Tunneling Splitting Research Articles

Multidimensional H-atom tunneling in the catecholate monoanion.

We present the catecholate monoanion as a new model system for the study of multidimensional tunneling. It has a symmetrical O-H double-well structure, and the H atom motion between the two wells is coupled to both low and high frequency modes with different strengths. With a view to studying mode-specific tunneling in the catecholate monoanion, we have developed a full (33) dimensional potential energy surface in transition state (TS) normal modes using a Distributed Gaussian Empirical Valence Bond (DGEVB) based approach. We have computed eigenstates in different subspaces using both unrelaxed and relaxed potentials based on the DGEVB model. With unrelaxed potentials, we present results up to 7D subspaces that include the imaginary frequency mode and six modes coupled to it. With relaxed potentials, we focus on the two most important coupling modes. The structures of the ground and vibrationally excited eigenstates are discussed for both approaches and mode-specific tunneling splitting and their trends are presented.

Measurement of Donor-Acceptor Interchange Tunnelling in Ar(H2O)2 using Rotational Spectroscopy and a Re-look at Its Structure and Bonding

In this work, we have measured the donor-acceptor interchange tunnelling splitting in the ground vibrational state of Ar(H2O)2. In the previous investigations, the donor-acceptor tunnelling splitting in fully deuterated species, Ar(D2O)2, was measured to be about 106 MHz. It could not be measured for Ar(H2O)2, as the splitting was expected to be several GHz. Besides, for (D2O)2, both A1 and B1 states have non-zero statistical weight appearing on both sides of the E1 state, having a semi-rigid rotor type spectrum making their observation easier, compared to that of (H2O)2, which has zero statistical weight for the B1 state. With the help of a fourfold periodic potential, we have accurately predicted the fingerprints of b-dipole A1+↔A1- transitions and observed them using a pulsed nozzle Fourier transform microwave spectrometer. Measurement of these transitions enabled us to determine the donor-acceptor tunnelling splitting of 4257.41(4) MHz in Ar(H2O)2. More detailed structural parameters of Ar(H2O)2 have been evaluated in this work and critically compared with (H2O)2. Atoms in Molecules (AIM), non-covalent interactions (NCI) index and natural bond orbital (NBO) analysis have also been carried out for both (H2O)2 and Ar(H2O)2, and these calculations confirm that the presence of argon (Ar) does not affect the structure and bonding in (H2O)2.

Higher-Energy Hexafluoroisopropanol···Water Isomer and Its Large Amplitude Motions: Rotational Spectra and DFT Calculations.

Rotational spectra of the 1,1,1,3,3,3-hexafluoro-2-propanol (HFIP)···water complex were measured using a chirped pulse Fourier-transform microwave spectrometer. The spectral analyses, aided by density functional theory calculations, reveal two HFIP···water isomers: one previously reported, trans HFIP (HFIPt)···water (Phys. Chem. Chem. Phys. 2015, 119, 5650-5657), and a new isomer, gauche HFIP (HFIPg)···water. To confirm the identity of the new isomer, rotational spectra of seven of its deuterated species were also measured and analyzed. Both the experimental and theoretical pieces of evidence indicate that the intermolecular interaction with water preferentially stabilizes the HFIPg monomer configuration over the global minimum configuration, HFIPt. The relative energy difference between these monomeric forms is 4.1 kJ mol-1 and decreases to 2.5 kJ mol-1 in the respective monohydrates at the B3LYP-D3(BJ)/def2-QZVP level of theory. Both rigid and relaxed potential energy surface scans were carried out to gain insights into the large-amplitude water motions in HFIPg···water. The nonobservation of a water tunneling splitting in HFIPt···water has been explained to be a result of a barrier-less (after zero-point-energy correction) pathway for the water motion, whereas in HFIPg···water, a relatively large water tunneling barrier was identified as the cause of barely resolved water tunneling splittings. Noncovalent interaction and quantum theory of atoms and molecule analyses were used to evaluate the changes in HFIPg···water when going from the minimum to the transition state in terms of attractive interactions such as the OH···H and OH···F contacts. The effect of fluorination is discussed by comparing the vastly different binding topologies of isopropanol···water and HFIP···water.

Inside Cover: Magnetization Dynamics on Isotope‐Isomorphic Holmium Single‐Molecule Magnets (Angew. Chem. Int. Ed. 52/2021)

Nuclear-spin driven relaxation collaborating with the ligands in lanthanide single molecule magnets (SMMs) remains ambiguous. In their Communication on page 27282, Liviu Ungur, Jun-Liang Liu, and co-workers report two HoIII isotopic-isomorphic SMMs, whose magnetization dynamics reveals a switch in the relative relaxation rates at low/high temperatures. A theoretical analysis suggests that the noticeable isotope effect arises from the tunnel splitting and the super-hyperfine interaction related to the distinct nuclear spin numbers of 1H and 2H.

Disentangling 4f -radical coupling and dissipative Landau-Zener quantum tunneling in a continuously measured single-ion magnet spin transistor

Both radical-lanthanide exchange coupling and crystal-field tunnel splitting are responsible for the ground to first-excited Kramers doublet energy gap $\mathrm{\ensuremath{\Delta}}{E}_{\text{KD}}$ in an Ising $4f$ single-ion magnet coupled to a noninnocent ligand. However, these two mechanisms cannot be easily disentangled via spectroscopy alone because their interplay results in a single absorption line. Here we present a microscopic theory for the continuous measurement of the dissipative Landau-Zener transitions recently discussed for a ${\mathrm{TbPc}}_{2}$ spin transistor [Phys. Rev. Lett. 118, 257701 (2017)] and show how this quantum transport experiment can be used to disentangle the magnetic coupling and the tunnel splitting contributions to $\mathrm{\ensuremath{\Delta}}{E}_{\text{KD}}$. In addition, our model elucidates the microscopic origin of the decoherent spin tunneling observed in the ${\mathrm{TbPc}}_{2}$ spin transistor, by explicitly exposing the role played by phonon-mediated spin relaxation and sequential electron transport through the noninnocent ligand, in a time-dependent magnetic field.

Magnetization Dynamics on Isotope-Isomorphic Holmium Single-Molecule Magnets.

Here we reported the deuteration of the metal-binding equatorial water molecules in a reported HoIII single-molecule magnet (SMM) with pentagonal-bipyramidal geometry, from [Ho(CyPh2 PO)2 (H2 O)5 ]3+ to [Ho(CyPh2 PO)2 (D2 O)5 ]3+ . The hyperfine structures originating from the nuclear spin of 165 HoIII can be clearly observed. Moreover, the resulting magnetization dynamics revealed the switch of the relative relaxation rates for the two isotope-isomorphic complexes-respectively faster/slower at low/high temperature. The noticeable isotope effect arises from not only the paramagnetic metal center but also the diamagnetic ligands, which can be explained by the ab initio calculated tunnel splitting and the involvement of the super-hyperfine interaction related to the difference in the nuclear spin number of protium (1 H, I=1 /2 ) and deuterium (2 H, I=1).

Magnetization Dynamics on Isotope‐Isomorphic Holmium Single‐Molecule Magnets

AbstractHere we reported the deuteration of the metal‐binding equatorial water molecules in a reported HoIII single‐molecule magnet (SMM) with pentagonal‐bipyramidal geometry, from [Ho(CyPh2PO)2(H2O)5]3+ to [Ho(CyPh2PO)2(D2O)5]3+. The hyperfine structures originating from the nuclear spin of 165HoIII can be clearly observed. Moreover, the resulting magnetization dynamics revealed the switch of the relative relaxation rates for the two isotope‐isomorphic complexes—respectively faster/slower at low/high temperature. The noticeable isotope effect arises from not only the paramagnetic metal center but also the diamagnetic ligands, which can be explained by the ab initio calculated tunnel splitting and the involvement of the super‐hyperfine interaction related to the difference in the nuclear spin number of protium (1H, I=1/2) and deuterium (2H, I=1).

Spectroscopic study of the tunneling dynamics in N2-water observed in the O-D stretch region.

The O-D stretch rovibrational spectra of N2-D2O and N2-DOH were measured and analyzed. A combination band involving the in-plane N2 bending vibration was also observed. These bands were recorded using a pulsed-slit supersonic jet expansion and a mid-infrared tunable optical parametric oscillator. The spectra were analyzed by considering the feasible tunneling motions, and transitions were fitted to independent asymmetric rotors for each tunneling state. The rotational constants of the four tunneling components of N2-D2O were retrieved for the excited vibrational states. A two order of magnitude increase in the tunneling splittings is observed for the asymmetric O-D stretch (ν3 in D2O) excitation compared to the symmetric stretch (ν1 in D2O) and to the ground vibrational state. This last finding indicates that the ν3 vibrational state is likely perturbed by a combination state that includes ν1. Finally, the observation of a local perturbation in the ν3 vibrational band, affecting the positions of few rovibrational levels, provides an experimental lower limit of the dissociation energy of the complex, D0 > 120cm-1.

A simplistic computational procedure for tunneling splittings caused by proton transfer

In this manuscript, we present an approach for computing tunneling splittings for large amplitude motions. The core of the approach is a solution of an effective one-dimensional Schrödinger equation with an effective mass and an effective potential energy surface composed of electronic and harmonic zero-point vibrational energies of small amplitude motions in the molecule. The method has been shown to work in cases of three model motions: nitrogen inversion in ammonia, single proton transfer in malonaldehyde, and double proton transfer in the formic acid dimer. In the current work, we also investigate the performance of different DFT and post-Hartree–Fock methods for prediction of the proton transfer tunneling splittings, quality of the effective Schrödinger equation parameters upon the isotopic substitution, and possibility of a complete basis set (CBS) extrapolation for the resulting tunneling splittings.

Theoretical Study of Proton Tunneling in the Imidazole–Imidazolium Complex

Proton tunneling in the hydrogen-bonded imidazole–imidazolium complex ion has been studied theoretically. Ab initio CASSCF/6-311++G(d,p) calculations concerning geometry optimization and vibrational frequencies have been carried out for equilibrium and transition state structures of the system. Two-dimensional double-well model potentials were constructed on the basis of ab initio results and used to analyze the proton dynamics in the hydrogen bond and the influence of the excitation of low-frequency hydrogen-bond vibrations on the proton tunneling splittings. The energy of tunneling-split vibrational sublevels of the high-frequency tunneling mode have been calculated for its ground and first excited vibrational state for the series of excitations of the coupled low-frequency intramolecular hydrogen-bond modes. The promoting and suppressing effect of the low-frequency modes on the proton splittings was shown in the ground and first excited vibrational state of the tunneling mode. The vibrational sublevels form the two separate semicontinuous bands between which the absorption transitions may occur. This mechanism explains the experimentally observed splitting and doublet-component broadening of the high-frequency N–H stretching infrared (IR) absorption band.

Rotational Coupling in Methyl-Tunneling Electron Spin Echo Envelope Modulation

Coherence between tunnel-split states of a methyl quantum rotor can be generated and observed in stimulated and spin-locked echo experiments, if hyperfine coupling of a nearby electron spin to the methyl protons breaks C_3 symmetry and is of the same order of magnitude as the tunnel splitting. Here, we consider the case of two methyl groups bound to the same sp^{3}-hybridized atom, which is important in the context of common nitroxide spin labels. For a simple form of the rotor-rotor coupling Hamiltonian, we provide an approach that allows for density operator computations of this system with 1152 quantum states with moderate computational effort. We find that, in the regime where the ratio between rotor-rotor coupling and rotational barrier is much smaller than unity, three-pulse ESEEM and hyperfine-decoupled ESEEM depend only on the tunnel splitting, but not on this ratio. This finding may simplify the treatment of tunnel-induced electron decoherence in systems where the methyl groups are bound to sp^{3}-hybridized atoms.

Coupling of torsion and OH-stretching in tert-butyl hydroperoxide. II. The OH-stretching fundamental and overtone spectra.

The vibrational spectra of gas phase tert-butyl hydroperoxide have been recorded in the OH-stretching fundamental and overtone regions (ΔvOH = 1-5) at room temperature using conventional Fourier transform infrared (ΔvOH = 1-3) and cavity ring-down (ΔvOH = 4-5) spectroscopy. In hydroperoxides, the OH-stretching and COOH torsion vibrations are strongly coupled. The double-well nature of the COOH torsion potential leads to tunneling splitting of the energy levels and, combined with the low frequency of the torsional vibration, results in spectra in the OH-stretching regions with multiple vibrational transitions. In each of the OH-stretching regions, both an OH-stretching and a stretch-torsion combination feature are observed, and we show direct evidence for the tunneling splitting in the OH-stretching fundamental region. We have developed two complementary vibrational models to describe the spectra of the OH-stretching regions, a reaction path model and a reduced dimensional local mode model, both of which describe the features of the vibrational spectra well. We also explore the torsional dependence of the OH-stretching transition dipole moment and show that a Franck-Condon treatment fails to capture the intensity in the region of the stretch-torsion combination features. The accuracy of the Franck-Condon treatment of these features improves with increasing ΔvOH.

Dissipation-induced symmetry breaking: Emphanitic transitions in lead- and tin-containing chalcogenides and halide perovskites

Lead- and tin-based chalcogenide semiconductors like PbTe or SnSe have long been known to exhibit an unusually low thermal conductivity that makes them very attractive thermoelectric materials. An apparently unrelated fact is that the excitonic bandgap in these materials increases with temperature, whereas for most semiconductors, one observes the opposite trend. These two anomalous features are also seen in a very different class of photovoltaic materials, namely, the halide perovskites such as CsPbBr3. It has been previously proposed that emphanisis, a local symmetry-breaking phenomenon, is the one common origin of these unusual features. Discovered a decade ago, emphanisis is the name given to the observed displacement of the lead or the tin ions from their cubic symmetry ground state to a locally distorted phase at high temperature. This phenomenon has been puzzling because it is unusual for the high-temperature state to be of a lower symmetry than the degenerate ground state. Motivated by the celebrated vibration-inversion resonance of the ammonia molecule, we propose a quantum tunneling-based model for emphanisis where decoherence is responsible for the local symmetry breaking with increasing temperature. From the analytic expression of the temperature dependence of the tunnel splitting (which serves as an order parameter), we provide three-parameter fitting formulas that capture the observed temperature dependence of the ionic displacements as well as the anomalous increase in the excitonic bandgap in all the relevant materials.

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.

Full-dimensional potential energy surface for acetylacetone and tunneling splittings.

We present a full-dimensional potential energy surface for acetylacetone (AcAc) using full and fragmented permutationally invariant polynomial approaches. Previously reported MP2/aVTZ energies and gradients are augmented by additional calculations at this level of theory for the fits. Numerous stationary points are reported as are the usual metrics to assess the precision of the fit. The electronic barrier height for the H-atom transfer is roughly 2.2 kcal mol-1. Diffusion Monte Carlo (DMC) calculations are used to calculate the ground state wavefunction and zero-point energy of acetylacetone. These together with fixed-node DMC calculations for the first excited-state provide the predicted tunneling splitting due to the barrier to H-transfer separating two equivalent wells. Simpler 1d calculations of this splitting are also reported for varying barrier heights including the CCSD(T) barrier height of 3.2 kcal mol-1. Based on those results the DMC splitting of 160 cm-1 with a statistical uncertainty of roughly 21 cm-1, calculated using the MP2-based PES, is estimated to decrease to 100 cm-1 for a barrier of 3.2 kcal mol-1. The fragmented surface is shown to be fast to evaluate.

Simple models for the quick estimation of ground state hydrogen tunneling splittings in alcohols and other compounds.

Models for the quick estimation of energy splittings caused by coherent tunneling of hydrogen atoms are evaluated with available experimental data for alcohols and improvements are proposed. The discussed models are mathematically simple and require only results from routine quantum chemical computations, i.e. hybrid DFT calculation of the equilibrium geometry and the transition state within the harmonic approximation. A benchmark of experimental splittings spanning four orders of magnitude for 27 alcohol species is captured by three evaluated models with a mean symmetric deviation factor of 1.7, 1.5 and 1.4, respectively, i.e. the calculated values deviate on average by this factor in either direction. Limitations of the models are explored with alcohols featuring uncommon properties, such as an inverted conformational energy sequence, a very light molecular frame, an elevated torsional frequency, or a coupling with a second internal degree of freedom. If the splitting of either the protiated or deuterated form of an alcohol is already experimentally determined, the one of the second isotopolog can be estimated by three additional models with a mean symmetric deviation factor of 1.14, 1.19 and 1.15, respectively. It is shown that this can be achieved with a novel approach without any quantum chemical calculation by directly correlating experimental splittings of isotopologs across related species. This is also demonstrated for other classes of compounds with hydrogen tunneling, such as amines, thiols, and phenols. Furthermore, it is found that the isotope effect can even be anticipated without any further knowledge about the system solely from the size of either splitting with a mean symmetric deviation factor of 1.3. This is based on an extensive sample of 77 pairs of splittings spanning eight orders of magnitude for isotopologs of chemically diverse compounds.

Multipole-moment effects in ion-molecule reactions at low temperatures: part I - ion-dipole enhancement of the rate coefficients of the He+ + NH3 and He+ + ND3 reactions at collisional energies Ecoll/kB near 0 K.

The energy dependence of the rates of the reactions between He+ and ammonia (NY3, Y = {H,D}), forming NY2+, Y and He as well as NY+, Y2 and He, and the corresponding product branching ratios have been measured at low collision energies Ecoll between 0 and kB·40 K using a recently developed merged-beam technique [Allmendinger et al., ChemPhysChem, 2016, 17, 3596]. To avoid heating of the ions by stray electric fields, the reactions are observed within the large orbit of a highly excited Rydberg electron. A beam of He Rydberg atoms was merged with a supersonic beam of ammonia using a curved surface-electrode Rydberg–Stark deflector, which is also used for adjusting the final velocity of the He Rydberg atoms, and thus the collision energy. A collision-energy resolution of about 200 mK was reached at the lowest Ecoll values. The reaction rate coefficients exhibit a sharp increase at collision energies below ∼kB·5 K and pronounced deviations from Langevin-capture behaviour. The experimental results are interpreted in terms of an adiabatic capture model describing the rotational-state-dependent orientation of the ammonia molecules by the electric field of the He+ atom. The model faithfully describes the experimental observations and enables the identification of three classes of |JKMp〉 rotational states of the ammonia molecules showing different low-energy capture behaviour: (A) high-field-seeking states with |KM| ≥ 1 correlating to the lower component of the umbrella-motion tunnelling doublet at low fields. These states undergo a negative linear Stark shift, which leads to strongly enhanced rate coefficients; (B) high-field-seeking states subject to a quadratic Stark shift at low fields and which exhibit only weak rate enhancements; and (C) low-field-seeking states with |KM| ≥ 1. These states exhibit a positive Stark shift at low fields, which completely suppresses the reactions at low collision energies. Marked differences in the low-energy reactivity of NH3 and ND3—the rate enhancements in ND3 are more pronounced than in NH3—are quantitatively explained by the model. They result from the reduced magnitudes of the tunnelling splitting and rotational intervals in ND3 and the different occupations of the rotational levels in the supersonic beam caused by the different nuclear-spin statistical weights. Thermal capture rate constants are derived from the model for the temperature range between 0 and 10 K relevant for astrochemistry. Comparison of the calculated thermal capture rate coefficients with the absolute reaction rates measured above 27 K by Marquette et al. (Chem. Phys. Lett., 1985, 122, 431) suggests that only 40% of the close collisions are reactive.

Semi-obstructed splitting behaviors of droplet in an asymmetric microfluidic T-junction

Asymmetric droplet splitting is a common method to obtain micro-droplets of different sizes. The study of droplet asymmetric splitting behaviors is of great significance to the fields of biomedicine, energy, chemical industry and food engineering. In this paper, the control flow is introduced into a branch of the T-shaped microchannel to control the pressure distribution in the channel and precisely control the size of the daughter droplets. The method is simple to operate and is a preferred method for asymmetric microfluidic splitting. Existing studies have analyzed droplet splitting modes, critical conditions for flow pattern transitions, and splitting dynamics, but the theoretical prediction of droplet asymmetric splitting behaviors needs to be strengthened. Moreover, compared with tunnel splitting and obstructed splitting, which are more abundantly studied, neither semi-obstructed splitting as an intermediate state of tunnel splitting nor obstructed splitting is analyzed sufficiently. Therefore, a microfluidic T-junction chip is designed and fabricated, with which asymmetrical splitting behaviors of droplets with a tunnel in a microfluidic T-junction are investigated experimentally. The influence of flow rate regulation on the droplet splitting ratio is studied. And a theoretical model is also established to predict the splitting ratio. The results are concluded as follows: 1) the process of asymmetrical droplet splitting is divided into three stages i.e. early squeezing, late squeezing and rapid pinch-off stage. In the early stage of squeezing, the radius of curvature of the droplet neck is sizable, and the additional pressure of interfacial tension is minor. Compared with the additional pressure that hinders neck contraction, the upstream continuous phase driving force is dominant, and the width of the neck changes linearly with time; in the process of late squeezing, the upstream pressure driving effect is still greater than the hindering effect of the additional tension, and the neck width changes exponentially with time; However, in the rapid pinch-off stage, the interfacial tension pointing to the center of the cross section of droplet neck dominates the pinch-off stage. Then, the droplet neck shrinks sharply. 2) Adjusting the flow rate of the branch channel can effectively control the asymmetric splitting ratio of the droplets, and under the current semi-obstructed asymmetric splitting of the droplets, the regulation effect is less affected by the size of the mother droplet, but more affected by the capillary number. 3) The prediction model of droplet splitting ratio based on the pressure drop model can effectively predict the droplet splitting ratio.

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.

Nuclear-Electronic Orbital Multistate Density Functional Theory.

Hydrogen tunneling is essential for a wide range of chemical and biological processes. The description of hydrogen tunneling with multicomponent quantum chemistry approaches, where the transferring hydrogen nucleus is treated on the same level as the electrons, is challenging due to the importance of both static and dynamical electron-proton correlation. Herein the nuclear-electronic orbital multistate density functional theory (NEO-MSDFT) method is presented as a strategy to include both types of correlation. In this approach, two localized nuclear-electronic wave functions obtained with the NEO-DFT method are combined with a nonorthogonal configurational interaction approach to produce bilobal, delocalized ground and excited vibronic states. By including a correction function, the NEO-MSDFT approach can produce quantitatively accurate hydrogen tunneling splittings for fixed geometries of systems such as malonaldehyde and acetoacetaldehyde. This approach is computationally efficient and can be combined with methods such as vibronic coupling theory to describe tunneling dynamics and to compute vibronic couplings in many types of systems.