J. Phys. Chem. A 2010, 114, 9749–9754
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˜ 2B1 ˜ 2A1 and X Renner-Teller Quantum Dynamics of NH(a1∆) + H Reactions on the NH2 A † Coupled Surfaces P. Defazio,*,‡ P. Gamallo,§ M. Gonza´lez,§ and C. Petrongolo‡,| Dipartimento di Chimica, UniVersita` di Siena, Via A. Moro 2, I-53100 Siena, Italy, Departament de Quı´mica Fı´sica i IQTC, UniVersitat de Barcelona, C/Martı´ i Franque´s 1, 08028, Barcelona, Spain, and Istituto per i Processi Chimico-Fisici del CNR, Via G. Moruzzi 1, I-56100 Pisa, Italy ReceiVed: March 8, 2010; ReVised Manuscript ReceiVed: April 22, 2010
Four reactions NH(a1∆) + H′(2S) are investigated by the quantum mechanical real wavepacket method, taking into account nonadiabatic Renner-Teller (RT) and rovibronic Coriolis couplings between the involved states. We consider depletion (d) to N(2D) + H2(X1Σ+g ), exchange (e) to NH′(a1∆) + H(2S), quenching (q) to NH(X3Σ-) + H′(2S), and exchange-quenching (eq) to NH′(X3Σ-) + H(2S). We extend our RT theory to a general AB + C collision using a geometry-dependent but very simple and empirical RT matrix element. Reaction probabilities, cross sections, and rate constants are presented, and RT results are compared with Born-Oppenheimer (BO), experimental, and semiclassical data. The nonadiabatic couplings open two new channels, (q) and (eq), and increase the (d) and (e) reactivity with respect to the BO one, when NH(a1∆) is rotationally excited. In this case, the quantum cross sections are larger than the semiclassical ones at low collision energies. The calculated rate constants at 300 K are k(d) ) 3.06, k(e) ) 3.32, k(q) ) 1.44, and k(eq) ) 1.70 in 10-11 cm3 s-1 compared with the measured values k(d) ) (3.2 ( 1.7), k(q + eq) ) (1.7 ( 0.3), and k(total) ) (4.8 ( 1.7). The theoretical depletion rate is thus in good agreement with the experimental value, but the quenching and total rates are overestimated, because the present RT couplings are too large. This discrepancy is probably due to our simple and empirical RT matrix element. 1. Introduction The nonadiabatic Renner-Teller (RT) effect in triatomic systems was theoretically predicted in 1934 by Renner1 and experimentally observed in 1959 by Dressler and Ramsay2 in ˜ 2B1 absorption spectrum of the NH2 radical. In ˜ 2A1 r X the A ˜ 2A1 are the degenerate ˜ 2B1 and A fact, the adiabatic states X 2 components of a linear Πu electronic species and are coupled ˜ 2A1〉 matrix element, where LˆHH is the ˜ 2B1|LˆHH|A to the 〈X component of the electronic angular momentum along the H-H axis. Because this matrix element is multiplied by a reducedmass factor that diverges at linearity, these states interact very strongly near a linear molecular configuration, and the Born-Oppenheimer (BO) approximation thus breaks down. After this first experiment work,2 the RT spectroscopy of NH2 was extensively investigated by many authors, both experimentally3 and theoretically,4-8 making the NH2 radical a prototypical example of RT effects in spectroscopy. From 1990, kinetic and dynamical properties of reactions of N(4S)9,10 and N(2D)11 with H2, and reactions of NH(X3Σ-)12,13 and NH(a1∆)14 with H were measured and calculated, including also their deuterated variants. The correlation diagram13 of Figure 1 shows that reactants and products of these reactions ˜ 2A′′ and A ˜ 2A′ species, whose correlate through the NH2 X potential energy surfaces (PESs) were calculated by means of multireference configuration-interaction methods.13,15-18 Recently, some authors have also investigated RT effects in chemical reactions involving NXY, where X and Y are H or D †
Part of the “Reinhard Schinke Festschrift”. * To whom correspondence should be addressed. E-mail:
[email protected]. ‡ Universita` di Siena. § Universitat de Barcelona. | Istituto per i Processi Chimico-Fisici del CNR.
atoms. Santoro et al.19 and Defazio and Petrongolo20 presented the RT dynamics of the reaction N(2D) + H2(X1Σg+), using semiclassical trajectory surface hopping (TSH) and quantum mechanical (QM) wavepacket (WP) methods, respectively. Then, in 2007, Adam et al.14 were the first to observe and measure RT effects in chemical dynamics, namely, in the reactions NX(a1∆) + Y(2S), where NX was in the ground vibrational state. In particular, they studied the following four processes for all possible choices of the X and Y atoms
NX(a1∆) + Y f N(2D) + XY,
(1)
NX depletion (d)
f NY(a1∆) + X, f NX(X3Σ-) + Y,
(2)
X-Y exchange (e)
(3)
NX quenching (q)
3 -
f NY(X Σ ) + X, X-Y exchange and NY quenching
(eq)
(4)
˜ 2A′ Figure 1 shows that reactions 3 and 4 start on the excited A 2 ˜ surface and finish on the ground X A′′ surface. Therefore, the (q) and (eq) channels are fully nonadiabatic, being closed within ˜ 2A′ PES and opened only by the BO approximation on the A ˜ 2A′ - X ˜ 2A′′ RT interactions. Performing laser-induced the A fluorescence experiments and TSH calculations, Adam et al.14 observed and calculated the nonadiabatic (q) and (eq) kinetics and dynamics induced by RT effects, showing, for example, that the (q) + (eq) contribution to the overall rate constant is equal to ∼35% at 300 K. In their TSH calculations, Adam et al.14 employed highly ˜ 2A′′ and A ˜ 2A′ PESs13 and adjusted a single RT accurate X
10.1021/jp102079n 2010 American Chemical Society Published on Web 05/07/2010
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Defazio et al. The Lˆz matrix element on the electronic basis is equal to
〈σ′|Lˆz |σ〉 ) δσ′,-σσiLσσ cos R
(6)
where Lσ′σ is a 3D and real function of the nuclear coordinates and cos R projects the matrix element from the X-Y direction to the z one. Using this equation and approximating the Lˆ2z matrix elements by δσ′σL2σ′σ cos2 R, the electronic, angular, and rotational ˆ is equal to representation of H
Figure 1. Adiabatic correlation diagram of the NH2 electronic states ˜ 2A′′ and A ˜ 2A′ from ref 14. Potential energies in electronvolts with X respect to NH(a1∆) + H.
empirical parameter to reproduce the observed NH(a1∆) + H total quenching rate. Using this parameter also for other isotopic combinations, they obtained a satisfactory agreement with the majority of experimental data. Akpinar et al.21 subsequently studied the BO dynamics of the reactions NH(a1∆) + H′ by a WP method that ignored the RT coupling and thus the (q) and (eq) RT processes. The (d) WP rate constant agreed well with the experimental and TSH data, after subtraction of the quenching rates from the total ones. In the present work, we extend the previous investigations14,21 and consider the collision NH(a1∆) + H, using a QM WP ˜ 2A′′ ˜ 2A′ - X method, taking explicitly into account the A 2 2 ˜ ˜ nonadiabatic RT coupling and using the X A′′ and A A′ PESs of ref 13, here called VX and VA, respectively. We follow our theory of RT effects in reaction dynamics20,22,23 and perform WP coupled channel (CC) calculations. However, we employ here a nuclear-geometry-dependent RT coupling that is partially adjusted for reproducing some of the observed rate constants.14 In fact, the constant value of previous works20,22,23 strongly overestimates the RT interactions, giving for (q) and (eq) much larger cross sections and rate constants than the experimental ones, as was found in ref 14 when the empirical parameter was not considered. This article is organized as follows: the theory and some computational aspects are described in Section 2, reaction probabilities, cross sections, and thermal rate constants are discussed in Section 3, and the main conclusions are presented in Section 4. 2. Theory and Calculations As in ref 23, for NH + H, we also employ reactant Jacobi coordinates R, r, and γ, a body- fixed reference frame with the ˆ that z axis along R, and a spinless molecular Hamiltonian H ˆ angular momentum contains both the total Jˆ and the electronic L ˜ 2A′ electronic states with ˜ 2A′′ and A operators. Let |σ〉 be the X reflection parity σ ) -1 with respect to the nuclear plane, respectively, |jK〉 be associated Legendre states, where K g 0 is the eigenvalue of Jˆz, and |Kσp〉 be symmetry-adapted Wigner states, where p ) (1 is the total parity. If B(R) ) (2µRR2)-1 and b(r) ) (2µrr2)-1, where µR and µr are reduced masses associated with R and r, respectively, then the RT coupling is due to the rovibronic operator
B(R) + b(r) ˆ ˆ JzLz TˆRT ) sin2 γ
(5)
which couples the vibrational motion to the total and electronic rotations.
ˆ |Kσp〉|jK〉|σ〉 〈σ′|〈j′K′|〈K′σ′p|H )δσ′σ{δK′K[δj′j(Tˆrad + (B + b)j(j + 1) + B(J(J + 1) - 2K2)) B cos2 γ + b 2 Lσσ cos2 R + Vσ |jK〉] sin2 γ + + - -δj′jB[δK′,K+1(1 + δK0)1/2λJK λjK + δK′,K-1(1 + δK1)1/2λJK λjK]} +〈j′K|
B cos2 γ + b Lσσ cos R|jK〉 sin2 γ cos γ + L cos R|jK〉 +δK′,K+1(1 + δK0)1/2BλJK 〈j′, K + 1| sin γ σσ cos γ +δK′,K-1(1 + δK1)1/2BλJK L cos R|jK〉] 〈j′,K - 1| sin γ σσ -δσ′,-σ[δKK2K〈j′K|
(7) where Tˆrad is the radial Hamiltonian, Vσ is the PES of the |σ〉 ( ) [l(l + 1) - m(m ( 1)]1/2. This expression state, and λlm extends those of refs 22 and 23 to a general AB + C collision with a geometry-dependent RT coupling. When the electronic states are equal, σ′ ) σ, note the effective potentials of the second term
Vσeff )
B cos2 γ + b 2 Lσ′σ cos2 R + Vσ 2 sin γ
(8)
which depend on the matrix element of Lˆz2 and thus generalize the usual BO PESs Vσ. When the electronic states are different, σ′ ) -σ, the last three terms of eq 7 describe the RT couplings: the first is K-diagonal and the other two are Coriolis couplings. When J ) 0, we have a generalized BO case because only the first two terms of eq 7 survive. As far as we know, Lσ′σ was very rarely calculated.24 In our previous work,23 we indeed approximated everywhere Lσ′σ by its linear value, Λ ) 1, where Λ is the electronic angular momentum quantum number. This approximation works quite well for N(2D) + H2, but it strongly overestimates the RT coupling for NH(a1∆) + H. In fact, WP snapshots23 of the reaction N(2D) + H2 show that RT jumps between the two PESs occur essentially inside the deep wells of both surfaces, where the approximation Lσ′σ ≈ Λ is accurate. On the other hand, the large exothermicity of the quenching process NH(a1∆) + H f NH(X3Σ-) + H gives rise to unphysical RT jumps from the ˜ 2A′′ PES if Lσ′σ ) Λ ˜ 2A′ PES to the ground X excited A everywhere. We then obtain very large RT effects, as Adam et al.14 also found in their TSH study without the empirical parameter. This implies that the RT coupling for the NH(a1∆) + H reaction must vary with the nuclear geometry, being appreciably different from zero only in the near-degeneracy region of the PESs and vanishing quickly outside this region. Because an ab initio Lσ′σ surface is not available for NH2, we define and employ an empirical expression that depends on the Jacobi coordinates R′, r′, and γ′ of the N(2D) + H2 channel and fulfills the above conditions. When NH2 is linear (γ′ ) 0),
RT Quantum Dynamics of NH(a1∆) + H
J. Phys. Chem. A, Vol. 114, No. 36, 2010 9751 TABLE 1: Parameters of the Calculationsa sinc initial WP translational energy center 0.2 eV R center and width R 9 and 6 smoothing parameter β 0.01 R range and no. of grid points 1-15 and 139 r range and numbers of grid points 1-13 and 139 no. of associated Legendre functions and of γ points 80 R and r absorption start at 12 and 10 absorption strength 0.01 flux analysis at r 9 asymptotic analysis at R 11 a
Values in au, unless otherwise specified.
Figure 2. Degeneracy and nondegeneracy regions of the collinear surfaces.
the PESs are degenerate or nondegenerate in the three regions shown in Figure 2. VX * VA or VX ) VA in the first two regions, respectively, where the latter is due the degeneracy of the N(2D) + H2 reactant channel. Therefore, we can safely assume that Lσ′σ vanishes in both regions (1) and (2). The two PESs are also degenerate in a third region, at small R′ and r′ values, and are associated with H-N-H or H-H-N arrangements on the left or right of the dotted green line, respectively, where Lσ′σ should be equal to Λ and decrease quickly when NH2 bends. Approximating region (3) by R′ e 1.93a0 and r′ e 4.31a0, we then define the following expression Lσ′σ ) Λ exp(-aR′2 sin2γ′) )0
R′ e 1.93 a0 and r′ e 4.31 a0 otherwise (9)
where the parameter a is partially adjusted for reproducing the observed (q) and (eq) rate constants14 and R′ sin γ′ is the distance from the N atom to the H-H axis. Equation 9 is very simple because it depends on just one empirical parameter that is only partially optimized. However, even a full WP optimization of a single parameter is impossible, taking into account the high computational cost of RT and CC calculations. Therefore, the use of a more flexible expression that depends on many parameters is outside the present computer facilities. For reactants NH(a1∆) + H′(2S) and product channels 1, 2, 3, and 4, we propagate initial WPs25 ψjJ0Kp0 taking into account the Coriolis couplings among all K values. The reactant NH(a1∆) is in the ground vibrational state and in the rotational state j0 g 2, owing to the ∆ electronic state, and K0 is the initial projection of both J and j0 along R. At the collision energy Ecol, initialstate-resolved probabilities, PjJ0Kp0 (Ecol), are then obtained via a flux analysis26 and the smaller internuclear distance technique21 for reactions (d), (e), and (eq) or via an asymptotic method25 for reaction (q), using a sinc initial WP27 and the parameters of Table 1. CC probabilities are calculated from J ) K0 up to 7 and then at J ) 15, 23, and 30 at j0 ) 2, 3, and 4 and at Ecol e 0.4 eV. Because these calculations are very expensive, partial wave probabilities are then interpolated for other J values, as described in ref 21. This approach gives converged cross sections σj0(Ecol) up to 0.4 eV and thermal rate constants kj0(T) up to a temperature T ) 500 K. 3. Results 3.1. Reaction Probabilities. The initial-state-resolved probabilities were calculated for different values of the parameter a
Figure 3. Reactions (d), (e), (q), and (eq). J ) 15, p ) -1, j0 ) 2, and K0 ) 0. BO (blue, ref 21) and RT (red) reaction probabilities, P1520 .
in eq 9. Because a controls the strength of the RT couplings in eq 7, its value was chosen to better reproduce the experimental total quenching rate at 300 K.14 After some tests, we used a ) 1000a0-2 for obtaining the probabilities of reactions 1-4, and we here report only a few examples. The interested reader can obtain all other probabilities from the authors. Figure 3 presents reaction probabilities P2150 at J ) 15, p ) -1, j0 ) 2, and K0 ) 0, showing BO results21 for (d) and (e) ˜ 2A′′ RT and RT probabilities for all channels. Because the X contribution to the depletion probabilities is very small, we here report only the sum on both surfaces. In addition to the resonances already discussed,21 we see that (d) and (e) RT probabilities are smaller than the BO ones at high collision energies, owing to the opening of the (q) and (eq) RT channels ˜ 2A′′. The lower RT reactivities with respect on the ground PES X to the BO ones could be also associated with the additional ˜ 2A′′ potential trapping of part of the RT WP into the deep X well. Overall, the reactivity of the (d) and (e) processes is larger than that of the (q) and (eq) processes, despite the larger exothermicity of the quenching reactions. The two PESs are barrierless and the probability thresholds thus depend on the centrifugal barrier BJ(J + 1). The scenario is similar if we consider an excited rotational level of NH(a1∆), as shown in Figure 4, which presents reaction probabilities P4150 at j0 ) 4. We find, indeed, only a decrease in the threshold energies by ∼0.025 eV with respect to the groundstate results, all other things being substantially unchanged. 3.2. Cross Sections. We compare BO and RT results in Figure 5, plotting depletion (top) and exchange (bottom) cross sections σj0 at two j0 values. We see that the (e) cross sections are generally larger than the (d) cross sections, save at j0 ) 2 and at low Ecol, when RT couplings are taken into account. At j0 ) 2, BO and RT σ2 are rather similar up to ∼0.03 eV,
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Figure 4. Reactions (d), (e), (q), and (eq). J ) 15, p ) -1, j0 ) 4, and K0 ) 0. BO (blue, ref 21) and RT (red) reaction probabilities, P1540 . Figure 6. Reactions (d), (e), (q), and (eq). j0 ) 2 (red), 3 (blue), and 4 (green). RT cross sections σ2, σ3, and σ4.
TABLE 2: Initial-State-Resolved RT Rate Constants, kj0, at T ) 300 K in 10-11 cm3 s-1
a
Figure 5. Reactions (d), above, and (e), below. j0 ) 2 (left) and 4 (right). BO (blue, ref 21) and RT (red) cross sections σ2 and σ4.
presenting small thresholds at ∼0.01 eV and sharp onsets at ∼0.02 eV. At higher collision energy, however, the RT couplings lower the cross sections in agreement with the probability results of Figures 3 and 4. Indeed, both (d) and (e) RT σ2 are nearly constant, on the average, above ∼0.1 eV, in agreement with the usual behavior of cross sections on barrierless PESs at high enough Ecol values, whereas the BO σ2 increase in this energy range. Moreover, the BO resonances are reduced in the coupled RT treatment, which opens and averages many more channels during the collisional dynamics. The right panels of Figure 5 refer to j0 ) 4 and show that the NH(a1∆) rotational excitation enhances the RT reactivity at low Ecol, with respect to the BO one. This is an interesting interplay between the NH(a1∆) rotational excitation and the nonadiabatic RT couplings, which reflects the threshold shifting of the reaction probabilities shown in Figure 4, but now it is more evident. Figure 6 presents RT cross sections σj0 of the four reactions at j0 ) 2, 3, and 4. The (d) and (e) cross sections increase with the NH(a1∆) rotational excitation, whereas the opposite holds for the (q) and (eq) channels. The near-threshold maximum values are thus equal to σ4(d) ) 10.6, σ4(e) ) 13.8, σ2(q) ) 3.52, and σ2(eq) ) 6.66 Å2 at Ecol ≈ 0.01 to 0.02 eV. Up to Ecol ≈ 0.04 eV, the present WP cross sections are rather different from the TSH results shown in Figure 4 of ref 14, noting that ji(TSH) ) j0(WP) - 2. In fact, at j0 ) 2, the (d) WP σ2 is smaller
j0
(d)
(e)
(q)
(eq)
2 3 4 5a
2.91 3.32 3.67 3.97
2.96 3.68 4.24 4.64
1.76 1.59 1.41 1.22
1.91 1.88 1.79 1.64
Estimates from parabolic fits to k2, k3, and k4.
than the TSH one near the threshold, and it is quite similar to the (eq) result, contrary to the TSH calculations14 that favor the depletion process by a factor of ∼4. Moreover, at j0 ) 4, the present (d) and (e) cross sections are much larger than the TSH ones, and the latter channel is preferred, whereas the opposite holds for semiclassical results. WP and TSH cross sections agree better at Ecol > 0.04 eV, where NH(a1∆) rotational and collision energy effects are small, and both calculations give the same ordering, that is, σ(d) ≈ σ(e) > σ(q) ≈ σ(eq) at any j0 value. 3.3. Rate Constants. Table 2 presents initial-state-resolved rate constants kj0 at 300 K of the four reactions here investigated, including the 1/2 electronic degeneracy factor. We report k2, k3, and k4 RT calculated rates, and k5 estimates from parabolic fits to the calculated ones. As expected, these rates reflect strictly the cross sections shown in Figure 6, that is, the j0 enhancement of the (d) and (e) reactivity and the j0 inhibition of the (q) and (eq) channels. Thermal rate constants at 300 K are reported in Table 3, where we compare RT and BO (in parentheses, ref 21) WP, experimental,14 and TSH14 results. The 1/2 factor and the k5 WP estimates are included. Experimentally, the two H atoms are indistinguishable in the product channels; (e) is, therefore, not observed, and only the sum (q) + (eq) can be measured. Subtracting the measured rates14 k(total) - k(q + eq), we obtain the experimental depletion rate, k(d) ) (3.2 ( 1.7) × 10-11 cm3 s-1, shown in Table 3, where we report the k(total) error because that of k(d) is unknown.14 The ordering of the WP RT rate constants is equal to that of the low-energy cross sections, that is, k(e) ≈ k(d) > k(eq) ≈ k(q). Comparing RT and BO WP results, we see that the RT couplings increase by ∼10% the (d) and (e) rates while opening the (q) + (eq) RT channels, which contribute to the total rate by 51%. Summing all four channels, the RT couplings increase
RT Quantum Dynamics of NH(a1∆) + H
J. Phys. Chem. A, Vol. 114, No. 36, 2010 9753
TABLE 3: Thermal RT Rate Constants k at T ) 300 K, in 10-11 cm3 s-1 a reaction
WP RT (BO)b
experimentc
TSHc
(d) (e) (q) (eq) (q) + (eq) totald
3.06 (2.77) 3.36 (3.02) 1.44 1.70 3.14 6.20
3.2 ( 1.7
2.97 2.44 0.96 0.68 1.64 4.62
1.7 ( 0.3 4.8 ( 1.7
a WP BO results from ref 21 are in parentheses. estimates. c Ref 14. d Exchange rates are not included.
b
Including k5
Figure 7. Reactions (d), blue; (e), red; (q), green; and (eq), black. RT thermal rate constants.
by ∼65% the overall rate, from the BO value equal to 5.79 × 10-11 to the RT value of 9.56 × 10-11 cm3 s-1. Table 3 shows that the WP RT depletion rate agrees well with the experimental and TSH rates14 and that the present total rate is within the large experimental error bar. However, the (q) and (eq) WP rate constants are larger than the previous rate constants of ref 14 by a factor of ∼2 because the present calculations overestimate the RT couplings. In fact, the WP (q) + (eq) weight of 51% is larger than that observed,14 equal to 35%. This finding is associated with the very simple expression of the Lσ′σ matrix element of eq 9, which was even impossible to optimize with our present computer facilities, contrary to the TSH calculations.14 According to some test calculations of reaction probabilities, we estimate that the a parameter of eq 9 should be increased to ∼1500a0-2 for halving the (q) and (eq) rate constants and thus obtaining a better agreement with the experimental data.14 RT rate constants are plotted in Figure 7 from 200 to 500 K, including rotational levels up to j0 ) 5. We see that (d) and (e) rates increase moderately with T up to 400 K, contrary to the rates of the (q) and (eq) channels that decrease moderately. The RT effects, which open the quenching channels, are thus less important at high temperature. 4. Conclusions We presented the QM dynamics of four NH(a1∆) + H′ reactions, namely, depletion (d) to N(2D) + H2, exchange (e) to NH′(a1∆) + H, quenching (q) to NH(X3Σ-) + H′, and exchange-quenching (eq) to NH′(X3Σ-) + H. The dynamics is ˜ 2A′′ ˜ 2A′ and X investigated on the nonadiabatically coupled A PESs of NH2, taking fully into account the RT couplings between these electronic states and the Coriolis couplings between different K values. In fact, the (q) and (eq) reactions can only occur through RT couplings. A general expression of the Hamiltonian matrix elements is presented, containing effective PESs and RT couplings. We employ a simple but
reasonable empirical expression of the RT matrix element, which depends on a single empirical parameter that is partially adjusted for reproducing the observed (q) + (eq) sum of the rate constants at 300 K. We consider the NH(a1∆) reactant in the ground vibrational state and in three rotational levels, and we propagate real WPs on both coupled surfaces obtaining reaction probabilities via flux or asymptotic analyses. When NH(a1∆) is in the ground rotational level, the (d) and (e) reactions proceed predominantly via an adiabatic, single˜ 2A′ surface mechanism; that is, they remain on the excited A surface. However, the NH(a1∆) rotational excitation enhances ˜ 2A′′ state and, therefore, the RT interaction with the ground X the reactivity of the (d) and (e) channels. The quenching processes are slightly preferred when j0 ) 2. The two-state treatment lowers the overall rotational enhancement of probabilities and cross sections at high collision energy, which is found within the BO approximation. Contrasting BO and RT results, we see that the RT couplings not only open two new quenching channels to NH(X3Σ-) but also increase slightly the depletion and exchange rate constants, thus enhancing remarkably the overall rate, k. The WP RT rate ordering is k(e) ≈ k(d) > k(eq) ≈ k(q), as in a previous experimental and TSH work, but the present calculations overestimate RT effects and, therefore, the total rate. This is probably due to the use of a too simple and not enough optimized expression of the RT matrix element, which should be further optimized or changed to an ab initio one, in a more accurate theoretical study. Acknowledgment. We thank R. Schinke and G. C. McBane for many stimulating discussions and we gratefully acknowledge R. Schinke for his PESs. This work was performed under the project HPC-EUROPA2 (project no. 228398), with the support of the European Community under the FP7 “Research Infrastructures Programme”. Many thanks are due to the CINECA supercomputer centre of Bologna, Italy. This work was also supported by IPCF-CNR of Pisa, by the Spanish Ministry of Education and Science and the Spanish Ministry of Science and Innovation (projects no. CTQ2005-09334-C02-01 and CTQ200806805-C02-01, respectively), and by the Generalitat of Catalonia (ref. no. 2009 SGR 17). References and Notes (1) Renner, R. Z. Phys. 1934, 92, 172. (2) Dressler, K.; Ramsay, D. A. Philos. Trans. R. Soc., A 1959, 251, 553. (3) Hadj Bachir, I.; Huet, T. R.; Destombes, J.-L.; Vervloet, M. J. Mol. Spectrosc. 1999, 193, 326, and references therein. (4) Jungen, C.; Hallin, K.-E. J.; Merer, A. J. Mol. Phys. 1980, 40, 25. (5) Dixon, R. N.; Irving, S. J.; Nightingale, J. R.; Vervloet, M. J. Chem. Soc., Faraday Trans. 1991, 87, 2121. (6) Gabriel, W.; Chambaud, G.; Rosmus, P.; Carter, S.; Handy, N. C. Mol. Phys. 1994, 81, 1445. (7) Jensen, P.; Kraemer, W. P.; Bunker, P. R. Mol. Phys. 2003, 101, 613. and references therein. (8) Petrongolo, C. J. Chem. Phys. 2005, 122, 234315. (9) Koshi, M.; Yoshimura, M.; Fukuda, K.; Matsui, H.; Saito, K.; Watanabe, M.; Imamura, A.; Chen, C. J. Chem. Phys. 1990, 93, 8703. (10) Davidson, D. F.; Hanson, R. K. Int. J. Chem. Kinet. 1990, 22, 843. (11) Suzuki, T.; Shihira, Y.; Sato, T.; Umemoto, H.; Tsunashima, S. J. Chem. Soc., Faraday Trans. 1993, 89, 995. (12) Adam, L.; Hack, W.; Zhu, H.; Qu, Z.-W.; Schinke, R. J. Chem. Phys. 2005, 122, 114301. (13) Qu, Z.-W.; Zhu, H.; Schinke, R.; Adam, L.; Hack, W. J. Chem. Phys. 2005, 122, 204313. (14) Adam, L.; Hack, W.; McBane, G. C.; Zhu, H.; Qu, Z.-W.; Schinke, R. J. Chem. Phys. 2007, 126, 034304. (15) Pederson, L. A.; Schatz, G. C.; Ho, T.-S.; Hollebeek, T.; Rabitz, H.; Harding, L. B.; Lendvay, G. J. Chem. Phys. 1999, 110, 9091.
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