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Investigation of the Structure and Spectroscopy of H5+ Using Diffusion Monte Carlo Zhou Lin and Anne B. McCoy* Department of Chemistry and Biochemistry, The Ohio State University, Columbus, Ohio 43210, United States S Supporting Information *

ABSTRACT: The results of diffusion Monte Carlo (DMC) calculations of the ground and selected excited states of H5+ and its deuterated analogues are presented. Comparisons are made between the results obtained from two recently reported potential surfaces. Both of these surfaces are based on CCSD(T) electronic energies, but the fits display substantial differences in the energies of low-lying stationary points. Little sensitivity to these features is found in the DMC results, which yield zero-point energies based on the two surfaces that differ by between 20 and 30 cm−1 for all twelve isotopologues of H5+. Likewise, projections of the ground state probability amplitudes, evaluated for the two surfaces, are virtually identical. By using the ground state probability amplitudes, vibrationally averaged rotational constants and dipole moments were calculated. On the basis of these calculations, all isotopologues are shown to be near-prolate symmetric tops. Further, in cases where the ion had a nonzero dipole moment, the magnitude of the vibrationally averaged dipole moment was found to range from 0.33 to 1.15 D, which is comparable to the dipole moments of H2D+ and HD2+. Excited states with up to three quanta in the shared proton stretch and one quantum in the in-phase stretch of the outer H2 groups were also investigated. Trends in the energies and the properties of these states are discussed.

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INTRODUCTION

lowest-energy transition states are shown in Figure 1 along with their relative energies. The presence of five hydrogen atoms and the associated large zero-point energy make vibrational calculations of H5 + challenging. At the same time, the presence of only four electrons makes highly accurate electronic structure calculations tractable for this system, and these calculated electronic energies can be used to construct a potential surface. Two such potentials have been reported recently. The first (hereafter referred to as PES-I) was developed by Xie, Braams, and Bowman7 and is based on a polynomial fit to more than 100 000 electronic energies that were evaluated at the CCSD(T)/aug-cc-pVTZ level of theory/basis. A second potential (PES-II) was developed by Roncero and co-workers.8 In that work, electronic energies evaluated at the CCSD(T)/ aug-cc-pVQZ level of theory/basis were fit to a model potential, the form of which is based on the triatomics-in-molecules formalism. In addition to detailed knowledge of the potential, a deeper understanding of the chemistry of H5+ requires the development of connections between the potential and the spectroscopy of the molecular ion. Several studies have explored the spectroscopy of H5+ on the basis of PES-I and PES-II with a variety of approaches. These include vibrational configuration-

Protonated hydrogen dimer, or H5+, is a molecular ion of longstanding interest since its first laboratory observation.1 This is due, at least in part, to its importance as an intermediate in the simplest H+ exchange reaction in the interstellar medium, specifically the exchange of a proton between molecular hydrogen and H3+.2 This reaction is thought to play a role in the nonstatistical H/D isotopic substitution and ortho/para distributions observed for H 3 + . 3,4 From a theoretical perspective, this molecular ion provides a challenge to standard approaches for studying vibrational energies and wave functions due to the presence of several low energy barriers for the exchange of the positions of two or more hydrogen atoms.5−8 The equilibrium structure of H5+, identified as 1-C2v in Figure 1, has two outer H2 units that lie in parallel planes and are rotated by a 90° torsion angle. In this structure, the fifth hydrogen atom is located on the axis that connects the centers of mass of two outer H2 units, and shifted so it is closer to one of the outer H2 units. This sets up a double minimum potential. The barrier, for the transfer of the central hydrogen atom between the two H2 groups, 2-D2d, is on the order of 60 cm−1,7 whereas the anharmonic frequency of this mode in H5+ is 379 cm−1.9 Likewise, there is a barrier in the torsion coordinate, 3-C2v and 4-D2h, of roughly 100 cm−1, whereas the calculated anharmonic frequency of this mode is between 80 and 90 cm−1. Finally, the barrier for exchange of the central hydrogen atom, with one of the outer ones, 5-C2v, is roughly 1550 cm−1.7,8 The minimum energy structure and the structures of the four © 2013 American Chemical Society

Special Issue: Curt Wittig Festschrift Received: February 9, 2013 Revised: April 4, 2013 Published: April 5, 2013 11725

dx.doi.org/10.1021/jp4014652 | J. Phys. Chem. A 2013, 117, 11725−11736

The Journal of Physical Chemistry A

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Figure 1. Structures and energies of five low-energy stationary points on the H5+ potential surface. The relative energies are from Xie et al.7 The columns are organized to illustrate the three types of isomerization processes: hydrogen transfer [(a) and (c)], torsion (b), and exchange of the central and an outer hydrogen atom (d).

the vibrational spectroscopy of H5+ and D5+ is extremely rich. In a recent publication, we investigated the assignments of the bands in the spectra obtained by Duncan and co-workers, proposing that some of the states that are optically accessible through single photon absorption correspond to large amplitude vibrations of the H2·H3+ complex.13 Given the importance of and interest in H5+ and D5+, in the present study we will revisit the ground state properties of H5+ evaluated by using two different potential surfaces. We will use the resulting vibrational wave functions to analyze the vibrational structure, by exploring the effect of deuteration on the frequency of the fundamental in the low-frequency proton transfer vibration. We will also evaluate the vibrationally averaged rotational constants using the ground state and several excited state wave functions. Finally, the vibrationally averaged dipole moments will be reported for the ground states of all of the isotopologues of H5+. The goal is to understand the sensitivity of the results to details of the potential surface, and to investigate how the large anharmonicity of H5+ is manifested in vibrational energies and wave functions.

interaction calculations (VCI), implemented by using the reaction path version of MULTIMODE,9,10 reduced dimensional treatments,11−13 multiconfigurational time-dependent Hartree (MCTDH),14,15 path integral Monte Carlo,16 fulldimensional variational calculations (FCI),17 and diffusion Monte Carlo (DMC) studies.10,13,18 Given the interest in and the importance of H5+, it is interesting to explore the ground state vibrational structure as well as the sensitivity of the calculated properties to the potential surface that has been employed. Further, astrochemical observation of this molecular ion requires knowledge of how the delocalized structure should be manifested in the rotational and ro-vibrational spectra.4 With two recently developed potential surfaces available for this system,7,8 we have the ingredients needed for such a study. Recently, Bowman and co-workers used DMC approaches to evaluate both the ground states energies and properties of all the possible isotopologues.18 They also evaluated the vibrational spectra of H5+ and D5+ by using VCI approaches based on a slightly modified fit to PES-I.10 Using the vibrationally averaged structures obtained from their DMC calculations, they evaluated approximate values for the ground state rotational constants and the dipole moments.19 Delgado-Barrio, Roncero, and their co-workers have performed a series of studies on H5+ with both of these surfaces. These include low-dimensional studies of the vibrational spectra above 2000 cm−1,11 as well as MCTDH studies of states of H5+ below 1050 cm−1.14 Very recently, they reported the results of a study that compared the two surfaces by using MCTDH to calculate the ground state energies of all possible deuterated variants of H5+ and the vibrationally excited states of H5+ and D5+.15 Song et al. used variational approaches to evaluate vibrational energies of H5+ up to 775 cm−1 above the vibrational ground state.17 Although they also reported zero-point energies for D5+, H4D+, and D4H+, excited state energies of these isotopologues were not provided. There has also been interest in the ro-vibrational spectroscopy of H5+. Bowman and Widicus-Weaver and their coworkers explored the feasibility of detecting rotational spectra of deuterated isotopologues of H5+.19 Building upon on the pioneering work of Okumura et al.,20 Duncan and co-workers explored the vibrational predissociation spectroscopy through single photon processes above 2000 cm−1.10 They also used multiphoton absorption to record transitions in H5+ and D5+ below 2000 cm−1.9 On the basis of these studies, it is clear that

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THEORY General Background. The study and associated methodologies follow earlier work in our group on CH5+ and its deuterated analogues21 and on H5+.13 Specifically, we use DMC in its simplest form to evaluate the ground state properties of H5+. The details of this approach and our implementation have been described in detail elsewhere,22,23 and we focus on the salient features. As mentioned above, in our work, we employ the simplest algorithm for the Monte Carlo solution to the imaginary-time time-dependent Schrödinger equation. This approach was initially described by Anderson.24,25 We base the treatment on an ensemble of N(τ) independently diffusing δ-functions or walkers. Each walker represents a separate configuration of the molecular ion, expressed in terms of its fifteen Cartesian coordinates. Over each time step, δτ, the walker is displaced along each of its coordinates. The size of the displacement is obtained from a Gauss-random distribution with a width in the ith coordinate of σi = 11726

δτ mi

(1) dx.doi.org/10.1021/jp4014652 | J. Phys. Chem. A 2013, 117, 11725−11736


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where mi is the mass associated with the ith coordinate. The value of the potential is used to determine the probability that a walker will survive the displacement. Specifically, we evaluate Sn = exp[− {V (rn) − Eref }δτ ]

corresponds to the isotopologue of interest, the walker is removed from the simulation. This choice restricts the motions of H5+. Because the barrier for this type of isomerization is high, and the tunneling splitting is small, we believe that this constraint should not affect the accuracy of the calculated vibrational frequency, when compared to observed properties. In fact, comparison between the results of our DMC calculation of the energy of this state in H5+ and the experimentally measured fundamental frequency shows a difference of 10 cm−1. Similar restrictions in previous DMC, VCI,10 and MCTDH14 calculations were either imposed explicitly or made through the choice of internal coordinates and the basis set used to expand the vibrational wave functions. The converged variational calculations of Song et al. do not appear to include this constraint,17 and comparison with the results of those calculations provide a way to investigate the effect of this constraint on the energies of vibrationally excited states. For more highly excited states or for states for which the node cannot be determined by symmetry, an adiabatic variant of DMC (ADMC) is used to identify the position of the node. The method is described in detail elsewhere.30,31 The underlying idea comes from the fact that when a node is properly placed, separate ground state calculations performed in the two parts of configuration space that are separated by the node will yield the same energy. In this study, we focus on three states where symmetry cannot be used to determine the locations of all of the nodal surfaces. Two of these are the vsp = 2 and 3 states. In these cases, the nodes between the outer nodal region and the central one are placed at a single value of the distance between the center of mass of one of the outer H2 groups and the central hydrogen atom, Ri. In the case of the vsp = 3 state, the third node is placed at R1 = R2, just as it was for the vsp = 1 state. The third state where the position of the node is evaluated by ADMC is the state with one quantum in the in-phase stretch of the two outer H2 groups (vH2 = 1). If we define the distance between the pairs of hydrogen atoms that form the two outer H2 units as r1 and r2, a nodal coordinate is

(2)

where rn represents the coordinates of the nth walker after the displacement and Eref provides an approximation to the zeropoint energy Eref (τ ) = V̅ (τ ) − α

N (τ ) − N (0) N (0)

(3)

Here V̅ (τ) is the average of the potential and N(τ) is the number of walkers at imaginary time τ. The integer value of Sn provides the number of walkers at the configuration of the nth walker at time τ + δτ, whereas the fractional part of Sn provides the probability that one additional walker is placed at that position. Both the diffusion of the walkers and the subsequent addition or removal of walkers based on the potential surface are accomplished by using Monte Carlo approaches. In addition to calculating the ground state energy and wave function, several other properties will be investigated. These include the vibrationally averaged rotational constants and the average values of the components of the dipole moment vector. On the basis of earlier studies on CH5+ 21,22 and H3+,26 we have found that even for very floppy ions, these quantities can be used to evaluate the rotational energies for low to moderate values of J. To evaluate these averages, we first shift the origin of each of the walkers to a center of mass frame. The coordinates of each of the walkers are then rotated to an Eckart frame by using wellknown algorithms.27,28 The elements of the inverse of the moment of inertia tensor and the components of the dipole moment vector are evaluated at the coordinates of the rotated walkers. Descendent weighting approaches29 are used to obtain the probability amplitude from the ground state wave function obtained from the DMC simulation. Using the probability amplitude, we can evaluate the vibrationally averaged rotational constants and dipole moments. Taken together these quantities can be used to predict the rotational spectrum of the deuterated analogues of H5+. Evaluation of Vibrationally Excited States. Vibrationally excited state of H5+ are evaluated by fixed-node DMC.25 For the calculations of the first excited vibrational state in the shared proton stretch (vsp = 1), we evaluate the lowest energy state that has a node when the distances between the central hydrogen atom and center of mass of each of the two outer H2 groups are equal (R1 = R2). When we implement this algorithm, we find that the energies that are calculated are the same as the ground state energy. Further analysis of the wave functions shows that the hydrogen atoms have rearranged, with the atom that had been in the central position being found in one of the four outside positions. To remove this possibility from the calculations, after each time step, we redefine the coordiantes so the hydrogen atom that is identified as being in the central position is the atom for which di =

s=

(5)

Once the locations of the node or nodes are identified by using ADMC, fixed-node DMC is used to evaluate the wave function in each of the nodal regions. These wave functions are then used to evaluate properties of these excited states. Although fixed-node DMC can be used to pick out individual excited states, it is not a very efficient approach. Further, from an experimental perspective, it is desirable to calculate the intensities as well as the frequencies of vibrational transitions. We recently proposed an alternative, albeit more approximate, approach for obtaining both the transition frequencies and intensities at a fraction of the computational effort. Specifically, we developed an algorithm that allows us to evaluate the frequencies and intensities of transitions from the ground state to states with two or fewer quanta of excitations based on the ground state probability amplitude evaluated by DMC. The details of the approach are described in earlier studies in which we compared the results of these calculations to the measured spectra for H3O2− 32 and H5O2+.33 Briefly, we use the ground state probability amplitude to evaluate a matrix of second moments in mass-weighted internal

∑ rij j≠i

1 (r1 + r2) 2

(4)

is the smallest. The summation in eq 4 is over the distances between the ith hydrogen atom and the other four hydrogen atoms. If after the atoms are reordered, the structure no longer 11727



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used to obtain the energies, knowledge of the relative amplitudes in the different nodal regions is needed to evaluate the expectation values. This is because the wave functions in different nodal regions are evaluated by independent DMC calculations. As a result, the relative normalization of the different nodal regions is not obtained directly from these DMC calculations. Several approaches have been proposed to address this problem. Buch and co-workers pointed out the requirement that the slope of the wave function on the two sides of the node must be equal.34,35 We modified this approach36 and focus on the continuity of the second derivative of the probability amplitude when projected onto the coordinate that was used to define the node. Specifically, we scale the amplitudes of the two parts of the wave function to obtain a second derivative of the projected probability amplitude that is continuous across the node. Because the amplitude is small over the range of the nodal coordinate used for the fit, the procedure introduces additional uncertainties to the expectation values evaluated for these excited states. To reflect this, we increased the uncertainties for reported rotational constants for the states when the node was determined using ADMC by an order of magnitude compared to those evaluated for the ground states and for the states with vsp = 1. Finally, these ADMC calculations were performed in two parts. For both parts, the values of Eref in eq 3 were evaluated as functions of the nodal coordinate on the basis of calculations in which the wave function was constrained to one side of the nodal surface. The resulting values of Eref as functions of the nodal coordinate were fit to cubic polynomials, and the value of the nodal coordinate at which the curves crossed was identified. This value of the coordinate provided the position of the node for subsequent calculations. In the first set of calculations, a pair of ADMC calculations was performed over a fairly large range of values of the nodal coordinate. In this part, relatively short propagations of 2750 time steps were used to sample Eref. The value of the position of the node that was obtained from the results of these ADMC calculations was used to determine the range of the nodal coordinate that was sampled in the second set of calculations. Specifically, Eref was evaluated over a 0.10 Å range of the nodal coordinate. This range was centered at the value of the node obtained from the first pair of ADMC calculations. The system was allowed to equilibrate at the initial value of the nodal coordinate for 20 000 time steps, and the simulation was run for an additional 30 800 time steps. Five independent pairs of ADMC calculations of this type were performed for each node in the wave function. The position of the node that was used in subsequent fixed-node DMC calculations of the energy and probability amplitude was obtained from the average of these five simulations.

coordinates. The eigenvectors of this matrix provide the vibrational coordinates that are used in the analysis. The product of one of these coordinates and the ground state wave function provides an approximation to the wave function for the excited state with one quantum of excitation in that mode. Within this approximation, we developed approximate expressions for the energy and matrix elements of the dipole moment operator in terms of the ground state probability amplitude. Similar approximations were developed for states that had either two quanta assigned in a single mode or two quanta distributed between two modes. The essential ingredient for this treatment is the choice of internal coordinates. Based on our work on H5O2+,33 we construct the internal coordinates from linear combinations of the atom−atom distances. For a molecule with five atoms there are ten such distances, but only nine independent internal coordinates. The redundancy is removed by taking linear combinations of the distances. Specifically, we combine the two distances between the outer hydrogen atoms on the same end of the ion (r1 and r2), the four distances between the outer and central hydrogen atoms, and the four distances between the pairs of hydrogen atoms at different ends of the molecule. The sums of the latter two sets of four coordinates each describe an overall breathing mode, and the redundancy is in this pair of coordinates. For the present study, we do not include the symmetric combination of the four distances between the outer and central hydrogen atoms. Numerical Details. The study was carried out with two potential energy surfaces for H5+, identified as PES-I7 and PESII.8 All DMC calculations were performed with an imaginary time step of δτ = 10 au and α = 0.05 hartree. The ground state calculations were initialized with N(0) = 20 000 walkers all at the 2-D2d stationary point geometry on PES-I, as depicted in Figure 1a. Excited state simulations started with ensembles of walkers obtained from equilibrated ground state simulations. In all DMC calculations, the walkers were allowed to equilibrate for 20 000 time steps before energies and wave functions were collected. To evaluate the probability amplitude, in each simulation the wave function was collected 80 times. For each of the eighty copies of the wave function, the number of descendants was collected over 35 time steps and the evaluations of the descendants were spaced by 350 time steps. The reported energies reflect averages over the 30 800 time steps during which the probability amplitude was obtained. The energies obtained from five independent simulations of each nodal region were averaged, and reported standard deviations are based on these averages. For example, for a v = 1 state, the reported average energy and standard deviation is based on the results of ten simulations, whereas for a v = 3 state the results of 20 simulations are used to evaluate the statistics. The average of the rotational constants and dipole moments were obtained from each of five independent simulations including all nodal regions, and the reported uncertainties reflect the standard deviations among these five results. As the reported uncertainties are based on the statistical differences between independent simulations, they do not account for any systematic errors. For example, the approximations invoked by the fixed-node treatments. The above description is sufficient for evaluating projections of the probability amplitude for the ground state, and for excited states where the node is known by symmetry so the integrated probability amplitude in the two nodal regions must be equal. For excited states where ADMC approaches were

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RESULTS Effects of Deuteration and Potential Surfaces on the Zero-Point Energy. Before discussing the results for partially deuterated H5+, we need to develop a notation for describing the location of the isotopic substitutions. As seen in Figure 1, the hydrogen atoms are numbered from one through five, with atoms one and two forming an outer H2 unit on one end and atoms four and five forming the other outer H2 unit. The central hydrogen atom is assigned the number three. To further simplify notation, and to help keep track of the location of the deuterium atoms, in the tables and figures the species will be described by using the [H2−H−H2]+ notation, where the first 11728



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Table 1. Zero-Point Energies (cm−1) Calculated by DMC for H5+ and Its Deuterated Isotopologuesa species [H2−H−H2] [H2−D−H2]+ [DH−H−H2]+ [DH−D−H2]+ [DH−H−DH]+ [D2−H−H2]+ [DH−D−DH]+ [D2−D−H2]+ [D2−H−DH]+ [D2−D−DH]+ [D2−H−D2]+ [D2−D−D2]+ +

PES-Ib

DMCc−PES-I

DMCd−PES-I

MCTDHe−PES-I

FCIf−PES-I

PES-IIb

MCTDHg−PES-II

7205.2(1.1) 6857.9(0.7) 6812.0(1.3) 6452.6(0.6) 6418.6(1.0) 6371.1(1.0) 6051.1(0.4) 5995.3(1.1) 5976.3(0.8) 5598.6(0.8) 5533.0(0.9) 5149.0(0.4)

3(4) 2(1) 4(2) 4(1) 1(3) 3(1) 4(1) 5(4) 4(1) 4(7) 0(2) 2(1)

8(9) 10(11) 4(7) 2(5) 3(8) 1(5) 4(7) 3(8) 3(6) 3(8) 2(7) 5(7)

−2.6 −3.8 −0.6 −2.7 3.2 −1.9 0.5 −3.3 2.7 −0.1 2.6 2.0

9.4 10.9

7234.1(0.6) 6887.9(1.1) 6840.2(0.8) 6480.7(1.0) 6445.0(0.7) 6396.4(0.8) 6079.1(0.2) 6022.4(1.1) 6001.1(0.4) 5624.1(0.4) 5555.5(0.7) 5171.7(0.9)

3.4 0.8

3.4 3.3

6.5 6.2

a

PES-I represents the potential surface developed by Xie et al.7 and PES-II represents the one developed by Aguado et al.8 bPresent study. Numbers in parentheses represent the standard deviation based on energies obtained from five statistically independent DMC simulations. cDifference between the results of importance sampling DMC18 and those of the present study, both evaluated based on PES-I.7 Uncertainties are those reported by Acioli et al.18 dDifference between the results of DMC without importance sampling18 and those of the present study, both evaluated based on PES-I.7 Uncertainties are those reported by Acioli et al.18 eDifference between the results of MCTDH calculations15 and those of the present study, both evaluated based on PES-I.7 fDifference between the results of full-dimensional variational calculations17 and those of the present study, both evaluated based on PES-I.7 gDifference between the results of MCTDH calculations15 and those of the present study, both evaluated based on PESII.8

Table 2. Comparison of Previously Reported Stationary Point Energies for H5+ a conformerb

CCSD(T)-R12c

CCSD(T)/atzd

CCSD(T)/aqze

PES-Id

PES-IIe

PES-I + ZPEf

1-C2v 2-D2d 3-C2v 4-D2h 5-C2v

−2.531794 61.5 96.6 182.2

−2.527994 48.4 96.4 162.8 1526.8

−2.530509 64.0 95.5 181.9 1543.5

−2.528015 52.2 103.0 160.2 1565.9

−2.530526 135.4 92.7 229.4 1552.9

−2.493197 −286.6 −17.5 −306.5 1338.7

a

Stationary point structures are provided in Figure 1. bThe energy of the 1-C2v conformer is reported in hartrees. Energies of the other conformers are reported relative to the 1-C2v conformer and are given in cm−1. cMuller and Kutzelnigg.6 dCCSD(T)/aug-cc-pVTZ calculations and fit surface.7 e CCSD(T)/aug-cc-pVQZ calculations and fit TRIM surface.8 fHarmonic zero-point corrected potential energies for H5+, obtained from normalmode analysis using PES-I.

the zero-point energies for all of the species in Table 1 with PES-I and for four of the species with PES-II. The differences between the zero-point energies evaluated by MCTDH and the values obtained in the present study are reported in Table 1. As seen, the differences are smaller than 5 cm−1 for all but the D5+ and the D4H+ species, calculated using PES-II. Song et al. reported zero-point energies from variational calculations (FCI) for four of the isotopologues. These energies are consistently larger than those obtained in the present study, as is expected. The differences are roughly 10 cm−1 when there are four or five hydrogen atoms and closer to 3.3 cm−1 in the isotopologues with zero or one hydrogen atom. Finally, Bowman and co-workers used a slightly modified version of PES-I to obtain both DMC and VCI (using the reaction path version of their MULTIMODE program) energies for H5+ and D5+. Their reported DMC energies of 7210 and 5152 cm−1 are basically unchanged from the results they obtained using the version of PES-I used in the present study. The VCI energies are slightly higher (7244 and 5174 cm−1, respectively).10 The larger values of the zero-point energies calculated by VCI likely reflect challenges in converging these energies using this basis set approach. Larger differences are observed when we compare the results obtained using PES-I and PES-II. This is not entirely surprising given the difference between the two surfaces. Specifically, although both surfaces are based on electronic energies obtained at the CCSD(T) level of theory, the energies of the

H2 represents atoms 1 and 2, the central hydrogen atom is atom 3, and the second H2 provides the isotopes of hydrogen for atoms 4 and 5. With this notation in place, we start by considering the zeropoint energies of H5+ and its deuterated analogues. To obtain these results, we use DMC to evaluate the ground state wave function of the species of interest. The resulting zero-point energies are reported for the two potential surfaces in the columns labeled PES-I7 and PES-II8 in Table 1. As shown, the zero-point energies we obtained using PES-I are in good agreement with those reported in an earlier study of Acioli et al.18 using the same surface. In that study, two DMC approaches were used to evaluate the ground state wave functions and properties. One set of calculations employed algorithms similar to those used in the present study, whereas the other set of DMC calculations employed importance sampling based on a trial wave function.37 In most cases the differences between the results of the present calculations and the two previously reported DMC calculations are smaller than 5 cm−1. The difference exceeds 5 cm−1 in only two cases, and in both cases the differences are 1 cm−1 smaller than the uncertainties reported for the earlier calculations. The calculated zero-point energies can be further compared to the zero-point energies obtained using other computational approaches. Delgado-Barrio and co-workers used MCTDH approaches to evaluate the energies and wave functions for the 20 lowest energy states of H5+ and D5+.15 They also reported 11729



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minimum energy conformer. These results provide a qualitative justification for why we can use DMC to evaluate properties of all the species of partially deuterated H5+, and why it is appropriate to consider these species in separate MCTDH calculations. It should be noted that the ground state of ions or molecules that undergo large amplitude motions in the ground state are not always associated with one partially deuterated species. For example, in CH5+, the barriers are all low, and when zero-point energy is considered the calculated ground state wave function is found to have amplitude in all of the isotopically distinct minima.21 On the other hand, as with H5+, we were able to obtain DMC results for each of the partially deuterated species of H5O2+ and H3O2−.33,38 We can further investigate the effects of deuteration by focusing on projections of the ground state probability amplitude onto the isomerization coordinates that take the system across the transition states depicted in Figure 1. In Figure 2, we project the probability amplitude onto the

various stationary point structures depicted in Figure 1 differ. The stationary point energies obtained from PES-I are provided in Figure 1. For comparison, the corresponding energies from PES-II along with the electronic energies on which both surfaces were based are provided in Table 2. We also include the results of CCSD(T)-R12 calculations as a benchmark for the electronic energies.6 Though the electronic energies are generally very similar, the energy of the 2-D2d and 4-D2h saddle points are appreciably higher on PES-II than on PES-I. Despite these differences, the zero-point energies evaluated by using the two surfaces differ by between 23 and 30 cm−1 for all twelve species. As expected, the more highly deuterated the ion, the smaller the difference. When the uncertainties in the DMC energies are taken into account, the range reflects the expected decrease in the zeropoint energy by 2−1/2 when all of the hydrogen atoms are replaced by deuterium atoms. Similar differences are seen when the results of recently reported MCTDH calculations are compared between two surfaces. The above observations demonstrate the relative insensitivity of these quantum mechanical results on changes in the heights of barriers on the potential surface that are small compared to the zero-point energies in the modes that connect these transition states to the minima they separate. This conclusion was also noted by Valdés et al. on the basis of their MCTDH studies.15 Ground State Probability Distributions. If we consider H5+ and its deuterated analogues, the electronic energies (exclusive of vibrational zero-point energy) must be the same for all of the equivalent minima. Once zero-point energy is introduced, the minima may no longer be equivalent. This quantum effect results from the difference in the zero-point energy for stretching and bending motions involving hydrogen and deuterium atoms. The frequencies associated with vibrations of the central hydrogen are much smaller than those associated with one of the outer hydrogen atoms. Consequently, for partially deuterated isotopologues of H5+, the zero-point energies associated with structures with a hydrogen atom in the central position are lower than those in which a deuterium atom is in the central position. Further, when there are two hydrogen and two deuterium atoms in the outer positions, it is energetically more favorable for the system to form D2 and H2 at the two ends rather than two HD units. This can be rationalized by the relative reduced masses of H2, HD, and D2. Although D2 has a reduced mass that is twice as large as the reduced mass for H2, the reduced mass of HD is only 1.33 times larger than the reduced mass of H2. The relative energies of all of the isotopologues of the partially deuterated H5+ can be anticipated by their harmonic zero-point energies. The above analysis, and the fact that we report two or three separate results for the zero-point energies of the various species of partially deuterated H5+ in Table 1, assume that these species are separated by sufficiently large barriers that the probability for tunneling is small. For example, in Table 1 there are separate entries for the two species that correspond to H4D+ ([H2−D−H2]+ and [DH−H−H2]+). The assumption that a single chemical entity may exist as two or more distinct species is supported by the harmonic zero-point corrected energies for the stationary point structures reported in Table 2. The results for PES-I are provided for H5+ in the right column of this table. As seen, the energies of conformers 2−4 all lie below the minimum energy conformer once zero-point energy is considered at the harmonic level. In contrast, the 5-C2v conformer remains 1339 cm−1 higher in energy than the

Figure 2. Two-dimensional projections of the ground state DMC probability amplitudes onto the distances between the central proton and the center of mass of H1H2 (R1) and the center of mass of H4H5 (R2). The structures in the insets are described as [H1H2−H3−H4H5]+.

distances between the central hydrogen atom (hydrogen atom 3 in Figure 1) and the center of mass of the H2 unit formed from hydrogen atoms 1 and 2, R1, and the distance between the hydrogen atom that is numbered three and the center of mass of the H2 unit formed by hydrogen atoms 4 and 5, R2. There are several notable features. First, the plotted probability distributions are spread over values of R1 and R2 that range from 0.75 to 1.75 Å. Further, the largest amplitude in H5+ and D5+ correspond to R1 = R2, and not at the minimum in the potential. The curvature of the plotted probability distributions implies that R1 and R2 are strongly coupled. Deuteration of the outer H2 groups leads to a shift in the distribution so that the central hydrogen or deuterium atom is more likely to be closer to the end of the molecule that contains the larger number of deuterium atoms. The shift will also affect the vibrationally 11730



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Plots of the projections of the probability amplitude for the five possible positions of the deuterium atom in H4D+ onto θ1 and θ2 allow us to explore the extent to which there is amplitude near the 5-C2v saddle point. As shown in Figure 1, the 5-C2v saddle point is the one that separates the species with the deuterium in position three and the species with the deuterium in one of the four other positions. The resulting probability distributions are plotted in Figure 4. As anticipated

averaged dipole moment, as will be discussed below. Deuteration in the central position decreases the width of the probability distribution, as one would anticipate by usual zeropoint effects, but does not affect the overall shape of the probability distribution. The above analysis was also performed with PES-II, and the resulting plots are indistinguishable from those in Figure 2. Although the relevant barrier heights on the two surfaces differ in some cases by as much as a factor of 2, the fact that the zeropoint energies of the various species calculated using PES-I and PES-II differ by less than 30 cm−1 indicates that the quantum mechanics is relatively insensitive to the differences between the two surfaces, at least in the configurations sampled by the ground state wave function. In Figure 3, we plot projections of the ground state probability amplitude onto the torsion coordinate, τ. Here τ

Figure 4. Two-dimensional projections of the ground state DMC probability amplitudes onto θ1 and θ2 for the five deuteration locations of H4D+, as indicated in the diagram above the plot. θ1 and θ2 are defined as the angles between one of the pairs of hydrogen atoms and the vector that connects the centers of mass of the two pairs of hydrogen atoms.

Figure 3. Fits of the projections of the ground state probability amplitudes onto τ defined in the inset for H5+ (black dotted line), [D2−H−H2]+ (gray solid line), [D2−D−H2]+ (red dashed line), and D5+ (green dot-dashed line). Note, the curves for [D2−H−H2]+ and [D2−D−H2]+ lie on top of each other.

by the discussion above, the projections of the probability amplitude for the ground states of the five species fall into nonoverlapping regions that are separated by the 5-C2v saddle point. This is consistent with the small tunneling splitting associated with this transition state. It also provides further justification for our treatment of [H2−D−H2]+ and [DH−H−H2]+ as distinct species. Dipole Moments and Rotational Constants. Although the above discussion provides a qualitative understanding of the zero-point structure of H5+ and how it is affected by deuteration, it is also of interest to consider how these properties manifest themselves in experimental observables. In Tables 3 and 4, we report the calculated vibrationally averaged rotational constants and averaged dipole moments for the species listed in Table 1, evaluated using PES-I. The results obtained using PES-II are provided as Supporting Information. Further, although rotational constants are reported for all species, we only report the average dipole moments for the species for which the dipole moment is not zero by symmetry. For comparison, we also report the values of the rotational constants and components of the dipole moment for the 2-D2d saddle point structure on PES-I, rotated into a principal axis frame. We choose to focus on this saddle point structure as it approximates the vibrationally averaged structure for H5+. It also provided the reference structure for the Eckart embedding. We start with the rotational constants, reported in Table 3. In most cases, the off-diagonal elements of the inverse moment of inertia tensor are zero, and where they are not, we have

is defined in the inset. To make it easier to distinguish among the curves, the probability distributions were fit to P(τ ) = P0 − P1 cos(2τ )

(6)

and the resulting fit parameters and plots showing the fits are provided in the Supporting Information. Here, the ground state samples the full range of the coordinate. Although there are fluctuations in the amplitude as a function of τ, there is significant amplitude found at all values of τ. As these plots illustrate, the amplitudes of these fluctuations in the probability distributions increase with deuteration in the outer positions but are not affected by deuteration of the central hydrogen atom. This is consistent with the lowering of the zero-point energy of this mode with deuteration in the outer positions. Finally, we project the probability amplitude onto θ1 and θ2 for H4D+. These angles are determined by first constructing the vector that lies along the bond that connects two hydrogen atoms that form one of the H2 groups, r1⃗ . The vector that connects the remaining two hydrogen atoms is r2⃗ . A third vector, R⃗ , connects the centers of mass of r1⃗ and r2⃗ . With these vectors defined, cos θi =

R⃗ · ri ⃗ Rri

(7)

Based on this definition, the values of θ1 and θ2 do not depend on the coordinates of the deuterium atom. 11731



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Table 3. Ground State Vibrationally Averaged Rotational Constants (cm−1) of H5+ and Its Deuterated Isotopologues, Evaluated Using PES-I,7 Compared to the Rotational Constants at the 2-D2d Saddle Point on the Same Potential Surface species [H2−H−H2] [H2−D−H2]+ [DH−H−H2]+ b [DH−D−H2]+ b [DH−H−DH]+ c [D2−H−H2]+ [DH−D−DH]+ c [D2−D−H2]+ [D2−H−DH]+ c [D2−D−DH]+ [D2−H−D2]+ [D2−D−D2]+ +

⟨A⟩a

⟨B⟩a

⟨C⟩a

A(D2d)

B(D2d)

C(D2d)

25.358(14) 24.902(13) 21.816(12) 21.341(12) 19.176(14) 17.144(7) 18.788(8) 16.808(11) 15.507(2) 15.245(4) 13.034(6) 12.878(3)

3.058(1) 3.047(2) 2.582(2) 2.568(1) 2.076(1) 2.348(1) 2.070(1) 2.324(1) 1.848(1) 1.841(1) 1.572(1) 1.568(1)

3.058(1) 3.047(2) 2.516(2) 2.498(1) 2.070(1) 2.203(1) 2.063(1) 2.179(1) 1.786(1) 1.780(1) 1.572(1) 1.568(1)

27.125 27.125 23.194 23.117 20.244 18.080 20.109 18.080 16.239 16.192 13.559 13.559

3.510 3.510 2.925 2.911 2.340 2.631 2.340 2.598 2.064 2.059 1.754 1.754

3.510 3.510 2.872 2.858 2.336 2.509 2.334 2.479 2.013 2.009 1.754 1.754

Numbers in parentheses represent the standard deviation based on five statistically independent calculations of the wave function. bThere are nonzero values for 0.5⟨IAB−1⟩ for [DH−H−H2]+ (0.038(3) cm−1) and [DH−D−H2]+ (0.057(1) cm−1). cThere are nonzero values for 0.5⟨IAC−1⟩ for [DH−H−DH]+ (0.026(1) cm−1), [DH−D−DH]+ (0.031(2) cm−1), and [D2−H−DH]+ (0.008(2) cm−1). a

Table 4. Ground State Vibrationally Averaged Dipole Moments (Debye) of Partially Deuterated H5+, Evaluated Using PES-I and the Dipole Moment Surfaces from Bowman and Co-workers,7,10 Compared to the Dipole Moment at the 2-D2d Saddle Point on the Same Potential Surface species

⟨μa⟩a,b

⟨μb⟩a,b

⟨μc⟩a,b

[DH−H−H2]+ [DH−D−H2]+ [DH−H−DH]+ [D2−H−H2]+ [DH−D−DH]+ [D2−D−H2]+ [D2−H−DH]+ [D2−D−DH]+

0.698(13) 0.443(8) 0 −1.145(11) 0 −0.732(10) −0.480(12) −0.298(13)

−0.271(1) −0.246(1) −0.373(2) 0 −0.331(1) 0 0 0

0 0 0 0 0 0 −0.247(1) −0.214(1)

(D2d)

μa

0.865 0.742 0 −1.449 0 −1.268 −0.618 −0.549

(D2d)

μb

−0.255 −0.217 −0.381 0 −0.334 0 0 0

(D2d)

μc

0 0 0 0 0 0 −0.276 −0.245

a Numbers in parentheses represent the standard deviation based on five statistically independent calculations of the wave function. bWhere the values are reported as 0, they are less than half of the statistical uncertainty.

at the center of charge. This is reflected in the reported values of the dipole moment for the D2d geometries. Including zeropoint vibrational energy lowers the magnitude of the average dipole moment. The decrease in the a-component of the dipole moment once vibration is considered reflects the shift of the central hydrogen toward the more deuterated end of the molecule, thereby shifting the center of charge closer to the center of mass of the ion. The latter effect was also observed in the earlier study of the effects of deuteration on the dipole moment of deuterated H5+. In that work it appears that they did not include the effect of deuteration on the location of the center of mass.19 This resulted in an underestimation of the value of the dipole moment by roughly a factor of 5 in their reported results. Smaller changes in the dipole moment along the b- or c-axis are seen when zero-point vibrations are taken into account. These shifts are much smaller than the decrease in the acomponent, and usually the b-component sees a slight increase in magnitude, whereas the magnitude of the c-component typically decreases. These trends correlate to whether the central hydrogen will be shifted toward an outer HD unit or toward an outer D2 unit. The dipole surface used in the present calculations was evaluated by Bowman and co-workers at the B3LYP/aug-ccpVTZ level of theory and basis set.10 One might expect that the calculated value of the dipole moment will be sensitive to the

reported the values in footnotes to the table. As we examine the trends in the rotational constants, we find that the vibrationally averaged values are smaller than the values evaluated at the 2-D2d saddle point structure, as expected. The fractional decrease of the B and C constants are much larger than the corresponding A constant, reflecting the very large amplitude motion of the central hydrogen atom along the symmetry axis as well as the strong coupling between this mode and the coordinate that represents the distance between the two outer H2 units. This is clearly shown in the projections of the probability amplitude onto R1 and R2, plotted in Figure 2. The primary contributor to the decrease in the A constants is the extension of the outer H2 bonds even at the zero-point level. Prediction of the microwave spectra of deuterated H5+ also requires knowledge of the components of the dipole moment. For many of the species of H5+, the dipole moment is zero by symmetry. Consequently, only results for the asymmetrically deuterated species are reported in Table 4. Further, we only report the results obtained using PES-I. The differences between the two potential surfaces lead to small changes in the calculated average dipole moment. The corresponding vibrationally averaged dipole moments evaluated using PES-II are provided as Supporting Information. Overall, the trends in the results based on both potentials are the same. The leading contribution to these vibrationally averaged dipole moments arises from the fact that in the 2-D2d structure, asymmetric deuteration shifts the center of mass of the ion so it is no longer 11732



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proton normal mode at the 2-D2d saddle point, we find that in H5+, the motion of the shared proton accounts for 79% of this normal mode. Of the other motions that contribute significantly to this mode, the most notable is the out-ofphase stretch of the outer H2 groups. The effect of this secondary contribution is illustrated by a small decrease in the harmonic frequency of the shared proton stretch, evaluated at the minimum energy structure, with deuteration in the outer H2 groups. Although the trend in the harmonic frequencies is consistent with the energies reported in Table 5 when a deuterium atom is in the central position, the opposite trend is found when a hydrogen atom is in the central position. We believe that the difference in the trends reflects the complexity of these vibrational states, and the relatively weak dependence of the frequencies on the deuteration of the outer hydrogen atoms. As seen in the results presented in Table 6, when we introduce more quanta of vibrational excitation into the shared

level of theory and basis set used to evaluate it. In the case of asymmetric deuteration of H5+, the leading contribution to the dipole moment comes from the shift in the center of charge from the center of mass in the D2d geometry. As this leading term is purely classical, the dipole moment for asymmetrically deuterated H5+ is anticipated to be relatively insensitive to the level of theory used to evaluate it. To calibrate the sizes of the calculated dipole moments for the asymmetrically deuterated species of H5+, which range from 0.33 to 1.15 D, we compare them to those for H2D+ and HD2+. As both H2D+ and HD2+ have C2v symmetry, only one component of the vibrationally averaged dipole moment is nonzero. The magnitude of the dipole moment for H2D+ was estimated by Dalgarno et al. to be 0.6 D,39 and Foster et al. put the dipole moment for HD2+ at 0.486 D.40 These values are comparable to and, in most cases, slightly smaller than our calculated values for the dipole moments of asymmetrically deuterated H5+. In summary, the partially deuterated forms of H5+ have dipole moments that should be sufficiently large for the microwave spectrum to be observed, provided sufficient abundance. Excited States. Shared Proton Stretch. Given the large coupling between R1 and R2 found in the ground state probability amplitudes, shown in Figure 2, a question of interest is how this coupling is manifested in the excited states in the shared proton stretch. We are also interested in how the differences between the potential surfaces affect the frequency of this mode in H5+ and D5+. In a recent publication, we showed that DMC provides an energy for this state in H5+ that is in good agreement with both VCI and MCTDH calculations performed by using the same or similar potential surfaces. Likewise, the D5+ energy calculated by DMC was in good agreement with the VCI calculations.10,13,15 At about the same time as ref 13 was published, Valdés et al. published the results of MCTDH calculations for H5+ and D5+ based on both PES-I and PES-II. The results of our calculations of the energy for the vsp = 1 states of symmetrically deuterated species are provided in Table 5. The first observation is that despite the strong coupling between R1 and R2, the frequency of the shared proton stretching vibration is only weakly affected by deuteration in the outer positions. This is consistent with the normal mode picture, which characterizes this mode as primarily shared proton in character. Specifically, if we decompose the shared

Table 6. Calculated Energies (cm−1) for the vsp = 2 and 3 States in the Shared Proton Stretch Mode and the Fundamental in the In-Phase H2 Stretch for H5+ and Its Deuterated Isotopologuesa [H2−H−H2] [H2−D−H2]+ [DH−H−DH]+ [DH−D−DH]+ [D2−H−D2]+ [D2−D−D2]+ +

[H2−H−H2] [H2−D−H2]+ [DH−H−DH]+ [DH−D−DH]+ [D2−H−D2]+ [D2−D−D2]+ f +e

PES-Ia,b

PES-IIa,c

⟨Evsp=1⟩d

369(2) 250(1) 373(2) 244(1) 380(1) 241(1)

356(1) 237(1) 360(2) 229(1) 367(1) 226(2)

364 260 358 248 357 232

d,e

673(2) 519(1) 653(1) 484(1) 655(1) 473(1)d,e

vsp = 3b f

982(1) 772(1) 962(1) 726(4) 968(1) 713(1)f

⟨Evsp=3⟩c

vH2 = 1b

1068 860 1108 753 954 771

3535(2) 3508(2) 3106(2) 3079(2) 2575(2) 2556(1)

a

All results are evaluated based on potential surface developed by Bowman and coworkers (PES-I).7 bNumbers in parentheses represent the combined standard deviations of the ground and excited states. c Evaluated using expectation values as described in the text. dMCTDH results for the vsp = 2 state of H5+, 679.5 cm−1, and D5+, 458.5 cm−1,15 both evaluated based on PES-I.7 eFCI results for the vsp = 2 state of H5+, 642.6 cm−1,17 evaluated based on PES-I.7 fMCTDH results for the vsp = 3 state of H5+, 1072.8 cm−1, and D5+, 687.6 cm−1,15 both evaluated based on PES-II.8

proton stretch, up to vsp = 3, the energies remain most sensitive to whether a hydrogen or deuterium atom is in the central position, although the atoms in the outer positions also have an effect. For the species investigated here the energy of the vsp = 1 state was determined by using a fixed-node treatment, where the node is at R1 = R2. The evaluation of the vsp = 2 and vsp = 3 states requires the use of the ADMC approach30,31 to identify the positions of the nodal surfaces. Although the central node in the vsp = 3 wave function is defined in the same manner as the node in the vsp = 1 wave function, the outer nodes are defined to occur at specific values of R1 and R2. A complete description of these calculations was provided previously.13 Projections of the probability amplitude onto R1 and R2 for H5+ are plotted in Figure 5, and the positions of the nodes for all species reported in Table 6 are provided in the Supporting Information. Curvature in the exact nodes for these states makes them harder to describe accurately using fixed-node DMC. To calibrate these calculations, we provide the results of recent MCTDH studies on H5+ and D5+ 15 and variational calculations on H5+.17 As seen, the agreement is very good, although certainly not exact. The good agreement between the results of the FCI variational calculations and those obtained by

Table 5. Calculated Energies (cm−1) for the vsp = 1 State of H5+ and Its Deuterated Isotopologues species

vsp = 2b

species

a

Numbers in parentheses represent the combined standard deviations of the ground and the first excited states. bPotential surface developed by Xie et al.7 cPotential surface developed by Aguado et al.8 dEvaluated based on PES-I7 with expectation values as described in the text. eExpt: 379 cm−1.9 VCI: 382 cm−1.10 FCI: 354.4 cm−1.17 MCTDH: 365.4 cm−1 (PES-I) and 353.5 cm−1 (PES-II).15 fVCI: 257 cm−1.10 MCTDH: 238.1 cm−1 (PES-I) and 225.0 cm−1 (PES-II).15 11733



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illustrate that the approach is less quantitative than the fixednode DMC, but the computer time needed to evaluate the ⟨Evsp=1⟩ is less than 2 min on our local workstations once we have obtained the ground state probability amplitudes. The accuracy of the approach for these fundamentals allows us to obtain at least a qualitative description of the spectrum not only of the states in Table 5 where the node can be determined by symmetry but also for the states where this is not the case. These results are provided in Table 7. The comparison between the fixed-node calculations and the ⟨Evsp=3⟩ values show larger differences than when vsp = 1. One source of the larger differences is the greater severity of the approximation of treating energies as expectation values over ground state probability amplitude as more quanta are put into the system. In addition, in the wave function that corresponds to the ⟨Evsp=3⟩ energy, one node is put into the coordinates that correspond to R1 + R2 and R1 − R2 in Figure 2, whereas in the fixed-node DMC calculation of the vsp = 3 state, one node corresponds to R1 = R2, and the second node is at a value of R1 or R2 determined by using ADMC. This difference in the nodal structure will certainly affect the calculated energies. We have also calculated the vsp = 1 energies with PES-II. Comparing the results for the two surfaces, we find that the vsp = 1 energies calculated using the two surfaces differ by between 12 and 16 cm−1, and there is no systematic dependence on the isotopic substitution. This further confirms that the two potentials provide very similar descriptions of the vibrational structure of H5+ despite the differences in the stationary point energies. Bowman and co-workers have also reported vsp = 1 energies for a potential that is similar to, albeit not identical to, PES-I. Their VCI calculations give vsp = 1 energies for H5+ and D5+ of 382 and 257 cm−1.10 On the basis of the relatively small differences between the energies obtained from PES-I and PESII, it is anticipated that these results would not change significantly if the same version of PES-I were used in the present calculations as was used by Cheng et al.10 Using the excited state wave functions obtained by performing DMC calculations on PES-I, we evaluated the rotational constants for all of the reported states. As mentioned above, due to numerical difficulties in generating the relative amplitude in the different nodal regions, the uncertainties of these quantities are larger for the vsp = 2 and vsp = 3 excited states than for the ground state and the state with vsp = 1. The results are provided in Supporting Information. As expected, on the basis of the strong coupling between the shared proton stretch and the H2−H2 distance, the rotational constants decrease with increased vibrational excitation. Outer H2 Stretches. A significant amount of attention has been placed on the spectrum of H5+ above 2500 cm−1 . The fundamental that is optically accessible in this spectral region is the out-of-phase H2 stretch. We have attempted to investigate the outer H2 stretches using DMC. In Table 6, we provide the resulting energies for the in-phase H2 stretch. Though this mode is optically dark, the combination of this mode with the shared proton stretch has been observed in the spectra of H5+ and D5+. The measured transition frequencies (3904 and 2815 cm−1)10 are in good agreement with the sum of the calculated frequencies of these two modes (3904 and 2797 cm−1). We made several attempts to obtain the out-of-phase stretch energy and the wave function but found that we were unable to evaluate this state using DMC. In retrospect the sources of the difficulty in obtaining this state could have been anticipated.

Figure 5. Two-dimensional projections of the DMC probability amplitudes onto R1 and R2 for the ground and three excited states in the shared proton stretch of H5+.

using DMC, VCI, and MCTDH provides justification for the constraints placed in the latter three methods that the wave function does not sample configurations that correspond to exchange of the outer and central hydrogen atoms. As mentioned above, two drawbacks of the DMC calculations are that they only provide the energies, and not the intensities, and that they also require separate sets of calculations for each state and for each nodal region of a given state. We have been developing an approximation that can be used to evaluate energies and intensities of transitions, based on the DMC ground state probability amplitude. The results for the vsp = 1 and the vsp = 3 states of the shared proton stretch are presented in Tables 5 and 6 for the states where fixed-node calculations are performed. The results for the remaining six species are reported in Table 7. In all cases, the calculated intensity of transitions to the vsp = 3 state is roughly 30% the intensity of the fundamental transition. For the vsp = 1 states, the energies obtained by this approximate approach are typically within 10 cm−1 of the fixed-node DMC results, and the differences are never larger than 25 cm−1 . These differences Table 7. Expectation Values of the Energies (cm−1) of the First and Third Excited States of the Shared Proton Stretch (vsp = 1 and 3) for Partially Deuterated Species of H5+ That Are Not Reported in Tables 5 and 6a species [H2−H−DH] [H2−H−D2]+ [DH−H−D2]+ [H2−D−DH]+ [H2−D−D2]+ [DH−D−D2]+

+

⟨Evsp=1⟩b

⟨Evsp=3⟩b

354 346 358 265 260 252

997 1108 1011 824 753 775

a

All results are evaluated based on the potential surface developed by Bowman and co-workers (PES-I).7 bEvaluated using expectation values as described in the text. 11734



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averaged rotational constants from excited state calculations; positions of the nodes obtained from the ADMC calculations; zero-point averaged values for the hydrogen distances for PES-I and PES-II; details of the fits reported in Figure 3. This material is available free of charge via the Internet at http://pubs.acs. org/.

Experimentally, the fundamental in the out-of-phase H2 stretch is observable because its energy is above the dissociation energy for H5+ to form H2 and H3+. As such, this state is a resonance state that is embedded in the dissociation continuum. From the perspective of the DMC calculations, when we impose the requirement that one of the H2 distances is larger than the other one, as would need to be the case when we put a node in the wave function when r1 = r2, we find that the state that is generated is a complex of H2 with H3+. The energies we obtain for these states are 2324 cm−1 for H5+ and 1624 cm−1 for D5+. The fact that DMC provides the lowest energy state with a particular nodal structure is both a strength and weakness of the approach. This problem does not occur for the in-phase stretch, which is why we were able to evaluate these states. Examination of the trends in the frequencies shows the anticipated global trend that the frequency is largest when there are no deuterium atoms and smallest when the outer groups are both D2. Intermediate energies are obtained when the outer groups are HD. The energies of states with the same deuteration in the outer positions are 19 to 27 cm−1 larger when there is a hydrogen atom in the central position, further indicating a coupling between the shared proton motion and that of the outer H2 groups.

■

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Support through grants from the Chemistry Division of the National Science Foundation (CHE-0848242 and CHE1213347) is gratefully acknowledged. We thank Professors Joel M. Bowman and Octavio Roncero for providing us with the codes used to generate the potential and dipole surfaces for H5+. We also thank Dr. Andrew S. Petit for helpful discussions. This work was supported in part by an allocation of computing time from the Ohio Supercomputer Center.

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■

CONCLUSIONS In this work, we explored the ground state properties for H5+ and its deuterated analogues. We investigated how deuteration affects the large amplitude motions, particularly the shared proton stretch and the torsion modes. We also investigated the sensitivity of these results to the choice of potential surface by comparing the results obtained using two different potential surfaces.7,8 We found that despite differences in the energies of several low-energy saddle points, the energies are very similar and the wave functions are virtually indistinguishable. Consistent with earlier studies, we find that the lowest energy isomers of partially deuterated species of H5+ have a hydrogen atom in the central position. When there is asymmetric partial deuteration in the outer positions, the central hydrogen atom will be shifted toward the heavier end of the ion. This shift leads to a slight decrease in the vibrationally averaged dipole moment as it leads to a shift of the center of charge to a location that is closer to the center of mass. Even after accounting for this reduction in the dipole moment, the average values of the components of the dipole moment for asymmetrically deuterated species of H5+ are comparable to or in many cases larger than those for H2D+ and HD2+. This provides evidence that the microwave spectrum could be recorded for asymmetrically deuterated species of H5+, particularly for the [DH−H−H2]+ species where the magnitude of the vibrationally averaged dipole moment is 1.15 D. One aspect of the dynamics of H5+ that has not been captured in quantum mechanical studies of the vibrational spectroscopy of H5+ to date is how the amplitude of the internal rotation of the H3+ group is affected by excitation of the shared proton stretch mode. We are presently developing DMC approaches that will allow us to investigate this aspect of the dynamics41 as it is anticipated to play an important role in the hydrogen/deuterium exchange reactions between H3+ and H2.

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AUTHOR INFORMATION

REFERENCES

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S Supporting Information *

Expectation values of the rotational constants and elements of the dipole moment, evaluated using PES-II; vibrationally 11735



Article

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