Localized orbitals. Population analysis

Electron distributions are ascertained by means of a Mulliken population analysis of the ir-electron atomic populations (in the case of complete n-delocalization, each atom in the 67r-electron five-membered ring would have 1.20 7r-electrons associated with it) by determining the spatial extent of the localized orbitals of both the out-of-plane lone pair of the heteroatom and the C=C double bonds as well as through comparing the total electron density plots in planes parallel to the molecular plane. 

Another group of compounds that have a twisted double bond are the bicyclic compounds with bridgehead double bonds such as 1,2-norbomene (9) and 1,7-norbornene (10). " It has been found that many compounds, such as 11, which is based on trawi-cyclooctene, may be isolated whereas those based on smaller trauj -cycloalkenes are usually quite unstable. Some evidence for the formation of 9 has been obtained by trapping the product of the dehalogenation of 1,2-diha-lonorbornanes." Here, the simplest view is that the two p orbitals that form the double bond in 9 and 10 are roughly perpendicular to each other. However, pyr-amidalization and rehybridization also are involved. One indication is the reduced localized 7i-orbital population found in the NBO analysis. Whereas normal alkenes have 71 populations of 1.96 e, for 9 with OS = 57 kcal/mol, it is 1.921, and for 10 with OS = 86 kcal/mol, it is 1.896. With 9, the deviations of the a and n orbitals from the line of centers are 24° and 19°, respectively, and with 10, the deviations are 34° and 29°. 

From a theoretical point of view XES furthermore provides a very strong basis for the evaluation of methods for population analysis, i.e., the decomposition of the molecular orbitals into atomic contributions [10]. Many different schemes subdividing the charge density into contributions assigned to respective atoms have been proposed, but the lack of means to directly measure the atomic populations in different orbitals has made all techniques somewhat arbitrary and a matter of taste. Due to the strongly localized character of the intermediate core-excited state, however, in combination with the direct dependence of the XES transition probability on the amount of local -population (assuming a Is core hole), XES provides a very sensitive tool to directly measure this atomic population. In all, we have an atom-specific tool, which can be used to address important questions regarding... 

This localization scheme permits tire assignment of hybridization both to the atomic lone pairs and to each atom s contributions to its bond orbitals. Hybridization is a widely employed and generally useful chemical concept even though it has no formal basis in the absence of high-syrnmetry constraints. Witli NBO analysis, tire percent s and p character (and d, f, etc.) is immediately evident from tire coefficients of tire AO basis functions from which the NAO or NBO is formed. In addition, population analysis can be carried out using the NBOs to derive partial atomic charges (NPA, see Section 9.1.3.2). 

Lovasz-Pelikan index spectral indices (0 eigenvalues of the adjacency matrix) LOVIs = LOcal Vertex Invariants local invariants Lowdin population analysis quantum-chemical descriptors Lowest-Observed-Effect Level biological activity indices (0 toxicological indices) lowest unoccupied molecular orbital quantum-chemical descriptors lowest unoccupied molecular orbital energy quantum-chemical descriptors LUDI energy function scoring functions Lu index —> hyper-Wiener-type indices... 

The net charges and net spin densities obtained from Mulliken population analysis are listed in Table 3. The net charges of An are all ca. +2.9 except for Pa, for which the charge is +3.5. For all of the complexes, the charges on the C atoms are all about -0.2, and the H atoms are essentially neutral, consistent with the picture that the charge transfer occurs primarily between the Cpn orbitals to the An orbitals, as expected. The net spin density increases from Th to Am, consistent with the increase in the number of metal-localized electrons in the series of actinocenes. 

An external localization method was proposed by Magnasco and Perico [20,21]. The localization criterion equals the maximization of the atomic population, obtained by using the Mulliken population analysis. The number of localized orbitals as well as their association to a given atom should be defined in advance. 

When the fiillerene was allowed to relax, conductance is reduced drastically. Only the LUMO orbital preserves its extended character, while the rest of the orbitals become localized at the interfaces. A Mulliken population analysis and the integration of the local density of states show that the excess transferred charge ( 3.3 e) concentrates mostly at the interface hexagons. Such a finding confirms that the details of the interactions between the metal-molecule contacts are the most important factors which determine the transport properties of the molecular bridge. 

By looking at the maximum localized Wannier functions and using different population analysis methods. Buhl et al. suggested that there is a donor-acceptor interaction between the lone pairs of Q and the anti-bonding population analysis revealed that the amount of charge transferred from anions to cations ( 0.2 5e) fluctuated very little during the M D trajectory. 

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