Theoretical Reaction Kinetics Astride the Transition between Moderate

Article pubs.acs.org/JPCA

Theoretical Reaction Kinetics Astride the Transition between Moderate and Deep Tunneling Regimes: The F + HD Case S. Cavalli* Dipartimento di Chimica, Biologia e Biotecnologie, Università di Perugia, 06123 Perugia, Italy

V. Aquilanti Dipartimento di Chimica, Biologia e Biotecnologie, Università di Perugia, 06123 Perugia, Italy Instituto de Fisica, Universidade Federal da Bahia, 40210-340 Salvador, Brazil

K. C. Mundim Instituto de Química, Universidade de Brasília, Caixa Postal 4478, 70904-970 Brasília, Brazil

D. De Fazio Istituto di Struttura della Materia - CNR, 00016 Roma, Italy ABSTRACT: For the reaction between F and HD, giving HF + D and DF + H, the rate constants, obtained from rigorous quantum scattering calculations at temperatures ranging from 350 K down to 100 K, show deviations from the Arrhenius behavior that have been interpreted in terms of tunneling of either H or D atoms through a potential energy barrier. The interval of temperature investigated extends from above to below a crossover value Tc, a transition temperature separating the moderate and deep quantum tunneling regimes. Below Tc, the rate of the H or D exchange reaction is controlled by the prevalence of tunneling over the thermal activation mechanism. In this temperature range, Bell’s early treatment of quantum tunneling, based on a semiclassical approximation for the barrier permeability, provides a reliable tool to quantitatively account for the contribution of the tunneling effect. This treatment is here applied for extracting from rate constants properties of the effective tunneling path, such as the activation barrier height and width. We show that this is a way of parametrizing the dependence of the apparent activation energy on temperature useful for both calculated and experimental rate constants in an ample interval of temperature, from above to below Tc, relevant for modelization of astrophysical and in general very low-temperature environments.

1. INTRODUCTION

In traditional chemical kinetics, the dependence of rate constants with temperature is described by the Arrhenius’s law, predicting a linear plot of the logarithm of the reaction rate against reciprocal temperature. In the presence of tunneling, the rate coefficient is larger than this law would predict and concave Arrhenius plots are observed. For all chemical reactions where the exchange of light species, such as the proton or the hydrogen atom, represents the rate-determining step, the deviations can be pronounced and dramatically enhance reactivity.9,10

Hydrogen atom transfer processes are ubiquitous in many areas of chemistry, ranging from reactions in space1,2 to the applications of biological interest3 where recently their relevance has been emphasized. Classical approaches, which are often successful in providing qualitative and quantitative understanding of reaction rates, fail to describe deviations observed because of quantum mechanical effects, such as tunneling and resonances (see, for example, the review paper4 and references therein). Although chemical reactivity for any reaction with an energy barrier can be often considered at low enough temperatures a manifestation of quantum mechanical molecular tunneling5 (see also ref 6), it is now well established that tunneling can be relevant even at room temperature7 and under physiological conditions, with important implications for enzymology, chemical catalysis and interstellar chemistry.8 © 2014 American Chemical Society

Special Issue: Franco Gianturco Festschrift Received: April 8, 2014 Revised: June 3, 2014 Published: June 3, 2014 6632

dx.doi.org/10.1021/jp503463w | J. Phys. Chem. A 2014, 118, 6632−6641

The Journal of Physical Chemistry A

Article

The dependence of the apparent activation energy on temperature will be computed from rate constants, obtained from previous rigorous quantum scattering calculation7 including all contributing partial waves and a fine energy grid Boltzmann averaging, in an ample interval of temperatures on two potential energy surfaces (namely the FXZ PES of ref 42 and PES V of ref 7) modeling the dynamics and kinetics of the F + HD reaction. The procedures to carry out quantum scattering calculations and to compute rate constants from the scattering matrix are briefly summarized in section 2.1. Bell’s treatment is described in section 2.2, with an emphasis placed on applying the formulas used in section 3 to the rate constants obtained from quantum close coupling scattering theory and the activation energies calculated for the first time for this system in this paper. Results are presented and discussed in section 3, and conclusions follow in section 4.

A detailed understanding of the tunneling phenomenon in chemical reactions requires quantum or semiclassical mechanics. There exist many ways of calculating the rate coefficients and accounting for tunneling, such as improvements on the transition state theory,11 the instanton theory12 and the ring polymer dynamics.13 For quantum scattering calculations, see, e.g., refs 5 and 14, and for approaches based on the evaluation of the flux autocorrelation functions, see ref 15 (for a summary of these approaches, see ref 16). The extent of the tunneling effect can also be estimated quantitatively by approximate onedimensional methods such as those of Eckart17 and Wigner,18 and Bell’s treatment,19,20 well summarized by Caldin.21 For related approaches, see refs 22 and 23. When the variation of the rate coefficient is analyzed as the temperature decreases, three different regions can be distinguished: the classical region with no tunneling contribution, one with a moderate tunneling, and one with large tunneling contribution.24,25 The separation of the moderate and deep quantum tunneling regimes is often individuated by a characteristic value Tc, known as the crossover temperature.24−26 Below Tc tunneling is the dominant reaction mechanism, whereas at temperatures larger than about 2Tc the tunneling contribution becomes small and the rate is controlled by the thermal activation mechanism. The intermediate temperature range is referred to as that where the tunneling correction is moderate. Typically, approximate methods account quantitatively for tunneling correction only for temperatures ranging in limited intervals. For example, the Wigner correction18 holds close to the top of the potential barrier, whereas the istanton theory applies only below the crossover temperature.27 We consider to be very important to look for uniform treatments whose validity extends over an interval of temperature as large as possible. The reaction of the fluorine atom with the HD molecule, which has been the object of extensive experimental28−35 and theoretical investigations7,34,36,37 ranging from chemical kinetics to crossed molecular beam data, is a good case study to test approximate methods to be used for investigating the kinetics of more complex hydrogen and proton transfer reactions. For this system, reactivity at lower than thermal energies deviates from Arrhenius behavior because of the progressively large role played by quantum mechanical tunneling, as the temperature decreases. The under barrier reactivity of the F + HD reaction and its isotopic variants is also enhanced by the resonance phenomenon.38,39 The case of cold and ultracold temperature regimes, where quantum mechanical resonances dominate the rate behaviors, will be considered elsewhere. In recent papers,9,10 we have shown that the deformed Arrhenius law (d-Arrhenius) proposed in refs 40 and 41 provides a satisfactory description of the quantum tunnel effect on reactivity in the case of small tunneling. However, this treatment becomes less satisfactory when the tunneling is the dominant reactive mechanism. In this paper, to assess the role that specific factors, such as width and shape of the potential energy barrier separating the reactant and product channels, play in controlling the effect of quantum tunnel, an analysis based on Bell’s early treatment19 of the concave Arrhenius plots will be applied. We show that this is a simple and accurate way of describing the dependence of the apparent activation energy on temperature that can be applied to both calculated and experimental rate constants in an ample interval from room to below the crossover temperature Tc.

2. THEORETICAL BACKGROUND 2.1. From Quantum Dynamics to Chemical Kinetics. Reaction rate constants can be calculated at the level of different accuracy, by methods based on classical, semiclassical, or quantum mechanics.16 Among them, quantum reactive scattering calculations provide the most rigorous approach, the results being exact for a given potential energy surface within a given numerical tolerance. The results presented here are based on this last approach (see refs 7, 36, and 43). Having to solve the complete state-to-state reactive scattering problem, these calculations are computationally very demanding and have been therefore limited to systems involving up to four atoms, and some approximations in general have to be introduced for treating more complicated chemical systems.44 The reader interested in the theories of reactive scattering is referred to ref 14 for a recent overview article. Within the framework of the hyper-spherical coordinates approach to the study of quantum reaction dynamics, we solve the time-independent Schrödinger equation for the motion of the three nuclei on an electronically adiabatic potential energy surface.45−48 The hyper-radius, playing the role of a reaction coordinate along the kinetic paths,49 is separated from the other variables and partitioned into several hundred sectors. First, as many eigenvalue problems as the number of sectors are first solved; then a set of coupled equations in the hyper-radial variable is propagated using a log derivative method subject to scattering boundary conditions: this yields the full scattering matrix,50 whose elements are the fundamental quantities linking a reactant initial state to a product final state. The thermal rate constant, k, as a function of the temperature is obtained by summing the state-to-state reaction probabilities, i.e., the square moduli of the scattering matrix elements, over the reactant states (properly weighted to take into account their relative population) and product states and next by averaging over the Maxwell−Boltzmann distribution of the initial translational energy. The procedure and the formulas used to calculate the thermal rate constant from the scattering matrix elements have already been given in ref 5 and for sake of brevity will not be repeated here. For most of chemical reactions, the variation of the rate constant with the temperature follows the Arrhenius law, traditionally written as k(T ) = A exp( −Ea /RT ) 6633

(1)



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where the parameters Ea and A are called the activation energy and the preexponential factor, respectively. Arrhenius law predicts a linear dependence of the logarithm of the reaction rate against reciprocal temperature, with −Ea/R being the slope of the straight line. However, nonlinear Arrhenius curves have been repeatedly observed. Reactions that proceed faster at low T than would be expected on the basis of the Arrhenius law exhibit a concave plot. Specifically, if there is tunneling, the apparent activation energy Ea = −R

d ln k (d 1/T )

(2)

deviates from constancy causing a concave curvature of the Arrhenius plot. On the other hand, when the observed reaction rate is lower than expected, the Arrhenius plot is convex. As discussed in ref 51, the reactions that show convex Arrhenius plots are considered rare. A differential measure of the curvature plot is sometimes referred to as LaMer’s heat capacity of activation.52 2.2. Bell’s Formulas. Bell’s seminal papers19,20 on the quantum mechanical tunneling in chemistry (see also ref 24) have inspired several subsequent theoretical studies; see, for example, refs 53−55. In his earlier paper,19 an expression is provided for the quantum tunneling correcting factor based on the Wentzel, Kramers, and Brillouin (WKB) semiclassical treatment for the barrier permeability, whereas in the later one,20 Bell obtains an expression that is exact for a parabolic barrier. To estimate the correcting factor accounting for the contribution of the quantum tunneling effect in chemical kinetics, we use both of Bell’s treatments.19,20 The main formulas used in the analysis of the rate constants are briefly sketched in the following. The simplest way to evaluate a tunneling correcting factor is for a parabolic barrier potential for motion along the effective reaction path. We consider an asymmetrical truncated parabolic barrier V (x ) =

B⎛ x 2 ⎞ ΔH ⎛⎜ x⎞ 1+ ⎟ ⎜1 − 2 ⎟ + ⎝ a⎠ 4⎝ 2 a ⎠

Figure 1. Schematic profile of an asymmetrical truncated parabolic barrier, showing the tunneling path for an exothermic reaction and illustrating the parameters appearing in eq 3.

where ℏ is the Planck’s constant divided by 2π and ν* is the magnitude of the barrier imaginary frequency. This expression is valid when the energy is smaller than the height of the barrier, whereas, when E ≥ V*, it is assumed G = 1. This approximate treatment has the advantage of leading to the following simple expression Q=

−a≤x≤a

ν* =

where ΔH is the reaction enthalpy and B = ( V* +

V * − ΔH )

∫0

∞

G (E ) exp[(V * − E)/RT ] dE RT

(4)

w=

(7)

1 2πa

B 2m

(8)

2 πν*

V* 2m

(9)

and, when the reaction is thermo-neutral, the expression w = 2a is recovered. With an Arrhenius-type law for the classical rate coefficient, it follows that ⎡ RT exp( −V */RT ) − ℏν* exp(−V */ℏν*) ⎤ k(T ) = A⎢ ⎥ ⎣ ⎦ RT − ℏν* (10)

(5)

where A is the pre-exponential factor assumed here to be a constant parameter. This formula bridges the low- and hightemperature limits of the rate of a chemical reaction. Toward the absolute zero, T → 0, eq 10 leads correctly to a nonvanishing and temperature independent expression

where G is the probability of a particle passing through a barrier of height V*. In the earlier formulation,19 use was made of the WKB approach to evaluate the tunneling probabilities G from barrier penetration integrals. In the particular case of a parabolic barrier, G = exp[(E − V *)/ℏν*]

RT − ℏν*

with m being the mass of the tunneling particle. The width of the barrier, w, at the lowest tunneling energy is

with V* and V* − ΔH being the barrier heights for direct and reverse reactions, respectively. The case of an exothermic reaction is illustrated in Figure 1. The parabolic barrier is more convenient than other types of barriers for kinetic calculations because the tunneling correcting factor takes a very simple analytic expression. In Bell’s approach,19,56 the rate constant is obtained multiplying the Arrhenius rate by a tunneling correcting factor Q=

( RT1 − ℏν1 * )⎤⎦

for the dependence of the quantum tunneling correcting factor on the temperature, with V* and ν* acting as parameters. For a given reaction, the barrier frequency, ν*, is related to the parameters a and B in eq 3 as follows

(3)

2

RT − ℏν* exp⎡⎣V *

k T → 0 = A exp( −V */ℏν*)

(6)

(11)

or alternatively 6634


The Journal of Physical Chemistry A ⎡ ⎤ −2πa 2m V * k T → 0 = A exp⎢ ⎥ ⎣ ℏ( V * + V * − ΔH ) ⎦

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b= (12)

However, to evaluate the accuracy of the physical parameters obtained, we have also used the expressions of Bell’s later treatment. In the case of eq 15 also a very simple formula for Ea can be obtained:

whereas at high temperature the Arrhenius rate expression (1) is recovered. The temperature at which the two are equal Tc =

ℏν* R

(13)

Ea = V * + 1/RT + π ℏν* cot(π ℏν*/RT )

25

is the crossover temperature, delimiting the deep (T < Tc) and moderate (Tc ≲ T ≲ 2 Tc) tunneling regimes. According to ref 26, at the temperature T = 2Tc, the tunneling and overbarrier mechanisms play the same role. In his later treatment,20 Bell obtains an expression for Q that is exact for a truncated parabolic barrier: Q=

u exp(α − β) πu − F(1,1−u ,2−u ;−exp( −β)) sin(πu) 1−u

where u = ℏv*/RT = Tc/T, α = V*/RT, β = V*/ℏv*, and F denotes a hyper-geometric function that can be expressed as a converging series. Well above the crossover temperature Tc all terms of the series can be neglected, and the tunneling correcting factor takes the following very simple expression

πu sin(πu)

1/ d ⎡ ⎛ ε ⎞ ε ⎤ ⎟ ≡ A 1 − d k(T ) = A expd ⎜ ⎢ ⎥ ⎣ ⎝ RT ⎠ RT ⎦

(19)

where the deformation parameter d, the pre-exponential factor A, and the energy ε are phenomenological quantities independent of temperature. In particular, ε and d are related to the apparent activation energy by

(15)

1 1 d ≡ − Ea ε RT

widely used in the literature; see, for example, refs 21, 24, and 22. However, as discussed in Appendix C of ref 24, eq 15 diverges when u is an integer number, i.e., when Tc is an integer multiple of T. More accurate expressions are obtained by retaining the first or additional terms of the hyper-geometric series21,23 that avoid the singularities. To minimize inaccuracies in the calculations, it is advisable to use these approximate expressions only far from these values. However, when the crossover temperature is not known a priori, it is not clear the range of temperature where eq 15 can be safely used to fit the data, and/or how many terms of the hyper-geometric series must be retained to avoid divergent behaviors. Retaining for example just the first term, we have the following expression for k:

(20)

and in the limit d → 0, when the expd function coincides with the exponential function, ε is equal to the activation energy, Ea. Taking the natural logarithm of k(T) in eq 19, we obtain the following expression ln k(T ) = ln A +

ε ⎟⎞ 1 ⎛⎜ ln 1 − d ⎝ d RT ⎠

(21)

which has been used in a previous work to fit the rate constants of the F + H2 reaction. In ref 10, it is shown that expanding 1/Ea obtained by eq 18 in a power series of 1/RT and retaining only the first linear term of the series expansion make it possible to establish an approximate connection of the ε and d parameters with V* and ν* 9

⎡ π exp( −V */RT ) exp( −V */ℏν*) ⎤ k(T ) = Aℏν*⎢ − ⎥ ⎣ RT sin(π ℏν*/RT ) RT − ℏν* ⎦ (16)

ε ≈ V*

where only the singularity at T = Tc is removed. In the present paper, to provide a general functional form as simple as possible to describe experimental or theoretical data, we have preferred to use the expression of the tunneling correcting factor given in eq 7, which is valid for all values of T for which tunneling is relevant, as shown, e.g., in ref 21 and confirmed by the results presented in section 3. As shown in ref 26, eq 7 represents a very satisfactory approximation below the crossover temperature Tc and leads to a simple expression for the apparent activation energy, eq 2. In fact, using the quantum mechanical expression of k given in eq 10 b + 1 − eb Ea = V * b(1 − (Tc/T )eb)

(18)

whereas applying eq 2 to eq 16 gives an expression too complicated to be given here explicitly but is shown in plots to be presented later. In section 3, the rate coefficients of the F + HD reaction calculated by quantum scattering calculations (see the previous subsection) will be compared to eqs 10 and 16 and the fitting parameters will be plugged in the previous equations to calculate the related apparent activation energies. 2.3. Phenomenological Description of the Rate Constants. In refs 9 and 10 we have used a deformed exponential function, expd, to generalize the Arrhenius law

(14)

Q=

V* ⎛ 1 1⎞ ⎜ − ⎟ R ⎝T Tc ⎠

d≈−

1 ⎛ hν* ⎞ ⎜ ⎟ 12 ⎝ V * ⎠

2

(22)

It is interesting to compare the limiting behaviors at high and low T of Ea obtained from eq 17 with those obtained from eq 20. At low enough T such that eb ≫ 1 the dependence of the reciprocal of apparent activation energy from eq 17 is

1 1 ℏν* =− + Ea RT (RT )2

(23)

involving the square of the reciprocal of the temperature. Otherwise, in the limit of high temperatures such that eb ≪ 1 (17)

1 ℏν* − RT 1 = ≈ Ea V *(ℏν* − RT ) + ℏν*RT V * − ℏν*

where 6635

(24)



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Comparison with eq 20 shows that both limiting behaviors differ in the two cases, suggesting relevant differences when the data are fit at very low and high temperatures with the two approaches.

Table 1. Values of the Fitting Parameters V*, hν*, and A Appearing in Eq 10 and Related Quantities A and Tc (Eqs 9 and 13, Respectively) for the Production of the HF and DF Molecules on the FXZ and PESV Potential Energy Surfacesa

3. A CASE STUDY: THE F + HD REACTION The quantum tunnel permeability depends mainly on masses of the involved atoms, and on height, width and shape of the activation barrier along the tunneling path. So, in chemical reactions the tunneling process is most relevant at low temperature for light atoms, such as hydrogen and deuterium. In this section, we use the treatment sketched in section 2.2 to analyze the relevance of these factors in accounting for the tunneling kinetics in the F + HD reaction. In particular, we will obtain interesting information on the effects of the barrier width: as the temperature decreases, this parameter becomes the decisive factor for the kinetic control. To this purpose, we consider the exchange reactions leading to the production of both the HF and the DF molecules on two potential energy surfaces, namely, the FXZ PES and PES V. The FXZ PES, recently proposed in the literature,42 has been obtained by ab initio quantum chemistry calculations, whereas PES V has been developed by us on the basis of modifications at the entrance channel of the FXZ PES (for details, see ref 7). A phenomenological entrance channel,57 taking into account the open-shell nature and the spin−orbit effects of the fluorine atom,46 was merged to the FXZ PES in a way so to leave unmodified the height of the barrier; for details on the merging procedure, see refs 5 and 47 and Figure 1 of ref 43. For the two surfaces taken into account, the energy profile along the reaction coordinate shows the same classical barrier (i.e., without zero point energy) of relatively low height, 1.77 kcal/ mol.7 The entrance channel features of the FXZ PES and PESV at perpendicular configuration have been compared explicitly in Figure 2 of ref 7: the van der Waals well for PES V is shallower and located at larger internuclear distance than for the FXZ PES. Correspondingly, the barrier in PES V is wider than that in the FXZ PES. The reaction dynamics and kinetics of the F + HD reactions on the two PESs have been investigated in a previous paper.7 Rigorous quantum scattering calculations have been carried out on a very fine grid of collision energies using a parallelized version of the ABC code;50 for the computational details we refer the interested reader to ref 7. In the range 100−350 K, 251 values of the rate constants for the H-transfer and Dtransfer reactions have been obtained summing the reaction probabilities over all the contributing partial waves and averaging over the Maxwell−Boltzmann distribution of the initial translational energies at a given temperature, as described in section 2.1. The accuracy of the rates was estimated to be around 1%.7 Concave Arrhenius plots have been observed indicating that the quantum tunneling effect plays a decisive role at the investigated temperatures. In the present paper, the rate constants of ref 7 have been fitted to eqs 10, 19, and 16 varying simultaneously all the parameters. The three-parameter best fits have been made using the “nonlinear curve fitting” resource as implemented in the free package GRACE (Graphing, Advanced Computation and Exploration of data). The values of the parameters obtained from each fit are listed respectively in Tables 1−3 for both PESs and for both reactive channels, together with the maximum and the mean percent differences between fitted values and close coupling data.

PES FXZ PES PES V FXZ PES PES V

V*

hν*

mean error (%)

A

HF Channel 5.82 × 10−11 0.26

1.60

778

1.45

519

1.56

459

0.40 5.50 × 10−11 DF Channel 4.36 × 10−11 0.45

1.48

367

4.06 × 10−11

0.42

max error (%)

w

Tc

3.4

0.50

178

1.8

0.70

119

7.2

0.58

105

1.9

0.71

84

Mean and maximum percent error of the fits are also reported. V* in kcal/mol, hν* in cm−1, A in cm3 molecule−1 s−1, w in Å, and Tc in K. a

Table 2. Values of the Fitting Parameters ε, d, and A Appearing in Eq 19, and Mean and Maximum Percent Error of the Fits, for the Production of the HF and the DF Molecules on the FXZ and PESV Potential Energy Surfacesa PES FXZ PES PES V FXZ PES PES V a

ε

d

5.08

−0.485

1.58

−0.171

1.58

−0.0993

1.40

−0.0474

mean error (%)

max error (%)

1.46

11.1

7.07 × 10−11 DF Channel 5.16 × 10−11

1.67

17.8

1.14

13.0

4.30 × 10−11

1.25

13.1

A HF Channel 2.56 × 10−10

ε in kcal/mol and A in cm molecule−1 s−1. 3

Table 3. As in Table 1 for the Fitting Formula in Eq 16a PES FXZ PES PES V FXZ PES PES V

V*

hν*

A

mean error (%)

HF Channel 6.70 × 10−11 0.25

1.44

656

1.30

431

1.42

384

5.88 × 10−11 0.74 DF Channel 4.61 × 10−11 0.24

1.36

294

4.18 × 10−11

0.77

max error (%)

w

Tc

0.8

0.56

150

6.0

0.81

88

1.0

0.66

99

6.2

0.84

67

V* in kcal/mol, hν* in cm−1, A in cm3 molecule−1 s−1, w in Å, and Tc in K. a

From the data reported, one can note that all the three formulas fit satisfactory well the close coupling data in the range of temperatures studied with a mean percent error within or close to the numerical accuracy of the data. Nevertheless, it is evident that eqs 10 and 16 reproduce much better the data (a factor near to five) with respect to eq 19; also the maximum error is lower (within 3−4%); additionally, the fitting parameters obtained appear to exhibit in all cases more physical appeal when comparison is made with the features of the PES investigated and summarized above. Less satisfactory are the parameters obtained fitting the data with eq 19 (Table 2), where the largest percent errors are above 10% and the ε obtained is for the case of the HF channel FXZ PES extremely large with respect to the classical barrier. All the found 6636



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Figure 2. Arrhenius plots for the production of HF (higher panels) and DF (lower panels) molecules. Filled circles indicate the results obtained from close coupling quantum scattering calculations7 employing FXZ PES (left panels) and PES V (right panels), whereas lines are obtained by inserting the values of the parameters given in Tables 1−3 in the formulas of section 2: red solid lines for Bell’s 1935 formula, eq 10; dashed blue lines for the d-Arrhenius formula, eq 19; green solid line for Bell’s 1959 formula, eq 16. For graphical reasons just one point out of each ten is reported for the close coupling data. The dot-dashed black lines are the classical Arrhenius results and the zero temperature limits of eq 11 obtained by applying the parameters of Table 1 (Ea = V*): the point where the two lines cross is the crossover temperature Tc also reported into the panels.

chooses to cut corners and takes a shorter way with a higher barrier. A possible physical reason that could explain these numerical evidence is that in PES V the long-range forces and the entrance channel van der Waals well (in perpendicular configuration) located at larger internuclear distances permit us to more efficiently drive the system to cross the lower bent transition state. The extent of “corner cutting” is also influenced on the reaction skew angle, which is the kinematic angle between the reactant and product valleys and depends only on masses; see, for example, ref 60. For the H-transfer reaction the skew angle is 37.3° whereas for the D-transfer reaction it is 56.7°: the larger the skew angle, the smaller is the curvature of the tunneling path.59 The smaller curvature favors the corner cutting for the production of the HF molecule on PES FXZ (w(HF) < w(DF) and V*(HF) > V*(DF)). These conclusions are also in agreement with our previous stereodynamical studies61 of this reaction, where a comparison between stereodirected62 matrix elements of the two channels clearly indicates a major propensity of the DF channel to cross the lower bent transition state of the system. Plots of ln k, for the production of both HF and DF, versus the inverse of absolute temperature are shown in the upper and lower panels of Figure 2, respectively. Also reported in the panels are the crossover temperatures Tc obtained applying eq 13. From Figure 2, we see that, when the temperature is larger than 2Tc the Arrhenius plots are strictly linear, indicating that the reactions proceed mainly via the thermal activation mechanism with a small contribution from tunneling. We note that this is the case of the DF channels at room temperature in both the PESs employed, whereas for the HF channel a significant tunneling effect is present also at room

imaginary barrier frequencies reflect the expected values on the basis of their physical meaning (sections 2.2 and 2.3) with ν*(HF) > ν*(DF) and ν*(FXZ PES) > ν*(PES-V), accordingly to differences in widths between the two PESs (see above). For the fitting using eq 10, the values obtained for the pre-exponential factor A depend on the isotopic substitution but are nearly independent (within about 5%) of the PES used. The comparison between the data of Tables 1 and 3 shows that eq 10, notwithstanding its simplicity, is able to give physical parameters within around 10%, indicating that the WKB approximation only slightly affects the quality of the results, when employed to fit low-temperature data. The analysis of the differences among the values shown in Table 1 supplies interesting mechanistic information. In fact, the differences in the entrance channel region between PES V and FXZ PES and the effect of the isotopic substitution directly influence the choice of the effective tunneling path. Among all available paths, the reaction follows the one whose features are a compromise between the height and width of the barrier.58,59 As far as PES V is concerned, the results show that V*(DF) is slightly larger than V*(HF) and the ratio ν*(HF)/ν*(DF) ≈ (mD/mH)1/2, so that w(DF) ≈ w(HF). Otherwise, for the FXZ PES, V*(HF) is slightly higher than V*(DF) and both values are larger than those obtained for PES V (a result supported also by the data in Tables 2 and 3). The situation is also different for the comparison of the barrier frequencies, where the ratio ν*(HF)/ν*(DF) has a value of 1.70 (w(DF) > w(HF)), compared to the value of (mD/mH)1/2 ≈ 1.41, expected on the basis of the isotopic substitution. These data suggest that on PES V the reacting system follows a longer path with a lower barrier whereas on the FXZ PES the system 6637



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Figure 3. Apparent activation energies (eq 2) in kcal/mol for the production of the HF (higher panels) and DF (lower panels) molecules as a function of the absolute temperature. Line styles and colors are as in Figure 2: red solid lines for Bell’s early treatment, eq 17; dashed blue lines for the d-Arrhenius treatment, eq 20; green solid lines for Bell’s later treatment (details in the main text). The optimized fitting parameters of Tables 1−3 are used in each respective formula.

Figure 4. As in Figure 3, but in a larger range of temperatures and with alternative parametrizations. Line styles and colors are as in Figure 3: red solid lines for Bell’s early treatment, eq 17; dashed blue lines for the d-Arrhenius treatment, eq 20; green solid lines for Bell’s later treatment (details in the main text). For all the curves, the same parameters of Table 3 estimated as those of highest accuracy are employed to exhibit the limiting behavior (section 2). Also reported in the panels, as dot-dashed black and red lines, are the high-temperature asymptotic limits given by eqs 20 and 24, respectively.

range mostly covered in our present study except for the HF channel of the FXZ PES, where the crossover temperature is significantly larger than the minimum temperature here studied (100 K). As the temperature decreases below the crossover

temperature in the case of the FXZ PES. Deviations from the Arrhenius law are more evident in the temperature range between 2Tc and Tc, where tunneling becomes competitive with the classical, over barrier reaction pathway. This is the 6638



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most cases satisfactory results at thermal and subthermal temperatures.

values, quantum tunneling becomes the dominant reaction mechanism and the rates exhibit an attenuated dependence on temperature. In this region of deep tunneling we can observe the largest deviations between “exact” and fitted values, especially for the fits employing eq 19. As discussed at the end of section 2.3, this is an expected result due to the different low-temperature limit behaviors (eqs 20 and 23) of the two approaches. Extrapolating the least-square fit curves, we can observe that, although the reaction rates extrapolated by eqs 10 and 16 seem to approach the physical correct constant behavior imposed by the Wigner’s law (see, e.g., ref 5), this is not the case for the rates extrapolated by eq 19, suggesting that the dArrhenius approach can be hardly applied to fit the behaviors of rates in the deep tunneling region. We can also note some relevant differences between the extrapolated trends of eqs 10 and 16, with the latter giving rates slightly lower: this could seem surprising, because the WKB approximation should be more accurate in the low collision energy range; we will go back to the origin of these differences later, when considering Figure 4. Calculations of the rate constants below 100 K are in progress and the results will be shown elsewhere. An “exact” dependence of the apparent activation energy, Ea, on temperature has been calculated as the negative of the logarithmic derivative of the close coupling rates with respect to the inverse of absolute temperature, eq 2. To obtain a smooth behavior for the activation energy dependences, the logarithm of the rate constants has been expanded in a polynomial series and the first derivatives have been calculated analytically: specifically, a polynomial of order six has been employed to reproduce the original data within numerical errors (maximum and mean difference 0.5% and 0.05% respectively). In Figure 3, we compare the results of these calculations with values obtained inserting data of Tables 1−3 into the formulas of sections 2.2 and 2.3 in the range 100 < T < 350 K. As can be seen by comparison of Figures 2 and 3, the apparent activation energies are much more sensitive than rate constants to the quality of the fitting functions, so that larger differences among the various sets of data are observed in Figure 3. In particular eq 20 hardly describes the concavity of the plots in the full range of temperatures and the higher accuracy of Bell’s formulas is now evident in all cases studied. The apparent activation energies are larger in the full range of T for the DF cases in both PESs and nearly constant for temperatures larger than 2Tc. Moreover the apparent activation energies are consistently larger in PES V than in FXZ PES, confirming the important role of the width of the barrier in controlling the quantum mechanical permeability of the barrier. To better understand the insight coming from the results, we found it convenient to compare in Figure 4 the various approaches by using in all the corresponding formulas the values of A, hν*, and V* from Table 3, estimated to be the most physically meaningful. Also, a wider temperature range is shown to illustrate both the low- and high-T behavior. It is interesting to note that, except for the PES-V DF case, the high-T asymptotic limit (eq 24) is far to be reached, demonstrating that the tunneling effect is relevant in the full temperature range considered. It can also be seen that Bell’s early treatment (eq 17) satisfactorily accounts for the apparent activation energy, particularly in the deep tunneling regime, and deviates from the later treatment at higher T, because it neglects the quantum over barrier reflection. The performances of the d-Arrhenius formulation appear to run in the opposite direction, giving in

4. CONCLUSIONS In this work, we have tested and compared the abilities of different approaches to describe rate constants for the F + HD reaction at the borderline between moderate and deep tunneling regimes. We limited our attention to formulations involving only one parameter more than those in the Arrhenius law to suggest simple and accurate parametrizations of experimental and/or theoretical kinetic data for use in modeling. To investigate the role of tunneling, we have used recent accurate quantum close coupling scattering results,7 obtained by employing two PESs differing only by the entrance channel of the reaction, having the same barrier heights but different shapes. The comparison between the reaction rates on the two PESs provides information on the role that the variation of the barrier width can play on the tunneling rates. The different approaches studied are a phenomenological one recently proposed9,10 and two older ones based on a parabolic approximation of the reaction barrier originally proposed by Bell in refs 19 and 20. As discussed previously,40,41 the advantage of the phenomenological method is its versatility to describe the rate constant behaviors irrespective of the dominant reaction mechanism, exhibiting either sub-Arrhenius (concave Arrhenius plot) or super-Arrhenius (convex Arrhenius plot) behavior. In contrast, the Bell approaches apply only to reactions with a barrier for which tunneling plays a relevant role. Regarding the definition of the quantum tunneling regimes, relevant is the concept of crossover temperature, directly connected to the imaginary frequency of the barrier (eq 13) and naturally obtained by the Bell early approach. For this system the results indicate that the investigated temperature range (100−350 K) belongs mainly to the moderate tunneling regime. In this regime all the approaches describe satisfactorily the rate constant dependence on temperature (Tables 1−3 and Figure 2), although the results using the Bell formulas are more accurate and physically motivated. A detailed analysis of the obtained parameters provides information on tunneling paths and gives mechanistic insight, confirming our previous stereodynamical analysis61,63 performed on the same system on the same potential energy surfaces. We have also calculated “exact” apparent activation energies from the quantum close coupling results and compared them with those obtained by the investigated formulas. The comparison has shown that the activation energies are much more sensitive than rate constants with respect to the accuracy of the approach employed. In particular, the values found using the fitting parameters of eq 10 allow us to obtain an excellent estimate of the dependence of the activation energy on temperature (Figure 3) and a good estimate (within about 10%) of the features of the barriers. In the range of temperature investigated, minor discrepancies have been observed only at the highest and at the lowest temperature values. As seen in the discussion of Figure 4, this is due to the neglect of the over barrier reflection, affecting the high-temperature behavior. The elimination of this deficiency20 introduces singularities and complexity in the analytical form that contrast with the requirement of simplicity and stability of the fitting functions searched. In general, Bell’s venerable approach turns out as very useful for extending the study of reactivity at very low temperatures, 6639



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describing a dependence of rate constants in agreement with Wigner threshold law. We also note the reliability of the proposed fitting formulas in spite of the simplified onedimensional models and the parabolic approximation on which they are based. As also documented in a recent comparison with the often used Koij−Arrhenius formula for different classes of reactions,64 this work promotes the use of simple and more accurate alternatives, especially for the astrophysical models involving low-temperature environments.

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AUTHOR INFORMATION

Corresponding Author

*S. Cavalli: e-mail, [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS S.C. and D.D.F. acknowledge the Italian MIUR for financial support by PRIN 2010/2011 grant N. 2010ERFKXL. Computational time has been supplied by CINECA (Bologna) under ISCRA project N. HP10CJX3D4. V.A. is grateful to CAPES for supporting him as Pesquisador Visitante Especial at Universidade Federal da Bahia.

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REFERENCES

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