Figure 4 – uploaded by Eduardo Chamorro

See full PDF downloadDownload figure

ttractor), and you will look at just one surface containing all the system. Increasins he value of the ELF, at the point of the minima, the basins separate and more tha me f-domain appears. This is called a bifurcation, and it is clearly visualized by « ifurcation diagram like the one in Figure 4. This is the bifurcation, diagram of the unction depicted in Figure 3. The first bifurcation corresponds to the separation of the lisynaptic basin and the core basin of point B and occurs at the global minimum of thi unction. The second bifurcation appears later on and corresponds to the separation o he central disynaptic basin and the core basin associated with the point A. Hence, the yoints of bifurcation correspond to the minima of the function. In the case of the ELF he lower the bifurcation point the more localized are the corresponding basins. This kind of analysis based on bifurcations is connected to the concept of synapti: der previously defined,” and it has been applied to the study of electron localizatior n some simple chemical systems* as it will be noted below. A recent applicatior howing the usefulness of this type of analysis has been reported by Silvi concernins he hondino nature of the VOx and VOx + (vy — 1 —4) molecular cveteme 22 — Figure 4 ttractor), and you will look at just one surface containing all the system. Increasins he value of the ELF, at the point of the minima, the basins separate and more tha me f-domain appears. This is called a bifurcation, and it is clearly visualized by « ifurcation diagram like the one in Figure 4. This is the bifurcation, diagram of the unction depicted in Figure 3. The first bifurcation corresponds to the separation of the lisynaptic basin and the core basin of point B and occurs at the global minimum of thi unction. The second bifurcation appears later on and corresponds to the separation o he central disynaptic basin and the core basin associated with the point A. Hence, the yoints of bifurcation correspond to the minima of the function. In the case of the ELF he lower the bifurcation point the more localized are the corresponding basins. This kind of analysis based on bifurcations is connected to the concept of synapti: der previously defined,” and it has been applied to the study of electron localizatior n some simple chemical systems* as it will be noted below. A recent applicatior howing the usefulness of this type of analysis has been reported by Silvi concernins he hondino nature of the VOx and VOx + (vy — 1 —4) molecular cveteme 22

Related Figures (21)

Figure 1 H,0O isosurfaces calculated at different levels of theory

Table 1 The ELF shell radii and electron numbers

Figure 2 The atomic shell structure determined by ELF

3D they can be a point, a surface or a volume, so are also the minima. In 1D, every attractor is surrounded by two minima (suppose the limit values at zero and infinity are minima), which encloses a line (see Figure 3); in 3D they are surrounded by a surface that encloses a volume. In 1D, mathematically one characterizes the basins saying that it is the line formed by all the points around the attractor such that the point plus the first derivative is closer to the attractor. In 3D, one follows the gradient of the function saying that the basin is formed by all the points enclosed in the volume formed by the gradient lines ending up in the attractor. One can try to visualize it: look at the 1D function in Figure 3, rotate around the y - axes in 360° and you will have a 2D function and the basin indicates in the Figure 3 will be a surface. One more rotation will produce a 3D function and the basin will be a volume. The concept of isosurface has also its correspondence in 1D. Look in Figure 3 at the points where the function has the value f(x) = Xo. The lines joining the points are the equivalent to the volume enclosed by an isosurface. They are called the f-domain. In Figure 3 there are three f-domains at f(x) = x9. However, if one goes to a lower value of the function, close to f(x) = 0, one will have only one f-domain. There is an important difference in both cases. In the first case, at f(x) = xo, the f-domains enclose only one attractor each, but in the second case, close to f(x) = 0, the only f-domain encloses the three attractors. The f-domains are said to be reducible when they contain more than one attractor ie ag

Understanding and using the ELF Figure 5 ELF isosurfaces of the para-benzine in singlet and triplet states breaking, which means that the wave function obtained from standard single-determinant methods fails to transform as an irreducible representation of the molecular point group. Crawford et al. published a thorough study on this problem.** From Figure 5 one can see that the ELF, and ELF, are different. One electron with spin a is located on the carbon atom at the bottom, whereas the other electron with B spin appears to be located on the carbon atom at the top of the figure.

Figure 7 ELF isosurfaces for some molecules with typical geometries predicted by VSEPR model. Understanding and using the ELF

P. Fuentealba et al. Figure 8 ELF isosurfaces of some classical organic molecules

Scheme 2 Transition structures for the thermal electrocyclization of (Z)-1,2,4,6-heptatetraene (X=CH,) and its heterosubstituted analogues, (2Z)-2,4,5-hexatrienal (K=O) and (2Z)-2,4,5- hexatrien-1-imine (X=NH)

Scheme 3 Analysis of the pericyclic and pseudopericyclic nature of bonding for some thermal decarbonilations

Scheme 4 [1s,3a]hydrogen, [1a,3s]methyl and [1a,3s]fluorine sigmatropic rearrangements in the allyl system. ELF analysis reveals a concerted interaction in the first case, a biradical interaction in the second one and a ion-pair interaction in the last one On the one hand, the fluctuation of electron density forms a cyclic and characteristic antarafacial pattern of bonding in the first case, while on the other, monosynaptic

Scheme 5a_ Total energy along the IRC for the [la,3s]F sigmatropic shift The reaction mechanism of the Bergman cyclization of the (Z)-hexa-1,5-diyne-3-ene to yield p-benzyne (Scheme 7 and Figure 9) has been studied recently in the framework

Figure 9 IRC profile of the Bergman cyclization of (Z)-hexa-1,5-diyn-3-ene

Figure 11 Evolution of the ELF,,ELF, and average ELF,-ELF, along the IRC path of Bergman cyclization

Understanding and using the ELF Figure 12 IRC profile of the trimerization of acetylene

Figure 13 ELF isosurfaces for: (a) acetylenes moieties, (b) step II (TS), (c) step III, (d) step IV and (e) step V (benzene)

Figure 14 Evolution of the ELF,, ELF, and average ELF,-ELF, along the IRC path of trimerization of acetylene

Figure 15 ELF isosurface of Li, and Li,H, A special branch of the atomic cluster research is the study of clusters of the metal transition atoms, especially because of their catalytic properties. However, for the tran- sition metal atoms, the application of the ELF deserves to be carefully analysed. One