Probing the Relationship Between Large-Amplitude Motions in H5+

Article pubs.acs.org/JPCA

Probing the Relationship Between Large-Amplitude Motions in H5+ and Proton Exchange Between H3+ and H2 Zhou Lin† and Anne B. McCoy*,‡ Department of Chemistry and Biochemistry, The Ohio State University, Columbus, Ohio 43210, United States S Supporting Information *

ABSTRACT: Understanding the spectroscopy and dynamics of H5+ is central in gaining insights into the H3+ + H2 → H5+ → H2 + H3+ proton transfer reaction. This molecular ion exhibits large-amplitude vibrations, which allow for the transfer of a proton between H3+ and H2 even in its ground vibrational state. With vibrational excitation, the number of open channels for permutations of protons increase. In this work, the minimized energy path variant of diffusion Monte Carlo is used to investigate how the energetically accessible proton permutations evolve as H5+ is dissociated into H3+ + H2. Two mechanisms for proton permutation are investigated. The first is the proton hop, which correlates to largeamplitude vibrations of the central proton in H5+. The second is the exchange of a pair of hydrogen atoms between H3+ and H2. This mechanism requires several proton hops along with a 120° rotation of H3+ within the H5+ molecular ion. This analysis shows that while there is a narrow region of configuration space over which both isomerization processes are energetically accessible, full permutation of the five protons in H5+ more likely occurs through a stepwise mechanism. Such full permutation of the protons becomes accessible when the shared proton stretch is excited to the vpt = 2 or 3 excited state. The effects of deuteration and rotational excitation of the H2 and H3+ products are also investigated. Deuteration inhibits permutation of protons, while rotational excitation has only a small impact on these processes.

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barrierless when zero-point motion is considered,13−17 while the barrier to internal rotation exceeds 1300 cm−1.13 In contrast near the dissociation limit, the H3+ product is free to rotate, while the proton hop would require the dissociation of H3+ into H2 + H, requiring 35 000 cm−1.18−20 The inclusion of the possibility for permutations of protons that are associated with these large-amplitude motions introduces challenges to theoretical studies. One of the difficulties lies in identifying a set of internal coordinates that provides efficient descriptions of intramolecular vibrations at all energetically relevant molecular geometries. Most theoretical studies that have been reported used a set of Jacobi coordinates that separates the central proton from two outer H 2 diatoms.14,16,21−23 These coordinates are denoted as {r}⃗ in Figure 2. They provide an efficient set of coordinates for calculations that focus on small-amplitude displacements from one of the minimum-energy structures as well as excitation of the large-amplitude proton transfer and torsion vibrations. However, larger basis sets will be required if the permutations brought about by internal rotation of H3+ are to be considered as well. In their study, Song et al. used a second set of Jacobi coordinates that separates H5+ into H3+···H2 complex (Figure 2b).24 These coordinates are denoted as {s}⃗ and

INTRODUCTION One of the simplest but most important proton transfer reactions in the interstellar medium is the barrierless reaction between H3+ and H21−5 H+3 + H 2 ↔ H+5 ↔ H 2 + H+3

(1)

which proceeds through a long-lived H5+ intermediate complex.6 Although the reactants and products in reaction 1 appear identical, they may represent different nuclear spin states brought about by the exchange of a proton between H3+ and H2. Partial deuteration will also lower the symmetry of the reaction. As a result, reactions that lead to exchange of protons either between the outer H2 groups or from H3+ to H2 become of interest. These are illustrated in Figure 1 and are referred to as the hop and exchange pathway, respectively, in the discussion that follows.6−12 As illustrated in Figure 1, some of these permutations are achieved by the large-amplitude motions in the H 5 + intermediate. In the present study we focus on the two largeamplitude motions that lead to the two-proton exchange pathway. The first of these is the proton transfer (pt) vibration, which involves displacement of the central proton along the axis that connects the centers of mass of the two outer H2 units. Large displacements along this coordinate lead to the proton hop process. The second involves the internal rotation (ir) of H3+ about the axis perpendicular to its molecular plane, which corresponds to the exchange pathway. These are illustrated with solid and dashed arrows in Figure 1. Analysis of energetics shows that near the equilibrium the hop exchange is effectively © 2015 American Chemical Society

Special Issue: Dynamics of Molecular Collisions XXV: Fifty Years of Chemical Reaction Dynamics Received: June 16, 2015 Revised: August 3, 2015 Published: August 5, 2015 12109

DOI: 10.1021/acs.jpca.5b05774 J. Phys. Chem. A 2015, 119, 12109−12118

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The Journal of Physical Chemistry A

The other issue we will explore is the effect of deuteration on these processes. This is of interest because asymmetrically deuterated isotopologues of H5+ have nonzero dipole moments and can be detected using rotational spectroscopy. In the cold interstellar medium of interest, the relative abundance of a deuterated species is higher than the relative abundance of the deuterium atom. For example, Roberts et al. reported ratio of abundance, [H2D+]/[H3+] ≃ 0.1 in the interstellar medium, while the universal ratio [D]/[H] ≃ 1.5 × 10−5.29 An explanation that is commonly agreed upon is that these deuterated species are more thermodynamically stable due to their lower zero-point energies.30−32 In this study, we will discuss the deuteration effect on the energies, the wave functions, and the permutations of protons or deuterons (D+) in partially and fully deuterated H5+.

Figure 1. Hop (solid arrows) and exchange (solid and dashed arrows) reaction pathways for H3+ + H2 → H5+ → H2 + H3+.

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THEORY Diffusion Monte Carlo. The implementation of DMC used in this study is based on the approach described by Anderson.33 This approach draws from the parallel between the diffusion equation and the time-dependent Schrö dinger equation when the time variable, t, is replaced by τ = it/ℏ.33,34 Note that while τ appears to be an imaginary quantity, within the context of the simulations it is treated as a real number, and all calculations are performed using real arithmetic. Diffusion Monte Carlo provides a powerful tool for studying ground-state properties of molecular systems, including molecules or molecular ions like H5+ that undergo largeamplitude motions even in their ground vibrational state. It also scales favorably with the system size as long as the electronic energy can be evaluated efficiently.33,35−38 In this section a brief summary of the DMC methodology is provided. More detailed descriptions of our implementation of DMC are provided elsewhere.36,37 In DMC, the imaginary-time time-dependent Schrödinger equation

Figure 2. Two sets of Jacobi coordinates that define the configuration of H5+ in this study. The wedge arrow in (b) represents the body-fixed z-axis (C3 axis) for H3+ and is perpendicular to the plane of H3+.

provide an efficient description for the large-amplitude internal rotation but lose the symmetry associated with the proton hop as they treat the two outer H2 units in an unbalanced way. A third set of coordinates, which has been used in several studies,25,26 replaces S⃗ in Figure 2b with the vector that connects H(5) with the center of mass of s2⃗ . While these coordinates represent a compromise between the {r}⃗ and {s}⃗ sets of coordinates, they do not contain a vector that could be considered as a dissociation coordinate. Further the Hamiltonian contains kinetic couplings. While we have demonstrated that the diffusion Monte Carlo (DMC) approaches used in the present study can handle these kinetic couplings,27 the lack of a dissociation coordinate makes this a less attractive coordinate choice for the present study, and we focus the discussion that follows on the two coordinate sets illustrated in Figure 2. The effects of the choice of coordinates on the calculated results becomes even more pronounced in reduced-dimensional treatments. In a recent study we used a reaction path variant of DMC to investigate the evolution of the wave function of H5+ as it dissociates into H3+ and H2.28 For these calculations, the {r}⃗ set of coordinates provide a better description of the system near the equilibrium geometry, while the {s}⃗ coordinates provide the better description near the dissociation limit. In this study, we extend that work, focusing on several aspects of the dissociation dynamics of H5+. One central question involves the extent to which the two isomerization pathways shown in Figure 1 are available to states that are accessed by linear spectroscopic techniques. A second issue that is of interest is whether there are states that access both pathways, thereby allowing full permutation of the protons. We will explore the effect of rotation of H3+ on the dynamics as the lowest energy state of this ion that is consistent with Pauli’s principle has j = |k| = 1 rather than j = |k| = 0. These states correspond to J = 0 levels of H5+ as well as J = 1 states with |K| = 1.

∂Ψ = −(T̂ + V̂ − Eref )Ψ ∂τ

(2)

is solved in a basis of δ-functions or walkers N (τ )

Ψ( r ⃗ , τ ) =

∑ δ(d)( r ⃗ − rk⃗ (τ))

(3)

k=1

where each of the walkers is localized at a specific configuration of the molecule of interest. In these expressions, the dimension of the system is represented by d, N(τ) is the total number of walkers at time τ, and Eref provides a shift to the zero in energy. The long-time solution to eq 2 is given by lim Ψ( r ⃗ , τ ) → c0 exp[−(E0 − Eref )τ ]Φ0( r ⃗)

τ→∞

(4)

where Φ0 is the ground-state solution to the Hamiltonian, and E0 provides the zero-point energy. Since Eref is used to maintain constant amplitude of Ψ(r,⃗ τ), it can be equated with E0. In the DMC calculations described below, this is achieved by using Eref (τ + δτ ) = V̅ (τ ) − α

N (τ ) − N (0) N (0)

(5)

where α = 1/(2δτ) and V̅ (τ) represents the ensemble averaged potential energy. 12110


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⎡ 2d(τ )d(τ + δτ ) μ (τ )μrecross (τ + δτ ) ⎤ recross ⎥ Precross = exp⎢ − ⎢ ⎥ δτ ⎣ ⎦

The simulation is performed in a series of small time steps, δτ. In each time step the kinetic energy contribution to the Hamiltonian, exp(−T̂ δτ), causes the diffusion of each of the walkers. Specifically, the displacement of a walker in each of the d Cartesian coordinates is determined by a Gaussian distribution with σi =

δτ Mi

(8)

Here, d(τ) represents the distance of the walker from the node, and μrecross is the mass associated with d(τ). In the DMC calculations, the value of Precross is compared to a random number between 0 and 1. If Precross is larger than this random number, the walker is removed from the simulation. Since the nodal surface that is used to define the excited-state wave function is based on a wave function that is obtained from a reduced-dimensional calculation, we use the magnitude of this wave function in place of d in eq 8. Details of this approach are provided in a recent study,15 and the expressions for μrecross used in this study are provided in the Supporting Information. Minimized Energy Path Diffusion Monte Carlo. To understand the H+3 + H2 reaction, we use DMC to scan the energies and wave functions as functions of either the distance between H3+ and H2, S, or the distance between the outer H2 groups, R. These coordinates are illustrated in Figure 2. This is achieved using the minimized energy path variant of DMC.42 In this approach, Eref is scanned as a function of R or S using the adiabatic DMC approach.43 In this way, Eref provides the sum of the potential energy from the full-dimensional potential energy surface and the zero-point energies from all other degrees of freedom. As such, Eref will be referred to as the one-dimensional “zero-point corrected potential energy surface,” and is denoted as VZPE. More details of this approach and its application to H5+ have been reported previously.28,42 The scans of VZPE(R) or VZPE(S) provide one-dimensional potential surfaces in the scanned coordinate. We can approximate the solution to the full-dimensional system within an adiabatic approximation by calculating the energy levels and wave functions within a discrete variable representation (DVR).44 In addition, we can explore how the wave function in the other degrees of freedom change as R or S is altered by performing DMC calculations for chosen values of R or S. One challenge in applying this approach to the evaluation of excited states of H5+ arises from the fact that in some cases the lowest-energy state with the specified node involves permutations of the protons. This most often occurs through the exchange of a pair of protons between the two outer H2 units when the reaction coordinate is defined by S or from rotation of the H+3 unit when R is the reaction coordinate. To minimize the effect of this undesired permutation, we evaluate the sum of the distances from any hydrogen H(k) to all of the other four hydrogen atoms,28

(6)

where Mi is the mass associated with the ith coordinate. Following the diffusion the potential energy is evaluated at the new position of the walker, and the value of Wk = exp{− [V ( rk⃗ ) − Eref ]δτ }

(7)

determines the contribution of the kth walker to the ensemble. The integer part of Wk provides the number of walkers that are placed at the new geometry of the kth walker, and the decimal part of Wk provides the probability that one additional walker will be added to the ensemble at the geometry of the kth walker. At the end of each time step, Eref is updated according to eq 5. Once the system is fully equilibrated, Eref fluctuates around E0, and the distribution of walkers converges to the groundstate wave function Φ0(r)⃗ . To obtain the probability amplitude associated with the ground state, the descendent weighting scheme discussed by Suhm et al. is used for this study.35 Calculations of Excited States. As is described above, the DMC approach can be used to obtain the zero-point energy and the ground-state wave function. To evaluate energies and wave functions for excited states, we employ the fixed-node approximation introduced by Anderson.39 This approach is based on the observation that an excited-state wave function goes to zero with a finite slope at a node in the same way as a wave function approaches zero near an infinite potential barrier. If the shape of the nodal surface is known, the potential energy surface is modified by making V(r)⃗ infinite in all but one of the nodal regions. Ground-state DMC calculations are performed using this modified potential energy surface, and the calculations are repeated for each of the nodal regions. Because the amplitude of the wave function must be zero in regions where the potential is infinite, any walker that moves out of the nodal region of interest is removed from the ensemble. For these studies, the nodal surfaces can be determined by symmetry and by using the reduced-dimensional calculations.40,41 A useful check of the validity of the nodal surface used for the calculation comes from the requirement that the energies obtained in all nodal regions must be equal. The fact that the simulations use finite time steps introduces an additional complication. Specifically, even in cases where the walker is in the nodal region of interest before and after the coordinates of the atoms are displaced, there is a probability that the walker would have moved out of this nodal region and back into it had smaller time steps been taken. The probability of such a recrossing event occurring, Precross, is largest when the walker is near the node and dies off as the distance of the walker from the node increases. To address this, a recrossing correction needs to be introduced. This is achieved by calculating the probability for recrossing,39

rkT =

∑ rjk j≠k

(9)

The central proton (which should be H(5)) should correspond to the smallest rTk . In the calculations using {r}⃗ any walker for which rT5 is not the smallest is removed from the ensemble, while in those using {s}⃗ any walker for which rT3 or rT4 is the smallest is removed from the ensemble. Numerical Details. All of the DMC calculations on H+5 and its isotopologues are performed with δτ = 10 au, α = 0.05 hartree, mH = mp + me = 1.00782510 amu and mD = mp + mn + me = 2.01649006 amu, using the potential energy surface developed by Xie et al.13 This is one of two potential surfaces that have been reported based on CCSD(T) calculations. The other is a triatomicsin-molecules potential developed by Aguado et al.45 Both surfaces have similar energies at the relevant stationary points, and both give 12111


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The Journal of Physical Chemistry A similar energy levels for calculations with J ≤ 2.14−17 This leads us to expect that both surfaces would yield similar results for this study. Our choice to use the potential developed by Xie et al. is based on the large difference in the computational resources required to run DMC calculations using these two surfaces. For each of the calculations for fixed-R or fixed-S the ensemble of walkers is equilibrated for 20 000 time steps, and the energies are averaged over the next 61 600 time steps. The ground-state calculations are initialized with all walkers placed at a C2v geometry with the terminal H2 groups rotated by 90° and with the reaction coordinate, R or S, adjusted to the chosen value. The excited-state calculations are initialized with the equilibrated distribution of walkers obtained at the end of the corresponding ground-state calculations. To obtain the probability amplitude associated with a particular value of R or S, a total of 16 copies of weights are collected using descendent weighting.35 Each weighting process is performed for 35 time steps, and two weighting processes are separated by 3815 time steps. For each of the adiabatic scans, the ensemble of walkers is also equilibrated for 20 000 time steps. The value of R or S is then varied by a small amount during each of the next 61 600 time steps. To obtain VZPE, we performed five scans in which R or S starts in the dissociation limit (R or S = 6.0 Å) and decreases to its value near the equilibrium structure (R or S = 1.5 Å), and another five scans start near the equilibrium structure and end in the dissociation limit. The VZPE(R) curve is obtained by averaging the results of these 10 scans. In contrast, the VZPE(S) curve is obtained from the average of the scans with increasing S for S < 2.50 Å and from the average of the scans with decreasing S when S > 2.50 Å. This is done to avoid sampling geometries in which the hydrogen atoms have permuted. This approach differs from that taken in our earlier study28 where the VZPE(S) curve was obtained by decreasing S from 6 to 1.5 Å. As a result, we also reanalyzed the results from that study using the revised form of VZPE(S). In the cases where the wave function in two nodal regions, namely, ΨT > 0 and ΨT < 0, can be related by symmetry, five DMC calculations are performed for ΨT > 0 to obtain the statistics, and one calculation is performed for ΨT < 0 to validate the nodal surface. In the cases where ΨT > 0 and ΨT < 0 cannot be related by symmetry, five calculations are performed in each of the nodal regions. In these cases, the full wave function is constructed by scaling the relative amplitudes in the two nodal regions to ensure continuity of its first derivative.46 All of the DVR calculations are performed based on an evenly spaced grid of 616 points over a range from 1.5 Å < R or S < 6.0 Å. The one-dimensional potential energy surface VDVR(R) and VDVR(S) are obtained by averaging VZPE(R) and VZPE(S), respectively, over results in groups of 100 time steps.

Figure 3. Minimized energy paths (VZPE, solid curves) and approximate zero-point energies (E0, dotted lines) calculated for the ground state of [H2−H−H2]+ (black), [H2−H−D2]+ (red), [D2−D− H2]+ (blue), and [D2−D−D2]+ (green), as functions of (a) R and (b) S [see Figure 2b].

Table 1. Ground-State Energies (in cm−1) for Four Isotopologues of H5+ Based on the Potential Scans Plotted in Figure 2 a E(R) 0 V(R) ∞ b D(R) 0 (S) E0 V(S) ∞ b D(S) 0 (DMC) d E0 b D(DMC) 0

[H2−H−H2]+

[H2−H−D2]+

[D2−D−H2]+

[D2−D−D2]+

7069.7(0.9) 9347.5(0.5) 2277.8(1.0)c 6925.6(1.0) 9431.0(0.4) 2505.5(1.0)c 7205.2(1.1) 2225.9(1.2)c

6257.1(1.6) 8648.2(0.2) 2391.1(1.6) 6141.3(1.4) 8800.6(0.7) 2659.3(1.6) 6371.1(1.0) 2429.5(1.2)

5859.0(1.5) 8152.9(0.1) 2294.0(1.5) 5743.2(1.2) 8180.0(0.5) 2437.3(1.3) 5995.3(1.1) 2185.2(1.2)

5046.6(1.5) 7469.5(0.3) 2422.6(1.5) 4946.1(0.9) 7550.1(0.5) 2604.0(1.0) 5149.0(0.4) 2401.1(0.7)

a Statistical uncertainties based on five independent DMC calculations are provided in parentheses. bThe numbers in parentheses provide combined uncertainties in E0 and V∞. cThe measured D0 is 2400 cm−1.47,48 dResults of full-dimensional DMC calculations.17

as functions of R and S, respectively, using the minimized energy path DMC approach for the four isotopologues of interest. The results are plotted as solid curves in Figure 3. The zero-point energies, E(R/S) , are evaluated based on these one-dimensional 0 potential curves as described in the previous section. They are shown with dotted lines in Figure 3. The calculated values of the E0’s, the asymptotic values of VZPE, V∞, and their differences, D0, are provided in Table 1. We approximate V(R/S) by the value ∞ of V(R/S) ZPE evaluated when R or S has reached a value of 6 Å as this quantity does not change when R or S is increased from 6 to 10 Å.13 The calculated zero-point and dissociation energies obtained from previously reported17 full-dimensional calculations are also provided in Table 1, and are identified as DMC. Comparing the results for the two sets of coordinates, the E(R) 0 values are closer to the full-dimensional values and show smaller variability (102−136 cm−1) than the reported E(S) 0 (230−280 cm−1). This reflects the fact that near the equilibrium configuration, H5+ is better represented by two H2 units that interact equivalently with the excess proton. This symmetry is captured in the {r}⃗ coordinates, but not the {s}⃗ ones. In contrast, at R = 6 Å or S = 6 Å the system has reached the H3+ + H2 asymptote, and the {s}⃗ coordinates provide the more separable description of the molecular motions in this region of the potential. Deviations between D(R) 0 and the results of the fulldimensional calculations reflect the poorer description of the wave function near the H3+ + H2 asymptotes provided by {r}⃗ . That said, both {r}⃗ and {s}⃗ capture the general trend in the dissociation energy upon isotopic substitution providing further validation of the approaches.

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RESULTS Ground-State Properties for H5+ and Its Deuterated Isotopologues. To understand the effect of deuteration on the proton exchange process, we perform the studies for different isotopologues of H5+. In this work we focus on the isotopologues where the structures can be generically described as [B2−B−A2]+ near the equilibrium and which dissociate to B3+ + A2. Here we use A and B to represent hydrogen or deuterium atoms. Ground-State Minimized Energy Paths. The ground-state zero-point corrected potential energy curves, V(R/S) ZPE , are calculated 12112


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The Journal of Physical Chemistry A Effects of Deuteration on Proton Transfer Vibration. We are interested in the effects of deuteration on the proton permutation dynamics, focusing on the two large-amplitude motions, the proton transfer vibration, and the internal rotation of H3+ illustrated in Figure 1. In our earlier study28 the proton transfer vibration was described using the Z-component of r3⃗ in Figure 2a, r3,Z. Because the origin of r3⃗ is at the center of mass of the outer H2 groups, the primary changes in the projection of the probability amplitude onto r3,Z upon asymmetric deuteration of the outer H2 units reflect the shift in the origin of this vector. This can be accounted for by shifting the origin of the plots of the projections of probability amplitude onto r3,Z by the position of the node in the first excited-state wave function for this isotopologue.17 This shift is chosen based on the observation that the node in the wave function lies close to the transition state for proton transfer. This corresponds to a shift of +0.472 Å for [H2−H−D2]+ and −0.426 Å for [D2−D−H2]+. The transition states for the proton transfer vibration in [H2−H− H2]+ and [D2−D−D2]+ are located at r3,Z = 0, while the transition states in [H2−H−D2]+ and [D2−D−H2]+ are shifted to the side closer to the H2 end. The resulting projections of the probability amplitude onto r3,Z are plotted for 2.2 < R < 3.0 Å in Figure 4.

namics. Here, rts is the value of r3,Z at the transition state, and δr = 0.04 Å. Although the two minima along the proton transfer vibration are equivalent on PES, the introduction of zero-point energy in other vibrations breaks this symmetry in [H2−H−D2]+ and [D2−D−H2]+. On the basis of the ground-state probability amplitude for these two isotopologues obtained in our earlier study,49 we expect that as R increases the central hydrogen or deuterium atom will be localized closer to the D2 end. This is further consistent with the relative asymptotic energies (evaluated using {s}⃗ ) of H2 + HD2+ compared to H3+ + D2 (8630.3(0.2) cm−1 versus 8800.6(0.7) cm−1) and D2 + H2D+ compared to D3+ + H2 (8417.6(0.4) cm−1 versus 8180.5(0.5) cm−1). This is also generally consistent with the results plotted in Figure 4. Note that, while DMC faithfully reproduces the probability amplitude when the wave function is localized in a single region of the PES, this is no longer the case for the relative amplitude when the wave function is localized in two or more geometrically separated regions with zero amplitude in the transition state between them. For [H2−H−H2]+ and [D2−D− D2]+ symmetry is used to correct the plots shown in Figure 4a,d as we require that P(r3,Z) = P(−r3,Z) in these cases. For R > 2.50 Å in Figure 4b,c, P(r3,Z = rts) ≃ 0, and the relative amplitude at positive and negative values of r3,Z needs to be interpreted with caution. Finally, as we compare the probability amplitude near r3,Z = rts, the transition state for the proton transfer, we find it is strongly correlated to the identity of the central atom. Specifically, when a hydrogen atom is in the central position, the amplitude is larger than when the central atom is a deuterium. Further this amplitude is insensitive to the deuteration of the outer hydrogen atoms. This effect and the double well potential lead to the unusual observation that the deuteration of the central hydrogen atom broadens P(r3,Z). Effects of Deuteration on Internal Rotation of H3+. The second large-amplitude motion we consider is the internal rotation of H3+ described by χ3. In these systems, the minima along the potential energy cut correspond to χ3 = 0 and ±2π/3, and the transition states are located at χ3= ±π/3 and ±π. In Figure 5 we present the projection of the probability amplitude onto χ3 for specific values of S for all four isotopologues of interest. On the basis of these plots we find that as the value of S is increased, the distributions become more isotropic and that the probability amplitude at the transition states increases. The probability amplitude near the transition state can be used to characterize the feasibility of permutations of the atoms that make up B3+ in B3+···A2. In this study δχ = 3° is used to define

Figure 4. Projections of probability amplitude onto the coordinate r3,Z defined in Figure 2a for different values of R (see inset) and each isotopologue of interest: (a) [H2−H−H2]+, (b) [H2−H−D2]+, (c) [D2−D−H2]+, and (d) [D2−D−D2]+. For [H2−H−D2]+ and [D2− D−H2]+ the origin was shifted by +0.472 and −0.426 Å, respectively, to facilitate comparisons among the plots.

Pts(ir)(S) =

rts + δr

∫r −δr ts

P(r3, Z ; R )dr3, Z

P(χ3 ; S)dχ3

(11)

As is seen for the proton transfer vibration, deuteration of B3+ lowers the probability amplitude near the transition state as we expected, and the amplitude is most sensitive to whether B3+ is H3+ or D3+ and not the identity of A2. This is also seen by the approximately equal values of P(ir) ts (S) for the pair of [H2− H−H2]+ and [H2−H−D2]+ and the pair of [D2−D−H2]+ and [D2−D−D2]+. Permutation of the Protons. Taken together, the results described above show that for smaller values of R or S, the permutations resulting from transfer of the central proton are important, but permutations that are associated with internal rotations, (e.g., permuting the central and an outer proton) are

Focusing on a single isotopologue, we find that, as R is increased, the amplitude at the transition state decreases. This is expected as the barrier to proton transfer between the two outer H2 units increases from 0 cm−1 to roughly 9200 cm−1 over this range of R.28 The probability amplitude near the transition state Pts(pt)(R ) =

π /3 + δχ

∫π /3−δχ

(10)

is used to determine the range of R over which the proton transfer is expected to play an important role in the dy12113


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as the value of S below which the tunneling splitting associated with internal rotation is below the threshold value. For H5+, −4 (pt) these definitions correspond to P(pt) and ts (Rpt) = P0 = 2 × 10 (ir) (ir) −3 Pts (Sir) = P0 = 1 × 10 . For the other isotopologues, we define Rpt and Sir to be the values of R and S for which P(pt) ts and + P(ir) ts have the same values as we use for H5 . In all cases, these correspond to smaller tunneling splittings compared to H5+. and P(ir) The values of P(pt) 0 0 are depicted with dotted lines in Figure 6, and the associated values of Rpt and Sir are reported in Table 2. Table 2. Ranges of R and S over which Proton Transfer and Internal Rotation Are Expected to Affect the Dynamics Are Reported along with the Tunneling Splitting Associated with the Corresponding Large-Amplitude Vibration species [H2−H−H2] [H2−H−D2]+ [D2−D−H2]+ [D2−D−D2]+

+

Figure 5. Projections of probability amplitude onto the coordinate χ3 defined in Figure 2b for different values of S and each isotopologue of interest: (a) [H2−H−H2]+, (b) [H2−H−D2]+, (c) [D2−D−H2]+, and (d) [D2−D−D2]+.

Rpt (Å)

νpt (cm−1)a,b

Sir (Å)

νir (cm−1)a,b

2.59 2.59 2.50 2.52

1.8(0.8) 0.6(0.4) 0.0c 0.7(0.6)

2.41 2.38 2.57 2.57

1.5(0.7) 1.4(0.6) 0.0c 0.0c

a

Tunneling splitting, obtained as the difference between the groundand excited-state energies. bThe numbers in the parentheses are the combined uncertainties of ground state and the first excited state. cThe magnitude is smaller than the uncertainty.

not energetically relevant. In contrast, at large values of R or S, the atoms can permute within the H3+ and H2 fragments, but cannot be exchanged between them. Here we focus on exploring the range of R/S over which each type of permutation is likely to occur, and whether the mechanism for both types of permutations taking place is occurring in a concerted or stepwise manner. To explore these questions, we plot P(pt) ts (R) (pt) and P(ir) ts (S) in Figure 6. As discussed above Pts (R) decreases with R, while P(ir) ts (S) increases with S.

Spectroscopic Implications of Proton Permutations. An interesting spectroscopic question that is raised by the above analysis is whether states for which full permutation of the protons is taking place can be probed experimentally. Intense transitions to the fundamental in the proton transfer vibration along with excitation of high overtones were observed in the infrared spectra of H5+.50 In an effort to understand this spectrum, we demonstrated that the excited states of H5+ with vpt ≥ 2 have significant probability amplitude that extends into the dissociation channel and far from the minimum energy geometry.49 The large amplitude of this vibration leads us to expect that H5+ samples regions of the PES where both the proton hop and internal rotation saddle points that lead to proton permutation should be accessible even at the relatively low levels of vibrational excitation. To address this question, we project the probability amplitude obtained from full-dimensional DMC calculations17 onto R and S. For [H2−H−H2]+ and [D2−D−D2]+, this analysis is performed for the ground state as well as the excited states with up to three quanta in the proton transfer vibration (vpt = 0−3). Because of the lowered symmetry of the partially deuterated isotopologues, only states with vpt = 0 or 1 are considered for [H2−H−D2]+ and [D2−D−H2]+. When an asymmetrically deuterated H5+ ion dissociates, the central proton/deuteron can attach to either of the outer H2/ D2 groups. As such two sets of dissociation products are possible. When we study the partially deuterated H5+, we focus on the isotopologues that correspond to the [B2−B−A2]+ equilibrium structure and B3+···A2 in the dissociation channel. In this arrangement, the central proton, H(5), is defined as the one that has the smallest total distance to other four atoms, as defined in eq 9, while H(1) H(2) is defined as the H2 unit that is closer to H(5). The resulting probability distributions for [H2− H−H2]+ are provided in the Supporting Information. For asymmetrically deuterated isotopologues [H2−H−D2]+ and [D2−D−H2]+, only the contribution to the probability amplitudes that corresponds to the central proton being

Figure 6. Probability amplitude near the transition state as functions of (a) R and (b) S for the species of [H2−H−H2]+ (black), [H2−H−D2]+ (red), [D2−D−H2]+ (blue), and [D2−D−D2]+ (green). P’s are the probabilities defined in eqs 10 and 11.

To set cutoff values for the range of R or S over which the proton hop and exchange can occur, we focus on the tunneling splittings associated with the proton transfer and internal rotation, respectively. We define the cutoff values for R and S for H5+ to correspond to the values of these coordinates when tunneling splittings are twice the size of the uncertainties obtained from the DMC calculations. This choice is dictated by the statistics of the DMC simulation and provides a semiquantitative measure of the importance of the various isomerization processes for the ground and selected excited states of H5+. For this analysis, we define Rpt as the value of R below which the tunneling splitting associated with the proton transfer vibration exceeds the threshold. Likewise, Sir is defined 12114


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Table 3. Probability Amplitude Sampled by the Ground and Excited States in the Proton Transfer Vibration in Each Region of the Configuration Space along with Their Energies in cm−1 a,b species [H2−H−H2]+

[H2−H−D2]+ [D2−D−H2]+ [D2−D−D2]+

state g.s. vpt = 1 vpt = 2 vpt = 3 g.s. vpt = 1c g.s. vpt = 1c g.s. vpt = 1 vpt = 2 vpt = 3

energy 7205.2(1.1) 369.1(1.5) 673.2(2.3) 982.1(1.4) 6371.1(1.0) 382.8(0.8) 5995.3(1.1) 265.7(1.4) 5149.0(0.4) 240.7(1.1) 472.6(1.0) 712.6(1.0)

I 1.7% 6.6% 12.5% 10.7% 2.2% 4.1% 6.1% 8.2% 2.5% 9.3% 22.2% 42.4%

II 97.2% 86.8% 67.0% 43.2% 95.6% 89.7% 93.8% 91.5% 97.5% 90.5% 72.9% 54.1%

III 1.1% 6.6% 20.2% 45.6% 2.1% 6.1% 0.1% 0.3% 0.0% 0.2% 0.9% 3.5%

IV 0.0% 0.0% 0.3% 0.5% 0.0% 0.2% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%

The zero-point energies are reported for the ground states, and the numbers in parentheses provide the statistical uncertainties from five independent DMC calculations. The excited-state energies are reported relative to the zero-point energies, and the numbers in parentheses are the combined uncertainties from the ground and excited states.17 bThe energetically feasible permutations are discussed in the text for each region. cThe nodal structure for the vpt = 1 states of [H2−H−D2]+ and [D2−D−H2]+ are determined using the ADMC approach. a

closer to H2 or the central deuteron being closer to D2 is plotted. In the discussion that follows, configuration space is divided into four regions based on the accessible proton permutations. These regions are determined by the values of R and S compared to the values of Rpt and Sir reported in Table 2. The probability amplitudes provided in Table 3 are obtained from an analysis of the probability amplitudes provided in the Supporting Information. We begin by considering Region II (small values of R and S). Configurations in this region correspond to structures near the minimum energy geometry, where the central proton is free to exchange between the two outer H2 groups, while the barrier to internal rotation is high. In all but one of the calculated states, this region represents the majority of the probability amplitude. For the same state, the probability amplitude in this region is generally larger when there is a deuterium rather than hydrogen atom in the central position. Region III corresponds to large values of both R and S where the system more closely resembles the dissociated products and the rotation of H3+ is close to barrierless, while the barrier to proton transfer is high. In the ground state there is a small amount of amplitude in this region. As before, changes in this amplitude with deuteration reflect changes in the values of Rpt and Sir with deuteration along with differences in the amplitude of the motions. In general, the amplitude in Region III increases with vibrational excitation and decreases by more than an order of magnitude with deuteration of the shared proton. The relative magnitude of the probability amplitude in Regions II and III depends on the definitions of Rpt and Sir. To check the above observations we explored the extent to which permutations of the outer and central proton were observed in full-dimensional DMC simulations in which the walkers were initiated in the minimum energy geometry for ground-state calculations and taken from a ground-state distribution for excited-state calculations. We focus on how many of the protons are found in the central (H(5)) position. In the groundstate calculations, all of the walkers show the same proton in this position. This is consistent with the very small amplitude in Region III for all of the isotopologues. In the DMC calculations on H5+, we obtain the state with one quantum into the shared proton stretch by placing a node in the wave function when H(5) is equidistant from the centers of mass of the two outer H2

units. While this choice clearly corresponds to the state of interest, there is a lower energy state that also satisfies this criterion. It is one in which H3+ has rotated by 120°, and there is excitation in the internal rotation. In previous studies, we circumvented this problem by requiring the central proton to be H(5).17,28,49 Removing this constraint and following the dynamics in the DMC simulations allow us to interrogate the accessibility of this isomerization in excited states. Specifically, we evaluate the time at which the nature of the excited state evolves from excitation in the proton transfer vibration to the internal rotation. In H5+, this occurs after roughly 1000 time steps, while for D5+ the isomerization occurs in only two of 10 simulations of 50 000 time steps, and in both cases it happens after more than 10 000 time steps. Corresponding behavior is seen for [H2−H−D2]+ and [D2−D−H2]+. These results are consistent with the amplitude in Region III for the vpt = 1 states of these isotopologues reported in Table 3. Note that earlier studies of H5+ and its deuterated analogues did not fully explore the role of the internal rotation on the vibrational energy levels.14,16,21−24 On the basis of the above analysis, we believe that there could be significant coupling between the proton transfer and internal rotation even at vpt = 1. Region IV (R ≤ Rpt and S ≥ Sir) is potentially the most interesting of the four regions as it corresponds to both largeamplitude motions to be energetically important. In this region, the permutation between any pair of protons can be achieved. The fact that the amplitude in this region is very low does not dismiss the possibility that full permutation of the protons can occur. Rather it points to a stepwise mechanism for this process in states, like the excited states in vpt, where there is significant amplitude in both Regions II and III. Excited-State Properties for H5+. In this section, we will discuss the effect of rotational excitation of H3+ and H2 on the proton transfer reaction illustrated in Figure 1. We will focus on the effects on the evolution of energies, wave functions, and the energetically accessible permutations of protons. As we analyze the results of these calculations, we focus on two issues. First, we explore the types of vibrational excitation of H5+ that correlate to the asymptotic nodal structures that are used to define these excited-state wave functions. Second, we investigate how introducing nodes that correspond to rotational excitation of H3+ affects how the wave function samples the four regions of the potential, discussed above. 12115


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coordinate associated with internal rotation of H3+, but the small splitting is not resolvable by the DMC calculations. Finally, we note that the states obtained by excitation of H3+ and H2 are not degenerate, and this reflects the broken symmetry of this coordinate system. The second issue involves how rotational excitation of H3+ affects the J = 0 results presented in Table 3. Intuitively, exciting the rotations, particularly the |k3| = 1 levels, should correspond to excitation of the coordinate that corresponds to internal rotation. Likewise, this excitation will shift the central proton away from the axis that connects the centers of mass of the two outer H2 units and will increase the size of the effective barrier to proton transfer. These results are quantified in Figure 8 in which we plot

Figure 7. Minimized energy paths (VZPE; solid lines) and approximate zero-point energies (E0; dotted lines)44 calculated as functions of S as defined in Figure 2b for the following |j3,|k3|,m3,j2,m2⟩prpm states: (a) 0, 0, 0, 0, 0 ++ (black), 1, 0, 0, 0, 0 ++ (red), 1, 0, 1, 0, 0 ++ (blue), 1, 1, 0, 0, 0 ++ (green), and 1, 1, 1, 0, 0 ++ (gray); (b) 1, 0, 0, 1, 0 ++ (dark yellow), 1, 0, 1, 1, 1 ++ (dark cyan), 1, 0, 1, 1, –1 ++ (wine), and 1, 0, 1, 1, –1 +– (violet).

Pts′ (α) =

Pts(α)(e.s.) Pts(α)(g.s.)

(12)

Here α represents either proton transfer or internal rotation, and (ir) P(pt) ts and Pts are defined in eqs 10 and 11. The reported results are consistent with the expectations, except for excitation of the torsion (m2 or m3 = 1 and k3 = 0), which has no appreciable effect on P(pt) ts .

We start by analyzing VZPE(S), plotted in Figure 7, for states with rotational excitation in the H3+ fragment. Since we introduced one or more nodes in the wave functions, the ground-state energies that are obtained by solving the onedimensional Schrödinger equation based on VZPE(S) provide an approximation to the excited-state vibrational energy within an adiabatic separation of S from the remaining vibrational degrees of freedom. The results of this analysis are reported in Table 4 along with the energies for states with rotational excitation of H2 obtained from the VZPE reported previously.28 As is seen, the agreement between the calculated energies and those reported previously14,15,24,50 is generally very good. The assignments, which were based on the nodal structure of the wave function, point to the fact that rotational excitation of the H3+ or H2 products correlates to excitation of the torsion (vγ > 0) or in-plane bend (vb > 0) of the outer H2 units. In some cases the resulting state reflects the upper state in the tunneling doublet associated with the internal rotor, which is indicated by a superscript u in Table 4. In these cases, analysis of the wave function indicates that there is a node in the

Figure 8. Relative probability amplitude at the transition state, (a) P′ts(pt) and (b) Pts′(ir), plotted as functions of S for the |j3 , |k 3| , m3 , j2 , m2⟩p p r m

states: 0, 0, 0, 0, 0 ++ (black), 1, 0, 0, 0, 0 ++ (red), 0, 0, 0, 1, 0 ++ (purple), 1, 0, 0, 1, 0 ++ (dark yellow), 1, 1, 0, 0, 0 ++ (green), 1, 1, 0, 0, 0 –+ (pink), and 1, 1, 1, 0, 0 ++ (gray).

Table 4. Energies (in cm−1) for the Selected Excited States Based on DVR Calculationsa asymptotic stateb

equilibrium assignmentc

0, 0, 0, 1, 0⟩++

0, 0, 0,

0, 0, 0, 1, 1⟩++

1, 0, 0, 1,

1, 1, 1, 1,

1, 1, 1, 0,

0, 0, 1, 0,

0, 0, 0, 0,

0⟩–+ 0⟩++ 0⟩++ 0⟩++

1, 0, 0, 1,

1, 0, 0, 1,

0, 0, 0,

1, 0, 1, 0, 0⟩++

1, 1, 1,

1, 0, 0, 1, 0⟩++

0, 0, 0,

1, 0, 1, 1, 1⟩++

2, 2, 0,

1, 0, 1, 1, –1⟩++

0, 0, 2,

1, 0, 1, 1, –1⟩+–

0, 0, 2,

l 1⟩++ l 0⟩++ u 0⟩++ u 0⟩++ u 0⟩++ l 1⟩++ l 0⟩++ l 2⟩++ l 0⟩++ l 0⟩++ l 0⟩+–

E0d 770.2(2.2)

DMCe

49.0(1.4)

59.1 (0.8)

0.0 0.0 59.1(1.1) 663.3(1.5)

VCI50 701/704

FCI24

MCTDH23 831.1/834.7

59.1

58.1(1.0)

701/704

831.1/834.7

1184

1012.6/1029.1

59.1

1005.1(1.9) 108.2(1.9)

118.2(0.9)

85.7(0.7)

85.6

66

87.3

92.6

119.1(1.7)

124.0

185

138.7

133.3

a

These energies are all reported relative to the value of E0 evaluated in the ground state. bThe asymptotic states are described using the rotational states of H3+ and H2, |j3,|k3|,m3,j2,m2⟩prpm as defined in the Supporting Information. cThe assignment of the states in the equilibrium structure of H5+ that correspond to the asymptotic states, described using |J , |K |, |vγ |, vb⟩up/ lp . dThe rotational energy relative to the X- and Y-axes, Exy, where Exy = R T

⟨B⟩[J(J + 1) − K2]and ⟨B⟩= 3.058 cm−1,17 is added to the result evaluated from the minimized energy path DMC calculation to account for the rotational contribution to the energy. eFull-dimensional fixed-node DMC calculations reported in our earlier study.15 12116


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with the codes used to evaluate the potential surface for H+5 . This work was supported in part by an allocation of computing time from the Ohio Supercomputer Center. Z.L. thanks the Graduate School at The Ohio State University for fellowship support.

CONCLUSION In this study, we explored the evolution of the energies, the wave functions, and the possible permutations of the protons as H5+ dissociates into H3+ and H2. We focused on two pathways, namely, the proton transfer in which a proton is transferred between H3+ and H2, which is a low-barrier process near the equilibrium geometry of H5+, and exchange of a pair of protons between H2 and H3+, which requires the internal rotation of H3+. This motion has a high barrier near the equilibrium structure, which vanishes as H5+ dissociates. These studies were performed using the minimized energy path DMC approach, which was used to make connections between the rovibrational energy levels of the H5+ intermediate and the rotational excited states of the dissociation products, H3+ and H2. To study the excited states with complicated nodal surfaces, we developed a generalized recrossing scheme. A central question that is addressed in this study is the extent to which the transition states associated with the permutation of the protons are accessed by various ground and excited vibrational states of H5+. We are also interested in how these results are affected by deuteration. We find that deuteration, particularly of the central proton, reduces the amplitude of the wave function near or beyond the transition-state region for proton permutations. Because of the large couplings between the proton transfer vibration and the dissociation coordinate, excitation of this mode increases the extent of permutations, and the full (G240) permutation-inversion symmetry of H5+ should be considered when studying states with as few as one quanta of excitation in this mode. In contrast, excitation of the rotation of the H2 or H3+ subunits that make up H5+ have at most a modest effect on the accessibility of permutations of the protons beyond what is sampled by the ground state. These findings imply increased complexity of the rotation/vibration energy structure of H5+ compared to what is expected for a rigid molecular ion that cannot undergo such permutations.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.5b05774. Definition of the masses used for the recrossing corrections; projections of full-dimensional probability amplitude onto R and S for H5+ and its isotopologues; validation of the generalized recrossing scheme; the |j3 | = |k 3| = |m3| = 1 state. (PDF)

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REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Addresses †

Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139 USA. ‡ Department of Chemistry, University of Washington, Seattle, WA 98195 USA. 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-1213347) is gratefully acknowledged. We thank Prof. J. M. Bowman for providing us 12117


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Probing the Relationship Between Large-Amplitude Motions in H5+

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