Partitioning of space

To be on the safe side, we should note that this kind of partitioning of space was used by Descartes in 1644 [139], and its origin can possibly be found in ancient times [141]. The further development was proposed by Gauss [142], Dirichlet [139], and others, but the most detailed and thorough mathematical description was given by, namely, Voronoi and Delaunay [143,144]. 

The electrostatic properties of the molecule may be used as a criterion for judging the MEM enhancement. Using the uniform prior density, the MEM molecular dipole moment derived by the discrete boundary partitioning of space (chapter 6) is only 1.3 D, compared with 9.1 D based on the experimental density,... 

There have been many attempts to define coordination number using a simple criterion to decide when two atoms are bonded (Brunner and Laves 1970). Rules have been proposed based on bond lengths, ionic radii, and topological properties such as the Voronoi partitioning of space, but none has proved entirely satisfactory. In this book the coordination number is determined by setting an arbitrary, though reasonable, lower threshold for the experimental bond valence (Altermatt and Brown 1985). 

Figure 9.5 AIM partitioning of space in a plane containing four atoms of CH3N2+ (the other two hydrogen atoms are symmetrically above and below the plane)...

The Voronoi deformation density approach, is based on the partitioning of space into the Voronoi cells of each atom A, that is, the region of space that is closer to that atom than to any other atom (cf. Wigner-Seitz cells in crystals see Chapter 1 of Ref. 202). The VDD charge of an atom A is then calculated as the difference between the (numerical) integral of the electron density p of the real molecule and the superposition of atomic densities SpB of the promolecule in its Voronoi cell (Eq. [42]) ... 

Another way to define ionic charges consists in partitioning space into elementary volumes associated to each atom. One method has been proposed by Bader [240,241]. Bader noted that, although the concept of atoms seems to lose significance when one considers the total electron density in a molecule or in a condensed phase, chemical intuition still relies on the notion that a molecule or a solid is a collection of atoms linked by a network of bonds. Consequently, Bader proposes to define an atom in molecule as a closed system, which can be described by a Schrodinger equation, and whose volume is defined in such a way that no electron flux passes through its surface. The mathematical condition which defines the partitioning of space into atomic bassins is thus ... 

The myosin heads are helically distributed and the actin molecules form helical double strands. There are also additional helical elements attached to the actin threads, but they can be ignored in this context. The cross-sectional arrangement of actin and myosin shown in Fig. 8.8 is consistent with the Q surface. The myosin molecviles are centred on the 62 axes and actin on 3l axes, which occur in the proportion 2 1. The Q surface partitioning of space into helical channel systems corresponds to the position of the myosin threads. There is thus no connection between adjacent channel systems, i.e. between neighbouring myosin threads. To vmderstand the connections between these channels, we can consider the rectangular nets, which span this surface, shown in Fig. 8.9. The channels exhibit four-coordination alternatively we can regard the vmits as four-armed. 

The partitioning of space into two regions is shown in Figure 8.6. 

Figure 5. Partitioning of space in embedding process (s) super system (ex. H3SiOHAlH3...NH3), frontier region and bulk (s ).

Delocalization indices have been also evaluated for the topological basins of the electron localizability indicator (ELI), whose topology defines partition of space into basins, representing various elements of chemical bonding, emerged from atomic shells cores, penultimate shells, lone pairs and bonds. 

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. 

From the latter formulation, one can immediately see that for this wave function, one has >1 = 1 when Qi contains either the space associated to one proton, H, or that for two of them, and He, cf. Fig. 10.5. One has a system//...//. This separation can be done, of course, also by isolating H, or He. These solutions can be obtained by a rotation along the threefold axis. Notice that there are three solutions, but as there are only two electrons of same spin these solutions overlap significantly, and do not form a partition of space, as it is given by Symmetry thus provides... 

MPDs do not Always Provide an Exact Partition of Space... 

Electronic systems well described by a single Lewis structure produce MPDs which correspond roughly to a partition of space which permits an association to bonds, lone pairs, or cores. However, our models have shown that this partition is not strict. A further point is that symmetry can produce alternative solutions. 

In this perturbing operator, uhf is the centro-symmetric Hartree-Fock (H-F) potential. Similarly as in the case of the crystal field potential included as a perturbation, also here the inter-shell interactions via the Vcon- are taken into account, thus the same partitioning of space in equation (10.26). However, for a certain order of three operators, D g Vcryst, and Vcorr in the final third-order contributions, it is also possible to take into account the interactions via the perturbing operator within the Q-space. This is why additional operators, QVcrystQ and QVQ, are included in the Hamiltonian of equation (10.26). 

From a physical point of view this relativistic model is also based on the perturbation approach, and at the second order, similarly as in the case of the standard J-O Theory, the crystal field potential plays the role of a mechanism that forces the electric dipole/ t—>f transitions. The only difference is that now the transition amplitude is in effectively relativistic form, as determined by the double tensor operator, but still of one particle nature. Furthermore, the same partitioning of space as in non-relativistic approach is valid here. The same requirements about the parity of the excited configurations are expected to be satisfied. As a final step of derivation of the effective operators, the coupling of double inter-shell tensor operators has to be performed. This procedure is based on the same rules of Racah algebra as presented in the case of the standard J-O theory. However, the coupling of the inter-shell double tensor operators consists of two steps, for spin and orbital parts separately. Thus, the rules presented in equations (10.15) and (10.16) have to be applied twice for orbital and spin momenta couplings, resulting in two 3j— and two 6j— coefficients. 

The gradient field of ELF yields a partition of space into basins of attractors which correspond to cores, lone pairs, and bond regions [36, 37]. As noted by Gillespie and Robinson This function (ELF) exhibits maxima at the most probable positions of localized electron pairs and each maximum is surrounded by a basin in which there is an increased probability of finding an electron pair. These basins correspond to the qualitative electron pair domains of the VSEPR model and have the same geometry as the VSEPR domains [50]. 

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