Double-well System Research Articles

Resonances crossing and electric field quantum sensors* *Andrea Sacchetti is member of Gruppo Nazionale per la Fisica Matematica of Istituto Nazionale di Alta Matematica (GNFM-INdAM). This work is partially supported by the Next Generation EU—Prin 2022CHELC7 project ‘Singular Interactions and Effective Models in Mathematical Physics’ and the UniMoRe-FIM project ‘Modelli e metodi della Fisica Matematica’.

We propose a theoretical model for a quantum sensor that can determine in a very simple way whether the intensity of an electric field has an assigned value or not. It is based on the fact that when an exact crossing of the imaginary parts of the resonances occurs in a double-well quantum system subject to an external DC electric field, a damped beating phenomenon occurs, which is absent if there is no such a crossing. This result is then tested numerically on an explicit one-dimensional model.

Observation of Pairwise Level Degeneracies and the Quantum Regime of the Arrhenius Law in a Double-Well Parametric Oscillator

By applying a microwave drive to a specially designed Josephson circuit, we have realized a double-well model system: a Kerr oscillator submitted to a squeezing force. We have observed, for the first time, the spectroscopic fingerprint of a quantum double-well Hamiltonian when its barrier height is increased: a pairwise level kissing (coalescence) corresponding to the exponential reduction of tunnel splitting in the excited states as they sink under the barrier. The discrete levels in the wells also manifest themselves in the activation time across the barrier which, instead of increasing smoothly as a function of the barrier height, presents steps each time a pair of excited states is captured by the wells. This experiment illustrates the quantum regime of Arrhenius’s law, whose observation is made possible here by the unprecedented combination of low dissipation, time-resolved state control, 98.5% quantum nondemolition single shot measurement fidelity, and complete microwave control over all Hamiltonian parameters in the quantum regime. Direct applications to quantum computation and simulation are discussed. Published by the American Physical Society 2024

Numerical simulation on converting abandoned wells into double-well open-loop geothermal system

Converting abandoned oil wells into geothermal systems holds immense potential and has garnered international attention. However, research on converting abandoned oil wells into Open-Loop Geothermal Systems (OLGS) remains insufficient and requires further exploration. To address this gap, this study utilizes COMSOL software employing the Finite Element Method (FEM) to establish a numerical model of a double-well OLGS with 3D multi-physics field coupling, investigating geothermal extraction through numerical simulation. The findings indicate that in double-well OLGS, utilizing high flow rates in the early production stage and lower flow rates in the later stages enhances system thermal power. Additionally, employing lower inlet temperatures in the early production stage and higher inlet temperatures in the later stages favors geothermal extraction. Higher geothermal gradients and larger well spacings further benefit geothermal extraction. Increasing the length of the production well's insulating layer in the early production stage helps minimize heat loss. The outlet temperature exhibits high sensitivity to flow rate and geothermal gradient throughout the production phase, while sensitivity shifts towards inlet temperature and well spacing in the later stages. In the five-spot OLGS, outlet temperature and thermal power increase with a reduction in the number of production wells; moreover, assuming identical outlet temperatures (100 ℃) after 50 years of production, the thermal power of the five-spot OLGS exceeds that of the double-well OLGS by an average of 20937.88 kW per year.

Data-driven discovery of invariant measures

Invariant measures encode the long-time behaviour of a dynamical system. In this work, we propose an optimization-based method to discover invariant measures directly from data gathered from a system. Our method does not require an explicit model for the dynamics and allows one to target specific invariant measures, such as physical and ergodic measures. Moreover, it applies to both deterministic and stochastic dynamics in either continuous or discrete time. We provide convergence results and illustrate the performance of our method on data from the logistic map and a stochastic double-well system, for which invariant measures can be found by other means. We then use our method to approximate the physical measure of the chaotic attractor of the Rössler system, and we extract unstable periodic orbits embedded in this attractor by identifying discrete-time periodic points of a suitably defined Poincaré map. This final example is truly data-driven and shows that our method can significantly outperform previous approaches based on model identification.

Chaotic dynamics of a double-well Bose-Einstein condensate.

We study the dynamics of a double-well BEC system subjected to oscillating dissipation. The macroscopic model is described within the mean-field approximation while noise effect due large reservoir fluctuation has been averaged out to zero. We chose a simple dissipative memory kernel to produce a time-dependant oscillating dissipation. Our numerically calculated phase-space and Lyapunov exponents shows enhances route to chaos as one increases driven dissipations amplitude.

Optimal Reaction Coordinates and Kinetic Rates from the Projected Dynamics of Transition Paths.

Finding optimal reaction coordinates and predicting accurate kinetic rates for activated processes are two of the foremost challenges of molecular simulations. We introduce an algorithm that tackles the two problems at once: starting from a limited number of reactive molecular dynamics trajectories (transition paths), we automatically generate with a Monte Carlo approach a sequence of different reaction coordinates that progressively reduce the kinetic rate of their projected effective dynamics. Based on a variational principle, the minimal rate accurately approximates the exact one, and it corresponds to the optimal reaction coordinate. After benchmarking the method on an analytic double-well system, we apply it to complex atomistic systems: the interaction of carbon nanoparticles of different sizes in water.

Study on the intrinsic mechanisms underlying enhanced geothermal system (EGS) heat transfer performance differences in multi-wells

The effectiveness of the multi-well arrangement mode in enhanced geothermal systems in large-scale fields has been well established. However, the mechanisms that lead to variations in heat transfer performance between different numbers of well arrangements have remained unclear. In this study, the multi-well pattern was segmented into several double-well units and a mathematical model was constructed to investigate seepage heat transfer in enhanced geothermal systems using numerical simulation techniques. The lifespan and heat transfer characteristics of both multi-well and corresponding double-well units were examined, with supplementary experimental validation. Additionally, a flow reduction coefficient for multi-well systems was proposed to explain the differences in heat transfer performance caused by the pressure superposition of multiple-well seepage fields. It was found that heat extraction from a multi-well enhanced geothermal system is equivalent to the linear superposition of corresponding double-well units. Furthermore, the lifespan of multi-well systems is significantly longer than that of corresponding double-well systems due to the influence of the multi-well flow reduction coefficient. In fact, this difference can exceed threefold in nine-well arrangements, resulting in reduced efficiency of multi-well heat exchange. This work provides a novel perspective on exploring the intrinsic mechanisms that underlie differences in heat transfer performance in enhanced geothermal systems from a flow field perspective. It contributes to a better understanding of optimal well arrangement modes for enhanced geothermal systems and supports the development of large-scale geothermal fields.

Correction to: ‘Mean-field theory for double-well systems on degree-heterogeneous networks’ (2023) by Kundu et al.

Open AccessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Kundu Prosenjit, MacLaren Neil G., Kori Hiroshi and Masuda Naoki 2023Correction to: ‘Mean-field theory for double-well systems on degree-heterogeneous networks’ (2023) by Kundu et al.Proc. R. Soc. A.4792023017120230171http://doi.org/10.1098/rspa.2023.0171SectionOpen AccessCorrectionCorrection to: ‘Mean-field theory for double-well systems on degree-heterogeneous networks’ (2023) by Kundu et al. Prosenjit Kundu Prosenjit Kundu http://orcid.org/0000-0002-0442-6784 Google Scholar Find this author on PubMed Search for more papers by this author , Neil G. MacLaren Neil G. MacLaren http://orcid.org/0000-0002-8478-8530 Google Scholar Find this author on PubMed Search for more papers by this author , Hiroshi Kori Hiroshi Kori http://orcid.org/0000-0003-2899-7896 Google Scholar Find this author on PubMed Search for more papers by this author and Naoki Masuda Naoki Masuda http://orcid.org/0000-0003-1567-801X Google Scholar Find this author on PubMed Search for more papers by this author Prosenjit Kundu Prosenjit Kundu http://orcid.org/0000-0002-0442-6784 Google Scholar Find this author on PubMed Search for more papers by this author , Neil G. MacLaren Neil G. MacLaren http://orcid.org/0000-0002-8478-8530 Google Scholar Find this author on PubMed Search for more papers by this author , Hiroshi Kori Hiroshi Kori http://orcid.org/0000-0003-2899-7896 Google Scholar Find this author on PubMed Search for more papers by this author and Naoki Masuda Naoki Masuda http://orcid.org/0000-0003-1567-801X Google Scholar Find this author on PubMed Search for more papers by this author Published:19 April 2023https://doi.org/10.1098/rspa.2023.0171This article corrects the followingResearch ArticleMean-field theory for double-well systems on degree-heterogeneous networkshttps://doi.org/10.1098/rspa.2022.0350 Prosenjit Kundu, Neil G. MacLaren, Hiroshi Kori and Naoki Masuda volume 478issue 2264Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences31 August 2022Proc. R. Soc. A478, 20220350. (Published online 31 August 2022) (https://doi.org/10.1098/rspa.2022.0350)Equation (3.5) was replaced with: Θ∗=1N⟨k⟩[∑k<k~kp(k)xℓ(k)+∑k≥k~kp(k)xu(k)].0.1Equations (4.4) and (4.5) were replaced with: uc,ℓ=y~(1)−DkmaxΘ∗,ℓ0.2and uc,u=y~(1)−DkminΘ∗,u,0.3respectively, also: Θ∗,ℓ=1N⟨k⟩∑kkp(k)xℓ(k)0.4and Θ∗,u=1N⟨k⟩∑kkp(k)xu(k).0.5The beginning of the sentence right after equation (4.5), i.e. ‘Therefore, except for regular networks (i.e. those in which all nodes have the same degree),’ has been replaced with the following text:‘We obtain kmax≫kmin in most networks. In contrast, the values of Θ∗,ℓ and Θ∗,u are comparable although Θ∗,ℓ<Θ∗,u. Therefore, except when kmax and kmin are close to each other, including the case of regular networks (i.e. those in which all nodes have the same degree),’.In the last sentence in the same paragraph, ‘(u−y~(2))/kmax’ and ‘(u−y~(1))/kmin’ has been replaced with ‘(y~(1)−u)/(kmaxΘ∗,ℓ)’ and ‘(y~(1)−u)/(kminΘ∗,u)’, respectively.In the seventh from last line on p. 9, ‘proportional’ has been replaced with ‘roughly proportional’.In the fourth from last line on p. 9, ‘y~(2)/kmin−y~(1)/kmax’ has been replaced with ‘1/kmin−1/kmax’.In the third and second from last lines on p. 9, the sentence‘Note that D∈((u−y~(2))/kmax,(u−y~(1))/kmin) implies that the size of the range of D is proportional to y~(2)/kmin−y~(1)/kmax.’has been replaced with:‘Note that D∈((y~(1)−u)/(kmaxΘ∗,ℓ),(y~(1)−u)/(kminΘ∗,u)) implies that the size of the range of D is roughly proportional to 1/kmin−1/kmax.’.Figure 5b has been replaced with the following figure. Previous Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsRelated articlesMean-field theory for double-well systems on degree-heterogeneous networks31 August 2022Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences This IssueApril 2023Volume 479Issue 2272 Article InformationDOI:https://doi.org/10.1098/rspa.2023.0171Published by:Royal SocietyOnline ISSN:1471-2946History: Manuscript received08/03/2023Manuscript accepted20/03/2023Published online19/04/2023Published in print26/04/2023 License:© 2023 The Authors.Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. Citations and impact Subjectscomplexity

Fourier-flow model generating Feynman paths

As an alternative but unified and more fundamental description for quantum physics, Feynman path integrals generalize the classical action principle to a probabilistic perspective, under which the physical observables' estimation translates into a weighted sum over all possible paths. The underlying difficulty is to tackle the whole path manifold from finite samples that can effectively represent the Feynman propagator dictated probability distribution. Modern generative models in machine learning can handle learning and representing probability distribution with high computational efficiency. In this study, we propose a Fourier-flow generative model to simulate the Feynman propagator and generate paths for quantum systems. As demonstration, we validate the path generator on the harmonic and anharmonic oscillators. The latter is a double-well system without analytic solutions. To preserve the periodic condition for the system, the Fourier transformation is introduced into the flow model to approach a Matsubara representation. With this novel development, the ground-state wave function and low-lying energy levels are estimated accurately. Our method offers a new avenue to investigate quantum systems with machine learning assisted Feynman Path integral solving.

Harmonic-Gaussian double-well potential stochastic resonance with its application to enhance weak fault characteristics of machinery

Noise would give rise to incorrect filtering frequency-band selection in signal filtering-based methods including fast kurtogram, teager energy operators and wavelet packet transform filters and meanwhile would result in incorrect selection of useful components and even mode mixing, end effects, etc., in signal decomposition-based methods including empirical mode decomposition, singular value decomposition and local mean decomposition. On the contrary, noise in stochastic resonance (SR) is beneficial to enhance weak signals of interest embedded in signals with strong background noise. Taking into account that nonlinear systems are crucial ingredients to activate the SR, here we investigate the SR in the cases of overdamped and underdamped harmonic-Gaussian double-well potential systems subjected to noise and a periodic signal. We derive and measure the analytic expression of the output signal-to-noise ratio (SNR) and the steady-state probability density (SPD) function under approximate adiabatic conditions. When the harmonic-Gaussian double-well potential loses its stability, we can observe the antiresonance phenomenon, whereas adding the damped factor into the overdamped system can change the stability of the harmonic-Gaussian double-well potential, resulting that the antiresonance behavior disappears in the underdamped system. Then, we use the overdamped and underdamped harmonic-Gaussian double-well potential SR to enhance weak useful characteristics for diagnosing incipient rotating machinery failures. Theoretical and experimental results show that adjusting both noise intensity and system parameters can activate overdamped and underdamped harmonic-Gaussian double-well potential SR in which there is a bell-shaped peak for the SNR. Additionally, the underdamped harmonic-Gaussian double-well potential SR is independent of frequency-shifted and rescaling transform to process large machine parameter signals and outperforms the overdamped one. Finally, comparing the advanced robust local mean decomposition (RLMD) method based on signal decomposition and the wavelet transform method based on noise cancellation or infogram method based on signal filtering, the overdamped or underdamped harmonic-Gaussian double-well potential SR methods characterize a better performance to detect a weak signal. Fault characteristics in the early stage of failures are successful in improving the incipient fault characteristic identification of rolling element bearings.

Multifunctional effects in magnetic nanoparticles for precision medicine: combining magnetic particle thermometry and hyperthermia.

An effective combination of magnetic hyperthermia and thermometry is shown to be implementable by using magnetic nanoparticles which behave either as a heat sources or as temperature sensors when excited at two different frequencies. Noninteracting magnetite nanoparticles are modeled as double-well systems and their magnetization is obtained by solving rate equations. Two temperature sensitive properties derived from the cyclic magnetization and exhibiting a linear dependence on temperature are studied and compared for monodisperse and polydisperse nanoparticles. The multifunctional effects enabling the combination of magnetic hyperthermia and thermometry are shown to depend on the interplay among nanoparticle size, intrinsic magnetic properties and driving-field frequency. Magnetic hyperthermia and thermometry can be effectively combined by properly tailoring the magnetic properties of nanoparticles and the driving-field frequencies.

Noise-dissipation Correlated Dynamics of a Double-well Bose-Einstein Condensate-reservoir System

Abstract: In this work, we study the dissipative dynamics of a double-well Bose-Einstein condensate (BEC) out-coupled to reservoir at each side of its trap. The sub-system comprises of a simple Bose-Hubbard model, where the interplay of atom-tunneling current and inter-particle interaction are the main quantum features. The contact with two separate heat baths causes dissipation and drives the system into a non-equilibrium state. The system is well described by the Generalized Quantum Heisenberg-Langevin equation. We considered two Markovian dissipative BEC systems based on (i) the mean-field model (MF), where the internal noise has been averaged out and (ii) the noise-correlated model (FDT). Physical quantities, such as population imbalance, coherence and entanglement of the system, are computed for the models. The two-mode BEC phases, such as the quantum tunneling state and the macroscopic quantum-trapping state, evolved into complicated dynamics by controlling the non-linear interaction and dissipation strengths. We found that many important quantum features produced by the noise-correlated FDT model are not captured by the mean-field model. Keywords: Double-well BEC, Dissipation, Noise, Markovian, Non-Markovian, Fixed points. PACS: 03.75 Lm, 03.65 Yz, 03.75 Gg, 05.

Numerical simulation of gas extraction performance from hydrate reservoirs using double-well systems

Multi-well systems can increase the drainage area and hence enhance the productivity of natural gas hydrate (NGH) production. Fundamentally, the double-well system is the key and basis of understanding the multi-well system and formulating field strategies for gas extraction from hydrate-bearing sediments. However, production performance and multi-well production strategies for recovering NGH by using depressurization have not been treated in detail. Here we develop a three-dimensional model to describe double-well production characteristics from the low-permeability hydrate reservoir and analyze well-deployment strategies. The productivity of this double-well system is 1.5–2.5 higher than that of the single well under the same conditions, which is triggered by enlarging the drainage area. The well spacing alters flow characteristics and corresponding geophysical fields, which is the key variable for the double-well system in gas recovery. Its range can be determined based on the combination of productivity and hydrate dissociation front. Besides, due to the limited enhancement in productivity using the double-well system, the multi-well system assisted with reservoir reconstruction technologies and stimulation methods is of significant practical value in realizing commercial exploitation of NGH.

Stochastic Lag Time Parameterization for Markov State Models of Protein Dynamics.

Markov state models (MSMs) play a key role in studying protein conformational dynamics. A sliding count window with a fixed lag time is widely used to sample sub-trajectories for transition counting and MSM construction. However, sub-trajectories sampled with a fixed lag time may not perform well under different selections of lag time, which requires strong prior practice and leads to less robust estimation. To alleviate it, we propose a novel stochastic method from a Poisson process to generate perturbative lag time for sub-trajectory sampling and utilize it to construct a Markov chain. Comprehensive evaluations on the double-well system, WW domain, BPTI, and RBD-ACE2 complex of SARS-CoV-2 reveal that our algorithm significantly increases the robustness and power of a constructed MSM without disturbing the Markovian properties. Furthermore, the superiority of our algorithm is amplified for slow dynamic modes in complex biological processes.

Laser-controlled electronic symmetry breaking in a phenylene ethynylene dimer: Simulation by the hierarchical equations of motion and optimal control

The ability to prepare a specific superposition of electronic excited states leading to a transitory symmetry breaking of the electronic density in complex systems remains a challenging concern. We investigate how an initial coherence can be controlled by laser fields. The selected molecular system is a symmetric dimer of phenylene ethynylene presenting different interesting properties: Two bright nearly degenerate excited states coupled through a conical intersection are addressable by orthogonal transition dipole moments and are well separated from neighboring states. Creating a superposed state with equal weights corresponds to a right or left electronic localization as in the double-well system followed by a transitory oscillation between the two wells. To ensure a spectral bandwidth typically smaller than 0.25 eV, the pulse duration is in the tens of femtoseconds range, so nuclear motion cannot be neglected. Optimal control theory (OCT) is applied with guess fields that effectively create the target coherence in the absence of dephasing due to the vibrational baths. We analyze the field reshaping proposed by the control and we further fit a sequence of pulses on the optimal field. The overall result is efficient and robust disymmetry control over reasonable timescales of few tens of femtoseconds, exceeding the pulse duration. The monotonically convergent algorithm is combined with the hierarchical equations of motion (HEOM) able to treat strongly coupled non-Markovian dynamics. We also check the implementation of the combined OCT-HEOM approach in the tensor-train representation with propagation using the time-dependent variational method.

Quantum engineering of a synthetic thermal bath for bosonic atoms in a one-dimensional optical lattice via Markovian feedback control

We propose and investigate a scheme for engineering a synthetic thermal bath for a bosonic quantum gas in a one-dimensional optical lattice based on Markovian feedback control. The performance of our scheme is quantified by the fidelity between the steady state of the system and the effective thermal state. For double-well and triple-well systems with non-interacting particles, the steady state is found to be an exact thermal state, which is attributed to the fact that the transfer rates between all pairs of coupled eigenstates satisfy detailed balance condition. The scenario changes when there are more lattice sites, where the detailed balance condition does not hold any more, but remains an accurate approximation. Remarkably, our scheme performs very well at low and high temperature regimes, with the fidelity close to one. The performance at intermediate temperatures~(where a crossover into a Bose condensed regime occurs) is slightly worse, and the fidelity shows a gentle decrease with increasing system size. We also discuss the interacting cases. In contrast to the non-interacting cases, the scheme is found to perform better at a higher temperature. Another difference is that the minimal temperature that can be engineered is nonzero and increases with the interaction strength.

Mean-field theory for double-well systems on degree-heterogeneous networks

Many complex dynamical systems in the real world, including ecological, climate, financial and power-grid systems, often show critical transitions, or tipping points, in which the system’s dynamics suddenly transit into a qualitatively different state. In mathematical models, tipping points happen as a control parameter gradually changes and crosses a certain threshold. Tipping elements in such systems may interact with each other as a network, and understanding the behaviour of interacting tipping elements is a challenge because of the high dimensionality originating from the network. Here, we develop a degree-based mean-field theory for a prototypical double-well system coupled on a network with the aim of understanding coupled tipping dynamics with a low-dimensional description. The method approximates both the onset of the tipping point and the position of equilibria with a reasonable accuracy. Based on the developed theory and numerical simulations, we also provide evidence for multistage tipping point transitions in networks of double-well systems.

CaO2-based electro-Fenton-oxidation of 1,2-dichloroethane in groundwater

It has been well recognized that the Fenton reaction requires a rigorous pH control and suffers from the fast self-degradation of H2O2. In an effort to resolve the technical demerits of the conventional Fenton reaction, particular concern on the use of CaO2-based Fenton reaction was paid in this study. To realize the practical use of CaO2 in the Fenton reaction for groundwater remediation, it could be of great importance to control its reaction rate in the subsurface. As such, this study laid great emphasis on the combined process of electrochemical oxidation and CaO2-based Fenton oxidation, using 1,2-dichloroethane (1,2-DCA) as a model compound. It was hypothesized that the reaction rate is also highly contingent on the formation of Fe(II) (stemmed from iron anode oxidation). Eighty percent of 1,2-DCA were degraded by the CaO2-based Fenton reaction. The final pH was neutral, inferring that the reaction could be a viable option for the subsurface environment. Moreover, the supply of electric current in an iron anode expedited 1,2-DCA degradation efficiency from 35 % to 62 % via electrically generated Fe(II), which donated electrons to H2O2, producing more hydroxyl radicals. An anode-cathode configuration from the single-well system enhanced the degradation of 1,2-DCA, with less amount of energy consumption than the double-well system. Based on results, CaO2-based electro-Fenton oxidation can remove well 1,2-DCA in groundwater and can be a strategic measure for groundwater remediation.

Quantifying Energetic and Entropic Pathways in Molecular Systems.

When examining dynamics occurring at nonzero temperatures, both energy and entropy must be taken into account to describe activated barrier crossing events. Furthermore, good reaction coordinates need to be constructed to describe different metastable states and the transition mechanisms between them. Here we use a physics-based machine learning method called state predictive information bottleneck (SPIB) to find nonlinear reaction coordinates for three systems of varying complexity. SPIB is able to correctly predict an entropic bottleneck for an analytical flat-energy double-well system and identify the entropy- and energy-dominated pathways for an analytical four-well system. Finally, for a simulation of benzoic acid permeation through a lipid bilayer, SPIB is able to discover the entropic and energetic barriers to the permeation process. Given these results, we thus establish that SPIB is a reasonable and robust method for finding the important entropy, energy, and enthalpy barriers in physical systems, which can then be used to enhance the understanding and sampling of different activated mechanisms.

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.