Interfacial electron transfer (IET) is one of the crucial steps in the light-harvesting process that occurs in various assemblies for solar energy conversion, such as dye-sensitized solar cells or dye-sensitized photoelectrosynthesis cells. Computational studies of IET in dye–semiconductor assemblies employ a variety of approaches, ranging from phenomenological models such as Fermi’s golden rule to more complex methods relying on explicit solutions of the time-dependent Schrödinger equation. This work investigates IET in a model pyridine–TiO2 assembly, with the goals of assessing the validity of Fermi’s golden rule for calculation of the IET rates, understanding the importance of conformational sampling in modeling the IET process, and establishing an approach to rapid computational screening of dye-sensitizers that undergo fast IET into the semiconductor. Our results suggest that IET is a two-step process, in which the electron is first transferred into the semiconductor surface states, followed by diffusion of the electron into the nanoparticle bulk states. Furthermore, while Fermi’s golden rule and related approaches are appropriate for predicting the initial IET rate (i.e., the initial transfer of an electron from the dye into the semiconductor surface states), they are not reliable for prediction of the overall IET rate. The inclusion of conformational sampling at room temperature into the model offers a more complete picture of the IET process, leading to a distribution of IET rates with a median rate faster than the IET rate obtained for the fully-optimized structure at 0 K. Finally, the two most important criteria for determination of the initial IET rate are the percentage of electron density on the linker in the excited state as well as the number of semiconductor acceptor states available at the energy of the excited state. Both of these can be obtained from relatively simple electronic structure calculations at either ab initio or semiempirical levels of theory and can thus be used for rapid screening of dyes with the desired properties.

A computational study of a series of [Fe(tpy)2]2+ (tpy = 2,2′:6′,2′′-terpyridine) complexes is reported, where the tpy ligand is substituted at the 4, 4′, and 4′′ positions by electron donor (furan, thiophene, selenophene, NH2) and acceptor (carboxylic acid, NO2) groups. Using DFT and TD-DFT calculations, we show that the substitution of heterocyclic π donor groups onto the tpy ligand scaffold leads to marked improvement of the [Fe(tpy)2]2+ absorption properties, characterized by increased molar extinction coefficients, shift of absorption energies to longer wavelengths, and broadening of the absorption spectrum in the visible region. The observed changes in the light absorption properties are due to destabilization of ligand-centered occupied π orbital energies, thus increasing the interactions between the metal t2g (HOMO) and ligand π orbitals. Substitution of extended π-conjugated groups, such as thienothiophene and dithienothiophene, further destabilizes the ligand π orbital energies, resulting in a fully ligand-localized HOMO (i.e., HOMO inversion) and additional improvement of the light absorption properties. These results open up a new strategy to tuning the light absorption properties of Fe(II)-polypyridines.

Rational design of artificial light-harvesting molecular architectures entails building systems that absorb strongly in the visible and near-IR region of the electromagnetic spectrum and also funnel excited state energy to a single site. The ability to model nonadiabatic processes, such as excited-state energy transfer (EET), that occur on a picosecond time scale can aid in the development of novel artificial light-harvesting arrays. A combination of density functional theory (DFT), time-dependent DFT, tight-binding molecular dynamics, and quantum dynamics is employed here to simulate EET in the ZnFbΦ dyad, a model artificial light-harvesting array that undergoes EET with an experimentally measured rate constant of (3.5 ps)−1 upon excitation at 550 nm in toluene [Yang et al. J. Phys. Chem. B 1998, 102, 9426−9436]. We find that to successfully simulate the EET process, it is important to (1) include coupling between nuclear and electronic degrees of freedom in the QD simulation, (2) account for Coulomb coupling between the electron and hole wavepackets, and (3) parametrize the extended Hückel model Hamiltonian employed in the QD simulations with respect to the DFT.

Tetrapyrrole-based pigments play a crucial role in photosynthesis as principal light absorbers in light-harvesting chemical systems. As such, accurate theoretical descriptions of the electronic absorption spectra of these pigments will aid in the proper description and understanding of the overall photophysics of photosynthesis. In this work, time-dependent density functional theory (TD-DFT) at the CAM-B3LYP/6-31G* level of theory is employed to produce the theoretical absorption spectra of several tetrapyrrole-based pigments. However, the application of TD-DFT to large systems with several hundreds of atoms can become computationally prohibitive. Therefore, in this study, TD-DFT calculations with reduced orbital spaces (ROSs) that exclude portions of occupied and virtual orbitals are pursued as a viable, computationally cost-effective alternative to conventional TD-DFT calculations. The effects of reducing orbital space size on theoretical spectra are qualitatively and quantitatively described, and both conventional and ROS results are benchmarked against experimental absorption spectra of various tetrapyrrole-based pigments. The orbital reduction approach is also applied to a large natural pigment assembly that comprises the principal light-absorbing component of the reaction center in purple bacteria. Overall, we find that TD-DFT calculations with proper and judicious orbital space reductions can adequately reproduce conventional, full orbital space, TD-DFT results of all pigments studied in this work.

Combined experimental and computational studies have revealed factors that influence the nondirected C–H activation in Cp*Ir complexes that contain carboxylate ligands. A two-step acetate-assisted pathway was shown to be operational where the first step involves substrate binding and the second step involves cleavage of the C–H bond of the substrate. A nonlinear Hammett plot was obtained to examine substituted arenes where a strong electronic dependence (ρ = 1.67) was observed for electron-donating groups, whereas no electronic dependence was observed for electron-withdrawing groups. Electron-donating substituents in the para position were shown to have a bigger impact on the C–H bond cleavage step, whereas electron-withdrawing substituents influenced the substrate-binding step. Although cleavage of the C–H bond was predicted to be more facile with arenes that contain substituents in the para position by DFT calculations, the cyclometalations of anisole and benzonitrile were observed experimentally. This suggests that these substituents, even though they are weakly directing, still result in cyclometalation because the barriers for activation at the ortho and para positions of arenes are comparable (24.3 and 26.5 kcal/mol, respectively). Incorporation of a weakly bound ligand was found to be necessary for facile reactivity. It is predicted by DFT calculations that the replacement of an oxygen atom with a nitrogen atom in the carboxylate ligand would lead to a dramatic reduction in the barrier for C–H activation, as the incorporation of formimidate and N-methylformimidate ligands leads to barriers of 23.4 and 21.7 kcal/mol, respectively. These values are significantly lower than the barrier calculated for the analogous acetate ligand (28.2 kcal/mol).

Over the past two decades, dye-sensitized solar cells (DSSCs) have become a viable and relatively cheap alternative to conventional crystalline silicon-based systems. At the heart of a DSSC is a wide band gap semiconductor, typically a TiO2 nanoparticle network, sensitized with a visible light absorbing chromophore. Ru(II)-polypyridines are often utilized as chromophores thanks to their chemical stability, long-lived metal-to-ligand charge transfer (MLCT) excited states, tunable redox potentials, and near perfect quantum efficiency of interfacial electron transfer (IET) into TiO2. More recently, coordination compounds based on first row transition metals, such as Fe(II)-polypyridines, gained some attention as potential sensitizers in DSSCs due to their low cost and abundance. While such complexes can in principle sensitize TiO2, they do so very inefficiently since their photoactive MLCT states undergo intersystem crossing (ISC) into low-lying metal-centered states on a subpicosecond time scale. Competition between the ultrafast ISC events and IET upon initial excitation of Fe(II)-polypyridines is the main obstacle to their utilization in DSSCs. Suitability of Fe(II)-polypyridines to serve as sensitizers could therefore be improved by adjusting relative rates of the ISC and IET processes, with the goal of making the IET more competitive with ISC.Our research program in computational inorganic chemistry utilizes a variety of tools based on density functional theory (DFT), time-dependent density functional theory (TD-DFT) and quantum dynamics to investigate structure–property relationships in Fe(II)-polypyridines, specifically focusing on their function as chromophores. One of the difficult problems is the accurate determination of energy differences between electronic states with various spin multiplicities (i.e., 1A, 1,3MLCT, 3T, 5T) in the ISC cascade. We have shown that DFT is capable of predicting the trends in the energy ordering of these electronic states in a set of structurally related complexes with the help of appropriate benchmarks, based either on experimental data or higher-level ab initio calculations.Models based on TD-DFT and quantum dynamics approaches have proven very useful in understanding IET processes in Fe(II)-polypyridine–TiO2 assemblies. For example, they helped us to elucidate the origin of “band selective” sensitization in the [Fe(bpy-dca)2(CN)2]–TiO2 assembly (bpy-dca = 2,2′-bipyridine-4,4′-dicarboxylic acid), first observed by Ferrere and Gregg [Ferrere, S.; Gregg, B. A. J. Am. Chem. Soc. 1998, 120, 843.]. They also shed light on the relationship between the linker group that anchors Fe(II)-polypyridines onto the TiO2 surface and the speed of IET in Fe(II)-polypyridine–TiO2 assemblies.More interestingly, our results show that the IET efficiency is strongly correlated with the amount of electron density on the linker group and that one can obtain insights into the IET in dye–semiconductor assemblies based on ground state electronic structure calculations alone. This may be useful for quick screening of a large number of complexes for use as potential sensitizers in DSSCs, especially if followed up by TD-DFT and quantum dynamics simulations for selected target compounds to confirm efficient sensitization. While our focus over the past few years has been exclusively on Fe(II)-polypyridines, the computational strategies outlined in this Account are applicable to a wide variety of sensitizers.

Copper(I) diimine complexes have emerged as low cost replacements for ruthenium complexes as light sensitizers and electron donors, but their shorter metal-to-ligand-charge-transfer (MLCT) states lifetimes and lability of transient Cu(II) species impede their intended functions. Two carboxylated Cu(I) bis-2,9-diphenylphenanthroline (dpp) complexes [Cu(I)(dpp-O(CH2CH2O)5)(dpp-(COOH)2)]+ and [Cu(I)(dpp-O(CH2CH2O)5)(dpp-(Φ-COOH)2)]+ (Φ = tolyl) with different linker lengths were synthesized in which the MLCT-state solvent quenching pathways are effectively blocked, the lifetime of the singlet MLCT state is prolonged, and the transient Cu(II) ligands are stabilized. Aiming at understanding the mechanisms of structural influence to the interfacial charge transfer in the dye-sensitized solar cell mimics, electronic and geometric structures as well as dynamics for the MLCT state of these complexes and their hybrid with TiO2 nanoparticles were investigated using optical transient spectroscopy, X-ray transient absorption spectroscopy, time-dependent density functional theory, and quantum dynamics simulations. The combined results show that these complexes exhibit strong absorption throughout the visible spectrum due to the severely flattened ground state, and a long-lived charge-separated Cu(II) has been achieved via ultrafast electron injection (<300 fs) from the 1MLCT state into TiO2 nanoparticles. The results also indicate that the TiO2-phen distance in these systems does not have significant effect on the efficiency of the interfacial electron-transfer process. The mechanisms for electron transfer in these systems are discussed and used to develop new strategies in optimizing copper(I) diimine complexes in solar energy conversion devices.

Dye-sensitized solar cells (DSSCs) often utilize transition metal-based chromophores for light absorption and semiconductor sensitization. Ru(II)-based dyes are among the most commonly used sensitizers in DSSCs. As ruthenium is both expensive and rare, complexes based on cheaper and more abundant iron could serve as a good alternative. In this study, we investigate Fe(II)-bis(terpyridine) and its cyclometalated analogues, in which pyridine ligands are systematically replaced by aryl groups, as potential photosensitizers in DSSCs. We employ density functional theory at the B3LYP/6-31G*,SDD level to obtain the ground state electronic structure of these complexes. Quantum dynamics simulations are utilized to study interfacial electron transfer between the Fe(II) photosensitizers and a titanium dioxide semiconductor. We find that cyclometalation stabilizes the singlet ground state of these complexes by 8–19 kcal/mol but reduces the electron density on the carboxylic acid attached to the aryl ring. The results suggest that cyclometalation provides a feasible route to increasing the efficiency of Fe(II) photosensitizers but that care should be taken in choosing the substitution position for the semiconductor anchoring group.

Fe(II) polypyridines are an important class of pseudo-octahedral metal complexes known for their potential applications in molecular electronic switches, data storage and display devices, sensors, and dye-sensitized solar cells. Fe(II) polypyridines have a d6 electronic configuration and pseudo-octahedral geometry and can therefore possess either a high-spin (quintet) or a low-spin (singlet) ground state. In this study, we investigate a series of complexes based on [Fe(tpy)2]2+ (tpy = 2,2′;6′,2″-terpyridine) and [Fe(dcpp)2]2+ (dcpp = 2,6-bis(2-carboxypyridyl)pyridine). The ligand field strength in these complexes is systematically tuned by replacing the central pyridine with five-membered (N-heterocyclic carbene, pyrrole, furan) or six-membered (aryl, thiazine-1,1-dioxide, 4-pyrone) moieties. To determine the impact of ligand substitutions on the relative energies of metal-centered states, the singlet, triplet, and quintet states of the Fe(II) complexes were optimized in water (PCM) using density functional theory at the B3LYP+D2 level with 6-311G* (nonmetals) and SDD (Fe) basis sets. It was found that the dcpp ligand scaffold allows for a more ideal octahedral coordination environment in comparison to the tpy ligand scaffold. The presence of six-membered central rings also allows for a more ideally octahedral coordination environment relative to five-membered central rings, regardless of the ligand scaffold. We find that the ligand field strength in the Fe(II) polypyridines can be tuned by altering the donor atom identity, with C donor atoms providing the strongest ligand field.

Iron(II) polypyridine complexes have the potential for numerous applications on a global scale, such as sensitizers, sensors, and molecular memory. The excited-state properties of these systems, particularly the intersystem crossing (ISC) rates, are sensitive to the choice of ligands and can be significantly altered depending on the coordination environment. We employ density functional theory and Smolyak’s sparse grid interpolation algorithm to construct potential energy surfaces (PESs) for the photophysically relevant states (1A, 3,5MC, and 1,3MLCT) of the [Fe(tpy)2]2+ (tpy = 2,2′:6′,2″-terpyridine) complex, with the goal of obtaining a deeper understanding of the ground- and excited-state electronic structure of this system. The three dimensions that define our adiabatic PESs consist of equatorial and axial metal–ligand bond length distortions and a terpyridine ligand “rocking angle”, which has not previously been investigated. The intersection crossing seams and minimum energy crossing points (MECPs) between surfaces are also determined. Overall, we find that the PESs of all electronic excited states investigated are characterized by low-energy valleys along the tpy rocking-angle coordinate. This results in the presence of large low-energy areas around the MECPs on the intersection seams of different electronic states and indicates that inclusion of this third coordinate is crucial for an adequate description of the PESs and surface crossing seams of the [Fe(tpy)2]2+ complex. Finally, we suggest that tuning the energetics of the tpy ligand rocking motion could provide a way to control the ISC process in this complex.

Light-harvesting antennas are protein–pigment complexes that play a crucial role in natural photosynthesis. The antenna complexes absorb light and transfer energy to photosynthetic reaction centers where charge separation occurs. This work focuses on computational studies of the electronic structure of the pigment networks of light-harvesting complex I (LH1), LH1 with the reaction center (RC-LH1), and light-harvesting complex II (LH2) found in purple bacteria. As the pigment networks of LH1, RC-LH1, and LH2 contain thousands of atoms, conventional density functional theory (DFT) and ab initio calculations of these systems are not computationally feasible. Therefore, we utilize DFT in conjunction with the energy-based fragmentation with molecular orbitals method and a semiempirical approach employing the extended Hückel model Hamiltonian to determine the electronic properties of these pigment assemblies. Our calculations provide a deeper understanding of the electronic structure of natural light-harvesting complexes, especially their pigment networks, which could assist in rational design of artificial photosynthetic devices.

Porphyrin–perylene arrays are ideal candidates for light-harvesting systems capable of panchromatic absorption. In this work, we employ density functional theory (DFT) and time-dependent DFT to investigate the unique UV–vis absorption properties exhibited by a series of ethynyl-linked porphyrin–perylene arrays that were previously synthesized and characterized spectroscopically [ Chem. Commun. 2014, 50, 14512−5]. We find that the ethynyl linker is responsible for strong electronic coupling of porphyrin and perylene subunits in these systems. Additionally, these arrays exhibit a low barrier to rotation around the ethynyl linker (<1.4 kcal/mol per one perylene substituent), which results in a wide range of molecular conformations characterized by different porphyrin–perylene dihedral angles being accessible at room temperature. The best match between the calculated and experimental UV–vis spectra is obtained by averaging the calculated UV–vis spectra over the range of conformations defined by the porphyrin−perylene dihedral angles. Finally, our calculations suggest that the transitions in the lower energy region (550–750 nm) can be assigned to the excitations originating from the porphyrin subunit; the mid-energy region transitions (450–550 nm) are assigned to the perylene-centered excitations, while the high-energy transitions (350–450 nm) involve contributions from both porphyrin and perylene subunits.

Pigment-based molecular arrays, especially those based on porphyrins, have been extensively studied as viable components of artificial light harvesting devices. Unlike porphyrins, bacteriochlorins absorb strongly in the NIR, yet little is known of the applicability of covalently linked bacteriochlorin-based arrays in this arena. To lay the foundation for future studies of excited state properties of such arrays, we present a systematic study of the ground state electronic structure of zinc bacteriochlorin (ZnBC) molecular arrays with various linkers and linker attachment sites (meso vs β) employing density functional theory in combination with the energy-based fragmentation (EBF) method, and the EBF with molecular orbitals (EBF-MO) method. We find that the level of steric hindrance between the ZnBC and the linker is directly correlated with the amount of ground sate electronic interactions between the ZnBCs. Low steric hindrance between the ZnBC and the linker found in alkyne-linked arrays results in strongly interacting arrays that are characterized by a decrease in the HOMO–LUMO energy gaps, large orbital energy dispersion in the frontier region, and low ZnBC-linker rotational barriers. In contrast, sterically hindered linkers, such as aryl-based linkers, result in weakly interacting arrays characterized by increased orbital energy degeneracy in the frontier region and high ZnBC-linker rotational barriers. For all linkers studied, the level of steric hindrance decreases when the ZnBCs are linked at the β position. Hence, ZnBC arrays that exhibit strong, weak, or intermediate ground-state electronic interactions can be realized by adjusting the level of steric hindrance with a judicious choice of the linker type and linker attachment site. Such tuning may be essential for design of light harvesting arrays with desired spectral properties.

Iron(II) polypyridines represent a cheaper and nontoxic alternative to analogous Ru(II) polypyridine dyes successfully used as photosensitizers in dye-sensitized solar cells (DSSCs). We employ density functional theory (DFT) and time-dependent DFT (TD-DFT) to study ground and excited state properties of [Fe(bpy)(CN)4]2–, [Fe(bpy-dca)(CN)4]2–, and [Fe(bpy-dca)2(CN)2] complexes, where bpy = 2,2′-bipyridine and dca = 4,4′-dicarboxylic acid. Quantum dynamics simulations are further used to investigate the interfacial electron transfer (IET) between the excited Fe(II) dyes and a TiO2 nanoparticle. All three dyes investigated display two bands in the visible region of the absorption spectrum, with the major transitions corresponding to the metal-to-ligand charge transfer states. The calculated IET rates from the particle states created by the excitation of the lower-energy absorption band are comparable to or slower than the rate of the excited state decay into the nonemissive, metal-centered states of the Fe(II) dyes (∼100 fs), indicating that the IET upon the excitation of this band is unlikely. Several particle states in the higher-energy absorption band display IET rates at or below 100 fs, suggesting the possibility of the IET between the Fe(II)-sensitizer and TiO2 nanoparticle upon excitation with visible light. Our results are consistent with the previous experimental work on Fe(II) sensitizers (Ferrere, S. Chem. Mater. 2000, 12, 1083) and elucidate the band-selective nature of the IET in these compounds.

Covalently bound molecular arrays composed of porphyrins or related pigments have gained a lot of interest as components of artificial light-harvesting systems and molecular photonic devices. The large size of these arrays, however, makes their theoretical investigation employing the ab initio or density functional methodologies difficult. Energy-based fragmentation methods (EBF) represent a set of conceptually simple approaches to theoretical investigation of large systems and were therefore chosen as a tool to study these systems. Here a new approach to EBF, EBF-MO, is introduced that enables one to obtain orbitals and orbital energies and to perform population analysis and excited-state calculations of large systems composed of hundreds of atoms. This approach was implemented into a parallel program, JETT, and the benchmark calculations have shown its accuracy and applicability to the ground- and excited-state calculations of systems containing transition metals and extended π-conjugation. EBF-MO was then applied to the density functional theory (DFT) and the time-dependent density functional theory (TDDFT) calculations of ground- and excited-state properties of a porphyrin-based molecular photonic wire composed of 472 atoms and 4265 basis functions at the B3LYP/LANL08,6-31G* level. The TDDFT calculations have revealed the character of the excited states, and the unidirectionality of the excitation energy transfer across the array relevant to its signal transfer function. The computational approaches introduced here have widened the applicability of the ab initio and density functional methodologies to calculations of extended systems such as natural and artificial light-harvesting systems and molecular photonic devices.

Pseudo-octahedral complexes of iron find applications as switches in molecular electronic devices, materials for data storage, and, more recently, as candidates for dye-sensitizers in dye-sensitized solar cells. Iron, as a first row transition metal, provides a weak ligand-field splitting in an octahedral environment. This results in the presence of low-lying 5T excited states that, depending on the identity of iron ligands, can become the ground state of the complex. The small energy difference between the low-spin, 1A, and high-spin, 5T, states presents a challenge for accurate prediction of their ground state using density functional theory. In this work, we investigate the applicability of the B3LYP functional to the ground state determination of first row transition metal complexes, focusing mainly on Fe(II) polypyridine complexes with ligands of varying ligand field strength. It has been shown previously that B3LYP artificially favors the 5T state as the ground state of Fe(II) complexes, and the error in the energy differences between the 1A and 5T states is systematic for a set of structurally related complexes. We demonstrate that structurally related complexes can be defined as pseudo-octahedral complexes that undergo similar distortion in the metal–ligand coordination environment between the high-spin and low-spin states. The systematic behavior of complexes with similar distortion can be exploited, and the ground state of an arbitrary Fe(II) complex can be determined by comparing the calculated energy differences between the singlet and quintet electronic states of a complex to the energy differences of structurally related complexes with a known, experimentally determined ground state.

We employ density functional theory to investigate ground state hole transfer in covalently linked oxidized zinc–zinc porphyrin ([ZnZn]+) and zinc–free-base porphyrin ([ZnFb]+) dyads in both coplanar and noncoplanar (tilted) conformations. We obtain reactant, product, and transition state (TS) for the hole transfer reaction in the [ZnZn]+ system. The hole is localized on a single porphyrin unit in the reactant and product states while delocalized in the TS, implying the dominance of superexchange mechanism in the hole transfer reaction. A metastable as well as stable states are located for the [ZnFb]+ system while no TS is found, indicating a barrierless hole transfer reaction. The hole lifetimes are calculated to be 15.80 and 0.034 ns for [ZnZn]+ in the coplanar and tilted conformation, respectively, and 14.45 and 0.313 ns for [ZnFb]+. The hole transfer rates are found to be several orders of magnitude faster in the tilted conformation than in the coplanar conformation for both dyads, showing the importance of noncoplanar conformation between the two porphyrin pigments in facilitating the hole transfer process. We also show that inclusion of solvent effects in calculations plays an important role in the proper ground state hole localization in oxidized dyads. These results provide an unconventional insight into the hole transfer mechanism in porphyrin arrays and are relevant to design of artificial photoharvesting materials.

•Review of spin-crossover chemistry and photochemistry relevant to Fe(II) complexes.•Comparison of popular techniques in modern electronic structure theory.•Survey of computational work on Fe(II) spin-state energetics and photochemistry.Effective strategies for designing Fe(II) coordination complexes with specifically tailored spin-state energetics can lead to advances in many areas of inorganic and materials chemistry. These include, but are not limited to, rational development of novel spin crossover complexes, efficient chromophores for photosensitization of dye-sensitized solar cells, and multifunctional materials. As the spin-state ordering of transition metal complexes is strongly rooted in their electronic structures, computational chemistry has naturally played an important role in assisting experimental work in this area. Unfortunately, despite many advances, accurate determination of the spin-state energetics of Fe(II) complexes still poses a remarkable challenge for virtually all applicable forms of electronic structure theory due to being controlled by a delicate balancing between correlation and exchange effects. This review focuses on some of the more notable successes and failures of modern electronic structure theory in properly describing these systems in the absence of solid-state effects. The strengths and weaknesses of using traditional wavefunction based methods and density functional theory are considered, and illustrative examples are provided to demonstrate that the modern computational chemist should make use of experimental data whenever possible and expect to utilize a combination of methods to obtain the best results. The review closes by briefly surveying some of the many interesting combined computational and experimental studies of Fe(II) chemistry that have lead to greater fundamental insight and practical understanding of this challenging class of systems.