Symmetry of opposites

I. Hargittai, M. Hargittai, Symmetry of Opposites—Antisymmetry. Math. Intell. 1994, 16, 60-66. 

Sometimes, the terms asymmetry, dissymmetry, and antisymmetry are confused in the literature although the scientific meaning of these terms is in complete conformity with the etymology of these words. Asymmetry means the complete absence of symmetry, dissymmetry means the derangement of symmetry. and antisymmetry means the symmetry of opposites (see Section 4.6). 

Atoms, linear molecules, and non-linear molecules have orbitals which can be labeled either according to the symmetry appropriate for that isolated species or for the species in an environment which produces lower symmetry. These orbitals should be viewed as regions of space in which electrons can move, with, of course, at most two electrons (of opposite spin) in each orbital. Specification of a particular occupancy of the set of orbitals available to the system gives an electronic configuration. For example,... 

If a eenter of symmetry is present, all of the states arising from are gerade however, the states arising from 71% ean be gerade if n and 71 are both g or both u or ungerade if n and 71 are of opposite inversion symmetry. 

According to one classification (15,16), symmetrical dinuclear PMDs can be divided into two classes, A and B, with respect to the symmetry of the frontier molecular orbital (MO). Thus, the lowest unoccupied MO (LUMO) of class-A dyes is antisymmetrical and the highest occupied MO (HOMO) is symmetrical, and the TT-system contains an odd number of TT-electron pairs. On the other hand, the frontier MO symmetry of class-B dyes is the opposite, and the molecule has an even number of TT-electron pairs. 

Note that a conjugated diene and a conjugated triene react with, opposite stereochemistry. The diene opens and doses by a conrotatory path, whereas the triene opens and closes by a disrotatory path. The difference is due to the different symmetries of the diene and triene HOMOs. 

Thermal and photochemical electrocyclic reactions always take place with opposite stereochemistry because the symmetries of the frontier orbitals are always different. Table 30.1 gives some simple rules that make it possible to predict the stereochemistry of electrocyclic reactions. 

Cycloaddition reactions are those in which two unsaturated molecules add together to yield a cyclic product. For example, Diels-AJder reaction between a diene (four tt electrons) and a dienophile (two tt electrons) yields a cyclohexene. Cycloadditions can take place either by suprafacial or antarafacial pathways. Suprafacial cycloaddition involves interaction between lobes on the same face of one component and on the same face of the second component. Antarafacial cycloaddition involves interaction between lobes on the same face of one component ancl on opposite faces of the other component. The reaction course in a specific case can be found by looking at the symmetry of the HOMO of one component and the lowest unoccupied molecular orbital (LUMO) of the other component. 

Overbeek and Booth [284] have extended the Henry model to include the effects of double-layer distortion by the relaxation effect. Since the double-layer charge is opposite to the particle charge, the fluid in the layer tends to move in the direction opposite to the particle. This distorts the symmetry of the flow and concentration profiles around the particle. Diffusion and electrical conductance tend to restore this symmetry however, it takes time for this to occur. This is known as the relaxation effect. The relaxation effect is not significant for zeta-potentials of less than 25 mV i.e., the Overbeek and Booth equations reduce to the Henry equation for zeta-potentials less than 25 mV [284]. For an electrophoretic mobility of approximately 10 X 10 " cm A -sec, the corresponding zeta potential is 20 mV at 25°C. Mobilities of up to 20 X 10 " cmW-s, i.e., zeta-potentials of 40 mV, are not uncommon for proteins at temperatures of 20-30°C, and thus relaxation may be important for some proteins. 

Since S is a symmetric matrix equal to Q(0), these equalities show that the off-diagonal blocks must vanish at x = 0, and hence that there is no instantaneous coupling between variables of opposite parity. The symmetry or asymmetry of the block matrices in the grouped representation is a convenient way of visualizing the parity results that follow. 

Increasing the reducing agent/W ratio and using Na-naphthalenide led to the isolation of the two-electron-reduced, diamagnetic compound 19. The C2v symmetry of the h NMR spectrum and the X-ray analysis are in agreement with the cen-trosymmetric structure sketched in Scheme 2 for 19 [W = W, 2.614(1) A]. The six-coordination of the metal and the inclusion of the alkali metal cation removes the planarity ofthe 04 core and the cone conformation of the calix[4]arene. The sodium cation within the calix[4]arene cavity is r 3-bonded to two opposite arene rings. 

The difference in stereochemical outcome of these reactions is determined, therefore, by the relative phase of the lobes—at the terminal carbon atoms—of the MOs of these (and other similar) mre systems by the symmetry of their orbitals, that is. As we have seen, the orbital lobes, at the two terminal carbon atoms, have the same phase in the HOMO (tf 3) of the triene (67re), and in the HOMO after irradiation (i/i3) of the diene (4ire) while these orbital lobes have opposite phases in the HOMO (i/r2) of the diene and in the HOMO after irradiation (ifi4) of the triene. Two such terminal lobes with the same phase require disrotatory movement before bond-making/bond-breaking can occur, while two terminal lobes with... 

For Comparison Notions of Normal Scattering. As the electron density is assumed to be a real quantity, it directly follows the central symmetry of scattering patterns known by the name Friedel s law [244], Friedel pairs are Bragg reflections hkl and hkl that are related by central symmetry. Concerning their scattering amplitudes, Friedel pairs have equal amplitude Aha = A-g and opposite phase (phki = -scattering intensity the phase information on the structure factor is lost. 

Dependence on Metallocene Symmetry of E-Z Selectivity for 2-Butene Copolymerizations. We have seen in the Section 3.1.3 that opposite enantiofaces are favored for primary and secondary propene insertion on C2-symmetric metallocenes, whereas the same enantioface is favored for primary and secondary insertion on Cv-symmetric metallocenes. In this framework, if the same steric interactions which rule the enantioselectivity of primary and secondary propene insertions hold for 2-butene, the insertion of... 

This idea is elegant for its simplicity and also for its usefulness. While often in phenomenological theories of materials, control of parameters with molecular structure would provide useful properties, but the parameters are not related in any obvious way to controllable molecular structural features. Meyer s idea, however, is just the opposite. Chemists have the ability to control enantiomeric purity and thus can easily create an LC phase lacking reflection symmetry. In the case of the SmC, the macroscopic polar symmetry of this fluid phase can lead to a macroscopic electric dipole, and such a dipole was indeed detected by Meyer and his collaborators in a SmC material, as reported in 1975.2... 

For polyenes in their ground state, the highest occupied MO (HOMO) will be symmetric with respect to m2 for 2, 6, 10. .. n electron systems and antisymmetric for 4, 8, 12. .. n electron systems. The lowest unoccupied MO (LUMO) has the symmetry opposite to that of HOMO. In the first excited state, the LUMO of the ground state will become singly occupied because of the promotion of an electron and it will thus become the new highest occupied orbital. Therefore in the circumstances the symmetry of the highest occupied orbital is opposite to that of the ground state. 

Figure 8.3 Model of two Fur dimers binding to opposite sides of canonical B-DNA. The twofold symmetry between the Fur dimers follows the symmetry of the DNA sequence. (From Pohl et al., 2003. Reproduced with permission of Blackwell Publishing Ltd.)...

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