Hole size, definition

If the notion that 15-crown-5 is selective for Na" " is correct, then Ks for the interaction 15-crown-5 Na+ should be greater than the interaction 15-crown-5 K+. We see from Table I that log Ks for the sodium interaction is 3.24 (1,740) compared to 3.43 (2,690) for potassium. An alternate definition of Na+-selectivity might be that 15-crown-5 binds Na" " more strongly than 18-crown-6 binds Na" ". The log Ks values for this pair of interactions is, respectively, 3.24 (1,740) vs. 4.35 (22,400). Clearly, 15-crown-5 is not selective for Na" " and the simple "hole-size rule does not correctly account for the results. 

The hole-size relationship between cations and crown ethers has been a part of the lore in the cation binding area for nearly two decades. Although, to our knowledge, no formal definition of this principle has ever been offered, the general concept seems to be that cation binding will be optimized when the cation diameter and macrocycle cavity size are identical. A simple consequence of this concept is the notion that 15-crown-5 is selective (binds more strongly) for Na+ over K+. We have measured the homogeneous (equilibrium) stability constants for the reaction... 

This chapter deals with certain nonhelicenic (geometrically planar) polyhexes, which may be simply connected (without holes) or multiply connected with a varying number of corona holes. It is recalled that a corona hole by definition should have a size of at least two hexagons. 

We may begin by describing any porous medium as a solid matter containing many holes or pores, which collectively constitute an array of tortuous passages. Refer to Figure 1 for an example. The number of holes or pores is sufficiently great that a volume average is needed to estimate pertinent properties. Pores that occupy a definite fraction of the bulk volume constitute a complex network of voids. The maimer in which holes or pores are embedded, the extent of their interconnection, and their location, size and shape characterize the porous medium. 

The detection of sharp plasmon absorption signifies the onset of metallic character. This phenomenon occurs in the presence of a conduction band intersected by the Fermi level, which enables electron-hole pairs of all energies, no matter how small, to be excited. A metal, of course, conducts current electrically and its resistivity has a positive temperature coefficient. On the basis of these definitions, aqueous 5-10 nm colloidal silver particles, in the millimolar concentration range, can be considered to be metallic. Smaller particles in the 100-A > D > 20-A size domain, which exhibit absorption spectra blue-shifted from the plasmon band (Fig. 80), have been suggested to be quasi-metallic [513] these particles are size-quantized [8-11]. Still smaller particles, having distinct absorption bands in the ultraviolet region, are non-metallic silver clusters. 

The map can be finite or infinite and some holes can be i-gons with i e R. If R = r, then the above definition corresponds to (r, q)gen-polycycles. If an (R, q)gen-polycycle is simply connected, then we call it an (R, )-polycycle those polycycles can be drawn on the plane with the holes being exterior faces. (R, )-polycycles with R = r are exactly the (r, interior faces is that polycyclic hydrocarbons in Chemistry have a molecular formula, which can modeled on such polycycles, see Figure 7.1. The definition of (R, < )-polycycles given here is combinatorial we no longer have the cell-homomorphism into r, q). We will define later on elliptic,... 

The interest in semiconductor QD s as NLO materials has resulted from the recent theoretical predictions of strong optical nonlinearities for materials having three dimensional quantum confinement (QC) of electrons (e) and holes (h) (2,29,20). QC whether in one, two or three dimensions increases the stability of the exciton compared to the bulk semiconductor and as a result, the exciton resonances remain well resolved at room temperature. The physics framework in which the optical nonlinearities of QD s are couched involves the third order term of the electrical susceptibility (called X )) for semiconductor nanocrystallites (these particles will be referred to as nanocrystallites because of the perfect uniformity in size and shape that distinguishes them from other clusters where these characteriestics may vary, but these crystallites are definitely of molecular size and character and a cluster description is the most appropriate) exhibiting QC in all three dimensions. (Second order nonlinearites are not considered here since they are generally small in the systems under consideration.)... 

The hole model for molecular liquids was elaborated by Furth [12], who supposed that the free volume of a liquid is not distributed uniformly between its molecules like in crystals, but is concentrated like some holes which can disappear in one place and appear in another place. These holes are in permanent motion, so that the situation is different from the jumps of the holes in a crystal. The appearance and disappearance of the holes in a liquid are a result of the fluctuations connected with thermal movements. These holes in liquids have no definite shape and size they can increase or decrease spontaneously. Furth [12] tried to calculate a large number of properties of the liquids viscosity, compressibility, thermal expansion, thermal conductivity, but the results were not successful. However, Furth obtained a precise result of the calculation of the volume change by melting and the entropy of melting. 

A particularly difficult problem appeared to be the systems of two active metals [27,28]. While, in several cases [27], the product patterns of the catalytic reaction show the presence of both active metals (Pt-Re, Pt-Co, Pt-Ir, Pd-Ni) in the surface, the chemisorption data, such as e.g. IR spectra of adsorbed CO, are less definite on this point. Recently Joyner and Shipiro [28] even speculated that — at least with Pt alloys — it is only Pt which forms the surface. Important information on the last mentioned problem has been supplied by single-crystal experiments, in which one metal (B) is covered by one, two or more monolayers of the second metal (A). It appeared [29] that, to see the bulk properties of a metal A, with regard to XPS and/or CO chemisorption, at least two or three layers of A should be laid down on metal B. This means that an ensemble of three or four contiguous surface A atoms must also have the A atoms underneath (atoms in the next layer, filling the holes of the first layer), to behave like corresponding ensembles of A in bulk metal A. This could be one of the reasons why the size of the necessary ensemble formally derived from the overall kinetic and the topmost layer composition is sometimes unreasonably large. 

We define an operator as closed , if its action on any model function G P produces only internal excitations within the IMS. An operator is quasi-open , if there exists at least one model function which gets excited to the complementary model space R by its action. Obviously, both closed and quasi-open operators are all labeled by only active orbitals. An operator is open , if it involves at least one hole or particle excitation, leading to excitations to the g-space by acting on any P-space function. It was shown by Mukheijee [28] that a size-extensive formulation within the effective Hamiltonians is possible for an IMS, if the cluster operators are chosen as all possible quasi-open and open excitations, and demand that the effective Hamiltonian is a closed operator. Mukhopadhyay et al. [61] developed an analogous Hilbert-space approach using the same idea. We note that the definition of the quasi-open and closed operators depends only on the IMS chosen by us, and not on any individual model function. 

In semiconductors, the existence (even in the bulk) of the characteristic band gap requires a better understanding of the conduction mechanism, which differs entirely from that of metals, to describe their quantum size effects [61-63]. By definition, the band gap is the energy necessary to create an electron e ) and hole (h ). Thus, the ionization potential E of a semiconductor can be treated as E = Eg-i-E, where Eg is the band gap energy and E is the ionization energy from the edge of the conduction band [61]. 

In connection with Observation 1, suppose that a coronoid C has a corona hole larger than naphthalene. Then we can imagine that a closer packing of the hexagons of C is possible by a partial filling of the corona hole so that the total number of internal vertices increases. In Observation 2 the crucial term ("perfect extremal coronoid") conforms with Definition 3.5 of Par. 3.3.4. It is reasonable to imagine that there is a critical smallest size for an extremal benzenoid, say A, so that A can be perforated with g naphthalene holes, which is taken to be the necessary condition for creating a perfect extremal tuple coronoid. 

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