Localized orbital locator

Several methods have been used for analyzing the electron density in more detail than we have done in this paper. These methods are based on different functions of the electron density and also the kinetic energy of the electrons but they are beyond the scope of this article. They include the Laplacian of the electron density ( L = - V2p) (Bader, 1990 Popelier, 2000), the electron localization function ELF (Becke Edgecombe, 1990), and the localized orbital locator LOL (Schinder Becke, 2000). These methods could usefully be presented in advanced undergraduate quantum chemistry courses and at the graduate level. They provide further understanding of the physical basis of the VSEPR model, and give a more quantitative picture of electron pair domains. 

The ELF has been extensively applied to a large number of systems and has also been used to quantify chemical concepts like the strength of the hydrogen bond [26] and aromaticity [27, 28]. Moreover, other interesting ELF-like scalar fields have been developed, such as the localized orbital locator (LOL) [29], based on the comparison of the local noninteracting kinetic energy density with that of the uniform electron gas, and the electron localizabihty indicator (ELI) [30-32], derived directly from the electron pair density without any reference to the uniform electron gas. 

In this arena, energetic properties are usually derived by integrating densities over real space domains, and not by examining appropriate scalar or vector fields. Exceptions to this rule exist the localized orbital locator (LOL) focuses on the topological properties of a kinetic energy density [34], and the QTAIM virial (t ) and energy density (J ) fields are commonly examined at critical points (CPs) of the density. The latter are however computed from the density and its derivatives through the QTAIM s local virial theorem [1], that depends on an arbitrary choice of the kinetic stress tensor [9]. 

The reason for this becomes apparent when one compares the shapes of the localized it orbitals with that of the ethylene 7r orbital. All of the former have a positive lobe which extends over at least three atoms. In contrast, the ethylene orbital is strictly limited to two atoms, i.e., the ethylene 7r orbital is considerably more localized than even the maximally localized orbitals occurring in the aromatic systems. This, then, is the origin of the theoretical resonance energy the additional stabilization that is found in aromatic conjugated systems arises from the fact that even the maximally localized it orbitals are still more delocalized than the ethylene orbital. The localized description permits us therefore to be more precise and suggests that resonance stabilization in aromatic molecules be ascribed to a "local delocalization of each localized orbital. One infers that it electrons are more delocalized than a electrons because only half as many orbitals cover the same available space. It is also noteworthy that localized it orbitals situated on joint atoms (n 2, it23, ir l4, n22 ) contribute more stabilization than those located on non-joint atoms, i.e. the joint provides more paths for local delocalization. 

This brings us to the last factor to consider—the size and direction of the atomic vector contributions - The size can only be significant if the coefficients of the 2p orbitals located on atom n in at least one of the most localized orbitals Xa nd Xh ( nd indeed, in the first approxi-... 

The main contribution to the total SOC value comes from the x component perpendicular to the molecular plane. The operator rotates the orbitals within the molecular (yz) plane, and since in the face-to-face conformation (a = p = 90°) both localized orbitals are located in this plane, this is the most favored conformation for spin-orbit coupling (cf Section 4.1). 

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