Isotopic Effects on the Dynamics of the CH3+ + H2 â CH5+ â CH3+ +

Article pubs.acs.org/JPCA

Isotopic Effects on the Dynamics of the CH3+ + H2 → CH5+ → CH3+ + H2 Reaction Charlotte E. Hinkle and Anne B. McCoy* Department of Chemistry, The Ohio State University, Columbus, Ohio 43210, United States S Supporting Information *

ABSTRACT: Diffusion Monte Carlo is used to investigate the anharmonic zero-point energy corrected energies for the CH3+ + H2→ CH5+ → CH3+ + H2 process as a function of the center of mass separation of the two fragments. In addition to the title reaction, all possible deuterated and several tritiated (CH4T+ and CH3T2+) analogues of this reaction are investigated. As anticipated, the replacement of one or more of the hydrogen atoms with deuterium or tritium atoms lowers the zero-point energy of the system. Further, in the partially deuterated or tritiated isotopologues, the lowest energy configuration generally has the heavy atoms in the CH3+ fragment. Analysis of the wave functions allows us to study how zero-point energy influences the approach geometries sampled during low-energy collisions between CH3+ and H2, and to gain insights into how the dynamics is affected by the substitution of heavier isotopes for one or more of the hydrogen atoms. Differences between quantum and classical descriptions of the title reaction are also discussed. has an energy that exceeds 1 kcal/mol. This makes CH5+ a good candidate for studies using Diffusion Monte Carlo approaches as these approaches have the flexibility required to evaluate the zero-point energy and ground state properties of such a fluxional molecule.16,18,22−24 Although the high resolution spectrum of CH5+, when assigned, will aid in the detection of CH5+ in planetary ionospheres or dense molecular clouds, it is also important to understand the factors that affect the dynamics of the CH3+ + H2 → CH5+ → CH3+ + H2 process as it plays an important role in the observed abundance of the various isotopologues of CH3+. Several experimental studies have sought to shed light on this reaction. Ion-trap experiments were performed to investigate the branching ratios for H/D exchange following the reaction of CH3+ with HD.25 Complementary dissociative recombination studies have investigated the products obtained from neutralization of CH5+. The most recent of these studies was carried out by Contenetti and co-workers in which they focused on the products formed after an electron exchanging collision between Cs atoms and CH5+ and isotopically substituted variants of CH5+.26,27 Though the dissociation takes place on the neutral surface, the greater propensity for H and HD loss from partially deuterated CH5+, compared to loss of D or D2, in both experiment and calculations suggests that the structure of CH5+ and its dissociation dynamics may be playing a role in the initial stages of the process.27 In the present study, we focus on the process studied by Gerlich, and co-workers:25

1. INTRODUCTION Protonated methane is believed to represent an important intermediate in astrochemical reactions,1,2 and in reactions within planetary ionospheres.3 Due to its high fluxionality and essentially flat potential surface, CH5+ has proved to be a challenging molecule for both experimental and theoretical studies. The first high resolution spectrum of CH5+ was reported in 1999,4 nearly 50 years after its observation was first reported on the basis of mass spectrometry experiments.5 A second high resolution spectrum was reported at lower rotational temperatures seven years later.6 Interestingly, assignments were not reported for either of these rotationally resolved spectra of CH5+, and to date the spectra remain only partially assigned. A third set of studies was carried out by Asvani et al.7,8 In these studies, they used a free electron laser to obtain a lower resolution action spectrum of CH5+ as well as its deuterated isotopologues. At this level of resolution, they and others were able to assign the bands to the CH stretch and HCH bending vibrations in CH5+.6−10 At the same time, CH5+ has been the subject of a broad range of theoretical studies. These include group theoretical treatments,11,12 direct dynamics and path integral studies,8,13,14 as well as large scale variational calculations of the vibrational energies.15 Independently, Two groups developed potential surfaces for CH5+.16−18 One was developed by Jordan and coworkers, and was interpolated on the basis of Diffusion Monte Carlo calculations, using the GROW algorithm of Collins and co-workers.19,20 A second potential, which is used in the present study, was developed by Bowman and co-workers using an expansion in symmetry invarient polynomials to fit the electronic energies.17,18 As anticipated by earlier electronic structure work,21 these surfaces both display the unusual feature that any two of the 120 equivalent minima on the potential can be connected through a series of saddle points, none of which © 2012 American Chemical Society

Received: February 12, 2012 Revised: April 22, 2012 Published: April 24, 2012 4687

dx.doi.org/10.1021/jp3014157 | J. Phys. Chem. A 2012, 116, 4687−4694

The Journal of Physical Chemistry A

Article

CH3 − nDn+ + HD → CH4 − nDn + 1+ → CH 2 − nDn + 1+ + H 2

In earlier studies, we showed that we can introduce a parametric dependence to the Hamiltonian and use DMC to evaluate how the zero-point energy is affected by changes to this parameter. This is achieved by first obtaining an equilibrated ensemble for a fixed value of the parameter and then slowly varying the parameter as the simulation proceeds.29,35 By ensuring that the change to the spectrum of the Hamiltonian over a single time step is small compared to the statistical fluctuations inherent in the DMC simulation, one can map out the energy profile of this parametrized system. Further by setting the parameter to a particular value of interest, standard descendant weighting approaches can be used to evaluate the properties associated with the wave function under the imposed constraint. We initially developed this functionality to locate nodal surfaces for excited state calculations when the position of the node could not be determined by symmetry.35 In the present application, we parametrize the Hamiltonian in terms of R, the center of mass separation of CH3+ and H2. The choice of scanned coordinate was made to ensure that the simple structure the Hamiltonian has when it is expressed in Cartesian coordinates is retained in this approach. Specifically, our implementation of DMC requires that the kinetic energy operator takes the form

(1)

The reaction is driven to the right when CH3+ and HD are combined in a low-temperature ion trap.25 They report a net rate constant of 1.6 × 10−9 cm3 s−1 at 15 K, and the rate constants are found to be independent of the value of n in eq 1. They also report that after three seconds, 99% of the CH3+ has been converted to CD3+. This is attributed to the exothermicity of the CH3+ + HD reaction when zero-point energies are taken into account. Interestingly quasiclassical trajectory studies, which do not account for this exothermicity, show roughly 60% isotope exchange. Because pure statistical arguments would predict that 50% of the reactive collisions would lead to H/D exchange additional dynamical factors must be at play.28 It is interesting to probe the extent to which the dynamical effects that lead to deuteration are quantum mechanical in origin, and how much can be explained by classical mechanics. Recently, we developed a variant of Diffusion Monte Carlo (DMC) that allows us to investigate the anharmonic zero-point corrected energies of a system as a function of the center of mass separation of the dissociating fragments (R) as well as the R dependence of the corresponding probability amplitudes.29 In contrast to standard treatments, which are often implemented in electronic structure packages where the energy of the system is minimized subject to one or more constraints and the harmonic zero-point energy is calculated for the active vibrational degrees of freedom, the DMC approach allows us to take into account the fully anharmonic zero-point energy. This is important as it also allows vibrations to evolve into hindered rotations and eventually free rotation as the fragments separate. In our previous study,29 we focused on the effects of singly deuterating CH5+. Here we expand on these studies for all deuterated isotopologues of CH5+ and for those with up to two tritium atoms, looking for trends in the dynamics of these systems and the role of quantum effects in the observed kinetics and dynamics of the title reaction.

T̂ = −

i=1

i=1

⎡ ⎤ 1 ⎢ ∂2 ∂2 ∂2 ⎥ + + mi ⎢⎣ ∂xi 2 ∂yi 2 ∂zi 2 ⎥⎦

E(R ) = ⟨V ⟩(R ) − α

(3)

N (τ ) − N (0) N (0)

(4)

using α = 0.1 hartree, and N(τ) represents the number of walkers at time τ N(0) = 20 000.31 In most cases, the calculations are performed both for R decreasing from 6.0 to 0.0 Å and R increasing from 0.0 to 6.0 Å, with a step size of ΔR = 0.0002 Å over each 10 au time step. In instances where the diatomic fragment is HT, R(τ=0) = 0.2 Å rather than 0.0 Å. For comparison, we also evaluated the minimum energy (without zero-point energy added) and corresponding structure as a function of R using the same potential surface. To distinguish between these two energies, we will refer to the minimum energy without zero-point energy as Eele(R) and the energy with zero-point energy included as E(R) in the discussion that follows.

n

∑ δ(r − rn(τ))

N

∑

This structure is retained when the kinetic energy operator is expressed in the Jacobi coordinates that connect the centers of mass of atoms or groups of atoms. The scanned coordinate, R, used in this study is the Jacobi coordinate that connects the centers of mass of the two fragments. The approach used in the present study is similar to the approach employed in the work of Gregory and Clary where they used the intrinsic reaction coordinate for single point DMC calculations of the OH + H2 reaction along the OH···H2 separation coordinate.36 All of the results reported here are based on calculations that are performed for 30 000, 10 au time steps using the global extension to the Bowman surface,17,18 reported by Jin et al.37 In addition, we evaluate an approximation to the zero-point corrected energy

2. THEORY The Diffusion Monte Carlo approach for obtaining the ground state solution to the Schrödinger equation was first proposed by Fermi,30 and the approach that we use to investigate chemical problems is based on the work of Anderson.31,32 For the purpose of the present study, we note that DMC provides the ground state solution to the Schrö dinger equation by employing statistical approaches to solve the time-dependent Schrödinger equation when it is cast in terms of an imaginary time variable, τ = it/ℏ. Further, the wave function is represented by an ensemble of n δ functions (or walkers) as Ψ(r,τ ) =

ℏ2 2

(2)

The positions of the walkers at time τ (rn(τ)) diffuse through the 3N-dimensional configuration space that represents the possible geometries of the N-atomic molecule of interest. As a result, in the infinite τ limit, this ensemble of δ functions provides a representation of the ground state wave function, and the corresponding energy is obtained by imposing the requirement that the size of the ensemble remains constant. Properties associated with |Ψ|2 can be obtained by using descendant weighting.33 A more detailed description of DMC and its application to studies of CH5+ can be found elsewhere.29,34

3. RESULTS 3.1. R-Dependent Energies. We have calculated the anharmonic zero-point energy corrected energy as a function of 4688



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Figure 1. Anharmonic zero-point energy corrected energies as a function of fragment separation for the (a) CH5+, (b) CH4D+, and (c) CH3D2+ systems. Where there are multiple curves in the same graph, they represent the energies for the different possible asymptotic fragmentation patterns. For these plots, the zero in energy is the minimum of the CH5+ potential.

R, E(R), from eq 4 for R decreasing from 6.0 to 0.0 Å and increasing from 0.0 to 6.0 Å. The results for the first of these calculations are plotted in Figure 1. In cases of partial deuteration, separate calculations were performed for each asymptotic channel leading to two curves for the CH4D+ system and three curves for CH3D2+. Plots for the other isotopologues are provided in Figures S1 and S2 of the Supporting Information. At the scale of the plots, the curves that are calculated when R is increased from 0.0 to 6.0 Å are virtually identical to those plotted. Differences are seen for 0.4 < R < 1.2 Å. These differences reflect the simulations getting temporarily trapped in local minima on the potential. All of the curves display the same basic trends. To aid in making comparisons among systems, in Table 1, we report the energies of all systems studied here for R = 1.0 and 5.0 Å. In Table 1, we also report the zero-point energies of these systems

when all internal degrees of freedom are considered, and from that the energy difference between the energy of the system when R = 5 Å and the zero-point energy. This provides an approximation to the dissociation energy. Closer examination, and comparison of the energies for CH5+ and CH4D+ evaluated when R = 5.0 and 15.0 Å indicate that the energy increases by an additional 200 cm−1 when R is increased from 5 to 15 Å. On the other hand, the energy differences between the various isotopologues change by much less. For example, when we compare the energy difference between the CH3+ + HD and the CH2D+ + H2 systems at 5.0 and 15.0 Å, we find that this difference changes by 10 cm−1 . As the potential used in this work does not take into account the effects of basis set superposition error,37 it is anticipated that the dissociation energy obtained by this DMC approach provides an upper bound for the true dissociation energy of CH5+. As the focus of the present work is on comparisons among the energetics of the various isotopologues, we focus this comparison on 5.0 Å separations. This choice is further supported by earlier studies of CH5+ and CH4D+ using the same potential, which indicated that the asymptotic limit is reached at 10 a0.37 Comparing the results reported in Table 1 to those of Jin et al.,37 we find that the energies reported here are roughly 50 cm−1 higher in the cases where they reported values. The 50 cm−1 deviation between the present results and those reported in ref 37 are believed to reflect differences in the way the two calculations were performed. When we follow the procedure outlined in ref 37, our results come into closer agreement with the previously reported energies. For values of R between 1.3 and 3.0 Å, the energy decreases more rapidly. Interestingly, the energy ordering of the isotopically substituted varieties of the same chemical species changes in this range of R. This region will be the focus of the discussion that follows. For shorter distances, the E(R) curves flatten and display small fluctuations. This region represents the potential minima. Because the minimum energy structure of CH5+ has Cs symmetry and only two of the five hydrogen atoms are related by reflection in a plane of symmetry, as R is decreased through this range, the systems sample a series of different minima on the potential surface. Similar behavior is seen when we minimize the potential energy for CH5+ and CH4D+ as functions of R, plotted in Figure 2. For R ≤ 0.5 Å, the energy increases. When the H2 fragment contains two identical atoms, the energy reaches a local maximum at R = 0.0 Å, and the value of this maximum is consistent with the energy of the D3h saddle point on the potential surface, which has an energy of 3850 cm−1 .29 When the H2 fragment contains two atoms with different masses R = 0.0 Å no longer corresponds to

Table 1. Energies for the CH3+···H2 Complexes with R = 1.0 Å and R = 5.0 Åa system CH3+ + H2 CH3+ + HD CH2D+ + H2 CH3+ + HT CH2T+ + H2 CH3+ + D2 CH2D+ + HD CHD2+ + H2 CH3+ + T2 CH2T+ + HT CHT2+ + H2 CH2D+ + D2 CHD2+ + HD CD3+ + H2 CHD2+ + D2 CD3+ + HD CD3+ + D2

E(5.0 Å) − E0 (cm−1)

E(1.0 Å) (cm−1)

E(5.0 Å) (cm−1)

9578 ± 1 9188 ± 2 8842 ± 2

25285 ± 1 24998 ± 2 24713 ± 1

10916 ± 5 10304 ± 5

14369 14694 14409

9091 ± 1 8589 ± 1

24890 ± 1 24472 ± 1

10043 ± 5

14847 14429

8920 ± 1 8519 ± 1

24656 ± 1 24425 ± 1

9696 ± 5

14960 14729

8191 ± 2

24131 ± 1

8632 ± 1 8029 ± 1

24378 ± 1 24077 ± 1

7718 ± 1

23639 ± 1

8172 ± 1

24084 ± 1

7885 ± 1

23842 ± 1

7632 ± 1 7511 ± 1

23537 ± 1 23500 ± 1

8563 ± 5

14443 14937

7361 ± 1 6947 ± 1

23251 ± 1 22909 ± 1

8045 ± 5

14688 14864

E0 (cm−1)

14435 9169 ± 5

15209 14908 14470

9094 ± 5

14990 14748

a

For comparison the zero-point energy E0 from ref 24 are reported along with the difference between E(5.0 Å) − E0, which provides an estimate of D0, as described in the text. 4689



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atom, which is much larger than the mass of any of the naturally occurring isotopes of hydrogen. 3.2. R Dependence of the Probability Amplitudes. The large slopes of E(R) and Eele(R) for 1.3 < R < 3.0 Å, seen in Figures 1 and 2, hints at the fact that partial deuteration may lead to a slight shift in how the H2 fragment is oriented relative to the CH3+ fragment. To test this hypothesis and to further address the question of how quantum effects are reflected in the dynamics, we turn our attention to an analysis of the probability amplitudes for fixed values of R. These results are obtained by running a DMC simulation in which the value of R is held constant. The resulting probability amplitudes can be projected onto the coordinate or coordinates of interest. To start, we focus on θH2 or the angle between R and the H2 bond axis (see inset in the left panel of Figure 3). When R = 1.0 Å, this distribution is narrow and is localized near 90°. As R is increased, the distribution broadens. When the H2 fragment contains identical isotopes of hydrogen, the distribution remains symmetric about θH2 = 90°, as it must be by symmetry. This is seen in the left and right panels of Figure 3. For the case when the system dissociates into CH3+ + HD (shown in the center panel), the symmetry is lost and the distributions are shifted to larger values of θH2. On the basis of the definition of θH2, this corresponds to the hydrogen atom of the HD fragment being closer to the CH3+ fragment than the D atom is. Similar behavior is seen in the other partially deuterated systems. For all three systems shown in Figure 3, we find that by 3 Å the H2 group is sampling all possible orientations (e.g., when the sin θH2 volume is taken into account, the probability amplitude is nonzero at all values of θH2). We have also investigated projections of the probability amplitude onto the two other angles that define the relative orientations of the fragments, θCH3+ and ϕH2 The results are presented in the Supporting Information (Figures S3 and S4). The rapid evolution of the H2 fragment from feeling the potential anisotropy to being a free rotor is also reflected by the projections of the probability amplitude onto the torsion angle between the two fragments. This distribution is found to be isotropic when R = 1.5 Å, as is seen in the results presented in Figure S3 (Supporting Information). The distributions that correspond to the tumbling of the CH3+ fragment also rapidly become isotropic, whereas in the case of CH2D+, strong orientational effects are seen beyond 3.0 Å. This is illustrated in the results that are plotted in Figure S4 (Supporting Information).

Figure 2. Plots of Eele(R) as functions of R based on the global surface of Jin et al.,37 for the CH5+ (black curve) and CH4D+ (red and blue dashed curves) systems. For these plots, the zero in energy is the lowest energy point on each of the curves.

the D3h saddle point and both Eele(R) and E(R) diverge as R approaches zero. In the present study, we focus on the behavior of dissociating CH5+ or combining CH3+ and H2 fragments over distances between the equilibrium structure (R = 1.1 Å) and the asymptotic region (R > 3.0 Å). Over this range of R, the energy increases rapidly with increasing R. Interestingly, whereas the energy ordering of the various fragment channels of the same species is the same at 1.0 and 5.0 Å (Table 1), at intermediate distances the order may shift. For example, as CH4D+ is allowed to dissociate, the CH2D+ + H2 channel is lower in energy than the CH3+ + HD channel for values of R between 1.85 and 2.25 Å, whereas the CH3+ + HD channel is the lower energy one for all other values of R. There are two possible origins for the energy reordering. One involves differences in the quantum zero-point energies of the systems as R is decreased. The other reflects differences in the definitions of R for the different dissociation channels. For a system that dissociates with one of the fragments being H2, D2, or T2, the center of mass of this fragment is at the center of the bond. In contrast, for the mixed isotopic species, the center of mass is shifted toward the atom with the heavier mass. This effect does not require quantum mechanics and is seen in Figure 2 where one can see that the blue dotted (CH3+ + HD) curve does not lie on top of the other two curves between 1.5 and 2.5 Å. This effect was explored briefly in our earlier study of this system and will be analyzed further in the discussion in the following subsection. It is interesting to note that although isotopic substitution in the CH3+ fragment will also shift the position of the center of mass, this does not appear to affect Eele(R) as dramatically. This is because the position of the center of mass of the CH3+ fragment is primarily determined by the mass of the carbon

Figure 3. Probability distributions projected onto θH2 (shown in the inset of the left most plot) for CH3+ + H2 [left], CH3+ + HD [center], and CH2D+ + H2 [right] when R = 1.0 to 3.0 Å. In the case of the CH3+ + HD system, values of θH2 that are larger than 90° correspond to the hydrogen atom being closer to the CH3+ fragment. 4690



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Figure 4. Comparison of projections of the probability distributions for CH3+ + HD onto rCH/D [(a)−(c)] and rHH/D [(d)−(f)] for R = 1.1, 1.3, and 1.5 Å. Here the black curves provide the sums of the red and blue curves. For comparison, the values of these distances obtained from the minimum energy structure at these values of R are plotted with vertical lines at the top. The relative length of the lines represents the number of distances with that value.

Figure 5. Projections of the probability amplitude for CH3+ + H2 [left], CH3+ + HD [center], and CH2D+ + H2 [right] onto rCH or rCD [top] and rHH or rHD [bottom] plotted as functions of R. The symbols represent the average value of the quantity of interest, whereas the error bars provide the width of the distribution.

We next turn to the changes of the geometries of the fragments as R is increased. Here we focus on projections of the probability amplitude onto the CH or HH distances. Because R is constrained to remain constant in these calculations, the hydrogen atoms are not equivalent, and the distributions that are plotted in Figure 4 reflect this inequivalence as CH4D+ dissociates to form CH3+ + HD. Each distribution is plotted with a different color, as indicated in the keys in panels (a) and (d). For comparison, we also indicate the values of these distances based on the minimum energy structures at the same values of R. In Figure 4, the results are plotted for R = 1.1 Å, which is near the potential minimum, 1.3 and 1.5 Å.

Focusing on the CH distance distributions at 1.1 Å, all three of the CH/D distributions overlap. On average the distances between the hydrogen atoms in the H2 fragment and the carbon atom are larger than the CH distances within the CH3+ fragment, but both are inside the envelope of the CH distance distribution for the CH3+ fragment. The HH distance distribution is more structured. When the four distributions are added together (black curve), there is a small peak at roughly 1.2 Å and a larger one at 1.9 Å. Further, the distribution closely resembles the distribution of HH distances that is obtained by plotting the projections of the ground state probability amplitude for CH4D+ onto rCH/D or rHH/D.18 The HD distance in the HD fragment contributes primarily to the 4691



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Figure 4, for CH3+ + H2 and for CH2D+ + HD when the origin of R is placed at the center of mass of HD and the center of mass of H2, respectively. The results are presented in Figures 6

peak at the smallest HH distances, although all four HH/D distance distributions have finite amplitude in this region. At separations of 1.3 or 1.5 Å, the HD distance in the HD fragment becomes the sole contributor to the amplitude in the peak at shortest HH/D distances. This peak also shifts to shorter values of rHD as R is increased, and approaches the asymptotic value of 0.8 Å at infinite separation. Likewise the CH distance distribution becomes bimodal with the peak at larger values of rCH, reflecting the distances between the hydrogen and deuterium atom in the HD fragment and the hydrogen atoms in the CH3+ fragment, whereas the peak at shorter HH distances results from the distances between the hydrogen atoms within the CH3+ fragment. To make the trends easier to visualize, and to allow for comparison among multiple systems and over a larger range of R, in Figure 5 we plot the average values of rCH/D and rHH/D as functions of R. The error bars provide the widths of the distributions. Here we focus on the CH5+ and CH4D+ systems. Analogous plots for the other isotopologues are provided in Figures S5−S10 of the Supporting Information, and display similar trends. To start, consider the CH distances (top panels). The blue triangles represent the distribution of the CH/D distances in CH3+ or CH2D+. This distribution shifts to slightly smaller CH/D distances as R increases. At the same time the width of the distribution is relatively independent of R. The red circles give the average distance between the hydrogen or deuterium atoms in the H2 fragment and the carbon atom. These values and the widths of these distributions both increase with R. The increased width reflects the extended rotational motion of this fragment as the system dissociates. Further, for the HD fragment, the CH distance remains shorter than the CD distance for a given value of R. Similar trends can be seen in the HH distance distributions. To analyze the anisotropy of the CH2D+ tumbling motion, compared to that of CH3+ at the same values of R, we consider the evolution of the HH and HD distance distributions shown in Figure 5(e,f) for R ≥ 3 Å. In panel (e) the two distributions that involve the distance between the hydrogen and deuterium atoms on two different fragments (identified as “rHH both” and “rDH both” in the legend) lie on top of each other. The distribution for rHH is slightly broader, reflecting the larger amplitude motion of the H atom in the HD fragment. In contrast in panel (f) these two distributions diverge at intermediate distances, and it is only at R > 5 Å that they lie on top of each other. Similar trends are seen for the other isotopically substituted variants of CH5+. On the basis of this, and the projections of the probability amplitudes onto the coordinate that describes the tumbling motion of the CH3+ fragment, we conclude that there is a propensity for the deuterium atoms in the CH3+ fragment to lie further from the dissociating H2 fragment. While this validates the emerging picture of the dissociation dynamics, the question remains whether the observed shorter CH distances compared to the CD ones reflects the dynamics of the system or the choice of coordinates used in the study. As indicated above, the definition of R is altered by partial deuteration of the H2 fragment because the origin of R moves from the center of the H2 fragment to a position closer to the deuterium atom. As E(R) decreases with R from 3.0 to 1.3 Å, this change in definition leads to the observed lowering of the electronic contribution to E(R) when the deuterium is placed in the H2 fragment, making it HD. To sort this out, we repeated the analysis of the CH/D distance distributions, shown in

Figure 6. Projections of the probability amplitude for CH3+ + HD [top] and CH3+ + H2 [bottom] onto the CH/D distances for R = 1.5 Å. As illustrated in the insets, in the left panels, the simulations were performed with origin of R placed at center of mass of HD, whereas in the right panels, it is at the center of the H2 bond axis.

Figure 7. Projections of the probability amplitude for CH3+ + HD [top] and CH3+ + H2 [bottom] onto the θH2 angle for R ranging from 1.0 to 3.0 Å. As illustrated the insets, in the left panels, the simulations were performed with origin of R placed at center of mass of HD, whereas in the right panels, it is at the center of the H2 bond axis. The second inset in panel (a) illustrates the definition of θH2, which is defined so that a value of 0° corresponds to the deuterium (blue) atom being closer to the CH3+ group.

and 7, where the resulting probability amplitudes are projected onto the CH/D distances with R = 1.5 Å, and onto θH2 for R ranging from 1.1 to 3.0 Å, respectively. Plots of the projections of the probability amplitude onto the CH/D distances when R = 1.3 and 1.9 Å are provided in Figures S11 and S12 of the Supporting Information. To start, we consider the CH distance distributions, presented in Figure 6 for R = 1.5 Å. The system that is investigated is depicted schematically in the insets of the panels, 4692



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depend on mass. Although challenging, it would be interesting to explore which of these two effects, asymptotic energetics vs orientation of HD as the molecule dissociates, play the larger role in leading to the CH3+ + HD reaction being driven toward CD3+ formation at low temperatures.

and the colors of the hydrogen and deuterium atoms in the H2 group are the same as the color of the CH distance distribution curves that involve that atom. The CH distance distributions involving the hydrogen atoms in the CH3+ group are represented by the dark red line. These distributions are, for the most part, the same in the four panels. The bright red solid and blue dashed lines used to present the results for the CH/D distance distributions involving the hydrogen/deuterium atoms in the H2 group are affected by the mass of the atom shown in blue and the position of the origin of R. The results obtained for CH4D+ (Figure 6(a)) are identical to those provided in Figure 4(b), and display a CD distance distribution that is shifted to slightly larger values of rCH/D compared to the CH distance distribution, plotted in red. In the case of CH5+ (Figure 6(d)), these two distributions are identical. On the other hand, when the origin for R in the CH3+ + HD calculation is shifted to the center of the HD bond, the CH distribution shifts to larger CH distances than the CD one. This is shown in Figure 6(b). This result can be interpreted to mean that as CH4D+ dissociates to form CH3+ + HD, the hydrogen atom tends to leave first. This is also seen in the θH2 distributions plotted in Figure 7. As anticipated by the above results, when the origin of R in the CH3+ + H2 system is shifted from the center of the H2 bond toward the hydrogen atom displayed in blue in Figure 6(c), the CH distance distribution involving the blue hydrogen atom is shifted to longer CH separation compared to the distribution obtained when the red atom in the H2 fragment is used. Taken together, we conclude that the shorter average CH distance (compared to the CD distance) evidenced in Figure 6(a) reflects the position of the origin of R in this system, and when we correct for this, we find that there is a slight propensity the HD fragment to be oriented so that the deuterium end is closer to the CH3+ fragment. As is seen in the distributions for 1.9 Å, provided in Figure S12 (Supporting Information), this propensity is lost at larger value of R. This result is consistent with the dissociative recombination experiments in which there is a clear propensity for loss of hydrogen atoms over deuterium atom loss.27

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ASSOCIATED CONTENT

S Supporting Information *

Plots of the anharmonic corrected energies for the CH2D3+, CHD4+, CD5+, CH4T+, and CH3T2+ systems as well as projections of their probability amplitudes onto various coordinates, and projections of the probability amplitudes for CH5+ and CH4D+ onto two additional coordinates that describe the orientations of the fragmenting partners are provided. This material is available free of charge via the Internet at http:// pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Support through grants from the Chemistry Division of the National Science Foundation (CHE-0848242) is gratefully acknowledged. We thank Professor Joel M. Bowman for many discussions about the DMC studies in ref 37, and for providing us with the codes used to evaluate the potential used in this work. This work was supported in part by an allocation of computing time from the Ohio Supercomputer Center.

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REFERENCES

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4. CONCLUSIONS In this work, we investigated the quantum zero-point energy effects on the association and dissociation of CH3+ + H2 and CH5+. Overall, the energetics follow the electronic energies, and when the system is partially deuterated or tritiated, the energy ordering of the various isotopologues is the same for the complex and the fragments. Further, as has long been recognized, once zero-point energy is considered, the different product channels obtained upon fragmentation of isotopically substituted CH5+ will have different energies, and the channel with the lowest energy is the one with the largest number of hydrogen atoms in the H2 group. This has been previously used to explain the observation that reactions of CH3+ with HD will eventually lead to the formation of CD3+. The quantum nature of the present study allowed us to further investigate the geometries sampled as the partially deuterated system dissociates to explore if these dynamical effects might also contribute to this propensity. Careful analysis of the probability amplitude indicates that indeed when CH4D+ falls apart to form CH3+ and HD, the hydrogen atom tends to be further from the CH3+ fragment than the deuterium atom. This orientation effect, like the energetic one, is quantum mechanical in origin, because classically the minimum energy geometries cannot 4693



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Isotopic Effects on the Dynamics of the CH3+ + H2 â CH5+ â CH3+ +

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Isotopic Effects on the Dynamics of the CH3+ + H2 â CH5+ â CH3+ +