfied and a QM gradient must be computed.This problem is evident for the interpolated point at 40 QM

gradients shown in Fig. 7. The estimate of the energy at thispoint brings the calculation closer to the actual TS, althoughthe next point is not vertically below it, as one might expect.The reason for this behavior is that at the next point a QMgradient is computed since the interpolation criterion is notmet.

The final energy profile of the MEP calculated by bothmethods is shown in Fig. 8. The energies of all interpolatedpoints in the final string obtained using the modified GSMwere updated by carrying out single-point energy calcula-tions. Table I shows the energies and dihedral angles for thereactant �C5�, product �C7AX�, and the estimated TS obtainedusing both methods. In both cases, the geometry of the TSand the activation energy agree well with one another, andwith the previous work of Perczel et al.44 The close agree-ment between the GSM and the modified GSM indicates thatthe accuracy of the estimated TS is not compromised by theuse of the modified GSM. As a final test, a TS search wasinitiated using the geometry corresponding to the highest en-ergy in each string calculation. Both calculations convergedto the same TS structure and the difference in the imaginaryfrequency was only 0.05 cm−1.

An investigation was carried out as well to establishwhether the search for the TS could be initiated using astring first grown at a low level of theory and then refined ata higher level of theory. The calculation approach for thisstrategy is shown in Fig. 2. The string was first grown usingHF/STO-3G and then converged using the modified GSM.This process required 324 QM gradients calculations. Thecalculated reaction path from this level of theory was thenused as the starting string for the calculation of the string atthe B3LYP /6-31G� level, a process which required an addi-tional 216 QM gradient calculations. The advantage of thisstrategy is that a force calculation at the HF/STO-3G leveltook �10–15 s, whereas a force calculation at the

FIG. 9. �Color online� H abstraction in methanol oxidation on VOx /SiO2.

10 100 10000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

RM

SE

rror

QM Gradients

GSMmod-GSMInterpolated Pt

FIG. 10. �Color online� Error in the estimate of the highest saddle pointfrom the true saddle point for H abstraction in methanol oxidation onVOx /SiO2. Results are shown at 11 nodes for the GSM and modified GSM.The open circles represent points where an interpolation step was performedin the modified GSM.

0

5

10

15

20

25

30

35

40

45

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Normalized Arclength

Ene

rgy

(kca

l/mol

)

GSM

mod-GSM

FIG. 11. �Color online� Final energy profile of the MEP for H abstraction inmethanol oxidation on VOx /SiO2 using the GSM and the modified GSM for11 nodes. The reaction path shown for the modified GSM is after single-point QM calculations have been performed on interpolated points that existin the final string.

174109-8 Goodrow, Bell, and Head-Gordon J. Chem. Phys. 129, 174109 �2008�

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B3LYP /6-31G� level took �8 min. The final TS estimateobtained from the string at the B3LYP /6-31G� level con-verged to the same TS structure as that determined previ-ously. As shown in Table II, a modified GSM calculation andTS search, both using B3LYP /6-31G�, took a total of 46 h todetermine the TS, whereas the time required for the hybrid,low-level/high-level approach was 32 h. The total CPU timein Table II was calculated as the CPU time to grow the stringplus the CPU time to optimize the TS estimate to the final TSstructure. The CPU time needed to grow the string is theproduct of the number of QM gradients and the time per QMgradient. In the case of the hybrid low-level/high-level ap-proach, the CPU time to grow the string is the sum of theCPU time for growing both the HF/STO-3G andB3LYP /6-31G� strings. Thus, while the total number of gra-dient calculations computed using the hybrid approach�324+216=540� is more than the number required if thewhole process were performed at the B3LYP /6-31G� levelof theory �339�, it represents a significant reduction in theoverall computational time because a force calculation at theHF/STO-3G level is an order of magnitude faster than one atcarried out at the B3LYP /6-31G� level.

C. H-abstraction in isolated vanadate sites supportedon silica

The third test case investigated was the rate-limiting stepfor methanol oxidation occurring on isolated vanadate spe-cies supported on silica, VOx /SiO2.48 This reaction systeminvolves 34 atoms. The rate-limiting step is the abstraction ofa H atom from an adsorbed methoxy group bonded to V tothe vanadyl O, as shown in Fig. 9. This test case was chosenbecause of the large number of atoms involved, including atransition metal, and the long computational time needed todetermine the TS using the GSM. The GSM and modifiedGSM were carried out at the B3LYP /6-31G� level. TheLANL2DZ effective core potential was used for vanadium.

Figure 10 shows the number of gradients required for con-vergence for both methods. The modified GSM only required381 QM gradients as compared to 651 for the original GSM.The reduction in the number of QM gradient calculationsrepresents a significant savings in computational time since asingle gradient �i.e., force� calculation for this system at theB3LYP /6-31G� /LANL2DZ level took �80–120 min. A to-tal of 76 points were interpolated. It is observed that once asufficient number of points have been determined on the po-tential energy surface by means of QM calculations, the in-terpolation scheme was able to estimate many points withreasonable accuracy, as seen from the end of the calculation.

As done previously, an analysis of the relative contribu-tion of each improvement to the GSM was performed. Theoriginal version of the GSM algorithm required 651 QMgradient calculations. Replacement of the Cartesian coordi-nates by internal coordinates reduced the number of QM gra-dient calculations by 8% to 599. This reduction is less thanthe 34% seen in alanine dipeptide isomerization because thereaction mechanism involves movement of only a single Hatom. The replacement of the steepest decent algorithm bythe conjugate gradient algorithm reduced the number of QMgradient calculations to 467. Finally, use of the PES interpo-lation scheme, further reduced the number of QM gradientcalculations to 381.

The final energy profile of the MEP obtained by bothmethods is shown in Fig. 11. Single-point energy calcula-tions were performed on the points on the final string ob-tained from the modified GSM that had been determined byinterpolation. The single-point energy calculations changedthe energy of several of the interpolated points in the finalstring by as much as 2–3 kcal/mol. At theB3LYP /6-31G� /LANL2DZ level, the change in energy forthe overall reaction shown is �E=38.1 kcal /mol and theactivation energy is �E‡=40.8 kcal /mol. These values com-pare well with those reported by Goodrow and Bell,48 �E

TABLE III. Comparison of TS geometries from final string calculations for H abstraction in methanol oxidationon VOx /SiO2. The strings were grown using the functional/basis sets listed. All TS estimates were refined atB3LYP /6-31G� /LANL2DZ.

Calculation method RC–H �Å� RO–H �Å� vimg �cm−1� �Eact �kcal /mol�

GSM Goodrow and Bella �B3LYP /6-31G�� 1.63 1.11 −870.9 40.3Modified GSM �B3LYP /6-31G�� 1.63 1.11 −870.9 40.8Modified-GSM string grown at HF/STO-3G,refined at B3LYP /6-31G� 1.63 1.11 −871.9 40.8

aReference 48.

TABLE IV. Time required to determine the TS for H transfer in VOx /SiO2.

Calculation methodFunctional/basis

setNo. of QM

gradientCPU time/gradient

�min�CPU time for tsoptimization �h�

Total CPU time�h�a

GSM string grown at B3LYP /6-31G� TS search atB3LYP /6-31G� B3LYP /6-31G� 651 �80–120 140 1225Modified-GSM string grown at B3LYP /6-31G� TS search atB3LYP /6-31G� B3LYP /6-31G� 339 �80–120 134 700

Modified-GSM string grown at HF/STO-3G, refined atB3LYP /6-31G� TS search at B3LYP /6-31G�

HF/STO-3G 330 �4–6 114 393B3LYP /6-31G� 151 �80–120

aTotal CPU time=CPU time for growing string+CPU time for optimizing TS estimate.

174109-9 Modified growing string method J. Chem. Phys. 129, 174109 �2008�

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=37.1 and �E‡=39.8 kcal /mol, respectively, obtained usingthe LACV3P��+ +basis set. A TS search calculation usingthe estimated geometry from the modified GSM convergedto the same energy and geometry as the GSM.

The modified GSM was used at the HF/STO-3G level toidentify the initial string. While the energetics were not prop-erly captured by this low level of theory, the topology of thepotential energy surface was correct. The converged reactionpath from this calculation was reasonably close to the exactreaction path determined at a higher level of theory. Thereaction path from HF/STO-3G was refined by beginninganother string calculation at the B3LYP /6-31G� /LANL2DZlevel. A comparison of the final TS structures obtained fromeach method are shown in Table III. The C–H and O–H bondlengths, as well as the imaginary vibrational frequency char-acteristic of the TS, agree well for all three cases. Table IVlists the CPU time required for each type of calculation. It isimportant to note that the times listed are for a single CPU,whereas, in practice, each gradient and TS search calculationwas performed using 2 CPUs. Using this hybrid method, 330QM gradients were calculated with HF/STO-3G and an ad-dition 151 QM gradients were calculated for convergence ofthe string at B3LYP /6-31G� /LANL2DZ, for a total of 481QM gradient calculations. The time required to identify theTS in this case was 393 h, whereas using the modified GSMto grow the entire string at the B3LYP /6-31G� /LANL2DZlevel required 700 h.

D. C–H bond activation in the oxidative carbonylationof toluene to p-toluic acid

A theoretical investigation of the oxidative carbonylationof toluene to p-toluic acid by the complex Rh�CO�2�TFA�3

�TFA=trifluoroacetate� has recently been reported by Zhengand Bell.49 As shown in Fig. 12, the rate-limiting step for thereaction is C–H bond activation of coordinated toluene. Thisparticular reaction was chosen because it involves the collec-tive movement of many atoms and the TS for the reactioncould not be found using the GSM by itself. Instead, theGSM had to be used in association with an iterative partialoptimization approach �IPOA� that involved the freezing andunfreezing of the Cartesian coordinates.49 While this ap-proach proved to be successful, the IPOA involved a lot ofhuman intervention and required over 1800 h of CPU time toconverge to the correct TS structure.

The strategy that we have used to determine the TS con-sisted of three steps: �1� growth of the string using the modi-

fied GSM at a low level of theory �HF/STO-3G�, �2� refine-ment of the string calculation at a higher level of theory�B3LYP /6-31G� and LANL2DZ for Rh�, and �3� utilizationof the TS estimated from the refined string to determine thefinal TS geometry. HF/STO-3G was used as a low level oftheory to gain an initial estimate of the reaction pathway.This method was chosen because the change in energy forthe reaction ��EHF=0.12 kcal /mol� is comparable to thatcomputed using a more accurate method ��EB3LYP

=0.43 kcal /mol for gas-phase species, �EB3LYP

=0.9 kcal /mol for the inclusion of solvent effects49�.A total of 852 gradients were computed and 166 points

were interpolated using the modified GSM, shown in Fig. 13.Growth and convergence of the initial string at the HF/STO-3G level required 738 gradients, with each calculation

FIG. 12. �Color online� C–H bond activation in toluene on a Rh complexcoordinated with two CO species and three TFA solvent ligands.

10 100 10000.0

0.5

1.0

1.5

2.0

2.5

3.0mod-GSMInterpolated Pt

RM

SE

rror

QM Gradients

FIG. 13. �Color online� Error in the estimate of the highest saddle pointfrom the true saddle point for C–H bond activation in toluene over Rh/TFAcomplex. The result shown is at 11 nodes for the modified GSM. The GSMis not shown because it was unable to converge. The open circles representpoints where an interpolation step was performed in the modified GSM.

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

14

Ene

rgy

(kca

l/mol

)

Normalized Arclength

mod-GSM E(HF)mod-GSM E(B3LYP)

FIG. 14. �Color online� Final energy profile of the MEP for C–H bondactivation in toluene over Rh/TFA complex using the modified GSM for 11nodes. The modified GSM was first used using HF/STO-3G, and then re-fined using B3LYP/6-31G/LANL2DZ. The points in each curve are joinedwith a spline.

174109-10 Goodrow, Bell, and Head-Gordon J. Chem. Phys. 129, 174109 �2008�

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taking �9.5 min, and 134 interpolated points. The stringwas then refined at the B3LYP /6-31G� /LANL2DZ level,which required the calculation of an additional 114 gradients,each calculation taking �100–120 min, and 32 interpolatedpoints. Figure 13 shows a gap between when the last pointinterpolated in the HF/STO-3G string �at 737 QM gradients�and the first point interpolated in theB3LYP /6-31G� /LANL2DZ string �at 793 gradients�. Thereason for this gap is that points computed in the HF/STO-3G string cannot be used in theB3LYP /6-31G� /LANL2DZ string because the energieswere very different, even though the geometry was correct.As a result, it was necessary to perform 56 QM calculationsat the B3LYP /6-31G� /LANL2DZ level before the criterionfor interpolating a point was satisfied.

An analysis of the relative contribution of various im-provements to the GSM was carried out; however, the calcu-lations did not converge. The original version of the GSMdid not converge after 1000 QM gradients usingB3LYP /6-31G� /LANL2DZ and the calculation was termi-nated after 1800 h. Replacement of the Cartesian coordinatesby internal coordinates and replacement of the steepest de-cent algorithm by the conjugate gradient algorithm were in-sufficient for the GSM to converge to a TS after 1000 QMgradients. Only by incorporating the PES interpolationscheme with the previous modifications was the GSM able toconverge, requiring 852 QM gradient calculations. All themodifications to the GSM work together in a synergisticfashion and are required for this reaction system because ofthe complexity involved in the movement of a large numberof atoms. In particular, the interpolation method is relied onheavily, with the fraction of points interpolated versus QMgradients calculated at nearly 20%. Lastly, the hybrid ap-proach of using a low-level/high-level method provides asignificant reduction in computational time for this reaction.

The converged strings using HF/STO-3G and

B3LYP /6-31G� /LANL2DZ are shown in Fig. 14. The esti-mate of the activation energy is 12.8 kcal/mol using HF and11.9 kcal/mol using B3LYP. A TS search was then initiatedusing the estimate from the final B3LYP string in order toconverge to the correct TS structure. The activation energyfor the final converged TS from the modified GSM was 9.4kcal/mol, in close agreement to the previously computedvalue of 9.1 kcal/mol ��E‡=7.7 kcal /mol when accountingfor solvent effects�.49 A comparison of the geometries of theTS determined by different approaches is shown in Table V.The TS from the converged string grown at the HF/STO-3Glevel provides a reasonable estimate of the actual TS. Thegeometry of the final converged TS at the B3LYP levelagrees closely with that reported by Zheng and Bell.49 Asshown in Table VI, the CPU time required for these threesteps are 117, 209, and 228 h, respectively, for a total CPUtime of 554 h. This strategy represents a large reduction incomputational time and human intervention over the previ-ous IPOA �1800 h�. It also demonstrates that a low level oftheory �in this case HF� can be used to generate a reasonabledepiction of the MEP and a good estimate of the TS, whichcan then be refined using a higher level of theory.

IV. CONCLUSIONS

Several modifications were made to the GSM with theaim of decreasing the computational requirements of themethod: �1� the Cartesian coordinates were replaced by in-ternal coordinates, �2� the steepest descent method used forthe minimization of orthogonal forces at each node along thestring was replaced by the conjugate gradient method, and�3� a PES interpolation scheme was used to generate an es-timate of the energy when possible. All three modificationscontributed to a reduction in the number of calls made to theQM code. The degree to which a particular modification al-ters the number of QM code calls is problem specific but is,

TABLE V. Comparison of TS geometry using different functionals/basis sets for C–H bond activation intoluene on Rh�CO�2�TFA�.

Bond length ��

Estimated TSfrom string

�HF/STO-3G�

Estimated TSfrom string�B3LYP /6-31G��

Optimized TSfrom string�B3LYP /6-31G��

TS determinedby Zheng and Bella

C–H 1.40 1.33 1.28 1.30O–H 1.43 1.30 1.38 1.35Rh–C 2.12 2.19 2.18 2.20Rh–O 2.04 2.12 2.11 2.11

aReference 49.

TABLE VI. Time required to determine the TS for C–H activation in toluene over Rh�CO�2�TFA�.

Calculation method Functional/basis setNo. of QM

gradientCPU time/gradient

�min�CPU time

for ts optimization Total CPU timeb

IPOA by Zheng and Bella B3LYP /6-31G� �1000 �100–120 ¯ �1800 h did not convergeModified-GSM string grown at HF/STO-3G,refined at B3LYP /6-31G� TS searchat B3LYP /6-31G�

HF/STO-3G 738 �9–10

228 h 554 hB3LYP /6-31G� 114 �100–120

aReference 49.bTotal CPU time=CPU time for growing string+CPU time for optimizing TS estimate.

174109-11 Modified growing string method J. Chem. Phys. 129, 174109 �2008�

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in general, about two to three times lower than that for theoriginal version of the method. It is significant that the esti-mate of the TS geometry obtained using the modified GSMled to refined TS structures that were equivalent to thosefound using the original method. The idea of using a lowlevel of theory to determine the initial locations of the stringnodes on the PES followed by refinement of the string usinga higher level of theory was also explored. Implementationof this approach led to further reductions in computationaltime, without any loss in the quality of the final TS geometryor energy. Thus, a recommended strategy for determiningTSs is to initiate a calculation using the modified GSM at alow level of theory or with small basis sets and then refinethe calculation at a higher level of theory with larger basissets.

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