Tight-Binding Assumptions

The indices (n, 0) or (0, m), d = 0°, correspond to the zigzag tube (so-called because of the AA/ shape around the circumference, perpendicular to the tube axis). The indices (n, m) with n = m (6 = 30°) corresponds to the armchair tube (with a / / shape around the circumference, perpendicular to the tube axis). If one of the two indices n or m is zero, the tube is nonchiral it is superimposable on its mirror image. A general chiral nanotube (nonsuperimposable on its mirror image) occurs for all other arbitrary angles. Common nanotubes are the armchair (5, 5) and the zigzag (9, 0). 

These factors, in turn, are dependent on the diameter and helicity. It has been found that metallicity occurs whenever (2n + m) or (2 + 2m) is an integer multiple of three. Hence, the armchair nanotube is metallic. Metallicity can only be exactly reached in the armchair nanotube. The zigzag nanombes can be semimetallic or semiconducting with a narrow band gap that is approximately inversely proportional to the tube radius, typically between 0.5 -1.0 eV. As the diameter of the nanombe increases, the band gap tends to zero, as in graphene. It should be pointed out that, theoretically, if sufficiently short nanotubes electrons are predicted to be confined to a discrete set of energy levels along all three orthogonal directions. Such nanotubes could be classified as zero-dimensional quantum dots. 

The original LCAO method by Bloch (Bloch, 1928) is difficult to carry out with full rigor because of the large number of complicated integrals that must be computed. In their 

Of course, in an actual tight-binding calculation, all such integrals are evaluated only at the high symmetry points in the BZ. Fitted parameters are then used to interpolate the band structure between these points. 

---

In this chapter we consider the situation where this assumption is no longer valid, because the affinity of the inhibitor for its target enzyme is so great that the value of K w approaches the total concentration of enzyme ( / T) in the assay system. This situation is referred to as tight binding inhibition, and it presents some unique challenges for quantitative assessment of inhibitor potency and for correct assessment of inhibitor SAR. 

To establish a quantitative relation between F and G for the entire tip and the entire sample, we have to consider all the states in the tip and the sample. A rigorous treatment is complicated. The following treatment is based on the approximate additivity of atomic force and tunneling conductance with respect to the atoms of the tip. In other words, the force between the entire tip and the sample can be approximated as the sum of the force between the individual atoms in the tip and the entire sample, so does tunneling conductivity. Because the tip is made of transition metals, for example, W, Pt, and Ir, the tight-binding approximation, and consequently, additivity, are reasonable assumptions. Under this approximation, the total force is... 

All three assumptions can be violated in the case of CYP enzymes, depending on the design of the in vitro CYP inhibition study. The first assumption can be potentially violated if the drug being tested is a time-dependent inhibitor (e.g., one with a slow on rate see below). The potency of some inhibitors (e.g., the CYP3A inhibitors ketoconazole and clotrimazole) is such that the free concentration of the inhibitor tends to approach the concentration of the enzyme (40), a violation of the second assumption. In the case of such tight-binding inhibition, an apparent A) value (A i a ) )) can be estimated, as follows ... 

The naive structural assumptions of crystalline models have been eliminated with the development of methods to calculate the electronic states of realistic liquid structures. An early approach of this type was the multiband tight-binding model that Yonezawa and Martino (1976) applied to an assumed hard-core liquid structure. Later, it was recognized that solution of the many-body statistical mechanics to calculate the liquid structure automatically solves the corresponding quantum mechanical problem required for the electronic states (Logan and Winn, 1988 Xu... 

From this simplified but still realistic model, we are led to two conclusions. First, band bounds are smeared out in general, and therefore we must determine n(E) and the features of the wave functions explicitly for each class of materials or models. Second, in real materials a fairly large number of matrix elements is required for a tight-binding model which accurately represents the electronic structure. The various kinds of disorder present affect these matrix elements and the states near the different band edges in quite different ways. These two circumstances lead to the approximate validity of two simplifying assumptions which we shall explore in the following ... 

We now derive the expressions for the two first moments Mi/Mo and My Mo of the local density of states (LDOS), normalized to one atom and one spin direction, for an oxide M Om, in a tight-binding approach. For this purpose, we will not need the two basic assumptions of the alternating lattice model. We will note AA) and Cp) the anion and cation atomic orbitals, and e x and ec/i their energies. The basis set is assumed to be orthonormal. The first moment Mi of an anion LDOS reads ... 

---