Instanton theory of tunneling in molecules with asymmetric isotopic substitutions

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.

Numerical Accuracy Matters: Applications of Machine Learned Potential Energy Surfaces

The role of numerical accuracy in training and evaluating neural network-based potential energy surfaces is examined for different experimental observables. For observables that require third- and fourth-order derivatives of the potential energy with respect to Cartesian coordinates single-precision arithmetics as is typically used in ML-based approaches is insufficient and leads to roughness of the underlying PES as is explicitly demonstrated. Increasing the numerical accuracy to double-precision gives a smooth PES with higher-order derivatives that are numerically stable and yield meaningful anharmonic frequencies and tunneling splitting as is demonstrated for H2CO and malonaldehyde. For molecular dynamics simulations, which only require first-order derivatives, single-precision arithmetics appears to be sufficient, though.

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.

Heavy-Atom Quantum Tunnelling in Spin Crossovers of Nitrenes

We simulate two recent matrix-isolation experiments at cryogenic temperatures, in which a nitrene undergoes spin crossover from its triplet state to a singlet state via quantum tunnelling. We detail the failure of the commonly applied weak-coupling method (based on a linear approximation of the potentials) in describing these deep-tunnelling reactions. The more rigorous approach of semiclassical golden-rule instanton theory in conjunction with double-hybrid density-functional theory and multireference perturbation theory does, however, provide rate constants and kinetic isotope effects in good agreement with experiment. In addition, these calculations locate the optimal tunnelling pathways, which provide a molecular picture of the reaction mechanism. The reactions involve substantial heavy-atom quantum tunnelling of carbon, nitrogen and oxygen atoms, which unexpectedly even continues to play a role at room temperature.

Instanton theory for Fermi's golden rule and beyond

Instanton theory provides a semiclassical approximation for computing quantum tunnelling effects in complex molecular systems. It is typically applied to proton-transfer reactions for which the Born-Oppenheimer approximation is valid. However, many processes in physics, chemistry and biology, such as electron transfers, are non-adiabatic and are correctly described instead using Fermi's golden rule. In this work, we discuss how instanton theory can be generalized to treat these reactions in the golden-rule limit. We then extend the theory to treat fourth-order processes such as bridge-mediated electron transfer and apply the method to simulate an electron moving through a model system of three coupled quantum dots. By comparison with benchmark quantum calculations, we demonstrate that the instanton results are much more reliable than alternative approximations based on superexchange-mediated effective coupling or a classical sequential mechanism. This article is part of the theme issue 'Chemistry without the Born-Oppenheimer approximation'.

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.

Hydrogen Delocalization in an Asymmetric Biomolecule: The Curious Case of Alpha-Fenchol

Rotational microwave jet spectroscopy studies of the monoterpenol α-fenchol have so far failed to identify its second most stable torsional conformer, despite computational predictions that it is only very slightly higher in energy than the global minimum. Vibrational FTIR and Raman jet spectroscopy investigations reveal unusually complex OH and OD stretching spectra compared to other alcohols. Via modeling of the torsional states, observed spectral splittings are explained by delocalization of the hydroxy hydrogen atom through quantum tunneling between the two non-equivalent but accidentally near-degenerate conformers separated by a low and narrow barrier. The energy differences between the torsional states are determined to be only 16(1) and 7(1) cm-1hc for the protiated and deuterated alcohol, respectively, which further shrink to 9(1) and 3(1) cm-1hc upon OH or OD stretch excitation. Comparisons are made with the more strongly asymmetric monoterpenols borneol and isopinocampheol as well as with the symmetric, rapidly tunneling propargyl alcohol. In addition, the third-in contrast localized-torsional conformer and the most stable dimer are assigned for α-fenchol, as well as the two most stable dimers for propargyl alcohol.

Double Proton Transfer Across a Table: The Formic Acid Dimer-Fluorobenzene Complex

Proton transfer via tunneling is a fundamental quantum‐mechanical phenomenon. We report rotational spectroscopy measurements of this process in the complex of the formic acid dimer with fluorobenzene. The assignment of the spectrum indicates that this complex exists in the form of a π–π stacked structure. Each rotational transition of the parent isotopologue exhibits splitting. Isotopic substitution experiments show that the spectral splitting results from double‐proton transfer tunneling in the formic acid dimer. Presence of fluorobenzene as a neighboring molecule does not quench the double proton transfer in the formic acid dimer but decreases its tunneling splitting from 341(3) MHz to 267.608(1) MHz. Calculations suggest that the presence of the weakly bounded fluorobenzene does not influence the activation energy of the proton transfer. The fluorobenzene is reoriented with respect to the formic acid dimer during the course of the reaction, slowing down the proton transfer motion.

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.