Energetics and Predissociation Dynamics of Small Water, HCl, and

Feb 3, 2016 - ABSTRACT: This Review summarizes recent research on vibrational predissociation. (VP) of hydrogen-bonded clusters. Specifically, the foc...
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Energetics and Predissociation Dynamics of Small Water, HCl, and Mixed HCl−Water Clusters Amit K. Samanta,† Yimin Wang,‡ John S. Mancini,‡ Joel M. Bowman,*,‡ and Hanna Reisler*,† †

Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482, United States Department of Chemistry and Cherry L. Emerson Center for Scientific Computation, Emory University, Atlanta, Georgia 30322, United States

‡

ABSTRACT: This Review summarizes recent research on vibrational predissociation (VP) of hydrogen-bonded clusters. Specifically, the focus is on breaking of hydrogen bonds following excitation of an intramolecular vibration of the cluster. VP of the water dimer and trimer, HCl clusters, and mixed HCl−water clusters are the major topics, but related work on hydrogen halide dimers and trimers, ammonia clusters, and mixed dimers with polyatomic units are reviewed for completion and comparison. The theoretical focus is on generating accurate potential energy surfaces (PESs) that can be used in detailed dynamical calculations, mainly using the quasiclassical trajectory approach. These PESs have to extend from the region describing large amplitude motion around the minimum to regions where fragments are formed. The experimental methodology exploits velocity map imaging to generate pair-correlated product translational energy distributions from which accurate bond dissociation energies of dimers and trimers and energy disposal in fragments are obtained. The excellent agreement between theory and experiment on bond dissociation energies, energy disposal in fragments, and the contributions of cooperativity demonstrates that it is now possible, with state-of-the-art experimental and theoretical methods, to make accurate predictions about dynamical and energetic properties of dissociating clusters.

CONTENTS 1. Introduction, scope, and background 1.1. Introduction and scope 1.2. Brief recap of vibrational predissociation of HF and HCl homodimers 1.3. Representative examples of vibrational predissociation of polyatomic dimers 2. Theoretical methods 2.1. Potential energy surfaces for noncovalent interactions 2.1.1. Composite many-body representations of noncovalent interactions for clusters 2.1.2. Strategies for fitting 2.2. Vibrational analyses of clusters 2.3. Quasiclassical trajectory calculations 3. Experimental methods 4. Water dimer and trimer 4.1. (H2O)2 and (D2O)2 4.2. Water trimer 5. HCl clusters 5.1. HCl trimer 5.2. Larger HCl clusters 6. (HCl)m−(H2O)n clusters 6.1. HCl−H2O 6.2. Larger clusters of HCl and water: vibrational energies, dissociation, and fingerprints of ionization 7. Ammonia dimer and trimer © 2016 American Chemical Society

8. Perspective and prospects for extensions Author Information Corresponding Authors Notes Biographies Acknowledgments References

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1. INTRODUCTION, SCOPE, AND BACKGROUND

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1.1. Introduction and scope

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“Noncovalent interactions” is a widely used phrase that describes the class of all chemical interactions that do not involve the making and breaking of chemical bonds. These interactions include the weak, so-called “van der Waals” interactions and the stronger and ubiquitous hydrogen-bonded (H-bonded) interactions.1−3 The latter are the focus of this Review. Perhaps the most widely and thoroughly studied substance where hydrogen bonds (H-bonds) plays a central role is water in its different forms, from clusters to the bulk.4−9 Numerous Reviews describe the electronic structure of these interactions, with many focusing on a decomposition of the various components of such interactions, e.g., electrostatic, Special Issue: Noncovalent Interactions

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induced, charge-transfer, and exchange-repulsion.10−16 This is not the focus of the theoretical work reviewed here. Instead, generating accurate potential energy surfaces (PESs) that can be used in detailed dynamics calculations is the focus here. More precisely, the focus is on breaking of H-bonds following excitation of an intramolecular vibration, a process known as vibrational predissociation (VP). Among noncovalently bonded systems, the water dimer is arguably the most studied. This is understandable given the role that water plays in living systems. Indeed, interest in the water dimer continues, as evidenced by a 2015 “Frontiers Article” devoted specifically to experimental studies of this dimer.17 The dissociation energy and VP dynamics of the water dimer and trimer are major topics of this Review, as well as recent research on HCl clusters and mixed water and HCl clusters. From the experimental point of view, up to a few years ago there have been only a few cases of dimers that include polyatomic subunits for which the intermolecular-bond dissociation energies (D0) have been determined with sufficient accuracy for comparison with the high-level calculations available today. (Hereafter we use “bond” instead of “intermolecular bond” unless the distinction needs to be made.) A particularly glaring omission was the D0 value of the water dimer, the most fundamental unit for comparisons with theory. Likewise, D0 for HCl−H2O was unknown. The pioneering work of Roger Miller and his group on VP dynamics and translational spectroscopy of small clusters, which is summarized in a 2001 Review, has shown that this information can be obtained in favorable cases.18 The experimental and theoretical research published since that time is described in detail in this Review. In recent years, the velocity map imaging (VMI) technique has been exploited successfully to determine accurate D0 values for a considerable number of H-bonded dimers.19−29 Recently, it has been extended to include the trimers of H2O, NH3, and HCl.30−34 Moreover, the measured pair-correlated fragment state distributions, when combined with dynamics calculations, have helped elucidate dissociation mechanisms.27,31−33 In fact, it is only with the help of theory that a unique interpretation could be achieved. The joint work on these clusters is highlighted in this Review, while examples of state-selected studies of VP of other small dimers and trimers carried out by other groups, although sometimes less comprehensive, are included for completion and comparison. From the theoretical point of view, H-bonded clusters present challenging systems for existing computational approaches and also for advancing them. Naturally, the PES is a critical quantity to be obtained from theory. Given the focus here on the water dimer and trimer, as well as HCl clusters and mixed HCl−water clusters, it is clear that the potentials for these systems are both high dimensional and highly anharmonic. Furthermore, to study the VP dynamics, the PESs have to extend from the region describing large amplitude motion around the minimum to regions where the fragments are formed. In all examples presented here, full-dimensional PESs have been developed. These PESs are precise mathematical fits to tens of thousands of ab initio electronic energies, typically obtained using the coupled-cluster-single− double and perturbative-triple excitations (CCSD(T)) theory with large basis sets. The techniques for generating these PESs are described below. Once the PESs are in hand, dynamics calculations can be done with them. This Review focuses on using the quasiclassical trajectory approach to the dynamics.

State-specific effects are expected in VP of dimers and trimers of small molecules because of the disparity between the intermolecular vibrational frequencies of the clusters and the intramolecular vibrational frequencies of the subunits, which makes intramolecular vibrational redistribution (IVR) inefficient.18,35−38 The bond dissociation energies of dimers are usually in the range of 400−2000 cm−1, and therefore excitation of CH, NH, or OH stretch fundamentals leads to VP. However, for the small clusters discussed in this Review, dissociation is usually slow due to the weak coupling between intramolecular and intermolecular vibrations, with lifetimes ranging from several picoseconds to nanoseconds, and this presents a challenge for trajectory calculations. Spectroscopic studies provide essential information on cluster geometry and frequencies of intermolecular and intramolecular vibrational modes, but they center mostly near the global minimum and do not address the VP dynamics. In the absence of quantitative theoretical predictions for VP, several propensity rules have been developed early on to aid in interpretation, and they continue to serve as useful guidelines. For example, it was noted that there is a direct correlation between the red-shift in the H-bonded intramolecular vibrational transition and the rate of VP.39,40 In addition, Ewing has developed semiempirical propensity rules based on the energy (momentum) gap law that predict the relative VP rates of dimers and provide guidelines for energy disposal in monomer fragments.35−37 These propensity rules predict a preference for fragment channels in which the number of transferred quanta in the dissociation is minimized, resulting in a preference for fragment vibrational excitation and minimization of translational energy (Et) release. The model describes correctly the VP rate of many dimers, as well as propensities in energy distributions in fragments. The latter are most clearly revealed in pair-correlated rotational distributions.18,23,25,29 The model does not address state-specific effects observed in vibrational state distributions and does not include constraints due to angular momentum conservation. A complementary model that includes angular momentum conservation was described by McCaffery and co-workers and was employed successfully to simulate rotational state distributions in the VP of weakly bound dimers.21,24,41,42 It is based on linear-to-angular momentum conversion induced by vibrational motion, which is described as an internal collision. It recognizes that, in order to explain high fragment rotational excitation, the repulsive, hard-shaped part of the PES must participate. The model identifies the principal geometries and impact parameters from which dissociation occurs by fitting to experimental results. Physical arguments based on the direction of initial impulse upon vibrational excitation and the constrained relative motion of the dimer’s subunits limit the number of possible exit channels. The experimentally determined vibrational excitation in products is used as input of the calculations of rotational state distributions. However, energy transfer in the exit channel, which can modify the initial distributions, is not included. For a description of statistical-like product state distributions, phase space theory (PST) is often used. It does not require knowledge of the PES, assumes that fragments states are populated as per their degeneracies, and applies conservation of energy as well as angular momentum.43,44 Ideally, quantum dynamics calculations of lifetimes and final state distributions using accurate, full-dimensional PESs, should be undertaken. Unfortunately, rigorous quantum dynamics 4914

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1.2. Brief recap of vibrational predissociation of HF and HCl homodimers

calculations are currently out of reach for the systems we review here. However, owing to recent progress made by Bowman’s group14,45−57 and others,12,13,58−89 accurate full-dimensional PESs are now available and can and have been used in extensive quasiclassical trajectory calculations, which, as opposed to “direct dynamics”, can be carried out for hundreds of picoseconds and for ensembles of thousands of trajectories. These studies, which are described in detail in this Review, relate observables such as dissociation energies and product state distributions to the properties of the PES. The excellent agreement between theory and experiment on bond dissociation energies, energy disposal in fragments, and the contributions of cooperativity demonstrates that state-of-theart theory is coequal to precise experiments in making accurate predictions about the properties of the clusters. In addition, we discuss recent theoretical research on larger HCl clusters, as well as mixed HCl−H2O clusters, which demonstrates the generality of this approach. We offer some discussion on calculations of fundamental monomer excitations in these clusters and comparisons with experiments. The experimental studies described in this Review rely on high-resolution spectroscopy results, which provide the geometries of the ground state of the clusters and their vibrational, rotational, and tunneling levels. These essential studies are not covered in this review, but we refer to them as needed. For example, the elegant work of Saykally’s group on vibrational− rotational−tunneling (VRT) states of the water dimer and trimer, which has been crucial in comparisons with PESs near the global minimum, is summarized in previous review articles to which the reader is referred.9,17 The present Review focuses on VP of small H-bonded clusters in which theory and experiment are combined synergistically to provide vibrational level energies above the dissociation threshold, accurate bond dissociation energies, energy flow pathways leading to H-bond breaking, and energy distributions in fragments. The time period covered is the past 10−15 years, after the last Chemical Reviews thematic option volume was published in 2000.2 During this time, considerable advances in theory and experiment have made possible detailed studies of VP dynamics, which have led to a much deeper understanding of VP in clusters. This Review is organized as follows. In section 1.2 we describe briefly previous work on HX (X = F, Cl) dimers for which comparisons with theory are available, specifically regarding D0 and rotational excitation in fragments. In section 1.3 we present examples of dimers of polyatomic molecules for which detailed experimental results on VP are published but comparisons with high-level theory are not yet available. Instead, the experimental results, which demonstrate vibrational state specificity in fragments, are rationalized or simulated by the approximate models described above. Sections 2 and 3 describe, respectively, theoretical and experimental methods. Sections 3−6 are devoted to discussion of VP of H-bonded dimers and trimers with water and/or HCl subunits for which detailed experimental results and high-level theoretical calculations exist. For completion, we describe in section 7 recent studies on the VP of the ammonia dimer and trimer. We conclude (section 8) with a perspective and prospects for future studies.

Extensive experimental and theoretical studies of the VP dynamics of HX homo dimers (X = F, Cl) make these dimers benchmark case studies.18,33,56,59,81,82,89−120 Much of the work on the HF dimer has been published by Miller’s group and is summarized in a Review;18 therefore, only a brief summary is included here. Calculations of final state distributions based on a pseudo atom−diatom approximation show that the monomer fragment of the H-bonded HF dimer is produced in the highest energetically accessible rotational state.93 This result is consistent with impulsive dissociation in which the hydrogen recoils off its partner, represented in the model by a spherical atom, generating a large torque on the hydrogen. In experimental measurements of product state distributions for the HF dimer, Miller and co-workers found a clear preference for the production of pairs in which one monomer fragment in highly rotationally excited while the other has low excitation.90,92,98,121 In addition, correlated pairs that require large kinetic energy release (KER) had low probability, in agreement with the energy (or momentum) gap law and the Ewing model.35−37 The same propensity, although slightly less extreme, was found in the predissociation of the HCl dimer,110 and this motif persists, to some degree, in dimers of polyatomic molecules.30 It was argued that, in order to excite high rotational states, the dissociating dimer must access the repulsive wall, where the anisotropy of the potential is large. The measured dissociation energy (1062 cm−1)90 was used to refine the PES.82 The preferential rotational excitation of one fragment originates in the L-shape geometry of the HF dimer. When the intermolecular bend of the dimer is excited together with the stretch in a combination band, the large amplitude bending motion makes the two HF molecules appear more equivalent, and the rotational energy is shared more evenly between the two fragments.90,121 The large rotational constant of HF minimizes energy exchange in the exit channel. Not unexpectedly, the HF dimer has also attracted theoretical attention. For example, Bačić and co-workers112 and later van der Avoird and co-workers113 performed 6dimensional, quantum coupled-channel calculations using previously published 6D-PESs.81,82 These studies predicted bound state properties of the HF dimer and its isotopologues, including their dissociation energies, intermolecular vibrational frequencies, tunneling splitting, line widths, and rotational distributions in the HF fragments following HF stretch excitation. The calculations reproduced bound state energies and obtained reasonable fits to experimental rotational state distributions and their dependence on the dimer’s vibrational state. Isotopic substitutions highlight another theme in the VP dynamics of clusters,18,122,123 namely, the limit on the magnitude of rotational angular momentum of the fragments. For DF−HF dissociation, the results are similar to those of HF−HF when the HF is H-bonded. In contrast, when DF is the bonded stretch, lower than expected rotational excitation in DF is observed, and HF carries away more internal energy than expected.18,122,123 Because of the relatively smaller rotational constant of DF, higher J states of the DF fragment must be excited in order to carry away the same amount of energy, and this requires generation of too high angular momentum. Constraints on fragment angular momentum have been observed in both diatomic and polyatomic dimers.41,42 4915

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internal energy is disposed in bending levels of acetylene.19 Predissociation is slow, and when the dimer finally falls apart, it exhibits state-specific effects in rotational and vibrational excitation of the fragments. In the absence of trajectory calculations on accurate PESs, understanding the evolution of these slow processes is a challenge. D0 of acetylene−HF is 1088 cm−1.129 Excitation of the HF stretch of the dimer results primarily in the production of HF (J = 11), with a small amount in J = 10. These states correlate with the ground vibrational state of the acetylene cofragment. In addition, a channel corresponding to bending excitation of the acetylene fragment is also observed. A rather similar behavior is observed upon excitation of the OH stretch of the acetylene−OH dimer.130−132 When exciting the overtone of the OH stretch, OH(ν = 1) was the main product, and the authors concluded that VP proceeds via two mechanisms: vibrational to rotational energy transfer and intermolecular vibrational energy transfer.130 From the highest observed rotational level of the OH fragment, an upper limit to the dissociation energy was set at 956 cm−1. A model that accounts for the partial quenching of the electronic angular momentum of the OH monomer in this dimer has been developed.131,132 Reisler and co-workers used velocity map imaging (VMI) to obtain pair-correlated product state distributions in the predissociation of the acetylene−HCl (DCl) dimer.19,21 The rotational state distributions in the acetylene cofragments are determined from images of selected HCl(J) and DCl(J) photofragments, and the D0’s of the dimers were measured to be 700 ± 10 and 755 ± 10 cm−1, respectively, in excellent agreement with theory.74,133 Following excitation of the dimer’s asym-CH stretch, the global H(D)Cl rotational distributions show large relative populations in high rotational states,19,21,127 while acetylene retains one quantum of CC stretch, as in the predissociation of its dimers with HF and OH.127−132 The remaining energy is distributed nearly statistically among acetylene rotational levels and relative translation. The DCl complex behaves similarly,21 but even though enough energy is available to populate DCl in ν = 1, such excitation is not found. Excitation of the dimer’s HCl stretch vibration reveals a different energy flow pattern.19 Predissociation populates only excited bending levels in acetylene, and the Et distribution is largely independent of HX(J). This suggests a constraint in converting linear to angular momentum.21,41,42 Although the calculated bond dissociation energies of these dimers agree fairly well with experiment, there are no dynamical calculations that address energy flow patterns. Instead, McCaffery and co-workers were able to reproduce the observed HX rotational excitations for X = F, Cl, and O by applying their angular momentum model for cases in which the HX stretch of the dimer is excited.24 The calculations, which use hard-shaped 3D ellipsoid potentials,41,42 demonstrate that the dimer is bent at the moment of dissociation, and several geometries lead to H-bond breakage via a clearly identified set of fragment quantum states. The results suggest a hierarchy in the disposal of excess energy and angular momentum between fragment vibration, rotation, and recoil. Deposition of the largest portion of energy into a C2H2 vibrational state sets an upper limit on the magnitude of energy and angular momentum remaining for C2H2 rotation and fragment recoil for each HCl(J) fragment. The extent of rotational excitation is constrained by the hardshaped repulsive potential, dimer geometry, and angular momentum conservation.24

The HCl dimer has also been studied in detail by both experiment105−111 and theory.56,114−116,118,119 It has the same equilibrium structure as the HF dimer, but it is bound less strongly (439110 vs 106290 cm−1). It is also floppier, and its VP lifetime is longer than that of the HF dimer. Pair-correlated rotational state distributions were measured following excitation of the HCl stretch fundamental (H-bonded and free), and like (HF)2, they show a strong propensity for channels that minimize translational energy release,110 pairing levels with high and low rotation. In addition, Wittig and co-workers studied the predissociation of the HCl dimer upon excitation of the HCl stretch overtone at 5650 cm−1.117 In this study, Hphotofragments generated by photodissociation of the dimer or the monomer fragment were monitored, and internal energy distributions were inferred from time-of-flight spectra. The investigators concluded that, surprisingly, the dominant predissociation pathway was the loss of two HCl stretch quanta to generate one HCl fragment with very high rotational excitation (up to J = 22, the maximum allowed by energy), while the second fragment had little rotational excitation. This intriguing result still awaits explanation by theory and merits further experimental investigation with direct HCl monomer fragment detection. From the theoretical side, several ab initio full-dimensional PESs have been reported for this dimer.56,114−116,118,119 New PESs for HCl clusters, based on one-, two-, and three-body interactions have been recently developed by Bowman and coworkers.56 These are described later in this Review. The rigorously calculated D0 value for the HCl dimer is in excellent agreement with experiments.110 Six-dimensional quantum calculations of predissociation lifetimes and product state distributions for monomer stretch excited states were reported,120 using the ES1-EL potential.107 More discussion of this potential is given in section 5.1. 1.3. Representative examples of vibrational predissociation of polyatomic dimers

Acetylene−HX dimers (X = F, Cl, O) are representative examples of dimers that exhibit vibrational state specificity in VP. Here the HX moiety is perpendicular to the C−C bond, with the H atom pointing toward the π-bond electron density of the triple bond. In these dimers, distinct patterns of vibrational energy flow and fingerprints in VP dynamics are observed. Energy disposal following VP reflects the extent of vibrational couplings within each subunit and/or with the other subunits mediated by intermolecular modes. In all cases, IVR is incomplete, giving rise to nonstatistical product energy distributions. Roger Miller was one of the first to extend the spectroscopic work on polyatomic dimers to state-to-state VP dynamics,18,40 using the acetylene−HF dimer as a benchmark.124−126 In later work it was found that there are common motifs in the VP of dimers in which acetylene serves as the Lewis base in a Tshaped structure with HF, HCl, and OH.18,19,24,40,124−132 For example, these three dimers show a strong propensity to populate the CC stretch of the acetylene fragment following excitation in the asym-CH stretch, while the remaining energy is disposed in high rotational levels of HX.24 This energy flow pattern is apparently promoted by a near resonance in the acetylene subunit between one quantum of the asym-CH stretch and the CC stretch plus two quanta of bend.128 On the other hand, when the stretch vibration of HX is excited, there is no CC stretch excitation in the acetylene fragment, and instead 4916

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structure, electronic dissociation energy (De), and experimental observables, such as the zero-point averaged structure, experimental bond dissociation energy (D0), product rotational and vibrational energies, VP dynamics, etc. The state-of-the-art in representing such PESs in 1980, during the heyday of van der Waals molecules consisting of two monomers, e.g., dimers of small molecules, is given in the Review by van der Avoird et al.68 The approach then and largely still taken today, albeit with more sophisticated treatments of the various components of noncovalent interactions,13,60 is to make use of the known results from multipole theory for rigid monomers to represent the important long-range part of the PES and to incorporate ab initio calculations into the leading order terms in that theory. This typically takes the form of representing the PES in a product of spherical harmonics times corresponding radial functions, which are determined numerically from ab initio energies. This approach has been widely applied, although almost exclusively treating the monomers as rigid, which for many applications, e.g., scattering, microwave spectroscopy, etc., is quite appropriate. This restriction to rigid monomers has extended the reach of the above approach to rigid polyatomic monomers, e.g., CH4−H2O.57 One relatively recent exception to the rigid monomer restriction is a full dimensional PES for H2−H2O.139 This PES is expressed as a product of coupled spherical harmonics in the angular variables times a secondorder expansion in intramolecular variables at each value of the distance between the centers of mass of H2O and H2. The fit was to 350 000 CCSD(T) and 812 CCSD(T)-R12 energies with intramolecular coordinates importance sampled using the zero-point wavefunction of each monomer. Below we describe a different approach to obtain PESs in full-dimensionality for noncovalent interactions, and although the focus is on water clusters, HCl clusters, and mixed H2O−HCl clusters, we mention at the end of this section recent work on fulldimensional potentials for H2−H2O, H2−H2O−H2O, CH4− H2O, and CH4−H2O−H2O. 2.1.1. Composite many-body representations of noncovalent interactions for clusters. The total potential of a cluster of N monomers, e.g., HCl and H2O, can be formally written as the sum of one-body (monomer), two-body, threebody, etc. interactions. For simplicity we write this schematically as

It is instructive to compare the state-specific behavior of these T-shaped dimers with those of dimers of ammonia with acetylene and water, which have the more traditional linear Hbond.20,22,134−138 In both of these cases, the H-bonded stretch vibration was excited (C−H and O−H for the acetylene and water, respectively). The intermolecular bond of ammonia− acetylene is fairly weak due to the low acidity of acetylene (D0 = 900 ± 10 cm−1, in agreement with theory).135,138 Following asym-CH stretch excitation, both fragments are generated with vibrational excitation distributed in specific ways. NH3 is always excited vibrationally with one or two quanta in the umbrella (symmetric bending) mode, ν2, while bending levels (ν4 and ν5) that minimize translational energy release are excited in C2H2.20 Although energy disposal follows the general guidelines proposed by Ewing,35,36 predissociation is state-specific with regard to vibrational energy disposal. The main predissociation channel is NH3 (1ν2) + C2H2 (2ν4 or 1ν4 + 1ν5), and a minor channel, NH3 (2ν2) + C2H2 (1ν4), is also observed, i.e., only channels with some energy transfer across the hydrogen bond. Other combinations of fragment states that provide pathways with low Et release are not populated (see Figure 1). As in the

Figure 1. Fragment energy levels relevant to the VP of the ammonia− acetylene dimer following excitation of the asym-CH stretch (ν3) vibration of the dimer.

other cases discussed, energy transfer from the high-frequency CH stretch to the intermolecular modes is inefficient and can take place only at specific orientations and impact parameters spanned by motions in the dimer. This restricts energy flow and results in slow VP rate and state-specific vibrational distributions in the fragments. As expected, the ammonia−water complex is more strongly bound (D0 = 1540 ± 10 cm−1), but it also exhibits state-specific behavior.22 In this case, the ammonia fragment always has one or two quanta in the ν2 mode, but the near-isoenergetic water bend vibration is not excited at all, which is surprising. Clearly in these systems vibrational energy distribution is far from statistical. The rotational distributions show a greater propensity to populate higher rotational levels that minimize Et release, subject to the constraints of the anisotropy of the angular potential and angular momentum conservation. Similar vibrational state specificity in fragment excitation is observed in the VP of the ammonia dimer, as described in section 7.

V = V (1)(i) + V (2)(i , j) + V (3)(i , j , k) + ...

where the summations over the monomer indices, i, j, k..., are implied. We take it as a given that a quantitative representation of V must contain V(3)(i, j, k); hence, we have included that term explicitly. (There are, of course, cases where V(3)(i, j, k) terms can be neglected, but those are not the subject of this article.) Note that, in the limit of infinite separation of all monomers, all n-body terms in this expansion vanish for n > 1. Also, it is clear that the dimensionality of the higher-body interactions grows significantly. In the example of water clusters, V(1)(i) depends on three variables, V(2)(i, j) depends on 12 variables, and V(3)(i, j, k) depends on 21 variables. So, a mathematical representation of V(2)(i, j) and V(3)(i, j, k) is clearly a major challenge. (As an aside, we note that V(2)(i, j) is often represented in the literature by, for example, LennardJones or Buckingham exp/6 expressions.) For polar molecules additional long-range terms, e.g., dipole−dipole, are added. These are often given by partial charge representations, which

2. THEORETICAL METHODS 2.1. Potential energy surfaces for noncovalent interactions

PESs describing noncovalent interactions are at the heart of theoretical and computational studies of the equilibrium 4917

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approximate anharmonic treatments to obtain excitation energies, in particular of OH and HCl fundamentals. The former are used to determine rigorous dissociation energies, D0. For this purpose the diffusion Monte Carlo (DMC) procedure is ideal, provided a full-dimensional PES is available. We follow the “unbiased” DMC procedure described in the pioneering paper by Anderson143 as elaborated both pedagogically by Kosztin144 and with many important details by McCoy.145 Details of these calculations for the water dimer and trimer, as well HCl and mixed HCl−H2O clusters, have been given in the literature.14,47−49,56,140,141,146,147 Here the results of these calculations for D0 values are simply given. However, the corresponding zero-point wavefunctions provide important insights into the degree of delocalization of the clusters, which often are not well represented by the global, static equilibrium structure. This is illustrated below for several HCl clusters. Standard normal-mode analyses are also performed using the PESs developed for the cluster of interest here. These provide the usual, if approximate, information on vibrational fundamentals of both inter- and intramolecular modes of the clusters. They are also the input for quasiclassical trajectory calculations of the VP, which we review briefly in the next subsection. Approximate, but realistic, anharmonic calculations of high-frequency intramolecular fundamentals have also been performed, mainly for HCl and mixed HCl−H2O clusters using local monomer anharmonic extensions of the normal-mode analysis.56,140

of course add to zero for neutral monomers. Expressions for V(3)(i, j, k) are less common. The physical origin of the longrange part of the three-body interactions is ascribed to threebody induction, which needs at least the introduction of the polarizability of the monomer. The real issue, in our opinion, is the lack of a physically motivated form for the short-range repulsion. 2.1.2. Strategies for fitting. The approach of Bowman and co-workers has been to use general mathematical representations of V(2)(i, j) and V(3)(i, j, k) that contain linear parameters to be determined by least-squares fits to databases of electronic energies. These representations are manifestly permutationally invariant and were developed to describe reactive interactions, where atom exchange occurs. Therefore, they mathematically describe the “redundancy” of electronic energies in the high dimensional space that is the major challenge for fits. Details of these permutationally invariant fitting bases are given elsewhere,45,14,46,52−54 and we refer the interested reader to those references. These methods were applied to fit V(2)(i, j) and V(3)(i, j, k) for clusters of H2O,14,49,51−53 and HCl,48,56 mixed clusters of H2O and HCl, 140,141 CH 4 −(H 2 O) n clusters, 54,57 and H 2 −(H 2 O) n clusters.142 More discussion of these will be given in the sections below. For mixed clusters, e.g., (HCl)m−(H2O)n, the many-body representation is more complex. For a general XnYm cluster, the full potential is given by V = Vx(1)(i) + Vy(1)(i) + Vxx(2)(i , j) + Vyy(2)(i , j)

2.3. Quasiclassical trajectory calculations

+ Vxy(2)(i , j) + Vxxx(3)(i , j , k) + Vxxy(3)(i , j , k) (3)

A major component of the joint experimental/theoretical work we have done is the study of VP, following the excitation of a high-energy stretch of the cluster, e.g., the H-bonded OH stretch in the water dimer and trimer and the HCl stretch in HCl clusters. Given the complexity of the clusters studied, a quantum approach is not feasible, and so quasiclassical trajectory calculations were done instead. The “quasiclassical” aspect of these calculations describes how initial and final conditions of the trajectories are analyzed according to quantum mechanical quantization. The general procedures for how these are done can be found in Reviews by Hase148,149 and in the original papers from us,27,31−33,56 so only a brief recap of these procedures is given here. Initial conditions that correspond to the experimental excitation of a high-frequency intramolecular stretch amount to giving the normal mode of that stretch, as determined from the PES, the energy of the experimental excitation. Other normal modes of the cluster are given (harmonic) zero-point energy (ZPE). Then thousands of trajectories are propagated using initial conditions sampled randomly from the classical phase space corresponding to this initial quantized state. The trajectories are propagated for sufficiently long times for the fragmentation(s) to occur. Typically this is in the range of tens of picoseconds to nanoseconds, depending on the cluster. The internal vibrational and rotational energies of the products as well as the relative translational energy distributions of the products, e.g., H2O, HCl, (H2O)2, etc., are then determined using standard methods. The well-known issue of violation of the zero-point energy is monitored and trajectories that produce fragments with less than ZPE are discarded using “hard” and “soft” constraints, as described in detail in our papers31,150 and demonstrated in section 4.2 for the water trimer.

(3)

+ Vxyy (i , j , k) + Vyyy (i , j , k)···

where the notation should be obvious and where all terms up to three-body interactions are shown. It is worth noting that, for n = 1, e.g., a clathrate hydrate such as CH4−(H2O)m, there are no two- and three-body terms involving CH4. Clearly, the level of electronic structure theory for the various terms in these many-body representations does not have to be the same. Typically, existing, often spectroscopically accurate monomer PESs have been used for the one-body terms, CCSD(T)-triple-ζ theory has been used for the various twobody terms, and either CCSD(T)-double-ζ or MP2-triple-ζ theory has been used for the three-body terms. This “composite” approach is very much in the spirit of approaches in electronic structure theory to obtain high accuracy that have been widely used for many years. The difference is it is used here to obtain full-dimensional PESs. Finally, we note that a many-body representation of the dipole moment can also be done.49,52,53,55,56 This is less common, and Bowman’s group may have been the first to consider this in 2009 when they represented the dipole moment of the water cluster as the sum of the water monomer dipole moment plus a two-body dipole moment. This is described in ref 53 and in more recent work using a spectroscopically accurate monomer dipole moment in ref 55. (A somewhat related approach for water was also done very recently by Paesani and co-workers.71,72) 2.2. Vibrational analyses of clusters

The vibrational analysis of clusters ranges from rigorous, fulldimensional calculations of the zero-point energies (and wavefunctions) of clusters and the various fragments to more 4918

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3. EXPERIMENTAL METHODS This section gives a brief description of the experimental methods used to study the VP of the dimers and trimers described here, with experiments carried out mostly at the paircorrelated level. We do not cover techniques for generation of clusters or those used to study vibrational and rotational spectroscopy. Rather, we describe methods that provide stateselected detection of products and measurements of Et release, which are needed to interrogate the VP dynamics. All the experiments described herein were carried out in molecular beams, and cluster selection was accomplished by its unique vibrational fingerprint. For these experiments to be successful, several requirements must be met: (1) The IR spectrum of the desired cluster must be separated from the monomer and from other clusters. (2) The cluster must be generated in sufficient concentration in the supersonic expansion. (3) A reliable state-selected detection method must exist for at least one of the monomer fragments. (4) To determine D0 accurately, a method that provides velocity/Et resolution sufficient to observe distinct structures and/or clear energy cutoffs in the distributions should be used. The most common methods for state-specific detection of products are laser-induced fluorescence (LIF), which was used, for example, to detect the OH product from the acetylene−OH dimer described above, and resonance enhanced multiphoton ionization (REMPI), which was used in the velocity map imaging (VMI) studies carried out at the pair-correlated level. IR detection of product states can also be used.18 The two methods used to determine kinetic energy release (KER) of recoiling fragments from VP of clusters are measurements of angular distributions by using a bolometer detector18 and determinations of velocity distributions by VMI using a position-sensitive detector.30 Other methods, such as detection of two products generated in coincidence,151−155 are not described here, although, when feasible, they can generate even more detailed state-to-state data sets. They are, however, more difficult to implement. The Miller group pioneered the determination of paircorrelated distributions using angular distribution of products.92 The detection scheme, which is described in a previous Review,18 exploits the fact that the various internal states of the molecular fragments become spatially separated in time because they have different translational energies. Spatial resolution is obtained by scanning the photofragment angular distributions in a molecular beam apparatus, and the velocity distribution reflects the angular distribution. Fragment signal intensity as a function of recoil angle is recorded by a bolometer detector, but because the bolometer is not a state-specific detector, a probe laser is necessary to record the angular (velocity) distributions of individual fragment states.156 In addition, a large electric field was used in some cases to orient the parent molecules by brute force (pendular state method),157−159 in such a way that the two fragments recoiled in opposite directions in the laboratory frame and could be detected separately. In the velocity map imaging (VMI) technique,160 REMPI detection provides state selectivity for product states, and paircorrelated product state distributions are routinely measured by monitoring fragments for which a good REMPI scheme exists. The velocity distributions determined from the images allow for accurate determination of D0. For best accuracy, the velocity distributions should exhibit structures that are uniquely assigned to internal states of the cofragment. Images obtained

by monitoring different internal states of the fragment must all be fit with a single D0, and this constrains the fit to an almost unique value. The VMI technique provides the best resolution when the recoil velocity is low, as is the case in most VP of clusters that tend to minimize Et release. A typical VMI arrangement consists of an ion-acceleration assembly, a fieldfree drift tube, and a microchannel plate (MCP) detector coupled to a phosphor screen that is monitored by a chargecoupled device (CCD) camera.160 An example of the molecular beam arrangement in Reisler’s laboratory161 is shown schematically in Figure 2, and the

Figure 2. Typical experimental arrangement used for VMI experiments of VP of dimers and trimers.

experimental method is demonstrated for the HCl−water dimer,23,25 which will be further described in section 6. The HCl stretch vibration of the dimer is excited by IR laser radiation at ∼2723 cm−1, and HCl and H2O fragments are produced in a distribution of rotational states. Fragments in selected rotational states are ionized by tunable UV radiation, and the ionized fragments are detected by a position-sensitive detector. The detected 2-dimensional projections are reconstructed to generate the 3-dimentional velocity (speed) distributions (using the BASEX method)162 from which the center-of-mass (c.m.) Et release is derived. Because the VP lifetime is quite long, the angular distributions are isotropic. Three separate experiments are carried out: (1) By selecting specific rovibrational states of one monomer fragment and scanning the wavelength of the IR laser, partial absorption spectra correlated with the monitored states are obtained (i.e., action spectra). (2) By tuning the IR laser wavelength to the maximum in the absorption spectrum of the dimer and scanning the UV laser wavelength, the fragment’s REMPI spectrum is obtained from which its rovibrational state distribution is derived. (3) By recording images of fragments in selected rovibrational levels, velocity (speed) distributions are obtained from which the corresponding Et distributions are derived. Figure 3 shows an example of a velocity distribution in the VP of HCl−H2O obtained from an image of H2O JKaKc = 423 monitored via the C̃ 1B1 (000) ← X̃ 1A1 (000) transition.25 Speed distributions are obtained by summing over the angular distribution for each radius in the reconstructed image and are converted to c.m. Et distributions using momentum conservation, the appropriate Jacobian (∝Et−1/2), and calibration constants. For example, in the VP of HCl−H2O, the internal energy distributions of the HCl cofragments correlated with each H2O (JKaKc) level are derived from the Et distribution of the H2O fragment, and D0 of HCl−H2O is obtained as well. (The large spike near zero in the velocity distribution shown in Figure 3 is due to monomer water in the molecular beam.) 4919

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In this Review we focus on joint theoretical and experimental investigations of H-bond breaking in the water dimer and trimer, which provide information on bond strength and energy relaxation. These, in turn, are important in understanding the behavior of complex water networks, such as liquids, solids, water chains, and other unusual noncovalently bound structures. It is clear from the structure of dimer that the dipole−dipole interaction is largely irrelevant in determining the structure. This also implies that developing accurate PESs for H-bonded clusters is clearly a challenge because a functional form based on dipole−dipole interactions is not realistic. Below we describe the major progress in doing this from the essentially fully ab initio approach described in section 2. In addition, in the context of the many-body representation of clusters, potentials for the dimer and trimer lead immediately to intrinsic two-body and three-body potentials that can be used in a manybody representation of much larger clusters.

Figure 3. Example of fragment velocity distribution obtained in the VP of HCl−H2O recorded by monitoring JKaKc = 423 of H2O.25 The peaks correspond to J levels of the HCl cofragment. Reprinted with permission from ref 25. Copyright 2011 American Chemical Society.

4.1. (H2O)2 and (D2O)2

Because of its importance, there is vast literature concerning the water dimer, covering experimental and theoretical aspects of its structure, spectroscopy, PES, etc. The reader is referred to several review articles, the most recent published in 2015.17,166,167 Here we present a combined experimental and theoretical picture of the bond breaking and VP dynamics of the water dimer (Scheme 1) following excitation of the Hbonded OH stretch of the donor.

Fittings of velocity distributions such as those shown in Figure 3 are accomplished by assigning a Gaussian-shaped curve to each rotational state of HCl with a width characteristic of the experimental resolution. The positions of the Gaussians are determined by varying D0 until a best fit is obtained, and the separation between peaks is fixed by the known rotational constant of HCl. In the present example, vibrational levels of the fragment are energetically inaccessible; therefore, Evib(HCl) and Evib(H2O) are set to zero and the energy conservation equation is

Scheme 1. Water Dimer

E int(HCl−H 2O) + hν = D0 + Et + Erot(HCl) + Erot(H 2O; JK aK c )

where Eint(HCl−H2O) is the internal energy of the dimer prior to excitation, estimated to be 1 cm−1 from T = 5 K in the molecular beam. The photon energy used for vibrational excitation of the dimer (2723 cm−1) is hν, Et is the (measured) c.m. translational energy, Erot(H2O; JKaKc) is the rotational energy of the monitored H2O fragment, and Erot(HCl) is the rotational energy of HCl, which is related to Et by energy conservation. State selection in the REMPI detection defines Erot(H2O; JKaKc), and Et is determined from the images. This procedure was followed for several H2O (JKaKc) levels and gave unique fits for D0 = 1334 ± 10 cm−1.23,25

The first ab initio PES for the water dimer, albeit rigid monomers, was reported in 2000 based on fitting roughly 2500 electronic energies using symmetry-adapted perturbation theory (D0 = 1165 ± 54 cm−1).13 This rigid monomer PES, denoted CC-pol, was updated in 2007 by refitting the 2510 configurations using a combination of MP2 and CCSD(T) electronic energies at the triple-ζ and including long-range polarization.60 In 2006, an ab initio PES for the water dimer with flexible monomers was reported, based on fitting roughly 30 000 CCSD(T)/aug-cc-pVTZ energies.168 The fit is a purely mathematical one, using permutationally invariant polynomials.46,169 This PES, denoted “HBB”, was further refined by adding roughly 10 000 additional CCSD(T)/aVTZ energies and was denoted “HBB1”.51 The most recent version of this full-dimensional PES (denoted “HBB2”) was published in 2009,14 and its long-range behavior is described accurately by the long-range part of the so-called TTM3-F potential. These PESs reproduce the 10 known low-lying stationary points170 to within several wavenumbers. These full-dimensional, ab initio PESs are given in terms of 15 internuclear distances (actually transformed to Morse variables) and are invariant with respect to the 4!2! = 48 permutations of like atoms. Even if the minimum of 12 variables was used to represent this PES, this is twice the

4. WATER DIMER AND TRIMER Water and its H-bond network are an exceedingly complicated system that is still not understood completely. The well-known unique properties of water are mainly due to the intermolecular forces that act among water molecules both in pairwise and nonpairwise additive interactions. As the smallest cluster, the water dimer serves as a benchmark for both experiment and theory. In addition, an accurate value of D0 of the dimer is crucial in assessing the contributions of the water dimer to its absorption in the atmosphere.163,164 Noncovalent interactions are important also in amorphous solid water,165 which is the most common form of water in the universe. Similarly, the cyclic water trimer is the smallest water cluster with a complete H-bond network,9 and it serves as a prototype for examination of cooperative (nonadditive) three-body interactions. 4920

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number, 6, used in rigid monomer versions of the PES. However, using the permutational invariance directly incorporated into the fitting basis, this high-dimensional PES could be precisely fit with a relatively small number of electronic energies. To conclude this brief recap of water dimer PESs, we note that the CCpol-8s rigid water pair potential was extended in 2012 and again in 2015 to include monomer flexibility (CCpol8sf).17,171,172 Also in 2013, another ab initio potential using a reduced set of permutationally invariant polynomials and CCSD(T)/CBS energies was reported.71 There are now several highly accurate potentials for the water dimer. The HBB2 PES was used in dynamics calculations of the VP dynamics of the water dimer in collaboration with experiment.27 This work is described next. Upon vibrational excitation of the H(D)-bonded OH(D) stretch fundamental of the dimers, two channels are energetically open for (H2O)2 and (D2O)2: (000) + (000) and (000) + (010), where (000) and (010) are the ground and first bending levels of the water fragment, respectively. As shown in Figure 4, 2 + 1 REMPI spectroscopy was used for state-specific detection of H2O and D2O fragments via the C̃ 1B1 (000) ← X̃ 1A1 (000 and 010) transitions.26,27 We note that the C̃ 1B1 excited state of H2O is more predissociative than that in D2O, making the H2O spectra less intense and thus noisier. On the other hand, the D2O spectra are more congested due to smaller spacing between rotational levels. The fragment REMPI spectra shown in parts a and c of Figure 4 were simulated fairly well with rotational temperatures of 250 and 150 K, respectively. No temperature gave a good fit to the spectrum in Figure 4b, because several prominent peaks include unusually large contributions from J > 11. Interestingly, the energies of these high rotational levels lie within 100 cm−1 of the 1178 cm−1 JKaKc= 00,0 level of D2O (010). As suggested elsewhere,27 the (000) + (000) channel may result in part from processes in which the excited dimer samples the repulsive part of the PES in an impulsive interaction, converting all the vibrational excitation into fragment rotation and translation, thereby accessing high fragment rotational levels. This, however, is a minor channel. Isolated transitions from state-selected H2O(010), D2O(000), and D2O(010) fragments were used for imaging, and accurate determination of D0 was achieved by using velocity distributions from multiple images. Representative velocity distributions obtained by monitoring H2O and D2O in several rovibrational states are displayed in Figure 5. The constraint that all the structures in these velocity distributions must be fit with a single D0 value leads to unique values of D0 = 1105 ± 10 and 1244 ± 10 cm−1 for (H2O)2 and (D2O)2, respectively.26,27 These values are in excellent agreement with the theory values of 1104 ± 5 and 1244 ± 5 cm−1, demonstrating the high quality of the water dimer PES calculated by Bowman’s group.14,146 Excellent agreement with the measured D0 value is also obtained in the recent calculation of Leforestier and co-workers (1108 cm−1), who used their new PES with flexible monomers (CCpol-8sf).171 Figure 5a shows a velocity distribution obtained by monitoring H2O (000) JKaKc = 32,1, where the cofragments are formed in both the (000) and (010) states. The best fit to this image was obtained with a ratio of (000):(010) = 1:2 in the cofragments. In addition, we note that the velocity distributions derived from some of the D2O (000) images, as seen, for example, in Figure 5c, confirm that the intense peaks in the

Figure 4. Measured REMPI spectra of water fragments from the VP of (H2O)2 and (D2O)2 (top traces) and corresponding simulations (bottom traces). Reprinted with permission from ref 27. Copyright 2012 American Chemical Society.

REMPI spectrum that could not be fit by a 250 K rotational temperature include large contributions from high J’s (prominent low-velocity features). The inability to observe a comparable excess population in high J levels in the REMPI spectrum of H 2 O (000) may be due to the faster predissociation of the C̃ 1B1 state of H2O.173 The quasiclassical trajectory (QCT) calculations give the overall rotational distribution for each water fragment channel.27 Vibrationally state-specific rotational distributions for (H2O)2 and (D2O)2 are shown in Figure 6. The rotational distributions for the (000) + (010) channel (Figure 6c and d) are significantly colder than those for the (000) + (000) channel (Figure 6a and b), as expected from the reduced available energy. The distributions are broad, and those corresponding to the donor and acceptor fragments in each channel are similar. There is also an equal likelihood that the bending excitation resides in the donor or acceptor fragments.27,146 This indicates that energy is shared between the 4921

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Figure 6. Calculated vibrationally state-specific rotational distributions in the (000) and (010) levels of the water fragments generated in VP of the water dimer.27 Reprinted with permission from ref 27. Copyright 2012 American Chemical Society.

example, there is a near-resonance between the bound OH stretch of (H2O)2 (3601 cm−1) and two quanta of intramolecular bend plus an intermolecular bend. The initial intramolecular bending excitation may be shared between the two water subunits or concentrated initially in only one of the water moieties. At least one quantum of bend must decay to the dissociative degree of freedom, a process that can still leave a quantum of bend excitation in one fragment. IVR among the intermolecular modes may accompany the coupling of bending quanta to the intermolecular modes, and this may facilitate, in turn, exchange of donor and acceptor prior to dissociation. This exchange can explain the final broad and similar rotational state distributions in the fragments. The coupling to the dissociation coordinate must be inefficient, as both experiment and theory show dimer lifetimes >10 ps.146,174 However, even though there is sufficient time for IVR, trajectory calculations demonstrate that only restricted paths lead to dissociation. The exchange between donor and acceptor commences at 8/cm−1 for water dimer fragments that have low rotational and translational energies.31 A best fit D0 and uncertainty were obtained for each image, and the final value was determined to be 2650 ± 150 cm−1,31 in good agreement with the calculated value of 2726 ± 30 cm−1.147 The cooperative contribution due to nonpairwise interactions can be evaluated in two ways. The D0 value for breaking two hydrogen bonds of the trimer can be compared with 1105 × 2 cm−1 = 2210 cm−1, i.e., twice the D0 value of the water dimer, which gives a cooperative contribution of 450− 500 cm−1. We can also examine the difference between the D0 value for breaking all the hydrogen bonds of the trimer, which is calculated ab initio as 3855 ± 20 cm−1,147 and the D0 for the H2O + (H2O)2 channel. The D0 value for breaking three Hbonds is 1129 cm−1 higher than the corresponding value for breaking only two, which is similar to the value of breaking a single hydrogen bond in (H2O)2. We conclude that most of the cooperative contribution is captured by the (H2O)3 → H2O + (H2O)2 dissociation, which involves breaking the cyclic structure, and that there is good agreement between theory and experiment on its value. The trajectories show, as expected, much internal isomerization of the trimer prior to dissociation.31 Snapshots of one trajectory, shown in Figure 9, illustrate this. The ring opens early in the trajectory, indicating the breaking of one H-bond. It reforms and breaks and reforms and breaks (often with different H-bonds breaking) until finally the second H-bond breaks and fragmentation is complete. The evolution described by the water trimer trajectories is similar to what was inferred from spectroscopic studies of the trimers of HF, DF, and HCl (see also section 5).187−189 The internal energy distribution of the water dimer fragment from the VP of the trimer is a challenge for QCT calculations. Figure 10 shows how much the results change depending on the type of constraint imposed, starting with no constraint. In this case, all trajectories that lead to the fragments are used to 4923

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below the zero-point energy of the dimer (which is roughly 10 150 cm−1 in the harmonic approximation) and extends to roughly 5000 cm−1 above the dimer ZPE. The “soft ZPE” constraint corresponds to discarding those trajectories where the sum of internal energies of the dimer plus monomer is below the total (harmonic) ZPE of the dimer and monomer. This constraint gives the correct threshold energy for the relative translational energy distribution, but as seen, it fails to give the correct threshold for the dimer internal energy. Next, the “hard” constraint on the monomer discards trajectories that do not give the monomer with at least its harmonic ZPE. This result does make a significant change in the high-energy portion of the dimer energy distribution; however, there is no correction on the threshold behavior. The “hard” ZPE constraint on the dimer does enforce the threshold but leads to too much internal excitation owing to the lack of the monomer constraint. Finally, the hard constraint on both dimer and monomer leads to the much narrower distribution shown in the bottom panel. The use of the hard constraint does in fact lead to very good consistency with the experimental observables, which are the relative Et distribution and the monomer rotational state distribution.

Figure 9. Snapshot of a trajectory (∼100 ps) in the VP of the water trimer (1−6 in increasing time). Reprinted with permission from ref 31. Copyright 2013 American Chemical Society.

5. HCl CLUSTERS Our recent joint theory/experiment work on the HCl trimer33 is described below in detail and includes comparisons to the VP of the HF trimer. Previous work on the HCl dimer is summarized in section 1.2. At the end of this section, we briefly review recent theoretical work on larger clusters. 5.1. HCl trimer

Experimental and theoretical studies of the VP of cyclic (HCl)3 induced by HCl stretch excitation at 2810 cm−1 allow for a detailed examination of the two fragmentation channels: HCl + (HCl)2 (channel I) and 3HCl (channel II). Total fragmentation is possible in this case because the D0 of (HCl)2 is small (439 cm−1).110 The extensive dynamics calculations of Mancini and Bowman show that the first step in the VP of the HCl trimer via channel I is opening of the ring, in accordance with the conclusions of previous spectroscopic studies of HF, DF, and HCl trimers.187−189 In addition, these calculations demonstrate that subsequent steps take place at much longer time and proceed via a saddle point, as discussed below. Clearly, for the trimer PES the many-body representation truncated at the three-body term is exact. Specifically, the surface is constructed from previous highly accurate, semiempirical monomer189 and dimer surfaces118 and a new ab initio three-body potential.56 The dimer potential is the ES1-EL potential energy surface for the dimer.118 This is a semiempirical surface based on the ES1 potential of Elrod and Saykally107 with the addition of an electrostatic interaction potential.118 The original ES1 potential was generated from least-squares fits to spectroscopic data and guided by an earlier ab initio potential energy surface.114 The three-body potential is a permutationally invariant fit to 51 466 three-body CCSD(T)F12a/aug-cc-pVDZ energies.56 Details of this fit are given in ref 56, and we refer the interested reader there. A summary of the key characteristics of the new trimer PES, namely, three stationary points and electronic dissociation energies (De) to the two channels of interest, are given in Table 1. To test the accuracy of the PES, corresponding direct CCSD(T)-F12a/aVDZ and CCSD(T)-F12a/aVTZ energies are also given. As seen, the PES gives results that are much

Figure 10. (H2O)2 internal energy distribution from QCT calculations using constraints to account for zero-point energy in the fragments H2O + (H2O)2. See text for details. Reprinted with permission from ref 31. Copyright 2013 American Chemical Society.

obtain the observable of interest, in this case the dimer internal energy.31 As seen, the result with no constraints on the dimer or the monomer is very broad and begins with a value much 4924

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dimer wavefunction is highly delocalized across the two equivalent global minima, resulting in two banana-shaped proton distributions. Relative to the dimer, the trimer is much more localized with the hydrogen bonds remaining unbroken in the ground state. The protons are still delocalized about the global minimum geometry forming three “mushroom-shaped” proton distributions. From this we can conclude that the trimer is more tightly confined than the dimer and also that tunneling to equivalent structures is negligible. To establish benchmark D0 values, complete basis set (CBS) extrapolations were performed to determine accurate De values for the HCl trimer system; details can be found in ref 56, and the results are summarized in Table 2, along with results from the PES. The PES De uncertainties are a reflection of a small issue with the dimer PES De as discussed elsewhere.56

Table 1. Relative Energies of Three Low-Lying Minima and De (cm−1) of the HCl Trimer from the Indicated Sources; Corresponding Structures Are Also Shown56

Table 2. De (cm−1) of (HCl)3 and (HCl)2 to Indicated Products56 system

De [PES]

De [CBS]

(HCl)3→ 3(HCl) (HCl)3→HCl + (HCl)2 (HCl)2→ 2(HCl)

2336 ± 45 1643 ± 31 692 ± 14

2373 1675 699

Extending the experimental investigations to trimers is challenging because the velocity distributions usually have no structure, as seen above for the water trimer. In trimers made of diatomic molecules, however, the relevant fragments densities of states are lower and structured velocity distributions may be observed. Figure 12 shows several velocity distributions of HCl(J) fragments following HCl stretch excitation of the trimer at 2810 cm−1.33 For J = 11, the available energy is sufficient to produce only monomer + dimer (channel I), whereas for J < 11 both channels I and II (three HCl fragments) are allowed. The structural features observed in the J = 11 velocity distribution arise because the available energy is small and only a few internal states of (HCl)2 are correlated with the monitored state. Fitting this velocity distribution provided a D0 value for channel I. This value was then used as input in fitting the velocity distributions of HCl fragments in J < 11, which included contributions from both channels, and these fittings allowed us to determine D0 for channel II accurately. As shown in Table 3, there is excellent agreement between experiment and theory, both when using the PES directly and when using the CBS De values. It is interesting to note that the experimentally determined trimer D0 values obtained for channels I and II and the dimer’s D 0 of 439 cm−1 place the cooperative (nonadditive) contribution at ∼250 cm−1, in good agreement with the theoretical values of ∼251 and ∼271 cm−1.33 The corresponding value for (H2O)3 is ∼450 cm−1.31 These are 22% and 19% of the total binding energy of the monomer to the dimer for HCl and H2O, respectively. The large contributions of cooperativity are somewhat surprising when considering that there are large deviations from the dimer geometries (T-shaped and linear for (HCl)2 and (H2O)2, respectively) in the cyclic trimers, which induce significant ring strain. Clearly, the greatest contributions to cooperativity come from the binding of the monomer to the dimer. Understanding the sources of the contributions to cooperativity is a topic of great theoretical interest.11,15,16 While the relative contributions of different interactions to the nonpairwise interactions depend on the theoretical treatment, there is general agreement that the three-

closer to the higher-level CCSD(T)-F12a/aVTZ ones than to the CCSD(T)-F12a/aVDZ ones. The De values from the PES cannot be directly compared to experiment; however, rigorous D0 values have been calculated from the PES, and the comparison with experiment is described below. It is of interest to note that the De values for the dimer are 692 (PES), 756 (CCSD(T)-F12a/aVDZ), and 707 (CCSD(T)-F12a/aVTZ) cm−1, and these values are significantly less than half the corresponding De values for the trimer to dissociate to the monomer plus the dimer (which requires breaking two H-bonds). Thus, as with the water trimer, there is a substantial three-body enhancement to the binding energy of the trimer. Rigorously calculated ZPEs of the trimer and dimer were obtained using the PES in DMC calculations.56 For the monomer, a spectroscopically accurate monomer potential was used to get the exact ZPE. The anharmonic ZPEs are 1483, 3235 ± 1, and 5260 ± 1 cm−1, for the monomer, dimer, and trimer, respectively. Relative to harmonic ZPEs, the anharmonic values are red-shifted 14 cm−1 in the monomer, 81 cm−1 in the dimer, and 106 cm−1 in the trimer. Isosurface representations of the HCl dimer and trimer zeropoint wavefunctions are given in Figure 11.33 As seen, the

Figure 11. Isosurface representations of the HCl dimer (left) and trimer (right) ground-state wavefunctions.33 Reprinted with permission from ref 33. Copyright 2014 American Chemical Society. 4925

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Figure 12. Velocity distributions obtained in VMI experiments of (HCl)3 by monitoring HCl in the indicated J levels.33 Reprinted with permission from ref 33. Copyright 2014 American Chemical Society.

Table 3. HCl Trimer and Dimer D0 Values in Wavenumbers33 system

D0 [PES]

D0 [CBS]

experiment

(HCl)3→ 3(HCl) (HCl)3→HCl + (HCl)2 (HCl)2→ 2(HCl)

1526 ± 46 1102 ± 33 425 ± 29

1564 ± 1 1133 ± 2 431 ± 1

1545 ± 10 1142 ± 20 439 ± 1

the major channel, monomer + dimer. Experiment and theory agree for J = 10, but they deviate progressively more as J decreases. This result is rationalized by realizing that some of the dimer fragments possess internal energies in excess of their dissociation energy. When these dimers dissociate further, the Et’s of the resulting HCl fragments extend from the maximum allowed by the excess energy down to near-zero. This secondary dissociation channel, [(HCl)2]** → HCl + HCl, is denoted channel Ia. Its contribution is expected to increase in going from J = 8 to 5, as the latter gives rise to a larger fraction of dimer cofragments with sufficient internal energies to further dissociate. Indeed, as shown elsewhere, the best agreement between theory and experiment is obtained for J = 10.33 The theoretical calculations show that the rate-limiting step in the VP is the relaxation of the HCl stretch excitation to the intermolecular modes of the trimer.33 Energy transfer occurs in two steps, as correctly surmised from earlier spectroscopic studies of the trimer.189 The first one, which occurs in roughly 250 ps, is energy transfer to the other HCl units in the ring. Then on a much longer time scale, of the order of 10 ns, this excitation relaxes to the low-frequency intermolecular modes. Following this relaxation, one of the H-bond breaks and the ring transitions to a stable open-chain conformer, which lies 737 cm−1 higher in energy.33 The minimum energy path for this ring opening is shown in Figure 14. The open-chain conformer lives for sufficiently long time to allow the excitation energy to fully localize in the H-bonds, leading to dissociation. Ultimately, two H-bonds are broken, and a monomer and dimer are formed.33 Both theory and experiment indicate that the (HCl)2 fragments from channel I have broad, statistical-like distributions of rovibrational energies. In addition, a small contribution from channel II is observed in the experiments, i.e., dissociation to three monomers. As discussed in detail in ref 33, the

body interaction is the main contributor. This topic is beyond the scope of this review. The experimental HCl fragment rotational distributions were determined by REMPI and agree with calculations (Figure 13). Both are broad and encompass all the allowed rotational levels. The pair-correlated Et distributions were compared to the corresponding distributions obtained by QCT calculations for

Figure 13. Comparison between the experimentally and theoretically determined rotational distributions of the HCl fragment in the VP of (HCl)3. Reprinted with permission from ref 33. Copyright 2014 American Chemical Society. 4926

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Figure 14. Minimum energy path to the open-chain configuration of (HCl)3.33 Reprinted with permission from ref 33. Copyright 2014 American Chemical Society.

Figure 15. Equilibrium structures of some larger clusters of HCl.

modes was performed (all other normal modes set to zero) using the planar configuration of the tetramer as the reference. This was argued to be the relevant configuration because the energy of this configuration is only 13 cm−1. The calculation determined the fundamental frequency of the degenerate infrared active mode to be 2789 cm−1, which is within 11 cm−1 of the gas-phase experimentally reported range for the trimer, 2774−2778 cm−1.189,192 More details of this calculation and more results of other calculations are given in ref 48.

experimental signature of channel II is the appearance of distinct structures in the Et distributions, which are correlated with specific J levels of the HCl cofragments. Data analysis suggests that channel II does not proceed by stepwise breaking of H-bonds but instead takes place in a single, concerted step in which all three H-bonds are broken.33 The contribution of this channel, however, is estimated at