advanced review geometry optimization - dept of chemistry...

Advanced Review

Geometry optimizationH. Bernhard Schlegel∗

Geometry optimization is an important part of most quantum chemical calcu-lations. This article surveys methods for optimizing equilibrium geometries, lo-cating transition structures, and following reaction paths. The emphasis is onoptimizations using quasi-Newton methods that rely on energy gradients, andthe discussion includes Hessian updating, line searches, trust radius, and rationalfunction optimization techniques. Single-ended and double-ended methods arediscussed for transition state searches. Single-ended techniques include quasi-Newton, reduced gradient following and eigenvector following methods. Double-ended methods include nudged elastic band, string, and growing string methods.The discussions conclude with methods for validating transition states and fol-lowing steepest descent reaction paths. C© 2011 John Wiley & Sons, Ltd. WIREs Comput MolSci 2011 1 790–809 DOI: 10.1002/wcms.34

INTRODUCTION

G eometry optimization is a key component ofmost computational chemistry studies that are

concerned with the structure and/or reactivity ofmolecules. This chapter describes some of the meth-ods that are used to optimize equilibrium geometries,locate transition structures (TSs), and follow reac-tion paths. Several surveys and reviews of geometryoptimization are available.1–14 Rather than an exten-sive review of the literature, this article is directed to-ward practical geometry optimization methods thatare currently in use. In particular, it is concerned withgeometry optimization methods applicable to elec-tronic structure calculations.15,16 Because electronicstructure calculations can be lengthy, geometry opti-mization methods need to be efficient and robust. Forinexpensive molecular mechanics calculations, sim-pler methods may be adequate. Most major electronicstructure packages have a selection of geometry op-timization algorithms. By discussing the componentsof various optimization methods we hope to aid thereader in selecting the most appropriate optimizationmethods and in overcoming the occasional difficultiesthat may arise. Unconstrained nonlinear optimizationmethods have been discussed extensively in numeri-cal analysis texts (e.g., Refs 17–20). These methodsare designed to find a local minimum closest to thestarting point. Global optimization and conforma-tional searching are more difficult problems and will

∗Correspondence to: [email protected]

Department of Chemistry, Wayne State University, Detroit, MI,USA

DOI: 10.1002/wcms.34

be reviewed elsewhere in this series. Global mappingof reaction networks21,22 is outside the scope of thischapter. Transition path sampling and reaction pathson free energy surfaces are also not covered.23 Dis-cussions of optimization methods to find conical in-tersections and minimum energy seam crossings canbe found elsewhere.22,24–36 Here, we focus on findingthe equilibrium geometry of an individual molecule,locating a TS for a reaction, and following the reac-tion path connecting reactants through a TS to prod-ucts. The geometry optimization methods discussedin this chapter use energy derivatives and depend onfactors such as the choice of coordinates, methods forcalculating the step direction, Hessian updating meth-ods, line search, and step size control strategies. Someof these methods have been compared recently37 formodest set of test cases for optimizing equilibriumgeometries38 and TSs.39

POTENTIAL ENERGY SURFACES

The structure of a molecule can be specified by giv-ing the locations of the atoms in the molecule. For agiven structure and electronic state, a molecule has aspecific energy. A potential energy surface describeshow the energy of the molecule in a particular statevaries as a function of the structure of the molecule.A simple representation of a potential energy surfaceis shown in Figure 1, in which the energy (verticalcoordinate) is a function of two geometric variables(the two horizontal coordinates).

The notion of molecular structure and po-tential energy surfaces are outcomes of theBorn–Oppenheimer approximation, which allows us

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WIREs Computational Molecular Science Geometry optimization

FIGURE 1 | Model potential energy surface showing minima,transition structures, second-order saddle points, reaction paths, and avalley ridge inflection point (Reprinted with permission from Ref 5.Copyright 1998 John Wiley & Sons.)

to separate the motion of the electrons from the mo-tion of the nuclei. Because the nuclei are much heav-ier and move much more slowly than the electrons,the energy of a molecule in the Born–Oppenheimerapproximation is obtained by solving the electronicstructure problem for a set of fixed nuclear positions.Because this can be repeated for any set of nuclearpositions, the energy of a molecule can be describedas a parametric function of the position of the nuclei,thereby yielding a potential energy surface.

A potential energy surface, like the one shown inFigure 1, can be visualized as a hilly landscape, withvalleys, peaks, and mountain passes. Even thoughmost molecules have many more than two geomet-ric variables, most of the important features of a po-tential energy surface can be represented in such alandscape.

The valleys of a potential energy surface repre-sent reactants, intermediates, and products of a reac-tion. The position of the minimum in a valley repre-sents the equilibrium structure. The energy differencebetween the product valley and reactant valley min-ima represents the energy of the reaction. Vibrationalmotion of the molecule about the reactant and prod-uct equilibrium geometries can be used to computezero-point energy and thermal corrections needed tocalculate enthalpy and free energy differences.40 Thelowest energy pathway between the reactant valleyand the product valley is the reaction path.41 Thehighest point on this lowest energy reaction path isthe TS for the reaction, and the difference betweenthe energy of the TS and the reactant is the energybarrier for the reaction. A TS is a maximum in onedirection (the direction connecting reactant and prod-

uct along the reaction path) and a minimum in allother directions (directions perpendicular to the reac-tion path). A TS is also termed as a first-order saddlepoint. In Figure 1, it can be visualized as a mountainpass connecting two valleys. A second-order saddlepoint (SOSP) is a maximum in two directions and aminimum in all the remaining directions. If a reactionpath goes through an SOSP, a lower energy reactionpath can always be found by displacing the path awayfrom the SOSP. An n-th order saddle point is a max-imum in n directions and a minimum in all the otherdirections.

For a thermally activated reaction, the energy ofthe TS and the shape of the potential energy surfacearound the TS can be used to estimate the reaction rate(see other reviews in this series). The steepest descentreaction path (SDP) from the TS down to the reactantsand to the products is termed the minimum energypath (MEP) or the intrinsic reaction coordinate (IRC;the MEP in mass-weighted coordinates). The reactionpath from reactants through intermediates (if any) toproducts describes the reaction mechanism.41 A moredetailed description of a reaction can be obtainedby classical trajectory calculations42–45 that simulatemolecular dynamics by integrating the classical equa-tions of motion for a molecule moving on a potentialenergy surface. Photochemistry involves motion onmultiple potential energy surfaces and transitions be-tween them (see Ref 46).

ENERGY DERIVATIVES

The first and second derivatives of the energy withrespect to the geometrical parameters can be used toconstruct a local quadratic approximation to the po-tential energy surface:

E(x) = E(x0) + gT0 �x + 1/2�xT H0�x (1)

where g0 is the gradient (dE/dx) at x0, H0 is the Hes-sian (d2E/dx2) at x0, and �x = x−x0. The gradientand Hessian can be used to confirm the character ofminima and TSs. The negative of the gradient is thevector of forces on the atoms in the molecule. Be-cause the forces are zero for minima, TSs, and higher-order saddle points, these structures are also termedstationary points. The Hessian or matrix of secondderivatives of the energy is also known as theforce constant matrix. The eigenvectors of the mass-weighted Hessian in Cartesian coordinates corre-spond to the normal modes of vibration (plus five orsix modes for translation and rotation).47 For a struc-ture to be characterized as a minimum, the gradient

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must be zero and all of the eigenvalues of the Hessiancorresponding to molecular vibrations must be pos-itive; equivalently, the vibrational frequencies mustbe real (the vibrational frequencies are proportionalto the square root of the eigenvalues of the mass-weighted Hessian). For a TS, the potential energysurface is a maximum in one direction (along thereaction path) and a minimum in all other perpen-dicular directions. Therefore, a TS is characterized bya zero gradient and a Hessian that has one (and onlyone) negative eigenvalue; correspondingly, a TS hasone and only one imaginary vibrational frequency. Ann-th order saddle point (also called a stationary pointof index n) has a zero gradient and is a maximum inn orthogonal directions and hence has n imaginaryfrequencies. For a TS, the vibrational mode corre-sponding to the imaginary frequency is also knownas the transition vector. At the TS, the transition vec-tor is tangent to the reaction path in mass-weightedcoordinates.

Most methods for efficient geometry optimiza-tion rely on first derivatives of the energy; some alsorequire second derivatives. For most levels of theoryused routinely for geometry optimization, the firstderivatives can be calculated analytically at a costcomparable to that for the energy. Analytic secondderivatives are also available for several levels of the-ory, but the cost is usually considerably higher thanfor first derivatives. With the possible exception ofoptimization of diatomic molecules, derivative-basedgeometry optimization methods are significantly moreefficient than energy-only algorithms. If analytic firstderivatives are not available, it is possible to use sim-plex and pattern search methods,48–50 but these be-come less efficient as the number of degree of freedomincreases.51 Thus, it may be more efficient to computegradients numerically and to use a gradient-basedoptimization algorithm than to use an energy-onlyalgorithm.

COORDINATES

In principle, any complete set of coordinates can beused to represent a molecule and its potential energysurface. However, choosing a good coordinate systemcan significantly improve the performance of geome-try optimizations. Inspection of the Hessian used inthe local quadratic approximation to the potentialenergy surface, in Eq. (1), can reveal some favorableaspects of a good coordinate system. For example, anoptimization will be less efficient if there are some verystiff coordinates and some very flexible coordinates.This corresponds to a mixture of very large and very

small eigenvalues of the Hessian, (i.e., the Hessian isan ill-conditioned matrix). Strong coupling betweencoordinates can also slow down an optimization. Thiscorresponds to off-diagonal Hessian matrix elementsthat are comparable in magnitude to the diagonal el-ements. Strong anharmonicity can seriously degradethe performance of an optimization. If the Hessianchanges rapidly when the geometry of the molecule ischanged, or if the valley around a minimum is stronglycurved, then the quadratic expression in Eq. (1) is apoor approximation to the potential energy surfaceand the optimization will be slow to converge. Thenature of the Hessian and the anharmonicity of thepotential energy surface will be directly affected bythe choice of the coordinate system.

There are a number of coordinate systems thatare typically used for geometry optimization. Carte-sian coordinates are perhaps the most universal andthe least ambiguous. An advantage is that most energyand derivative calculations are carried out in Carte-sian coordinates. However, they are not well suitedfor geometry optimization because they do not reflectthe ‘chemical structure’ and bonding of a molecule.The x, y, and z coordinates of an atom are stronglycoupled to each other and to the coordinates of neigh-boring atoms.

Internal coordinates such as bond lengths andvalence angles are more descriptive of the molecularstructure and are more useful for geometry optimiza-tion. Bond stretching requires more energy than anglebending or torsion about single bonds. More impor-tantly, the coupling between stretches, bends, and tor-sions are usually much smaller than between Carte-sian coordinates. In addition, internal coordinates aremuch better than Cartesians for representing curvilin-ear motions such as valence angle bending and rota-tion about single bonds. For an acyclic molecule withN atoms, it is easy to select set of 3N−6 internal coor-dinates to represent the molecule (3N−5 coordinatesfor a linear molecule). Z-matrix coordinates are an ex-ample of such a coordinate system.52 It is straightfor-ward to convert geometries and derivatives betweenCartesian and Z-matrix internal coordinates.53

For acyclic molecules, the set of all bonds, an-gles, and torsions represents the intrinsic connectiv-ity and flexibility of the molecule. However, for acyclic molecule, this introduces more than the 3N−6coordinates required to define the geometry ofthe molecule. Such a coordinate system has acertain amount of redundancy in the geometricparameters.53–68 Because only 3N−6 of these redun-dant internal coordinates can be transformed backto Cartesian coordinates in three dimensions, cer-tain combinations of the redundant internals must



be constrained during the optimization. The set of allbonds, valence angles, and torsions (if necessary, aug-mented by out-of-plane bends and linear bends) con-stitutes a primitive redundant coordinate system.53,58

In some cases, it may be advantageous to form lin-ear combinations of the primitive redundant inter-nals to form natural or delocalized redundant internalcoordinates54–57 or symmetry-adapted redundant in-ternal coordinates. For periodic systems such as solidsor surfaces, unit cell parameters need to be added66–68

(either explicitly or implicitly via coordinates thatcross the boundaries of the unit cell). For moleculesin nonisotropic media, additional coordinates areneeded to specify the orientation of the molecule. Forsystems containing more than one fragment, addi-tional coordinates are required to specify the positionsof the fragments relative to each other. The union ofthe redundant internal coordinates for the reactantsand products is usually a good coordinate system forTS optimization.58 The transformation of Cartesiancoordinates and derivatives to redundant internals isstraightforward, but the back transformation of a fi-nite displacement of redundant internals to Cartesianusually is solved iteratively.53–68

NEWTON AND QUASI-NEWTONMETHODS

As described in standard texts on optimization,17–20

most nonlinear optimization algorithms are based ona local quadratic approximation of the potential en-ergy surface; Eq. (1). Differentiation with respect tothe coordinates yields an approximation for the gra-dient, given by:

g(x) = g0 + H0�x. (2)

At a stationary point, the gradient is zero, g(x) =0; thus, in the local quadratic approximation to thepotential energy surface, the displacement to the min-imum is given by:

�x = −H−10 g0. (3)

This is known as the Newton or Newton–Raphsonstep.

Newton and quasi-Newton methods are themost efficient and widely used procedures for opti-mizing equilibrium geometries and can also be usedeffectively to find TSs. For each step in the Newtonmethod, the Hessian in Eq. (3) is calculated at the cur-rent point. For quasi-Newton methods, Eq. (3) is usedwith an approximate Hessian that is updated at eachstep of the optimization (see below). Because actualpotential energy surfaces are rarely quadratic, several

Newton or quasi-Newton steps are required to reach astationary point. For minimization, the Hessian musthave all positive eigenvalues (i.e., positive definite).If one or more eigenvalues are negative, the step willbe toward a first or higher-order saddle point. Thus,without some means of controlling the step size anddirection, simple Newton steps are not robust. Sim-ilarly, if the aim is to optimize to a TS, the Hessianmust have one and only one negative eigenvalue, andthe corresponding eigenvector (i.e., the transition vec-tor) must be roughly parallel to the reaction path.Methods for ensuring that the step is in the desireddirection for minimization or TS optimization are dis-cussed in the section dealing with step size control.

At each step, Newton methods require the ex-plicit calculation of the Hessian, which can be rathercostly. Quasi-Newton methods start with an inex-pensive approximation to the Hessian. The differencebetween the calculated change in the gradient andthe change predicted with the approximate Hessianis used to improve the Hessian at each step in theoptimization.17–20

Hnew = Hold + �H (4)

For a quadratic surface, the updated Hessianmust fulfill the Newton condition,

�g = Hnew�x, (5)

where �g = g(xnew) − g(xold) and �x = (xnew − xold).However, there are an infinite number of ways toupdate the Hessian and fulfill the Newton condition.One of the simplest updates is the symmetric rankone (SR1) update,17–20 also known as the Murtagh–Sargent update.69

�HSR1 = (�g − Hold�x) (�g − Hold�x)T

(�g − Hold�x)T�x(6)

This formula can encounter numerical problemsif |�g − Hold�x| is very small. The Broyden familyof updates70 avoids this problem while ensuring thatthe Hessian update is positive definite. The Broyden–Fletcher–Goldfarb–Shanno (BFGS) update70–73 is themost successful and widely used member of thisfamily:

�HBFGS = �g �gT

�gT�x− Hold�x �xTHold

�xTHold�x. (7)

For TS optimization, it is important that theHessian has one and only one negative eigenvalue.This should be checked at every step of the opti-mization. If the Hessian does not have the correctnumber of negative eigenvalues, the eigenvalues needto be shifted or one of the methods for step size control



needs to be used (see Step Size Control). The initial es-timate of the Hessian for TS optimizations must haveone negative eigenvalue and the associated eigenvec-tor should be approximately parallel to the reactionpath. The Hessian update should not be forced tobe positive definite. The Powell-symmetric-Broyden(PSB) update18 fulfills this role:

�HPSB = (�g − Hold�x)�xT + �x (�g − Hold�x)T

�xT�x

− (�xT(�g − Hold�x)) �x �xT

(�xT�x)2. (8)

Bofill74 found that a combination of the PSB andSR1 updates performs better for TS optimizations:

�HBofill = φ�HSR1 + (1 − φ)�HPSB,

where φ = ((�g − Hold�x)T�x) 2

|�g − Hold�x|2 |�x|2 . (9)

Farkas and Schlegel75 constructed a similar up-date for minimization:

�HFS = φ1/2�HSR1 + (1 − φ1/2)�HBFGS, (10)

where φ is same as in Eq. (9). Bofill76 has recently re-viewed Hessian updating methods and proposed somepromising new formulas suitable for minima and TSs.For controlling the step size by trust radius (τ ) meth-ods or rational function optimization (see Step SizeControl), diagonalizing the Hessian can become acomputational bottleneck for large systems. Updat-ing the eigenvectors and eigenvalues of the Hessiancan be considerably more efficient.77

Quasi-Newton methods require an initial esti-mate of the Hessian. A scaled identity matrix maybe sufficient in some cases, but a better starting Hes-sian can be obtained from knowledge of the structureand connectivity of the molecule. Simple empiricalestimates of stretching, bending, and torsional forceconstants are usually satisfactory.78–80 Calculation ofan initial Hessian by molecular mechanics or semi-empirical electronic structure methods can provide abetter estimate. If the molecule has been optimized ata lower level of theory, the updated Hessian from theoptimization or the Hessian from a frequency calcu-lation at a lower level of theory are even better initialestimates. For harder cases, the rows and columnsof the Hessian can be calculated numerically for afew critical coordinates. For more difficult cases, thefull Hessian can be calculated at the beginning of theoptimization and recalculated at every few step if nec-essary. Recalculating the Hessian at each step corre-sponds to the Newton method.

Standard quasi-Newton methods store and in-vert the full Hessian. For large optimization problems,

this may be a bottleneck. The updating methods canbe reformulated to update the inverse of the Hessian.For example, the BFGS formula for the update of theinverse Hessian is:

�BBFGS = �x�xT

�xT�g− Bold�g�gTBold

�gTBold�g, (11)

where B = H−1, and the updated inverse Hessianobeys �x = B�g. Limited memory quasi-Newtonmethods such as L-BFGS80–89 avoid the storage of thefull Hessian or its inverse which would require O(n2)memory for n variables. Instead, they start with a di-agonal inverse Hessian, and store only the �x and �gvectors from a limited number of previous steps; thus,the storage is only O(n). The inverse Hessian is writ-ten as a diagonal Hessian plus the updates using thestored vectors. For the Newton step, xnew = xold −B gold, the product of the updated inverse Hessianand the gradient involves only O(n) work because itcan be expressed in terms of dot products betweenvectors.

CONJUGATE GRADIENT METHODS

Conjugate gradient methods are suitable for verylarge systems because they require less storage thanlimited memory quasi-Newton methods. The conceptbehind conjugate gradient methods is to choose anew search direction that will lower the energy whileremaining at or near the minimum in the previoussearch direction. If the Hessian has coupling betweenthe coordinates, the optimal search directions are notorthogonal but are conjugate, in the sense that �xnew

H�xold = 0. Two of the most frequently used con-jugate gradient methods are Fletcher–Reeves90 andPolak-Ribiere91:

si = − gi + gTi gi

gTi−1gi−1

si−1 (12)

si = − gi + (gi − gi−1)Tgi

gTi−1gi−1

si−1, (13)

where si and si−1 are the current and previous searchdirections, respectively, and gi and gi−1 are the currentand previous gradients, respectively. Note that onlythree vectors need to be stored. Unlike quasi-Newtonmethods, conjugate gradient methods require a linesearch at each step in order to converge.

GDIIS

Direct inversion of the iterative subspace (DIIS) con-structs a solution to a set of linear equations suchas Eq. (3) by expanding in a set of error vectors or



residuals from prior iterations. Geometry optimiza-tion by DIIS (GDIIS) applies this approach to theoptimization of equilibrium geometries and transi-tion structures.92–95 In the numerical analysis liter-ature, DIIS is known as generalized minimum of theresidual (GMRES).96 It has its origins in methods likethe Krylov,97 Lanczos,98 and Arnoldi99 algorithms.In GDIIS, the goal is to construct a new geometry asa linear combination of previous geometries so as tominimize the size of the Newton step [i.e., the residualin the iterative solution of Eq. (3)].

x∗ =∑

i

ci xi g∗ =∑

i

ci g(xi )∑

i

ci = 1 (14)

Minimize |xnew − x∗|2 with respect to ci ,

where xnew = x∗ − H−1g∗ (15)

The GDIIS coefficients are determined by minimiz-ing the residual and can be obtained by solving thefollowing:[

A 11 0

] [cλ

]=

[01

],

where Ai j = (H−1gi )T(H−1g j ). (16)

The GEDIIS method95 uses Aij = (gi − gj) (xi −xj). If A becomes singular, the oldest geometry is dis-carded and the solution is attempted again. The ap-proximate Hessian can be held constant, but betterperformance is obtained if it is updated. If the mini-mization wanders into a region with a negative eigen-value, it may converge to a TS. For a TS optimization,GDIIS sometimes converges not to the desired reac-tion barrier but to the nearest saddle point. One wayto control GDIIS optimizations is to compare the pre-dicted geometry with the one from a Newton step.94

If the angle is too large, it is safer to take the New-ton step (provided the Hessian has the correct num-ber of negative eigenvalues). In the final stages of aminimization, GDIIS sometimes converges faster thanquasi-Newton methods; a hybrid of quasi-Newtonand DIIS methods can be more efficient.95

LINE SEARCH

Newton, quasi-Newton, and GDIIS methods will con-verge without a line search if the surface is quadratic.However, life is not quadratic. For anharmonic sur-faces, the Newton step (Eq. (3)) may be too large ortoo small. Near an inflection point, Newton steps canget into an infinite hysteresis loop. Conjugate gradientmethods need a line search because they generate onlya search direction but not a step length. Thus, a linesearch is a good idea for any method, especially if it

can be done with little or no extra cost. The line searchneed not be exact but must fulfill the Wolfe condition,�E < α gT�x and gnew T�x > β gold T�x, reducing thefunction value and the magnitude of the gradient (α ≈0.1 and β ≈ 0.5 have been found to be practical forquasi-Newton optimizations18). A simple approachis to fit a polynomial to the energy and gradient atthe current and the previous geometry (a cubic poly-nomial or a quartic polynomial constrained to haveno negative second derivatives100). The minimum isfound on the polynomial, the gradient is interpolatedto the minimum, and the interpolated gradient is usedfor the next quasi-Newton step. If there is a largeextrapolation to the minimum, it is safer to explic-itly search for the minimum by doing additional en-ergy (and gradient) calculations to obtain a sufficientreduction in the energy (and the magnitude of thegradient).

STEP SIZE CONTROL

The quadratic approximation to the potential energysurface is satisfactory only for a small local region,usually specified by a trust radius, τ . Steps outside thisregion are risky and optimizations are more robust ifthe step size does not exceed τ . An initial estimate of τ

can be updated during the course of an optimizationbased on how well the potential energy surface canbe fit by a quadratic expression. A typical updatingrecipe is as follows18:

ρ = �E/(gT�x + 1/2 �xT H0 �x).

If ρ > 0.75 and 5/4 |�x| > τ old,

then τ new = 2 τ old.

If ρ < 0.25, then τ new = 1/4 |�x|.Otherwise, τ new = τ old. (17)

The simplest approach to step size control is toscale the Newton step back if τ is exceeded. A betterapproach is to minimize the energy under the con-straint that the step is not larger than τ .18,20 In thetrust radius method (TRM), this is done by using aLagrangian multiplier, λ, and corresponds to min-imizing E(x)−1/2 λ (�x2−τ 2). With the usualquadratic approximation for E(x), this yields:

g0 + H0�x − λ�x = 0

or �x = −(H0 − λI)−1g0 (18)

where I is the identity matrix.For minimizations, λ must be chosen so that all

the eigenvalues of the shifted Hessian, H − λ I, arepositive [i.e., λ must be smaller (more negative) thanthe lowest eigenvalue of H]. Thus, even if H has one or



FIGURE 2 | Plot of the displacement squared,|�x |2 = |(H − λ I )−1g|2, as a function of the Hessian shiftparameter λ (Reprinted with permission from Ref 102. Copyright 1983ACS Publications.). The singularties occur at the eigenvalues of theHessian, hi . (a) For displacement to a minimum with a trust radius ofτ 1, shift parameter λ1 must be less than the lowest eigenvalue of theHessian, h1. (b) For a displacement to a transition structure, the shiftparameter must be between h1 and 1

2 h2. For τ 1, the lower of the twosolutions is chosen. For a smaller trust radius τ 2, there is no solution;λ2 is chosen as the minimum in the curve [or λ2 = 1

2 (h1 + 12 h2) if λ2

> 12 h2] and the displacement is scaled back to the trust radius (see

Refs 101–104 for more details).

more negative eigenvalues, this approach can be usedto take a controlled step downhill toward a minimum.A plot of the square of step size as a function of λ isshown in Figure 2(a) (the singularities occur when λ

is equal to one of the eigenvalues of H).For TS optimization, the shifted Hessian must

have one negative eigenvalue (i.e., λ must be largerthan the lowest eigenvalue of H but smaller than thesecond lowest eigenvalue). As shown in Figure 2(b), ifthere are two solutions for a given step size, λ is usu-ally chosen to be closer to the lowest eigenvalue; ifthere are no solutions for a given step size, λ is chosenas the minimum between the lowest and second low-est eigenvalue and the resulting step is scaled back tothe τ .101–104 This procedure then allows a controlledstep to be taken toward the TS even when the Hes-sian does not have the correct number of negativeeigenvalues.

An alternative approach for TS optimization isto use the eigenvectors of the approximate Hessian todivide the coordinates into two groups; the optimiza-tion then searches for a maximum along one eigen-vector and for a minimum in the remaining space.The step size in this approach can be controlled byusing two Lagrangian multipliers, one for the step up-hill along one eigenvector and the other for the step

downhill in the remainder of the space. Because thisapproach allows one to follow an eigenvector uphilleven if it does not have the lowest eigenvalue, it isalso known as eigenvector following.74,101–109 Alter-natively, a separate λ can be chosen for each eigen-vector such that the step varies from steepest descentfor large gradients to a Newton step with a shiftedHessian for moderate gradients to a simple Newtonstep for small gradients.2 For all of these approachesto transition structure optimization, the Hessian musthave a suitable eigenvector that resembles the desiredreaction path. Instead of focusing on a single eigen-vector, the reduced potential surface approach selectsa small subset of the coordinates for the transitionstructure search and minimizes the energy in the re-maining space.110,111

The rational function optimization (RFO)method is a closely related approach for con-trolling the step size for both minimization andtransition structure searching that replaces thequadratic approximation by a rational functionapproximation.105

�E(x) = gT�x + 1/2�xTH�x1 + �xTS�x

= 12

[�x 1][

H ggT 0

] [�x1

]

[�x 1][

S 00 1

] [�x1

] (19)

Minimizing �E with respect to �x by settingd�E/d�x = 0 leads to an eigenvalue equation involv-ing the Hessian augmented by the gradient.[

H ggT 0

] [�x1

]= 2 �E

[S 00 1

] [�x1

](20)

Typically, S is chosen as a constant times theidentity matrix [in this case the top row of Eq. (20) re-duces to �x = −(H0 − λI)−1g0, as in TRM; Eq. (18)].The lowest eigenvalue and eigenvector of the aug-mented Hessian are used for minimizations, whereasthe second lowest eigenvalue and eigenvector are cho-sen for transition structure optimizations. The �xobtained by the RFO approach is reduced if it islarger than the τ .105,106,108,109 A partitioned RFOmethod may be better for saddle point searches if theHessian does not have the correct number of neg-ative eigenvalues.105 In this case, one eigenvector ischosen to be followed uphill to a maximum and theremaining eigenvectors are chosen for minimization;the RFO method is used for optimization in both sub-spaces. It is possible to reach other transition struc-tures by following an eigenvector other than the onewith the lowest eigenvalue.105–109



CONSTRAINED OPTIMIZATION

Under a variety of circumstances, it may be necessaryto apply constraints while optimizing the geometry(e.g., scanning potential energy surfaces, coordinatedriving, reaction path following, etc.). For nonre-dundant coordinate systems and simple constraints,the coordinates being held constant can be easily re-moved from the space of variables being optimized.For more general constraints and/or redundant inter-nal coordinate systems, constraints can be applied bypenalty functions, projection methods, or Lagrangianmultipliers.

In the penalty function method, the constraintsCi (x) = 0 are imposed by adding an extra term,1/2

∑αiCi (x)2, to the energy in Eq. (1) and the

energy is minimized as usual. Because the α′i s need

to have suitably large values so that the constraintsare approximately satisfied at the minimum, the opti-mization may converge much slower than the corre-sponding unconstrained optimization.

The preferred method for including con-straints in an optimization is by using Lagrangianmultipliers.

L(x) = E(x) +∑

i

λiCi (x) (21)

At convergence of the constrained optimization,the derivative of the Lagrangian L(x) with respectto the coordinate x and the Lagrangian multiplier λi

must be zero. In the LQA,

∂L(x)∂x

= g0 + H0�x +∑

i

λi∂Ci (x)

∂x= 0

and∂L(x)∂λi

= Ci (x) = 0. (22)

Because the Lagrangian multipliers are opti-mized along with the geometric variables, this methodgenerally converges much faster than the penalty func-tion method and the constraints are satisfied exactly.In the special case in which the constraint is linearfunction of the displacements, cT�x = c0, the La-grangian multiplier problem can be solved by usingan augmented Hessian,

g0 + H0�x + λc = 0 and cT�x = c0

or[

H ccT 0

] [�xλ

]=

[−g0

c0

](23)

Another way of applying linear constraints isthe projection method, in which a projector P is usedto remove the directions in which the displacementsare constrained to be zero.

Pg0 + PH0P�x + α(I − P) = 0,

P = I −∑

i

cicTi /|ci |2, (24)

where the ci are a set of orthogonal constraint vec-tors and α > 0. For redundant internal coordinates,the projector needs to remove the coordinate redun-dancies as well as the constraint directions.58

SPECIAL CONSIDERATIONS FORTRANSITION STRUCTUREOPTIMIZATION

Finding a minimum is comparatively easy because thenegative of the gradient always points downhill. Bycontrast, a transition structure optimization must stepuphill in one direction and downhill in all other or-thogonal directions. Often, the uphill direction is notknown in advance and must be determined during thecourse of the optimization. As a result, numerous spe-cial methods have been developed for transition struc-ture searching and many of them are closely related.They can be loosely classified as single-ended anddouble-ended methods.2 Single-ended methods startwith an initial structure and displace it toward thetransition structure. These algorithms include quasi-Newton and related methods (as discussed above andin Quasi-Newton Methods for Transition Structuresand Related Single-Ended Methods). Double-endedmethods start from the reactants and products andwork from both sides to find the transition struc-ture and the reaction path. These include the nudgedelastic band (NEB) method, string method (SM), andgrowing string method (GSM; see Chain-of-States andDouble-Ended Methods).

The success of a given transition structure op-timization method depends on the topology of thesurface as well as the starting structure(s), initial Hes-sian(s), and coordinates. The two dimensional con-tour plots in Figure 3 illustrate some of the impor-tant features that can be found in higher dimensionalsurfaces.112,113 If the change from reactant to productis dominated by a single coordinate, the potential en-ergy surface will have an approximately linear valley,as shown in Figure 3(a). Hindered rotation about asingle bond is an example of such an I-shaped sur-face. All methods should be able to find the transitionstructure on this type of surface without difficulty.Making one bond and breaking the other results inan L- or V-shaped surface, as shown in Figure 3(b).Because the reaction path is strongly curved, findingthe transition structure can be a bit more difficult.A T-shaped surface involves a transition structure



FIGURE 3 | Various classes of reaction channels near thetransition structure on reactive potential energy surfaces: (a) I-shapedvalley, (b) L- or V-shaped valley, (c) T-shaped valley, and (d) H- orX-shaped valley. (Reprinted with permission from Ref 113. Copyright2009 ACS Publications.)

in a hanging valley at the side of a main valley,Figure 3(c). A method that follows the main valleywill miss the transition structure, but will reach thetransition structure if it starts from the other valley.The H- or X-shaped surface in Figure 3(d) is the mostdifficult because following the valley floor from eitherthe reactant or the product side will miss the transi-tion structure.

Quasi-Newton and single-ended methods arediscussed first. These methods differ primarily in theway they try to get close to the quadratic region oftransition structure. Double-ended methods start withthe reactants and products, and optimize the reactionpath as well as the transition structure.

Quasi-newton Methods for TransitionStructuresNewton and quasi-Newton algorithms are the mostefficient single-ended methods for optimizing transi-tion structures if the starting geometry is within thequadratic region of the transition structure. With suit-able techniques for controlling the optimization stepssuch as TRM, RFO and eigenvector following (seeStep Size Control), these methods will also convergeto a transition structure even if they start outside the

quadratic region. Many of the related methods differprimarily in the techniques they use to get close to thequadratic region. There are three main differences inusing quasi-Newton methods to optimize transitionstructures compared with minima:

1. The Hessian update must allow for nega-tive eigenvalues (Newton and Quasi-NewtonMethods). BFGS (Eq. (7)) yields a positivedefinite update and is not appropriate. SR1and PSB updates are suitable (Eqs. (6) and(8)); the Bofill update74 (Eq. (9)) combinesSR1 and PSB yielding better results.

2. Line searches (Step Size Control) are gener-ally not possible because the step toward thetransition structure may be uphill or down-hill. If the coordinates can be partitioned apriori into a subspace spanning the coordi-nates involved in the transition vector andthe remaining subspace involving only coor-dinates to be minimized, then a line searchcan be used in the latter subspace.

3. Controlling the step size and direction (seeStep Size Control) are keys to the success ofquasi-Newton methods for transition struc-tures, especially if the initial structure isnot in the quadratic region. For the TRM(Eq. (18)), the shifted Hessian, H − λI, isrequired to have one negative eigenvalueand λ must be chosen so that the step sizedoes not exceed τ . This is illustrated inFigure 2(b) and discussed in the section StepSize Control. For the partitioned rationalfunction optimization and eigenvector fol-lowing methods,105–109 Eqs. (19) and (20),two different values of λ are used, one toshift the eigenvalue for the maximization di-rection and the other to shift the eigenvaluesfor minimization in the remaining directions.

A quasi-Newton optimization of a transitionstructure requires an initial geometry and an initialestimate of the Hessian. The initial geometry must besomewhere near the quadratic region of the transitionstructure. This can be challenging at times becauseour qualitative understanding of transition structuregeometries is not as well developed as for equilibriumgeometries. There are some general rules-of-thumbthat may be useful. For example, the bonds that arebeing made or broken in a reaction are, in manycases, elongated by ca 50% in the transition struc-ture. The Hammond’s postulate114 is useful for esti-mating the position of the transition structure alongthe reaction path. It states that the transition structure



is more reactant-like for an exothermic reaction andmore product-like for an endothermic reaction. Or-bital symmetry/phase conservation rules115 may indi-cate when a reaction should occur via a non-least-motion pathway. The choice of coordinates for theoptimization is also very important. Often the unionbetween the (redundant) internal coordinates of thereactants and products provides a good coordinate setfor the transition structure. Sometimes extra dummyatoms may need to be added to avoid problems withthe coordinate system.

When a transition structure for an analogous re-action is not available as an initial guess, then brack-eting the search region for the transition structure op-timization can be useful. In the QST2 approach,116 astructure on the reactant side and one on the productside are used to provide bounds on transition struc-ture geometry and to approximate the direction ofthe reaction path through the transition structure.The QST3 input adds a third structure as an initialestimate of the transition structure. A series of stepsis taken along the path connecting these two or threestructures until a maximum is reached. Then, a full di-mensional quasi-Newton optimization is carried outto find the transition structure.

Quasi-Newton methods require an initial esti-mate of the Hessian that has a negative eigenvaluewith a corresponding eigenvector that is roughlyparallel to the desired reaction path. The empir-ical rules for estimating Hessians for equilibriumgeometries78–80 do not provide suitable estimates ofthe required negative eigenvalue and eigenvector ofthe Hessian. Thus, quasi-Newton transition structureoptimizations are typically started with an analyticalor numerical calculation of the full Hessian (or at leastnumerical calculation of the most important compo-nents of the Hessian needed to obtain the transitionvector). For QSTn methods,116 updating the Hessianduring initial maximization steps along the reactionpath is usually sufficient to produce a suitable nega-tive eigenvalue and corresponding eigenvector.

An empirical valence bond (EVB) model117 ofthe surface can be useful in obtaining a starting ge-ometry and an initial Hessian for a quasi-Newtontransition structure optimization. A quadratic surfaceis constructed around the reactant minimum and an-other around the product minimum. These two sur-faces intersect along a seam. The lowest point alongthis seam is a good estimate of the transition structuregeometry118,119 (provided that there are no additionalintermediates or transition structures along the reac-tion path). An initial estimate of the Hessian can beobtained from the interaction of the reactant surfaceand the product surface using a 2 × 2 EVB Hamilto-

nian with a suitable guess for the interaction matrixelement.120

If more information about the potential energysurface is needed to find the quadratic region of thetransition structure, then one can calculate the energyfor a series of points along the linear synchronoustransit121 (LST) path between reactants and productsto find a maximum along this approximate path (LSTscan); internal or distance matrix coordinates are bet-ter than Cartesian coordinates for an LST scan. Formore complex systems, two or more directions mayneed to be scanned to find the ridge separating re-actants and products. These scans could be carriedout with or without optimizing the remaining coor-dinates (relaxed vs rigid surface scan). The reducedsurface method110,111 selects a set of coordinates tospan a two- or three-dimensional surface and mini-mizes the energy with respect to all of the remainingcoordinates. This reduced dimensional surface can beinterpolated using distance-weighted interpolants andsearched exhaustively for the lowest energy transitionstructures and reaction paths.

Related Single-Ended MethodsIn principle, one should be able to start at the re-actants (or products) and go uphill to the transi-tion structure. However, all directions from a mini-mum are uphill and most will end at higher energytransition structures or higher-order saddle points.One approach is to choose a coordinate that willcarry the molecule from reactant to product and stepalong this coordinate while minimizing all other di-rections. This is known as coordinate driving andinvolves a series of constrained optimizations (seeConstrained Optimization). Coordinate driving hasits pitfalls.122–125 For example, if the dominant co-ordinate changes along the reaction path, this ap-proach may not work (e.g., Figure 3(c) and (d)).Alternatively, one can choose a direction and per-form a constrained optimization of the componentsof the gradient perpendicular to this direction. This isknown variously as line-then-plane,126 reduced gradi-ent following,127–129 and Newton trajectories.130–132

Growing string methods (GSM)131–136 are coordinatedriving or reduced gradient following/Newton trajec-tory methods that start from both the reactant andproduct side. Because GSMs are double-ended meth-ods closely related to chain-of-states methods likeNEB method and SM; they are discussed in the nextsection.

An alternative to coordinate driving and re-duced gradient following is to step uphill alongthe lowest eigenvector of the Hessian—this amounts



to the walking up valleys or eigenvector followingmethod described above.101–109 Stepping toward thetransition structure is handled by a quasi-Newton ap-proach with TRM and RFO, Eqs. (18)–(20), to con-trol the step.

The dimer method137–140 is a variant of theeigenvector following approach that uses gradientscalculated at two closely spaced points to keep trackof the direction to search for a maximum. At eachstep, the dimer is rotated to minimize the sum of thetwo energies (this can be the more costly step in thedimer method). The minimum in the dimer energyoccurs when the vector between the points is alignedwith the Hessian eigenvector with the lowest eigen-value. The midpoint of the dimer is displaced uphilltoward the transition structure in a manner similar tothe eigenvector following method.

Another way to walk uphill along the shal-lowest ascent path is to follow a gradient extremalpath.141–143 Although steepest descent reaction pathscan only be followed downhill, gradient extremalpaths are defined locally (the gradient is an eigenvec-tor of the Hessian) and can be followed uphill as wellas downhill. A number of algorithms have been de-veloped for following gradient extremal paths.144–146

Gradient extremals do pass through minima, transi-tion structures, and higher-order saddle points; how-ever, they have a tendency to wander about the sur-face rather than follow the more direct route thatsteepest descent paths take.146,147 This makes gradi-ent extremal path following less attractive for transi-tion structure optimization.

The image function or associated surfaceapproach148–150 converts a transition structure searchinto a minimization on a modified surface that hasa minimum that coincides with the transition struc-ture of the original function. In a representation inwhich the Hessian is diagonal, this is accomplishedby choosing an eigenvector to follow and invert-ing the sign of the corresponding component of thegradient and Hessian eigenvalue. A TRM approach,Eq. (18), is then used to search for the minimum onthe image function or associated surface. The transi-tion structure can also be optimized as a minimumon the gradient norm surface, |g(x)| (Refs 151–153);however, the gradient norm can have additional min-ima at nonstationary points in which the gradient isnot zero. Because of the many minima and smallerradius of convergence, minimizing the gradient normis less practical for transition structure optimizations.

Bounds on the transition structure can be foundby approaching it from both the reactant and productside, providing progressively tighter limits.142,154,155

This is closely related to the GSM131–136 (see below).

FIGURE 4 | An example of a double-ended reaction pathoptimization on the Muller–Brown surface.154 The path optimizationwith 11 points starts with the linear synchronous transit path (black);the first two iterations are shown in blue and green, respectively. Thepath after 12 steps is in red and can be compared with the steepestdescent path in light blue.

Given two points on opposite sides of the ridgeseparating reactants from products, the ridgemethod156,157 can also be used to optimize the tran-sition structure. A point on the ridge is obtained byfinding a maximum along a line connecting a pointin the reactant valley and a point in the product val-ley. Two points on opposite sides of the ridge pointare generated by taking small steps along this line.Then, a downhill step is taken from these two pointsto generate two new points; the maximum along theline between these new points is a lower energy ridgepoint. The process is repeated until a minimum isfound along the ridge; this minimum is a transitionstructure.

Chain-of-States and Double-EndedMethodsInstead of optimizing a single point toward the transi-tion structure for a reaction, an alternative approachis to optimize the entire reaction path connectingreactants to products. Typically this is done by repre-senting the path by a series of points, that is, a chain-of-states. An example of a path optimization on theMuller–Brown surface154 is shown in Figure 4. Thevarious methods of this type differ primarily by howthe points are generated, what function of the points isminimized, and what constraints are imposed to con-trol the optimization. Many of the path optimization



methods are based on minimizing the integral ofthe energy along the path, normalized by the pathlength158:

Epath = 1L

∫E(x(s)) ds, (25)

where L is the total length of the path. Although thisdoes not yield a steepest descent path,159 it provides agood approximation to it. The elastic band or chain-of-states method replaces the integral by a discreteset of points, and the path is found by a constrainedminimization of the sum of the energies of the pointson the path from reactants to products,

Vpath =∑

E(xi ), (26)

where xi are the points on the path, which are requiredto remain equally spaced. Additional potentials aresometimes introduced to prevent the path from kink-ing or coiling up in minima.160 The number of pointsneeded depends on the nature of the path (e.g., num-ber of intermediates and transition states, curvatureof the path, etc.) and can range from less than 10 tomore than 50. Typically, several thousands of stepsare required for convergence, primarily because themotions of adjacent points are strongly coupled. Thenudged elastic band (NEB) method161–170 and stringmethod (SM)171–176 are two current approaches thatoffer some improvement in the convergence of thepath optimization.

In the NEB method,161–163 the points are keptequally spaced by adding a spring potential betweenthe points (c.f. penalty function method, ConstrainedOptimization):

Vspring = 12

kspring∑

(xi − xi−1)2. (27)

The gradient for a point has contributions fromthe potential energy surface and from the spring po-tential, which can be projected into components par-allel and perpendicular to the path:

gi = dVpath

dxig‖

i = (gT

i τ i)τ i g⊥

i = gi − g‖i , (28)

gi = dVspring

dxig‖

i = (gT

i τ i)τ i g⊥

i = gi − g‖i , (29)

where τ i is the normalized tangent to the path at xi.The tangent can be calculated by central difference(of xi−1 and xi+1) or by forward difference using thepoint uphill from xi. The NEB method161–163 uses thegradient of the spring potential to displace (nudge)the points along the reaction path, and uses the gra-dient of the unmodified potential energy surface fordirections perpendicular to the path.

gNEBi = g⊥

i + g‖i (30)

The doubly NEB method164 includes a portionof the spring gradient perpendicular to the path (e.g.,a second nudge).

gDNEBi = g⊥

i + g‖i + (

g⊥i − (

g⊥Ti g⊥

i

)g⊥

i /|g⊥i |2) (31)

This accelerates the optimization in the early stages,but may inhibit full convergence in the later stages.168

The original NEB method used a quenched ordamped Velocity Verlet method to propagate thepoints toward the path.161–163 After each step, mostof the velocity is removed. Alternatively, a quasi-Newton method can be used for the constrained op-timization, preferably by moving all of the pointssimultaneously.164,165,167,168 Because the dimensionof the optimization problem (number of coordinatetimes the number of points) can be rather large, lim-ited memory quasi-Newton methods such as L-BFGSand ABNR have been used.164,165,167,168 Currently,the NEB/L-BFGS method is the most efficient and sta-ble nudge elastic band method.168

The spring potential used to maintain uniformspacing in the NEB method introduces additional cou-pling between the points on the path that slows downthe convergence of the optimization. An early pathoptimization method avoided the spring potential bymoving the points so that the gradient would lie onthe steepest descent path.173 The recently developedstring method171,172 also minimizes the perpendiculargradient, g⊥

i in Eq. (28), for the points on the path.A cubic spline is used to redistribute the points tomaintain equal spacing and the fourth-order Runge–Kutta method is used to evolve the path.171,172 Inthe quadratic string method, the points move on lo-cal quadratic approximations to the surface usingupdated Hessians.174,175 Alternatively, the L-BFGSmethod can be used to propagate all of the pointsat the same time.168 The efficiency of the best stringmethods appears to be similar to the best NEB meth-ods, requiring from hundreds to thousands of gradi-ent calculations to optimize the path.168,176

The GSM133–136 reduces the total effort by gen-erating the points one at a time starting from thereactants and products. This is similar to the line-then-plane method126 and reduced gradient follow-ing/Newton trajectories131,132 from both ends of thepath. The GSM is also closely related to earlier meth-ods that stepped from the reactant and product to-ward the transition structure.142,154,155 As the stringgrows toward the center, the endpoints of the stringprovide better brackets for the transition structure.The search can be completed by interpolating to the



maximum along the path and optimizing the transi-tion structure.136 If the ends of the growing string aretoo far apart, a NEB method or SM can be used tooptimize of the entire path. To reduce the cost, a lowlevel of theory can be used to grow the string and ahigher level of theory can be used to refine the stringor to optimize the transition structure.134,135

Chain-of-state methods such as NEB, SM, andGSM are still more costly than quasi-Newton meth-ods for transition structures, and there is considerableroom for improving the efficiency of these methods.However, these methods do map out the entire re-action path. This can be advantageous if the pathcontains several transition states and intermediates.Furthermore, chain-of-states methods are more read-ily parallelized than quasi-Newton and other single-ended methods. The development of these methods isongoing.

Characterization of an OptimizedTransition StructuresOnce a transition structure has been optimized, itis necessary to confirm that it is indeed a transi-tion structure and is appropriate for the reaction un-der consideration. A vibrational frequency calculationneeds to be carried out for the transition structureto confirm that it has one and only one imaginaryfrequency (equivalently, the full dimensional Hessianmust have one and only one negative eigenvalue). Itis not sufficient to examine the updated Hessian usedin the quasi-Newton optimization process because itmay be subject to numerical noise and the optimiza-tion may not have explored the full coordinate space(e.g., modes that lower the symmetry of the molecule).Firstly, an accurate Hessian must be calculated analyt-ically or numerically and used for the vibrational fre-quency analysis. Secondly, the transition vector, thatis, the vibrational normal mode associated with theimaginary frequency must be inspected to make surethat the motion corresponds to the desired reaction.This is most easily done with graphical software thatcan animate vibrational modes. For complex reactionnetworks, reaction path following may be needed toverify that the transition structure connects the de-sired reactants and products. Chain-of-states meth-ods yield good approximations to the reaction pathas a part of the optimization. For quasi-Newton andother methods that converge on a single structure, fol-lowing the steepest descent path (see Reaction PathFollowing) will indicate the reactants and productsconnected by the transition structure. Reaction pathfollowing may also reveal other stationary points onthe path.

REACTION PATH FOLLOWING

For quasi-Newton and related methods that locateonly the transition structure, the steepest descent re-action path can be obtained by following the gra-dient downhill. Chain-of-states approaches such asNEB method, SM, and GMS (see Chain-of-States andDouble-Ended Methods) provide an approximationto the reaction path as a part of the optimization.Methods such as variational transition state theory177

and reaction path Hamiltonian178 may require a moreaccurate path. For this case, rather than trying toobtain very tight convergence with a chain-of-statesmethod, it is much more efficient to calculate an ac-curate path by a reaction path following method.

In the steepest descent reaction path (SDP) orminimum energy path (MEP), x(s) is defined as:

dx(s)ds

= −g(x(s))|g(x(s))| , (32)

where s is the arc length along the path. When mass-weighted coordinates are used, the path is the IRCof Fukui179 and corresponds to a classical particlemoving with infinitesimal kinetic energy. A numberof reviews are available for reaction path followingmethods.8,180 In principle, the reaction path can beobtained by standard methods for numerical integra-tion of ordinary differential equations.181 However,in some regions of the potential energy surface, thesemethods produce paths that zigzag across the valleyunless very small step sizes are used. This is charac-teristic of stiff differential equations for which specialtechniques such as implicit methods are needed.182

The Ishida, Morokuma, Komornicki (IMK)method183 uses an Euler step followed by a line searchto step back toward the path. The LQA methodof McIver and Page,184,185 and subsequent improve-ments by others186–188 obtain a step along the pathby integrating Eq. (32) analytically on a quadraticenergy surface using a calculated or updated Hes-sian. Because it makes use of quadratic information,LQA can take large steps than IMK. The second-ordermethod of Gonzalez and Schlegel189,190 (GS2) is animplicit trapezoid method that combines an explicitEuler step with an implicit Euler step. The latter is ob-tained by a constrained optimization. The larger stepsize and greater stability of the GS2 method morethan compensates for the few extra gradient calcula-tions needed in the optimization. The implicit trape-zoid method has been generalized and extended byBurger and Yang191,192 to a family of implicit–explicitmethods for accurate and efficient integration of re-action paths.



The reaction path can be thought of as an av-erage over a collection of classical trajectories or atrajectory in a very viscous medium. The dynamic re-action path method193 and the damped Velocity Ver-let approach194 obtain a reaction path by calculat-ing a classical trajectory in which almost all of thekinetic energy has been removed at every step. TheHessian-based predictor–corrector method for reac-tion paths195,196 adapts an algorithm used for classi-cal trajectory calculations. An LQA predictor step istaken on a local quadratic surface; a new Hessian isobtained at the end of the predictor step (by updat-ing or by direct calculation); a corrector step is ob-tained on a distance-weighted interpolant using thelocal quadratics at the beginning and end of the pre-dictor step.

A reaction path calculated at a lower level oftheory can be used to provide an estimate of the transi-tion structure optimization at a higher level of theorywhen full optimization at the higher level is not prac-tical. The potential energy surface is more likely tochange along the path than perpendicular to the pathbecause reaction energies and bond making/breakingcoordinates are typically more sensitive to the ac-curacy of the electronic structure calculation. In theIRCMax method, single point high-level energy cal-culations along the low-level path are interpolated togive an estimate of the transition structure and energyat the high level of theory.197

Reaction paths can also be formulated as a vari-ational method by finding the path that minimizes thefollowing integral159,198–201:

I =∫ √

g(x(t))T g(x(t))√

(dx(t)/dt)T(dx(t)/dt) dt

=∫ √

g(x(s))T g(x(s)) ds, (33)

where x(t) is the reaction path parameterized by anarbitrary variable t, whereas x(s) is the reaction pathas a function of the arc length (see Eq. (32)). This isclosely related to obtaining a classical trajectory byfinding the path that minimizes the classical action,and may hold promise for future developments.

SUMMARY

Methods for optimizing equilibrium geometries, lo-cating transition structures, and following reac-tion paths have been outlined in this chapter.Quasi-Newton methods are very efficient for ge-ometry optimization when used with Hessian up-dating, approximate line searches, and trust radiusor rational function techniques for step size con-trol. Single-ended and double-ended methods fortransition structure searches have been summarized.Procedures for approaching transitions structures aredescribed. Quasi-Newton methods with eigenvectorfollowing and rational function optimization are effi-cient for finding transition structures. Double-endedtechniques for transition structures are discussed andinclude nudged elastic band, string and growing stringmethods. The chapter concludes with a discussion ofvalidating transition structures and a description ofmethods for following steepest descent paths.

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