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Ab Initio Investigation of the Abstraction Reactions by H and D from Tetramethylsilane and Its Deuterated Substitutions I. Oueslati,†,‡ B. Kerkeni,*,†,¶ A. Spielfiedel,‡ W.-Ü L. Tchang-Brillet,‡ and N. Feautrier‡ †

Faculté des Sciences de Tunis, Département de Physique, (LPMC), Université de Tunis El Manar, 2092 Tunis, Tunisia LERMA, UMR8112-CNRS, Observatoire de Paris, Université Pierre et Marie Curie, 5 place Jules Janssen, 92195 Meudon Cedex, France ¶ Institut Supérieur des Arts Multimédia de la Manouba, Université de la Manouba, 2010 la Manouba, Tunisia ‡

S Supporting Information *

ABSTRACT: Thermal rate constants for chemical reactions using the corrections of zero curvature tunneling (ZCT) and of small curvature tunneling (SCT) methods are reported. The general procedure is implemented and used with high-quality ab initio computations and semiclassical reaction probabilities along the minimum energy path (MEP). The approach is based on a vibrational adiabatic reaction path and is applied to the H + Si(CH3)4 → H2 + Si(CH3)3CH2 reaction and its isotopically substituted variants. All of the degrees of freedom are optimized, and harmonic vibrational frequencies and zero-point energies are calculated at the MP2(full) level with the cc-pVTZ basis set. Single-point energies are calculated at a higher level of theory with the same basis set, namely, CCSD(T,full). The influence of the basis set superposition error (BSSE) on the energetics is tested. The method is further exploited to predict primary and secondary kinetic isotope effects (KIEs and SKIEs, respectively). Rate constants computed with the ZCT and SCT methods over a wide temperature range (180−2000 K) show important quantum tunneling effects at low temperatures when compared to rates obtained from the purely classical transition-state theory (TST) and from the canonical variational transition state theory (CVT). For the H + Si(CH3)4 reaction, they are given by the following expressions: k(TST/ZCT) = 9.47 × 10−19 × T2.65 exp(−2455.7/T) and k(CVT/SCT) = 7.81 × 10−19 × T2.61 exp[(2704.2/T) (in cm3 molecule−1 s−1). These calculated rates are in very good agreement with those from available experiments.

1. INTRODUCTION Since their detection in the interstellar medium (ISM),1 in the gas phase of molecular clouds,2 and in the circumstellar envelope (CSE) of carbon stars,2−7 silicon carbide molecules and grains8 have been the subject of growing scientific interest. The CSE of carbon-rich stars shows rich silicon chemistry9 despite the fact that silicon participates in dust formation in the ISM through depletion.10 MacKay and Charnley7 have reported some silicon-bearing species (SiC, SiC2, SiC4, SiH2, and SiH4) observed in the CSE of the carbon-rich star IRC + 10216 (Cw Leo) at the temperature of 2200 K. The inner layers of this CSE are sites where a variety of processes influence the chemical composition of this medium. SiC is believed to be a © 2014 American Chemical Society

significant constituent of the dust around C-rich asymptotic giant branch (AGB) stars, and its well-known infrared (IR) peak at ∼11.3 μm is observed in many C star spectra.11 In this paper, we choose to study the abstraction reaction by hydrogen atoms from tetramethylsilane (TMS) as a prototype to study molecular hydrogen formation on silicon carbide grains in carbon star envelopes.12,13 The Si(CH3)4 molecule could be a precursor of SiC grains; it was used to model (βSiC) cubic structure polytypes.14 The 21 μm feature is seen clearly in Received: July 23, 2013 Revised: December 20, 2013 Published: January 10, 2014 791

dx.doi.org/10.1021/jp407310c | J. Phys. Chem. A 2014, 118, 791−802

The Journal of Physical Chemistry A

Article

the nano-SiC spectra and appears in the granular βSiC spectra.15 Silicon carbide is also used in the industry of semiconductors.16 Silane (SiH4) and its methyl-substituted variants are employed in chemical vapor deposition of silicon and silicon compounds.17 Over the past 15 years, the study of the dynamics and kinetics of H atom abstraction reactions by H atoms from polyatomic molecules has evolved rapidly. The reaction with methane18−23 has become a benchmark system from which an appreciable amount of our current understanding of polyatomic elementary reactivity now derives. The study of the reactivity of larger saturated hydrocarbons24−30 with H atoms revealed strong similarities with the methane reaction. For most chemical reactions, the only quantum effects are vibrational quantization and moderate tunneling. The current study will be useful for further understanding of H abstraction reactions from a large molecule, with emphasis on its kinetic features. In order to achieve this goal, we use state of the art ab initio calculations, followed by reliable dynamics and kinetics treatments to recover crucial quantum effects involved in any H atom transfer reaction. The combination of methods that we use is suitable for reactions involving a large number of atoms because multidimensional quantum chemistry development of potential energy surfaces (PESs) might appear prohibitively expensive to develop. We perform semiclassical tunneling calculations on a ground-state adiabatic minimum energy path (MEP) potential. The kinetic isotopic effects (KIEs) are also investigated. To our best knowledge, no theoretical attention has been paid to rate constants for H atom reactions with silicon compounds, apart from those containing Si−H.17,31,32 Moreover, there are only a few experimental results33−37 concerning the abstraction reaction from TMS. For the H + Si(CH3)4 reaction, rate constants have been studied experimentally by Arthur et al.;33 they were measured with the pulsed photolysis resonance absorption technique and are given by the following expression 7.5 ± 4 × 10−11 exp[−3990 ± 300/T] cm3 molecule−1 s−1 for the 425−570 K temperature range. In this work, we focus on the H abstraction reaction from TMS and on the KIEs due to deuterium substitution. The rate constants of the following reaction, denoted reaction R1.a H + Si(CH3)4 → H 2 + Si(CH3)3 CH 2

the procedure employed to compute the reaction probabilities and to determine the rate constants. The results pertaining to structure optimization, vibrational frequencies, and energetics are given in section 3. In section 4, we present the rate constants and the KIEs, and we discuss the results and make comparison with the available experimental results from the literature. Finally, conclusions are displayed in section 5.

2. METHODOLOGY 2.1. Electronic Structure Calculations. In the present work, the equilibrium geometries of all of the stationary points (reactants, products, hydrogen-bonded complexes, and saddle points) were optimized by the second-order Møller−Plesset (MP2) perturbation theory (where MP2 denotes RMP2 or UMP239,40 depending on the number of electrons of the system), with the correlation-consistent polarized valence basis set of Dunning,41,42 namely, the cc-pVTZ. This approach satisfactorily predicts the geometry and the IR spectra.43 We have also investigated the reliability of the M06-2X functional44 in the optimization of the fragments’ optimized geometries. Vibrational frequencies were computed at the MP2/cc-pVTZ level to determine the nature of the stationary geometries using the second-order derivatives of the energy with respect to the Cartesian nuclear coordinates and then transforming to massweighted coordinates. To ensure adequate convergence and reliability of the frequencies, the “tight” option was used. At the same level, the MEP was computed using the intrinsic reaction coordinate (IRC) method45 with a gradient step size of 0.01 (amu)1/2 a0. IRC calculations require initial force constants of the transition state. Then, the first- and second-order energy derivatives were obtained to calculate the harmonic vibrational frequencies along the reaction path. For all fragments and all of the points along the MEP, singlepoint energy calculations at the CCSD(T) level (coupled cluster with single, double and perturbative triple excitations)46 with the same basis set, using the MP2 geometries, were carried out in order to obtain more accurate energies and barrier heights and to perform the following dynamics calculations. All electrons have been included in correlation treatments. Furthermore, the counterpoise method proposed by Boys and Bernadi47 was applied to correct the effect of basis set superposition error (BSSE) on the energies for the identified van der Waals complexes. The CPU time required for MP2 geometry optimizations and frequency calculations with GAUSSIAN 0948 is 100 h using one processor for each fragment, while 50 h were needed for single-point energy computations at the CCSD(T) level with the MOLPRO 2010 package.49 2.2. Semiclassical Reaction Probabilities. Light atoms such as hydrogen tunnel through barriers that cannot be surmounted classically. This quantum effect can be important at low temperatures. The exponential damping of wave tunneling through a given barrier leads to the tunneling probability P(E) at total energy E50

(R1.a)

are calculated over the temperature range of 180−2000 K. The KIEs for the same temperature range are investigated for the following reactions, denoted reactions R1.b, R1.c, R1.d, and R1.e: D + Si(CH3)4 → HD + Si(CH3)3 CH 2

(R1.b)

D + Si(CD3)4 → D2 + Si(CD3)3 CD2

(R1.c)

H + Si(CD3)4 → HD + Si(CD3)3 CD2

(R1.d)

H + Si(CH3)3 CH 2D → HD + Si(CH3)3 CH 2

(R1.e)

Investigation of KIEs is an important tool as it can be used to diagnose chemical bond breakage and formation and to test the shape of the PES, especially the barrier width and the vibrational force constants near the transition-state region.18 In the past, KIEs have been investigated only in abstraction reactions from Si−H compounds.17,38 This paper is organized as follows. In section 2, we thoroughly describe the electronic structure calculations and

P(E) = exp[− 2θ(E)]

(1)

where θ represents the amount of exponential decay. 2.2.1. Zero Curvature Tunneling (ZCT) Method. When the reaction path curvature can be neglected, the proper tunneling path is the MEP. For tunneling along the MEP, θ(E) is the imaginary action integral given by 792


The Journal of Physical Chemistry A θ(E) = ℏ−1

∫s

s1

Article

ds {2μ[V aG(s) − E]}1/2

constants from transition state theory (TST) and to canonical variational transition state theory rate constants (CVT) in order to assess significance of quantum tunnelling and variational effects, respectively. The TST and TST/ZCT calculations are performed using a code developed by B. Kerkeni.55 The CVT and the CVT/SCT rate constants are computed using the POLYRATE program, version 2010-A.56 For a bimolecular reaction, the purely classical TST rate constants are given by21

(2)

0

VGa (s)

with = VMEP(s) + ZPE(s), where VMEP is the classical potential energy of the MEP and ZPE is the zero-point energy correction relevant to the frequencies orthogonal to the reaction path. The s0 and s1 are the reaction coordinates of the classical turning points in the reactant and product valleys, respectively, and μ is the reduced mass. This approach is called the zero curvature tunneling (ZCT) method.51 2.2.2. Small Curvature Tunneling (SCT) Method. Liu et al.52 have reported that the tunneling probabilities may increase because of the corner-cutting effect, which is not accounted for in the ZCT method. More advanced tunneling methods are based on the inclusion of deviations between the tunneling path and the reaction path.53 In the present work, the centrifugal dominant small curvature semiclassical adiabatic ground state (CD-SCSAG) tunneling method52,54 is applied. In this method, referred to as the small curvature tunneling method (SCT),54 the effect of the reaction curvature is included by replacing the reduced mass μ by an effective mass μeff.52 The transmission probability at energy E is 1 P(E) = 1 + exp−2θ(E) (3)

k TST(T ) =

∫s

s1

ds {2μeff [V aG(s) − E]}1/2

(4)

0

⎪

⎪

⎪

⎪

θ (E ) = ℏ

ξ2

∫ξ1

dξ

{2μ[V aG(s(ξ))

(10)

⎛ μk T ⎞3/2 Q trans = ⎜ B 2 ⎟ ⎝ 2π ℏ ⎠

(11)

T3 ΘaΘ bΘc

π σ

(12)

where Θi = ℏ2/(2IikB), with Ii is the corresponding inertia moment. The inertia moments take the values Ia = Ib = Ic = 583.90658 au for Si(CH3)4 and 575.13025, 621.82027, and 634.31393 au for TS(1.a) of reaction R1.a. The symmetry numbers σ are 12 for Si(CH3)4 and 1 for the transition state. All vibrational modes are treated as separable harmonic oscillators

(5)

In terms of ξ, θ(E) is defined as −1

Q tot = Q transQ rotQ vibQ elec

Q rot =

Let the distance along the small curvature path be ξ, the curvature at s be u(s), and let t(s) be the distance between the considered tunneling path and the MEP. The element of length along the tunneling path, dξ, is given by 1/2 ⎧ ⎡ dt ( s) ⎤ 2 ⎫ 2 dξ = ⎨[1 − u(s)t(s)] + ⎢ ⎥ ⎬ ds ⎣ ds ⎦ ⎭ ⎩

(9)

where ΔVa is the adiabatic vibrational barrier height, h the Planck constant, and kB the Boltzmann constant. Q#, QA, and QB are, respectively, the transition state (TS) and the reactant partition functions, which can be expressed as a product of the partition functions of the translational motion (expressed in per unit volume), the internal motions (vibration, rotation), and the electronic distribution. These can be calculated within the rigid rotor harmonic oscillator approximation

where θ(E) is the imaginary action integral along the tunneling path θ(E) = ℏ−1

Q #(T ) kBT e−ΔVa / kBT hQ A(T )Q B(T )

m

Q vib(T ) =

1/2

− E]}

⎪

⎪

⎪

⎪

i

B

(13)

i=1

(6)

where m is the number of vibrational modes and ωi is the harmonic frequency. For the torsional mode, the hindered rotor approximation introduced by Chuang and Truhlar57 was used. The corresponding partition function QHin is defined as a correction of the harmonic oscillator partition function Qvib(T)

Combining eqs 4, 5, and 6, the effective reduced mass is given by ⎧ ⎡ dt ( s) ⎤ 2 ⎫ μeff (s) = μ⎨[1 − u(s)t(s)]2 + ⎢ ⎥⎬ ⎣ ds ⎦ ⎭ ⎩

∏ [1 − exp−(ℏω /k T)]−1

(7)

⎡ω ⎛ ⎞1/2 ⎤ j 2πI Q Hin = Q vib(T ) tanh⎢ ⎜ red ⎟ ⎥ ⎢⎣ σj ⎝ kBT ⎠ ⎥⎦

However, to make this approach valid even when t(s) is greater than the radius of curvature, the reduced mass is calculated as52 μeff (s) = μ

(14)

where σj is the periodicity of the torsional potential, ωj is the torsional harmonic frequency, and Ired is the reduced moment of inertia for the internal rotation of the subgroup HCH3 with respect to the Si−C bond (oz axis) for reaction R1.a (see Figure 1). The expression of Ired is approximated by58 Ired = I1 × I2/(I1 + I2), where I1 and I2 denote the moments of inertia for Si(CH3)3 and HCH3, respectively. Computed moments of inertia Ired for each transition state are reported in Table 1. The torsional barrier height Wj can be approximated57 from ωj as

⎧ ⎡ 2⎤ ⎪ exp⎢ − 2u(s)t(s) − [u(s)t(s)]2 + ⎛⎜ dt(s) ⎞⎟ ⎥ × min⎨ ⎢⎣ ⎝ ds ⎠ ⎥⎦ ⎪ ⎪ ⎩1 (8)

2.3. Rate Constant Calculations. In order to calculate thermal rate constants and KIEs over a wide temperature range, the ZCT51 and SCT52 approaches were applied. The thermal rate constants for reaction R1.a and its isotopic analogues are reported and are also compared to purely classical rate 793



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k CVT(T ) = σ

⎡ −ΔG(T , s*,CVT ) ⎤ kBT 0 K exp⎢ ⎥ h kBT ⎣ ⎦

(16)

,CVT

where s* is the location of the canonical transition state on the reaction path, ΔG(T, s*,CVT) is the free energy of activation, and K0 is the reciprocal of the standard-state concentration, taken as 1 molecule cm−3. In this work, special attention was given to the influence of quantum tunneling on the rate constants because tunneling is important for reactions involving transfer of a light particle, such as a hydrogen atom. In the ZCT and SCT approaches, tunneling is calculated from the semiclassical (i.e., WKB) ground-state transmission coefficient κ(T)52

Figure 1. Optimized geometries of the transition state TS(1.a) at the MP2/cc-pVTZ level (Si atom: yellow; C atoms: blue; H atoms: gray).

Table 1. Calculated Reduced Moments of Inertia for Si(CH3)4 and the Transition States of All of the Studied Reactions species

I1(au)a

I2 (au)b

Ired (au)

TS(1.a) Si(CH3)4 TS(1.b) TS(1.c) TS(1.d) TS(1.e)

1045630 1044672 1045630 1349682 1349682 1045630

51391 20693 75877 98810 74235 59585

48983 20291c 70743 92070 70365 56372

a

I1 corresponds to ISi(CH3)3. bI2 corresponds to IHCH3. experimental value for Si(CH3)4 is 20745 au.59

⎛ ωj ⎞2 Wj = 2 × Ired⎜⎜ ⎟⎟ ⎝ σj ⎠

c

k ZCT(T ) = κZCT(T ) × k TST(T )

(17)

k SCT(T ) = κSCT(T ) × k CVT(T )

(18)

where κ(T) is given by ∞

κ (T ) =

∞

∫V

a(s = 0)

The

( ) exp(− ) dE

∫0 P(E) exp − kBET dE E kBT

(19)

where P(E) is either given by eq 1 for ZCT or by eq 3 for SCT. Thermal rate constants for reaction R1.a and deuterated analogues reported in this work were obtained from ZCT and SCT calculations. A comparison to TST and CVT results was performed in order to highlight the importance of quantum and variational effects, respectively.

(15)

3. ELECTRONIC STRUCTURE RESULTS 3.1. Stationary Points. Table 2 reports the geometries, optimized at the MP2 level, of the reactant (Si(CH3)4), products (Si(CH3)3CH2, H2), and transition state (TS(1.a)) involved in reaction R1.a. As can be seen from the table, the obtained parameters compare well with those that we obtained using the M06-2X functional and with the available experimental61−63 and theoretical17,64,65 values. The geometry of the transition state TS(1.a) is almost linear (∠C−H−H = 175.8°) (Figure 1). Two van der Waals complexes are found for

However, in this paper, we use our ab initio calculated value Wj = 1.3 kcal/mol for reaction R1.a at the CCSD(T)/cc-pVTZ level. The variational transition-state theory (VTST)52,54,60 was used to estimate the rate constants and kinetic isotope effects (KIEs). In the present work, the CVT52,54,60 is applied. At the temperature T, kCVT is calculated by minimizing the generalized TST rate constants, kG(T), as a function of s60 k CVT(T ) = mins kG(T , s)

Table 2. Molecular Geometry Parameters of the Fragments (Bond Lengths in Å, and Angles in °) Involved in Reaction R1.a at the MP2/cc-pVTZ Levela species Si(CH3)4

Si(CH3)3CH2

H2

TS(1.a)

a

methods/basis

SiC

CHa

CH

∠HCHa

∠HCH

∠SiCHa

HaHb

∠CHaHb

MP2/cc-pVTZ M062X/cc-pVTZ MP2/6-31 + G(d,p)64 MP2/6-311 + G(2d,2p)65 expt (2000)61 expt (1970)62 MP2/cc-pVTZ M062X/cc-pVTZ MP2/6-31 + G(d,p)64 MP2/6-311 + G(2d,2p)65 MP2/cc-pVTZ M062X/cc-pVTZ UCCSD(T)/cc-pVQZ17 expt (1979)63 MP2/cc-pVTZ M062X/cc-pVTZ

1.883 1.879 1.886 1.881 1.877 ± 0.004 1.875 ± 0.002 1.860 1.850 1.864 1.859 − − − − 1.873 1.870

1.090 1.091 1.092 1.088 1.11 ± 0.003 − − − − − − − − − 1.381 1.372

1.090 1.091 − 1.088 − − 1.082 1.084 1.084 1.081 − − − − 1.090 1.090

107.8 107.7 107.7 107.8 − 109.8 ± 0.8 − − − − − − − − 100.5 101.3

107.9 107.8 − 107.8 − − 114.0 114.0 113.9 114.0 − − − − 107.9 107.8

111.1 111.1 111.2 111.1 111.0 ± 0.2 109.2 ± 0.8 − − − − − − − − 103.4 103.1

− − − − − − − − − − 0.738 0.737 0.742 0.741 0.892 0.898

− − − − − − − − − − − − − − 175.8 177.0

Ha is the abstracted atom, while Hb is the incoming atom. 794



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complexes are 0.176 kcal/mol (61.55 cm−1) and 0.238 kcal/ mol (83.24 cm −1), respectively, at the CCSD(T) level. When taking the BSSE correction into account, the energies of weakly bonded complexes become 0.101 and 0.163 kcal/mol lower than the energies of reactants and products, respectively, at the MP2/cc-pVTZ level; the two complexes are thus stable. The computed energetics are the classical barrier height (ΔV), the vibrational adiabatic barrier height (ΔVa) including the difference of zero-point vibrational energy between the transition state and the reactants. The energy change ΔE for the reaction excludes vibrational ZPEs, which are included in the calculation of the enthalpy ΔH of reaction. In Table 5, we compare the energies at 298.15 K calculated at the MP2 and CCSD(T) levels for the isotopic reactions. All

reaction R1.a, CR(1.a)R on the reactant side and CR(1.a)P on the product side (Table 3). Table 3. Optimized Geometries (Bond Lengths in Å, and Angles in °) of TS(1.a), CR(1.a)R, and CR(1.a)Pa species CHa CH HaHb ∠HaCH ∠HCH a

methods/basis

TS(1.a)

CR(1.a)R

CR(1.a)P

MP2/cc-pVTZ M062X/cc-pVTZ MP2/cc-pVTZ M062X/cc-pVTZ MP2/cc-pVTZ M062X/cc-pVTZ MP2/cc-pVTZ M062X/cc-pVTZ MP2/cc-pVTZ M062X/cc-pVTZ

1.381 1.378 1.090 1.092 0.892 0.901 100.5 101.2 111.5 107.9

1.090 1.091 1.090 1.090 3.221 3.210 107.8 107.8 107.9 107.9

3.077 2.977 1.080 1.090 0.738 0.740 90.3 90.0 114.0 114.1

Table 5. Energetics of the Isotopic Reactions at the MP2 and CCSD(T)//MP2 Levels with the cc-pVTZ Basis Set (in kcal/mol)

Ha is the abstracted atom, while Hb is the incoming atom.

The H···H bond at the transition state is longer than the equilibrium value of 0.737 Å for H2 by 21%, while the breaking bond C···H is elongated by 27% for TS(1.a). This indicates that TS(1.a) is product-like, and thus, reaction R1.a will proceed via a late reaction transition state and is endothermic. This result is consistent with Hammond’s postulate66 at the MP2 level of theory. 3.2. Vibrational Analysis. Isotope effects involving deuterated species lead to differences in the vibrational frequencies and hence in zero-point vibrational energies. Therefore, frequency calculations have been obtained for all of the isotopic variants with a minimal computational cost because these only require application of the adequate mass multiplication with the Hessian matrix of the original fragments. The frequencies relevant to reactants, vdW complexes, and products involved in the reactions show excellent agreement with the available experimental63,67,68 and theoretical values; 65 they are listed in Tables S1−S3 (Supporting Information). The frequencies of the transition state TS(1.a) are listed in Table 4.

ΔV R1.a R1.b R1.c R1.d R1.e ΔVa R1.a R1.b R1.c R1.d R1.e ΔE R1.a R1.b R1.c R1.d R1.e ΔH R1.a R1.b R1.c R1.d R1.e

Table 4. Calculated Frequencies (cm−1) for the Saddle Point and the ZPE for Reaction R1.a at the MP2/cc-pVTZ Level frequency (cm−1) TS(1.a)

1663i, 93, 147, 165, 168, 173, 178, 204 224, 224, 330, 553, 605, 677, 693, 700 716, 718, 746, 825, 837, 871, 881, 884, 1134, 1161, 1279, 1282, 1289, 1410, 1463, 1466 1466, 1476, 1476, 1484, 1790, 3065, 3066, 3066 3129, 3158, 3161, 3161, 3163, 3163, 3165, 3218

ZPE (hartree)

MP2

CCSD(T)

Si(CH3)4 + H Si(CH3)4 + D Si(CD3)4 + D Si(CD3)4 + H Si(CH3)3CH2D + H

16.66 16.66 16.66 16.66 16.66

11.72 11.72 11.72 11.72 11.72


15.42 14.47 15.69 16.66 16.57

10.48 9.53 10.75 11.73 11.64


3.74 3.74 3.74 3.74 3.74

−1.02 −1.02 −1.02 −1.02 −1.02


1.67 0.81 2.12 3.15 2.73

−3.08 −3.95 −2.64 −1.61 −2.03

0.146494

reactions are found to be endothermic at the MP2 level and exothermic at the CCSD(T) level, which is indicative of the importance of high-level correlation treatments. At both levels of theory, the Si(CH3)4 + D reaction R1.b is found to have the smallest vibrational adiabatic barrier, whereas the Si(CD3)4 + H reaction R1.d has the highest one. The calculated adiabatic vibrational barrier height 10.48 kcal/mol for (R1.a) is larger than the experimental value (7.93 ± 0.6 kcal/mol).33 This difference could be explained by the fact that in the experiment,33 the activation barrier is obtained from a fit of the temperature-dependent kinetic data to an Arrhenius functional form. 3.4. Determination of the Reaction Path. We have checked that TS(1.a) correlates the already optimized reactants and products. The MEP was obtained from IRC calculations, and the frequencies of all points along the MEP were also computed at the MP2/cc-pVTZ level. The energies were further refined at the CCSD(T) level using the same basis set

The calculated frequencies for H2, D2, and Si(CH3)4 are within ∼7% of the experimental values. The normal-mode analysis confirms the existence of a single transition state that has a single imaginary frequency (1663i cm−1 for reaction R1.a) corresponding to the motion of a H atom transferring between C and H atoms. 3.3. Energetics. In order to acquire reliable energetics of the hydrogen-bonded complexes, the BSSE correction47 is taken into account. The energies of reactants and products are considered as the reference for the van der Waals complexes on the reactant side (CR(1.a)R) and on the product side (CR(1.a)P). The well depths of CR(1.a)R and CR(1.a)P 795



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and the MP2 geometries. Figure 2 shows the variation of the classical potential energy (VMEP), the vibrationally adiabatic

Figure 3. Comparison of the computed k1.a(TST), k1.a(TST/ZCT), k1.a(CVT), and k1.a(CVT/SCT) rate constants to experimental values for the Si(CH3)4 + H reaction. Figure 2. Classical potential energy (VMEP), vibrational adiabatic potential energy (VGa ), and ZPE as functions of s (amu1/2 a0) at the CCSD(T)/cc-pVTZ//MP2/cc-pVTZ level for the Si(CH3)4 + H reaction.

4. RATE CONSTANTS As noted above, the TST with ZCT correction (TST/ZCT) and the CVT with SCT correction (CVT/SCT) were used to take into account the tunneling and variational effects in the calculation of the rate constants. These approaches are illustrated below. All reactions were studied with the CCSD(T)/cc-pVTZ//MP2/cc-pVTZ level of model chemistry. 4.1. Si(CH3)4 + H Reaction. Table 6 displays the TST/MP2 results of the rate constants and the CCSD(T) results relevant to k1.a(TST), k1.a(TST/ZCT), k1.a(CVT), and k1.a(CVT/SCT), as well as the available experimental data. As can be seen in column 2 of the table, MP2 calculations cannot predict correctly the rate constants. This is a consequence of their inability to predict the barrier height correctly, as already shown in Table 5. The logarithm of the calculated k1.a(TST), k1.a(CVT), k1.a(TST/ZCT), and k1.a(CVT/SCT) rate constants and the available experimental data are plotted against 1000/T in Figure

ground-state potential energy (VGa ), and the ZPE as a function of the reaction coordinate (s). The position of the maximum of the VMEP corresponds to the optimized saddle point geometry obtained at the MP2/ccpVTZ level. The ZPE curve presents a small drop near the saddle point. This behavior is in accordance with other hydrogen abstraction reactions from TMS, 65 NH 3 , 69 (C2H5)2SiH2,70 and SiH4.71 For reaction R1.a and all isotopic reactions, the classical potential energy and the vibrational adiabatic potential energy (VMEP and VGa ) have the same shape far from the cusp.

Table 6. Comparison between Rate Constants Calculated at the MP2 and CCSD(T) Levels with Experimental Values for Reaction R1.a (cm3 molecule−1 s−1)a

a

T (K)

k1.ab(TST/MP2)

k1.a(TST)

k1.a(CVT)

k1.a(TST/ZCT)

k1.a(CVT/SCT)

exp.33

180 300 425 450 475 500 525 550 570 1000 1500 2000

6.124(−29) 1.967(−21) 4.473(−18) 1.269(−17) 3.240(−17) 7.559(−17) 1.632(−16) 3.296(−16) 5.544(−16) 3.235(−13) 6.911(−12) 3.610(−11)

6.040(−23) 7.767(−18) 1.546(−15) 3.171(−15) 6.054(−15) 1.087(−14) 1.853(−14) 3.018(−14) 4.332(−14) 3.879(−12) 3.621(−11) 1.250(−10)

4.39(−23) 5.32(−18) 1.02(−15) 2.08(−15) 3.95(−15) 7.07(−15) 1.20(−14) 1.95(−14) 2.79(−14) 2.42(−12) 2.25(−11) 7.75(−11)

1.132(−20) 6.888(−17) 5.894(−15) 1.098(−14) 1.929(−14) 3.221(−14) 5.149(−14) 7.922(−14) 1.091(−13) 6.290(−12) 4.944(−11) 1.573(−10)

3.88(−19) 1.31(−16) 4.87(−15) 8.35(−15) 1.37(−14) 2.16(−14) 3.36(−14) 5.26(−14) 7.25(−14) 3.99(−12) 2.95(−11) 9.76(−11)

− − 6.28(−15) 1.06(−14) 1.69(−14) 2.57(−14) 3.76(−14) 5.31(−14) 6.84(−14) − − −

Powers of 10 in parentheses. bComputed k1.a(TST) with the MP2/cc-pVTZ barrier. 796



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Figure 4. KIEs computed from TST (a) and TST/ZCT (b) methods for Si(CH3)4 + H (k1.a/k1.b, blue curve), Si(CD3)4 + D (k1.a/k1.c, red curve), Si(CD3)4 + H (k1.a/k1.d, green curve), and Si(CH3)3CH2D + H (k1.a/k1.e, dashed black curve).

3. For high temperatures, calculated rate constants are nearly the same. Very similar results are obtained over the whole

temperature range from CVT and TST approaches. The ratio k1.a(CVT)/k1.a(TST) takes the values 0.73, 0.66, 0.62, and 0.61 797



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The KIEs from TST and TST/ZCT calculations are shown versus the temperature T in Figures 4a and b, respectively. One can see in Figure 4a that the TST KIEs are lower than 1 in the case of the D + Si(CH3)4 R1.b reaction for temperatures below 800 K. This inverse KIE for reaction R1.b is due to its small vibrational adiabatic barrier (see Table 5). Reactions R1.c, R1.d, and R1.e are slower than reaction R1.a because of their high vibrational adiabatic barriers; thus, they have normal KIEs for the whole temperature range. The behavior of KIEs calculated from TST may be associated with the effect of the ZPE on the activation barriers. In fact, for reactions R1.a and R1.b, the ZPE of the transition state TS(1.a) (0.146496 hartree) is larger than that of TS(1.b) (0.144984 hartree); therefore, the incoming H atom for the first one is more delocalized and is less strongly bound than the incoming D atom. One can expect that reaction R1.b is faster than reaction R1.a because the forming HD bond (the ZPE value is 0.008938 hartree) is stronger than the H2 bond (the ZPE value is 0.010320 hartee). For reactions R1.c, R1.d, and R1.e, the broken C−D bond, which has the lowest ZPE compared to the broken C−H bond in reaction R1.a, affects significantly the rate constants, and therefore, the adiabatic barriers of these reactions are the highest. From Figure 4b, which displays the TST/ZCT rate constants, we note that for temperatures lower than 800 K, the KIEs of reactions R1.c, R1.d, and R1.e are always normal because of the ZPE and tunneling contributions that favor reaction R1.a. In this last reaction, quantum effects are more efficient for the lighter H atom. For higher temperatures, the computed KIEs for reactions R1.c and R1.d are slightly greater than 1, which is partly due to the adiabatic barrier heights and to the different reduced masses through the ratio of the translational partition functions. For reaction R1.b, the KIEs show an interesting behavior because for temperatures lower than 220 K, a normal effect is seen with a KIE that reaches 1.39 at 180 K. It indicates that the tunneling effect prevails over the effect of ZPE in a TST/ZCT calculation, contrary to a TST calculation. The KIEs increase at low temperatures because the abstraction by H is favored over abstraction by D. Indeed, the transmission coefficients for reactions R1.a and R1.b at 180 K are 187.34 and 19.46, respectively. This behavior was also found in the study of the H abstraction reaction from methanol first by Kerkeni and Clary73 and later by Goumans and Kåstner.74 These authors found that at high temperatures, the isotopic reaction that has the lower vibrational adiabatic barrier gives rise to an inverse KIE, whereas at low temperature (135 K), the reaction involving H atoms is favored. We further notice from Figure 4b, that, for the R1.b reaction in the temperature range of 220−800 K, an inverse KIE appears, owing to its lower adiabatic barrier. For temperatures higher than 800 K, the KIE becomes normal, as expected, because the ZCT transmission coefficients κ(T) at 840 K are 1.80 and 1.50 for reactions R1.a and R1.b, respectively. Our investigations showed that the CVT KIEs have the same behavior as TST KIEs. For the whole temperature range, we found an inverse KIE for reaction R1.b and a normal KIE for reactions R1.c, R1.d, and R1.e. Figure 5 displays results of the CVT/SCT KIEs. These KIEs are larger than 1 for the whole temperature range in the case of Si(CD3)4 + D (reaction R1.c), Si(CD3)4 + H (reaction R1.d), and Si(CH3)3CH2D + H (reaction R1.e), similar to the case of TST/ZCT. This normal effect is explained by the quantum contributions (ZPE and tunneling effects). Because of its weaker vibrational adiabatic barrier, the R1.b reaction shows an

Figure 5. KIEs computed from the CVT/SCT method for Si(CH3)4 + D (k1.a/k1.b), Si(CD3)4 + D (k1.a/k1.c), Si(CD3)4 + H (k1.a/k1.d), and Si(CH3)3CH2D + H (k1.a/k1.e).

at 180, 400, 1000, and 2000 K, respectively. This result indicates that the variational effects are rather small. For low temperatures, the tunneling effects are more pronounced. The transmission coefficients for reaction R1.a are 187.34, 3.81, 2.52, and 1.62 for ZCT and 9078.40, 4.94, 2.60, and 1.65 for SCT at 180, 425, 570, and 1000 K, respectively. The marked increase of the transmission coefficient at low temperatures implies that the quantum effect increases significantly the rate constants. This observation was expected because the tunneling contribution is important even for ambient temperature.72 Table 6 and Figure 3 show a good agreement between our computed k1.a(TST/ZCT) and k1.a(CVT/SCT) rate constants and the experimental values for the temperature range (425− 570 K).33 At low temperature (T = 180 K), we observe a difference of about 1 order of magnitude between the computed TST/ZCT and CVT/SCT rate constants; this points out the high sensitivity of the calculated rate constants to the quantum and variational effects at low temperatures. The H abstraction reaction from TMS is faster than that from CH4 by a factor of 1500 at 300 K. The C−H bond in TMS is easier to break because the barrier height is lower, 10.48 (kcal/mol) for TMS (this work) and 14.20 (kcal/mol) for CH4,21 in good agreement with the experimental values 7.9333 and 13.78,19 respectively. Therefore, the silicon atom has an important effect on the reactivity of the adjacent C−H bonds. 4.2. KIEs. Isotopic substitution is a useful technique to probe reaction mechanisms and to analyze tunneling and variational effects.17,73,74 The isotope change may affect the reaction rate in a number of ways, providing clues to the pathway of the reaction. The advantage of isotopic substitution is that this is the least disturbing structural change in a molecule. In the present work, calculated KIEs are defined as the ratio of the rate constant of the unsubstituted reaction R1.a to the rate constants of the isotopic reactions with heavier mass, reactions R1.b, R1.c, R1.d, and R1.e, that is, KIE = kH/kD. With this convention, KIEs greater than 1 are called “normal”, and those less than 1 are called “inverse”. 798



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Figure 6. Thermal rate constants for the Si(CH3)4 + H reaction and deuterium-substituted reactions calculated by TST/ZCT (a) and CVT/SCT (b) methods: k1.a blue curve, k1.c red curve, k1.d black curve, and 12 × k1.e green curve.

Comparing reactions R1.c and R1.d to reation R1.e, deuterium substitution could give clues about the secondary kinetic isotope effects (SKIEs) (see Figure 6). The rate constants of the H + Si(CH3)3CH2D reaction R1.e compared to those pertaining to the H + Si(CD3)4 reaction R1.c are

inverse CVT/SCT KIE for the temperature range 340−2000 K, while a normal KIE is found from the TST/ZCT calculations at high temperatures (800−2000 K). This difference can be attributed to variational effects, which outweight the effective reduced mass contributions effects. 799



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the KIEs. In particular, for the R(1.b) reaction, the ratio k1.a/k1.b is found to be very sensitive to the way the tunneling and variational corrections are taken into account at low temperatures. In order to achieve a further understanding of the H abstraction reaction from TMS, we are currently investigating state-selected dynamics using reduced dimensionality quantum dynamics calculations.76

reduced by a factor of 12 because of their different reactant rotational symmetry numbers, which increases the rotational partition function for reaction R1.e. Multiplying k1.e by a factor of 12, a comparison excluding rotational contributions can be made with k1.c. Even though the barrier for reaction R1.c is lower than that for reaction R1.e, 12 × k1.e is still larger than k1.c because of the tunneling contribution effect. At T = 180 K, the ZCT transmission coefficients (eq 19) are 552.13 for reaction R1.e and 36.00 for reaction R1.c. Indeed, this coefficient depends on the semiclassical transmission probability P(E) (eq 1) and consequently on the reduced mass μ, which is 1817 au for reaction R1.e and 3599 au for reaction R1.c. At the same temperature, the SCT transmission coefficients are 9849.30 for reaction R1.e and 495.28 for reaction R1.c. In addition to tunneling, the variational effects contribute to the reduction of the k1.c(CVT/SCT) rate constants because the variational contributing factors, kCVT/kTST, at 180 K are 0.7 and 1 for reactions R1.c and R1.e, respectively. We notice for reactions R1.d and R1.e, which have almost the same reduced masses and adiabatic barriers, that 12 × k1.e is larger than k1.d (Figure 6) because of the ZPE contribution that renders the shape of the MEP narrower for reactions R1.e. At T = 180 K, the ZCT and SCT transmission coefficients are respectively 501.19 and 8307.4 for reaction R1.d. At this temperature, the ratio kCVT/kTST is 0.29 and 1 for reactions R1.d and R1.e, respectively; hence, the variational effect is more pronounced for reaction R1.d. Therefore, secondary isotope effects contribute only minimally in comparison to the primary effect to a decrease of the rate constants at low temperatures. As expected, all isotopic reaction rates are lower than k1.a because tunneling is more pronounced for the transfer of H atoms over that of D atoms. As noted by other studies,75 sophisticated TST interpretations of experimental KIEs are a powerful method to test models of abstraction reaction dynamics and to derive transition-state structures and properties.

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ASSOCIATED CONTENT

S Supporting Information *

Harmonic vibrational frequencies of reactants, products, and van der Waals complexes for reactions R1.a−R1.e are reported in Tables S1−S3. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS B.K. is thankful to Université de Pierre et Marie Curie Paris VI for a 1 month invited professorship. This work was fully supported by the European Community FP7-ITN Marie Curie Programme (LASSIE project, Grant Agreement No. 238258). The calculations were performed at the IDRIS and CINES French National Computer centres under Projects 2012046838 and 2013046838. The authors would like to thank Professor Donald G. Truhlar for providing the POLYRATE program, version 2010-A.

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REFERENCES

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5. CONCLUSION In this paper, we have studied the abstraction reactions of a hydrogen from TMS and its isotopic substituted reactions using ab initio methods to gain the MEP in order to predict the ZCT and SCT rate constants. All coordinates were fully relaxed in the optimization steps, and ZPEs orthogonal to the reaction path were calculated. The electronic properties were obtained at the CCSD(T)/cc-pVTZ//MP2/cc-pVTZ level. The adiabatic barrier height of reactions R1.a (Si(CH3)4 + H) and R1.b (Si(CH3)4 + D) are 10.48 and 9.53 kcal/mol, respectively, while the exothermicity of these reactions are −3.08 and −3.95 kcal/mol. The BSSE counterpoise correction on identified hydrogen-bonded complexes was included to improve their energetics. The rate constants are reported over a wide temperature range (180−2000 K). The reaction mechanism is identified as a direct hydrogen abstraction one. The Si(CH3)4 + H calculated rate constants k1.a(TST/ZCT) and k1.a(CVT/SCT) are in good agreement with experimental data and highlight the importance of tunneling corrections at low temperature where differences by 2 and 3 orders of magnitude compared to, respectively, TST and CVT are obtained at T = 180 K. We have also investigated the KIEs due to H → D exchange in the abstraction reaction of H + Si(CH3)4. For all four isotopic reactions, quantum effects are pointed out as they can explain the temperature variation of 800



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