Application of electronic structure and transition state theory: Reaction of hydrogen with silicon radicals

Application of Electronic Structure andTransition State Theory: Reaction ofHydrogen With Silicon Radicals

LONNIE D. CROSBY, HENRY A. KURTZDepartment of Chemistry, Computational Research on Materials Institute, CROMIUM,University of Memphis, Memphis, Tennessee 38152

Received 10 March 2006; accepted 22 May 2006Published online 4 August 2006 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.21154

ABSTRACT: The reaction of molecular hydrogen with silicon radicals is investigatedthrough direct dynamics calculations. Previous studies of this reaction with E′ centers varyin their description of the reaction thermodynamics and kinetics. A survey of methods isconducted to illuminate the best method for describing these properties. This is achievedvia a similar reaction with a silyl radical, in which both the forward and reverse directionsare considered, discussed in terms of theory, basis, and the importance of tunnelingcorrections. The methods studied include Hartree–Fock (SCF), two density functionals(B3LYP and MPW1K), Møller–Plesset second-order perturbation (MP2), andcoupled-cluster singles and doubles (CCSD) with a variety of all electron basis sets rangingfrom 6-31G to aug-cc-pVTZ. Two methods are identified as good candidates: a densityfunctional theory (DFT) method (MPW1K) and a dual-level method including MP2, andCCSD theories. These methods yield rate constants that agree within one order ofmagnitude of experiment and activation energies within 1 kcal/mol over the 293–683 Ktemperature range. The B3LYP and SCF methods do not perform as well and arediscounted early. Basis sets that perform well are found to include diffuse and polarizingfunctions on all atoms, including hydrogen. These include the 6-31++G** andaug-cc-pVDZ basis sets. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem 106: 3149–3159, 2006

Key words: direct dynamics; hydrogen reactions; MOSFET devices

Introduction

T he reaction of molecular hydrogen withdefects in Si/SiO2 materials is of consider-

able importance in the manufacture of metal oxide

Correspondence to: H. Kurtz; e-mail: [email protected]

semiconductor field effect transistor (MOSFET)devices. Hydrogen is commonly used to quench thedevastating effects of defect formation in these mate-rials. These defects may be formed in the oxide, suchas the E′ center, or at the Si/SiO2 interface Pb cen-ters. Both defects contain a dangling bond or siliconradical. The reaction of interest is where molecularhydrogen quenches the dangling bond producing

International Journal of Quantum Chemistry, Vol 106, 3149–3159 (2006)© 2006 Wiley Periodicals, Inc.

CROSBY AND KURTZ

atomic hydrogen. The reverse reaction may alsoinduce defect formation from hydrogenated precur-sors:

≡ Si · + H2 � ≡ Si − H + H · . (1)

Experimental kinetic studies of the forward reac-tion yield activation energies within the range of6.9–9.3 kcal/mol, for the E′ center [1, 2]. Theoreticalcalculations of this reaction barrier have consistentlygiven barriers larger than those obtained in experi-ment. Kurtz and Karna [3] estimated a forward acti-vation energy using self-consistent field (SCF) theoryand direct dynamics calculations of ∼17 kcal/molwithout tunneling and of 12.41 kcal/mol with tun-neling included. Another study, by Vitiello et al. [4],yielded a forward barrier of ∼13 kcal/mol usingthe B3LYP DFT hybrid functional. These DFT cal-culations did not include tunneling and are basedon an adiabatic reaction path. Both estimates arehigher than experiment by 3–6 kcal/mol. If tun-neling is as important in the B3LYP method as inSCF (∼4 kcal/mol), this would bring the activationenergy in line with experiment. However, differ-ences in the reaction energies and barriers betweenthese methods makes this assessment difficult. TheB3LYP method shows a much more endothermicreaction than the SCF or Møller–Plesset second-orderperturbation (MP2) methods of Kurtz and Karna [3].The SCF method demonstrates an essentially neu-tral reaction energy that is improved upon in theMP2 method to be endothermic by 4.44 kcal/mol.This result is ∼8 kcal/mol less endothermic than theB3LYP estimate. The effects of this reaction energydifference are apparent in the forward and reversereaction barriers. The forward barriers between MP2and B3LYP methods are similar, decreasing in B3LYPby ∼4 kcal/mol. However, the reverse barrier is dra-matically different between these methods, decreas-ing in B3LYP by ∼12 kcal/mol.

Our goal is to predict accurately the activationenergies for the reaction of molecular hydrogen witha variety of dangling bond defects and the “stan-dard” methods above are clearly not able to do so.The goal of this study is to define such a method byvirtue of an experimentally well-studied test reac-tion: the reaction of molecular hydrogen with thesilyl radical. The method should produce accuratelycalculated energies, barriers, and kinetics for thesetypes of reactions. Any method chosen will needto provide acceptable accuracy with an understand-ing of the limitations. An important requirement isthat the method be cost effective enough to apply to

systems at least three to four times larger than theone described in this study.

Transition State TheoryCalculations

Rate constants for a bimolecular reaction, A+B →{AB}‡ → products, can be calculated via transitionstate theory (TST) [5] if the properties of the reac-tants (A and B) and the activated complex {AB}‡ areknown. The activated complex is a saddle point onthe reaction coordinate with one imaginary vibra-tional frequency and is the transition state betweenthe reactants and products. The TST rate constant iscalculated via

ktst(T) = k0Th

Z′‡

ZAZBexp

[−ε‡

k0T

], (2)

where k0 and h are the Boltzmann and Planck con-stants, respectively. Equation (2) contains the ratioof canonical partition functions (Zi) of the transitionstate to the reactants and the energy difference (ε‡) ingoing from the reactants to the transition state. Thecanonical partition functions include translational,vibrational, rotational, and electronic terms. Theseterms (partition functions) are usually evaluatedvia several approximations such as particle in thebox for translational, harmonic oscillator for vibra-tional, and rigid rotor for rotational motion. The Z

′

nomenclature denotes the removal of an imaginaryfrequency from the vibrational partition functions.The ε‡ energy difference, called the classical bar-rier hereafter, includes zero-point vibrational energy(ZPE) corrections.

The TST rate constant only requires knowledge ofthree points, if both the forward and reverse reac-tions are considered, along the reaction coordinate:the reactants, transition state, and products. To eval-uate Eq. (2), geometries, energies, and vibrationalfrequencies are required at each of these points,along with additional information, such as electronicdegeneracies, masses, and symmetry factors. Cor-rections to TST, however, require more informationabout the reaction coordinate. Canonical variationaltransition state theory (CVT) [5] corrects Eq. (2) bysubstituting the transition state properties with thoseof a generalized transition state that maximizes thefree energy of activation. This generalized transi-tion state exists close to the previous transition stateand is located on the reaction coordinate, hereafter,called the minimum energy path (MEP), connecting

3150 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 106, NO. 15

ELECTRONIC STRUCTURE AND TST APPLICATION TO H REACTION WITH Si RADICALS

the reactants and products through the transitionstate. Therefore, this method requires knowledgeof the MEP close to the transition state. Correc-tions for TST with regard to nonadiabatic effects,such as tunneling, include zero-curvature tunneling(ZCT) [6] and small-curvature tunneling (SCT) [7]approximations. These methods correct Eq. (2) bya multiplicative constant defined as the ratio of theBoltzmann averaged tunneling probability to theBolzmann averaged classical transmission probabil-ity, i.e., ≥1. The tunneling probability calculationsrequire the MEP over a much greater range, includ-ing all values between which tunneling can occur.ZCT allows for tunneling along the MEP, whereasSCT includes regions off the MEP, corner cutting.Another tunneling approximation is large curva-ture tunneling (LCT) method. This method is notincluded in this study because of its computationalexpense. The MEP required for these corrections toTST requires the same information as the previouslydiscussed single points, at each point along the MEP,plus reaction path curvature components.

The quantities required to solve for the rate con-stants can be obtained by two methods. These arevia an analytical potential energy surface (PES) orby electronic structure calculations. The goals ofthese methods are the same: to obtain geometries,energies, gradients, and hessians at the stationarypoints (reactants and products) and along the MEP(transition state, and generalized transition states).Considering the difficulty in producing a PES for allbut the smallest systems, electronic structure calcu-lations are more practical. Using electronic structurecalculations for TST or its corrections is commonlycalled direct dynamics.

Theoretical Methodology

Rate constants are calculated for the reaction ofsilyl radical with molecular hydrogen. The reversereaction is the hydrogen abstraction reaction fromsilane with atomic hydrogen:

H2 + ·SiH3 � SiH4 + ·H. (3)

Rate constants are calculated with the POLYRATE9.3.1 [8] program, using TST, CVT, CVT/ZCT,and CVT/SCT. The electronic structure informa-tion is supplied using NWChem 4.7 [9] and theDIRDYVTST module [10] for the calculation of theMEP. The MEP is calculated via a steepest decentalgorithm in mass weighted coordinates from the

transition state. The step size used is 0.01 amu12 bohr

with hessians calculated every 10 steps. This hes-sian grid is interpolated to the smaller grid forsubsequent calculations. POLYRATE is also used toperform dual-level calculations using higher-levelcalculations for the stationary points and lower-levelcalculations for the MEP. The MEP is corrected tothe higher level, using interpolated optimized cor-rections (IOC) with double Eckart corrections forthe energy and interpolated corrections via loga-rithms (ICL) for the frequencies [11]. These proce-dures correct the energy, frequency, and geometriesof the MEP via a zero-order approach, where thehigher-level quantities are known only at the threestationary points and interpolated to the rest of theMEP.

Unless otherwise noted, geometry and saddle-point optimizations are carried out with the respec-tive theory, using the 6-31++G** basis [12]. Asingle or double slash notation is used in denot-ing dual-level calculations. For single slash, thenotation is MEP level/stationary level with thestationary level correcting the energies, geome-tries, and frequencies of the MEP. The doubleslash notation is used here to denote a split inthe stationary level between that used to cor-rect geometries and frequencies (stationary level 1)and that used to correct energies (stationary level2), MEP level/stationary level 1/stationary level 2.In the level descriptions, theories are given with orwithout basis set; if not given, the basis is 6-31++G**.

Results

ELECTRONIC STRUCTURE ANDTHERMODYNAMICS

Since the rate constants for the forward andreverse reactions are related by the equilibrium con-stant, and therefore to the reaction free energy (Keq =exp[−�Grxn

RT ]), a good place to begin is with the ther-modynamics of this reaction. Simply, to describeboth the forward and reverse reactions accurately,the thermodynamics must be correct. The change inenthalpy at 0 K [�Hrxn(0 K)] is equal to the changein Gibbs free energy at 0 K [�Grxn(0 K)] and is cal-culated by single-point energy calculations at thereactants, products, and transition state, includingZPE corrections. The transition state is not requiredfor the reaction enthalpy change; however, its inclu-sion will allow the calculation of the forward andreverse classical barriers.

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CROSBY AND KURTZ

TABLE IEnthalpies of reaction at 0 K and reaction barriers inkcal/mol by theory.

Theory �H(0 K) V ‡f V ‡

r

SCF 7.735 21.96 14.23B3LYP 15.54 16.63 1.092MPW1K 12.55 15.62 3.072MP2 10.93 20.33 9.396CCSD 13.53 20.77 7.235

Five different theories are examined with the 6-31++G** basis. This basis is chosen based on itssmall size and inclusion of polarization and diffusefunctions on hydrogen. These additional functionsshould be important given the nature of the stud-ied reaction. Geometries, energies, and hessians arecalculated with Hartree–Fock (SCF), MP2, two den-sity functionals (B3LYP [13] and MPW1K [14]), andcoupled-cluster singles and doubles (CCSD). Thesetwo density functionals are chosen on the basis oftheir applicability to thermodynamic and kineticproperties. The B3LYP functional was parameterizedprimarily for thermodynamics, while the MPW1Kfunctional was parameterized for kinetics. Table Igives the reaction energy including ZPE corrections,also forward, and reverse classical reaction barriers.The forward direction of this reaction is endother-mic with the experimental �Hrxn(0 K) equal to 11.9±2.4 kcal/mol [15, 16]. Only three methods, MPW1K,MP2, and CCSD, agree within experimental error.SCF and B3LYP give reaction energies that are muchtoo low and high, respectively. This result points tothe importance of electron correlation and param-eterization of the density functionals. The SCF andB3LYP methods cannot give consistent results in thiscase, and are removed from further consideration.

The remaining three methods are therefore goodcandidates for rate calculations. However, the for-ward and reverse barriers are quite different amongthese methods. The barriers are highest with MP2,followed by CCSD; MPW1K has the lowest barriers.

These three methods are also examined withregard to the choice of basis set for the energycalculations. The basis sets include 6-31G, 6-31G**,6-31++G**, cc-pVDZ, cc-pVTZ, aug-cc-pVDZ, andaug-cc-pVTZ [12, 17]. The energies reported inTable II are given without ZPE corrections dueto their low magnitude, �0.1 kcal/mol. Thereaction energies are once again compared withthe experimental �Hrxn(0 K) value of 11.9 ±2.4 kcal/mol [15,16]. The forward and reverse barri-ers are expected to be too large without ZPE cor-rections, ∼0.1–−0.8 kcal/mol. All basis sets in allmethods agree within experimental error, except forthe 6-31G basis, which is the only basis set that doesnot contain polarizing functions. The inclusion ofpolarizing functions is obviously important in thereaction energy calculations. For the forward andreverse barriers, a general trend of decreasing bar-rier height emerges as the basis sets progress fromthe 6-31G-type sets, to the Dunning double andtriple zeta sets, and finally to the augmented Dun-ning double and triple zeta basis sets. The MPW1Kmethod breaks this trend due to a reduced basis setdependence as compared with the other methods.

GEOMETRY CONSIDERATIONS

The theory and basis set dependence of the ther-modynamics of the reaction have been examinedat their respective theories and 6-31++G** geome-tries. It is expected that there will be more geometrydependence at the transition state as opposed tothe other various minima, reactants, and products.Analyzing the geometries of selected reactants (H2),

TABLE IIReaction and barrier energies in kcal/mol by basis, without zero-point energy.

MPW1K MP2 CCSD

Basis set Reaction energy V ‡f V ‡

r Reaction energy V ‡f V ‡

r Reaction energy V ‡f V ‡

r

6-31G 16.73 19.24 2.513 16.39 26.52 10.14 18.86 26.98 8.1296-31G** 13.26 16.93 3.668 11.47 22.31 10.85 13.95 22.74 8.7876-31++G** 12.66 16.17 3.509 11.00 21.15 10.15 13.50 21.51 8.012cc-pVDZ 11.89 15.18 3.291 9.512 18.88 9.370 12.11 19.13 7.021aug-cc-pVDZ 12.00 15.12 3.127 9.375 17.35 7.975 12.16 17.43 5.263cc-pVTZ 12.82 16.30 3.477 10.25 18.65 8.400 12.85 18.76 5.911aug-cc-pVTZ 12.79 16.22 3.434 10.02 17.29 7.270 12.15 NA NA



TABLE IIIReactants, products, and transition state geometry.

Bond length (Å)

Theory Si H (SiH4) Si Ha (TS) Ha Hb (TS) H H (H2) ∠Si Ha Hb (TS)(◦) Imaginary freq (cm−1)

MPW1K 1.479 1.561 1.244 0.739 180.00 818.85MP2 1.472 1.615 1.069 0.734 180.00 1663.06CCSD 1.474 1.597 1.127 0.739 179.99 1443.22

products (SiH4), and the transition state (H2SiH3),the silane Si H bond length, hydrogen H H bondlength, and the corresponding bond lengths inthe transition state can be observed. Table III dis-plays data corresponding to these bond lengths inthe reactants, products, and transition states. TheH H Si bond angle and imaginary frequencies arealso presented. The Si H bond length in silane hasan experimental value of 1.4798 Å, and the H Hbond length in molecular hydrogen has a valueof 0.7414 Å; both are within 0.008 Å of all theo-retical geometries using the 6-31++G** basis [18,19]. The close agreement between these theoreti-cal geometries and experiment demonstrates theirquality.

The transition state is much more dependent ontheory than are these minima.All transition states arelinear with respect to the H H Si bond angle. How-ever, their proximity to products is different, as evi-denced by the H H and Si H bond lengths. LongerH H bond lengths correlate well with shortenedSi H bond lengths. The MP2, CCSD, and MPW1Kmethods have progressively later transition states.This trend explains the change in their classicalbarriers, which decrease along this same trend, con-sistent with Hammond’s postulate. Another strik-ing change in these methods is in their imaginaryfrequencies. The MPW1K imaginary frequency is≥600 cm−1 smaller than the other two methods.

These transition states have also been optimizedusing the aug-cc-pVDZ basis. This has minimaleffects on the calculated energies, �0.1 kcal/molwith the aug-cc-pVDZ basis, and bond lengths,�0.03 Å at each level of theory. These geometries stillexhibit linear H H Si bond angles; however, theimaginary frequencies decrease by ∼100 cm−1. As aresult, we assume that the transition state geometriesare not greatly affected by an improvement of basisset and that the 6-31++G** basis is adequate. How-ever, these geometries are theory dependent withnoticeable differences between the three methodsunder investigation.

MINIMUM ENERGY PATHS

Apart from the previously discussed thermo-dynamic properties of the reaction, an importantcomponent for CVT and tunneling calculations isthe MEP. The MEP has two main components thatdetermine the adiabatic energy profile. These are theenergy and ZPE corrections along the path. Otherrequired information includes geometries and gra-dients. This is all the information needed for CVTand ZCT/SCT calculations, except for the reactionpath curvature, which is derived from the derivativeof the gradient.

The MEPs calculated with the 6-31++G** basis atthe MPW1K, MP2, and CCSD levels of theory arequantitatively different. Previously discussed dif-ferences include the reaction energies and classicalbarriers. Another difference is the broadness of thereaction barrier, shown in Figure 1. The y-axis is inarbitrary energy units of kcal/mol normalized at the

FIGURE 1. MEPs calculated with the 6-31++G** basisare shown as a function of energy and displacementfrom the transition state (s = 0) for four theories. All arenormalized at the transition state to E = 25 kcal/mol.Only the energy component of the MEPs is shown.


CROSBY AND KURTZ

FIGURE 2. Hydrogen and silane bond lengths of thetransition state are plotted along the MEP for fourtheories using the 6-31++G** basis. Placement of thesetransition states (TS) is also shown.

transition state (s = 0) at 25 kcal/mol. The x-axis is inunits of s (amu

12 bohr) and refers to displacements in

mass weighted coordinates from the transition state.The sign convention for s is negative toward reac-tants and positive toward products. By inspection,the MPW1K theory produces a broader barrier thanthe other theories. The MEP progressively becomesless broad from CCSD to MP2. As expected, thisbehavior is consistent with the imaginary frequencyat the transition state. The lower-magnitude imagi-nary frequencies produce broader barriers than dothe higher-magnitude frequencies. This variationaffects the CVT and tunneling calculations by incor-porating greater variability into the location of theadiabatic maxima and by making tunneling lessimportant, for a broader barrier. This result con-nects the properties of the transition state to that ofthe MEP.

The transition states, as previously discussed,occur at different geometries for these methods. Thisis not readily seen in Figure 1 due to the normal-ization of s = 0 for each transition state. Since thegeometries along each MEP are linear with respectto the H H Si bond angle, the MEP may be plottedtwo-dimensionally (2D), not by energy vs s, but bythe H H and H Si distances. This view illustratesthe reaction path curvature and relative placement

of the transition state (see Fig. 2). The MEPs arevery similar in the region of space they occupy. Theregions toward reactants and products are essen-tially straight (i.e., no curvature) at constant values ofH H and H Si distance, respectively. These regionscorrespond to the H2 approach and H departure fromthe reaction site. The reaction takes place in a smallregion of high reaction path curvature within whichthe transition states are located. The transition statesfor the various MEPs appear in this region at dif-fering positions. All transition states are closer toproducts than reactants and are progressively latertransition states from MP2, CCSD, to MPW1K. Thisis the same conclusion that was reached by analysisof the transition state geometries.

Dual-level calculations can be employed wherethe MEP calculated is corrected with higher-levelsingle-point calculations. These are useful when thefull MEP calculation at the higher level is not feasi-ble, but where single-point calculations at the threestationary points are desirable. For example, theillustrations of the CCSD MEP only include theenergy component. The numerous hessian calcula-tions required for the full MEP are too costly. Itis necessary to approximate the full CCSD MEPwith a dual-level calculation. This would require alower-level MEP (MPW1K or MP2) and single-pointCCSD calculations, which are more cost effective andapplicable to larger models. Figure 3 shows MEPscorrected to the CCSD level of theory from MP2

FIGURE 3. MEPs calculated with the 6-31++G** basisare shown as a function of energy and displacementfrom the transition state for CCSD and two MEPscorrected to this level. Only the energy component of theMEPs is shown, and reactants are set to E = 0 kcal/mol.



and MPW1K and a normally calculated CCSD MEP.The plots go to the same asymptotic reactant andproduct values and share the same transition stateenergy. The values between the stationary points,however, deviate from the CCSD values dependingon the lower-level method. The MP2/CCSD MEP isslightly too broad, and the MPW1K/CCSD MEP issignificantly narrow. This situation is due to the scal-ing of the lower-level MEP’s imaginary frequencyto the higher-level value. From MP2 to CCSD, theimaginary frequency decreases by ∼200 cm−1 andincreases by ∼600 cm−1 from MPW1K to CCSD.Another consequence of the imaginary frequencyadjustment is the good fit of all MEPs near thetransition state.

Because of the path independence of TST andthe fact that all MEPs share CCSD single-pointproperties, the TST calculations will be equivalent.The CVT calculations depend only on the MEPnear the transition state; therefore, the CVT calcu-lations will be largely equivalent. However, smalldeviations will persist due to differences in thevibrational frequencies calculated by each lower-level method. The tunneling calculations will beaffected by the changes in broadness of these MEPs.The MP2/CCSD method will underestimate andthe MPW1K/CCSD will overestimate the tunnel-ing corrections compared with CCSD. The region(<14 kcal/mol) of large deviation between CCSDand MPW1K/CCSD methods is not included in thetunneling calculations due to zero tunneling prob-ability. This will lessen the overestimation that theMPW1K/CCSD method will provide. These predic-tions were verified by rate constant calculations at293 K for the forward and reverse reactions. The TSTand CVT rate constants were equivalent for the twodual-level MEPs. The CVT/SCT rate constants differby ∼1 order of magnitude, and the MPW1K/CCSDmethod consistently produced larger rate constantsthan the MP2/CCSD method. This result illustratesthe realization that the lower- and higher-level theo-ries need to be compatible with one another. In otherwords, the correction needs to be small to main-tain consistency in the results. The MPW1K/CCSDmethod is therefore of little use in this regard dueto the large changes present between MPW1K andCCSD theories. The suggested method to approx-imate the CCSD MEP is MP2/CCSD. However, iflarger systems are to be used, geometry and hessiancalculations even at the stationary points are likelyto be too costly. A similar method MP2/MP2/CCSDmay be used to reduce the cost by correcting theMP2 MEP with only CCSD energies. If the changes in

geometries and frequencies are truly small betweenmethods, this simplification will not adversely affectthe consistency of results.

RATE CONSTANTS

Rate constants are calculated for each of three the-ories (MPW1K, MP2, and CCSD) using energies andgeometries from the 6-31++G** basis. The MEP usedfor each calculation, except CCSD, are based on thesame theory and basis. For CCSD, the MEP used iseither MP2/CCSD or MP2/MP2/CCSD. The massesof all atoms correspond to the most abundant iso-tope. Electronic degeneracies, in the ground state, areset to singlets for molecular hydrogen and silane, andto doublets for atomic hydrogen, silyl radical, andthe transition state. The forward and reverse reac-tions are calculated with overall symmetry factorsof 2 and 4, respectively. These reaction rates are cal-culated completely in the ground state and over the293–683 K temperature range.

Experimental kinetic studies have been studiedextensively for the reverse hydrogen extraction reac-tion in the gas phase [20–31]. Only three have focusedon the temperature dependence of the rate con-stant [20–22]. Arthur et al. [21] measured Arrheniusparameters of A = 2.3 ± 0.3x10−11 cm3s−1 and Ea =2.77 ± 0.07 kcal/mol over 294–487 K. Goumri andYuan and colleagues [22] measured a larger preex-ponential factor, A = 1.78 ± 0.11x10−10 cm3s−1, and asomewhat larger activation energy, Ea = 3.82 ± 0.05kcal/mol, over the larger temperature range of 290–658 K. Finally, Arthur and Miles [20] measured apreexponential factor of 7.5±0.5x10−11 cm3s−1 and anactivation energy of 3.20±0.05 kcal/mol over a sim-ilar temperature range, 298–636 K. Variability in thepreexponential factor of less than one order of mag-nitude and in the activation energy of ∼1 kcal/molis evident from this experimental evidence in thecombined temperature range of 290–658 K.

The forward reaction rates have not been studiedexperimentally; however, these rate constants canbe computed from the reverse rates via an equilib-rium constant. The data used for this equilibriumconstant include the standard heats of formation,entropy, and constant pressure heat capacity at 298 Kfor the reactants and products [15, 16]. Owing tothe error associated with the enthalpy of reaction,these equilibrium constants are known only to atemperature-dependent factor, 100 at 293 K and 7.6at 683 K.

MPW1K TST rate constants are within oneorder of magnitude of experiment over the entire


CROSBY AND KURTZ

FIGURE 4. Arrhenius plots for the MPW1K 6-31++G** method is shown for both the forward and reverse directions ofthe studied reaction. Experimental results are from references [20–31]. The forward experimental results are obtainedfrom balance with the reverse.

temperature range for both the forward and reversereaction. CVT/SCT calculations have little effect onthese rate constants. From TST to CVT/SCT calcu-lations, the rate constant only increases by a factorof ∼4 at 293 K. Tunneling is obviously not impor-tant with the MPW1K method. Figure 4 shows theArrhenius plots of the forward and reverse reactionrate constants calculated by TST, CVT, CVT/ZCT,and CVT/SCT calculations.

The MP2/MP2/CCSD aug-cc-pVDZ method alsoproduces rate constants within one order of mag-nitude of experiment over the entire temperaturerange. However, this method requires the inclu-sion of tunneling via the CVT/ZCT or CVT/SCTapproximations, especially at the lower tempera-tures in the range. Figure 5 shows theArrhenius plotsfor this method for both the forward and reversereactions.

FIGURE 5. Arrhenius plots for the MP2/MP2/CCSD aug-cc-pVDZ method is shown for both the forward and reversedirections of the studied reaction. Experimental results are from references [20–31]. The forward experimental resultsare obtained from balance with the reverse.



The choice of the aug-cc-pVDZ basis set for thisapproximate CCSD method is made as a necessity.Smaller basis sets, such as the 6-31++G** basis, pro-duce rate constants that are two to three orders ofmagnitude too low. This result details a consistentimprovement upon improvements in basis. The MP2method also consistently improves upon improve-ments in basis. However, this method, using thebest basis included in the study, aug-cc-pVTZ, fellabout one order of magnitude too low for the reversereaction.

ARRHENIUS PARAMETERS

The rate constant data from the MPW1K andMP2/MP2/CCSD aug-cc-pVDZ methods are usedto extract Arrhenius parameters over three tem-perature ranges. These temperature ranges corre-spond to roughly those used in experimental workand include 293–483 K, 293–633 K, and 293–653 K.Table IV shows these Arrhenius parameters withexperimental values for the forward and reverse

reactions. Experimental data are only available forthe reverse reactions, so the equilibrium constantscan be used to approximate the forward reactionArrhenius parameters from the reverse. In eachcase, the forward activation energies are derived bythe reverse activation energies plus 11.9 kcal/mol.The forward preexponential factors are similarlyacquired by the reverse values times 6.60x10−2. TheseArrhenius parameters were obtained similarly toexperimental methods by a least-squares fit of therate constants.

All calculated preexponential factors agree withinone order of magnitude of experiment. The acti-vation energies, however, do not all agree within1 kcal/mol of experimental values. For MPW1K,all calculated activation energies agree with exper-iment. The MP2/MP2/CCSD aug-cc-pVDZ methodprovides activation energies within 1 kcal/mol ofexperiment with CVT/ZCT and CVT/SCT approx-imations. However, at higher temperatures, the TSTand CVT calculations also agree with experiment.This illustrates the different effects of tunneling

TABLE IVArrhenius parameters for the MPW1K and MP2/MP2/CCSD aug-cc-pVDZ methods over three temperature ranges.

Activation energy (kcal/mol) Preexponential factor (cm3s−1)

Parameter method 293–483 K 293–633 K 293–653 K 293–483 K 293–633 K 293–653 K

MPW1K

Forward TST 15.36 15.49 15.51 6.04E-12 7.31E-12 7.50E-12CVT 15.30 15.42 15.44 5.37E-12 6.37E-12 6.52E-12CVT/ZCT 14.82 15.00 15.02 3.46E-12 4.43E-12 4.58E-12CVT/SCT 14.49 14.69 14.72 2.83E-12 3.76E-12 3.90E-12Experiment 14.67 15.10 15.72 1.52E-12 4.95E-12 1.17E-11

Reverse TST 3.421 3.578 3.598 1.97E-10 2.45E-10 2.52E-10CVT 3.362 3.505 3.524 1.75E-10 2.14E-10 2.19E-10CVT/ZCT 2.885 3.083 3.107 1.13E-10 1.49E-10 1.54E-10CVT/SCT 2.554 2.779 2.807 9.21E-11 1.26E-10 1.31E-10Experiment [20–22] 2.770 3.200 3.820 2.30E-11 7.50E-11 1.78E-10

MP2/MP2/CCSD aug-cc-pVDZ

Forward TST 16.14 16.28 16.30 6.11E-12 7.45E-12 7.65E-12CVT 16.13 16.26 16.28 5.45E-12 6.57E-12 6.74E-12CVT/ZCT 14.89 15.10 15.13 2.64E-12 3.59E-12 3.73E-12CVT/SCT 14.68 14.90 14.93 2.42E-12 3.31E-12 3.44E-12Experiment 14.67 15.10 15.72 1.52E-12 4.95E-12 1.17E-11

Reverse TST 4.797 4.963 4.984 1.98E-10 2.50E-10 2.57E-10CVT 4.787 4.945 4.966 1.77E-10 2.21E-10 2.27E-10CVT/ZCT 3.542 3.784 3.814 8.58E-11 1.20E-10 1.25E-10CVT/SCT 3.334 3.582 3.612 7.85E-11 1.11E-10 1.16E-10Experiment [20–22] 2.770 3.200 3.820 2.30E-11 7.50E-11 1.78E-10


CROSBY AND KURTZ

between these calculations. Tunneling is not requiredto obtain acceptable activation energies from theMPW1K method. Tunneling is required at lowertemperatures for the MP2/MP2/CCSD aug-cc-pVDZ method.

Conclusion

The early dismissal of B3LYP as a method of inter-est due to its highly endothermic reaction enthalpysuggests that this method, as previously used byVitiello et al. [4] for the E′ center underestimatesthe reverse barrier. The SCF method used by Kurtzand Karna [3] for this same center is found to over-estimate this barrier suggested by the SCF reactionenergy as not being endothermic enough. Two meth-ods have been identified in this work as being ableto calculate accurately the kinetics of our experimen-tally well-studied prototype reaction of hydrogenand a silyl radical. These are MPW1K 6-31++G**and MP2/MP2/CCSD aug-cc-pVDZ. Both methodsproduce rate constants within 1 order of magnitudeand activation energies within 1 kcal/mol of experi-ment for the hydrogen silyl radical reaction over the293–683 K temperature range. The reaction thermo-dynamics also agree within experimental error forthese methods. The effect of tunneling on the rateconstants calculated with these methods differs fromslight (i.e., a factor of 4) to an order of magnitude at293 K.

Both methods will be applied to various othersilicon radical reactions with hydrogen, includingthe E′ center. As mentioned in the Introduction, ourgoal is to study systems with three to four times thenumber of heavy atoms. From this point of view,the MPW1K 6-31++G** method has great promisedue to its lower computational costs. Both methodswill still be used in future studies to examine poten-tial differences when more electronegative atoms areconnected to the central silicon radical atom.

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Application of electronic structure and transition state theory: Reaction of hydrogen with silicon radicals