Carbon electronic structure

There was much affinity between Coulson and Barriol, not only because of the many subjects they shared, but also because of their similar way of proceeding and thinking. They both conceded a high value, in many respects, to the determination of dipole moments. Both worked on methane (CH4) and more particularly on the dipole moment of the C-H group, for which Coulson gave a direction when Barriol s simple model could not. [25] It is highly interesting to compare the way how the two authors express themselves to show that experience or physical and chemical evidence had to correct the false inferences or deductions that square in no way with reality the description of the carbon electronic structure fails to account for four equivalent bonds. We have to admit that the C-orbitals that are... 

To improve upon die mean-field picture of electronic structure, one must move beyond the singleconfiguration approximation. It is essential to do so to achieve higher accuracy, but it is also important to do so to achieve a conceptually correct view of the chemical electronic structure. Although the picture of configurations in which A electrons occupy A spin orbitals may be familiar and usefiil for systematizing the electronic states of atoms and molecules, these constructs are approximations to the true states of the system. They were introduced when the mean-field approximation was made, and neither orbitals nor configurations can be claimed to describe the proper eigenstates T, . It is thus inconsistent to insist that the carbon atom... 

In TT-complexes formed from aromatic compounds and halogens, the halogen is not bound to any single carbon atom but to the 7r-electron structure of the aromatic, though the precise geometry of the complexes is uncertain. The complexes with silver ions also do not have the silver associated with a particular carbon atom of the aromatic ring, as is shown by the structure of the complex from benzene and silver perchlorate. ... 

The electronic structure of a trimethine asymmetrical cyanine, controls the attack of a ketomethylene (Scheme 54). There is a condensation of the nucleophilic carbon on the electrophilic central carbon atom of the methine chain, leading to a neutrodimethine cyanine and simultaneously elimination of the more basic nucleus. 

Electron delocalization m allylic carbocations can be indicated using a dashed line to show the sharing of a pair of rr electrons by the three carbons The structural formula IS completed by placing a positive charge above the dashed line or by adding partial pos itive charges to the carbons at the end of the allylic system... 

Consider what happens if, for example, an ensemble of carbon atoms is subjected to X rays of 1486.6 eV energy (the usual X-ray source in commercial XPS instruments). A carbon atom has 6 electrons, two each in the Is, 2s, and 2p orbitals, usually written as C Is 2s 2p. The energy level diagram of Figure la represents this electronic structure. The photoelectron process for removing an electron from the... 

Structurally, carbon nanotubes of small diameter are examples of a onedimensional periodic structure along the nanotube axis. In single wall carbon nanotubes, confinement of the stnreture in the radial direction is provided by the monolayer thickness of the nanotube in the radial direction. Circumferentially, the periodic boundary condition applies to the enlarged unit cell that is formed in real space. The application of this periodic boundary condition to the graphene electronic states leads to the prediction of a remarkable electronic structure for carbon nanotubes of small diameter. We first present... 

Experimental measurements to test the remarkable theoretical predictions of the electronic structure of carbon nanotubes are difficult to carry out because... 

Abstract—The fundamental relations governing the geometry of carbon nanotubes are reviewed, and explicit examples are pre.sented. A framework is given for the symmetry properties of carbon nanotubes for both symmorphic and non-symmorphic tubules which have screw-axis symmetry. The implications of symmetry on the vibrational and electronic structure of ID carbon nanotube systems are considered. The corresponding properties of double-wall nanotubes and arrays of nanotubes are also discussed. 

Of particular importance to carbon nanotube physics are the many possible symmetries or geometries that can be realized on a cylindrical surface in carbon nanotubes without the introduction of strain. For ID systems on a cylindrical surface, translational symmetry with a screw axis could affect the electronic structure and related properties. The exotic electronic properties of ID carbon nanotubes are seen to arise predominately from intralayer interactions, rather than from interlayer interactions between multilayers within a single carbon nanotube or between two different nanotubes. Since the symmetry of a single nanotube is essential for understanding the basic physics of carbon nanotubes, most of this article focuses on the symmetry properties of single layer nanotubes, with a brief discussion also provided for two-layer nanotubes and an ordered array of similar nanotubes. 

Regarding the electronic structure, the number of energy bands for ( ,0) zigzag carbon nanotubes is In, the number of carbon atoms per unit cell, with symmetries... 

Inspired by experimental observations on bundles of carbon nanotubes, calculations of the electronic structure have also been carried out on arrays of (6,6) armchair nanotubes to determine the crystalline structure of the arrays, the relative orientation of adjacent nanotubes, and the optimal spacing between them. Figure 5 shows one tetragonal and two hexagonal arrays that were considered, with space group symmetries P42/mmc P6/mmni Dh,), and P6/mcc... 

Key Words —Carbon nanotube, electronic properties, structural properties, strain energy, band gap, band structure, electronic structure. 

Before we can analyze the electronic structure of a nanotube in terms of its helical symmetry, we need to find an appropriate helical operator S>(h,ip), representing a screw operation with a translation h units along the cylinder axis in conjunction with a rotation if radians about this axis. We also wish to find the operator S that requires the minimum unit cell size (i.e., the smallest set of carbon atoms needed to generate the entire nanotube using S) to minimize the computational complexity of calculating the electronic structure. We can find this helical operator by first... 

We will now discuss the electronic structure of single-shell carbon nanotubes in a progression of more sophisticated models. We shall begin with perhaps the simplest model for the electronic structure of the nanotubes a Hiickel model for a single graphite sheet with periodic boundary conditions analogous to those im-... 

The previous analysis of the electronic structure of the carbon nanotubes assumed that we could neglect curvature effects, treating the nanotube as a single... 

Experimental measurements to test these remarkable theoretical predictions of the electronic structure of carbon nanotubes are difficult to carry out because of the strong dependence of the predicted properties on tubule diameter and chirality. Ideally, electronic or optical measurements should be made on individual single-wall nanotubes that have been characterized with regard to diameter and chiral angle. Further ex-... 

The precise description of geometrical structures of CNTs has been reported by lijima [1], who was the first discoverer of carbon microtubules. Electron diffraction (ED) results are presented in Chap. 3. In this chapter, the authors will focus on the electronic structures of CNTs from the viewpoint of EELS by using TEM equipped with an energy-filter in the column or under the column. 

Electronic Structures of Single-Walled Carbon Nanotubes... 

Studies on the electronic structure of carbon nanotube (CNT) is of much importance toward its efficient utilisation in electronic devices. It is well known that the early prediction of its peculiar electronic structure [1-3] right after the lijima s observation of multi-walled CNT (MWCNT) [4] seems to have actually triggered the subsequent and explosive series of experimental researches of CNT. In that prediction, alternative appearance of metallic and semiconductive nature in CNT depending on the combination of diameter and pitch or, more specifically, chiral vector of CNT expressed by two kinds of non-negative integers (a, b) as described later (see Fig. 1). 

In this chapter the results of detailed research on the realistic electronic structure of single-walled CNT (SWCNT) are summarised with explicit consideration of carbon-carbon bond-alternation patterns accompanied by the metal-insulator transition inherent in low-dimensional materials including CNT. Moreover, recent selective topics of electronic structures of CNT are also described. Throughout this chapter the terminology "CNT stands for SWCNT unless specially noted. 

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