Two-Dimensional Electronic Spectroscopy Reveals Dynamics and Mechanisms of Solvent-Driven Inertial Relaxation in Polar BODIPY Dyes

Ultrafast Coherent Delocalization Revealed in Multilayer QDs under a Chiral Potential

In recent years, it was found that current passing through chiral molecules exhibits spin preference, an effect known as Chiral Induced Spin Selectivity (CISS). The effect also enables the reduction of scattering and therefore enhances delocalization. As a result, the delocalization of an exciton generated in the dots is not symmetric and relates to the electronic and hole excited spins. In this work utilizing fast spectroscopy on hybrid multilayered QDs with a chiral polypeptide linker system, we probed the interdot chiral coupling on a short time scale. Surprisingly, we found strong coherent coupling and delocalization despite having long 4-nm chiral linkers. We ascribe the results to asymmetric delocalization that is controlled by the electron spin. The effect is not measured when using shorter nonchiral linkers. As the system mimics light-harvesting antennas, the results may shed light on a mechanism of fast and efficient energy transfer in these systems.

Experimental and computational characterisation of an artificial light harvesting complex

Photosynthesis has been shown to be a highly efficient process for energy transfer in plants and bacteria. Like natural photosynthetic systems, the artificial light harvesting complex (LHC) BODIPY pillar[5]arene exhibits Förster resonance energy transfer (FRET). However, extensive characterisation of the BODIPY pillar[5]arene LHC to determine its suitability as an artificial LHC has yet to occur. In this paper we experimentally and computationally investigate the photophysical properties of the LHC by comparing the light absorption of the BODIPY LHC to individual BODIPY chromophores. Our results show evidence for quantum coherence, with oscillation frequencies of 100 cm^-1 and 600 cm^-1, which are attributable to vibronic, or exciton-phonon type coupling. Computational analysis suggests strong couplings of the molecular orbitals of the LHC resulting from the stacking of neighbouring BODIPY chromophore units. Interestingly, we find a 40% reduction in the absorbance of light for the BODIPY LHC compared to the individual chromophores which we attribute to electronic interactions between the conjugated π-systems of the BODIPY chromophores and the pillar[5]arene backbone.

Aggregation behavior and spectroscopic properties of red-emitting distyryl-BODIPY in aqueous solution, Langmuir-Schaefer films and Pluoronic® F127 micelles

Red-emitting distyryl substituted BODIPY dyes are among the most promising luminophors for bioimaging and optics applications. However, the practical application of BODIPYs is limited due to their high hydrophobicity and tendency to aggregate in aqueous organic solutions and solid phase. In this article, we propose an elegant solution to this problem. To this end, we carried out the detailed experimental and quantum-chemical study of the structural and spectral features of BF₂-ms-phenyl-5,5'-bis(4-dimethylaminostyryl)-3,3'-dimethyl-2,2'-dipyrromethene (distyryl-BDP). The particular attention was paid to analysis of high sensitivity of the distyryl-BDP spectral characteristics to the solvent properties, and also the aggregation behavior features both in water-organic media and in mono- and multilayer Langmuir-Schaefer films. We selected the best conditions to obtain the hydrophilic micellar structures of distyryl-BDP with Pluronic® F127 having a high efficiency of dye solubilization. This method increasing the solubility improves the distyryl-BDP transport efficiency in physiological aqueous media. The aqueous solutions of distyryl-BDP-Pl micelles show the intense fluorescence in the phototherapy window region (λ_fl = 739 nm).

Coherent Vibrational Spectrum via Time-Resolved Fluorescence for Molecular Dynamics and Identification of Emitting Species-Application to Excited-State Intramolecular Proton Transfer

Time-resolved fluorescence (TF) with high-enough resolution enables recording of a coherent vibrational spectrum (CVS). Because a CVS attained via TF (CVSF) is descended from the frequency modulation of the fluorescence spectrum, it gives the vibrational spectrum of the emitting state. Therefore, CVSF can be a powerful tool for the identification of an emitting state along with the investigation of molecular dynamics in excited states. Herein, we report CVSF of a Schiff base salicylaldehyde azine (SAA) that has two possible excited-state intramolecular proton transfer (ESIPT) sites. The ESIPT time of SAA in dichloromethane is determined to be 22 fs. Quantitative agreement between the experimental CVSF and calculated CVSF of the mono-keto isomer demonstrates that ESIPT indeed occurs in SAA only on one side. More importantly, we show that a CVSF can be utilized to identify an emitting species and its state with the help of quantum chemical calculations. Implications of the CVSF obtained by assuming impulsive excitation of vibrations are discussed in terms of the molecular mechanism of ESIPT and the generation of nuclear wave packets in the product state.

Deciphering Photoinduced Charge Transfer Dynamics in a Cross-Linked Graphene-Dye Nanohybrid

The search for synthetic materials that mimic natural photosynthesis by converting solar energy into other more useful forms of energy is an ever-growing research endeavor. Graphene-based materials, with their exceptional electronic and optical properties, are exemplary candidates for high-efficiency solar energy harvesting devices. High photoactivity can be conveniently achieved by functionalizing graphene with small molecule organic semiconductors whose band-gaps can be tuned by structural modification, leading to interactions between the π-conjugated electronic systems in both the semiconductor and graphene. Here we investigate the ultrafast transient optical properties of a cross-linked graphene-dye (diphenyl-dithiophenediketopyrrolopyrrole) nanohybrid material, in which oligomers of the organic semiconductor dye are covalently bound to a random network of few-layer graphene flakes, and compare the results to those obtained for the reference dye monomer. Using a combination of ultrafast transient absorption and two-dimensional electronic spectroscopy, we provide substantial evidence for photoinduced charge transfer that occurs within 18 ps in the nanohybrid system. Notably, subpicosecond photoinduced torsional relaxation observed in the constituent dye monomer is absent in the cross-linked nanohybrid system. Through density functional theory calculations, we compare the competing effects of covalent bonding, increasing conjugation length, and the presence of multiple graphene flakes. We find evidence that the observed ultrafast charge transfer process occurs through a superexchange mechanism in which the oligomeric dye bridge provides virtual states enabling charge transfer between graphene-dye covalent bond sites.

Coherent Two-Dimensional and Broadband Electronic Spectroscopies

Over the past few decades, coherent broadband spectroscopy has been widely used to improve our understanding of ultrafast processes (e.g., photoinduced electron transfer, proton transfer, and proton-coupled electron transfer reactions) at femtosecond resolution. The advances in femtosecond laser technology along with the development of nonlinear multidimensional spectroscopy enabled further insights into ultrafast energy transfer and carrier relaxation processes in complex biological and material systems. New discoveries and interpretations have led to improved design principles for optimizing the photophysical properties of various artificial systems. In this review, we first provide a detailed theoretical framework of both coherent broadband and two-dimensional electronic spectroscopy (2DES). We then discuss a selection of experimental approaches and considerations of 2DES along with best practices for data processing and analysis. Finally, we review several examples where coherent broadband and 2DES were employed to reveal mechanisms of photoinitiated ultrafast processes in molecular, biological, and material systems. We end the review with a brief perspective on the future of the experimental techniques themselves and their potential to answer an even greater range of scientific questions.

The Influence of Electronic Polarization on Nonlinear Optical Spectroscopy

The environment surrounding a chromophore can dramatically affect the energy absorption and relaxation process, as manifested in optical spectra. Simulations of nonlinear optical spectroscopy, such as two-dimensional electronic spectroscopy (2DES) and transient absorption (TA), will be influenced by the computational model of the environment. We here compare a fixed point charge molecular mechanics model and a quantum mechanical (QM) model of the environment in computed 2DES and TA spectra of Nile red in water and the chromophore of photoactive yellow protein (PYP) in water and protein environments. In addition to simulating these nonlinear optical spectra, we directly juxtapose the computed excitation energy correlation function to the dynamic Stokes shift function often used to analyze environment dynamics. Overall, we find that for the three systems studied here the mutual electronic polarization provided by the QM environment manifests in broader 2DES signals, as well as a larger reorganization energy and a larger static Stokes shift due to stronger coupling between the chromophore and the environment.

Supramolecular BODIPY based dimers: synthesis, computational and spectroscopic studies

The synthetic procedures for the preparation of supramolecular BODIPY dimers decorated with complementary patterns able to induce the formation of a triple hydrogen bond through mutual interactions are here reported. The BODIPY and styryl-equipped BODIPY species have been suitably functionalized in meso position with 2,6-diacetamido-4-pyridyl and 1-butyl-6-uracyl moieties. Dimers and monomers have been subjected to computational and photophysical investigations in solvent media. Various peculiarities concerning the effects of the interaction geometry on the stability of the H-bonded systems have also been investigated. The combination of modelling and experimental data provides a paradigm for improving and refining the BODIPY synthetic pathway to have chromophoric architectures with a programmable supramolecular identity. Furthermore, the possibility of assembling dimers of different dyes through H-bonds could be appealing for a systematic investigation of the principal factors affecting the dynamics of the energy migration and possibly driving coherent transfer mechanisms. Our work highlights how the chemical versatility of these dyes can be exploited to design new BODIPY-based supramolecular architectures.

Ultrafast Fluctuations in PM6 Domains of Binary and Ternary Organic Photovoltaic Thin Films Probed with Two-Dimensional White-Light Spectroscopy

We present two-dimensional white-light spectroscopy (2DWL) measurements of binary and ternary bulk heterojunctions of the polymer donor PM6 mixed with state-of-the-art nonfullerene acceptors Y6 or IT4F. The ternary film has a shorter lifetime and faster spectral diffusion than either of the binary films. 2D line shape analysis of the PM6 ground state bleach with a Kubo model determines that all three films have similar amplitudes of fluctuations (Δ = 0.29 fs^-1) in their transition frequencies, but different relaxation times (ranging from 102 to 24 fs). The ternary film exhibits faster dynamics than either of the binary films. The short lifetime of the ternary blend is consistent with increased photoexcitation transfer and the fast frequency fluctuations are consistent with structural dynamics of aliphatic side chains. These results suggest that the femtosecond fluctuations of PM6 are impacted by the choice of the acceptor molecules. We hypothesize that those dynamics are either indicative, or perhaps the initial source, of structural dynamics that ultimately contribute to solar cell operation.

2D Electronic Spectroscopic Techniques for Quantum Technology Applications

2D electronic spectroscopy (2DES) techniques have gained particular interest given their capability of following ultrafast coherent and noncoherent processes in real-time. Although the fame of 2DES is still majorly linked to the investigation of energy and charge transport in biological light-harvesting complexes, 2DES is now starting to be recognized as a particularly valuable tool for studying transport processes in artificial nanomaterials and nanodevices. Particularly meaningful is the possibility of assessing coherent mechanisms active in the transport of excitation energy in these materials toward possible quantum technology applications. The diverse nature of these new target samples poses significant challenges and calls for a critical rethinking of the technique and its different realizations. With the confluence of promising new applications and rapidly developing technical capabilities, the enormous potential of 2DES techniques to impact the field of nanosystems, quantum technologies, and quantum devices is here delineated.

Vibronic and Environmental Effects in Simulations of Optical Spectroscopy

Including both environmental and vibronic effects is important for accurate simulation of optical spectra, but combining these effects remains computationally challenging. We outline two approaches that consider both the explicit atomistic environment and the vibronic transitions. Both phenomena are responsible for spectral shapes in linear spectroscopy and the electronic evolution measured in nonlinear spectroscopy. The first approach utilizes snapshots of chromophore-environment configurations for which chromophore normal modes are determined. We outline various approximations for this static approach that assumes harmonic potentials and ignores dynamic system-environment coupling. The second approach obtains excitation energies for a series of time-correlated snapshots. This dynamic approach relies on the accurate truncation of the cumulant expansion but treats the dynamics of the chromophore and the environment on equal footing. Both approaches show significant potential for making strides toward more accurate optical spectroscopy simulations of complex condensed phase systems.

The effect of hydrogen bonds on the ultrafast relaxation dynamics of a BODIPY dimer

The influence of hydrogen bonds (H-bonds) in the structure, dynamics, and functionality of biological and artificial complex systems is the subject of intense investigation. In this broad context, particular attention has recently been focused on the ultrafast H-bond dependent dynamical properties in the electronic excited state because of their potentially dramatic consequences on the mechanism, dynamics, and efficiency of photochemical reactions and photophysical processes of crucial importance for life and technology. Excited-state H-bond dynamics generally occur on ultrafast time scales of hundreds of femtoseconds or less, making the characterization of associated mechanisms particularly challenging with conventional time-resolved techniques. Here, 2D electronic spectroscopy is exploited to shed light on this still largely unexplored dynamic mechanism. An H-bonded molecular dimer prepared by self-assembly of two boron-dipyrromethene dyes has been specifically designed and synthesized for this aim. The obtained results confirm that upon formation of H-bonds and the dimer, a new ultrafast relaxation channel is activated in the ultrafast dynamics, mediated by the vibrational motions of the hydrogen donor and acceptor groups. This relaxation channel also involves, beyond intra-molecular relaxations, an inter-molecular transfer process. This is particularly significant considering the long distance between the centers of mass of the two molecules. These findings suggest that the design of H-bonded structures is a particularly powerful tool to drive the ultrafast dynamics in complex materials.

High-Efficiency FRET Processes in BODIPY-Functionalized Quantum Dot Architectures

Efficient FRET systems are developed combining colloidal CdSe quantum dots (QDs) donors and BODIPY acceptors. To promote effective energy transfer in FRET architectures, the distance between the organic fluorophore and the QDs needs to be optimized by a careful system engineering. In this context, BODIPY dyes bearing amino-terminated functionalities are used in virtue of the high affinity of amine groups in coordinating the QD surface. A preliminary QD surface treatment with a short amine ligand is performed to favor the interaction with the organic fluorophores in solution. The successful coordination of the dye to the QD surface, accomplishing a short donor-acceptor distance, provides effective energy transfer already in solution, with efficiency of 76 %. The efficiency further increases in the solid state where the QDs and the dye are deposited as single coordinated units from solution, with a distance between the fluorophores down to 2.2 nm, demonstrating the effectiveness of the coupling strategy.

Biomimetic Nanoarchitectures for Light Harvesting: Self-Assembly of Pyropheophorbide-Peptide Conjugates

The biological light-harvesting process offers an unlimited source of inspiration. The high level of control, adaptation capability, and efficiency challenge humankind to create artificial biomimicking nanoarchitectures with the same performances to respond to our energy needs. Here, in the extensive search for design principles at the base of efficient artificial light harvesters, an approach based on self-assembly of pigment-peptide conjugates is proposed. The solvent-driven and controlled aggregation of the peptide moieties promotes the formation of a dense network of interacting pigments, giving rise to an excitonic network characterized by intense and spectrally wide absorption bands. The ultrafast dynamics of the nanosystems studied through two-dimensional electronic spectroscopy reveals that the excitation energy is funneled in an ultrafast time range (hundreds of femtoseconds) to a manifold of long-living dark states, thus suggesting the considerable potentiality of the systems as efficient harvesters.

Exploiting Machine Learning to Efficiently Predict Multidimensional Optical Spectra in Complex Environments

The excited-state dynamics of chromophores in complex environments determine a range of vital biological and energy capture processes. Time-resolved, multidimensional optical spectroscopies provide a key tool to investigate these processes. Although theory has the potential to decode these spectra in terms of the electronic and atomistic dynamics, the need for large numbers of excited-state electronic structure calculations severely limits first-principles predictions of multidimensional optical spectra for chromophores in the condensed phase. Here, we leverage the locality of chromophore excitations to develop machine learning models to predict the excited-state energy gap of chromophores in complex environments for efficiently constructing linear and multidimensional optical spectra. By analyzing the performance of these models, which span a hierarchy of physical approximations, across a range of chromophore-environment interaction strengths, we provide strategies for the construction of machine learning models that greatly accelerate the calculation of multidimensional optical spectra from first principles.

Nonlinear spectroscopy in the condensed phase: The role of Duschinsky rotations and third order cumulant contributions

First-principles modeling of nonlinear optical spectra in the condensed phase is highly challenging because both environment and vibronic interactions can play a large role in determining spectral shapes and excited state dynamics. Here, we compute two dimensional electronic spectroscopy (2DES) signals based on a cumulant expansion of the energy gap fluctuation operator, with specific focus on analyzing mode mixing effects introduced by the Duschinsky rotation and the role of the third order term in the cumulant expansion for both model and realistic condensed phase systems. We show that for a harmonic model system, the third order cumulant correction captures effects introduced by a mismatch in curvatures of ground and excited state potential energy surfaces, as well as effects of mode mixing. We also demonstrate that 2DES signals can be accurately reconstructed from purely classical correlation functions using quantum correction factors. We then compute nonlinear optical spectra for the Nile red and methylene blue chromophores in solution, assessing the third order cumulant contribution for realistic systems. We show that the third order cumulant correction is strongly dependent on the treatment of the solvent environment, revealing the interplay between environmental polarization and the electronic-vibrational coupling.

Relaxation Dynamics of Chlorophyll b in the Sub-ps Ultrafast Timescale Measured by 2D Electronic Spectroscopy

A thorough characterization of the early time sub-100 fs relaxation dynamics of biologically relevant chromophores is of crucial importance for a complete understanding of the mechanisms regulating the ultrafast dynamics of the relaxation processes in more complex multichromophoric light-harvesting systems. While chlorophyll a has already been the object of several investigations, little has been reported on chlorophyll b, despite its pivotal role in many functionalities of photosynthetic proteins. Here the relaxation dynamics of chlorophyll b in the ultrafast regime have been characterized using 2D electronic spectroscopy. The comparison of experimental measurements performed at room temperature and 77 K allows the mechanisms and the dynamics of the sub-100 fs relaxation dynamics to be characterized, including spectral diffusion and fast internal conversion assisted by a specific set of vibrational modes.

How the Protein Environment Can Tune the Energy, the Coupling, and the Ultrafast Dynamics of Interacting Chlorophylls: The Example of the Water-Soluble Chlorophyll Protein

The interplay between active molecules and the protein environment in light-harvesting complexes tunes the photophysics and the dynamical properties of pigment-protein complexes in a subtle way, which is not fully understood. Here we characterized the photophysics and the ultrafast dynamics of four variants of the water-soluble chlorophyll protein (WSCP) as an ideal model system to study the behavior of strongly interacting chlorophylls. We found that when coordinated by the WSCP protein, the presence of the formyl group in chlorophyll b replacing the methyl group in chlorophyll a strongly affects the exciton energy and the dynamics of the system, opening up the possibility of tuning the photophysics and the transport properties of multichromophores by engineering specific interactions with the surroundings.

Synergistic Approach of Ultrafast Spectroscopy and Molecular Simulations in the Characterization of Intramolecular Charge Transfer in Push-Pull Molecules

The comprehensive characterization of Intramolecular Charge Transfer (ICT) stemming in push-pull molecules with a delocalized π-system of electrons is noteworthy for a bespoke design of organic materials, spanning widespread applications from photovoltaics to nanomedicine imaging devices. Photo-induced ICT is characterized by structural reorganizations, which allows the molecule to adapt to the new electronic density distribution. Herein, we discuss recent photophysical advances combined with recent progresses in the computational chemistry of photoactive molecular ensembles. We focus the discussion on femtosecond Transient Absorption Spectroscopy (TAS) enabling us to follow the transition from a Locally Excited (LE) state to the ICT and to understand how the environment polarity influences radiative and non-radiative decay mechanisms. In many cases, the charge transfer transition is accompanied by structural rearrangements, such as the twisting or molecule planarization. The possibility of an accurate prediction of the charge-transfer occurring in complex molecules and molecular materials represents an enormous advantage in guiding new molecular and materials design. We briefly report on recent advances in ultrafast multidimensional spectroscopy, in particular, Two-Dimensional Electronic Spectroscopy (2DES), in unraveling the ICT nature of push-pull molecular systems. A theoretical description at the atomistic level of photo-induced molecular transitions can predict with reasonable accuracy the properties of photoactive molecules. In this framework, the review includes a discussion on the advances from simulation and modeling, which have provided, over the years, significant information on photoexcitation, emission, charge-transport, and decay pathways. Density Functional Theory (DFT) coupled with the Time-Dependent (TD) framework can describe electronic properties and dynamics for a limited system size. More recently, Machine Learning (ML) or deep learning approaches, as well as free-energy simulations containing excited state potentials, can speed up the calculations with transferable accuracy to more complex molecules with extended system size. A perspective on combining ultrafast spectroscopy with molecular simulations is foreseen for optimizing the design of photoactive compounds with tunable properties.

Role of specific solute–solvent interactions on the photophysical properties of distyryl substituted BODIPY derivatives

The role played by specific solute–solvent interactions on the spectroscopic properties of experimentally available BODIPY derivatives has been investigated.

On the simulation of vibrationally resolved electronic spectra of medium-size molecules: the case of styryl substituted BODIPYs

BODIPY dyes are used in a variety of applications because of their peculiar spectroscopic and photo-physical properties that vary depending on the stereochemistry of the functional groups attached to the boron-dipyrromethene core structure. In this work, we have applied several computational methods, adapted for semi-rigid molecules based on the Franck-Condon principle, for the study of the optical properties of BODIPY systems and for the understanding of the influence of functional groups on their spectroscopic features. We have analyzed the electronic spectra of two styryl substituted BODIPY molecules of technological interest, properly taking into account the vibronic contribution. For comparison with recently recorded experimental data in methanol, the vibrationally resolved electronic spectra of these systems were computed using both Time-Independent (TI) and Time-Dependent (TD) formalisms. The first step toward the analysis of optical properties of the styryl modified BODIPYs was a benchmark of several density functionals, to select the most appropriate one. We have found that all benchmarked functionals provide good results in terms of band shape but some of them show strong discrepancies in terms of band position. Beyond the issue of the electronic structure calculation method, different levels of sophistication can be adopted for the calculation of vibronic transitions. In this study, the effect of mode couplings and the influence of the Herzberg-Teller terms on the theoretical spectra has been investigated. It has been found that all levels of theory considered give reproducible results for the investigated systems: band positions and shapes are similar at all levels and little improvements have been found in terms of band shape with the inclusion of Herzberg-Teller effect. Inclusion of temperature effects proved to be challenging due to the important impact of large amplitude motions. Better agreement can be achieved by adopting a suitable set of coordinates coupled with a reduced-dimensionality scheme.

Optimization and selection of time-frequency transforms for wave-packet analysis in ultrafast spectroscopy

The analysis of quantum beats in time-resolved spectroscopic signals is becoming a task of primary importance because it is now clear that they bring crucial information about chemical reactivity, transport, and relaxation processes. Here we describe how to exploit the wide family of time-frequency transform methodologies to obtain information not only about the frequency but also about the dynamics of the oscillating components contributing to the overall beating signal. Several linear and bilinear transforms have been considered, and a general and easy procedure to judge in a non-arbitrary way the performances of different transforms has been outlined.

Spectral Filtering as a Tool for Two-Dimensional Spectroscopy: A Theoretical Model

Two-dimensional optical spectroscopy is a powerful technique for the probing of coherent quantum superpositions. Recently, the finite width of the laser spectrum has been employed to selectively tune experiments for the study of particular coherences. This involves the exclusion of certain transition frequencies, which results in the elimination of specific Liouville pathways. The rigorous analysis of such experiments requires the use of ever more sophisticated theoretical models for the optical spectroscopy of electronic and vibronic systems. Here we develop a nonimpulsive and non-Markovian model, which combines an explicit definition of the laser spectrum, via the equation of motion-phase matching approach (EOM-PMA), with the hierarchical equations of motion (HEOM). This theoretical framework is capable of simulating the 2D spectroscopy of vibronic systems with low frequency modes, coupled to environments of intermediate and slower time scales. In order to demonstrate the spectral filtering of vibronic coherences, we examine the elimination of lower energy peaks from the 2D spectra of a zinc porphyrin monomer upon blue-shifting the laser spectrum. The filtering of Liouville pathways is revealed through the disappearance of peaks from the amplitude spectra for a coupled vibrational mode.

Fluorescence-Detected Two-Quantum and One-Quantum-Two-Quantum 2D Electronic Spectroscopy

We demonstrate two-quantum (2Q) coherent two-dimensional (2D) electronic spectroscopy using a shot-to-shot-modulated pulse shaper and fluorescence detection. Broadband collinear excitation is realized with the supercontinuum output of an argon-filled hollow-core fiber, enabling us to excite multiple transitions simultaneously in the visible range. The 2Q contribution is extracted via a three-pulse sequence with 16-fold phase cycling and simulated employing cresyl violet as a model system. Furthermore, we report the first experimental realization of one-quantum-two-quantum (1Q-2Q) 2D spectroscopy, offering less congested spectra as compared with the 2Q implementation. We avoid scattering artifacts and nonresonant solvent contributions by using fluorescence as the observable. This allows us to extract quantitative information about doubly excited states that agree with literature expectations. The high sensitivity and background-free nature of fluorescence detection allow for a general applicability of this method to many other systems.