Most atomic scale transitions in the condensed phasuch as chemical reactions and diffusion, are activatedcesses, i.e., require surmounting a significant energy barWhile under typical conditions the thermal energy is onorder ofkBT50.025 eV, the barriers for such transitions atypically on the order of 0.5 eV or higher. A transition thoccurs thousands of times per second is so slow on the sof atomic vibrations that it would typically take several thosands of years of computational time on the fastest modcomputer to simulate a classical trajectory with reasonachance of observing a single transition. As a result, classdynamics simulations of activated transitions are faced wan impossible time scale problem. It is not possible toserve such transitions by simply simulating the classicalnamics of the system. Arbitrarily raising the temperaturethe system can lead to crossover to a different transimechanisms. The problem is to find a simulation algoritthat can be used to find which transition would occur andwhat rate, if the classical dynamics could be simulated folong enough time.
Within transition state theory~TST! the problem be-comes that of finding the free energy barrier for ttransition.1 This is a very challenging problem, especiawhen the mechanism of the transition is unknown. Withthe harmonic approximation to transition state theo~hTST!2,3 the problem becomes that of finding the sadpoint on the potential energy surface corresponding tmaximum along a minimum energy path that takes the s
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tem from one potential energy minimum to another. Thisstill a difficult problem when dealing with condensed matsystems because of the high dimensionality of the potenenergy surface. When both the initial and final states oftransition are known, minimum energy path~s! for the tran-sition can be found quite readily4 ~the problem of makingsure the path with lowest activation energy has been founstill a difficult problem!. When only the initial state of thetransition is known, the problem of finding the relevasaddle point~s! becomes that of navigating in high dimensional space—a very challenging task. If the set of all revant, low lying saddle points for transitions from a giveinitial state could be found reliably and the prefactor in thTST rate constant expression evaluated, then long timetivated dynamics of the system could, in principle, be simlated. This may be impossible for all but the simplest stems.
Recently, significant progress has been made towathis goal. In the activation-relaxation technique developedBarkema and Mousseau, the system is driven from onetential energy basin to another by inverting the componenthe force acting on the system along a line drawn frominstantaneous configuration to the initial configuration5,6 ~orto a trailing image of the system7!. The new potential energybasin is then accepted or rejected based on Monte Carlo spling. This has enabled equilibration of supercooled liqudown to much lower temperature than could be achiewith direct classical dynamics simulations. In principle tmethod could be used to estimate the rate of the transit
7011J. Chem. Phys., Vol. 111, No. 15, 15 October 1999 Finding saddle points on high dimensional surfaces
observed using harmonic transition state theory, but thegorithm does not always take the system through the cvicinity of the saddle point, making the estimate of the acvation energy uncertain. Furthermore, there is no guarathe activation-relaxation technique will give the same trantion as a long classical trajectory, i.e., the transition withlowest saddle point.
A very different approach to long time simulations hbeen developed by Voter, the so-called hyperdynammethod.8,9 There, a classical trajectory is generated formodified potential with a reduced well depth. The activatransitions are thus made more probable and may beserved during the short time interval accessable by classdynamics simulations. The relative rate of different transitmechanisms is preserved in the modified system~within thetransition state theory approximation! so the hyperdynamicstrajectory should reveal the most probable activated trations. It can, furthermore, provide an estimate of the trantion rate within the full, anharmonic transition state theoapproximation. In general, the hyperdynamics trajectsimulation still requires a large number of force evaluatiomore than can be handled at the present time withab initiomethods, and the acceleration of the transitions becosmaller as the system gets larger.
Powerful methods have been developed for climbingpotential energy surfaces from minima to saddle points inabinitio calculations of molecules. These methods have becpart of the standard tool kit for molecular calculations, bthey require the evaluation and inversion of the Hessiantrix at each point along the search so as to find the lonormal modes of the potential energy surface. The stratof following local normal modes to find saddle points wapparently first described by Crippen and Scheraga,10 andlater by Hilderbrandt.11 In these early algorithms, a smastep is taken up the potential along a particular mode,lowed by a step towards lower potential energy alongother modes. In the early 1980’s, these methods wereplaced by quasi-Newton methods, in which the eigenvalof the Hessian matrix are shifted to ensure that the potenis maximized along one chosen mode and minimized alall others. The shift parameters, or Lagrange multipliewere introduced by Cerjan and Miller12 and later modified bySimonset al.,13,14 and by Wales.15 A summary of the earlydevelopments is given in Ref. 16. These methods have bused extensively inab initio calculations of molecules anempirical potential calculations of atomic and molecuclusters.17,18 We will refer to these methods collectively amode following methods. They are derived by expandingpotential in a local quadratic form, and selecting one oflocal harmonic modes as the direction for the climb. If tsoftest mode is chosen, this is analogous to walking upslowest ascent of a valley. This does not necessarily leadsaddle point~as will be illustrated in an example below!, soan important property of these methods is the abilitysearch for a saddle point along different orthogonal moleading away from a given initial configuration. But, sinthe mode following methods require the evaluation andversion of the second derivative Hessian matrix, they scpoorly with the number of degrees of freedom in the syste
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Furthermore, second derivatives are only available at ralow levels ofab initio calculations, and are also not availabin plane wave based density functional theory~DFT! calcu-lations. The dimer method presented here captures the mimportant qualities of the mode following algorithms, whiusing only first derivatives of the potential energy.
An illustration of the importance of having a method thcan be used to systematically identify saddle points leadfrom a given initial state, is the discovery by Feibelman1990 that an Al adatom does not diffuse on the Al~100!surface by repeated hops from one site to another, aspreviously been assumed, but rather by a concerted exchmechanism involving concerted displacement of twatoms.19 This illustrates well how the preconceived notiona transition mechanism can be incorrect even for a simsystem. With the rapid increase in computational power aincreased sophistication of simulation software, the complity of simulated systems has increased greatly. For msystems that are actively being studied, even withab initiomethods, it is difficult and, in any case risky to simply guewhat the transition mechanism is.
II. THE DIMER METHOD
The method presented here for finding saddle pointsvolves working with two images~or two different replicas!of the system. We will refer to this pair of images as t‘‘dimer.’’ If the system hasn atoms, each one of the imageis specified by 3n coordinates. The two replicas have almothe same set of 3n coordinates, but are displaced slightly ba fixed distance. The saddle point search algorithm involmoving the dimer uphill on the potential energy surfacfrom the vicinity of the potential energy minimum of thinitial state up towards a saddle point. Along the way, tdimer is rotated in order to find the lowest curvature modethe potential energy at the point where the dimer is locatThe strategy of estimating the lowest curvature mode apoint without having to evaluate the Hessian matrix was psented by Voter in his hyperdynamics method. There, iused to construct a repulsive bias potential so as to acceleclassical dynamics of activated processes.9 In the methodpresented here, we use his dimer strategy to make the sefor saddle points more efficient.
A. Forces and energies
The dimer, depicted in Fig. 1, is a pair of images seprated from their common midpointR by a distanceDR. ThevectorN which defines the dimer orientation is a unit vectpointing from one image atR2 to the other image atR1 .When a transition state search is launched from an inconfiguration, with no prior knowledge of whatN might be,a random unit vector is assigned toN and the correspondingdimer images are formed
R15R1DRN
and
R25R2DRN. ~1!
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7012 J. Chem. Phys., Vol. 111, No. 15, 15 October 1999 G. Henkelman and H. Jonsson
Initially, and whenever the dimer is moved to a new locatiothe forces acting on the dimer and the energy of the dimare evaluated. These quantities are calculated from theergy and the force (E1 , F1 , E2 , andF2) acting on the twoimages. The energy of the dimerE5E11E2 is the sum ofthe energy of the images. The energy and the force actinthe midpoint of the dimer are labeled asE0 andFR and arecalculated by interpolating between the images. The forceFR
is simply the average force (F11F2)/2. The energy of themidpoint is estimated by using both the force and the eneof the two images. A relation forE0 can be derived from thefinite difference formula for the curvature of the potentialCalong the dimer:
C5~F22F1!•N
2DR5
E22E0
~DR!2 , ~2!
E0 can be isolated from this expression in terms ofknown forces on the images
E05E
21
DR
4~F12F2!•N. ~3!
It should be emphasized that all the properties ofdimer are derived from the forces and energy of the timages. There is no need to evaluate energy and force amidpoint between the two images. This is importantminimizing the total number of force evaluations requiredfind saddle points. An additional benefit of this strategythat the method can be efficiently parallelized over two pcessors, the energy and force on each image being calcuon a separate processor. Forab initio calculations, in whichforce evaluations typically take a very long time comparwith communication time, the execution time for each trasition state search is effectively halved if two processorsused.
B. Rotating the dimer
Each time the dimer is displaced, it is also rotated witsingle iteration towards the minimum energy configuratioThe practicality of the dimer method relies heavily on usian efficient algorithm for the rotation. Minimizing the dimeenergy,E, is equivalent to finding the lowest curvature mo
FIG. 1. Definition of the various position and force vectors of the dimThe rotational force on the dimer,F', is the net force acting on imageperpendicular to the direction of the dimer.
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at R. The energy,E0 , at the fixed midpoint of the dimer isconstant during the rotation. SinceDR is also constant, Eq~2! shows that the dimer energy,E, is linearly related to thecurvature,C, along the dimer. Therefore, the direction whicminimizes E is along the minimum curvature mode at thmidpoint R.
1. Modified Newton method for rotation within aplane
The dimer rotation will first be discussed in the conteof a modified Newton method. In the next section, tmethod will be extended to incorporate also a conjugate gdient approach. The dimer is rotated along the rotatioforce, F'5F1
'2F2' , where Fi
'[Fi2(Fi•N) N for i 51,2.The rotational force is taken to be the net force actingimage 1~see Fig. 1!. The rotation plane is spanned byF' andthe dimer orientationN. It is useful to define a unit vectorQ, within the plane of rotation, perpendicular toN. For themodified Newton method,Q is just a unit vector parallel toF'. The vectorsQ and N form an orthonormal basis whichspans the rotation plane. Given an angle of rotation,du,image 1 moves fromR1 to R1* ~see Fig. 2!
R1* 5R1~N cosdu1Q sindu!DR. ~4!
After image 1 is moved to the new pointR1* , the new dimerorientationN* is calculated, and image 2 is positioned atR2*according to Eq.~1!. The forcesF1* , F2* , and F* 5F1*2F2* are then computed. A scalar rotational forceF5F'
•Q/DR is used to describe the magnitude of the rotatioforce along the direction of rotation. Dividing byDR scalesthe magnitude ofF so that it is independent of the dimeseparation. A finite difference approximation to the chanin the rotational force,F, as the dimer rotates through thangledu is given by
F85dF
du'UF* •Q* 2F•Q
duU
u5du/2
. ~5!
This approximation most accurately estimates the derivafor the midpoint of the finite rotation atu5du/2.
.FIG. 2. Definition of the various quantities involved in rotating the dimAll vectors are in the plane of rotation. The dimer is first rotated abousmall angledu to give a finite difference estimate ofF8 @given by Eq.~5!#.The dimer is then rotated by a calculated angleDu @given by Eq.~13!# tozero the force within the plane of rotation.
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7013J. Chem. Phys., Vol. 111, No. 15, 15 October 1999 Finding saddle points on high dimensional surfaces
A reasonable estimate of the rotation,Du, required tobring F to zero can be obtained from Newton’s method
Du'~F•Q1F* •Q* !
22F8. ~6!
The rotation angles are illustrated in Fig. 2. UnfortunateNewton’s method systematically overestimates the angleDurequired to rotate the dimer to the minimum energy. An iprovement on Eq.~6! can be made if the form ofE(u), thedimer energy as a function of rotation angle, is known. Tcan be accomplished in the following way. The first terma Taylor expansion of the potentialU in the neighborhood ofR is a hyperplane through the pointU(R)5E0 . This termalone produces no rotational force on the dimer becausedimer energy in this case is independent of orientatiE(u)52E0 . The quadratic term in the Taylor expansion itroduces a rotational force. In order to write an analytic foof the quadratic approximation to the potential,x and y aredefined to be the normal modes of the potentialU within theplane of rotation. The plane is the two-dimensional subspspanned byQ andN. Including terms up to second orderthe Taylor expansion gives
U5E02~Fxx1Fy y!1 12 ~cxx21cy y2!. ~7!
The forcesFx and Fy are 2]U/]x and 2]U/]y wherexand y are distances alongx and y. The curvature of thepotential alongx and y are cx and cy , respectively. Thedimer energyE can be expressed within this quadratic aproximation as a function ofu:
E52E01~DR!2@cx cos2~u2u0!1cy sin2~u2u0!#, ~8!
whereu0 is some reference angle. As expected,Fx and Fy
which define the linear change inU do not contribute to theenergy of the dimer. Equation~8! can be rearranged usingtrigonometric identity:
E52E01 12 ~DR!2$~cx2cy!cos@2~u2u0!#
1~cx1cy!%. ~9!
The derivative of this potential yields an analytical expresion for the scalar rotational force on the dimer
F5A sin@2~u2u0!#. ~10!
The constantA5(cx2cy) does not, in practice, have to bevaluated. The energy and the rotational force on the diare invariant to rotations ofp. Equation~10! shows thatu0
can be interpreted as the angle at which the force ondimer is zero within the rotation plane. The differenceu2u0 is the necessary angle of rotation required to reaczero force. It is now possible to obtain an analytic formthe derivativeF8 defined in Eq.~5! within this harmonicapproximation to the energy
F85dF
du52A cos@2~u2u0!#. ~11!
In a simulation,F andF8 are evaluated at some orienttion of the dimer. If this point is labeled asu50, then theangle through which the dimer should be rotated to reac
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zero in the force isu0 . Let F0 andF08 be the values ofF andF8 evaluated atu50. Then the ratio of Eqs.~10! and ~11!:
F0
F085
1
2tan~22u0!, ~12!
yields a simple expression in which the desired rotatangle can be isolated in terms of known quantities
Du5u0521
2arctanS 2F0
F08D . ~13!
Equation~13! has better behavior than Eq.~6! for rotation inthe limits ofF→0 andF8→0. This approach to the minimization is similar to a method used to minimize the electrodegrees of freedom in plane wave based density functiotheory ~DFT! calculations.20
As an example, we will discuss an application of tmodified Newton method to a system representing an alunum adatom on an Al~100! surface. Here we focus on thproperties of the rotation only. The results of the saddle posearches for this system are presented in Sec. III B. A diis placed on the potential surface and incrementally rotathrough an angle of 2p. Figure 3 shows the force and energof the dimer. The energy shows the expected sinusoidalhavior in the local quadratic approximation to the potentiThe period ofp is due to the symmetry of the dimer. Thsinusoidal curve evaluated using Eq.~9! agrees well with theenergy data. The force is also well represented by the ssoidal Eq.~10!. This is to be expected becauseF is simply
FIG. 3. Illustration of the modified Newton’s method for orienting thdimer. The force and the energy of the dimer for an Al adatom on Al~100!are shown for a full rotation. The success of a sinusoidal fit to the dimenergy indicates that a quadratic approximation@Eq. ~9!# is a good approxi-mation. A fit @Eq. ~10!# to the force acting on the dimer yields a minimumdimer energy within the plane atu050.64937, in good agreement with thaobtained from Eq.~13! using onlyF andF8 calculated atu50. The dashedline shows the magnitude of the total rotational force. Atu5u0 this forcehas no component in the plane of rotation.
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7014 J. Chem. Phys., Vol. 111, No. 15, 15 October 1999 G. Henkelman and H. Jonsson
proportional to the derivative of the dimer energy with rspect tou. A fit to these data~shown in Fig. 3! gives a valueof u0'0.649 37. This value, indicated by the vertical dashline, corresponds well with the minimum in the dimer enerwithin the rotation plane. In a simulation,Du is determinedfrom Eq. ~13!. For this example,F0 was found to be4.453 12, andF08 was 22.4847, from whichDu was calcu-lated as 0.649 36, in good agreement with the observalue.
In summary, the quadratic approximation to the potenprovides a formula for rotating the dimer and zeroing trotational force within the plane. This is done by evaluatithe magnitude of the rotational forceF, the curvature of thedimer energyF8, and evaluatingDu by substituting into Eq.~13!. Figure 3 shows the magnitude of the total rotationforce. Once the dimer is rotated byDu, the rotational forcehas essentially no component within the plane of rotationis a bit disconcerting to see that the magnitude of the tforce drops by only 35% in the first iteration. This ratiotypical of the modified Newton method. This can be improved upon by considering conjugate gradients.
2. Conjugate gradient choice of rotation plane
Conjugate gradient methods tend to be more efficithan steepest descent methods because the force at bocurrent iteration and the previous iteration are used to demine an optimal direction of minimization.21 We use a con-jugate gradient algorithm for choosing the plane of rotatiwhile the minimization of the force on the dimer withinplane is carried out using the modified Newton methodscribed in the previous section. The conjugate gradmethod as described in Ref. 21 cannot be applied directlthe dimer energy minimization. For rotation, the dimemidpoint and separation must be held fixed, which addsadditional constraint on the system. In this section, the trational conjugate gradient method is first reviewed, and themodification for the constrained dimer rotation is describ
The first step in the conjugate gradient minimization isteepest descent step in which the direction of displacemis given by the gradient,~or force!, F. The energy alongF isthen minimized. For subsequent iterations, the directiondisplacements,G is taken to be a linear combination of thcurrent force,Fi , and the force at the previous iteratioFi 21 . The vectorG at iterationi is defined recursively:
Gi5Fi1g iGi 21 , ~14!
whereg i is the weighting factor
g i5~Fi2Fi 21!•Fi
Fi•Fi. ~15!
In the dimer method, the traditional conjugate gradiemethod is modified in several ways to accommodate the cstraints implicit in the dimer rotation. Each minimization drection,G, becomes a plane of rotation spanned by the uvectorQ which is parallel toG', and the dimer orientationN. The line minimization step is implemented with the modfied Newton’s method of the previous section. The differenis that, for every step other than the first,Q is not along theforceF' as it was, but rather along the conjugate vectorG'.
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Equations analogous to Eqs.~14! and ~15! are used in theconjugate gradient rotation scheme. For the first iteratG'5F'. For thei th iteration
Gi'5Fi
'1g i uGi 21' uQ i 21** ~16!
where
g i5~Fi
'2Fi 21' !•Fi
'
Fi'•Fi
' , ~17!
is the weighting factor between the rotational forceFi' and
the old modified force vectorGi 21' . The vectorGi 21 in Eq.
~14! has been replaced by the vectoruGi 21' uQ i 21** ~Fig. 2
shows howQ** is found!. This difference is due to the facthat Gi 21
' was aligned alongQi 21 . As described in the previous paragraph, a vector which is perpendicular toNi withinthe old rotation plane is needed. This is simply a vecalongQi 21** , with a magnitude equal toGi 21
' . The conjugategradient approach@Eqs. ~16! and ~17!#, including the modi-fied Newton algorithm for line minimization@Eq. ~13!#, rep-resents a significant improvement over a straightforwardplication of the standard conjugate gradient constraint fordimer. The average number of force calls required to finsaddle point in the Al/Al~100! study was reduced by a factoof about 2.5.
C. Translating the dimer
Compared to rotation, the translation of the dimer is retively straightforward. The saddle point is a maximum alothe lowest curvature mode, the reaction coordinate, anminimum along all other modes. The dimer will orient itsealong the lowest curvature mode when the energy ofdimer is minimized by rotation. The net translational foracting on the two images in the dimer,FR , tends to pull thedimer towards a minimum, however. Therefore, a modififorce, F†, is defined where the force component along tdimer is inverted:
F†5FR22Fi. ~18!
Movement of the dimer along this modified force will brinit to a saddle point. This is illustrated in Fig. 4. In principlany optimization algorithm depending only on first derivtives can be used to move the dimer along the effective foto the saddle point.
We have used two different algorithms to translatedimer. The first is similar to an algorithm that has been usby others to find potential energy minima.22 We will refer tothis algorithm as the ‘‘quick-min’’ algorithm. A time stepsize, Dt, is selected. This should be as large as possiwhile still allowing the system to reach the convergence cteria for the saddle point. The system is propagated frominitial position using a classical dynamics algorithm, with tmodification that only the projection of the velocity at thprevious step along the current force is kept. Additionallythe dot product of the force and the velocity becomes netive, the cumulative velocity is set to zero
DV i5Fi†Dt/m ~19!
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7015J. Chem. Phys., Vol. 111, No. 15, 15 October 1999 Finding saddle points on high dimensional surfaces
V i5H DV i~11DV i•V i 21 /DV i2! if V i•Fi
†.0
DV i if V i•Fi†,0
. ~20!
There is a problem with the algorithm described thuswhen the dimer is started from a shallow minimum. If tlowest curvature mode is along a contour of the potenenergy basin, the dimer can take a very long time to leavebasin, or even possibly become trapped there forever. Alution to this problem is to treat regions where all modhave positive curvature, theconvex regions, differently fromthe regions where at least one mode has a negative curvathe nonconvex regions. The neighborhood of potentiaminima falls into the first category while the saddle poregion falls into the second category. Equation~18! is modi-fied in the following way to ensure that the dimer quickleaves convex regions:
F†5H 2Fi if C.0
FR22Fi if C,0, ~21!
where C is the minimum curvature. In the convex regioC.0, and the dimer follows this mode up the potential sface until the lowest curvature becomes negative. It is psible thatC never becomes negative~an example of that in atwo-dimensional case will be given below!, in which casethe dimer continues to climb up the potential forever. Tproblem is unlikely to occur in large atomic systems. Wnever encountered it in the Al/Al~100! calculations describedbelow.
The second method we tried for translating the dimwas the conjugate gradient method. This was found to pform better than quick-min in the Al/Al~100! calculations. Inthe initial step, the system is minimized along a line definby the initial force. Analogous to the rotation algorithm, tsystem is moved a small distance along the line~keeping thedimer orientation fixed!, and the derivative in the magnitudof the effective forces was calculated. Newton’s methodused to estimate the zero in the effective force along theand the dimer is moved to that point. If the effective forcethe line increases in the small step, the dimer is still in
FIG. 4. The effective forceF† acting on the center of the dimer is the tru
forceFR with the component along the lowest curvature modeN inverted. Inthe neighborhood of a saddle point, the effective force points towardssaddle point.
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minimum region, and Newton’s method calculates a sbackwards against the effective force, pulling the dimer bainto the minimum. In this case, the dimer is simply movwith the effective force a predefined step size. This algorittends to move the dimer out of the convex region quicand in practice speeds up convergence to a saddle pAfter each translation, the dimer is reoriented and thmoved along a direction conjugate to the previous lminimization.21
D. Selecting initial configurations
In most systems, there can be a large number of sapoints leading out of the potential energy basin of interestsingle saddle point search will generally not be enoughaddress the question of how the system tends to leavebasin. In general, it is necessary to knowall low lying saddlepoints ~to within a few kBT from the lowest energy saddlpoint! leading from a potential energy basin. While no exiing method can guarantee that all relevant saddle pointsbe found, reasonable progress may be made if there is ato search for new saddle points in a manner that minimithe number of duplications.
One simple approach is to start with a collection of intial configurations, scattered about the potential energy mmum in the initial basin. In order to avoid high energy cofigurations, which might be spatially near the potentminimum, a system can be evolved by classical dynamicsome finite temperature and configurations of the atomsresponding to maximal displacements from the potentialergy minimum can be saved as initial configurationssaddle point searches. In other words, different saddle pocan be found if the initial configurations are drawn from thigh potential energy images within a thermal ensemblethe potential energy basin. This approach turned out toquite successful. But, it is important to realize that sosaddle points can be systematically excluded when onlymethod is used. The configurations generated tend toalong low energy modes around the minimum and the dimsearches from these configurations tend to converge tosame saddle points, the saddle point lying at the end of acurvature mode. These are, however, often the lowest ensaddle points. Starting with a random set of images displafrom the minimum, for example, a Gaussian distributiondisplacements amounting to'0.1 Å in each coordinate, gava greater variety of saddle points and therefore better spling. The following section describes how different saddpoints can be found when starting from the same initial cfiguration.
E. Orthogonalization
The various mode following algorithms can convergea variety of saddle points starting from the same initial cofiguration by following different normal modes.12–14,16Thereis, however, no inherent relationship between the numbenormal modes and the number of saddle points. The imptant aspect of the mode following algorithms is the orthognality of the modes, which tends to lead the system in dferent directions towards different saddle points. This kind
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7016 J. Chem. Phys., Vol. 111, No. 15, 15 October 1999 G. Henkelman and H. Jonsson
orthogonality constraint can be built into the dimer methquite easily and with little increase in computational cost
From any initial configuration, the lowest curvatumode can be found with the dimer rotation method descriin Sec. II B. When the dimer is rotated to its minimum eergy configuration, the dimer orientation is along the lowcurvature modeN1 . Let the curvature along this mode bdenoted byC1 . The first saddle point search will typically blaunched with the dimer initially oriented along this mode.is also possible to find the next lowest mode by again roing a dimer to its minimum energy configuration but nomaintaining orthogonality to the vectorN1 . Orthogonaliza-tion is carried out by simply subtracting any componealongN1 from the vectorsN, F1 , andF2 , while rotating thedimer to a minimum energy. A new saddle point searchthen be launched from the same initial configuration withdimer initially oriented along the second lowest modeN2 . Inthis second search, the orthogonality condition betweencurrent orientation of the dimerN and the initial orientationin the first searchN1 is maintained until the curvatureCalong the dimer becomes lower than the curvature measalong the directionN1 . This is not the same as requiring ththe curvature alongN be lower thanC1 , the initial curvaturealongN1 . The requirement is thatN gives the lowest curvature mode at the point on the potential energy surface wthe orthogonality constraint is dropped. WhenN gives thelowest curvature mode, there is no need to maintain thethogonality condition to the vectorN1 because the dimer hano tendency to rotate into this now higher curvature dirtion.
It is straightforward to continue this procedure to follosystematically different directions. As long as the curvatalong the current lowest modeN is greater than the curvaturin one of the initial directions of earlier searches, thenorthogonality condition is maintained. These different sadpoint searches can be carried out in parallel after the inset of low lying modes (N1 ,N2 ,...) have been found. Thecost of these subsequent searches increases somewhacause as long as the dimer does not lie along the lowcurvature mode, the force and dimer energy must be cputed for every initially lower mode. Two things save thpotentially poor scaling. First, there is no need to compthe curvatures very often, and second, it has turned outthe dimer tends to escape the region where previous mhave lower curvature quite quickly. The combination ofdistribution of initial configurations and orthogonalizatioprovides an efficient, highly parallelizable, method fsearching for saddle points leading out of a given potenenergy basin.
III. RESULTS
The characteristics of the dimer method have been sied using two model potentials. The first is a twdimensional model potential. The second is a system resenting an Al adatom on an Al~100! surface—a systemcontaining 301 atoms.
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A. Two-dimensional test system
Simple potential surfaces which can be visualized eaare important test cases for any saddle point search metWe have chosen a LEPS potential coupled to a harmooscillator because it illuminates some of the problems tcan be encountered in saddle point searches. The anaform and justification for the potential are describelsewhere.4 The slowest ascent direction along2 y from theinitial basin centered at~0.7655, 0.2490! does not lead to asaddle point if a steepest ascent search is carried out athis mode. Without modification, the potential has a sinsaddle point between the two large basins defined apprmately by x,2 and x.2. We have added two Gaussiafunctions to this potential to increase the number of sadpoints leading out of the initial basin from one to four
Gi~x,y!5Aie2~x2x0i
!2/2sxie2~y2y0i!2/2syi. ~22!
The parameters of the Gaussian functions are given in TI. Figure 5 illustrates how the dimer method works on thpotential surface.
A classical trajectory calculation was used to generthree starting configurations for the saddle point searcheFig. 5~a!. Dimers are placed at these initial points~the dimerseparation is much too small to be resolved!, and the subse-quent saddle point searches following the lowest modeusing the quick-min algorithm are shown. There is a faisharp change in the two paths initially following the neavertical directions. These are the points at which the curture along the dimer has switched from positive to negat~the boundary of convex and nonconvex regions!, as dictatedby Eq.~21!. To begin with, in the convex region, the dimeronly moved up the potential surface along the lowest curture orientation. After entering the nonconvex region, tdimer is also being moved down along the force perpendlar to the dimer orientation.
Figure 5~b! shows the results of saddle point searchusing the orthogonalization algorithm. Two initial configurtions were used and for each one the two lowest normodes were used in saddle point searches. The two seaalong 1 x and 1 y quickly converge to saddle points. Thsearch following the softest mode along2 y goes down toabout y5210 before locking onto the negative curvatumode alongx and rotating by 90°. The minimization perpendicular to the dimer brings it back towards the saddle poThe second search from this same initial point bringsdimer in the2 x direction, where no saddle point exists. Itconvenient to specify both a maximum potential energy, a
TABLE I. Parameters for the Gaussian functions added to the tdimensional test potential.
i 1 2
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7017J. Chem. Phys., Vol. 111, No. 15, 15 October 1999 Finding saddle points on high dimensional surfaces
FIG. 5. Two algorithms for moving the dimer are compared, along with a mode following algorithm~Ref. 16!. The left-hand side figure shows a shomolecular dynamics trajectory~dashed line!, and three initial configurations generated from the trajectory. The quick-min algorithm was used to modimer along the lowest curvature direction. In the middle figure, two initial configurations were chosen on opposite sides of the minimum. Fconfiguration, two dimer search calculations were carried out, in one case following the lowest curvature direction, in the other following the neowestcurvature. The dimer moving off the plot to the right-hand side did not converge to a saddle point. The dimer which moved off the bottom of the plotedafter moving toy5210 and finally did converge to a saddle point. In this calculation, the conjugate gradient method was used to move the dimright-hand side figure shows results of a mode following method requiring the diagonalization of the Hessian matrix. Qualitatively, the results of te dimermethod with orthogonalization are very similar to the results of the mode following method. This comparison is only feasible in a few dimensions bese themode following algorithm requires inversion of the Hessian matrix. None of the methods located one of the saddle points~in the upper right-hand side corneof the basin!.
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a maximum number of allowed iterations so that a faisearch, such as this one, can be aborted without wastinunreasonable amount of effort.
Figure 5~c! shows the results of a mode followinmethod16 which relies on computing the Hessian matrix.performs very efficiently on this test potential, finding thrmodes in an average of 16 moves each. In two dimensithe cost of each move is small, but as the number of dimsions increase the cost of evaluating and inverting the Hsian matrix increases rapidly~asn3!. Even if the number ofsteps remains small, the total cost becomes prohibitivelarge systems. It is reassuring to see that the orthogonadimer searches mimic the attractive features of the mfollowing algorithm without having to evaluate the Hessimatrix.
It is instructive to identify which initial points lead thdimer to a given saddle point. Figure 6 shows the basinattraction for the various saddle points of the twdimensional test problem when only the lowest mode islowed. A test dimer was placed at an array of initial poinThe shaded areas are the regions of the potential in whichdimer rotated into a negative curvature mode, indicating tthere is at least one negative curvature mode at that pEach dimer was then moved in the direction of the effectforce to a saddle point. The regions are shaded accordin
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which saddle point the dimer moved to. In this way, tnegative curvature region is further divided up into four bsins of attraction, one for each saddle point. If the dimerstarted outside of the shaded region, it can still convergesaddle point as shown in Fig. 5, by moving up the potenalong the lowest~positive! curvature mode until it reachesbasin of attraction. Once the dimer reaches a basin of atttion of a saddle point, it is most likely to converge to thsaddle point. Therefore, images which start around theminimum will not reach saddle point 3.
It is fairly clear then why none of the images startaround the left minimum reached saddle point 3. They wohave to pass through basins 2 or 4 before getting to the baround saddle point 3. This illustrates a limitation of tdimer method. If a saddle point has a basin which is srounded or separated from the initial configuration by othsaddle point basins, it is very unlikely for the dimer methto find it. Another limitation of the method can be seenconsidering saddle point 3. This saddle point connectstwo minima on the left by a rather curved minimum enerpath. Dimers that are started from the minimum on the rifollowing the 1 y mode converge to saddle point 3. But this not a saddle point leading out of that minimum. This isproblem when the goal is to find all saddle points leadingof a given minimum. It is easy enough to check if a giv
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7018 J. Chem. Phys., Vol. 111, No. 15, 15 October 1999 G. Henkelman and H. Jonsson
saddle point is relevant for a given initial state, but the cof finding it can be a wasted effort. This effect is observedthe aluminum system as well as this two-dimensional pottial.
B. Al adatom on an Al „100… surface
We now turn to a study of transition mechanisms insystem where an Al adatom is initially sitting in the fourfohollow site on Al~100! surface. The goal is to find all mechanisms by which the system can escape from this initial swith an activation energy less than about 1 eV. An embedatom potential, similar to that of Voter and Chen,23 was usedto model the atomic interactions. The energy of the two loest saddle points predicted by this potential turn out toclose to the results of density functional theocalculations,19 indicating that the potential function is reasonably accurate. The substrate consists of 300 atoms, 5layer in six layers. The bottom two layers are held frozen athe top surface is left open to vacuum. A single Al adatomplaced on the surface bringing the total number of degreefreedom to 603.
The dimer method was run starting from 1000 randomchosen configurations around the minimum. A clustercluding the adatom and 25 nearby substrate atoms wasplaced according to a Gaussian probability distribution wa width of 0.1 Å along each coordinate. The conjugate gdient method was used both to rotate and translate the diValues of the finite difference steps for rotation and trans
FIG. 6. Regions of attraction around each saddle point. The shaded recorrespond to points with at least one negative curvature mode. The dent shades of gray indicate which saddle point the dimer will converge tlimitation of the method is apparent here. A dimer starting from the inibasin on the left-hand side will not be able to find saddle point 3, whrepresents one of the escape routes from the basin, without first visitinbasin around saddle point 2 or 4.
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tion were du51024 rad anddR51023 Å, respectively. Amaximum translation distance was set at 0.1 Å, anddimer separation was set atDR51022 Å. The tolerance forconvergence to a saddle point was set atuFu,1024 whereFis the 3n dimensional force vector.
A summary of the results is shown in Fig. 7. Of the 10dimer searches, 990 converged to saddle points below 2Three searches failed to converge within the imposed limi2000 iterations. Seven of the searches converged to thewavelength, high energy mode~with activation energy of'5eV! corresponding to a concerted shift of the 51 surfaceoms by one lattice constant. The energy of the saddle powere binned and the number of searches within each bishown in the histogram in Fig. 7. The average computatiocost to find the saddle points within each bin is giventerms of the number of force evaluations. The average nber of force evaluations needed to converge on a saddle pwas 400, that is 200 per image in the dimer. The three lowenergy saddle points attracted 78% of the 1000 dim~saddle points that are equivalent by symmetry are groutogether!. These are the exchange process involving twooms found by Feibelman,19 the hop, and a remarkably lowfour atom exchange process~see Fig. 8!. A three atom ex-change process has the fourth lowest saddle point. No tsition mechanisms were found with saddle point energythe range 0.44–0.76 eV, but above this energy gap therelarge number of different processes with saddle point eneup to about 1.5 eV. Figure 8 shows the ten lowest enetransitions. The initial state, saddle point, and final stateshown. The energies listed below each transition numberthe energy of the saddle point configuration with respect tconfiguration of an adatom on a flat surface.
Once the dimer has converged to a saddle point, i
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FIG. 7. The result of 1000 saddle point searches using the dimer methoan Al adatom on an Al~100! surface. Each search starts from a point radomly displaced from the potential energy minimum, using a Gaussiantribution with a width of 0.1 Å in each coordinate of 25 atoms including tadatom and its neighbors. A large number of saddle points was found.histogram shows how many of the dimers converged on saddle points ingiven energy range. The processes corresponding to the ten lowest esaddle points are shown in Fig. 8. A total of 60 processes below 2.0 eV wfound, most of them in the range of 0.8–1.5 eV. The filled circle showsaverage number of force evaluations required to converge to the sapoints within each bin. The bars show the range between the shorteslongest search. On average 400 force evaluations~200 per image in thedimer! were needed to converge to a saddle point~dashed line!.
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7019J. Chem. Phys., Vol. 111, No. 15, 15 October 1999 Finding saddle points on high dimensional surfaces
FIG. 8. The ten lowest energy transtion mechanisms found in 1000searches using the dimer method. Aon-top view of a small section of thesurface is shown in the initial state~left-hand side!, saddle point~middle!,and final state~right-hand side!. Theatoms are shaded by depth, and theoms that move the most are labeleThe energy of the saddle point configurations is given in eV with respecto an Al adatom in the fourfold hollowsite on the flat Al~100! surface. In ad-dition to the two atom exchange process~process 1, discovered by Feibeman! and the hop~process 2!, a fouratom exchange mechanism~process 3!and a three atom exchange mechanis~process 4! are found to be low energydiffusion mechanisms. Other processes correspond to an adatom gting buried in the surface~process 5!,vacancy formation~processes 6, 7, 9and 10!, and a four atom exchange diffusion mechanism involving a seconlayer atom~process 8!.
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easy to trace out the minimum energy path. The dimer isallowed to rotate into the lowest curvature mode, thestable mode, so that it is aligned along the reaction coonate. An image is placed on one side of the dimer alongdirectionN1 . The distance of the image from the midpointthe dimer~the saddle point! is chosen according to the desired resolution of the path. In a manner very similar toalgorithm used to rotate the dimer, the energy of this imais minimized while keeping its distance from the previoimage~in this case the saddle point! fixed. This procedure isrepeated, each time placing a new image initially alonglocal path ~the line between the two previous images! tominimize the number of function calls required to zero ttangential force on the new image. The process is stopwhen the minimum energy of an image is greater than thathe previous image. After the path is traced out in one dirtion from the saddle point to a minimum, the opposite diretion must be followed to complete the minimum energy paThis method was used for the ten saddle points of Fig. 8the energy paths are shown in Fig. 9. The reaction coordiis scaled so that the distance between the minimum andsaddle point is 1 unit. The paths which do not terminate bat an energy of zero, the energy of a single Al adatom onAl ~100! surface, lead to other local minima on the potentenergy surface. The final state of path 5 correspondsstable arrangement with the adatom buried in the surfacgroup of four atoms in the surface layer has rotated by 4The final states of paths 6, 7, and 10 at approximately 0.7are arrangements in which an addimer/vacancy pair hascreated. Path 8 corresponds to a four atom exchange dsion mechanism involving a second layer atom.
Several orthogonal dimer searches were then carriedfrom the 1000 initial configurations described above. F
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each initial configuration, a total of eight orthogonal dimsearches were carried out. In each successive run, the dorientation is orthogonalized to the initial orientation of thdimer in the previous runs. After each search, the sadpoint obtained was compared to those found in previosearches started from the same initial configuration. Inway, an average chance of finding a new saddle point isubsequent, orthogonal search was estimated. The resulshown in Fig. 10. On average, the second search fromgiven initial configuration led to a new saddle point 60%the time. A new saddle point was found in the third sea
FIG. 9. The minimum energy paths corresponding to the ten saddle poidentified and shown in Fig. 8. The reaction coordinate has been scalethat21 represents the initial minimum and 0 the saddle point. Transition6, 7, 9, and 10 lead to final state local minima which do not correspondsingle adatom on the Al~100! surface and are therefore asymmetric.
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7020 J. Chem. Phys., Vol. 111, No. 15, 15 October 1999 G. Henkelman and H. Jonsson
about 40% of the time. During the course of these simutions, many new transition mechanisms for the Al/Al~100!system were found. Some of the more interesting low enesaddle points are shown in Fig. 11. A transition involving tformation of a local hex reconstruction in the surface layethe saddle point was found. In the final state of the transita group of four surface atoms has rotated so as to exchplaces~process 1!. A similar rotation of four surface atomalso occurs in the second transition shown in Fig. 11.addimer/vacancy pair forms in several of the transitions~pro-
FIG. 10. For each one of the 1000 random initial configurations, a totaeight dimer searches were carried out with each subsequent search ortnalized to the initial orientation of the dimer in previous searches. Tsaddle point obtained in each run was compared to the saddle points fin previous searches started from the same initial configuration. The fishows the average fraction of searches which lead to a new saddle poinexample, after a saddle point was found by following the lowest curvadirection, the chance of finding a new saddle point when a second loworthogonal direction was followed is approximately 60%. The chancefinding a new saddle point when the third lowest mode is subsequefollowed is approximately 40%.
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cesses 3, 4, and 7!. For some of the transitions, neither thinitial nor the final state corresponds to an adatom on asurface~processes 5, 6, and 8!, illustrating that the dimerdoes not always converge on saddle points correspondinescape routes for the potential energy basin where the inpoint of the search is located.
C. Scaling with system size
The motivation behind the dimer method is to develan algorithm which scales well with system size. Most of tsavings over the mode following methods, such asCerjan–Miller method, is a result of not having to evaluaand invert the Hessian matrix. The dominant computatioeffort in the dimer method involves computation of the foron the atoms, so the effort can effectively be measured bynumber of force evaluations required to converge on a sapoint. The more degrees of freedom there are in the systhe more iterations are needed to orient and translatedimer. An important question is how the computationalfort scales with the number of degrees of freedom. The mamum number of degrees of freedom was obtained inAl/Al ~100! system by freezing only 55 atoms at the bottoof the substrate leaving the method to explore the remain738 degrees of freedom. In the other extreme, only the atom was allowed to move, making 3 the minimum numberdegrees of freedom. The average cost of finding an ensemof saddle points was computed for eight configurations wmore and more of the substrate atoms frozen. For the mrestricted systems with fewer than 70 degrees of freedomhop was the only process found. The dimer method cverged with about 70 force evaluations in these simulatioFor simulations with fewer frozen atoms the full rangesaddle points was found. This distribution was fairly insesitive to the number of degrees of freedom beyond 70. Taverage number of force evaluations for these runs isproximately 400, the same as what was shown in Fig. 7. T
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FIG. 11. Searches using orthogondirections led to many new transitionfor the Al/Al~100! system. Some ofthe more interesting low energy transtions are shown in the figure. The firstransition shows the formation of a local hex reconstruction in the surfacplane at the saddle point. In the end,rotation of a group of four surface atoms has occurred. A similar rotationalso occurs in the second transitionAn addimer/vacancy pair forms intransitions 3, 4, and 7. Neither the initial nor the final state in transitions 56, and 8 corresponds to an adatom oa flat surface, illustrating that thedimer does not always converge osaddle points corresponding to escaroutes for the potential energy basiwhere the initial point of the search islocated.
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7021J. Chem. Phys., Vol. 111, No. 15, 15 October 1999 Finding saddle points on high dimensional surfaces
data in Fig. 12 show that the method is relatively insensitto increasing the number of dimensions in the problem,long as all processes are available to the system. This isencouraging since it indicates that the dimer method shobe useful for finding saddle points in large and complex stems.
IV. DISCUSSION
Finding the mechanism and estimating the rate of avated transitions from a given initial state boils down tocating the low energy saddle points at the rim of the potenenergy basin corresponding to the initial state~if the har-monic approximation to transition state theory is a goodproximation!. For problems involving diffusion of atoms inand on crystals this problem is tractable, but far from triviPreconceived notions of the transition mechanism can becorrect, as exemplified by the diffusion of an adatom onAl ~100! surface, which was thought to occur by simple hoof the adatom from one surface site to another until Feibman discovered that an exchange mechanism is significalower in energy.19 A systematic procedure for finding saddpoints is needed for such systems.Ab initio calculations ofsmall molecules have over the last decade made use oefficient mode-following algorithm.12–14,16This method hasalso been applied to small clusters described by empirpotentials, but the mode following algorithm scales poowith size, making it inefficient for solid state applications.also requires knowledge of second derivatives of the potial energy with respect to the coordinates of the atoms, pventing its use in the highly successful plane wave ba
FIG. 12. The scaling of the computational effort for the dimer methodmeasured by number of force evaluations, as a function of the systemThese data are taken from the Al/Al~100! system. The substrate consisted300 atoms, 50 atoms per layer. Starting with all movable atoms, the numof degrees of freedom was gradually reduced by freezing more and mothe substrate atoms so they became effectively removed from the caltion. If an atom is frozen the dimer cannot be oriented in these degreefreedom, and the forces on frozen atoms do no affect the dimer. Forsmallest number of degrees of freedom, the only mechanism found wahop. When more than 20 atoms were unfrozen, the full range of sapoints was found. In this region, the plot shows a remarkably slow increin cost with system size, especially if compared to then3 scaling of modefollowing algorithms which involve the inversion of the Hessian matrix.
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DFT calculations of solids and surfaces of solids.~Some ex-amples of DFT studies of surface diffusion are Refs.24–27.! The dimer method presented here can be applielarge systems and since it only relies on first derivativesthe energy, it can be used in conjunction with plane wabased DFT calculations of the atomic forces. We are crently implementing the dimer method in a plane wave Dcode.
An essential aspect of the dimer method is a highly otimized algorithm for rotating the dimer into the lowest eergy orientation. This makes it feasible to use the methodconjunction withab initio atomic forces. The calculations fothe Al/Al~100! system took on average 400 force evaluatioto converge on a saddle point. Since the force calculationthe two images in the dimer are independent, the dimmethod parallelizes almost perfectly over two processorthe force evaluation is computationally intensive as inabinitio calculations. This means the computational effort200 force evaluations per processor. In order to find theof low lying saddle points, a few saddle point searches hto be carried out. In the Al/Al~100! system, approximatelyten searches would suffice to find the three to four lowsaddle points. Using either a collection of randomly chosinitial points or orthogonal searches from a given initial po~or a combination of both!, the different saddle poinsearches can be carried out in parallel. The algorithm wtherefore, be particularly useful when a cluster of processis available for the computations.
Note added in proof. After submitting our manuscript wehave learned of a new method by Munro and [email protected]. B 59, 3969 ~1999!# which also enables saddle poinsearches with only first derivatives.
ACKNOWLEDGMENTS
This work was funded by the National Science Foundtion, Grant No. CHE-9710995. We would like to thank PeFeibelman for helpful comments on the manuscript. Wwould also like to acknowledge Marcus Hudritsch for texcellent graphics library SceneLib which made the outof the simulations so much easier to visualize.
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