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Möbius (left) and Hückel (right) orbital arrays. The two orbital arrays in Figure 3 are just examples and do not correspond to real systems. In inspecting the Möbius one on the left, plus–minus overlaps are seen between orbital pairs 2-3, 3-4, 4-5, 5-6, and 6-1, corresponding to an odd number (5), as required by a Möbius system.
The Hückel method or Hückel molecular orbital theory, proposed by Erich Hückel in 1930, is a simple method for calculating molecular orbitals as linear combinations of atomic orbitals. The theory predicts the molecular orbitals for π-electrons in π-delocalized molecules , such as ethylene , benzene , butadiene , and pyridine .
In contrast to the rarity of Möbius aromatic ground state molecular systems, there are many examples of pericyclic transition states that exhibit Möbius aromaticity. The classification of a pericyclic transition state as either Möbius or Hückel topology determines whether 4N or 4N + 2 electrons are required to make the transition state aromatic or antiaromatic, and therefore, allowed or ...
Benzene, the most widely recognized aromatic compound with six delocalized π-electrons (4n + 2, for n = 1).. In organic chemistry, Hückel's rule predicts that a planar ring molecule will have aromatic properties if it has 4n + 2 π-electrons, where n is a non-negative integer.
Hückel is most famous for developing the Hückel method of approximate molecular orbital (MO) calculations on π electron systems, a simplified quantum-mechanical method to deal with planar unsaturated organic molecules. In 1930 he proposed a σ/π separation theory to explain the restricted rotation of alkenes (compounds containing a C=C ...
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In the extended Hückel method, only valence electrons are considered; the core electron energies and functions are supposed to be more or less constant between atoms of the same type. The method uses a series of parametrized energies calculated from atomic ionization potentials or theoretical methods to fill the diagonal of the Fock matrix.
Given the number of problems (55 in total), just a few are presented here. The test functions used to evaluate the algorithms for MOP were taken from Deb, [ 4 ] Binh et al. [ 5 ] and Binh. [ 6 ] The software developed by Deb can be downloaded, [ 7 ] which implements the NSGA-II procedure with GAs, or the program posted on Internet, [ 8 ] which ...