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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 ...
Hückel's rule can also be applied to molecules containing other atoms such as nitrogen or oxygen. For example, pyridine (C 5 H 5 N) has a ring structure similar to benzene, except that one -CH- group is replaced by a nitrogen atom with no hydrogen. There are still six π electrons and the pyridine molecule is also aromatic and known for its ...
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 ...
The artificial landscapes presented herein for single-objective optimization problems are taken from Bäck, [1] Haupt et al. [2] and from Rody Oldenhuis software. [3] 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 ...
Hundred-dollar, Hundred-digit Challenge problems — list of ten problems proposed by Nick Trefethen in 2002; International Workshops on Lattice QCD and Numerical Analysis; Timeline of numerical analysis after 1945; General classes of methods: Collocation method — discretizes a continuous equation by requiring it only to hold at certain points
Problems that involve many governing factors, where most of them cannot be expressed numerically can be well suited for morphological analysis. The conventional approach is to break a complex system into parts, isolate the parts (dropping the 'trivial' elements) whose contributions are critical to the output and solve the simplified system for ...