An ab initio study of tunneling splittings in the water trimer

Tunneling splittings in the vibrationally excited states of water trimer

Tunneling splitting (TS) patterns in vibrationally excited states of the water trimer are calculated, taking into account six tunneling pathways that describe the flips of free OH bonds and five bifurcation mechanisms that break and reform hydrogen bonds in the trimer ring. A tunneling matrix (TM) model is used to derive the energy shifts due to tunneling in terms of the six distinct TM elements in symbolic form. TM elements are calculated using the recently-developed modified WKB (Wentzel-Kramers-Brillouin) method in full dimensionality. Convergence was achieved for the lowest six excited vibrational modes. Bifurcation widths of the pseudorotational quartets are shown to be of comparable size to the ground-state widths, obtained using instanton theory, or increased for some particular modes of vibration. The largest increase is obtained for the excited out-of-phase flip of two adjacent water monomers with free OH bonds pointing in opposite directions relative to the ring plane. Bifurcation widths in (D2O)3 are found to be two orders of magnitude smaller than in (H2O)3. Geometrical arguments were used to explain the order of states in some TS multiplets in vibrationally excited water trimers.

Systematic analysis of electronic barrier heights and widths for concerted proton transfer in cyclic hydrogen bonded clusters: (HF)n, (HCl)n and (H2O)n where n = 3, 4, 5

The MP2 and CCSD(T) methods are paired with correlation consistent basis sets as large as aug-cc-pVQZ to optimize the structures of the cyclic minima for (HF)n, (HCl)n and (H2O)n where n = 3-5, as well as the corresponding transition states (TSs) for concerted proton transfer (CPT). MP2 and CCSD(T) harmonic vibrational frequencies confirm the nature of each minimum and TS. Both conventional and explicitly correlated CCSD(T) computations are employed to assess the electronic dissociation energies and barrier heights for CPT near the complete basis (CBS) limit for all 9 clusters. Results for (HF)n are consistent with prior studies identifying Cnh and Dnh point group symmetry for the minima and TSs, respectively. Our computations also confirm that CPT proceeds through Cs TS structures for the C1 minima of (H2O)3 and (H2O)5, whereas the process goes through a TS with D2d symmetry for the S4 global minimum of (H2O)4. This work corroborates earlier findings that the minima for (HCl)3, (HCl)4 and (HCl)5 have C3h, S4 and C1 point group symmetry, respectively, and that the Cnh structures are not minima for n = 4 and 5. Moreover, our computations show the TSs for CPT in (HCl)3, (HCl)4 and (HCl)5 have D3h, D2d, and C2 point group symmetry, respectively. At the CCSD(T) CBS limit, (HF)4 and (HF)5 have the smallest electronic barrier heights for CPT (≈15 kcal mol-1 for both), followed by the HF trimer (≈21 kcal mol-1). The barriers are appreciably higher for the other clusters (around 27 kcal mol-1 for (H2O)4 and (HCl)3; roughly 30 kcal mol-1 for (H2O)3, (H2O)5 and (HCl)4; up to 38 kcal mol-1 for (HCl)5). At the CBS limit, MP2 significantly underestimates the CCSD(T) barrier heights (e.g., by ca. 2, 4 and 7 kcal mol-1 for the pentamers of HF, H2O and HCl, respectively), whereas CCSD overestimates these barriers by roughly the same magnitude. Scaling the barrier heights and dissociation energies by the number of fragments in the cluster reveals strong linear relationships between the two quantities and with the magnitudes of the imaginary vibrational frequency for the TSs.

Tunneling splittings using modified WKB method in Cartesian coordinates: The test case of vinyl radical

Modified WKB theory for calculating tunneling splittings in symmetric multi-well systems in full dimensionality is re-derived using Cartesian coordinates. It is explicitly shown that the theory rests on the wavefunction that is exact for harmonic potentials. The theory was applied to calculate tunneling splittings in vinyl radical and some of its deuterated isotopologues in their vibrational ground states and the low-lying vibrationally excited states and compared to exact variational results. The exact results are reproduced within a factor of 2 in most states. Remarkably, all large enhancements of tunneling splittings relative to the ground state, up to three orders in magnitude in some excited mode combinations, are well reproduced. It is also shown that in the asymmetrically deuterated vinyl radical, the theory correctly predicts the states that are localized in a single well and the delocalized tunneling states. Modified WKB theory on the minimum action path is computationally inexpensive and can also be applied without modification to much larger systems in full dimensionality; the results of this test case serve to give insight into the expected accuracy of the method.

Accurate calculation of tunneling splittings in water clusters using path-integral based methods

Tunneling splittings observed in molecular rovibrational spectra are significant evidence for tunneling motion of hydrogen nuclei in water clusters. Accurate calculations of the splitting sizes from first principles require a combination of high-quality inter-atomic interactions and rigorous methods to treat the nuclei with quantum mechanics. Many theoretical efforts have been made in recent decades. This Perspective focuses on two path-integral based tunneling splitting methods whose computational cost scales well with the system size, namely, the ring-polymer instanton method and the path-integral molecular dynamics (PIMD) method. From a simple derivation, we show that the former is a semiclassical approximation to the latter, despite that the two methods are derived very differently. Currently, the PIMD method is considered to be an ideal route to rigorously compute the ground-state tunneling splitting, while the instanton method sacrifices some accuracy for a significantly smaller computational cost. An application scenario of such a quantitatively rigorous calculation is to test and calibrate the potential energy surfaces of molecular systems by spectroscopic accuracy. Recent progress in water clusters is reviewed, and the current challenges are discussed.

Equilibrium and Dynamical Characteristics of Hydrogen Bond Bifurcations in Water-Water and Water-Ammonia Dimers: A Path Integral Molecular Dynamics Study

We present path integral molecular dynamics results that describe the effects of nuclear quantum fluctuations on equilibrium and dynamical characteristics pertaining to bifurcation pathways in hydrogen bonded dimers combining water and ammonia, at cryogenic temperatures of the order of 20 K. Along these isomerizations, the hydrogen atoms in the molecules acting as hydrogen-bond donors interchange their original dangling/connective characters. Our results reveal that the resulting quantum transition paths comprise three stages: the initial and final ones involve overall rotations during which the two protons retain their classical-like characteristics. Effects from quantum fluctuation are clearly manifested in the changes operated at the intermediate passages over transition states, as the spatial extents of the protons stretch over typical lengths comparable to the distances between connective and dangling basins of attractions. Consequently, the classical over-the-hill path is replaced by a tunneling controlled mechanism which, within the path integral perspective, can be cast in terms of concerted inter-basin migrations of polymer beads from dangling-to-connective and from connective-to-dangling, at practically no energy costs. We also estimated the characteristic timescales describing such interconversions within the approximate ring polymer rate theory. Effects derived from full and partial deuteration are also discussed.

Tunnelling splitting patterns in some partially deuterated water trimers

We apply our recently developed semiclassical method for calculating tunnelling splittings (TS) in asymmetric systems to make the first characterization of the ground-state TS pattern of some partially deuterated water trimers. Similarly to homoisotopic water trimers, the ground-state TS patterns are explained in terms of six distinct rearrangement mechanisms. TS patterns in (D2O)(H2O)2 and (H2O)(D2O)2 are composed of sextets induced by the dynamics of flips, and each of its levels is further finely split into a quartet of doublets and a doublet of quartets, respectively, due to various bifurcation dynamics. The TS pattern is obtained using 18 distinct tunnelling matrix elements. TS patterns of (HOD)(H2O)2 and (HOD)(D2O)2 each consists of two sextets, belonging to in-bond and out-of-bond substituted isomers. These sextet levels are further split into quartets by bifurcations. The TS pattern is computed in terms of 13 matrix elements. We also derive analytic expressions for bifurcation tunnelling splittings in terms of tunnelling matrix elements using symmetry. The present approach can be applied to other water clusters and also to the low-lying vibrationally excited states and should help in the interpretation and assignment of experimental spectra in the future.

Instanton calculations of tunneling splittings in chiral molecules

We report the ground state tunneling splittings (ΔE_± ) of a number of axially chiral molecules using the ring-polymer instanton (RPI) method (J. Chem. Phys., 2011, 134, 054109). The list includes isotopomers of hydrogen dichalcogenides H₂ X₂ (X = O, S, Se, Te, and Po), hydrogen thioperoxide HSOH and dichlorodisulfane S₂ Cl₂ . Ab initio electronic-structure calculations have been performed on the level of second-order Møller-Plesset perturbation (MP2) theory either with split-valance basis sets or augmented correlation-consistent basis sets on H, O, S, and Cl atoms. Energy-consistent pseudopotential and corresponding triple zeta basis sets of the Stuttgart group are used on Se, Te, and Po atoms. The results are further improved using single point calculations performed at the coupled cluster level with iterative singles and doubles and perturbative triples amplitudes. When available for comparison, our computed values of ΔE_± are found to lie within the same order of magnitude as values reported in the literature, although RPI also provides predictions for H₂ Po₂ and S₂ Cl₂ , which have not previously been directly calculated. Since RPI is a single-shot method which does not require detailed prior knowledge of the optimal tunneling path, it offers an effective way for estimating the tunneling dynamics of more complex chiral molecules, and especially those with small tunneling splittings.

Trajectory on-the-fly molecular dynamics approach to tunneling splitting in the electronic excited state: A case of tropolone

The semiclassical tunneling method is applied to evaluate the tunneling splitting of tropolone due to the intramolecular proton transfer in the electronic excited state, first time, in a framework of the trajectory on-the-fly molecular dynamics (TOF-MD) approach. To prevent unphysical zero-point vibrational energy transfer among the normal modes of vibration, quantum zero-point vibrational energies are assigned only to the vibrational modes related to intramolecular proton transfer, whereas the remaining modes are treated as bath modes. Practical ways to determine the tunnel-initiating points and tunneling path are introduced. It is shown that the tunneling splitting decreases as the bath-mode energy increases. The experimental tunneling splitting value is well reproduced by the present TOF-MD approach based on the Wentzel-Kramers-Brillouin (WKB) approximation.

Instanton theory of ground-state tunneling splittings with general paths

We derive a multidimensional instanton theory for calculating ground-state tunneling splittings in Cartesian coordinates for general paths. It is an extension of the method by Mil'nikov and Nakamura [J. Chem. Phys. 115, 6881 (2001)] to include asymmetric paths that are necessary for calculating tunneling splitting patterns in multi-well systems, such as water clusters. The approach avoids multiple expensive matrix diagonalizations to converge the fluctuation prefactor in the ring-polymer instanton (RPI) method, and instead replaces them by an integration of a Riccati differential equation. When combined with the string method for locating instantons, we avoid the need to converge the calculation with respect to the imaginary time period of the semiclassical orbit, thereby reducing the number of convergence parameters of the optimized object to just one: the number of equally spaced system replicas used to represent the instanton path. The entirety of the numerical effort is thus concentrated in optimizing the shape of the path and evaluating hessians along the path, which is a dramatic improvement over RPI. In addition to the standard instanton approximations, we neglect the coupling of vibrational modes to external rotations. The method is tested on the model potential of malonaldehyde and on the water dimer and trimer, giving close agreement with RPI at a much-reduced cost.

Upper and lower bounds for tunneling splittings in a symmetric double-well potential

Ground state tunneling gaps: solid circles are mean of eigenvalues and lower bound gaps.

Tunneling Splittings in Water Clusters from Path Integral Molecular Dynamics

We present calculations of tunneling splittings in selected small water clusters, based on a recently developed path integral molecular dynamics (PIMD) method. The ground-rotational-state tunneling motions associated with the largest splittings in the water dimer, trimer, and hexamer are considered, and we show that the PIMD predictions are in very good agreement with benchmark quantum and experimental results. As the tunneling spectra are highly sensitive to both the details of the quantum dynamics and the potential energy surface, our calculations are a validation of the MB-Pol surface as well as the accuracy of PIMD. The favorable scaling of PIMD with system size paves the way for calculations of tunneling splittings in large, nonrigid molecular systems with motions that cannot be treated accurately by other methods, such as the semiclassical instanton.

Locating Instantons in Calculations of Tunneling Splittings: The Test Case of Malonaldehyde

The recently developed ring-polymer instanton (RPI) method [J. Chem. Phys. 2011, 134, 054109] is an efficient technique for calculating approximate tunneling splittings in high-dimensional molecular systems. The key step is locating the instanton tunneling-path at zero temperature. Here, we show that techniques previously designed for locating instantons in finite-temperature rate calculations can be adapted to the RPI method, where they become extremely efficient, reducing the number of potential energy calls by 2 orders of magnitude. We investigate one technique that employs variable time steps to minimize the action integral, and two that employ equally spaced position steps to minimize the abbreviated (i.e., Jacobi) action integral, using respectively the nudged elastic band (NEB) and string methods. We recommend use of the latter because it is parameter-free, but all three methods give comparable efficiency savings. Having located the instanton pathway, we then interpolate the instanton path onto a fine grid of imaginary time points, allowing us to compute the fluctuation prefactor. The crucial modification needed to the original finite-temperature algorithms is to allow the end points of the zero-temperature instanton path to describe overall rotations, which is done using a standard quaternion algorithm. These approaches will allow the RPI method to be combined effectively with expensive potential energy surfaces or on-the-fly electronic structure methods.

Experimental and theoretical investigations of the dissociation energy (D0) and dynamics of the water trimer, (H2O)3

We report a joint experimental-theoretical study of the predissociation dynamics of the water trimer following excitation of the hydrogen bonded OH-stretch fundamental. The bond dissociation energy (D0) for the (H2O)3 → H2O + (H2O)2 dissociation channel is determined from fitting the speed distributions of selected rovibrational states of the water monomer fragment using velocity map imaging. The experimental value, D0 = 2650 ± 150 cm(-1), is in good agreement with the previously determined theoretical value, 2726 ± 30 cm(-1), obtained using an ab initio full-dimensional potential energy surface (PES) together with Diffusion Monte Carlo calculations [ Wang ; Bowman . J. Chem. Phys. 2011 , 135 , 131101 ]. Comparing this value to D0 of the dimer places the contribution of nonpairwise additivity to the hydrogen bonding at 450-500 cm(-1). Quasiclassical trajectory (QCT) calculations using this PES help elucidate the reaction mechanism. The trajectories show that most often one hydrogen bond breaks first, followed by breaking and re-forming of hydrogen bonds (often with different hydrogen bonds breaking) until, after many picoseconds, a water monomer is finally released. The translational energy distributions calculated by QCT for selected rotational levels of the monomer fragment agree with the experimental observations. The product translational and rotational energy distributions calculated by QCT also agree with statistical predictions. The availability of low-lying intermolecular vibrational levels in the dimer fragment is likely to facilitate energy transfer before dissociation occurs, leading to statistical-like product state distributions.

Pathways for hydrogen bond switching in a tetrameric methanol cluster

Computational techniques (second order Møller-Plesset MP2 perturbation theory in conjunction with medium and large size basis sets) are applied to explore structural aspects of a hydrogen-bonded tetrameric cluster of methanol molecules, based geometrically on a tetrahedral arrangement of the four oxygen atoms of the cluster. The hydrogen-bonded structures that represent minima on the potential energy surface are established, and the complete set of pathways that allow interconversion between these structures through "switching" of the hydrogen bonding arrangement are elucidated. The implications of these results in terms of dynamic properties of the cluster are discussed.