Quadratic String Method for Locating Instantons in Tunneling Splitting Calculations

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.

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.

Exact tunneling splittings from symmetrized path integrals

We develop a new simulation technique based on path-integral molecular dynamics for calculating ground-state tunneling splitting patterns from ratios of symmetrized partition functions. In particular, molecular systems are rigorously projected onto their J = 0 rotational state by an "Eckart spring" that connects two adjacent beads in a ring polymer. Using this procedure, the tunneling splitting can be obtained from thermodynamic integration at just one (sufficiently low) temperature. Converged results are formally identical to the values that would have been obtained by solving the full rovibrational Schrödinger equation on a given Born-Oppenheimer potential energy surface. The new approach is showcased with simulations of hydronium and methanol, which are in good agreement with wavefunction-based calculations and experimental measurements. The method will be of particular use for the study of low-barrier methyl rotations and other floppy modes, where instanton theory is not valid.

Vibrational Tunneling Spectra of Molecules with Asymmetric Wells: A Combined Vibrational Configuration Interaction and Instanton Approach

A combined approach that uses the vibrational configuration interaction (VCI) and semiclassical instanton theory was developed to study vibrational tunneling spectra of molecules with multiple wells in full dimensionality. The method can be applied to calculate low-lying vibrational states in the systems with an arbitrary number of minima, which are not necessarily equal in energy or shape. It was tested on a two-dimensional double-well model system and on malonaldehyde, and the calculations reproduced the exact quantum mechanical (QM) results with high accuracy. The method was subsequently applied to calculate the vibrational spectrum of the asymmetrically deuterated malonaldehyde with nondegenerate vibrational frequencies in the two wells. The spectrum is obtained at a cost of single-well VCI calculations used to calculate the local energies. The interactions between states of different wells are computed semiclassically using the instanton theory at a comparatively negligible computational cost. The method is particularly suited to systems in which the wells are separated by large potential barriers and tunneling splittings are small, for example, in some water clusters, when the exact QM methods come at a prohibitive computational cost.

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.

Tunneling splittings of vibrationally excited states using general instanton paths

A multidimensional semiclassical method for calculating tunneling splittings in vibrationally excited states of molecules using Cartesian coordinates is developed. It is an extension of the theory by Mil'nikov and Nakamura [J. Chem. Phys. 122, 124311 (2005)] to asymmetric paths that are necessary for calculating tunneling splitting patterns in multi-well systems, such as water clusters. Additionally, new terms are introduced in the description of the semiclassical wavefunction that drastically improves the splitting estimates for certain systems. The method is based on the instanton theory and builds the semiclassical wavefunction of the vibrationally excited states from the ground-state instanton wavefunction along the minimum action path and its harmonic neighborhood. The splittings of excited states are thus obtained at a negligible added numerical effort. The cost is concentrated, as for the ground-state splittings, in the instanton path optimization and the hessian evaluation along the path. The method can thus be applied without modification to many mid-sized molecules in full dimensionality and in combination with on-the-fly evaluation of electronic potentials. The tests were performed on several model potentials and on the water dimer.

Instanton theory of tunneling in molecules with asymmetric isotopic substitutions

We consider quantum tunneling in asymmetric double-well systems for which the local minima in the two wells have the same energy, but the frequencies differ slightly. In a molecular context, this situation can arise if the symmetry is broken by isotopic substitutions. We derive a generalization of instanton theory for these asymmetric systems, leading to a semiclassical expression for the tunneling matrix element and hence the energy-level splitting. We benchmark the method using a set of one- and two-dimensional models, for which the results compare favorably with numerically exact quantum calculations. Using the ring-polymer instanton approach, we apply the method to compute the level splittings in various isotopomers of malonaldehyde in full dimensionality and analyze the relative contributions from the zero-point energy difference and tunneling effects.

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.

Quantum tunnelling pathways of the water pentamer

Five tunnelling rearrangement pathways in water pentamer are responsible for the ground-state tunnelling splitting pattern of 320 states.

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.

Path Integral Energy Landscapes for Water Clusters

The energy landscapes for a discretized path integral representation of the water dimer, trimer and pentamer are characterized in terms of the localized (classical) and delocalized minima and transition states. The transition states are finite-temperature approximations to the exact instanton path, and they are typically used to calculate the tunneling splittings or reaction rates. The features of the path integral landscape are explored, thus elucidating procedures that could usefully be automated when searching for instantons in larger systems. Our work not only clarifies the role of minima and transition states in path integral calculations but also enables us to analyze the quantum-to-classical transition.

Rotation-tunneling spectrum of the water dimer from instanton theory

A linearly-scaling path-integral method is developed to calculate rotation-tunneling spectra, and is applied to the water dimer.