Ground-state Tunneling Splitting Research Articles

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.

An Ab Initio Neural Network Potential Energy Surface for the Dimer of Formic Acid and Further Quantum Tunneling Dynamics.

We construct a full-dimensional ab initio neural network potential energy surface (PES) for the isomerization system of the formic acid dimer (FAD). This is based upon ab initio calculations using the DLPNO-CCSD(T) approach with the aug-cc-pVTZ basis set, performed at over 14000 symmetry-unique geometries. An accurate fit to the obtained energies is generated using a general neural network fitting procedure combined with the fundamental invariant method, and the overall energy-weighted root-mean-square fitting error is about 6.4 cm-1. Using this PES, we present a multidimensional quantum dynamics study on tunneling splittings with an efficient theoretical scheme developed by our group. The ground-state tunneling splitting of FAD calculated with a four-mode coupled method is in good agreement with the most recent experimental measurements. The PES can be applied for further dynamics studies. The effectiveness of the present scheme for constructing a high-dimensional PES is demonstrated, and this scheme is expected to be feasible for larger molecular systems.

Quantum dynamics calculations on isotope effects of hydrogen transfer isomerization in formic acid dimer

We present a quantum dynamics study on the isotope effects of hydrogen transfer isomerization in the formic acid dimer, and this is achieved by multidimensional dynamics calculations with an efficient quantum mechanical theoretical scheme developed by our group, on a full-dimensional neural network ab initio potential energy surface. The ground-state and fundamental tunneling splittings for four deuterium isotopologues of formic acid dimer are considered, and the calculated results are in very good general agreement with the available experimental measurements. Strong isotope effects are revealed, the mode-specific fundamental excitation effects on the tunneling rate are evidently influenced by the deuterium substitution of H atom with the substitution on the OH bond being more effective than on the CH bond. Our studies are helpful for acquiring a better understanding of isotope effects in the double-hydrogen transfer processes.

The tunneling splitting and the Kramers theory of activated processes

The study of tunneling splitting is fundamental to get insight into the dynamics of a multitude of molecular systems. In this paper, a novel approach to the analysis of the ground-state tunneling splitting is presented and explicitly applied to one-dimensional systems, with the aim of benchmarking its applicability in the simplest case. The isomorphism between the Fokker-Planck-Smoluchowski operator and the Born-Oppenheimer quantum Hamiltonian is the key element of this method. The localization function approach, used in the field of stochastic processes to study the Kramers problem, leads to a simple, yet asymptotically justified, integral approximation for the tunneling splitting. The comparison with exact values of the tunneling splittings shows a better accuracy than WKB semiclassical estimates for the family of considered model systems; demonstrating that the method may improve numerical estimates beside offering an alternative point of view on the features controlling the tunneling splitting in simple potential profiles.

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.

Double Proton Transfer in the Dimer of Formic Acid: An Efficient Quantum Mechanical Scheme.

Double proton transfer plays an important role in biology and chemistry, such as with DNA base pairs, proteins and molecular clusters, and direct information about these processes can be obtained from tunneling splittings. Carboxylic acid dimers are prototypes for multiple proton transfer, of which the formic acid dimer is the simplest one. Here, we present efficient quantum dynamics calculations of ground-state and fundamental excitation tunneling splittings in the formic acid dimer and its deuterium isotopologues. These are achieved with a multidimensional scheme developed by us, in which the saddle-point normal coordinates are chosen, the basis functions are customized for the proton transfer process, and the preconditioned inexact spectral transform method is used to solve the resultant eigenvalue problem. Our computational results are in excellent agreement with the most recent experiments (Zhang et al., 2017; Li et al., 2019).

Topological quantum oscillation of magnetization in the Josephson φ0 junction

Abstract We show that the ground state tunnel splitting of the magnetization can be topologically quenched in a nanomagnetic Josephson φ 0 junction. Due to the relative strength of the spin-orbit interaction the tunnel splitting is found to oscillate with the bias current applied to the junction as well as the magnetic field along the hard axis. Spin-coherent-state path integrals are used to obtain an analytic expression for the tunnel splitting controlled by topological interference of two tunnel paths of opposite windings and the condition in which the tunnel splitting is frozen. This feature is expected to be observable in existing experimental technique.

The hydrogen tunneling splitting in malonaldehyde: A full-dimensional time-independent quantum mechanical method.

The accurate time-independent quantum dynamics calculations on the ground-state tunneling splitting of malonaldehyde in full dimensionality are reported for the first time. This is achieved with an efficient method developed by us. In our method, the basis functions are customized for the hydrogen transfer process which has the effect of greatly reducing the size of the final Hamiltonian matrix, and the Lanczos method and parallel strategy are used to further overcome the memory and central processing unit time bottlenecks. The obtained ground-state tunneling splitting of 24.5 cm(-1) is in excellent agreement with the benchmark value of 23.8 cm(-1) computed with the full-dimensional, multi-configurational time-dependent Hartree approach on the same potential energy surface, and we estimate that our reported value has an uncertainty of less than 0.5 cm(-1). Moreover, the role of various vibrational modes strongly coupled to the hydrogen transfer process is revealed.

An ab initio potential energy surface for the formic acid dimer: zero-point energy, selected anharmonic fundamental energies, and ground-state tunneling splitting calculated in relaxed 1-4-mode subspaces.

We report a full-dimensional, permutationally invariant potential energy surface (PES) for the cyclic formic acid dimer. This PES is a least-squares fit to 13475 CCSD(T)-F12a/haTZ (VTZ for H and aVTZ for C and O) energies. The energy-weighted, root-mean-square fitting error is 11 cm-1 and the barrier for the double-proton transfer on the PES is 2848 cm-1, in good agreement with the directly-calculated ab initio value of 2853 cm-1. The zero-point vibrational energy of 15 337 ± 7 cm-1 is obtained from diffusion Monte Carlo calculations. Energies of fundamentals of fifteen modes are calculated using the vibrational self-consistent field and virtual-state configuration interaction method. The ground-state tunneling splitting is computed using a reduced-dimensional Hamiltonian with relaxed potentials. The highest-level, four-mode coupled calculation gives a tunneling splitting of 0.037 cm-1, which is roughly twice the experimental value. The tunneling splittings of (DCOOH)2 and (DCOOD)2 from one to three mode calculations are, as expected, smaller than that for (HCOOH)2 and consistent with experiment.

Full-dimensional quantum calculations of the vibrational states of ${\rm H}_5^+$H5+

Full-dimensional quantum calculations of the vibrational states of H5(+) have been performed on the accurate potential energy surface developed by Xie et al. [J. Chem. Phys. 122, 224307 (2005)]. The zero point energies of H5(+), H4D(+), D4H(+), and D5(+) and their ground-state geometries are presented and compared with earlier theoretical results. The first 10 low-lying excited states of H5(+) are assigned to the fundamental, overtone, and combination of the H2-H3(+) stretch, the shared proton hopping and the out-of-plane torsion. The ground-state torsional tunneling splitting, the fundamental of the photon hopping mode and the first overtone of the torsion mode are 87.3 cm(-1), 354.4 cm(-1), and 444.0 cm(-1), respectively. All of these values agree well with the diffusion Monte Carlo and multi-configuration time-dependent Hartree results where available.

Reduced-Dimensional Quantum Computations for the Rotational–Vibrational Dynamics of F––CH4 and F––CH2D2

Variational rotational-vibrational quantum chemical computations are performed for the F(-)-CH4 and F(-)-CH2D2 anion complexes using several reduced-dimensional models in a curvilinear polyspherical coordinate system and utilizing an accurate ab initio potential energy surface (PES). The implementation of the models is made practical by using the general rovibrational code GENIUSH, which constructs the complicated form of the exact rovibrational kinetic energy operator in reduced and full dimensions in any user-specified coordinates and body-fixed frames. A one-dimensional CF stretch, 1D(RCF), a two-dimensional intermolecular bend, 2D(θ,φ), and a three-dimensional intermolecular, 3D(RCF,θ,φ), rigid methane model provide vibrational energies for the low-frequency, large-amplitude modes in good agreement with full-dimensional MCTDH results for F(-)-CH4. The 2D(θ,φ) and 3D(RCF,θ,φ) four-well computations, describing equally the four possible CH-F(-) bonds, show that the ground-state tunneling splitting is less than 0.01 cm(-1). For the hydrogen-bonded CH stretching fundamental a local-mode model is found to have almost spectroscopic accuracy, whereas a harmonic frequency analysis performs poorly. The 2D(θ,φ) and 3D(RCF,θ,φ) rotational-vibrational computations on the Td-symmetric four-well PES reveal that in most cases F(-)-CH4 behaves as a semirigid C3v symmetric top. For the degenerate intermolecular bending vibrational states substantial splittings of the rigid rotor levels are observed. For F(-)-CH2D2 the rotational levels guide the assignment of the vibrational states to either F(-)-H or F(-)-D connectivity.

Low-temperature tunneling and rotational dynamics of the ammonium cations in (NH4)2B12H12

Low-temperature neutron scattering spectra of diammonium dodecahydro-closo-dodecaborate [(NH(4))(2)B(12)H(12)] reveal two NH(4)(+) rotational tunneling peaks (e.g., 18.5 μeV and 37 μeV at 4 K), consistent with the tetrahedral symmetry and environment of the cations. The tunneling peaks persist between 4 K and 40 K. An estimate was made for the tunnel splitting of the first NH(4)(+) librational state from a fit of the observed ground-state tunnel splitting as a function of temperature. At temperatures of 50 K-70 K, classical neutron quasi-elastic scattering appears to dominate the spectra and is attributed to NH(4)(+) cation jump reorientation about the four C(3) axes defined by the N-H bonds. A reorientational activation energy of 8.1 ± 0.6 meV (0.79 ± 0.06 kJ/mol) is determined from the behavior of the quasi-elastic linewidths in this temperature regime. This activation energy is in accord with a change in NH(4)(+) dynamical behavior above 70 K. A low-temperature inelastic neutron scattering feature at 7.8 meV is assigned to a NH(4)(+) librational mode. At increased temperatures, this feature drops in intensity, having shifted entirely to higher energies by 200 K, suggesting the onset of quasi-free NH(4)(+) rotation. This is consistent with neutron-diffraction-based model refinements, which derive very large thermal ellipsoids for the ammonium-ion hydrogen atoms at room temperature in the direction of reorientation.

Communication: Prediction of the rate constant of bimolecular hydrogen exchange in the water dimer using an ab initio potential energy surface

We report the properties of two novel transition states of the bimolecular hydrogen exchange reaction in the water dimer, based on an ab initio water dimer potential [A. Shank et al., J. Chem. Phys. 130, 144314 (2009)]. The realism of the two transition states is assessed by comparing structures, energies, and harmonic frequencies obtained from the potential energy surface and new high-level ab initio calculations. The rate constant for the exchange is obtained using conventional transition state theory with a tunneling correction. We employ a one-dimensional approach for the tunneling calculations using a relaxed potential from the full-dimensional potential in the imaginary-frequency normal mode of the saddle point, Q(im). The accuracy of this one-dimensional approach has been shown for the ground-state tunneling splittings for H and D-transfer in malonaldehyde and for the D+H(2) reaction [Y. Wang and J. M. Bowman, J. Chem. Phys. 129, 121103 (2008)]. This approach is applied to calculate the rate constant for the H(2)O+H(2)O exchange and also for H(2)O+D(2)O→2HOD. The local zero-point energy is also obtained using diffusion Monte Carlo calculations in the space of real-frequency-saddle-point normal modes, as a function of Q(im).

Multiconfigurational time-dependent Hartree calculations for tunneling splittings of vibrational states: Theoretical considerations and application to malonaldehyde

Full-dimensional multiconfigurational time-dependent Hartree calculations on the tunneling splitting of the vibrational ground state and the low lying excited states of malonaldehyde are presented. Methodological developments utilizing the symmetry of double well systems for the efficient calculation of tunneling splittings are described and discussed. Important aspects of the theory underlying the previously communicated results for the ground state tunneling splitting [M. D. Coutinho-Neto et al., J. Chem. Phys. 121, 9207 (2004)] are detailed and further developments facilitating the calculation of tunneling splittings for vibrationally excited states are introduced. Utilizing these developments, the 14 lowest vibrational states of malonaldehyde, i.e., seven tunneling splittings, have been computed. The tunneling splittings are found to vary significantly depending on the particular vibrational excitation. This results in a complex pattern of vibrational levels. Studying the dependence of the tunneling splittings on the vibrational excitation, good agreement with available experimental results is found and intuitive interpretations of the results can be given.

One-dimensional tunneling calculations in the imaginary-frequency, rectilinear saddle-point normal mode

We present tunneling calculations using the reaction path Hamiltonian in the zero-curvature approximation and a one-dimensional Hamiltonian in the imaginary-frequency, rectilinear normal mode of a saddle point, neglecting the vibrational angular momentum terms. This latter Hamiltonian was recently introduced and applied to the tunneling splitting in full-dimensional malonaldeyde [Y. Wang et al., J. Chem. Phys. 128, 224314 (2008)]. The results using the latter method are shown to be much more accurate than those using the former one for the ground-state tunneling splittings for H and D-transfer in malonaldehyde and for the D+H(2) reaction in three dimensions for zero total angular momentum.

Full-dimensional quantum calculations of ground-state tunneling splitting of malonaldehyde using an accurate ab initio potential energy surface

Quantum calculations of the ground vibrational state tunneling splitting of H-atom and D-atom transfer in malonaldehyde are performed on a full-dimensional ab initio potential energy surface (PES). The PES is a fit to 11 147 near basis-set-limit frozen-core CCSD(T) electronic energies. This surface properly describes the invariance of the potential with respect to all permutations of identical atoms. The saddle-point barrier for the H-atom transfer on the PES is 4.1 kcalmol, in excellent agreement with the reported ab initio value. Model one-dimensional and "exact" full-dimensional calculations of the splitting for H- and D-atom transfer are done using this PES. The tunneling splittings in full dimensionality are calculated using the unbiased "fixed-node" diffusion Monte Carlo (DMC) method in Cartesian and saddle-point normal coordinates. The ground-state tunneling splitting is found to be 21.6 cm(-1) in Cartesian coordinates and 22.6 cm(-1) in normal coordinates, with an uncertainty of 2-3 cm(-1). This splitting is also calculated based on a model which makes use of the exact single-well zero-point energy (ZPE) obtained with the MULTIMODE code and DMC ZPE and this calculation gives a tunneling splitting of 21-22 cm(-1). The corresponding computed splittings for the D-atom transfer are 3.0, 3.1, and 2-3 cm(-1). These calculated tunneling splittings agree with each other to within less than the standard uncertainties obtained with the DMC method used, which are between 2 and 3 cm(-1), and agree well with the experimental values of 21.6 and 2.9 cm(-1) for the H and D transfer, respectively.

Stereomutation Tunneling Switching Dynamics and Parity Violation in Chlorineperoxide Cl−O−O−Cl

In a search for efficient spectroscopic avenues toward experiments on molecular parity violation, we investigate the stereomutation tunneling processes in the axially chiral chlorine isotopomers of Cl2O2 by the quasi-adiabatic channel reaction path Hamiltonian (RPH) approach and the corresponding parity violating potentials by means of quantum chemical calculations including our recently developed Multiconfiguration linear response (MC-LR) approach to electroweak quantum chemistry. The calculated ground-state torsional tunneling splittings for all isotopomers of Cl2O2 are much smaller than the parity violating energy differences Delta(pv)E between the enantiomers of these molecules and therefore parity violation is predicted to dominate the quantum dynamics of stereomutation at low energies. We also compare these with torsional ground-state tunneling splittings and parity violating energy differences of the whole series of axially chiral HXYH(+) isotopomers (with X, Y= Cl(+), O, S, Se, Te). A comparison with our previous results for the homologous molecule Cl2S2 shows that for Cl2O2 a spectroscopic high-resolution analysis should be easier and the energy region of large tunneling splittings should be more easily accessible by IR excitation. We thus propose a scheme using "tunneling switching" with vibrational excitation in order to carry out the measurement of time-dependent parity violation in superposition states of initially well-defined parity. We discuss the advantages and drawbacks of such an experiment that can be carried out entirely in the IR spectral range (for Cl2O2 or related molecules).

Ground-state and vibrationally assisted tunneling in the formic acid dimer

The previously developed instanton theory [G. V. Mil'nikov and H. Nakamura, J. Chem. Phys. 122, 124311 (2005)] is applied to the calculation of vibrationally assisted tunneling splitting of the deuterated formic acid dimer (DCOOH)2 with all the degrees of freedom taken into account. The ground-state tunnel splitting is determined by the density-functional theory combined with coupled cluster level of quantum chemistry to be 0.0038 cm(-1) which is comparable to the experimental value of 0.0029 cm(-1). Further, the tunnel splittings of fundamental excitations are estimated for frequencies below 300 cm(-1). In this energy range it is found that the excitation modes may either enhance or suppress tunneling as compared to the ground state. For the higher-frequency modes a rapid growth of the tunnel splitting is observed. At frequencies above 1000 cm(-1) the semiclassical solution becomes unstable and no reliable tunneling splittings can be obtained. This is in vast contrast to the adiabatic approximation to the instanton theory in which the tunnel splittings can be retrieved up to 3000 cm(-1). We discuss this disparity from the viewpoint of the multidimensional character of tunneling in hydrogen bonds and the adiabatic approximation is concluded to be inaccurate.

Quantum Tunneling and Quantum Phase Interference in a [MnII2MnIII2] Single-Molecule Magnet

[Mn4(hmp)6(H2O)2(NO3)2](NO3)2.2.5H2O (1) has been synthesized from the reaction of 2-hydroxymethylpyridine (Hhmp) with Mn(NO3)2.4H2O in the presence of tetraethylammonium hydroxide. 1 crystallizes in the triclinic P space group with two crystallographically independent centrosymmetrical [Mn4(hmp)6(H2O)2(NO3)2]2+ complexes in the packing structure. Four Mn ions are arranged in a double-cuboidal fashion where outer Mn2+ are heptacoordinated and inner Mn3+ are hexacoordinated. dc magnetic measurements show that both Mn2+...Mn3+ and Mn3+...Mn3+ interactions are ferromagnetic with J(wb)/k(B) = +0.80(5) K, and J(bb)/k(B) = +7.1(1) K, respectively, leading to an S(T) = 9 ground state. Combined ac and dc measurements reveal the single-molecule magnet (SMM) behavior of 1 with both thermally activated and ground-state tunneling regimes, including quantum phase interference. In the thermally activated regime, the characteristic relaxation time (tau) of the system follows an Arrhenius law with tau0 = 6.7 x 10(-)(9) s and delta(eff)/k(B) = 20.9 K. Below 0.34 K, tau saturates indicating that the quantum tunneling of the magnetization becomes the dominant relaxation process as expected for SMMs. Down to 0.04 K, field dependence of the magnetization measured using the mu-SQUID technique shows the presence of very weak inter-SMM interactions (zJ'/k(B) approximately -1.5 x 10(-3) K) and allows an estimation of D/k(B) at -0.35 K. Quantum phase interference has been used to confirm the D value and to estimate the transverse anisotropic parameter to E/k(B) = +0.083 K and the ground-state tunnel splitting delta(LZ) = 3 x 10(-7) K at H(trans) = 0 Oe. These results rationalize the observed tunneling time (tau(QTM)) and the effective energy barrier (delta(eff)).

Numerical simulation on tunnel splitting of Bose-Einstein condensate in multi-well potentials

The low-energy-level macroscopic wave functions of the Bose-Einstein condensate(BEC) trapped in a symmetric double-well and a periodic potential are obtained by solving the Gross-Pitaevskii equation numerically. The ground state tunnel splitting is evaluated in terms of the even and odd wave functions corresponding to the global ground and excited states respectively. We show that the numerical result is in good agreement with the analytic level splitting obtained by means of the periodic instanton method.