Shape function electron density

The value of SI(II) also varies in the range 0 to 1. When pA = npB (n is constant and 0) is substituted into Eq. (15), the Carbo index will be unity, which means that the Carbo index represents the similarity of the shapes of electron density distributions but not of the magnitudes. When pA = npB is substituted into Eq. (16), the result will be SI(H) = 2n/(l + n2), which means that the Hodgkin index characterizes the similarity in the shape as well as in the magnitude of the electron density functions [104],... 

From a map at low resolution (5 A or higher) one can obtain the shape of the molecule and sometimes identify a-helical regions as rods of electron density. At medium resolution (around 3 A) it is usually possible to trace the path of the polypeptide chain and to fit a known amino acid sequence into the map. At this resolution it should be possible to distinguish the density of an alanine side chain from that of a leucine, whereas at 4 A resolution there is little side chain detail. Gross features of functionally important aspects of a structure usually can be deduced at 3 A resolution, including the identification of active-site residues. At 2 A resolution details are sufficiently well resolved in the map to decide between a leucine and an isoleucine side chain, and at 1 A resolution one sees atoms as discrete balls of density. However, the structures of only a few small proteins have been determined to such high resolution. 

The size and shape of an electron cloud is described by the electron density (the number of electrons per unit volume). Consider a graph of electron density in the hydrogen atom as a function of distance from the nucleus. 

The connection between a molecule s electron density surface, an electrostatic potential surface, and the molecule s electrostatic potential map can be illustrated for benzene. The electron density surface defines molecular shape and size. It performs the same function as a conventional space-filling model by indicating how close two benzenes can get in a liquid or crystalline state. 

The division of the molecular volume into atomic basins follows from a deeper analysis based on the principle of stationary action. The shapes of the atomic basins, and the associated electron densities, in a functional group are very similar in different molecules. The local properties of the wave function are therefore transferable to a very good approximation, which rationalizes the basis for organic chemistry, that functional groups react similarly in different molecules. It may be shown that any observable... 

The fact that features in the total electron density are closely related to the shapes of the HOMO and LUMO provides a much better rationale of why FMO theory works as well as it does, than does the perturbation derivation. It should be noted, however, that improvements in the wave function do not necessarily lead to a better performance of the FMO method. Indeed the use of MOs from semi-empirical methods usually works better than data from ab initio wave functions. Furthermore it should be kept in mind that only the HOMO orbital converges to a specific shape and energy as the basis set is... 

Figure 3.15 An sp hybrid orbital, (a) left, radial functions for the 2s and 2p atomic orbitals right, radial function for the sp hybrid orbital (b) left, the shapes of the 2s and 2p atomic orbitals as indicated by a single contour value right, the shape of the sp hybrid orbital as indicated by the same contour, (c) The shape of a surface of constant electron density for the sp hybrid orbital, (d) Simplified representation of (c). (Reproduced with permission from R. J. Gillespie, D. A. Humphreys, N. C. Baird, and E. A. Robinson, Chemistry, 2nd Ed., 1989, Allyn and Bacon, Boston.)...

Frequently at least one of the phases forms particles (e.g., crystalline lamellae). The shape and position of the ith particle from the irradiated volume is described by a shape function Yi (r). It is obvious that the scattering intensity of an ideal multiphase system can be expressed in terms of autocorrelations Y 2 (r) and cross-correlations of the shape functions and the average electron densities of each phase (cf. Sect. 2.5). 

According to some authors, the similarity between shape functions might give more interesting information than using the electron density [63]. In this context, it should be mentioned that the Carbo index retains the same value. Moreover, it has... 

The shape function, denoted as cr(r), is defined as the electron density per particle,... 

The shape function had a role in theoretical chemistry and physics long before it was named by Parr and Bartolotti. For example, in x-ray measurements of the electron density, what one actually measures is the shape function—the relative abundance of electrons at different locations in the molecule. Determining the actual electron density requires calibration to a standard with known electron density. On the theoretical side, the shape function appears early in the history of Thomas-Fermi theory. For example, the Majorana-Fermi-Amaldi approximation to the exchange potential is just [3,4]... 

The contribution of Parr and Bartolotti is not diminished by these precedents they were the first to recognize that o-(r) is a quantity of interest in its own right, separate from the electron density [1], They also deserve credit for coining the name, shape function, which captures the essence of the quantity and provides an essential verbal handle that facilitated future work. 

Notice that how the shape function naturally enters this discussion. Because the number of electrons is fixed, the variational procedure for the electron density is actually a variational procedure for the shape function. So it is simpler to restate the equations associated with the variational principle in terms of the shape function. Parr and Bartolotti have done this, and note that because the normalization of the shape function is fixed,... 

The use of the electron density and the number of electrons as a set of independent variables, in contrast to the canonical set, namely, the external potential and the number of electrons, is based on a series of papers by Nalewajski [21,22]. A.C. realized that this choice is problematic because one cannot change the number of electrons while the electron density remains constant. After several attempts, he found that the energy per particle possesses the convexity properties that are required by the Legendre transformations. When the Legendre transform was performed on the energy per particle, the shape function immediately appeared as the conjugate variable to the external potential, so that the electron density was split into two pieces that can be varied independent the number of electrons and their distribution in space. 

But the proof is deceptively simple [31]. Because the shape function is proportional to the electron density, it inherits the characteristic electron-nuclear coalescence cusps at the positions of the atomic nuclei [32,33]. The location of those cusps determines the positions of the nuclei, R the steepness of the cusps determines the atomic charges, Za. So the shape function determines the external potential for any molecular system [31]. 

This similarity indicator, in fact, precedes Parr and Bartolotti s introduction of the shape function terminology [59]. In general, it seems that the shape function is preferred to the electron density as a descriptor of molecular similarity whenever one is interested in chemical similarity. Similarity measures that use the electron density will typically predict that fluorine resembles chlorine less than it resembles sodium, oxygen, or neon using the shape function helps one to avoid conflating similarity of electron number with chemical similarity [53,57]. 

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