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An ab initio study of the energetic, structural, electronic, and optical absorption properties of the 26 lead nanowires, Pb n ͑n =1,18͒ having different m-gonal ͑m =1-8͒ cross sections has been made in the density functional theory in local density approximation considering also the spin-orbit coupling ͑SOI͒. There are four groups of the stable structures: planar, caged, pyramidal, and helical. The binding energy of a nanowire, in general, increases with the coordination number except in those systems where the nearest neighbors are comparatively far away. A 14-Pb hexagonal helical configuration has maximum stability followed by the heptagonal, other hexagonal, and pentagonal wires. All the nanowires are metallic. The exceptions are the 2-Pb and 3-Pb semiconducting nanowires. A large number of the conduction channels leading to high quantum ballistic conduction are seen for a number of the m-gonal ͑m =4-8͒ configuration wires. The calculated optical absorption without and with the SOI are quite different in terms of the number of the absorption peaks which are enhanced approximately by a multiplying factor of 2 by the SOI. The m-gonal ͑m =4-8͒ nanowires reveal multipeaked, strong, and extended optical absorption over the whole visible region. Our analysis of the experimental data for the Pb samples that have been fabricated by Romanov points towards the occurrence of the 2-Pb ladder chains.

The energetics, structural, electronic and optical absorption properties of the bismuth nanowires Bin with n = 1, 6 have been investigated using density functional theory (DFT) in the local density approximation (LDA) including the spin–orbit coupling (SOI). The inclusion of the SOI appreciably affects all the physical properties of the wires. The stable structures form four groups: the planar structures, the caged configurations, the pyramidal structures and the helical configurations. This finding may be a guide for the construction of atomic configurations of the nanowires possessing a larger number of atoms per unit cell. The most stable wire configurations are the 5-Bi pentagonal, and the 6-Bi hexagonal and 6-Bi triple zigzag wires, which should be seen in the experiments. All the wires are metallic. The behaviour of the electron states of the second category structures is quite near to that of a linear chain where the parabolic bands cross the EF, and the number of the channels available for the electric conduction is large. Thus, one should grow the wire structures falling into the second category for achieving high conduction. For the 5-Bi pentagonal and 6-Bi hexagonal cross-sectional wires, the number of channels available for the electric conduction are ten and twelve, respectively. The SOI drastically affects the calculated optical absorption, especially in the low energy region. The absorption peaks are different in terms of the number and the energy locations for the different wires, and may be used for the characterization of the structure of a wire. Our analysis of the calculated electronic structure and the optical data of all the studied structures supports the occurrence of the 4-Bi double and/or 6-Bi triple zigzag chains in the samples of Romanov.

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A comprehensive ab initio investigation of the effects of the relative orientation (RO) between the adjacent tubes in a rope on the stability, structural, electronic, optical and Raman-active properties has been performed for the ropes of small diameter carbon (6, 6) nanotubes. A number of new features not discussed earlier are observed in the present study. The symmetric rope with an RO of 0° is metallic in all directions, whereas the asymmetric ropes with a non-zero value of RO are semiconductors along the tube axis but semi-metallic normal to rope axis. The band gap increases with RO up to an angle of 15° and thereafter reveals oscillatory behaviour. No dips appear in the symmetric rope but they do exist in the asymmetric rope. Strong optical absorption appears along the axis in the energy range 2.4–4.2 eV in the isolated tube. On the other hand, for the ropes, the strong absorption extends up to the energy region 1.8–4.5 eV. Strong peaks also occur at 0.05 and 0.15 eV for the ropes with RO = 0° and 15°, respectively. The even-parity Raman-active radial breathing mode (RBM) frequencies calculated here for the isolated (n,n), n = 3–6 tubes are seen to deviate from the usual law (where d is the tube diameter). For small diameter tubes, this shows an approximate variation, ω = 1/d1/2. The RBM frequencies for the ropes are either greater or smaller compared to the isolated tube, depending on the value of RO. A cubic anharmonicity of about 14% is seen in the potential for the radial mode vibrations. The RBM frequencies calculated here for some ropes, which are lower compared to that of the isolated tube, concur with the available Raman data.

In this article, by making use of the q-Srivastava-Attiya operator, we introduce and investigate a new family SWΣ(δ,γ,λ,s,t,q,r) of normalized holomorphic and bi-univalent functions in the open unit disk U, which are associated with the Bazilevič functions and the λ-pseudo-starlike functions as well as the Horadam polynomials. We estimate the second and the third coefficients in the Taylor-Maclaurin expansions of functions belonging to the holomorphic and bi-univalent function class, which we introduce here. Furthermore, we establish the Fekete-Szegö inequality for functions in the family SWΣ(δ,γ,λ,s,t,q,r). Relevant connections of some of the special cases of the main results with those in several earlier works are also pointed out. Our usage here of the basic or quantum (or q-) extension of the familiar Hurwitz-Lerch zeta function Φ(z,s,a) is justified by the fact that several members of this family of zeta functions possess properties with local or non-local symmetries. Our study of the applications of such quantum (or q-) extensions in this paper is also motivated by the symmetric nature of quantum calculus itself.