TheRate: Program forab initio direct dynamics calculations of thermal and vibrational-state-selected rate constants

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TheRate: Program for Ab Initio DirectDynamics Calculations of Thermal andVibrational-State-SelectedRate Constants

WENDELL T. DUNCAN, ROBERT L. BELL, and THANH N. TRUONGDepartment of Chemistry, University of Utah, Salt Lake City, Utah 84112

Received 5 August 1997; accepted 10 February 1998

Ž .ABSTRACT: We introduce TheRate THEoretical RATEs , a completeŽ .application program with a graphical user interface GUI for calculating rate

constants from first principles. It is based on canonical variational transition-stateŽ .theory CVT augmented by multidimensional semiclassical zero and small

Ž .curvature tunneling approximations. Conventional transition-state theory TSTwith one-dimensional Wigner or Eckart tunneling corrections is also available.Potential energy information needed for the rate calculations are obtained fromab initio molecular orbital andror density functional electronic structure theory.Vibrational-state-selected rate constants may be calculated using a diabeticmodel. TheRate also introduces several technical advancements, namely thefocusing technique and energy interpolation procedure. The focusing techniqueminimizes the number of Hessian calculations required by distributing moreHessian grid points in regions that are critical to the CVT and tunnelingcalculations and fewer Hessian grid points elsewhere. The energy interpolationprocedure allows the use of a computationally less demanding electronicstructure theory such as DFT to calculate the Hessians and geometries, while theenergetics can be improved by performing a small number of single-pointenergy calculations along the MEP at a more accurate level of theory. TheCH q H l CH q H reaction is used as a model to demonstrate usage of the4 3 2program, and the convergence of the rate constants with respect to the numberof electronic structure calculations. Q 1998 John Wiley & Sons, Inc. J ComputChem 19: 1039]1052, 1998

Keywords: kinetics; thermal rate constants; direct dynamics; variationaltransition-state theory

Correspondence to: T. N. Truong; e-mail: [email protected]

( )Journal of Computational Chemistry, Vol. 19, No. 9, 1039]1052 1998Q 1998 John Wiley & Sons, Inc. CCC 0192-8651 / 98 / 091039-14

DUNCAN, BELL, AND TRUONG

Introduction

he fields of computational fluid dynamics,T process simulation and design, combustion,and atmospheric chemistry are just a few of theareas that need accurate rate constants of the un-derlying elementary chemical reactions.1] 3 Predict-ing rate constants is in fact a major goal of compu-

4 Žtational chemistry. The TheRate THEoretical.RATEs program seeks to bring together many of

the recent advances in computational chemistrymethods in a user-friendly environment to calcu-late elementary reaction rate constants from firstprinciples, and seeks to bridge the gap betweenchemistry and chemical engineering.

Calculations of rate constants require a delicatebalance between the accuracy of the dynamicaltheory and the efficiency in obtaining accuratepotential energy information. In the extreme ofrigorous dynamical treatment, accurate quantumdynamics calculations yield detailed state-to-statereactive cross-sections or rate constants with fullconsideration of quantum effects.5 ] 9 However,such calculations are currently limited to four-atomreacting systems with the use of global analyticalpotential energy functions. In the other extreme,

Ž .transition-state theory TST has been practical fora wide range of chemical processes due to itssimplicity. The basic model only requires potential

Ž .energy information at the reactant s and transitionstate, because it treats many dynamical effectsonly approximately. The major advantage of TSTis that such limited potential energy informationmay be obtained from accurate electronic structurecalculations without the need of an analytical po-tential energy function. Variational effects andrortunneling are important in many reactions, and forthese systems more accurate dynamical treatmentsare desirable. For instance, variational transition

Ž .10, 11state theory CVT provides a well-establishedmethodology to bridge this gap. In this case, moreinformation on the potential energy surface isneeded. Conventionally, the additional informa-tion is obtained by first constructing an analyticalpotential function which is fitted to results fromaccurate electronic structure calculations andavailable experimental data.12 ] 14 This becomes animpossible task as the size of the reacting system

Ž .increases, because: i the number of energy pointsrequired grows geometrically with the number of

Ž .internal coordinates; ii potential functional forms

Ž .are arbitrary; and iii fitting procedures do notguarantee convergence and correct global topol-ogy.12 The direct dynamics approach offers a vi-able alternative for studying the kinetics and dy-namics of complex systems. In the direct dynamicsapproach, all required energies, forces, and geome-tries needed to evaluate dynamical properties areobtained directly from electronic structure calcula-tions rather than from empirical analytical forcefields.15 ] 21

The direct dynamics approach, however, is lim-ited by the computational demand of electronicstructure calculations. For thermal rates, reactionpath Hamiltonian methods22 ] 25 such as the canon-

Ž .ical variational transition-state theory CVT areparticularly attractive due to the limited amountof potential energy and Hessian information that isrequired.26 Direct dynamics with CVT thus offersan efficient and cost-effective methodology. Fur-thermore, several theoretical reviews27, 28 have in-dicated that CVT plus multidimensional semiclas-sical tunneling approximations yield accurate rateconstants not only for gas-phase reactions but alsofor chemisorption and diffusion on metals. To fullyconverge thermal rate constants, however, existingmethodologies require a large number of Hessianpoints along the reaction coordinate. Computation-ally, it is expensive if these Hessians are to becalculated at an accurate level of ab initio molecu-lar orbital theory.

Several approaches have been proposed to re-duce this computational demand. One approach isto estimate rate constants and tunneling contribu-tions by using the Interpolated CVT when avail-able accurate ab initio electronic structure informa-tion is very limited.29 Another is to carry out CVTcalculations with multidimensional semiclassicaltunneling approximations using a semiempiricalmolecular orbital Hamiltonian at the neglect-of-di-

Ž .atomic-differential-overlap NDDO level as a fit-ting function in which NDDO parameters havebeen readjusted to represent accurately activationenergy for specific reactions.30, 31 Both of these ap-proaches have been applied successfully to variousgas-phase chemical reactions.17, 32 ] 44 However,many difficulties persist. In particular, in the for-mer interpolated CVT approach, it is difficult tocorrelate vibrational modes at the transition-stateregion to reactant and product asymptotes due tomode crossings that occur in most polyatomic sys-tems. In the latter, it may prove to be difficult toadjust the original NDDO parameters to describeaccurately the transition-state region if the originalNDDO potential energy surface differs signifi-

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cantly from the reference accurate ab initio surface.Furthermore, even if the parameters are adjustedto reproduce the barrier height and force constantsat the transition state,31 there is no guarantee thatthe reaction path curvature and coupling betweenvibrational modes and reaction coordinate are cor-rect. These factors directly affect the computedtunneling probability and product-state distribu-tions.45 Recent development in combining bothapproaches into what is known as dual-level dy-namics16, 19, 46 has shown some promise.

Our own efforts, however, have been to developmethodologies for reducing computational de-mand while carrying out CVT calculations withmultidimensional semiclassical tunneling correc-tions directly from a sufficiently accurate level of

Ž .ab initio molecular orbital MO andror densityŽ .functional theory DFT without invoking any po-

tential fitting procedure.17, 18, 37, 47 ] 51 We attack thisproblem from three separate fronts. First, we pro-pose the use of a nonlocal DFT functional52 fordetermining geometries, gradients, and Hessiansalong the reaction coordinates. In particular, thehybrid Becke half-and-half exchange53 with Lee]

54 Ž .Yang]Parr correlation functionals BHH-LYP , asimplemented in the Gaussian program,55 has beenfound to give quite accurate geometries and fre-quencies along the reaction path with a lowercomputational cost than correlated levels of abinitio MO theory and yet often yields comparableaccuracy, particularly for large systems. We havediscussed the accuracy of the BHH-LYP method atlength in previous studies.18, 47, 56 Note that devel-opment of DFT functionals is currency a veryactive research area. More accurate DFT function-als, as they become available, may be used forcalculating reaction path information. Second, wehave developed a focusing technique to minimizethe number of calculated Hessians along the reac-tion coordinate. Third, we have developed an in-terpolation approach for improving the accuracy ofthe energetics information along the reaction coor-dinate from a minimal number of additional sin-gle-point calculations at a higher level of theorywith a larger basis set. We will discuss the lattertwo efforts in this study.

In addition to being an advanced research tool,this program is also envisioned to be a usefulteaching tool for graduate-level kinetics courses.For this purpose, a brief synopsis of the underly-ing dynamical theories, namely transition-statetheory, canonical variational transition-state the-ory, and various semiclassical tunneling approx-imations, is provided. The accuracy and con-

vergence of the focusing technique and energyinterpolation procedure are then described. TheCH q H l CH q H reaction is used to ana-4 3 2lyze these technical advances, and to demonstratethe program usage.

Transition-State Theory

Transition state theory,57 ] 62 or activated com-plex theory, is a well-developed formalism forobtaining thermal rate constants by mixing theimportant features of the potential energy surfacewith a statistical representation of the dynamics.

In addition to the Born]Oppenheimer approxi-mation, TST is based on three assumptions: First,classically, there exists a surface in phase spacethat divides it into a reactant region and a productregion. It is assumed that this dividing surface islocated at the transition state which is defined asthe maximum value on the minimum energy pathŽ .MEP of the potential energy surface that connects

Ž . Ž .the reactant s and product s . Any trajectory pass-Ž .ing through the dividing surface or bottleneck

from the reactant side is assumed to form productseventually. This is often referred to as the nonre-crossing rule. Second, the reactant equilibrium isassumed to maintain a Boltzmann energy distribu-tion. Finally, activated complexes are assumed tohave Boltzmann energy distributions correspond-ing to the temperature of the reacting system.These activated complexes are defined as super-molecules having configurations located in thevicinity of the transition state.

Ž .The thermal rate constant, k T , for a bimolecu-lar reaction A q B ª C q D is given by

k T qXb ‡ ‡Ž . Ž .k s s exp yb DV 1h N q qa A B

where k is Boltzmann’s constant, T is the temper-bature, N is Avogadro’s number, h is Planck’saconstant, and b is 1rk T. DV ‡ is the classicalbbarrier height; that is, the potential energy differ-ence between the reactants and transition state. qAand q denote the respective total partition func-B

tions57, 58, 63 of the reactants with the translationalpartition functions expressed in per unit volume.The partition function of the activated complex isqX and does not include translational motion along‡

the reaction coordinate because this is treated sep-arately. The reaction symmetry number, s , repre-sents the number of indistinguishable ways the

JOURNAL OF COMPUTATIONAL CHEMISTRY 1041


reactants may approach the activated complex re-gion. We will discuss forward and reverse reactionsymmetry numbers in more detail later.

Variational Transition-State Theory

Canonical variational transition-state theoryŽ . 64 ] 67CVT is an extension of transition state theory.It minimizes the recrossing effects and provides aframework for a more accurate description ofquantum tunneling effects to be considered.

CVT minimizes the recrossing effects by effec-tively moving the dividing surface along the MEPbetween the reactants and products so as to mini-mize the rate. The reaction coordinate, s, is definedas the distance along the MEP calculated in mass-weighted Cartesian coordinates with the originlocated at the saddle point and is positive on theproduct side and negative on the reactant side. Fora canonical ensemble at a given temperature T , the

Ž .canonical variational theory CVT rate constantfor a bimolecular reaction is given by:

C V T Ž . GT Ž . Ž .k T s min k T , s 2s

where:

GT Ž .kT q T , sGT Ž . Ž Ž .. Ž .k T , s s s exp yb V s 3M EPh N q qa A B

GT Ž .In these equations, k T , s is the generalizedtransition-state theory rate constant where the di-viding surface is perpendicular to the MEP and

GT Ž .intersecting it at s. q T , s is the partition func-tion of the generalized transition state at s withthe motion along the reaction coordinate removed.

Ž . ŽV s is the classical potential energy theM EP.Born]Oppenheimer potential energy along the

MEP with the zero of energy at the reactants. Notethat if the generalized transition state is located at

Ž . Ž .the saddle point s s 0 , eq. 3 reduces to that ofconventional TST.

CVT yields hybrid rate constants that treat mo-tion along the reaction coordinate classically.Quantum effects in this degree of freedom areincluded by multiplying the CVT rate constant by

Ž .a ground-state transmission coefficient k T . Thus,the final CVT rate constant is given by:

Ž . Ž . C V T Ž . Ž .k T s k T k T 4

Forward and ReverseReaction Symmetries

The reaction symmetry number,62, 68 ] 70 whichappeared in the rate expression, is the ratio of therotational symmetry numbers and is also knownas the statistical factor. It describes the number ofsymmetry equivalent reaction paths. For bimolecu-lar reactions, the forward reaction symmetry num-ber is computed using the following formula:

Ž . Ž .NSYM Reactant 1 )NYSM Reactant 2Ž .s s 5f Ž .NSYM GTS

where NSYM is the symmetry number associatedwith the point group to which the consideredmolecular configuration belongs. Similarly, the re-verse symmetry number is given by:

Ž . Ž .NSYM Product 1 )NSYM Product 2Ž .s s 6r Ž .NSYM GTS

For optically active molecules there is an extracorrection because each optically active isomerrepresents distinct, but energetically equivalent,states. Each optically active configuration shouldbe given an extra factor of 1r2.68

We use the CH q HCl l CH q Cl reaction3 4as an illustrative example. From Table I71 we findthat the CH Cl complex belongs to the C point4 3vgroup with NSYM s 3. The CH reactant belongs3to the D point group with NSYM s 6. The HCl3hreactant belongs to the C point group with`v

TABLE I.Symmetry Number of Various Point Groups.71

Point group NSYM Point group NSYM

C , C , C 1 Atom 11 i sC , C , C 2 D , D , D 42 2v 2h 2 2d 2hC , C , C 3 D , D , D 63 3v 3h 3 3d 3hC , C , C 4 D , D , D 84 4v 4h 4 4d 4hC , C , C 5 D , D , D 105 5v 5h 5 5d 5hC , C , C 6 D , D , D 126 6v 6h 6 6d 6hC 7 C 87 8D 16 S 28h 4S 3 S 46 8C 1 D 2`v `hT, T , T 12 O, O 24d h hI 60h

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NSYM s 1. The Cl product atom belongs to thespherical orthogonal point group with NSYM s 1.The CH product belongs to the T point group4 dwith NSYM s 12. The forward reaction symmetryis then:

6)1Ž .s s s 2 7f 3

and the reverse reaction symmetry is:

12)1Ž .s s s 4 8r 3

For some reactions the transition state belongsto a higher symmetry point group than other pointsalong the MEP. Consider for example the S ClyqN2CH Cl reaction—the transition state has D sym-3 3hmetry, whereas points along the MEP only haveC symmetry. In this case, the lower symmetry3vpoint group should be used, even in TST calcula-tions, because the activated complexes in generalhave lower symmetry with the transition staterepresenting a special case.

Tunneling Methods

The current TheRate version includes four dif-ferent semiclassical methods for calculating the

Ž .transmission coefficients, k T ; namely the one-di-mensional Wigner and Eckart methods and, themultidimensional zero-curvature and centrifugal-dominant small-curvature methods.27, 34, 64, 72 Forconvenience, we label them as W, ECK, ZCT, andSCT, respectively.

For TST rates, the Wigner and Eckart transmis-sion probabilities are used. For CVT rates, the ZCTand SCT transmission probabilities are utilized.The Eckart method is a special case of the ZCTcalculation where the potential energy for tunnel-ing is fitted by an Eckart function. Also, the ZCTmethod is limiting case of the SCT method. Thesemethods are described in detail in what follows.

WIGNER TUNNELING CORRECTION

The Wigner correction for tunneling assumes aparabolic potential for the nuclear motion near thetransition state64 :

1‡ 2Ž . Ž .V x s V y mv x 9o 2

where V is the energy at the top of the barrier,oand v ‡ is the imaginary frequency of the transi-

Ž .tion state. The Wigner tunneling correction, k Tthen is given by:

1 2‡Ž . w x Ž .k T s 1 q "v b 1024

where b is 1rk T. Thus, the Wigner correctionbonly requires the imaginary frequency at the tran-sition state.

SMALL-CURVATURE TUNNELING

First, we approximate the effective potential fortunneling to be:

3Ny7 1Ž . Ž . Ž . Ž .V s s V s q q m "v s 11Ýd M EP i iž /2is1

where m is the vibrational state of mode i orthog-ional to the MEP. If all vibrational modes are in

Ž .their ground state, V s is equivalent to the vibra-dtionally adiabatic ground state potential curve

GŽ .V s , which is used in calculations of tunnelingacontributions to thermal rate constants. The trans-

Ž .mission coefficient, k T , is then approximated asthe ratio of the thermally averaged multidimen-sional semiclassical transmission probability,

GŽ .P E , to the thermally averaged classical trans-mission probability for scattering by the effective

Ž .potential, V s . If we denote the CVT transitiondC V T Ž .state at temperature T as s T , then the quasi-)� C V T Ž .4classical threshold energy, V s T , can be de-d )

Ž . Ž .noted as E T . The equation for k T then be-)comes67:

`yE r k TbŽ .P E e dEH

0Ž . Ž .k T s 12`

yE r k Tbe dEHŽ .E T)

Ž .Notice that the integral in the numerator of eq. 12Ž .involves E above E T , as well as tunneling)

energies below this. Thus, the semiclassical trans-Ž .mission probability, P E , accounts for both non-

classical reflection at energies above the quasiclas-Žsical threshold and nonclassical transmission i.e.,

.tunneling, at energies below that threshold . Be-Ž .cause of the Boltzmann factor in eq. 12 , tunneling

is by far the more important of these two quantumeffects.

The centrifugal-dominant small-curvature semi-73 Ž .classical approximation SCT is a generalization

of the Marcus]Coltrin74 approximation in which



the tunneling path is distorted from the MEP outto a concave-side vibrational turning point in thedirection of the internal centrifugal force. Insteadof explicitly defining this tunneling path, the cen-trifugal effect is included by replacing the reduced

Ž .mass by an effective reduced mass, m s , whiche f f

is used to evaluate the imaginary action integraland thereby tunneling probabilities. Note that, inthe mass-weighted Cartesian coordinate system,the reduced mass, m, is set equal to 1 amu. Thetransmission probability at energy E is:

1Ž . Ž .P E s 132u ŽE .� 41 q e

Ž .where u E is the imaginary action integral evalu-ated along the tunneling path:

s2p rŽ . Ž . < Ž . < Ž .u E s 2m s E y V s ds 14'H e f f dh sl

The integration limits, s and s , are the reaction-l rcoordinate classical turning points defined by:

w Ž .x w Ž .x Ž .V s E s V s E s E 15d l d r

Note that the ZCT results can be obtained byŽ . Ž .setting m s equal to m in eq. 14 . The effect ofe f f

the reaction-path curvature included in the effec-Ž .tive reduced mass m s is explained in whate f f

follows.The small-curvature tunneling amplitude corre-

sponds approximately to an implicit tunneling paththat follows the line of concave-side vibrational

Ž .turning point at a distance, t s , from the MEP inthe direction of the reaction-path curvature vector.Let the distance along the small-curvature tunnel-

Ž .ing path be j and the curvature at s be k s ; then,it can be shown by analytical geometry that:

2Ž .dt s2w Ž .x Ž .dj s 1 y a s q ds 16)½ 5ds

where:

Ž . < Ž . Ž . < Ž .a s s k s t s 17

The imaginary action integral along the small-curvature tunneling path is defined as:

2pŽ . < w Ž .x < Ž .u E s 2m E y V s j dj 18'H dh

Ž . Ž .By comparing eqs. 14 and 18 , the effective re-duced mass is given by:

2Ž .dt s2Ž . w Ž .x Ž .m s s m 1 y a s q 19e f f ½ 5ds

However, to make the method generally applica-Ž .ble, even when t s is greater than or equal to the

radius of cunature of the reaction path, we includeŽ .only the leading terms of eq. 19 , but not singular-

ities by the approximated form:

Ž .m s s m =e f f

22Ž . w Ž .x Ž .exp y2 a s y a s q dtrds� 4min ½ 1Ž .20

The magnitude of the reaction-path curvature,Ž . 75k s , is given by :

1r2Fy12Ž . w Ž .x Ž .k s s k s 21Ý i½ 5

is1

where the summation is over all generalized nor-Ž . Ž .mal modes i s 1, 2, 3, . . . , F y 1 , and k s isi

the reaction-path curvature component of mode igiven by:

= VTŽ . Ž .k s s yL F 22i i 2< <= V

where LT is the transpose of the generalized nor-imal mode eigenvector of mode i, F is the force

Ž .constant matrix Hessian matrix , and = V is thegradient. Finally, within the harmonic approxima-

49Ž .tion, t s is given by :

y1r421r2 Fy1 2w Ž .x Ž .k " Ý k s v si i iŽ . Ž .t s s 23ž / ½ 5Ž .m 1 q 2mi

where v is the generalized vibrational frequencyiof mode i in state m .i

ECKART TUNNELING CORRECTION

This methodology34 requires no ab initio calcu-lations at points other than reactants, products,and saddle point. It uses the zero-curvaturemethodology developed in the previous section.Ž .k T is evaluated with an approximated adiabatic

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ground-state potential energy curve, V G based onaan Eckart function.

First, we approximate the classical potential en-Ž .ergy, V s by an Eckart function whose param-M EP

eters are calculated from the classical potentialŽ . Ž .energies at the reactants R , saddle point ‡ , and

Ž .products P , and from the imaginary frequency.This fit yields the range parameter:

2‡Ž .m v B2 Ž .a s y 24‡ ‡Ž .2V V y A

where V ‡ is the classical barrier height, v ‡ is theimaginary frequency, m is the reduced mass,

Ž .A s V s s q` , andM EP

‡ ‡ ‡'Ž . Ž .B s 2V y A q 2 V V y A .

V G is then approximated by another Eckartafunction, which is assumed to have the same range

Ž . Ž .parameter a and location of the maximum s s 0Ž .as the classical V s approximation. This EckartM EP

function goes through the zero-point-corrected en-ergies at the reactants, saddle point, and products.This yields:

ay byG Ž . Ž .V s s q q c 25a 21 q y Ž .1 q y

where:

aŽ syso . Ž .y s e 26G Ž . G Ž . Ž .a s V s s q` y V s s y` 27a a

‡G ‡G ‡GŽ . Ž . Ž .'b s 2 DV y a q 2 DV DV y a 28a a a

G Ž . Ž .c s « s s y` 29int

and:

1 a q bŽ .s s y ln 30o ž /a b y a

Here, DV ‡G is the zero-point energy correctedabarrier at the saddle point, relative to reactants,

G Ž .and « s s y` is the sum of the zero-pointintenergies of the two reactants.

For most reactions, the zero-point energy correc-tion often lowers the barrier. In this case, the rangeparameter, a, for the V G curve should be larger;athat is, the width of the V G curve is wider thanathat of the V curve. Thus, the approximation ofM EP

using the same a for V and V G curves oftenM EP aoverestimates the tunneling probability. We havefound that, in many cases, this error is compen-

Žsated for by the ‘‘corner cutting’’ effects see SCT.methodology that are not included in this

methodology.

Vibrational-State-Selected Reactions

TheRate provides an option to calculate rateconstants with specific initial reactant vibrationalstates.47, 76 To do this, the vibrational modes alongthe reaction coordinate are first correlated as dis-

Žcussed later see ‘‘Technical Considerations’’ in.Appendix A . A statistical vibrationally diabetic

model,76, 77 which assumes that the vibrationalmodes preserve their characteristic motions as thecomplex moves along the reaction coordinate, isthen used. In the statistical]diabatic model theexpression for the vibrational-state-selected rateconstants differ from the statistical form for thethermal rate in the vibrational partition functionfor the selected mode and in the potential energy

wcurve used for the tunneling calculations see eq.Ž .x11 . The vibrational partition of the mode i in theselected state m is given by:

Ž . yŽ1 r2qm." v i r k bT Ž .q m , T s e 31i

This partition function replaces the thermal distri-bution partition function for the selected vibra-tional mode.

The statistical]diabatic model is expected toprovide only semiquantatitive state-selected rateconstants because vibrational]vibrational and vi-brational]rotational couplings were not included.Nevertheless, it gives useful insight into vibra-tional-state specificity of gas-phase polyatomic re-actions.

Focusing Technique

The main goal of the focusing technique is tominimize the computational cost of calculatingHessians along the MEP while maintaining a rea-sonable level of accuracy in the calculated rateconstants. This is done by focusing the computa-tional efforts to regions of the MEP that are mostsensitive to the rate calculations. For reactions withreasonable barriers, CVT rate constants are sensi-tive to regions near the transition state, whereasZCT and SCT tunneling contributions show strongdependencies not only on the barrier height, butalso on the barrier width and the reaction pathcurvature. It is important to point out that regions



with large reaction path curvature are often not inthe vicinity of the transition state. It is an empiricalfact borne out from our many applications usingthe reaction path Hamiltonian22 approach that suchregions coincide with those having large curvaturein the geometrical parameters as functions of thereaction coordinate.

Based on this observation, we have designed aweighting function based on the sum of the nor-malized curvatures of the bond distances, angles,and dihedral angles as functions of the reactioncoordinate and the potential curve, each with dif-ferent relative weightings. Thus, this functionweights the importance of different regions alongthe MEP so as to obtain both accurate CVT rateconstants and SCT tunneling contributions. Wefound that weight factors of 1.0 for bond distances,0.5 for angles and dihedral angle, and 2.0 for thepotential energy curve work well for many testedreactions. Users have the option to change thesefactors in TheRate to suit different types of reac-tions. For instance bond-breaking processes wouldweight bond distances more than angles, but thereverse is more appropriate for studying internalrotation processes. The curvatures of geometricalparameters as functions of the reaction coordinateare determined from piecewise quadratic fits tointervals of eight data points along the MEP usingsingle-value decomposition. This weighting func-tion is used to distribute a given number of pointsalong the MEP for further Hessian calculations.

Energy Interpolation Technique

The accuracy of the potential energy along theMEP is critical for quantitative prediction of rateconstants. However, it is often true that the Hes-sian and geometrical information along the MEPconverges with respect to the level of theory andbasis set faster than the energy. It is thereforedesirable to find the geometries and Hessians at alower level of theory and then improve the ener-getic information by performing single-point en-ergy calculations at a higher level of theory androra larger basis set. Because these energy calcula-tions are often computationally expensive, we havedeveloped an interpolation technique that only re-quires a small subset of the Hessian calculationpoints for more accurate single-point energy calcu-lations.

Typically, the energy points are interpolateddirectly from the single-point calculations. How-

ever, in our interpolation procedure, we insteadinterpolate the energy correction to the potentialenergy curve already obtained at the lower level oftheory. The energy difference between results fromthe two levels of theory is calculated at selectedpoints of the Hessian grid. This correction term iscubic spline interpolated, and then added backinto the potential energy result from the lowerlevel of theory. As discussed in what follows, byusing this technique, very few single-point energycalculations are needed.

An automated method is provided in the TheR-ate software for selecting a user-given number ofpoints along the MEP for single-point energy cal-culations. Six to ten points are typically sufficient.

Note that the location of the maximum on themore accurate potential curve is often shifted fromthe saddle point calculated from the lower level.The origin of the reaction coordinate, s, is thenshifted accordingly in CVT calculations, so propervariational effects are calculated.

Model Reaction

To illustrate the use and effectiveness of thesetechniques, we have performed calculations on theCH q H l CH q H reaction using different4 3 2Hessian and energy grids. Because the motivatingfactor of this work is to elucidate the technique,instead of predicting accurate thermal rate con-stants, a low level of electronic structure theorysuffices. We refer to our previous work for accu-rate rate calculations of this reaction.17, 47 Here weused the STO-3G basis set in conjunction with

Ž .unrestricted Hartree]Fock UHF theory to calcu-late the MEP information. The MEP was calculatedin mass-weighted Cartesian coordinates using astep size of 0.01 amu1r2 bohr for 100 points on eachside of the transition state using the Gonzalez]Schlegel method.78, 79

FOCUSING TECHNIQUE ANALYSIS

We used Hessian grids of 7, 10, 12, 15, 20, 25,30, 40, and 50 points to test our focusing tech-nique. Figure 1 shows how various geometricalparameters change along the reaction coordinatefrom the reactants to the products. Figure 2 showsa second derivative analysis of the geometries andplacement of the Hessian grids with various totals.There is some small noise in the graph near thetransition state due to inaccuracies in the MEP in

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this region. This is due to the linear first step offthe transition state that often causes some smalloscillatory motion for several subsequent stepsalong the MEP. Also shown is a normalized energygraph and the sum of the two energy and geome-try weightings. As described in the Theory section,the sum is used to bin weight distribute the Hes-sian grid points. Figure 2 also shows the pointsselected for further Hessian calculations with dif-fering selected numbers of total Hessians.

Figure 3 shows the change in frequencies calcu-lated in Cartesian coordinates of the reaction pathHamiltonian versus the reaction coordinate. Thefour graphs are constructed using 7, 15, 25, and 50total Hessian calculations. Fifteen points are suffi-cient to show all major features of the reaction,whereas 25 points are needed to provide smoothcurves from reactants to products. Typically, be-tween 20 and 30 frequency data points along anMEP are sufficient to obtain smooth correlatedfrequencies.

Figure 4 shows the curvature component of Bm f

coupling as a function of the reaction coordinateusing 50 Hessian calculations. The SCT ‘‘cornercutting’’ effects are large in regions with signifi-cant reaction path curvature. A comparison of theB curves with the weighting curves show thatm f

our weighting scheme has correctly identified theareas where the curvature components are large.

FIGURE 1. Plots of geometrical parameters of the H2+ CH ª H + CH reaction versus the reaction3 4coordinate. The dotted curve represents the HCH bond

( ) ( )angle degrees . Various bond lengths bohr are alsoshown.

( )FIGURE 2. a Plot of locations of the Hessian gridsselected by the focusing technique for each given

( )number of Hessian points. b Plot of the weightingcurves. The normalized potential energy, the normalizedsum of the geometrical second derivative components,and the total weighting are shown.

ŽNote the locations of the two peaks y0.30 and1r2 .y0.45 amu bohr in the second derivative of theŽ .geometries see Fig. 1 coincide with the peaks of

the B coupling.m fFigures 5 and 6 show the convergence of CVT

rates as a function of the number of Hessian calcu-lations. This reaction involves transfer of the lighthydrogen atom, so tunneling plays a significantrole. The CVT rate constants have converged to afinal result with only seven Hessian grid points.Likewise, the CVT rate constant with the zero-curvature tunneling correction also converges withonly seven Hessian grid points. SCT tunneling,



FIGURE 3. Generalized frequencies of the H + CH2 3reaction plotted versus reaction coordinate. The Hessiangrid for each graph is: upper left } 7; upper right } 15;lower left } 25; lower right } 50.

FIGURE 4. B coupling of the H + CH reactionmf 2 3plotted versus reaction coordinate.

FIGURE 5. Convergence of the CVT / SCT rate constantat 300 K with respect to the number of Hessian gridpoints. The solid line gives the percent difference fromthe 50-Hessian rate. The dashed line is a log plot of theactual rate constants.

however, requires at least 30 Hessian calculationsto converge to within 40%. An important implica-tion of this is that if the CVT dividing surfaceremains close to the transition state, and if tunnel-ing is not important, accurate results may be ob-tained with fewer Hessian calculations. As tunnel-ing becomes more important, more Hessian points

FIGURE 6. Similar to Figure 5 except for different levelsof theory. The lower curve is the CVT rate. The middlecurve is the CVT / ZCT rate. The upper dashed curve isthe CVT / SCT rate.

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THERATE PROGRAM

are needed to model the reaction path curvatureaccurately.

ANALYSIS OF ENERGYINTERPOLATION PROCEDURE

We have performed single-point energy calcula-Žtions using MP2 Mo/ ller]Plesset perturbation the-

.ory with the 6-31G basis set. Figure 7 showsa comparison of the UHFrSTO-3G and MP2r6-31GrrUHFrSTO-3G potential curves, whereasFigure 8 shows convergence of the MP2 energieswith increasing number of energy calculations. Therelative inaccuracy of the HF theory and STO-3Gbasis set used in the low-level calculations givesour energy interpolation scheme a severe test. TheHartree]Fock geometries are quite different fromthe geometry results of the more accurate MP2calculation. In a practical sense, this means that thepotential energy peak is shifted noticeably as en-ergy corrections are added. Even for this severe

Ž .case scenario, only 11 one third of Hessian pointsadditional energy calculations plus the stationarypoints are required to converge the rate calcula-tion. Figure 9 shows the rate convergence as thenumber of energy replacement points increase.Systems which use a better theoretical method forcalculating the MEP will require fewer energyreplacement points.

FIGURE 7. HF and MP2 // HF potential energies alongthe reaction coordinate. The MP2 // HF energies wereobtained at every Hessian grid point. Both plots use the30-point Hessian grid.

FIGURE 8. MP2 // HF potential energy curvesinterpolated from a different number of single-pointenergy calculations. The solid line represents an energyinterpolation from 7 single-point energies, the dotted linefrom 11 points, and the dashed line from 15 points.

Conclusion

Computational and theoretical advances haveprogressed sufficiently so that it is now possible tocalculate reaction rates directly from ab initio the-

FIGURE 9. Convergence of the CVT / SCT rate constantat 300 K with respect to the number of single-pointenergies. The solid line gives the percent difference fromthe 30-point rate. The dashed line is a log plot of theactual rate constant.



ory without the intervening use of an analyticalpotential energy surface. We have developed the

Ž .TheRate Theoretical Rates program to performsuch calculations efficiently. The number of calcu-lations required for a variational transition-statecalculation with semiclassical tunneling calcula-tions is gratifyingly small—fewer than would nor-mally be required in constructing a high-qualityanalytical potential energy surface. An intelligentchoice of points at which to calculate the frequen-cies should require fewer than 30 Hessian calcula-tions for the typical system. Improvement of theenergetics requires still fewer ab initio calculations,which is convenient as these often require robustlevels of theory to obtain accurate theoretical rates.

We have used the CH q H l CH q H reac-4 3 2tion to examine our Hessian focusing routine andenergy replacement technique. These results pro-vide a baseline for the number of Hessian calcula-tions and single-point energy calculations that arerequired.

Finally, TheRate provides a user-friendly envi-ronment making it a promising computational toolfor both teaching and research. For further infor-mation on obtaining TheRate, please see our webpage at http:rrwww.usi.utah.edurresearchr The-Rate.

Appendix A

TECHNICAL CONSIDERATIONS

Vibrational mode correlation. A new vibrationalmode correlation routine is utilized here. The tra-ditional way for correlating modes uses mode]mode overlap between adjacent points on theMEP.23 For the most part, this procedure is verysuccessful for small systems. For larger systems, inwhich multiple mode crossings may occur withina small interval of the reaction coordinate, thisoverlap criterion is not always sufficient to dis-criminate between modes. This problem is respon-sible for the apparent avoided crossings betweenmodes in plots of generalized frequencies versusreaction coordinate.

The new routine solves this problem by calcu-lating additional overlap matrices not only be-tween adjacent points but also between points upto two reaction coordinate positions apart. If theadjacent overlap fails to achieve minimum correla-tion levels, the routine attempts to correlate eitherforward or backward up to a maximum of tworeaction coordinate positions to make a betterchoice.

Focusing technique. This range of data points de-faults to 8 and may be changed by the user. Asmentioned earlier, the first few steps along theMEP from the transition state are somewhat er-ratic. Thus we interpolate data points in the transi-tion-state region rather than use the raw data. Thenumber of data points to be interpolated may bechanged by the user. The default value is 2.

MEP Calculations. As the MEP approaches thereactant and product regions, the gradient be-comes small. This directly affects the stability ofthe numerical techniques used to find the MEP,and as a result small oscillations in the geometricalparameters often occur. The numerically calcu-lated derivatives in the tunneling calculations areespecially susceptible to these errors. However, itis not necessary to integrate the MEP far into theseregions for rate calculations.

The range of the MEP needed for rate calcula-tions varies depending on the type of reaction. Fortypical reactions involving transfer of a light parti-cle, tunneling is often quite significant and largerMEP ranges are needed to ensure inclusion ofregions having large curvature. However, for reac-tions where tunneling is anticipated to be small,smaller ranges of MEP values are sufficient. Withthe use of the extrapolation to the reactant andproduct limits described next, the MEP range in-cluding the top half of the barrier is normallysufficient.

Extrapolations to reactants and products. Calcula-Ž .tion of the u E integral requires knowing the

classical turning points, s , and s , at the energy,r lE. As discussed in the previous subsection, it is notnecessary to integrate far into the reactant andproduct channels. However, to converge the tun-neling probability, the MEP is extrapolated to thereactant and product stationary states using anexponential form given by:

Ž g s . Ž .E s V 1 q ae 32

Ž .where V is the energy at the reactant or productand the variables g and a are adjusted so thecurve and its first derivative are continuous at theconnecting point.

Smoothing the m . Small oscillations in the nu-e f f

merically produced MEP may lead to noise in thecalculated reaction path curvature and, subse-

w Ž .xquently, in m eq. 20 . We compensate for thise f feffect by removing high-frequency oscillations in

VOL. 19, NO. 91050

THERATE PROGRAM

the m data. This is done using a fast Fouriere f f

transform low-pass filter. The endpoints are as-signed the value of m before the Fourier transformto resolve problems with the one-sided numericalderivatives at these points.

Integration. Numerical integrations are per-formed using a globally adaptive Gaussian]Kon-rad quadrature scheme.80 This same approach is

Ž . Ž .used for integration of both of eqs. 12 and 14 .The globally adaptive procedure starts with a 21-point Konrad quadrature integration and, if theestimated error is too large, progressively subdi-vides the intervals until the estimated error con-

Ž .verges below an acceptable precision of 1.0E-3 .

Appendix B

PROGRAM USAGE AND CONSIDERATIONS

Supported platforms and availability. TheRate isJava based and runs on any platform with JDK1.1.5, or better, which is freely available on mostUnix platforms, as well as Macintosh and PCcompatibles. A network connection is required.

Associated software. TheRate is currently config-ured to use Gaussian-94 as the basis for all elec-tronic structure calculations. Interfacing TheRatewith other electronic structure packages is easilydone by rewriting the parsing classes. If interested,please contact: [email protected]

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VOL. 19, NO. 91052

Documents

TheRate: Program forab initio direct dynamics calculations of thermal and vibrational-state-selected rate constants