Mode Specificity and Product Energy Disposal in Unimolecular

Article pubs.acs.org/JPCA

Mode Specificity and Product Energy Disposal in Unimolecular Reactions: Insights from the Sudden Vector Projection Model Jun Li and Hua Guo* Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New Mexico 87131, United States ABSTRACT: A simple model is proposed to predict mode specificity and product energy disposal in unimolecular dissociation reactions. This so-called Sudden Vector Projection (SVP) model quantifies the coupling of a reactant or product mode with the reaction coordinate at the transition state by projecting the corresponding normal mode vector onto the imaginary frequency mode at the saddle point. Due to the sudden assumption, SVP predictions for mode specificity are expected to be valid only when the reactant molecule has weak intermodal coupling. On the other hand, the sudden limit is generally satisfied for its predictions of product energy disposal in unimolecular reactions with a tight barrier. The SVP model is applied to several prototypical systems and the agreement with available experimental and theoretical results is satisfactory. unimolecular decay,10,14,15 there is still no simple and reliable means to quantify such a coupling. A rule-of-thumb in this respect is particularly desired for unimolecular decay of large molecules for which neither accurate PESs nor quantum dynamics is viable. Recently, we proposed a simple and reliable model to rationalize mode specificity and bond selectivity in activated bimolecular reactions.16−18 This so-called Sudden Vector Projection (SVP) model was based on the assumption that the collision time is significantly shorter than that of IVR in the collision partners, which holds for many bimolecular reactions involving molecules with low densities of states. The key insight provided by the SVP model is that the extent of coupling of a reactant mode, whether it is a vibration, rotation, or translation, with the reaction coordinate at the transition state dictates its capacity for enhancing reactivity. In the sudden limit, such coupling is proportional to the projection of the reactant normal mode vector onto the reaction coordinate vector at the transition state, underscoring the importance of the transition state in reaction dynamics. The SVP model has been successfully applied to both bimolecular16−19 and gas-surface reactions.20,21 It has also been found consistent with the venerable Polanyi’s rules22 in cases where they are applicable. In this work, we discuss the extension of the SVP model to unimolecular dissociation reactions, which requires some modifications of the original formulation. We note in passing that Bowman and co-workers have recently proposed a similar model for determining mode specificity in tunneling.23,24 The SVP model is employed here to rationalize mode specificity in

I. INTRODUCTION The study of decomposition reactions of isolated and energetic molecules has a long and distinguished history.1 Many such unimolecular processes can be treated with statistical theories, such as the Rice−Ramsperger−Kassel−Marcus (RRKM) theory 2 and the Statistical Adiabatic Channel Model (SACM).3 Such theories are based on the premise that efficient intramolecular vibrational energy redistribution (IVR) results in ergodicity in the phase space. In this limit, the decay rate can be readily modeled from equilibrium properties such as partition functions, underscoring the negligible role played by dynamics. For many unimolecular processes, the statistical treatment has been quite successful, particularly when averaged properties such as thermal rate constants are of interest. Thanks to experimental advances in preparing highly excited molecules with quantum state resolution, however, an increasing number of studies have revealed the existence of nonstatistical behaviors in unimolecular reactions.4−7 For these reactions, IVR is often incomplete and only parts of the phase space are coupled with the dissociation process and thus accessed.8 A well-known manifestation of such nonstatistical dynamics is mode and/or state specificities, namely the dependence of the decay rate on the vibrational mode and/or state of the reactant molecule. Mode and state specificities are typically associated with systems with a relatively sparse density of states,9 and many of such reactions involve hydrogen atoms, whose dynamics is accentuated by tunneling.10,11 Theoretical description of mode-specific unimolecular decay relies on the quantum mechanical characterization of metastable resonance states on global potential energy surfaces (PESs).9,12 Such calculations are only viable for small molecules.13 While it has long been recognized that the coupling of vibrational modes to the transition state is the key to understand mode-specific © 2014 American Chemical Society

Received: February 4, 2014 Revised: March 10, 2014 Published: March 11, 2014 2419

dx.doi.org/10.1021/jp501255t | J. Phys. Chem. A 2014, 118, 2419−2425

The Journal of Physical Chemistry A

Article

determined by a normal mode analysis. The projections (Pi′ = Q⃗ i′·Q⃗ RC ∈ [0,1]) thus give predictions on energy disposal in the products. The SVP model in predicting product energy disposal represents the opposite limit of the statistical theory for product energy distribution,1 and it is mostly suited for reactions with large energy release from the transition state. As discussed in our earlier publications,16−18 the SVP model is related to the Franck−Condon (FC) approximation.27−30 Both assume the sudden limit and are based on the direct overlap between the properties of the reactant/product and transition state. However, the FC model is a quantum mechanical formulation that relies on the wave function at the transition state, while the SVP model focuses on the corresponding reaction coordinate in the normal mode picture. As a result, the FC model is capable of providing more quantitative predictions, although it is computationally much more demanding. The relatively inexpensive SVP model, on the other hand, is more suited for offering a qualitative picture. It should also be pointed out that the FC model is inappropriate when the reactant/product mode is strongly coupled with the reaction coordinate.18 It is important to note that the determination of the vibrational normal modes and reaction coordinate does not require a global PES. Instead, only the vicinity of the stationary point is needed. Hence, these calculations can rely on direct ab initio calculations of the gradients and Hessians at the stationary points. To this end, we have used an explicitly correlated coupled-cluster method with singles, doubles, and perturbative triples excitations with the augmented correlation consistent polarized valence triple-ζ basis set with frozen cores (FCCCSD(T)-F12a/aug-cc-pVTZ), which has been shown to yield atomization energies, electron affinities, ionization potentials, equilibrium geometries, and harmonic frequencies for both close- and open-shell systems better than CCSD(T) with the augmented correlation-consistent polarized valence quintuple-ζ basis set (AV5Z).31−33 Since the SVP model is a qualitative model, a high-level ab initio calculation of the stationary points along the reaction pathway is in many cases not essential for computing the vectors. In addition to the high-level ab initio methods mentioned above, hence, we have also tested the density functional theory (DFT) method based on the B3LYP functional. The computation of the normal mode vectors is performed with the same protocol. As shown below, the DFT based SVP values are consistent with those obtained at the CCSD(T)-F12/AVTZ level, and they are sufficient in predicting trends in mode specificity and product energy disposal.

several prototypical systems, including some dominated by tunneling. In particular, we exploit an important feature of the SVP model, namely its requirement of only local PES properties near stationary points. To this end, both the equilibrium geometry of the reactant and product molecules and the dissociation transition state are computed using ab initio methods and the corresponding normal mode vectors are determined. The SVP predictions are compared with the available experimental and theoretical results. Our discussion cumulates with caveats on the applicability of this model.

II. SUDDEN VECTOR PROJECTION MODEL As mentioned above, the basic premise of the original SVP model16−18 is that a bimolecular reaction occurs in the sudden limit, in which the intramolecular vibrational energy redistribution (IVR) in the reactants is assumed to take much longer than the collision time. This is a reasonable assumption for many activated bimolecular reactions, particularly for those involving molecules with sparse densities of states. For unimolecular reactions, we assume in the same spirit that intermodal couplings of the reactant molecule are weak. In this limit, the molecular vibration is regular and assignable, even when excited.25 These assumptions define the applicability of the SVP model by excluding unimolecular reactions that can be described by the statistical models in the presence of strong IVR. It is further assumed that the unimolecular decay rate for a particular resonance depends on its coupling with the reaction coordinate at the dissociation transition state. To determine the coupling, we resort to the normal mode picture in which both vibrational modes of the reactant molecule and the reaction coordinate at the transition state are approximated by the corresponding normal mode vectors: Q⃗ i and Q⃗ RC. As a result, the projection of Q⃗ i onto Q⃗ RC, namely Pi = Q⃗ i·Q⃗ RC ∈ [0,1], provides a measure of the coupling. We note in passing that a local mode representation can also be used for this purpose, but we will focus here on the normal mode picture. The vector for the reaction coordinate (Q⃗ RC) at the transition state can be readily determined by a standard normal mode analysis, and it corresponds to the sole imaginary frequency at the first-order saddle point. (For barrierless reactions, this vector can be approximated by the translational vector between the two fragments, which corresponds to a transition state in the product asymptote.) However, the definition of the reactant normal modes is more challenging and contains some arbitrariness. In order to keep the dissociation system in the same frame of reference, an intrinsic reaction coordinate (IRC) analysis26 is performed to connect the transition state with the molecular equilibrium. The vibrational normal mode vectors can then be determined at the optimized geometry of the reactant molecule. Since the normal modes are used to describe the molecular vibration, systems with large intermodal coupling will diminish the predictability of the SVP model. The SVP model can also be used to predict product energy disposal in both unimolecular and bimolecular reactions, thanks to microscopic reversibility. To this end, the projection of the product vibrational, rotational, and translational modes onto the reaction coordinate at the transition state needs be determined. We adapt the same strategy used in bimolecular reactions.16 Specifically, the two dissociation fragments are separated from the saddle point along the scattering coordinate defined as the distance between the two centers of mass. At a sufficiently large distance, the fragments are optimized with minimal reorientation and their vibrational vectors (Q⃗ i′) are

III. RESULTS AND DISCUSSION In this work, we investigate several prototypical unimolecular reactions for which experimental and/or theoretical data exist:

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HCO → H + CO

(R1)

HN2 → H + N2

(R2)

H 2CO → H 2 + CO

(R3)

HFCO → HF + CO

(R4)

HOCO → H + CO2

(R5)

NH4 → H + NH3

(R6)



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These unimolecular reactions all have a dissociation barrier; thus, they are amenable to the SVP model. As discussed below, however, some have strong intermodal coupling and IVR, which render the SVP predictions less reliable or inapplicable. Figure 1 displays the energetics of the reactant molecules, transition states, and products for the six reactions. The

Figure 3. Geometries (Å and deg) of the transition states and the corresponding reaction coordinate vectors (without mass scaling) for the six prototypical unimolecular dissociation reactions.

on various PESs.35−39 These calculations are consistent with experimental results,40−42 and both indicate strong mode specificity in the decay rate. The SVP values in Table 1 suggest that the H−C stretching mode is strongly coupled with the reaction coordinate, while the couplings for the C−O stretching and H−C−O bending modes are much weaker. This is expected, as the reaction coordinate is largely along the H−C dissociation coordinate, as shown in Figure 3. These predictions are in accord with the calculated and measured widths of the resonances. The SVP model (Table 1) also predicts that the excess dissociation energy is largely released into the translational degree of freedom for the products. On the other hand, the dissociation is expected to impart little energy into the vibration of the CO product, but a torque at the transition provides some rotational excitation. These predictions are consistent with experimental observations43,44 and theoretical calculations.36,45−48 The tunneling facilitated decay of HN2 has been investigated theoretically by several authors.49−52 The lifetimes of several resonances have been determined.50,51 These calculations indicated that the excitation of the H−N stretch greatly enhances the dissociation, whereas the N−N stretch has a mild effect and the H−N−N bend is the least effective in promoting the reactions. These observations are consistent with the SVP values listed in Table 1 and the reaction coordinate vector in Figure 3. The SVP model further predicts that the dissociation will impart most energy into the translational mode, while some N2 vibrational and rotational excitations are also expected. So far, there has been neither experimental nor theoretical studies on the product state distribution. The dissociation of formaldehyde (H2CO) to H2 and CO has been extensively investigated by both experimentalists53−57 and theoreticians.58−64 The transition state, shown in Figure 3, is planar with an elongated C−H bond. Although quantum state resolved dissociation rates have been measured, little is known about their assignment.57 Indeed, the success of a random-matrix based theory in describing the lifetime distributions suggests significant mode mixing in H2CO,60 which undermines the validity of the SVP model in predicting the mode specificity in the dissociation reaction. Nevertheless, we have computed the SVP values and listed them in Table 1, and they suggest that excitation of every stretching and bending mode of this molecule helps to overcome a high and tight barrier. Among them, the H−C−H asymmetric stretching mode is predicted to have the largest coupling with the reaction

Figure 1. Energetics (in kcal/mol relative to the reactants) for six prototypical unimolecular dissociation reactions.

Figure 2. Geometries (Å and deg) of the reactant molecules for the six prototypical unimolecular dissociation reactions.

reactant geometries are shown in Figure 2, and the reaction coordinate vectors at the transition states are depicted in Figure 3 along with their geometries. The SVP values for mode specificity (Pi) and for product energy disposal (Pi′) are listed in Table 1 and depicted in Figure 4. For comparison, the SVP values were also generated using the B3LYP/AVTZ method and listed in the same table. Generally speaking, they are consistent with the corresponding FC-CCSD(T)-F12/AVTZ values. As shown in Figure 1, the dissociation of the formyl radical (HCO and DCO) from its ground electronic state involves a significant barrier.34−36 This system has served as a prototype for unimolecular decay reactions with strong mode specificity.9,12 Positions and widths of resonances have been calculated 2421



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Table 1. SVP Projections (Pi) of the Mode Specificity and Product Energy Disposal for Several Unimolecular Dissociation Reactions R1−R6 at the Levels of FC-CCSD(T)-F12a/AVTZ and B3LYP/AVTZa mode specificity reaction

mode

R1

HCO 1. H−C stretch 2. C−O stretch 3. HCO bend NH2 1. H−N stretch 2. N−N stretch 3. HNN bend H2CO 1. H−C symm stretch 2. C−O stretch 3. CH2 bend 4. out-of-plane bend 5. H−C asymm stretch 6. CH2 rock HFCO 1. C−H stretch 2. C−O stretch 3. HCO bend 4. C−F stretch 5. F−C−O bend 6. out-of-plane bend HOCO 1. H−O stretch 2. C−O stretch 3. HO′C bend 4. C−O′ stretch 5. O−C−O′ bend 6. out-of-plane bend NH4 1. symm stretch 2. umbrella bend 3. asymm stretch 4. rock bend

R2

R3

R4

R5

R6

a

Pi/CCSD(T)

product energy disposal Pi/DFT

0.997 0.045 0.051

0.996 0.056 0.016

0.967 0.243 0.070

0.972 0.231 0.073

0.276 0.329 0.444 0.008 0.752 0.224

0.274 0.317 0.487 0.008 0.738 0.218

0.627 0.250 0.706 0.158 0.144 0.000

0.628 0.239 0.701 0.182 0.156 0.000

0.839 0.052 0.008 0.475 0.259 0.006

0.799 0.079 0.016 0.493 0.306 0.000

0.596 0.028 0.433 0.000

0.460 0.018 0.410 0.002

Pi′/CCSD(T)

Pi′/DFT

0.040 0.271 0.936

0.025 0.248 0.951

0.252 0.206 0.923

0.206 0.176 0.946

0.516 0.076 0.441 0.280 0.507

0.562 0.088 0.443 0.191 0.516

HF + CO 1. HF stretch 2. CO stretch 3. CO rotation 4. HF rotation 5. translation

0.866 0.053 0.336 0.013 0.356

0.857 0.051 0.310 0.045 0.397

H + CO2 1. CO2 symm stretch 2. CO2 bend 3. CO2 asymm stretch 4. CO2 rotation 5. translation

0.137 0.239 0.064 0.248 0.622

0.147 0.258 0.041 0.318 0.735

H + NH3 1. NH3 symm. stretch 2. NH3 umbrella 3. NH3 asymm stretch 4. NH3 bend 5. NH3 rotation 6. translation

0.083 0.050 0.000 0.000 0.051 0.993

0.072 0.082 0.002 0.001 0.005 0.994

mode H + CO 1. CO stretch 2. CO rotation 3. translation H + N2 1. NN stretch 2. N2 rotation 3. translation H2 + CO 1. HH stretch 2. CO stretch 3. CO rotation 4. H2 rotation 5. translation

For degenerate modes, the SVP values are averaged.

barrier. Experimentally, state-specific dissociation rates near the dissociation barrier have been measured.66,67 While IVR is moderate in this system, mode specificity has been observed. In particular, the rates for the overtones of the out-of-plane bending (v6) mode have been found to be 1 order of magnitude slower than others. This observation is consistent with the SVP values as listed in Table 1, which suggests that the torsional mode of HFCO is orthogonal to the reaction coordinate. Other normal modes are strongly coupled with the reaction coordinate at the transition state, but the strong mode mixing in this system68−70 renders the SVP model less relevant. On the other hand, the SVP model (Table 1) predicts that the CO product is rotationally hot but vibrationally cold, which is in good accord with the experimental observations.71 In addition, this model also predicts that the HF product is excited in both the vibrational and rotational degrees of freedom, in accord with experimental data.72 These predictions on product energy disposal are also consistent with theoretical calculations.68,73,74 The cis-HOCO complex is an intermediate in the important combustion reaction HO + CO → H + CO2, and as a result it

coordinate. On the other hand, the out-of-plane bending mode is essentially orthogonal to the reaction coordinate, consistent with the planarity of both the reactant and saddle point. Given the large exothermicity of the dissociation, on the other hand, predictions concerning the product energy disposal by the SVP model are expected to be accurate. Indeed, the projection values in Table 1 suggest that a large fraction of the excess energy is likely disposed in the translational degree of freedom. In addition, the H2 product is vibrationally and rotationally hot, while CO is vibrationally cold but rotationally hot. These predictions are consistent with the experimentally measured product distributions53−55 and theoretical predictions.58,61,62,64 Note that there is a radical channel (to H + HCO) that opens immediately above this transition state and the emerging roaming channel produces different products than those via the tight transition state.65 Equation R6 (HFCO → HF + CO) represents another extensively studied unimolecular decay process. The transition state shown in Figure 3 features an elongated F−C bond, resembling its H2CO counterpart but with a significantly lower 2422



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relatively low level electronic structure theory, such as DFT, without introducing qualitative errors. Due to the assumptions intrinsic to the SVP model, however, its applicability in predicting mode specificity in unimolecular decay reactions is restricted to systems that have weak intermodel coupling and IVR. This, however, presents no significant drawback, as the strong IVR regime can be readily characterized by statistical theories. On the other hand, its ability in predicting product energy disposal is much more universal, as the post-transitionstate dynamics is typically fast and direct. Such rules-of-thumb could be very useful for unimolecular reactions of large molecules, for which quantum dynamics on a global multidimensional PES is not easily obtainable and DFT is the only viable method for determining the stationary points. The success of the simple SVP model seems to suggest that a more sophisticated model might be developed to quantify the coupling of individual resonances with the reaction coordinate at the dissociative saddle point. Nonlinear dynamics techniques are available to construct these resonances,25,90 but the quantification of the coupling will have to go beyond the harmonic oscillator approximation used in the SVP model.

Figure 4. Alignments of the reactant (R) and product (P) modes with the reaction coordinate (RC) for the six prototypical unimolecular dissociation reactions.

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has received much attention in the past.75 The decay dynamics of the cis-HOCO has been investigated via the photodetachment of the HOCO− anion, and tunneling through the cisbarrier (shown in Figure 3) has been found to be quite facile.76,77 The tunneling facilitated dissociation of cis-HOCO via a significant but thin barrier78 has been confirmed by quantum mechanical calculations, and significant mode specificity was uncovered.24,79,80 Specifically, the terminal C− O stretching (v4) and O−C−O bending (v5) modes, in addition to the O−H stretching mode (v1), were found to enhance the tunneling rates. These observations are consistent with the SVP values in Table 1. Furthermore, the SVP model predicts that the stretching and bending modes of the CO2 product are excited, which is consistent with both experimental81,82 and theoretical observations.83−85 Finally, we discuss the dissociation of the ammonium radical (eq R6). Due to the shallow well corresponding to this species, NH4 is heavily predissociative in the ground electronic state.86 Much longer lifetimes of the deuterated counterpart (ND4) strongly suggest tunneling.87 There has been neither experimental nor theoretical studies of the mode specificity in R6, but there is renewed interest in these issues.88,89 Thus, the SVP model serves here as a prediction. It is clear from Table 1 that all stretching modes of NH4 are strongly coupled with the reaction coordinate at the transition state. This makes intuitive sense, as the transition state has one elongated N−H bond. Other modes have weak couplings. On the product side, the SVP model predicts that the energy is largely imparted into the translation.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by Department of Energy (DE-FG0205ER15694). We thank Joel Bowman and Bill Hase for several stimulating discussions.

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REFERENCES

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IV. CONCLUSIONS In this work, we extend the SVP model to treat unimolecular dissociation reactions. The essence of the model is to quantify the coupling of a specific reactant or product mode with the reaction coordinate at the transition state by the projection of normal mode vectors. We demonstrate that this simple model, which requires information only in the vicinity of the stationary points, is capable of predicting qualitatively mode specificity and product energy disposal in several prototypical systems. It is further shown that the SVP values can be computed with a 2423



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