Cross polarization effect of donor–acceptor group on a potential single‐molecule transistor

Theoretical investigation of the polarization effect on a potential single‐molecule transistor has been studied with density functional theory. 4,4′‐(2‐Amino‐5‐nitro‐1,4‐phenylene)bis(ethyne‐2,1‐diyl) dibenzenethiol (AN‐OPE), containing a donor and an acceptor (D–A) crossed to its oligo(p‐phenylene‐ethynylene) (OPE) backbone, was used as a prototype for this study. Simulation results indicate that AN‐OPE has a higher on/off ratio on conductance than OPE because of the larger polarization along the D–A direction. This high on/off ratio was proved by the 20 times variation in molecular charge, 15 times variation in bond lengths, 49 times variation in polarizability, 9 times variation in the rotation angles, and 13 times variation in the highest occupied molecular orbital–lowest unoccupied molecular orbital gaps under the same gate using B3LYP/6‐31G (d, p). And results imply that conjugated molecules with a cross D–A structure could be a good direction for constructing a better single‐molecule field‐effect transistor. Copyright © 2014 John Wiley & Sons, Ltd.     

Fully conjugated push–pull dendrons with high dipole moments in excited state; synthesis and theoretical rationalization

Novel fully conjugated push–pull dendrons were synthesized by a divergent approach to evaluate the performance of non‐conventional architectures like dendritic one in charge separation processes associated with photovoltaic events. The dipole moments in excited state were estimated by the solvatochromic method, to be related to the charge separation efficiency. A 1:2 ratio of donor–acceptor groups (methoxy and nitro groups respectively) promotes the largest dipole moment in both ground and excited state (up to 17 D), due to the efficient electron density transfer over the entire molecule, through the π‐electron system. The synthesized dendrons induce charge transfer on excitation as follows from UV–vis absorption–emission analysis and theoretical calculations. Copyright © 2015 John Wiley & Sons, Ltd.     

Solvent effects on electronic absorption spectra of donor‐substituted 11,11,12,12‐tetracyano‐9, 10‐anthraquinodimethanes (TCAQs)

Solvent effects on the electronic absorption spectra of donor‐substituted 11,11,12,12‐tetracyano‐9, 10‐anthraquinodimethanes (TCAQs) 1 – 3 have been investigated in 32 well‐selected solvents. These compounds were chosen as model structures for charge‐transfer chromophores featuring second‐ and third‐order nonlinear optical properties. The resulting data were evaluated by means of theoretical models and (semi)empirical correlations determining the optical properties related to electron distribution and polarizability. We found that solvent effects on a polar D‐π‐A system do not depend on the donor/acceptor orientation (HOMO/LUMO localization) but especially on the length of the π‐system in between. The observed solvent effects are described with high accuracy by the applied theoretical models and linear combinations of physical quantities. Solvent polarization, permanent dipole moment, and molar volume substantially affect the longest‐wavelength absorption maxima. Solvent‐induced bathochromic shift resulting from the solvent polarity is described with high accuracy by the Born function. On the other hand, hypsochromic effects of the solvent permanent dipole moment are caused due to the slower reorganization of molecular dipoles compared with the rate of excitation. Solvent polarizability shifts the longest‐wavelength absorption maxima bathochromically with increasing length of the π‐conjugated system. Whereas this effect could be suitably described by the Onsager‐induced polarizability, the orientation polarizability was not found to be important. The solvent molar volume as a hypsochromic shift‐inducing factor is only relevant if the size of the solute and solvent molecules are comparable. If the size of the solute is considerably larger than that of the solvent molecules, the solvent behaves as a ‘shape continuum.’ Copyright © 2008 John Wiley & Sons, Ltd.     

Solvatochromic behavior of dyes with dimethylamino electron‐donor and nitro electron‐acceptor groups in their molecular structure

Six dyes with N,N‐dimethylaminophenyl and 4‐nitrophenyl or 2,4‐dinitrophenyl groups in their molecular structures were prepared and characterized. These compounds have different conjugated bridges (C?C, C?N, and N?N) connecting the electron‐donor and the electron‐acceptor groups. All compounds are solvatochromic, with reverse solvatochromism occurring. The solvatochromic band observed in each spectrum for the dyes is due to a π ? π* transition, of an intramolecular charge transfer nature, which occurs from the electron‐donor N,N‐dimethylaminophenyl group to the electron‐acceptor group in the molecules, which is reinforced by the structures of the compounds optimized by applying density functional theory, which exhibit high planarity. The reverse solvatochromism was explained considering two resonance structures. The benzenoid form is better stabilized in less polar solvents and characterizes the region displaying positive solvatochromism, while the dipolar form is better stabilized in more polar solvents, in the region of negative solvatochromism. The Catalán multiparametric approach was used to study the contribution of solvent acidity, basicity, dipolarity, and polarizability to the solvatochromism exhibited by the compounds. These compounds are good candidates for the investigation of the polarizability and, to a lesser extent, the dipolarity of the medium, with very little interference from specific interactions of the solvent through hydrogen bonding. Copyright © 2014 John Wiley & Sons, Ltd.     

Two-photon absorption in two-dimensional conjugated quadrupolar chromophores

We present ultrafast z-scan measurements of the two-photon absorption (TPA) spectra of a pair of two-dimensionally conjugated quadrupolar donor/acceptor (D/A) chromophores. The all-donor substituted species displays a peak TPA cross section [sigma(2)=520+/-30 GM] that is more than twice that of the D-A species [sigma(2)=240+/-20 GM]. Unlike previous structure-property studies that have evaluated TPA behavior for D/A molecules through the comparison of dipolar and quadrupolar compounds, both molecules investigated herein are quadrupolar, ultimately providing a more consistent evaluation of the effects of donor and/or acceptor substitution on the TPA of conjugated chromophores.     

Solvatochromism as an efficient tool to study N,N‐dimethylamino‐ and cyano‐substituted π‐conjugated molecules with an intramolecular charge‐transfer absorption

A representative data set has been gained by the measurement of the electronic absorption spectra of 12 systematically selected push–pull systems with an intramolecular charge‐transfer (CT) absorption and the general structure D–π–A (D = donor, A = acceptor) featuring electron‐withdrawing CN groups, electron‐donating N(CH₃)₂ groups, and various π‐conjugated backbones in 32 solvents with different polarities. The longest‐wavelength absorption maxima λ_max and the corresponding wavenumbers $\tilde {v}_{{\rm max}} $ were evaluated from the UV/Vis spectra measured in 32 well‐selected solvents. The D–π–A push–pull systems were further characterized by quantum‐chemical quantities and simple structural parameters. Structure–solvatochromism relationships were evaluated by multidimensional statistic methods. Whereas solvent polarizability and solvent cavity size proved to be the most important factors affecting the position of λ_max, the solvent polarity was less important. The most important characteristics of organic CT compounds are the energy of the LUMO, the permanent dipole moment, the COSMO (COnductor‐like Screening MOdel) area, the COSMO volume, the number, and ratio of N,N‐dimethylamino and cyano groups, and eventually the number of triple bonds (π‐linkers). A relation between the first‐order polarizability α, the longest‐wavelength absorption maxima λ_max, and the structural features has also been found. The higher‐order polarizabilities β and γ are not related to the observed solvatochromism. Copyright © 2010 John Wiley & Sons, Ltd.     

First order molecular hyperpolarizabilities and intramolecular charge transfer from the vibrational spectra of NLO material: ethyl‐3‐(3, 4‐dihydroxyphenyl)‐2‐propenoate

As part of the efforts for the design of new organic nonlinear optical(NLO) materials with high efficiency for present day technological requirements, a comprehensive investigation on the intramolecular charge transfer(CT) of an efficient π‐conjugated potential push–pull NLO chromophore, ethyl‐3‐(3,4‐dihydroxyphenyl)‐2‐propenoate(EDP) to a strong electron‐acceptor group through the π‐conjugated bridge has been carried out from their vibrational spectra. The first hyperpolarizabilities of caffeic derivatives are investigated by ab initio method. The NLO efficiency is experimentally measured by powder efficiency experiment. The strongest vibrational modes contributing to the electro‐optic effect from the simultaneous infrared(IR) and Raman activities of the ring CC stretching modes, in‐plane deformation modes, and the umbrella mode of the methyl groups have been identified and analyzed unambiguously. The influence of electronic effects, hyperconjugation and backdonation, on the C H stretching vibrations of both methyl and methylene groups causing the decrease of stretching wavenumbers and IR intensities has been extensively investigated. Copyright © 2009 John Wiley & Sons, Ltd.     

Theoretical study of one‐ and two‐photon absorption properties of expanded donor–acceptor calix[4]arenes

Two types of donor(D)–acceptor(A) calix[4]arenes have been theoretically studied using DFT//B3LYP/6‐31G(d) method and ZINDO/CISD method. The calculations show that the substitution of C? C by the conjugation bridge C?C and N?N plays an important part in altering one‐photon absorption (OPA) and two‐photon absorption (TPA) properties. The maximum OPA wavelengths of all studied compounds are less than 400 nm, which means high transparency. The geometry of the calixarenes strongly influences the TPA properties of the studied compounds. In addition, the nitro derivatives have a wider TPA response range than other non‐nitro derivatives. The tetrasubstituted calix[4]arenes (type B calixarenes) have a larger TPA cross‐section values than the bisubstituted calix[4]arenes (type A calixarenes). Copyright © 2009 John Wiley & Sons, Ltd.     

Intramolecular electron transfer and charge delocalization in bistable donor–acceptor systems based on perchlorotriphenylmethyl radicals linked to ferrocene and tetrathiafulvalene units

A comparative analysis of the physicochemical properties of different donor–acceptor dyads, based on a perchlorotriphenylmethyl radical acceptor subunit linked through a vinylene π‐bridge to ferrocene derivatives (1–3) or a tetrathiafulvalene (4) donor subunit, is presented. Intramolecular electron transfer and charge delocalization of these donor–acceptor dyads are discussed on the basis of data obtained in solution by cyclic voltammetry, UV–Vis near‐infrared, and electron spin resonance spectroscopies. Bistability in the crystalline phase is also discussed on the basis of the results provided by X‐ray diffraction analysis and temperature‐dependent Mössbauer spectra. Copyright © 2014 John Wiley & Sons, Ltd.     

Ab Initio Simulation of Charge Transfer at the Semiconductor Quantum Dot/TiO2 Interface in Quantum Dot‐Sensitized Solar Cells

Quantum dot‐sensitized solar cells (QDSSCs) have emerged as a promising solar architecture for next‐generation solar cells. The QDSSCs exhibit a remarkably fast electron transfer from the quantum dot (QD) donor to the TiO₂ acceptor with size quantization properties of QDs that allows for the modulation of band energies to control photoresponse and photoconversion efficiency of solar cells. To understand the mechanisms that underpin this rapid charge transfer, the electronic properties of CdSe and PbSe QDs with different sizes on the TiO₂ substrate are simulated using a rigorous ab initio density functional method. This method capitalizes on localized orbital basis set, which is computationally less intensive. Quite intriguingly, a remarkable set of electron bridging states between QDs and TiO₂ occurring via the strong bonding between the conduction bands of QDs and TiO₂ is revealed. Such bridging states account for the fast adiabatic charge transfer from the QD donor to the TiO₂ acceptor, and may be a general feature for strongly coupled donor/acceptor systems. All the QDs/TiO₂ systems exhibit type II band alignments, with conduction band offsets that increase with the decrease in QD size. This facilitates the charge transfer from QDs donors to TiO₂ acceptors and explains the dependence of the increased charge transfer rate with the decreased QD size.     

Theoretical studies of the structural,electronic and optical properties of carbazole‐based compounds

Carbazole derivatives have drawn increasing attention recently in organic electronic device applications because of their particular optoelectronic properties. An in‐depth theoretical investigation was elaborated in this paper to reveal the molecular structures, optoelectronic properties, and the structure‐property relationships of different carbazole‐linked functional groups. The geometric and electronic structures in ground and the mobility for the hole and electron are both calculated by density functional theory method. The excited‐state geometries of these compounds were obtained through Single‐excitation Configuration Interaction method, and time‐dependent density functional theory calculation results described the absorption and emission spectra properties, respectively. Some conclusions are as follows: (1) enlarging the π‐conjugated area, the corresponding spectra red shifted markedly; (2) by introducing the electron‐donor such as carbazole, the spectra blue shifted slightly; (3) compared with compound 1, the spectra for these compounds are hardly influenced by introducing an electron‐acceptor or heterocyclic substitution. On all accounts, these compounds are interesting optoelectronic functional materials. On the basis of their structural modifiability, the arylamine derivatives substituted carbazole compounds have great potential in the applications of organic light‐emitting diodes, organic solar cells, and sensors. Copyright © 2011 John Wiley & Sons, Ltd.     

On the nature of some biradicaloid and antiaromatic Vis/NIR‐chromophores

Dyes derived from the biradicalic oxyallyl and cyclopentadienylium were calculated by time‐dependent density functional theory (TDDFT) and the characteristics of the prominent low‐energy transitions revealed by graphical means. According to theoretical and experimental studies, 4‐aminophenyl‐substituted dyes absorb intensely at long wavelengths up to the near infrared. If the amino groups are removed the absorption wavelengths are changed little. As found in the previous studies on the squaraine and croconaine oxyallyl dyes, the substituents play only a minor role in the spectral excitation. Charge‐transfer‐type excitations do not occur between the donor aryl substituents and the central oxyallyl or cyclopentadienylium acceptor group. This behaviour is exceptional since donor–acceptor compounds tend to produce charge‐transfer‐ or polymethine‐type electronic transitions. The hitherto rarely used electron density difference (EDD) maps clearly unveiled the spectral excitation features. The spectral excitation of the title compounds is predominantly localized at the oxyallyl and cyclopentadienylium groups, respectively. Characteristics of simple chromophoric compounds and of conventional CT‐ and polymethine dyes are given for comparison. The biradicaloid character of these dyes is supported by the calculated low singlet–triplet splitting energies. Spin properties were characterized in terms of expectation values of the S²‐operator and antiaromatic properties in terms of nucleus‐independent chemical shifts (NICS). According to the ΔE_S/T and <S²> criteria compounds with the acyclic oxyallyl fragment are more biradicaloid. The parent compounds oxyallyl, thioxyallyl and cyclopentadienylium, display extremely large <S²> values. These compounds are triplets in the ground state. The absorption wavelengths of selected biradicaloid species were also calculated by the multi‐reference SORCI method. Copyright © 2010 John Wiley & Sons, Ltd.     

Ring strain energy in ether‐ and lactone‐containing spiro compounds

Ring strain energies (RSEs) have been calculated for oxygen‐containing spiro compounds using the group equivalent reaction (GER) formalism. The RSEs for all compounds studied were calculated from the energies of fully‐optimized structures at the MP2 + ZPE/cc‐pVDZ level and the more computationally costly G4(MP2) method. RSEs for selected compounds were also calculated with the CBS‐QB3 method, with less than 1 kcal/mol difference observed between G4(MP2) and CBS‐QB3. The difference between the less costly MP2 + ZPE and G4(MP2) methods was less than 1.5 kcal/mol. The highest RSEs were found for the compounds containing two three‐membered rings, and these compounds also exhibited the greatest excess strain energy (ESE) of about 12 kcal/mol. The RSEs of cyclic lactones vary with ring size differently than those of cyclic ethers. Cyclic ethers' RSEs decrease by a small amount from the three‐ to four‐membered rings then decrease drastically as the ring increases to 5 atoms, and approaches zero for the six‐membered ring, the same unexpected behavior as seen in cycloalkanes. Cyclic lactones' RSEs decrease linearly to almost zero from the three‐ to the five‐membered ring, then increase by 1–2 kcal/mol in the six‐membered ring. Lactone‐containing spiro compounds exhibit regularly diminishing ESE as the size of the lactone ring increases, down to about 3 kcal/mol in the δ‐lactone‐containing spiro compound. Substitution of methyl groups decreases RSE in these oxygen‐containing spiro compounds, while substitution of fluorine significantly increases RSE, as has been reported in other compounds. But RSE alone is shown to not correlate completely with chemical reactivity of these spiro compounds. Copyright © 2014 John Wiley & Sons, Ltd.     

Merocyanines based on 1,3‐indanedione: electronic structure and solvatochromism

A series of merocyanines derived from 1,3‐indanedione and heterocycles of various electron‐donating properties was studied in detail. Their solvatochromic properties were explored in a wide range of solvent polarities to reveal the dependences of their chromacity and electronic structure on the key structural parameters – the properties of a donor heterocycle and the polymethine chain length. Also the dyes were studied by NMR spectroscopy and by quantum chemical calculations, both with the semiempirical AM1 and the non‐empirical density functional theory/B3LYP method. The solvatochromic properties of the explored dyes are rather close to those of merocyanines derived from malononitrile as acceptor group. Appreciable distinctions were observed only in protic ethanol; obviously, they are connected with the formation of solvent–solute H‐bonds in the case of 1,3‐indanedione derivatives. The electron‐acceptor properties of 1,3‐indanedione were found to be somewhat stronger in comparison with those of malononitrile even in aprotic solvents, contrary to the known literature data. Analysis of the merocyanines' molecular orbitals and simulation of their electronic spectra were carried out both in vacuum and in the solvent matrix, and the absorption electronic transitions were analyzed. Static nonlinear optical properties were calculated for both the new merocyanines and the corresponding cationic and anionic cyanine dyes. Copyright © 2010 John Wiley & Sons, Ltd.     

Push–pull hyperbranched molecules. A theoretical study

The electronic properties of the ground state, unrelaxed and relaxed first excited states of push–pull hyperbranched molecules bearing amino and nitro terminal groups have been studied at BB1K/cc‐pvdz//HF/6‐31g(d), TD‐BB1K/cc‐pvdz//HF/6‐31g(d) and TD‐BB1K/cc‐pvdz//CIS/6‐31g(d) levels of theory, respectively. It was demonstrated that dendritic architecture of push–pull molecules favours the charge transfer in the excited state compared to linear molecules. The possibility of adopting a plane conformation is an important condition for the charge transfer in an excited state. According to the calculations 1:1 ratio of donor and acceptor groups is another important precondition for the manifestation of strong charge separation in the excited state. In case of excess of nitro groups over the amino, some of the excitations participating in the S0 → S1 transition favour the charge transfer in the excited state in the opposite directions, thus decreasing the charge separation. Copyright © 2008 John Wiley & Sons, Ltd.     

Spectroscopic study of polyphenylquinolines—materials with efficient intramolecular charge transfer

The energy structure and nature of intramolecular charge transfer have been determined for a series of synthesized 2,6-polyphenylquinolines (PPQs) containing resorcinol, oxygen, and phenylamine bridging groups between phenylquinoline cycles and possessing an arylene radical in the form of alkylated derivatives of carbazole or indolo[3,2-b]carbazole, as well as arylated derivatives of carbazole with nitro- and fluorine substituents. The analysis was based on data on their absorption spectra, photosensitivity, carrier photogeneration quantum efficiency, and luminescence. It is shown that the PPQ monomeric unit is a composite intramolecular charge-transfer complex (ICTC) between the donor arylene fragment and the acceptor quinoline cycle; the latter, in turn, is an ICTC between the donor bridging group and quinoline ring (when acceptor fluorine or nitro substituents are introduced into the carbazole structure, the complex becomes ternary), and the narrowest band gap (2.3 eV) corresponds to the PPQ having a phenylamine bridging group and a fragment of indolo[3,2-b]carbazole.     

Effectiveness of charge separation from the long-lived second excited state of donors

In molecular zinc-porphyrin-based donor–acceptor systems, the electron transfer from the second singlet excited state S₂ is accompanied by ultrafast recombination into the first excited state, resulting in a low quantum yield of the thermalized charge-separated state (20%). It is demonstrated that the quantum yield of ultrafast charge separation in donor–acceptor triads D–A₁–A₂ can be close to 100% in molecular systems with lifetimes of the S₂ state longer than 150 ps. As prototypes of such systems, donor–acceptor diads D–A₁ and triads D–A₁–A₂ are considered, wherein the xanthione molecule plays the role of a donor. The ranges of the model parameters are determined in which the efficiency of charge separation is high. The twostage photoinduced charge transfer is studied within the framework of a multichannel stochastic model that takes into account the reorganization of a polar solvent and a high-frequency intramolecular vibrational mode.     

Synthesis and photoinduced electron transfer characteristic of a bis (zinc porphyrin)‐perylene bisimide array

In this article, a donor–acceptor array consisting of two zinc porphyrin (ZnPOR) units attached to the 1,7‐positions of perylene‐3,4:9,10‐bis(dicarboximide) (PDI) was synthesized and characterized. This double‐wing molecule exhibits very broad absorption in the region from 300 to 900 nm. Especially, its lower energy absorption feature presumably arises from donor–acceptor interactions. The fluorescence emission spectra confirmed that photoinduced electron transfer occurred from POR to perylene bisimide moiety in this triad. In contrast to previously studied systems incorporating POR and PDI groups, this array shows the evidence of a relatively strong ground‐state electronic coupling between the donor and acceptor moieties. Additionally, highest occupied molecular orbital and lowest unoccupied molecular orbital values of the array were acquired by cyclic voltammetry (CV). The results showed that these energy values fulfill the energetic conditions required for the proposed electron transfer. More importantly, the photocurrent measurement showed that this molecule exhibits a high capacity to form a photoinduced charge‐separated state and to produce steady and prompt cathodic photocurrent responses. These results confirmed the role of this new array toward harvesting light energy and generating photocurrent during the operation of a photoelectrochemical cell. Copyright © 2011 John Wiley & Sons, Ltd.     

Nondoped blue fluorescent OLED based on cyanophenanthrimidazole‐styryl‐triphenylamine/carbazole materials

New 2‐(4′‐9H‐carbazole‐9‐yl)‐styryl‐1H‐phenathro[9,10‐d]imidazole‐1‐yl)benzonitrile (SPICN‐Cz) and 4‐(2‐(4‐(diphenylamino)phenyl‐styryl‐1H‐phenathro[9,10‐d]imidazole‐1‐yl)benzonitrile (SPICN‐TPA) have been synthesised, and their photophysical, electrochemical, and electroluminescent properties were analysed in comparison with their cyano‐free parent compounds, SPI‐Cz, and SPI‐TPA. Solvatochromic effects show the transformation of an excited state character from locally excited (LE) state to charge transfer (CT) state. Using time‐dependent density functional theory calculation, the excited state properties of these donor‐acceptor blue emissive materials have been analysed. Their excited state properties have been tuned by replacing the strong donor triphenylamine to weak donor carbazole to achieve the combination of high photoluminance efficiency locally excited (LE) component and high exciton‐utilizing CT component in one excited state. Hybridization processes between LE and CT components of SPICN‐Cz and SPICN‐TPA in the emissive state have been discussed. The nondoped organic light emitting diode device based on SPICN‐Cz exhibit better electroluminescent performances than those of SPICN‐TPA–based device: high external quantum efficiency of 2.58 %, current efficiency of 2.90 cd A^‐1, and power efficiency of 2.26 lm W^‐1 with Commission Internationale de l'Éclairage (CIE) coordinates of (0.15, 0.12). The excited state modulation and the composition of LE and CT states in the donor‐acceptor system could be useful to design low‐cost, high‐efficiency fluorescent organic light emitting diode materials.     

Photostability of D–π–A nonlinear optical chromophores containing a benzothiazolium acceptor

For the practical application of second‐order NLO materials, not only a high molecular quadratic hyperpolarizability β but also good thermal, chemical, and photochemical stabilities are required. Most of the state‐of‐the‐art chromophores with high NLO response cannot be put to use because they are photochemically highly unstable. Good thermal and photochemical stabilities with preserved high hyperpolarizabilities can be achieved by replacement of an aromatic ring with easily delocalizable heteroaromatics, e.g., with benzothiazole. Furthermore, desirable modifications of the benzothiazole fragment lead to improvement in β values. Here we report results of a comprehensive investigation of the photochemical stability of seven D–π–A push–pull molecules based on a N‐methylbenzothiazolium acceptor and a N,N‐dimethylaminophenyl donor with a different length of conjugated bridge and different acceptor strength. The quantum yield (Φ) and the kinetic parameters of photoreactions were determined for existing photodegradation pathways on irradiation at 300–850 nm in MeOH. Trans–cis photoisomerization is proposed as a fast but inefficient photobleaching mechanism for these irradiation wavelengths. Self‐sensitized photooxidation by ¹O₂ makes very slow parallel photodegradation pathway and, albeit to small value of Φ, plays a dominant role in the photodegradation of the compounds investigated. Both structural modifications (extension of conjugated bridge and an additional acceptor group bonded to heterocycle) resulting in an increase of NLO response led to a decrease in photostability due to the self‐sensitized ¹O₂ photooxidative attack. Thus a compromise should be found between an increase in NLO response and a decrease in photostability to make a choice of studied compounds for practical applications. Copyright © 2010 John Wiley & Sons, Ltd.