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Understanding the crystal-packing patterns of tetrapyrroles underpins a number of objectives in materials sciences. Here, the zinc chelate of 3,13-diacetyl-10-(pentafluorophenyl)-18,18-dimethylchlorin was prepared, crystallized in the presence of tetrahydrofuran (THF), and analyzed by single-crystal x-ray diffraction (SCXRD). The zinc chlorin (Zn1) afforded a herringbone pattern of parallel columnar stacks of chlorins rotated by 67.41${}^{\circ }$ with respect to each other. The chlorins within a column are stacked in pairs having interplanar distances of 3.289 Å within a pair and 4.407 Å between pairs. The chlorins in a pair are aligned in a “head-to-head” arrangement with cofacial $\pi$–$\pi$ overlap chiefly only of the respective ring A units; the overall composition is thus of columnar stacks of slipped-stacked pairs of chlorins. Each chlorin is enantiomeric due to the coordination at the Zn(II) center of a single THF molecule; the juxtaposition of chlorin $\alpha$–$\beta$ faces opposite the apically ligated THF molecules affords a racemate. All chlorins in a column face in the same direction, with 3,13-acetyl groups essentially coplanar with the macrocycle and oriented toward the respective 5,15-positions (affording an O$\cdot \cdot \cdot$H–C contact of 2.33 or 2.39 Å), and with interdigitation of alternating pentafluorophenyl groups in adjacent columns. A large number of SCXRD analyses of synthetic chlorins will likely be required to understand the relationship between structure and assembly patterns.

We study the effect of hotter or colder particles on the evolution of the chiral magnetic field in the early universe. We are interested in the temperature-dependent term in the chiral vortical effect (CVE). There are no changes in the magnetic energy spectrum at large length scales but in the Kolmogorov regime we do find a difference. Our numerical results show that the Gaussian peak in the magnetic spectrum becomes negatively skewed. The negatively skewed peak can be fitted with a beta distribution. Analytically, one can relate the non-Gaussianity of the distribution to the temperature-dependent vorticity term. The vorticity term is therefore responsible for the beta distribution in the magnetic spectrum. Since the beta distribution has already been used to model turbulent dispersion in fluids, hence it seems that the presence of hotter or colder particles may lead to turbulence in the magnetized plasma.

Within the effective quark model with chiral U(3) × U(3) symmetry we calculate the S-wave and P-wave amplitudes of the nonleptonic decay Σ⁺ → pπ⁰, the partial width and the dynamical polarization of the proton in the dependence on the polarization of the Σ⁺-hyperon. The theoretical results agree well with the experimental data.

We bring to light an electroweak model which has been reappearing in the literature under various guises.${}^{1-5}$ In this model, weak isospin is shown to act automatically on states of only a single chirality (left). This is achieved by building the model exclusively from the raising and lowering operators of the Clifford algebra $ℂl(4)$. That is, states constructed from these ladder operators mimic the behaviour of left- and right-handed electrons and neutrinos under unitary ladder operator symmetry. This ladder operator symmetry is found to be generated uniquely by $su{(2)}_L$ and $u{(1)}_Y$. Crucially, the model demonstrates how parity can be maximally violated, without the usual step of introducing extra gauge and extra Higgs bosons, or ad hoc projectors.

The method to obtain phonon dispersion of achiral single-wall carbon nanotubes (SWNTs) from 6×6 matrix proposed by Mahan and Jeon⁷ has been extended to chiral SWNTs. The number of calculated phonon modes of a chiral SWNT (10, 1) is much larger than that of a zigzag one (10, 0) because the number of atoms in the translational unit cell of chiral SWNT is larger than that of an achiral one even though they have relative similar radius. The possible application of our approach to other models with more phonon potential terms beyond Mahan and Jeon's model is discussed.

In this study, a dynamic chiral metamaterial (MTM) structure leading to an asymmetric electromagnetic (EM) wave transmission for linear polarization is presented. The structure is composed of square-shaped resonator with gaps on both sides of a dielectric substrate with a certain degree of rotation. The dynamic structure is adjustable via various parameters to fit any desired frequency ranges. Theoretical and experimental analysis of the proposed structure are conducted and given in detail. The suggested model can provide constant chirality over a certain frequency band and thus, it can be used to design myriad novel devices such as EM transmission and antireflection filters, and polarization rotators for desired frequency regimes.

Chiral metamaterial (MTM) researchers generally concentrate and aim to obtain large chirality with optical activity in certain frequencies. However, new generation planar chiral MTM which have small and constant/flat chirality over a certain frequency band have not queried by this time in literature. In fact, this area is mostly ignored by researchers. This study, first one according to best of our knowledge in the literature, is investigating the small and constant/fixed chirality and focuses on the new generation planar chiral MTM based on circular split ring resonators (SRRs), in details. It can be seen from the results that the proposed model can provide small and constant/flat chirality over a certain frequency band and thus it can be used to design myriad novel devices such as polarization rotators, and electromagnetic transmission and antireflection filters for several frequency regions.

There are many studies in literature on chiral metamaterials (MTMs) to obtain large chiralities with dynamic optical activities. With this regard, this new generation planar chiral MTM study focuses on a small, non-dispersive (constant/flat) chirality admittance over an indicated frequency band which has not been investigated so far in literature. This new generation planar chiral MTM provides a small and a constant/fixed chirality which is mostly ignored by the scientists. This study numerically and experimentally investigates and examines these new generation MTMs based on circular split ring resonators (SRRs) with an increased capacitance in details. Obtained results show that the suggested structure can provide a small and constant/flat chirality admittance over a certain frequency band and hence it can be used to design myriad novel electromagnetic (EM) devices such as transmission and antireflection filters, polarization rotators for any desired frequency regions.

Chiral metamaterial-based sensor is designed for sensing the change in substance ratio of chemical substances when combined with distilled water. These samples have been prepared with 10%, 50% and 90% volume fraction of methanol, ethanol, acetone, ammonia and isopropyl alcohol. The complex permittivity of the prepared samples has been measured by Agilent 85070E dielectric probe kit and compared with the simulations. In the proposed design, linear shifts with the frequency bands of more than 100 MHz are observed in different measurements. For different material sensing, pure samples have been used and their reflection coefficient measurement results have been presented. Unique side of this study is that the structure provides very clear and sensitive results and presents a new approach to the microfluidic sensor applications by using the sample holder.

Three disjoint rays in ℝ³ form Borromean rays provided their union is knotted, but the union of any two components is unknotted. We construct infinitely many Borromean rays, uncountably many of which are pairwise inequivalent. We obtain uncountably many Borromean hyperplanes.

A developed effective chiral Lagrangian approach is used to study the cold dense nuclear matter. Improved calculation of baryon number density as a function of chemical potential is given. The resonance appeared in a natural way as chemical potential is getting larger. Calculations of susceptibility and pressure including the resonance contribution are also given.

We report DFT calculations on potential intramolecular, enantioselective hydrogenation catalysts based around borenium-carbenes based on a camphor scaffold. Using the M06-2X meta-hybrid functional, we find frustrated Lewis pair (FLP) behavior with suitably chosen linkers that prevent association of Lewis bases with the borenium center. These intramolecular FLPs are predicted to be able to heterolytically dissociate H₂. Barriers to dissociation and the endo/exoergic nature of the reaction can be tuned by the nature of the base and substituent on B. The reactivity of the hydrogenated FLP catalyst with olefin and carbonyl substrates is then explored: we predict concerted reactions for all substrates considered with relatively low barriers and large exoergic character. Hydrogenation of both faces of a prochiral substrate is also examined, indicating a small but significant variation in reaction barrier in favor of the Si-face, ascribed to stronger interactions with the aromatic $\pi$-system in the TS compared to the Re-face.

As a historical remark the similarities are pointed out between a form of chiral action introduced by Schwinger and the formalism used in the noncommutative extensions of electromagnetism.

The present paper focuses on a certain class of Banach manifolds we call Rogers supermanifolds since they are indeed supermanifolds modeled on graded Banach spaces. Although the subject of holonomy is well-developed for superanalytic supermanifolds utilizing local ring formulations of supermanifolds this seems not to be the case for supermanifolds modeled on graded Banach manifolds in the sense of Rogers. The proof of our main result requires a partial development of these concepts for such supermanifolds. Our main result determines conditions under which a super connection on a superprincipal bundle induces a connection on a quotient superprincipal bundle where is a foliation of and is the induced foliation on . We also show how such a quotient formulation may be used to describe in a fully geometric fashion the so-called "conventional constraints" of super Yang–Mills theory. One consequence of our development is that instead of requiring two superconnections to describe Yang–Mills theory as is the case in some formulations, we describe the relevant concepts using a single superconnection and moreover we show that the "pregauge transformations" are simply ordinary gauge transformations on the appropriate quotient bundles.

The tin(IV) porphyrin complex with chiral amino acid ligands trans-bis(L-prolinato)[5,10,15,20-tetrakis(4-pyridyl)porphyrinato]tin(IV) (1) was synthesized and fully characterized. The X-ray crystal structural analysis of 1 reveals that the Sn(IV) center is octahedrally coordinated by the porphyrin occupying the square base and axial coordination of two L-prolinate ligands in trans-position with respect to each other. The pyrrolidine rings of the L-prolinate ligands in 1 exhibit a C_s-C^β-exo envelope conformation with C_s symmetry. The non-classical hydrogen bonding interactions mainly contribute to assemble 1 into a one-dimensional packed structure.

5-(8-ethoxycarbonyl-1-naphthyl)-10,15,20-triphenyl porphyrin (ENTPP) has been synthesized in a one-pot reaction, and the corresponding chiral crystalline samples have been obtained by spontaneous resolution. ¹H NMR spectrum suggests it is mononaphthyl substituted species and an ethyl group is over the porphyrin plane. The structure has been further confirmed by X-ray crystallography. ENTPP·C₆H₁₄ (C₅₇H₅₀N₄O₂): monoclinic, P2₁, a = 10.707(2) Å, b = 12.203(2) Å, c = 17.858(4) Å, β = 103.06(3)°, V = 2272.8(8) ų, Z = 2. The 8-position substituent, ester group, lies above the porphyrin plane and leads to the conformational chirality. The entire structure is built up with homochiral molecules, which leads to a chiral crystal through packing in P2₁ space group. Circular dichroism (CD) spectra have exhibited remarkable absorptions in the Soret band region, which further confirms the homochirality of the crystalline samples.

A novel bis-palladium(II) complex of doubly N-confused octaphyrin (Pd${}_2$-2) adopting a Möbius-twisted topological structure was synthesized and characterized. Due to the effective 36$\pi$-electronic delocalization over the Möbius-twisted octaphyrin scaffold, characteristic Soret and Q-like absorption features were observed in the near-infrared (NIR) region. However, the complex Pd${}_2$-2 exhibited weak aromatic character attributed to the distinct cross-conjugated resonance contribution as inferred from the ¹H NMR chemical shifts as well as theoretical assessments ($e.g.$ nucleus-independent chemical shift (NICS)). Since the bis-palladium(II) complexation of doubly N-confused octaphyrin 2 imparted significant conformational stability, topologically chiral enantiomers of Pd${}_2$-2 were successfully separated as ($P)$- and ($M)$-twisted forms. The resulting isomers revealed relatively large circular dichroism (CD) responses with an absorption anisotropy factor of $g_{abs}$$=$ 0.009 in the NIR region ($\lambda =$ 823 nm). In addition, the cyclic voltammogram of Pd${}_2$-2 revealed redox-rich properties due to its large $\pi$-conjugated system.

We investigated chirality transfer processes with two amide-linked zinc bisporphyrinates as hosts and chiral amino acid esters as guests. The linkers in these hosts contain a coordination site (pyridyl nitrogen or amino nitrogen). CD spectra were measured after titration of these zinc bisporphyrinates with amino acid esters. The CD spectra show that the signals were inverted during the titration. This result suggests that there is a two-step chirality induction process, which is most likely dominated by the corresponding 1:1 and 1:2 host-guest complexes. In the 1:1 complexes, the pyridyl nitrogen or amino nitrogen in the linkers is coordinated with zinc. NMR spectra confirmed such coordination interactions. Theoretical calculations also confirmed the corresponding chirality induction and inversion. This work provides a useful strategy to tune chirality transfer processes by introducing an extra coordination site in the linker.

A chiral bisporphyrin containing hydroxy groups in the linker H₄(L-TABis) was designed and synthesized. The CD spectra show that the intensity of the CD signal increases approximately 12-fold upon metalation with Zn${}^{2+}$ (up to 1450 cm${}^{-1}$M${}^{-1})$. Our investigation suggests that the Zn-OH coordination interactions lead to enhanced CD in [Zn₂(L-TABis)]. The DFT-optimized structure of [Zn₂(L-TABis)] indicates that the chirality of L-tartaric acid is transferred to the bisporphyrin to form an M configuration with a shorter Zn$\cdots$Zn distance and a negative C15-C5-C5^′-C15^′ torsional angle, which is consistent with the experimental results. This work presents a useful example of enhancing CD by adjusting the interactions between the substituents in the linker and the center metal ions.

The chirality (of Greek χείρ, ch[e]ir: hand) is a fundamental symmetry property of three-dimensional objects. Your two hands are not superimposable. They are mirror images of each other. They are therefore chiral (Figure 10.1). The chirality of your right hand can be seen by trying to put it in a glove for your left hand. The same can be seen by trying to put your left hand into a glove for your right hand. The same experience can be done while putting on shoes. Normally you are not comfortable after exchanging the shoe for the left foot with the one for the right foot…