Isosurface plot

Many functions, such as electron density, spin density, or the electrostatic potential of a molecule, have three coordinate dimensions and one data dimension. These functions are often plotted as the surface associated with a particular data value, called an isosurface plot (Figure 13.5). This is the three-dimensional analog of a contour plot. 

FIGURE 13.5 Isosurface plots, (a) Region of negative electrostatic potential around the water molecule. (A) Region where the Laplacian of the electron density is negative. Both of these plots have been proposed as descriptors of the lone-pair electrons. This example is typical in that the shapes of these regions are similar, but the Laplacian region tends to be closer to the nucleus. 

An isosurface plot of the electron density is shown in Figure 7.8. 

Some of the major packages are better at visualization than others. In any case, there are a host of third-party providers with software on offer. Here then is what you might like to do with the results of the calculations above (and I used HyperChem to produce the following screen grabs). First of all an isosurface plot of the electron density (Figure 10.16). Next are isosurface densities for the highest... 

The manifold energy splitting is accompanied by the formation of delocalized orbitals as shown by the contour and isosurface plots in Fig. 18 a... 

Fig. 10.3. Water molecules in a cavity of lysozyme. Only the surrounding residues are displayed. The isosurfaces of water oxygen (green) and hydrogen (pink) for the 3D distributions larger than 8 (left), the most probable model of the hydration structure reconstructed from the isosurface plots (center), and the crystallographic...

Figures 10.10a and b show, respectively, space-filling models and electron density isosurfaces plotted at 0.002 e/(tZo) for water, ammonia, and methane. The electron densities plotted here include all of the electrons in the molecule. They are calculated using state-of-the-art ab initio quantum chemical methods (see discussion in Chapter 6).

The heating in the bottom of cells A and B is driven by radiant energy from the flame, as demonstrated by a plot of the incident radiant flux in Figure 11.20. However, from this plot it can be deduced that peak heating near the center of the tube bank is not caused by radiation. It must, instead, be the result of convection. The net convective heat flux plot and the isosurface plot... 

Fig. 8a,b Some relevant metal bis(dithiolene) systems used in the search for molecular conductors. Shown in a are isosurface plots of the HOMO and LUMO calculated for Ni[tmdt]2. Reproduced with permission from [232]... 

Figure 8.4 CAS(x,y) SCF and DFT spin density difference distributions with respect to the DMRC(13,29)[2048] reference spin density determined for 2048 renormalized active-system states. X indicates the number of electrons correlated in y orbitals. An isovalue of 0.001 was chosen for isosurface plots. A blue surface indicates an excess of a-electron density, while a yellow surface encodes an excess of 3-electron density for the spin densities and vice versa for the spin density difference distribution. The CAS(x,y) SCF...

Fig. 21 Isosurface plot of V />(r) NHC SiCl2 (8) at the —0.53 eA level around Si (a), contour plots of -V p(r) in the C-Si-VSCC plane (b), orbital diagram for the dual dcmor-acceptor Sl-NHC bond (c), and in the Cl(inplane)-Si-VSCC (d). Local charge concentrations are depicted in blue, charge depletions in red. The contour values are at 0.2 x 10 , 0.4 x 10 , and 0.8 X 10" withn=-3, 2, 1...

Figure 4.3.3. A, gray scale NIR-CI acquired from a longitudinal section of a com kernel. B, NIR reflectance spectrum acquired at a single pixel position corresponding to the hard endosperm of the kernel. C, three-dimensional isosurface plot calculated from principal components of the NIR spectra of the kernel arising predominantly from oil (green) and different starches (red, yellow).

Thus one resorts to contour maps or isosurface plots to represent them. Unfortunately, they show only a part of the information contained in the function, since they depend on the contour (or isosurface) value choice one decides to plot. In order to have a more imambiguous way to analyze a three-dimensional (or higher dimension) function, one could use the framework of the topological analysis. In theoretical chemistiy, this has already been done in the pioneer works of Bader, which originated the Quantum Theory of Atoms in Molecules (QTAIM) [1]. Later this topological analysis was applied to interpret the Electron Localization Function [2-4], and lately it has been applied to the study of the Fukui function [5-7], which is namely the object of this chapter. 

While these comments pertain to orbital isosurface plots in general, the situation is particularly dire for diffuse electrons. In such cases, one should demand to know what fraction of the orbital density is encapsulated within a given isosurface plot. In other words, we need to know the fractional electron value. 

These authors then took b = 0hy fiat, citing studies of hemibonded cation radical systems in which the SIE, which arises primarily from an overly delocalized cation hole, was found to be roughly proportional to /[m(r)]. 7338 value a = 0.2 was then chosen by comparison to CCSD(T) calculations for some cation dimer radicals. Whether this rationale extends to anions is unclear, and in fact very different parameters a = 0.8 and b = 0.5) have been suggested on the basis of studies of other, non-hemibonded cation radicals. Nevertheless, the parameters a = 0.2 and b = 0 were adopted in the aforementioned (H20)J2 calculations, whereas in ah initio molecular dynamics simulations of e (aq) in bulk water, the value a = 0.3 (with b again fixed at zero) was found to provide better agreement with the experimental absorption spectrum. " In Ref. 330, isosurface plots of m(r) are presented for one particular isomer of (H20)J2, and the result obtained at the SIC-PBE level is seen to be qualitatively similar to the MP2 result. However, the SIC-PBE and RI-MP2 VDEs are rather different, and these differences do not appear to be systematic. (On the other hand, the RI-MP2/6-311G benchmarks in Ref. 330 could certainly be improved, in terms of the diffuseness of the basis set.) Moreover, the SIC is found to have qualitative effects on reactivity the aqueous-phase reaction H" " -I- e H, simulated inside of a water cluster, proceeds readily with the SIC but not without it. " In view of these issues, it seems that careful, systematic benchmark studies of SIC functionals for weakly-bound anions are probably warranted. 

Figure 1.16 (A) The projected density of state of the defective K-Ce2Zr20s. Isosurface plots of partial charge density of the gap states of K-Ce2Zr20s (B) and bulk Ce02 (C) In the presence of an O vacancy. They show that in both cases the two excess electrons In the presence of the 0 vacancy are localized in the 4f orbital of the two second nearest-neighbor Ce +, thus forming two Ce +. The circles In (B) and (C) show the relaxation pattern around the 0 vacancies which are at the centers of the corresponding circle. Adapted with permission from Wang et al. (2009b). Copyright (2009) John Wiley Sons, Inc.

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