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Minimum Energy Path Diffusion Monte Carlo Approach for Investigating Anharmonic Quantum Effects: Applications to the CHþ 3 þ H2 Reaction Charlotte E. Hinkle and Anne B. McCoy* Department of Chemistry, The Ohio State University, Columbus, Ohio 43210

ABSTRACT A method for evaluating anharmonic corrections to energies along a minimum energy path is developed and described. The approach is based upon the Diffusion Monte Carlo theory as initially developed by Anderson. Diffusion Monte Carlo has been shown to be effective for evaluating the ground-state properties of highly anharmonic systems. By using Jacobi coordinates, the evaluation of anharmonic corrections to the energies along a minimum energy path are straightforþ ward to implement using Diffusion Monte Carlo. In this work, the CHþ 3 þ H2 f CH5 reaction and its singly deuterated analogues are investigated. In addition to exploring how the energetics of this reaction change upon partial deuteration, projections of the probability amplitude onto various internal coordinates are evaluated and used to provide a quantum mechanical description of how deuteration affects the orientation of the two fragments as they combine to form the CH4Dþ molecular ion. SECTION Molecular Structure, Quantum Chemistry, General Theory

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quantum approaches essential, while the number of atoms (six) makes full-dimensional quantum treatments of the dynamics challenging, if not prohibitive.5 As mentioned above, an alternative approach is to define the reaction coordinate and investigate the evolution of the anharmonically corrected energy and the associated wave function as the reacting species are brought together. This will require a formalism for evaluating anharmonic zero-point energies as a function of a reaction coordinate. To address these issues, we use the Diffusion Monte Carlo (DMC) approach. This method can be traced to Fermi, Metropolis, and Ulam,6 and our implementation is based on the work of Anderson7,8 and of Buch.9 DMC has been shown to be an extremely effective approach for studying molecules,10 13 including CHþ11,12 and Hþ that undergo large-amplitude 5 5, motions. Several years ago, Clary and co-workers14 used DMC to perform reaction path studies for the OH þ H2 reaction. In their implementation, they used the MHA reaction path formalism to obtain a reaction coordinate s and evaluated anharmonic corrections by performing the DMC simulation at fixed values of s. In that work, they only allowed displacements of coordinates orthogonal to the reaction coordinate. In the present study, we pursue a different approach in which we use the adiabatic DMC formalism, developed in our group,15 to evaluate the anharmonic zero-point corrected energy as a

he reaction path formalism has seen considerable success in the development of statistical theories of reaction rates and reaction dynamics as well as in reduced-dimensional treatments of quantum dynamics.1 While understanding the minimum energy path that takes a system from one configuration to another is useful, the associated zero-point energy can change dramatically along the path. Miller, Handy, and Adams (MHA)2 developed a reaction path Hamiltonian in which the reaction coordinate has been projected out and the remaining degrees of freedom are treated as normal modes. This formalism has been used in a variety of applications, often to obtain harmonic zero-point corrections to the minimum energy path. Such harmonic evaluations of the zero-point energy correction can be made using a variety of electronic structure packages. In many systems, the vibrations are highly anharmonic at some points along the reaction path, and harmonic treatments cannot fully capture the zero-point correction to the minimum energy path. For example, in systems that dissociate into two polyatomic fragments, six of the vibrations evolve into six additional translations and rotations as the system goes from one molecule with 3N - 6 vibrations to two molecules with a total of 3N - 12 vibrations. There are many molecules for which the vibrations are highly anharmonic, even in their ground states. A particularly extreme example of this is CHþ 5 . When the anharmonic vibrational ground-state wave function is evaluated, it is found that there is comparable amplitude at the 120 equivalent minima and at the 180 low-energy saddle points that connect these minima.3,4 The highly delocalized nature of CHþ 5 makes

r 2009 American Chemical Society

Received Date: November 22, 2009 Accepted Date: December 15, 2009 Published on Web Date: December 29, 2009

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DOI: 10.1021/jz900300u |J. Phys. Chem. Lett. 2010, 1, 562–567

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function of the center of mass separation of the two fragments. As an initial application of this approach, we focus on the CHþ 3 þ H2 reaction and its singly deuterated analogues. These systems are of interest due to their potential importance in astrochemical reactions16,17 and for reactions within planetary ionspheres.18 In kinetics studies, Gerlich and coworkers19 determined that the reactions CH3 -n Dnþ þ HD f CH4 -n Dnþþ 1 f CH2 -n Dnþþ 1 þ H2

ð1Þ

have net rate constants of 1.6 10-9 cm3 s-1 at 15 K. These rate constants are independent of the value of n in eq 1. This value of the rate constant is slightly larger than the Langevin rate constant of 1.32 10-9 cm3 s-1, and after 3 s, more than þ 99% of the CHþ 3 has been converted to CD3 at 15 K. While the efficiency of these reactions can be rationalized by the exothermicity of the reactions due to the smaller zero-point energy of the products of reaction 1 compared to that of the reactants, the question remains if other dynamical effects are at play. Bowman and co-workers investigated the CHþ 3 þ HD reaction using classical mechanical approaches and obtained rates for the forward reaction that are consistent with experiment.20 However, classical mechanics is not expected to provide a complete picture as it is not sensitive to the zeropoint effects that are responsible for the exothermicity of the hydrogen/deuterium exchange reaction. In this work, we investigate changes to the energy and wave functions of the CHþ 5 system as it is formed from the þ CHþ 3 and H2 fragments. From earlier studies of CH5 , we and 3,4,11,12,21,22 found that the ground vibrational state has others equal probability amplitude at each of the 120 equivalent minima and has approximately the same amplitude at these minima and at the 180 saddle points that connect the minima.4,23 Despite the large fluxionality of CHþ 5 , even in its vibrational ground state, projections of the ground-state probability amplitude indicate that, on average, CHþ 5 can be described by a CHþ 3 subunit, connected to H2 by a twoelectron, three-centered bond.3,11,23-25 While all of the hydrogen atoms are equivalent in CHþ 5 , the introduction of a single deuterium atom lowers the symmetry. Specifically, the quantum mechanical ground state of CH4Dþ has the deuterium atom preferentially located in the CH2Dþ subunit.12 One of the strengths of the DMC approach is that it provides a way to obtain a Monte Carlo sampling of the ground-state wave function and, by extension, a statistical sampling of the ground-state probability amplitude that can be readily projected onto any coordinate or coordinates of þ interest. As we consider the formation of CHþ 5 from CH3 and H2, we focus on several projections of the probability amplitude. More specifically, we project the probability amplitude onto the CH and HH distances as well as the Euler angles that describe the orientations of the two subunits relative to the vector that connects their centers of mass, R, as depicted in Figure 1. As the molecular fragments are brought together, the Euler angles evolve from rotational coordinates into two bends and a torision. It should be noted that, unlike earlier studies, the five hydrogen atoms are not treated in an equivalent manner in these simulations. As a result, projections of


Figure 1. Jacobi vectors that define CHþ 5 . The hydrogen atoms in the CHþ 3 subunit are represented by the blue circles, those in the H2 subunit are red, and the carbon atom is represented by the black circle. The Euler angles that are used to define the geometries of the complex as it combines are defined as the angle between RB and B r CHþ3 ,3 (θCHþ3 ) and the angle between RB and B r H2 (θH2). The angle φH2 is defined as the torsion angle involving B r H2, RB, and one of the blue dashed lines, which connects one of the hydrogen atoms in the CHþ 3 subunit to the center of mass of the carbon atom and the other two hydrogen atoms.

the probability amplitudes will be divided into the pieces that depend on coordinates that only involve the atoms in the CHþ 3 subunit and the H2 subunit and those that require information about the atoms in both of the subunits. In this work, we use the Diffusion Monte Carlo approach, as described by Anderson,7,8 to obtain the energies and wave functions of CHþ 5 as functions R. The general approach and our implementation of it are described in detail elsewhere and will not be repeated here.10,26 This implementation of DMC requires that the kinetic energy operator be separable into n contributions that each depend on one Cartesian component of the momentum and that the coefficients be independent of the coordinates of the atoms, for example n p2 X j ð2Þ T¼ 2mj j ¼1 In developing a formalism for separating out a reaction coordinate and evaluating the zero-point energy and anharmonic wave function as a function of this coordinate, we wish to retain the above structure for the kinetic energy operator while introducing a parameter, R, which can be varied to allow the molecule to form from its two constituent fragments. A set of coordinates that provides such a structure for the kinetic energy operator is the Jacobi coordinates. In general, N vectors are needed to describe an N atomic system. One provides the location of the center of mass of the system. As the kinetic energy of the center of mass commutes with the Hamiltonian, the components of this vector can be ignored. A second vector can be defined to connect the centers of mass of the fragments and is denoted as R in Figure 1. The remaining N - 2 vectors provide information about the

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structure and orientation of the two fragments. They are set up either by connecting atoms to the center of mass of previously connected fragments or by connecting the centers of mass of two such fragments. As such, the mass associated with each vector is the reduced mass of the fragments that are connected by the vector. The definition of the vectors that define the separate r H2, which fragments is flexible. In the case of CHþ 5 , we use B is the vector that defines the relative positions of the two hydrogen atoms in the H2 portion (shown in red in Figure 1). The CHþ 3 part is described by three vectors. In the analysis of the probability amplitudes, we employ two sets of coordinates. The first includes the vector between two of the hydrogen atoms, B r CH3þ,1, the vector between the third hydrogen atom and the center of mass associated with B r CHþ3 ,1, r CHþ3 ,2, and the vector that connects the carbon atom to the B center of mass of the three hydrogen atoms, B r CHþ3 ,3. These vectors are identified in Figure 1. A second set of vectors is r CHþ3 ,1, the vector that used to define φH2. This set includes B connects the carbon atom to the center of mass associated with B r CH3þ,1, and the vector that connects the remaining hydrogen atom to the center of mass of B r CHþ3 ,2. The three choices for this last vector are depicted with blue dashed lines in Figure 1. Since we are interested in how the energy and the probability amplitudes depend on the center of mass separation of the fragments, R, and not the orientation of this vector, we take RB = (0,0,R). With the coordinates defined, we perform the DMC simulations in the usual way,7,10 except that the value of R is fixed and only the components of the B r R,i vectors are allowed to diffuse. While we could map out the R dependence of the energy in this manner, it rapidly becomes tedious. Instead, we employ the adiabatic Diffusion Monte Carlo approach, developed by Lee, Herbert, and McCoy.15 This approach allows us to determine how the lowest-energy eigenvalue of the Hamiltonian depends on a parameter in the Hamiltonian, in this case R. More specifically, we make R a linear function of the diffusion time, τ, with a slope chosen so that the change in energy resulting from changing the value of R at each time step in the simulation is small compared to the statistical fluctuations of the energy resulting from the Monte Carlo sampling. In the present work, we increase R by 0.0002 Å at each 10 au time step, and the potential of Jin et al. is used to evaluate the potential energy as a function of the Cartesian coordiantes of the six atoms.22 þ In Figure 2, we plot the energies of the CHþ 3 þ H2, CH3 þ þ HD, and CH2D þ H2 systems as functions of R, for R < 6 Å. In all cases, we have run the simulations in both directions, going from small to large values of R and from large to small values. Both simulations give numerically identical results, as they should. The dynamics represented by the curves in Figure 2 can be divided into four regions. At the smallest distances, R < 0.5 Å, the energy decreases with increasing R, and there is a finite maximum in the potential energy at R = 0 in the reactions of H2, while the energy diverges at R = 0 for the CHþ 3 þ HD system. The R = 0 configuration corresponds to the centers of mass of the two subunits occupying the same point in space. This can be achieved by a D3h saddle point structure, which was found to


Figure 2. Anharmonic zero-point corrected energies for the (a) þ þ þ þ CHþ 3 þ H2 f CH5 , (b) CH3 þ HD f CH4D (black), and CH2D þ H2 f CH4Dþ (red) collision complexes.

have an energy that is 3843 cm-1 above the global minimum on the potential of Jin et al.22 Such a structure can be achieved þ for the CHþ 3 þ H2 and CH2D þ H2 systems. When H2 is replaced by HD, the center of mass is no longer at the midpoint of the bond. As a result, the system cannot sample the D3h saddle point and is forced to sample higher-energy regions of the potential when R = 0. The second region spans from 0.5 to 1.3 Å. It is characterized by a number of local minima in the energy curves. By symmetry, when R is evaluated at the 120 minimum energy structures of CHþ 5 , seven unique values are obtained. We have correlated the local minima in the curves in Figure 2 to these seven unique values of R. The fact that all seven configurations are sampled indicates that the artificial partitioning of the hydrogen atoms into CHþ 3 and H2 subunits does not put any constraints on the possible minima that can be sampled in these simulations. In the case of the singly deuterated complex, we find that the CH2Dþ þ H2 curve lies lower in energy than the CHþ 3 þ HD curve. This is consistent with the localization of the deuterium in the CH2Dþ group in CH4Dþ.21 Next we consider large values of R (>2 Å). In this region, the system should be considered as separated CHþ 3 and H2 molecules. The calculated energy at the longest distance is consistent with this assertion. In the case of the CHþ 3 þ H2 system, the energies that we calculate are in agreement with the energy reported by Jin et al.22 For the CH4Dþ systems, the CH2Dþ þ H2 channel is lower in energy than the CHþ 3 þ HD

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þ þ Figure 3. Projections of the probability amplitudes for the CHþ 3 þ H2 (top), CH2D þ H2 (middle), and CH3 þ HD (bottom) systems onto the three Euler angles defined in the text for values of R ranging from 0.64 to 5.0 Å.

channel, as is anticipated by the zero-point energies of the fragments. The intermediate region is the most interesting of the four. It corresponds to 1.3 < R < 2 Å. In this region, the energy increases with increasing R. For the singly deuterated systems, the energies of the black and red curves cross in this region, placing the CHþ 3 þ HD system lower in energy when 1.8 < R < 2.2 Å. To understand the origins of this crossing and to determine the distance at which we should consider CHþ 5 as a molecule rather than a long-range ion-molecule complex, we turn to an investigation of the probability amplitudes. Specifically, we focus on how the distribution of r CH3þ,3 the values of the angles between the RB and B r H2 or B (Figure 1) depend on R. In Figure 3, we also plot the projections of the probability amplitudes onto the angle φH2. This angle is defined as the torsion angle involving B r H2, RB, and a vector that connects one of the hydrogen atoms in the CHþ 3 subunit to the center of mass of the carbon and two other hydrogen atoms. As described, there are three possible definitions for φH2, and the distributions plotted in Figure 3 represent the sum of the distributions obtained from these definitions. To start, we focus on the CHþ 3 þ H2 system, shown in the top row of Figure 3, where the projections of the probability distributions onto these three angles are plotted for 0.64 < R < 5.0 Å. Arrows indicate the shift in the distributions with R. The longest (5 Å, shown in violet) and shortest (0.64 Å, shown in black) distances provide an indication of the asymptotic behavior. At the longest distances, the three angular distributions are isotropic once one notes that the distributions in θR (R = H2 or CHþ 3 ) include sin θR volume elements. At the shortest distances, the distributions are strongly peaked. The distributions in the two zenith angles are peaked at values of r CHþ3 ,3 being collinear with B r H2 θR that correlate to RB and B perpendicular to these vectors. Likewise, the distribution in φH2 is strongly anisotropic for R < 1.1 Å. These distributions 3 are consistent with the minimum energy structure of CHþ 5 . At intermediate values of R, the values of θH2 and φH2 change smoothly between the two limits. The behavior of θCHþ3 is


more complicated. On the basis of these plots alone, it remains challenging to discern a particular value of R at which the system makes the transition from two separate interacting molecules to a complex. To address this question, we focus on how the projections of the probability amplitude onto the CH and HH distances evolve as R is increased. We find that a value of R = 1.3 Å corresponds to the transition from distributions that more þ closely resemble CHþ 5 to ones that can be described as CH3 þ H2. These projections are plotted in Figure 4 for R = 1.3 Å, and the corresponding distributions for R = 1.2 and 1.4 Å are plotted in Figure S1 and Figure S2 (in the Supporting Information). The distributions are divided into three contributions. Specifically those involving only the hydrogen atoms in the CHþ 3 subunit are plotted with blue dashed lines, those that involve only the hydrogen atoms in the H2 subunit are plotted with red dotted lines, and those that involve one hydrogen atom from each subunit are plotted with purple dot-dot-dashed lines. Darker shades indicate that the distances involve one deuterium atom, while the lighter shades are for distances that only involve hydrogen and carbon atoms. The sums of the distributions are plotted in black. When R g 1.3 Å, the summed CH distance distribution is bimodal. Likewise, at these distances, the HH distance distribution can be described as the sum of two nonoverlapping distributions. One involves the distance between the hydrogen atoms that are connected by B r H2, and the other involves the other nine hydrogen-hydrogen distances. At shorter distances, the two distributions coalesce. Next we consider the two CH4Dþ systems, plotted in the middle (CH2Dþ þ H2) and bottom (CHþ 3 þ HD) panels of Figures 3 and 4. In the case of the CH2Dþ þ H2, the distributions are similar to those for CHþ 3 þ H2. There are small differences. For example, the θCHþ3 distribution is shifted to larger values of the angle, while the φH2 distribution contains both solid and dashed lines. These differentiate between the situations when a hydrogen (solid) or a deuterium atom (dashed) in CH2Dþ is used to define φH2. In the case of CHþ 3 þ HD, the distributions show larger deviations from the other

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important, and classical studies, like those undertaken by Bowman and co-workers,20 provide a good model for reaction dynamics and kinetics. In this Letter, we presented a general approach for investigating reaction paths for highly quantum mechanical systems. We applied the approach to a study of the CHþ 3 þ H2 system and its single deuterated analogues. The plots of energy as a function of the dissociation coordinate show structure at short distances, which is indicative of the multiple minima on the potential surface. By using DMC in this study, we have also investigated how the wave functions change as the system evolves from reactants to complex formation. Analysis of these wave functions allowed us to determine the separation at which the transition from two separated reacting species evolves into a complex in which the hydrogen atoms exchange freely. This provides a way to determine the center of mass separation at which statistical models that are based on long-lived complex formation are expected to become applicable. The situation becomes more interesting when a single deuterium atom is introduced. Here, quantum mechanical zero-point effects lead to the nonstatistical entrance channel geometries, with the hydrogen end of HD approaching the CHþ 3 with greater probability. At low collision energies, this has interesting implications for the hydrogen/ deuterium exchange reactions. Since the full quantum dynamics studies of CHþ 5 are beyond our current capabilities, insights on this and other similar reactions, like HD þ Hþ 3 obtained from the reaction path DMC approach described here will be helpful in directing future classical, semiclassical, and statistical treatments of these important reactions.

Figure 4. Projections of the probability amplitudes for the CHþ 3 þ H2 (top), CH2Dþ þ H2 (middle), and CHþ 3 þ HD (bottom) systems onto the CH (left column) and HH distances for R = 1.3 Å. The colors indicate the locations of the hydrogen (lighter shades) and deuterium (darker shades). As mentioned in the text, the hydrogen atoms are not treated equivalently, and distances involving only the hydrogen atoms in the H2 subunit are plotted in red with dotted lines; those that involve only hydrogen atoms in the CHþ 3 subunit are plotted with dashed lines in blue, while HH distances that involve hydrogen atoms in the two subunits are plotted with dot-dot-dashed lines in shades of purple. The sums of the distributions are represented by the black solid lines.

SUPPORTING INFORMATION AVAILABLE Plots similar to Figure 4 for R = 1.2 and 1.4 Å and the HH and HD distances between the two subunits for the two forms of CH4Dþ at R = 1.3 and 2.0 Å. This material is available free of charge via the Internet at http://pubs.acs.org.

two sets. The changes in the θCHþ3 distribution can be understood by recognizing that the center of mass of HD is shifted from the center of the HD bond to a position that is closer to the deuterium atom. The changes in the θH2 distribution reflect a more fundamental change in the physics. This is also seen in the plots in Figure 4 and Figure S3 (Supporting Information). In this case, the partial deuteration leads to an energy difference between the configurations in which the hydrogen in the HD is closer to the CHþ 3 subunit and is lower in energy than the configurations in which the deuterium atom is closer. This effect is only seen when the zero-point energy is included and is strictly quantum mechanical in nature. There are several consequences of this behavior in the CHþ 3 þ HD system. First, because the center of mass of HD is closer to the deuterium atom, the distance between the center of the HD bond and the center of mass of the CHþ 3 subunit is smaller than R. This difference is responsible for the crossing of the black and red curves in Figure 2. A second and more interesting consequence of this behavior is the conclusion that when low-energy collisions occur between CHþ 3 and HD, the hydrogen end of HD is more likely to collide with the CHþ 3 molecular ion first. This is not a favored configurations for HD exchange reactions, and we expect that direct collisions are likely to be relatively inefficient for hydrogen/deuterium exchange reactions in eq 1. On the other hand, once the CH4Dþ complex is formed, this effect is likely to become much less


AUTHOR INFORMATION Corresponding Author: *To whom correspondence should be addressed. E-mail: mccoy@ chemistry.ohio-state.edu.

ACKNOWLEDGMENT Support through grants from the Chemistry Division of the National Science Foundation through Grants CHE-0515627 and CHE-0848242 is gratefully acknowledged. This work was supported in part by an allocation of computing time from the Ohio Supercomputer Center.

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