Tunneling Splitting Research Articles

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.

Nonrigid water octamer: Computations with the 8-cube.

A 8-cube model of the fully nonrigid water octamer is considered within the 8-dimensional hyperoctahedral wreath product group with 10,321,920 operations and 185 irreducible representations by employing computational and mathematical techniques. For the two lowest-lying isomers of (H2 O)8 with D2d and S4 symmetries of a rigid (H2 O)8 , correlation tables and nuclear spin statistics are constructed for the tunneling splittings of the rotational levels are computed by a computational matrix polynomial generating function technique combined with Möbius inversion, and the relationship to the 8-cube multinomials are pointed out. Multinomial generating functions combined with the induced representation techniques are employed to compute and construct the nuclear spin species, nuclear spin statistical weights and tunneling splittings of rovibronic levels. We have also computed the spin statistical weights and tunneling splittings of the rotational levels for a semirigid water octamer within the wreath product Oh [S2 ] consisting of 12,288 operations.

H2O-CO and D2O-CO complexes: Intra- and intermolecular rovibrational states from full-dimensional and fully coupled quantum calculations.

We present efficient yet rigorous, full-dimensional quantum bound-state calculations of the fully coupled J = 0 and one intra- and intermolecular rovibrational levels of H2O-CO and D2O-CO complexes. The new ab initio nine-dimensional (9D) potential energy surface (PES) [Y. Liu and J. Li, Phys. Chem. Chem. Phys. 21, 24101 (2019)] is employed. In the spirit of the recently introduced general procedure [P. M. Felker and Z. Bačić, J. Chem. Phys. 151, 024305 (2019)], the 9D rovibrational Hamiltonian is partitioned into a 5D (rigid-monomer) intermolecular Hamiltonian, two intramolecular vibrational Hamiltonians-one for the water monomer (3D) and another for the CO monomer (1D), and a 9D remainder term. The low-energy eigenstates of the three reduced-dimension Hamiltonians are used to build up the 9D product contracted basis, in which the matrix of the full rovibrational Hamiltonian is diagonalized. In line with the findings of our earlier study referenced above, the 5D intermolecular eigenstates included in the 9D bases extend up to at most 230 cm-1 above the lowest-energy state of the given parity, much less than the intramolecular fundamentals of the two complexes that span the range of energies from about 1200 cm-1 to 3800 cm-1. The resulting Hamiltonian matrices are small for the 9D quantum problem considered, ≈ 10 000 for J = 0 and 13 500 for J = 1 calculations, allowing for direct diagonalization. The 9D calculations permit exploring a number of features of the rovibrational level structure of H2O-CO and D2O-CO that are beyond the quantum 5D rigid-monomer treatments reported to date. These include the differences in the magnitudes of the hydrogen-exchange tunneling splittings computed in 9D and 5D, the sensitivity of the tunneling splittings to the intramolecular vibrational excitation, the frequency shifts of the intramolecular vibrational modes, which, depending on the mode, can be either blue- or redshifts, and the effects of the excitation of the intramolecular fundamentals on the low-lying intermolecular eigenstates. Also examined is the extent of the eigenstate delocalization over the two minima on the PES. Whenever possible, a comparison is made with the experimental data in the literature.

Accurate quantum mechanical calculations on deuterated vinylidene isomerization.

We present accurate quantum dynamics calculations on vinylidene-d2 (fully deuterated vinylidene) isomerization. This is achieved by large-scale parallel computations with an efficient theoretical scheme developed by us in which basis functions are customized for the deuterium migration process. Mode-specific permutation tunneling splittings of vinylidene-d2 states are reported here for the first time, and evident isotope effects are revealed. The A1 ', B2 ', A1 ″, and B2 ″ symmetric vinylidene-d2 states are calculated, and the peaks recently observed in the cryo-SEVI spectra are analyzed, with very good agreement between theory and experiment achieved for the energy levels. In addition, a few vinylidene-d2 states beyond the experimental energy region are reported. Our studies are helpful for acquiring a better understanding of isotope effects in the double-hydrogen migration processes.

Spin-phonon interaction induces tunnel splitting in single-molecule magnets

Quantum tunneling of the magnetization is a major obstacle to the use of single-molecule magnets (SMMs) as basic constituents of next-generation storage devices. In this context, phonons are often only considered (perturbatively) as disturbances that promote the spin system to traverse the anisotropy barrier. Here, we demonstrate the ability of phonons to induce a tunnel splitting of the ground doublet which then reduces the required bistability due to Landau-Zener tunneling of the magnetization. Harmful are those phonons that modify the spin Hamiltonian so that its rotational symmetry about the field axis is destroyed. In our calculations we treat spins and phonons on the same footing by performing quantum calculations of a Hamiltonian where the single-anisotropy tensors are coupled to harmonic oscillators.

Efficient Quantum Mechanical Calculations of Mode-Specific Tunneling Splittings upon Fundamental Excitation in the Dimer of Formic Acid.

The formic acid dimer (FAD) is an important benchmark system for understanding the double hydrogen transfer process. Most recently, Zhang et al. measured a few tunneling splittings upon fundamental excitation of FAD precisely (Zhang, Y. et al. J. Chem. Phys. 2017, 146, 244306); however, relevant theoretical studies are very limited. Here, we present a multidimensional quantum dynamics study on mode-specific tunneling splittings upon fundamental excitation in FAD with an efficient theoretical scheme developed by our group in which the process-oriented basis function customization strategy is combined with the preconditioned inexact spectral transform method. Various mode-specific tunneling splittings upon fundamental excitation are systematically calculated, and interesting mode-specific excitation effects on tunneling rate are identified. In particular, the calculated tunneling splittings for the ν22 and ν21 states are in good agreement with experiment, and the remarkable mode-specific suppression effects upon excitation should result from that the antisymmetric vibrational modes hinder the concerted double H-transfer. The present work is helpful to acquire a better understanding of the mode-specific excitation effects on tunneling processes.

Permutationally invariant polynomial potential energy surfaces for tropolone and H and D atom tunneling dynamics

We report permutationally invariant polynomial (PIP) fits to energies and gradients for 15-atom tropolone. These include standard, augmented, and fragmented PIP bases. Approximately, 6600 energies and their associated gradients are obtained from direct-dynamics calculations using DFT/B3LYP/6-31+G(d) supplemented by grid calculations spanning an energy range up to roughly 35 000 cm-1. Three fragmentation schemes are investigated with respect to efficiency and fit precision. In addition, several fits are done with reduced weight for gradient data relative to energies. These do result in more precision for the H-transfer barrier height. The properties of the fits such as stationary points, harmonic frequencies, and the barrier to H-atom transfer are reported and compared to direct calculations. A previous 1D model is used to obtain the tunneling splitting for the ground vibrational state and qualitative predictions for excited vibrational states. This model is applied to numerous fits with different barrier heights and then used to extrapolate the H and D atom tunneling splittings to values at the CCSD(T)-F12 barrier. The extrapolated values are 2.3 and 0.14 cm-1, respectively for H and D. These are about a factor of two larger than experiment, but within the expected level of agreement with experiment for the 1D method used and the level of the electronic structure theory.

Analytic Route to Tunneling Splittings Using Semiclassical Perturbation Theory.

We present an efficient, analytical, and simple route to approximating tunneling splittings in multidimensional chemical systems, directly from ab initio computations. The method is based on the Wentzel-Kramers-Brillouin (WKB) approximation combined with the vibrational perturbation theory. Anharmonicity and corner-cutting effects are implicitly accounted for without requiring a full potential energy surface. We test this method on the following three systems: a model one-dimensional double-well potential, the isomerization of malonaldehyde, and the isomerization of tropolone. The method is shown to be efficient and reliable.

Hyperfine and quadrupole interactions for Dy isotopes in DyPc2 molecules

Nuclear spin levels play an important role in understanding magnetization dynamics and implementation and control of quantum bits in lanthanide-based single-molecule magnets. We investigate the hyperfine and nuclear quadrupole interactions for 161Dy and 163Dy nuclei in anionic DyPc2 (Pc = phthalocyanine) single-molecule magnets, using multiconfigurational ab initio methods (beyond density-functional theory) including spin–orbit interaction. The two isotopes of Dy are chosen because the others have zero nuclear spin. Both isotopes have the nuclear spin I = 5/2, although the magnitude and sign of the nuclear magnetic moment differ from each other. The large energy gap between the electronic ground and first-excited Kramers doublets, allows us to map the microscopic hyperfine and quadrupole interaction Hamiltonian onto an effective Hamiltonian with an electronic pseudo-spin that corresponds to the ground Kramers doublet. Our ab initio calculations show that the coupling between the nuclear spin and electronic orbital angular momentum contributes the most to the hyperfine interaction and that both the hyperfine and nuclear quadrupole interactions for 161Dy and 163Dy nuclei are much smaller than those for the 159Tb nucleus in TbPc2 single-molecule magnets. The calculated separations of the electronic-nuclear levels are comparable to experimental data reported for 163DyPc2. We demonstrate that hyperfine interaction for the Dy Kramers ion leads to tunnel splitting (or quantum tunneling of magnetization) at zero field. This effect does not occur for TbPc2 single-molecule magnets. The magnetic field values of the avoided level crossings for 161DyPc2 and 163DyPc2 are found to be noticeably different, which can be observed from the experiment.

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.

Barriers to internal rotation and tunnelling splittings of the torsional states in the HO(CH2)OH, DO(CH2)OH and DO(CH2)OD molecules

Torsional states caused by vibrations of hydroxyl groups in the methanediol molecule and its two deuterated analogues – DO(CH2)OH and DO(CH2)OD were analysed at MP2/cc-pVTZ and CCSD(T)/cc-pVQZ levels of theory. In the first case, 2D PES and 2D surfaces of kinematic coefficients were calculated with geometry optimisation for all other geometric parameters, and in the second case, only the energy of optimised configurations at the MP2/cc-pVTZ level of theory was determined. Then 2D PES was recounted to the complete basis set (CBS) limit by extrapolating the results of calculations at the MP2/cc-pVTZ and MP2/cc-pVQZ levels of theory The calculated values were then averaged over four equivalent points on the coordinate plane. Hamiltonian matrices were constructed using DVR and Fourier methods. After their subsequent diagonalization, the energies of the stationary torsional states were computed. Their classification by C2V(M) and CS(M) molecular symmetry groups has been performed. The splitting values due to the tunnelling of the thirty most deeply located torsional states in the three studied molecules were also determined. The torsional states, internal rotation barriers, and tunnelling frequencies in the molecules of methanediol and hydrogen trioxide were compared.

Spectroscopic Signatures of Mode-Dependent Tunnel Splitting in the Iodide-Water Binary Complex.

The gas-phase vibrational spectrum of the isolated iodide-water cluster ion (I-·H2O), first reported in 1996, presents one of the most difficult, long-standing spectroscopic puzzles involving ion microhydration. Although the spectra of the smaller halides are well described in the context of an asymmetrical ground-state structure in which only one OH group is hydrogen-bonded to the ion, the I-·H2O spectrum displays multiplet structures with partially resolved rotational patterns that are additionally influenced by quantum nuclear spin statistics. In this study, this complex behavior is unraveled with a combination of experimental methods, including ion preparation in a temperature-controlled ion trap and spectral simplification through applications of tag-free, two-color IR-IR double-resonance spectroscopy. Analysis of the double-resonance spectra reveals a vibrational ground-state tunneling splitting of about 20 cm-1, which is on the same order as the spacing between the peaks that comprise the multiplet structure. These findings are further supported by the results obtained from a fully coupled, six-dimensional calculation of the vibrational spectrum. The underlying level structure can then be understood as a consequence of experimentally measurable, vibrational mode-dependent tunneling splittings (which, in the case of the ground vibrational state, is comparable to the rotational energy spacing between levels with Ka = 0 and 1), as well as Fermi resonance interactions. The latter include the hydrogen-bonded OH stretches and combination bands that involve the HOH bend overtones and soft-mode excitations of frustrated translation and rotation displacements of the water molecule relative to the ion. These anharmonic couplings yield closely spaced bands that are activated in the IR by borrowing intensity from the OH stretch fundamentals.

Effects of a dissipative coupling to the momentum of a particle in a double well potential

Double well potentials offer the possibility of coherent state preparation and therefore constitute important building blocks in the analysis of quantum information and quantum engineering devices. Here we present a study of the coherent tunneling in a parabolic double well potential in presence of different dissipative interactions. Specifically, we investigate the effects of an environmental coupling to the momentum and/or to the position of a particle in the potential. Using the semiclassical approximation to calculate instanton paths in Euclidean time, we find that momentum dissipation enhances the coherent tunnel splitting. In presence of both types of dissipation, the momentum dissipation shifts the critical coupling strength of the dissipative phase transition induced by the position dissipation.

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.

Local and global approaches to treat the torsional barriers of 4-methylacetophenone using microwave spectroscopy.

The Fourier transform microwave spectrum of 4-methylacetophenone recorded from 8 GHz to 18 GHz under jet-cooled conditions has revealed large tunneling splittings arising from a low barrier to internal rotation of the ring methyl group and small splittings from a high torsional barrier of the acetyl methyl group. The large splittings are especially challenging to model, while the small splittings are difficult to analyze due to the resolution limit of 120 kHz. The combination of two methyl groups undergoing internal rotations caused each rotational transition to split into five torsional species, which were resolved and fitted using a modified version of the XIAM code and the newly developed ntop code to a root-mean-square deviation close to measurement accuracy, providing an estimate of the V3 potential barriers of about 22 cm-1 and 584-588 cm-1 for the ring and the acetyl methyl groups, respectively. The assignment was aided by separately fitting the five torsional species using odd-power order operators. Only one conformer in which all heavy atoms are located on a symmetry plane could be identified in the spectrum, in agreement with results from conformation analysis using quantum chemical calculations.

Adjustable coupling and in situ variable frequency electron paramagnetic resonance probe with loop-gap resonators for spectroscopy up to X-band.

In standard electron paramagnetic resonance (EPR) spectroscopy, the frequency of an experiment is set and the spectrum is acquired using the magnetic field as the independent variable. There are cases in which it is desirable instead to fix the field and tune the frequency such as when studying avoided level crossings. We have designed and tested an adjustable frequency and variable coupling EPR probe with loop-gap resonators (LGRs) that works at a temperature as low as 1.8 K. The frequency is tuned by adjusting the height of a dielectric piece of sapphire inserted into the gap of an LGR; coupling of the microwave antenna is varied with the height of the antenna above the LGR. Both coupling antenna and dielectric are located within the cryogenic sample chamber, but their motion is controlled with external micrometers located outside the cryostat. The frequency of the LGR (∼4 GHz) can be adjusted by more than 1 GHz (>25%). To cover a wide range of frequencies, different LGRs can be designed to cover frequencies up to X-band. We demonstrate the operation of our probe by mapping out avoided crossings for the Ni4 molecular nanomagnet to determine the tunnel splittings with high precision.

Quantum tunnelling pathways of the water pentamer.

We apply the semiclassical instanton method to calculate all feasible tunnelling pathways in the water pentamer. Similarly to the water trimer, there are labile flip dynamics as well as a number of different bifurcation pathways coupled to flips. In contrast to the trimer, the puckering motion of the oxygen ring makes the ring-polymer instanton approach difficult to converge, a problem which is resolved by using a recently developed time-independent formalism of the method. We use the results to predict the complete ground-state tunnelling splitting pattern of 320 states, which should help in the continuing effort to assign the experimental spectrum. A comparison between the rearrangement pathways in the water trimer and pentamer sheds light on the many-body cooperative effects of hydrogen bonding which are important for a full understanding of the liquid state.

Berry-phase blockade and oscillations in single-molecule magnets Mn12 and Fe8 coupled to polarized leads

We have studied the Berry-phase interference in single-molecule magnets [Formula: see text] and [Formula: see text] coupled to polarized electrodes in the Coulomb blockade regime under external magnetic field. We show that for the oppositely fully polarized leads, the current is completely blockaded as a consequence of the topologically quenched tunnel splitting, which is called as Berry-phase blockade; whereas for the partially polarized leads, the current oscillates periodically with increasing external transverse magnetic field (Berry-phase oscillations). The reason is that different quantum spin tunneling trajectories can combine and give rise to either constructive or destructive interference effects.

Dispersion-driven conformational preference in the gas phase: Microwave spectroscopic and theoretical study of allyl isocyanate.

The conformations of allyl isocyanate (CH2=CHCH2N=C=O) were explored in the gas phase by combining theoretical calculations and Fourier transform microwave spectroscopy, including the chirped pulse and Balle-Flygare types. Three conformers (I, II, and III) were predicted using D3(BJ) dispersion-corrected B3LYP and MP2 methods; however, the lowest energy conformer (conf. I) was absent at the standard B3LYP level. The observed microwave spectra are consistent with the presence of both conf. I and III in the supersonic jet, and surprisingly, this is the first report of the global minimum conf. I both experimentally and theoretically. Rotational transitions from the parent species of both conformers as well as their minor isotopologues (13C, 15N, and 18O) in natural abundance were assigned allowing experimental geometries to be derived. For conf. I, in addition to the typical splitting pattern due to the 14N quadrupole nucleus, the transitions show a tunneling splitting which arises from the interconversion motion between its two mirror images. The experimental observation of conf. I and the absence of conf. II in the jet are rationalized using quantum-chemical calculations to explore the importance of electron correlation and in particular, demonstrate the necessity of including dispersion effects in density functional theory calculations even for seemingly small molecules.

Theoretical investigation of potential energy surface and bound states for the N2-OCS van der Waals complex.

In this work we report an ab initio intermolecular potential energy surface and theoretically spectroscopic studies for N2-OCS complex. A four-dimensional intermolecular potential energy surface (4D PES) is constructed at the level of single and double excitation coupled-cluster method with a non-iterative perturbation treatment of triple excitations [CCSD(T)] with aug-cc-pVTZ basis set supplemented with bond functions. A global minimum corresponding to a planar and nearly T-shaped structure, which has been observed experimentally, is located at R = 3.96 Å, θ1 = 8 or 172°, θ2 = 75°, φ = 180 or 0° with a well depth of 271.078 cm-1 on the potential energy surface. The local minimum corresponding to a linear geometry is also found at R = 5.06 Å, θ1 = 2 or 178°, θ2 = 179°, φ = 0 or 180° with a well depth of 224.743 cm-1. The bound state calculations have been performed for the complex by approximating the N2 and OCS molecules as the rigid rotors. The calculated structural parameters and rotational transition frequencies are in good agreement with the experimental observed values. Based on the ab initio PES, the tunneling splitting is calculated to be 0.052 cm-1 for the ground vibrational state, which can just reproduce 64% of experimental observation (0.082 cm-1). A refined method is used to calculate the tunneling splitting, and a much better result is obtained with the value of 0.094 cm-1.