Maximum probability domains

The Bond Analysis Techniques (ELF and Maximum Probability Domains) Application to a Family of Models Relevant to Bio-Inorganic Chemistry... 

Abstract Electron Localization Function (ELF) and Maximum Probability Domain (MPD) analyses have been applied to model metal-porphyrins and show compatible and complementary results. ELF basins are quite different from MPDs, but are a necessary starting point for optimizing them. The analyses of the bond between the metal and porphyrin do not show significant differences between nontransition and transition metals, fit all the cases considered, we find signatures characteristic of essentially ionic bonds. 

Keywords Density functional theory Electron localization function Maximum probability domains Molecular orbitals... 

Algorithms for Managing and Optimizing Maximum Probability Domains. 123... 

The Bond Analysis Techniques (ELF and Maximum Probability Domains)... 

Fig. 7 Cationic (/ ) and anionic right) maximum probability domains for crystalline MgO. Both the domains are optimized by maximizing the probability of containing ten electrons...

The chemically important Maximum Probability Domains for the same systems are shown in Figs. 7 and 8. For magnesium oxide, the cation and anion 10 electrons MPD have high probability 0.85 and 0.57 for cation and anion, respectively the volumes of two ions are very different, 19.4 and 90.5 cubic bohrs the MPD average net charges are, respectively, -i-l. 85 and -1.85 and these are very close to the formal ones. The two-electron MPD reported in Fig. 8 has a probability equal to 0.42, a volume of 56.0 cubic bohrs, and an average population of 2.04 electrons. 

In the present study, we consider the bond analysis in metal porphyrins. In the following discussion, the porphyrin will be often indicated as phy. We apply Electron Localization Function (ELF) analysis and Maximum Probability Domains (MPDs) analysis for understanding the eventual differences between transition metal and non-transition metals inserted into the porphyrin ring. As an example of nontransition metal, we consider Be, Mg, and Zn (a d metal, so d orbitals can be considered core-Uke). For transition metals, we consider Fe, Co, and Ni. Due to the difficulties in finding experimental examples of pure square planar coordination... 

Table 2 Numerical results about maximum probability domains in metal-porphyrins...

We have analyzed the metal-ligand bond in metal porphyrin models, using Electron Localization Function (ELF) and Maximum Probability Domains Analysis (MPDs). We do not find any relevant difference between non-transition and transition metal models. Our MPDs analysis shows that the metal-ligand bond is essentially ionic for all models, similar to the electride (charge separated structures) behavior of metal ions within cages [52]. In our MPDs analysis, we do not find clues of stabilization of bonds due to covalent interactions between the central metals and the porphyrin ring both in non-transition and in transition metal porphyrins. 

On the basis of results collected hitherto, the Maximum Probability Domains Analysis has been shown as a valuable tool for performing further analysis of bonds in other coordination compounds, without biases coming from historical concepts. 

Understanding Maximum Probability Domains with Simple Models... 

Abstract The paper presents maximum probability domains (MPDs). These are regions of the three dimensional space for which the probability to find a given number of electrons is maximal. In order to clarity issues hidden by numerical uncertainties, some simple models are used. They show that MPDs reproduce features which one would expect using chemical intuition. For a given number of electrons, there can be several solutions, corresponding to different chemical situations (e.g. different bonds). Some of them can be equivalent, by symmetry. Symmetry can produce, however, alternative solutions. The models show that MPDs do not exactly partition space, and they can also be formed by disjoint subdomains. Finally, an example shows that a partition of space, as provided by loge theory, can lead to situations difficult to deal with, not present for MPDs. 

In order to better understand the maximum probability domains (the region of space maximizing the probability to find a given number of electrons in it), we studied some simple model systems. 

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