Cn Rings Research Articles

On Σ-skew reflexive-nilpotents-property for rings

In this paper, we study the reflexive-nilpotents-property (briefly, RNP) for skew PBW extensions. With this aim, we introduce the Σ-skew CN and Σ-skew reflexive (RNP) rings. Under conditions of compatibility, we investigate the transfer of there flexive-nilpotents-property from a ring of coefficients to a skew PBW extension. We also consider this property for localizations on these families of noncommutative rings. Our results extend those corresponding presented by Bhattacharjee [9].

Cyclo[n]carbons Form Strong N → C Dative/Covalent Bonds with Piperidine.

The newly synthesized C18 ring is demonstrated as the smallest all-carbon acceptor that exhibits strong electron acceptance. This study provides a quantum-chemical investigation of the electron-acceptance behavior of monocyclic carbon rings with a particular emphasis on C18 through the formation of a dative bond with piperidine. The results show that Cn rings form strong dative bonds with piperidine, whereas the respective van der Waals (vdW) complexes are higher in energy. The main driving force is the release of angle strain of cyclo[n]carbons caused by the change in hybridization from sp to sp2 associated with the formation of the dative bond. On the contrary, other sp allotropes, diynes, favorably form vdW complexes. Molecular dynamics (MD) simulations support the stability of the dative bond throughout a simulation of 20 ps. This opens up the possibility of stabilizing highly reactive C18 through dative/covalent functionalization.

Aromaticity of Even-Number Cyclo[n]carbons (n = 6-100).

The recently synthesized cyclo[18]carbon molecule has been characterized in a number of studies by calculating electronic, spectroscopic, and mechanical properties. However, cyclo[18]carbon is only one member of the class of cyclo[n]carbons—standalone carbon allotrope representatives. Many of the larger members of this class of molecules have not been thoroughly investigated. In this work, we calculate the magnetically induced current density of cyclo[n]carbons in order to elucidate how electron delocalization and aromatic properties change with the size of the molecular ring (n), where n is an even number between 6 and 100. We find that the Hückel rules for aromaticity (4k + 2) and antiaromaticity (4k) become degenerate for large Cn rings (n > 50), which can be understood as a transition from a delocalized electronic structure to a nonaromatic structure with localized current density fluxes in the triple bonds. Actually, the calculations suggest that cyclo[n]carbons with n > 50 are nonaromatic cyclic polyalkynes. The influence of the amount of nonlocal exchange and the asymptotic behavior of the exchange–correlation potential of the employed density functionals on the strength of the magnetically induced ring current and the aromatic character of the large cyclo[n]carbons is also discussed.

Weakly local commutativity for rings with unity

A ring R is called a left weakly local commutative ring (WLC, for short) if for any a ∈ N (R) and b ∈ R, (ab)2 = ba2b, which is a proper generalization of CN rings. In this paper, we show that (1) a ring R is commutative if and only if (xy)2 = yx 2 y for each x, y ∈ SN (R) = {a ∈ R|a ∉ N (R)} (2) R is a left WLC ring if and only if xyx = yx 2 for each x ∈ N (R) and y ∈ SN (R); (3) R is a reduced ring if and only if T 2 (R) is a left WLC ring; (4) R is a CN ring if and only if V 2(R) is a left WLC ring; (5) R is a commutative reduced ring if and only if V 3(R) is a left WLC ring.

CN rings in full protoplanetary disks around young stars as probes of disk structure

Aims. Bright ring-like structure emission of the CN molecule has been observed in protoplanetary disks. We investigate whether such structures are due to the morphology of the disk itself or if they are instead an intrinsic feature of CN emission. With the intention of using CN as a diagnostic, we also address to which physical and chemical parameters CN is most sensitive. Methods. A set of disk models were run for different stellar spectra, masses, and physical structures via the 2D thermochemical code DALI. An updated chemical network that accounts for the most relevant CN reactions was adopted. Results. Ring-shaped emission is found to be a common feature of all adopted models; the highest abundance is found in the upper outer regions of the disk, and the column density peaks at 30−100 AU for T Tauri stars with standard accretion rates. Higher mass disks generally show brighter CN. Higher UV fields, such as those appropriate for T Tauri stars with high accretion rates or for Herbig Ae stars or for higher disk flaring, generally result in brighter and larger rings. These trends are due to the main formation paths of CN, which all start with vibrationally excited H\hbox{$_2^*$} molecules, that are produced through far ultraviolet (FUV) pumping of H2. The model results compare well with observed disk-integrated CN fluxes and the observed location of the CN ring for the TW Hya disk. Conclusions. CN rings are produced naturally in protoplanetary disks and do not require a specific underlying disk structure such as a dust cavity or gap. The strong link between FUV flux and CN emission can provide critical information regarding the vertical structure of the disk and the distribution of dust grains which affects the UV penetration, and could help to break some degeneracies in the SED fitting. In contrast with C2H or c-C3H2, the CN flux is not very sensitive to carbon and oxygen depletion.

NEC rings

A ring R is called NEC if for any a, b ? N(R), ab = ba. The class of NEC rings is a proper generalization of the class of CN rings. First, with the aid of NEC rings, some characterizations of CN rings and reduced rings are given. Next we extend many properties of CN rings to NEC rings such as we show that NEC rings are directly finite and left min-abel; NEC regular ring are strongly regular; a ring R is NEC if and only if every Pierce stalk of R is NEC; Also we discuss some properties of NEC exchange rings; Finally, we give some properities of MP-invertible elements.

Tunable electronic structure and spin splitting in single and multiple Fe-adsorbed g-C2N with different layers: A first-principles study

The electronic structure of Fe adsorbed g-C2N with different layers is investigated by first-principles calculations. The Fe1 and Fe2 represent the Fe adsorptions at CC and CN rings, and Fe11 and Fe121 adsorption sites are also considered. The Fe1 adsorbed g-C2N is metallic with layer from n = 1 to 4, and the maximum spin splitting is 515, 428, 46 and 133 meV. The band gap of Fe2 adsorbed g-C2N with different layers is 0, 0, 117 and 6 meV, and the maximum spin splitting is 565, 369, 195 and 146 meV, respectively. All of the Fe11 adsorbed g-C2N are metallic with layer from n = 1 to 4, and the maximum spin splitting is 199, 0, 83 and 203 meV. An indirect band gap of 215 meV appears in Fe121 adsorbed g-C2N at layer n = 3, and the maximum spin splitting is 283, 211, 304 and 153 meV, respectively. Our results show that the electronic structures of Fe adsorbed novel two-dimensional semiconductor g-C2N can be tuned by different layers. Moreover, the spin splitting of Fe2 adsorbed g-C2N decreases monotonically as g-C2N layer increases from n = 1 to 4, which will provide more potential applications in spintronic devices.

Tunable electronic structure of monolayer semiconductor g-C2N by adsorbing transition metals: A first-principles study

The electronic structure of transition metal (TM) adsorbed two-dimensional g-C2N is investigated by first-principles calculations. Most of TM-adsorbed g-C2N show metallic properties, but Ni and Zn adsorptions show a band gap of 0.6 and 1.6 eV. The maximum spin splitting of TM-adsorbed g-C2N from Sc to Zn is 110, 186, 117, 100, 0, 515, 147, 64, 95 and 0 meV, respectively. The magnetic moments of adsorbed structure from Sc to Co are in the range from 0.15 to 2.11 μB, while in Ni, Cu and Zn adsorbed cases no magnetic moments appear. In Fe-adsorbed g-C2N, the adsorption structure at site of Fe12 generates a 90 meV band gap, where the adsorption sites at CC and CN rings are marked as Fe1 and Fe2. The maximum spin splitting is 515, 412 and 750 meV as Fe atomic concentration is 5.56%, 11.11% and 16.67%. Our results can bring more significant basis on the design of spintronic and valleytronic devices.

Some notes on JTTC rings

A ring R is called JTTC if for any a∈N(R) and b∈R, (ab)2=ab2a, which is a proper generalization of CN rings. In this paper, we show that (1) a ring R is commutative if and only if (xy)2=xy2x for each x∈SN(R) and y∈SZr(R); (2) R is a JTTC ring if and only if xyx=x2y for each x∈N(R) and y∈SZr(R); (3) R is a reduced ring if and only if T3(R) is a JTTC ring; (4) R is a CN ring if and only if V2(R) is a JTTC ring; (5) R is a commutative reduced ring if and only if TV4(R) is a JTTC ring; (6) R is a commutative ring if and only if G3(R) is a JTTC ring; (7) If R is a JTTC ring containing a von Neumann regular maximal left ideal, then R is a strongly regular ring.

ON NILPOTENT POWER SERIES WITH NILPOTENT COEFFICIENTS

Antoine studied conditions which are connected to the question of Amitsur of whether or not a polynomial ring over a nil ring is nil, introducing the notion of nil-Armendariz rings. Hizem extended the nil-Armendariz property for polynomial rings onto power-series rings, say nil power-serieswise rings. In this paper, we introduce the notion of power-serieswise CN rings that is a generalization of nil power-serieswise Armendariz rings. Finally, we study the nil-Armendariz property for Ore extensions and skew power series rings.

Ground and Low-Lying Excited Electronic States of [3,3′] Bidiazirinylidene (C2N4)

The second-order generalized Van Vleck perturbation theory (GVVPT2) variant of multireference perturbation theory was used to investigate the ground and low-lying excited electronic states of [3,3'] bidiazirinylidene (C(2)N(4)). C(2)N(4) is arguably the simplest molecule with multiple CN(2) rings. Equilibrium geometries and adiabatic energy differences of the 1(1)A(g) ground state and 2 (1)A(g), 1(1)B(1g), 1(3)B(1g), 1(1)B(2g), 1(3)B(2g), 1(1)B(3g), 1(3)B(3g), 1(1)B(2u), 1(3)B(2u), 1(1)B(3u), and 1(3)B(3u) excited states of D(2h) symmetry were obtained. In addition, vertical excitation energies to and emission energies from the excited states were obtained. The 1(1)A(g) optimized geometry and harmonic frequencies for the CN(2) rings in C(2)N(4) are similar to those for the related difluorodiazirine molecule: R(CN), R(NN) bonds and the angleNCN angle are 0.012 and 0.006 A longer and 0.8 degrees smaller than that for the ring in F(2)CN(2). The NN stretch frequencies are 1470 and 1562 cm(-1), and CN stretches are 993 and 995 cm(-1). The similarity between the GVVPT2 results of C(2)N(4) and F(2)CN(2) suggests that C(2)N(4) might be sufficiently stable to be synthesized. The lowest-energy spin- and electric dipole-allowed vertical excitation, to the 1(1)B(2u) state, is predicted to occur in the visible (2.30 eV, 538.9 nm).

Characteristics of green thermo-electric tapes

The electronic structure of transition metal (TM) adsorbed two-dimensional g-C2N is investigated by first-principles calculations. Most of TM-adsorbed g-C2N show metallic properties, but Ni and Zn adsorptions show a band gap of 0.6 and 1.6 eV. The maximum spin splitting of TM-adsorbed g-C2N from Sc to Zn is 110, 186, 117, 100, 0, 515, 147, 64, 95 and 0 meV, respectively. The magnetic moments of adsorbed structure from Sc to Co are in the range from 0.15 to 2.11 μB, while in Ni, Cu and Zn adsorbed cases no magnetic moments appear. In Fe-adsorbed g-C2N, the adsorption structure at site of Fe12 generates a 90 meV band gap, where the adsorption sites at CC and CN rings are marked as Fe1 and Fe2. The maximum spin splitting is 515, 412 and 750 meV as Fe atomic concentration is 5.56%, 11.11% and 16.67%. Our results can bring more significant basis on the design of spintronic and valleytronic devices.

From small to large behavior: The transition from the aromatic to the Peierls regime in carbon rings

Results of local density approximation (LDA) and Hartree–Fock (HF) calculations for even numbered monocyclic rings are reported. Small Cn rings satisfying n=4N+2 show aromatic stability with equal bond-length structures, whereas rings of size n=4N show antiaromatic destabilization with bond-length alternation. For large rings a transition, the Peierls transition, from aromatic and antiaromatic to nonaromatic behavior, takes place. Above the Peierls transition, both n=4N and n=4N+2 rings show bond-length alternation and no differences in stability. The critical size for the transition to nonaromatic behavior depends on the electron-phonon coupling strength and therefore depends on the choice of ab initio method. HF predicts nonaromatic behavior for ring sizes n=14 and above. Fully optimized LDA results are presented up to n=42, which still has a cumulenic structure. Calculations based on periodic infinite ring systems show that within LDA the onset of nonaromatic behavior does not occur until n=82. Experimental results suggest that aromatic behavior exists in these ring systems to at least n=22. The force constant for in-plane angle bending may also be estimated from these calculations and was found to be 0.022 kcal mol−1 deg−2 per atom, not a strong function of size beyond n=22, and in good agreement with experimental estimates.

Valence one-electron and shake-up ionization bands of carbon clusters. II. The Cn (n=4,6,8,10) rings

The 1h (one-hole) and 2h-1p (two-hole; one-particle) shake-up bands in the valence ionization spectrum of small carbon rings (C4, C6, C8, C10) are investigated up to 40 eV, using the one-particle Green’s Function approach. Calculations have been performed using both the third-order algebraic diagrammatic construction [ADC(3)] scheme and the outer-valence Green’s function (OVGF) approximation. The obtained ADC(3) results indicate a major fragmentation of lines into complex sets of 2h-1p satellites, even for outer-levels of Π-character. The simulated ionization spectra provide striking structural signatures for the carbon rings vs. the carbon chains, that could be usefully exploited to discriminate these two types of clusters in plasma conditions. In general and in spite of the extent of the shake-up spreading, one can rather easily trace from our convolutions the energy degeneracies, and in the outer-valence region, the Σ–Π near-energy degeneracies which characterize the electronic structure of these doubly conjugated rings. In relationship to its doubly antiaromatic (i.e., polyynic) nature and lower symmetry, C8 is subject to stronger many-body effects than the other rings. ADC(3) calculations on first-order saddle point forms of C6 and C10 indicate that thermal averaging of conformations could have a substantial effect on the spectral bands, in particular the shake-up ones.

Some Boron-Containing Ring Systems

The synthesis and characterisation of series of new types of boron-containing ring systems are described. They include: (a) carborane systems containing C-H--X (X = O or N) hydrogen bonds; (b) systems in which Cn rings share C-C links with o-carborane units; (c) macrocycles containing 2,3 or 4 carborane icosahedra linked through benzene or pyridine rings, and (d) ‘new types of ‘carborazacycles’ containing one carbon, two boron and three nitrogen atoms in a single 6-membered ring system.