Potential Energy Surface Research Articles

Quantum study of the CH3+ photodissociation in full-dimensional neural network potential energy surfaces.

C H 3 + , a cornerstone intermediate in interstellar chemistry, has recently been detected for the first time by using the James Webb Space Telescope. The photodissociation of this ion is studied here. Accurate explicitly correlated multi-reference configuration interaction abinitio calculations are done, and full-dimensional potential energy surfaces are developed for the three lower electronic states, with a fundamental invariant neural network method. The photodissociation cross section is calculated using a full-dimensional quantum wave packet method in heliocentric Radau coordinates. The wave packet is represented in angular and radial grids, allowing us to reduce the number of points physically accessible, requiring to push up the spurious states appearing when evaluating the angular kinetic terms, through projection technique. The photodissociation spectra, when employed in astrochemical models to simulate the conditions of the Orion bar, result in a lesser destruction of CH3+ compared to that obtained when utilizing the recommended values in the kinetic database for astrochemistry.

BeM(CO)3- (M = Co, Rh, Ir) and BeM(CO)3 (M = Ni, Pd, Pt): Triply bonded terminal beryllium in zero oxidation state.

We perform detailed potential energy surface explorations of BeM(CO)3- (M = Co, Rh, Ir) and BeM(CO)3 (M = Ni, Pd, Pt) using both single-reference and multireference-based methods. The present results at the CASPT2(12,12)/def2-QZVPD//M06-D3/def2-TZVPPD level reveal that the global minimum of BeM(CO)3- (M = Co, Rh, Ir) and BePt(CO)3 is a C3v symmetric structure with an 1A1 electronic state, where Be is located in a terminal position bonded to M along the center axis. For other cases, the C3v symmetric structure is a low-lying local minimum. Although the present complexes are isoelectronic with the recently reported BFe(CO)3- complex having a B-Fe quadruple bond, radial orbital-energy slope (ROS) analysis reveals that the highest occupied molecular orbital (HOMO) in the title complexes is slightly antibonding in nature, which bars a quadruple bonding assignment. Similar weak antibonding nature of HOMO in the previously reported BeM(CO)4 (M = Ru, Os) complexes is also noted in ROS analysis. The bonding analysis through energy decomposition analysis in combination with the natural orbital for chemical valence shows that the bonding between Be and M(CO)3q (q = -1 for M = Co, Rh, Ir and q = 0 for M = Ni, Pd, Pt) can be best described as Be in the ground state (1S) interacting with M(CO)30/- via dative bonds. The Be(spσ) → M(CO)3q σ-donation and the complementary Be(spσ) ← M(CO)3q σ-back donation make the overall σ bond, which is accompanied by two weak Be(pπ) ← M(CO)3q π-bonds. These complexes represent triply bonded terminal beryllium in an unusual zero oxidation state.

Quantum states and static field ionization of a cylindrical confined hydrogen atom: A diffusion Monte Carlo study

AbstractIn this article, the diffusion Monte Carlo (DMC) method is applied to the study of the quantum states of a hydrogen atom confined into a cylindrical potential well. We present an independent reproduction of previous studies based on different methods, in particular the energy eigenvalues for ground and selected excited states and the polarizability of the ground state, both for finite and infinite cylinders. The static field ionization of ground and excited states of the confined atom is discussed, including the determination of the potential energy surface and equilibrium position of the proton. This study provides a further demonstration of the versatility of the DMC method for this and analogous problems.

GradNav: Accelerated Exploration of Potential Energy Surfaces with Gradient-Based Navigation.

Exploring the potential energy surface (PES) of molecular systems is important for comprehending their complex behaviors, particularly through the identification of various metastable states. However, the transition between these states is often hindered by substantial energy barriers, demanding prolonged molecular simulations that consume considerable computational resources. Our study introduces the gradient-based navigation (GradNav) algorithm, which accelerates the exploration of the energy surface and enables proper reconstruction of the PES. This algorithm employs a strategy of initiating short simulation runs from updated starting points derived from prior observations to effectively navigate across potential barriers and explore new regions. To evaluate GradNav's performance, we introduce two metrics: the deepest well escape frame (DWEF) and the search success initialization ratio (SSIR). Through applications on Langevin dynamics within Müller-type PESs and molecular dynamics simulations of the Fs-peptide protein, these metrics demonstrate GradNav's enhanced ability to escape deep energy wells and its reduced reliance on initial conditions, as denoted by the reduced DWEF values and increased SSIR values, respectively. Consequently, this improved exploration capability enables more precise energy estimations from simulation trajectories.

Kinetic energy dependence and potential energy surface of the spin-forbidden reaction Sm+ (8F) + N2O (1Σ+) → SmO+ (6Δ) + N2 (1Σg+).

The kinetic energy dependence of the title reaction is examined using guided ion beam tandem mass spectrometry. Because this reaction is spin-forbidden, crossings between octet and sextet hypersurfaces presumably must occur. Furthermore, Sm+ must transition from a 4f66s1 configuration in the reactant to 4f55d2 in order to have the orbital occupancy required to form the triple bond in SmO+ (6Δ). Despite being strongly exothermic (∼4eV), the reaction proceeds with low efficiency (18% ± 4%) via a barrierless process at low energies. Below ∼0.3eV, the cross section follows a kinetic energy dependence that roughly parallels that of the collision rate for ion-dipole reactions. At higher collision energies, the reaction cross section increases until it follows the trajectory cross section closely from 3 to 5eV, indicating that another pathway opens on the reaction hypersurface. Modeling this increase yields a threshold energy for this new pathway at 0.54 ± 0.05eV. Theoretical potential energy surfaces that do not include spin-orbit interactions for the reaction show that there is a barrier of height 1.19eV (MP2) or 0.49eV [CCSD(T)] to insertion of Sm+ into the N2-O bond and that there are several places where octet and sextet surfaces can intersect and interact. By considering the distribution of spin-orbit states generated in the ion source, the internal energy of the N2O reactant, and the influence of coupling between electronic, orbital, and rotational angular momentum, the low-efficiency, exothermic behavior as well as the increase in efficiency at higher energies can plausibly be explained.

Computational Investigations of the Detailed Mechanism of Reverse Intersystem Crossing in Inverted Singlet-Triplet Gap Molecules.

Inverted singlet-triplet gap (INVEST) materials have promising photophysical properties for optoelectronic applications due to an inversion of their lowest singlet (S1) and triplet (T1) excited states. This results in an exothermic reverse intersystem crossing (rISC) process that potentially enhances triplet harvesting, compared to thermally activated delayed fluorescence (TADF) emitters with endothermic rISCs. However, the processes and phenomena that facilitate conversion between excited states for INVEST materials are underexplored. We investigate the complex potential energy surfaces (PESs) of the excited states of three heavily studied azaphenalene INVEST compounds, namely, cyclazine, pentazine, and heptazine using two state-of-the-art computational methodologies, namely, RMS-CASPT2 and SCS-ADC(2) methods. Our findings suggest that ISC and rISC processes take place directly between the S1 and T1 electronic states in all three compounds through a minimum-energy crossing point (MECP) with an activation energy barrier between 0.11 to 0.58 eV above the S1 state for ISC and between 0.06 and 0.36 eV above the T1 state for rISC. We predict that higher-lying triplet states are not populated, since the crossing point structures to these states are not energetically accessible. Furthermore, the conical intersection (CI) between the ground and S1 states is high in energy for all compounds (between 0.4 to 2.0 eV) which makes nonradiative decay back to the ground state a relatively slow process. We demonstrate that the spin-orbit coupling (SOC) driving the S1-T1 conversion is enhanced by vibronic coupling with higher-lying singlet and triplet states possessing vibrational modes of proper symmetry. We also rationalize that the experimentally observed anti-Kasha emission of cyclazine is due to the energetically inaccessible CI between the bright S2 and the dark S1 states, hindering internal conversion. Finally, we show that SCS-ADC(2) is able to qualitatively reproduce excited state features, but consistently overpredict relative energies of excited state structural minima compared to RMS-CASPT2. The identification of these excited state features elaborates design rules for new INVEST emitters with improved emission quantum yields.

Theory of 2D electronic spectroscopy of water soluble chlorophyll-binding protein (WSCP): Signatures of Chl b derivate.

We investigate how electronic excitations and subsequent dissipative dynamics in the water soluble chlorophyll-binding protein (WSCP) are connected to features in two-dimensional (2D) electronic spectra, thereby comparing results from our theoretical approach with experimental data from the literature. Our calculations rely on third-order response functions, which we derived from a second-order cumulant expansion of the dissipative dynamics involving the partial ordering prescription, assuming a fast vibrational relaxation in the potential energy surfaces of excitons. Depending on whether the WSCP complex containing a tetrameric arrangement of pigments composed of two dimers with weak excitonic coupling between them binds the chlorophyll variant Chl a or Chl b, the resulting linear absorption and circular dichroism spectra and particularly the 2D spectra exhibit substantial differences in line shapes. These differences between Chl a WSCP and Chl b WSCP cannot be explained by the slightly modified excitonic couplings within the two variants. In the case of Chl a WSCP, the assumption of equivalent dimer subunits facilitates a reproduction of substantial features from the experiment by the calculations. In contrast, for Chl b WSCP, we have to assume that the sample, in addition to Chl b dimers, contains a small but distinct fraction of chemically modified Chl b pigments. The existence of such Chl b derivates has been proposed by Pieper et al. [J. Phys. Chem. B 115, 4042 (2011)] based on low-temperature absorption and hole-burning spectroscopy. Here, we provide independent evidence.

Unifying O(3) Equivariant Neural Networks Design with Tensor-Network Formalism

Abstract Many learning tasks, including learning potential energy surfaces from ab initio calculations, involve global spatial symmetries and permutational symmetry between atoms or general particles. Equivariant graph neural networks are a standard approach to such problems, with one of the most successful methods employing tensor products between various tensors that transform under the spatial group. However, as the number of different tensors and the complexity of relationships between them increase, maintaining parsimony and equivariance becomes increasingly challenging. In this paper, we propose using fusion diagrams, a technique widely employed in simulating SU($2$)-symmetric quantum many-body problems, to design new equivariant components for equivariant neural networks. This results in a diagrammatic approach to constructing novel neural network architectures. When applied to particles within a given local neighborhood, the resulting components, which we term "fusion blocks," serve as universal approximators of any continuous equivariant function defined on the neighborhood. We incorporate a fusion block into pre-existing equivariant architectures (Cormorant and MACE), leading to improved performance with fewer parameters on a range of challenging chemical problems. Furthermore, we apply group-equivariant neural networks to study non-adiabatic molecular dynamics of stilbene cis-trans isomerization and water adsorption cycles in porous materials, such as metal-organic frameworks. Our approach, which combines tensor networks with equivariant neural networks, suggests a potentially fruitful direction for designing more expressive equivariant neural networks.

Understanding the Photoinduced Desorption and Oxidation of CO on Ru(0001) Using a Neural Network Potential Energy Surface

Understanding the Photoinduced Desorption and Oxidation of CO on Ru(0001) Using a Neural Network Potential Energy Surface

Simulation of interlayer coupling for electroactive covalent organic framework design.

Porous, stacked two-dimensional covalent organic frameworks (2D COFs) bearing semiconducting linkers can support directional charge transfer across adjacent layers of the COF. To better inform the current and possible future design rules for enhancing electron and hole transport in such materials, an understanding of how linker selection and functionalization affects interlayer electronic couplings is essential. We report electronic structure simulation and analysis of electronic couplings across adjacent linker units and to encapsulated species in functionalized electroactive 2D COFs. The detailed dependence of these electronic couplings on interlayer interactions is examined through scans along key interlayer degrees of freedom and through configurational sampling from equilibrium molecular dynamics on semiempirical potential energy surfaces. Beyond affirming the sensitivity of the electronic coupling to interlayer distance and orientation, these studies offer guidance toward linker functionalization strategies for enhancing charge carrier transport in electroactive 2D COFs.

Effects of triaxial deformation on the fission barrier in the Z = 118 − 120 nuclei* *Supported by the National Natural Science Foundation of China (Grant Nos. 12205076 and 12047504), the China Postdoctoral Science Foundation (Grant No. 2020M670012), and the Launching Fund of Henan University of Technology (Grant No. 2021BS047).

By using potential energy surface (PES) calculations in the three-dimensional space (β 2, γ, β 4) within the framework of the macroscopic-microscopic model, the fission trajectory and fission barrier for Z = 118(Og), 119, 120 nuclei has been systematically investigated. The calculated PES includes macroscopic liquid-drop energy, microscopic shell correction and pairing correction. Taking the 294Og176 nucleus as an example, we discuss the next closed shell after Z = 82 and N = 126 with the calculated Woods–Saxon single-particle levels. Then, the results of PES in 294Og is illustrated from the (X, Y) scale to the (β 2, γ) scale. The γ degree of freedom reveals the shape evolution clearly during the fission process. The structure near the minimum and saddle point of the PES in the Z = 118, 119, 120 nuclei is demonstrated simultaneously. Based on the potential energy curves, general trends of the evolution of the fission barrier heights and widths are also studied. The triaxial deformation in these superheavy mass regions plays a vital role in the first fission barrier, showing a significant reduction in both triaxial paths. In addition, the model-dependent fission barriers of proton-rich nuclei 295Og, 296119, and 297120 are analyzed briefly. Our studies could be valuable for synthesizing the superheavy new elements in the forthcoming HIAF and other facilities.

Physical Prior Mean Function-Driven Gaussian Processes Search for Minimum-Energy Reaction Paths with a Climbing-Image Nudged Elastic Band: A General Method for Gas-Phase, Interfacial, and Bulk-Phase Reactions.

The climbing-image nudged elastic band (CI-NEB) method serves as an indispensable tool for computational chemists, offering insight into minimum-energy reaction paths (MEPs) by delineating both transition states (TSs) and intermediate nonstationary structures along reaction coordinates. However, executing CI-NEB calculations for reactions with extensive reaction coordinate spans necessitates a large number of images to ensure a reliable convergence of the MEPs and TS structures, presenting a computationally demanding optimization challenge, even with mildly costly electronic-structure methods. In this study, we advocate for the utilization of physically inspired prior mean function-based Gaussian processes (GPs) to expedite MEP exploration and TS optimization via the CI-NEB method. By incorporating reliable prior physical approximations into potential energy surface (PES) modeling, we demonstrate enhanced efficiency in multidimensional CI-NEB optimization with surrogate-based optimizers. Our physically informed GP approach not only outperforms traditional nonsurrogate-based optimizers in optimization efficiency but also on-the-fly learns the reaction path valley during optimization, culminating in significant advancements. The surrogate PES derived from our optimization exhibits high accuracy compared to true PES references, aligning with our emphasis on leveraging reliable physical priors for robust and efficient posterior mean learning in GPs. Through a systematic benchmark study encompassing various reaction pathways, including gas-phase, bulk-phase, and interfacial/surface reactions, our physical GPs consistently demonstrate superior efficiency and reliability. For instance, they outperform the popular fast inertial relaxation engine optimizer by approximately a factor of 10, showcasing their versatility and efficacy in exploring reaction mechanisms and surface reaction PESs.

Dynamic embedding of effective harmonic normal mode vibrations in all-atomistic energy gap fluctuations: Case study of light harvesting 2 complex.

Environmental effects in excitation energy transfer have mostly been modeled by baths of harmonic oscillators, but to what extent such modeling provides a reliable description of actual interactions between molecular systems and environments remains an open issue. We address this issue by investigating fluctuations in the excitation energies of the light harvesting 2 complex using a realistic all-atomistic simulation of the potential energy surface. Our analyses reveal that molecular motions exhibit significant anharmonic features, even for underdamped intramolecular vibrations. In particular, we find that the anharmonicity contributes to the broadening of spectral densities and substantial overlaps between neighboring peaks, which complicates the meaning of mode frequencies constituting a bath model. Thus, we develop a strategy to construct a minimally underdamped harmonic bath that has a clear connection to all-atomistic dynamics by utilizing actual normal modes of molecules but optimizing their frequencies such that the resulting bath model can best reproduce the all-atomistic simulation results. By subtracting the underdamped contribution from the entire fluctuations, we also show that identifying a residual spectral density representing all other contributions with overdamped behavior is possible. We find that this can be fitted well with a well-established analytic form of a spectral density function or, alternatively, modeled as explicit time dependent fluctuations with muti-exponential or power law type correlation functions. We provide an assessment and the implications of these possibilities. The approach presented here can also serve as a general strategy to construct a simplified bath model that can effectively represent the underlying all-atomistic bath dynamics.

Comprehensive Study of Equilibrium Structure of Trans-Azobenzene: Gas Electron Diffraction and Quantum Chemical Calculations

The geometrical re parameters of trans-azobenzene (E-AB) free molecule were refined by gas electron diffraction (GED) method using available experimental data obtained previously by S. Konaka and coworkers. Structural analysis was carried out by various techniques. First of all, these included the widely used molecular orbital constrained gas electron diffraction method and regularization method. The results of the refinements using different models were also compared—a semirigid model, three variants of one-dimensional dynamic models, and a two-dimensional pseudoconformer model. Several descriptions have been used due to the fact that E-AB has a shallow potential energy surface along the rotation coordinates of phenyl groups. Despite this, it turned out that the semirigid model is suitable for use for E-AB and allows good agreement with experimental data to be achieved. According to the results of GED structural analysis, coupled with the results of DLPNO-CCSD(T0) calculations, E-AB has a planar structure. Based only on GED data, it is impossible to unambiguously determine the rotational angle of the phenyl group due to the facts that (i) with rotation over a wide range of angles, the bonded distances in the molecule change insignificantly and (ii) potential function in a structural analysis within a dynamic model is not determined with the necessary accuracy. This work also examines the sensitivity of the GED method to structural changes caused by trans-cis isomerization. The paper also analyzes the applicability of different variants of density functional theory (DFT) calculations in GED structural analysis using E-AB as an example. There are not enough similar methodological works in the literature. This experimental and methodological information is especially important and relevant for planning and implementing GED experiments and corresponding processing of the results for azobenzene derivatives, in which the conformer and isomeric diversity are even more complicated due to the presence of different substituents.

Ab initio dispersion potentials based on physics-based functional forms with machine learning.

In this study, we introduce SAPT10K, a comprehensive dataset comprising 9982 noncovalent interaction energies and their binding energy components (electrostatics, exchange, induction, and dispersion) for diverse intermolecular complexes of 944 unique dimers. These complexes cover significant portions of the intermolecular potential energy surface and were computed using higher-order symmetry-adapted perturbation theory, SAPT2+(3)(CCD), with a large aug-cc-pVTZ basis set. The dispersion energy values in SAPT10K serve as crucial inputs for refining the abinitio dispersion potentials based on Grimme's D3 and many-body dispersion (MBD) models. Additionally, Δ machine learning (ML) models based on newly developed intermolecular features, which are derived from intermolecular histograms of distances for element/substructure pairs to simultaneously account for local environments as well as long-range correlations, are also developed to address deficiencies of the D3/MBD models, including the inflexibility of their functional forms, the absence of MBD contributions in D3, and the standard Hirshfeld partitioning scheme used in MBD. The developed dispersion models can be applied to complexes involving a wide range of elements and charged monomers, surpassing other popular ML models, which are limited to systems with only neutral monomers and specific elements. The efficient D3-ML model, with Cartesian coordinates as the sole input, demonstrates promising results on a testing set comprising 6714 dimers, outperforming another popular ML model, component-based machine-learned intermolecular force field (CLIFF), by 1.5 times. These refined D3/MBD-ML models have the capability to replace the time-consuming dispersion components in symmetry-adapted perturbation theory-based calculations and can promptly illustrate the dispersion contribution in noncovalent complexes for supramolecular assembly and chemical reactions.

Towards Modelling Mechanical Shaking Using Potential Energy Surfaces: A Toy Model Analysis

In this work, we formalize the effect of mechanical shaking by using various forms of an externally exerted force, which may be constant or may be position-dependent, and we examine the changes in the potential energy surfaces that quantify the chemical reaction. We use a simple toy model to model the potential energy surfaces of a chemical reaction, and we study the effect of a constant or position-dependent externally exerted force for various forms of the force. As we demonstrate, the effect of the force can be quite dramatic on the potential energy surfaces, which acquire new stationary points and new Newton trajectories that are distinct from the original ones that were obtained in the absence of mechanochemical effects. We also introduce a new approach to mechanochemical interactions, using a dynamical systems approach for the Newton trajectories. As we show, the dynamical system attractor properties of the trajectories in the phase space are identical to the stationary points of the potential energy surfaces, but the phase space contains much more information regarding the possible evolution of the chemical reaction—information that is quantified by the existence of unstable or saddle fixed points in the phase space. We also discuss how an experimental method for a suitable symmetric liquid solution substance might formalize the effect of shaking via various forms of external force, even in the form of an extended coordinate-dependent force matrix. This approach may experimentally quantify the Epstein effect of shaking in chemical solutions via mechanochemistry methods.

Pretraining of attention-based deep learning potential model for molecular simulation

Machine learning-assisted modeling of the inter-atomic potential energy surface (PES) is revolutionizing the field of molecular simulation. With the accumulation of high-quality electronic structure data, a model that can be pretrained on all available data and finetuned on downstream tasks with a small additional effort would bring the field to a new stage. Here we propose DPA-1, a Deep Potential model with a gated attention mechanism, which is highly effective for representing the conformation and chemical spaces of atomic systems and learning the PES. We tested DPA-1 on a number of systems and observed superior performance compared with existing benchmarks. When pretrained on large-scale datasets containing 56 elements, DPA-1 can be successfully applied to various downstream tasks with a great improvement of sample efficiency. Surprisingly, for different elements, the learned type embedding parameters form a spiral in the latent space and have a natural correspondence with their positions on the periodic table, showing interesting interpretability of the pretrained DPA-1 model.

Complexity of many-body interactions in transition metals via machine-learned force fields from the TM23 data set

This work examines challenges associated with the accuracy of machine-learned force fields (MLFFs) for bulk solid and liquid phases of d-block elements. In exhaustive detail, we contrast the performance of force, energy, and stress predictions across the transition metals for two leading MLFF models: a kernel-based atomic cluster expansion method implemented using sparse Gaussian processes (FLARE), and an equivariant message-passing neural network (NequIP). Early transition metals present higher relative errors and are more difficult to learn relative to late platinum- and coinage-group elements, and this trend persists across model architectures. Trends in complexity of interatomic interactions for different metals are revealed via comparison of the performance of representations with different many-body order and angular resolution. Using arguments based on perturbation theory on the occupied and unoccupied d states near the Fermi level, we determine that the large, sharp d density of states both above and below the Fermi level in early transition metals leads to a more complex, harder-to-learn potential energy surface for these metals. Increasing the fictitious electronic temperature (smearing) modifies the angular sensitivity of forces and makes the early transition metal forces easier to learn. This work illustrates challenges in capturing intricate properties of metallic bonding with current leading MLFFs and provides a reference data set for transition metals, aimed at benchmarking the accuracy and improving the development of emerging machine-learned approximations.

Non-adiabatic direct quantum dynamics using force fields: Toward solvation

Quantum dynamics simulations are becoming a powerful tool for understanding photo-excited molecules. Their poor scaling, however, means that it is hard to study molecules with more than a few atoms accurately, and a major challenge at the moment is the inclusion of the molecular environment. Here, we present a proof of principle for a way to break the two bottlenecks preventing large but accurate simulations. First, the problem of providing the potential energy surfaces for a general system is addressed by parameterizing a standard force field to reproduce the potential surfaces of the molecule’s excited-states, including the all-important vibronic coupling. While not shown here, this would trivially enable the use of an explicit solvent. Second, to help the scaling of the nuclear dynamics propagation, a hierarchy of approximations is introduced to the variational multi-configurational Gaussian method that retains the variational quantum wavepacket description of the key quantum degrees of freedom and uses classical trajectories for the remaining in a quantum mechanics/molecular mechanics like approach. The method is referred to as force field quantum dynamics (FF-QD), and a two-state ππ*/nπ* model of uracil, excited to its lowest bright ππ* state, is used as a test case.

Design, modeling and experiments of bistable piezoelectric energy harvester with self-decreasing potential energy barrier effect

It is difficult for conventional bistable energy harvesters to undergo interwell motion in weak excitation environment. In this study, a bistable energy harvester with self-decreasing potential energy barrier effect is proposed, which innovatively turns a fixed end of S-shaped generator beam into the movable end with a spring oscillator, which effectively decreases the potential energy barrier and makes the harvester susceptible to interwell motion. Furthermore, a theoretical analysis of the nonlinear driving magnetic force and the system governing equations is carried out, then a numerical simulation of potential energy surface of harvester is performed to demonstrate the effect of the self-decreasing potential energy barrier in detail. The effect of spring pre-compression and stiffness of the moveable end on the system potential energy is analyzed by numerical simulation. Finally, the key parameters affecting the output performance, such as spring pre-compression, magnets distance, and external load resistance, are analyzed. The harvester has a maximum effective bandwidth of 6.62Hz (4.61–11.23Hz), a maximum peak-to-peak voltage of 10.2V, an RMS voltage of 1.3V, and a maximum output power of 2.91 μW. The proposed harvester achieves efficient broadband energy harvesting in low frequency environments.