Double symmetry

This effect originates in the various f electron configurations of lanthanides and actinides. This effect is based on the correlation observed between the full pattern of the effect and the sequence of values of the L quantum number [52]. The correlation consists of the occurrence of the same double symmetry in (a) the series of L quantum number values of the ground terms of f element ions and (b) the sequence of relatively stabilized or destabilized f electron configurations (i.e.) the double-double effect. Accordingly... 

Furthermore, a rather bold but chemically more feasible mechanism can account for the proposed migration of the acetoxy group. This is double, symmetry allowed, suprafacial-suprafacial, Claisen-type, 3,3-sigmatropic shift. 

H is even with respect to the interchange of any particle index (double symmetry group for each pair). As a result of group theory we immediately derive that the corresponding eigenstates are either even or odd ... 

The importance of the analogy between the four segments f°—f3, f4—f7, f7—f10, and fu-f14, besides double symmetry as the integral parts of the double-double effect is emphasized. 

Relatively more stable configurations have even values of L-quantum number (S = 0 and 1 = 6 terms) while relatively unstable configurations correspond to odd values of the L-quantum number (F = 3 and H = 5 terms). It should be noticed that relatively stabilized configurations display a relatively smaller ability to complex formation (less negative values of AG°) and vice versa. In Fig. 1 it is shown that the sequence of L-values has the same double symmetry as the double-double effect. However, no linear correlation has been experimentally observed between changes in the free energy of complex formation or lattice parameters of f-element compounds and the values of the L-quantum number of appropriate f-ions. The lack of such linear correlation is explained in Sect. 5. 

Fig. 3a-c. Double symmetry displayed by the spin-pairing stabilization energy a) Symmetry with respect to the f7 configuration according to expression (2), the symmetry within the two subgroups according to expression (3). The values plotted are calculated assuming that both parameters D and E3 decrease by one percent b) combined stabilization c) combined stabilization under the assumption that parameter E3 decreases by one percent while D only by 0.25 percent. (-------------------------5 f,... 

Returning to coefficients of interelectronic repulsion parameters D or E1 = 8 D/9 and E3, the explanation of double symmetry reflected in the double-double effect has been advanced also by Nugent50. ... 

It is easy to derive the ooexistenoe ourve. Beeause of the symmetry, the double tangent is horizontal and the ooexistent... 

When Cj symmetry is present, irreducible representations of the double group (see Table II) so that = 0 by symmetry. In this case, there are only... 

Furthermore (f) 1 2 dichloroethene has a center of symmetry located at the mid t point of the carbon-carbon double bond This too tells us the molecule is achiral... 

Let us now examine the Diels-Alder cycloaddition from a molecular orbital perspective Chemical experience such as the observation that the substituents that increase the reac tivity of a dienophile tend to be those that attract electrons suggests that electrons flow from the diene to the dienophile during the reaction Thus the orbitals to be considered are the HOMO of the diene and the LUMO of the dienophile As shown m Figure 10 11 for the case of ethylene and 1 3 butadiene the symmetry properties of the HOMO of the diene and the LUMO of the dienophile permit bond formation between the ends of the diene system and the two carbons of the dienophile double bond because the necessary orbitals overlap m phase with each other Cycloaddition of a diene and an alkene is said to be a symmetry allowed reaction... 

The chemistry of propylene is characterized both by the double bond and by the aHyUc hydrogen atoms. Propylene is the smallest stable unsaturated hydrocarbon molecule that exhibits low order symmetry, ie, only reflection along the main plane. This loss of symmetry, which implies the possibiUty of different types of chemical reactions, is also responsible for the existence of the propylene dipole moment of 0.35 D. Carbon atoms 1 and 2 have trigonal planar geometry identical to that of ethylene. Generally, these carbons are not free to rotate, because of the double bond. Carbon atom 3 is tetrahedral, like methane, and is free to rotate. The hydrogen atoms attached to this carbon are aUyflc. 

Diene moieties, reactive in [2 + 4] additions, can be formed from benzazetines by ring opening to azaxylylenes (Section 5.09.4.2.3). 3,4-Bis(trifluoromethyl)-l,2-dithietene is in equilibrium with hexafluorobutane-2,3-dithione, which adds alkenes to form 2,3-bis-(trifluoromethyl)-l,4-dithiins (Scheme 17 Section 5.15.2.4.6). Systems with more than two conjugated double bonds can react by [6ir + 2ir] processes, which in azepines can compete with the [47t + 27t] reaction (Scheme 18 Section 5.16.3.8.1). Oxepins prefer to react as 47t components, through their oxanorcaradiene isomer, in which the 47r-system is nearly planar (Section 5.17.2.2.5). Thiepins behave similarly (Section 5.17.2.4.4). Nonaromatic heteronins also react in orbital symmetry-controlled [4 + 2] and [8 + 2] cycloadditions (Scheme 19 Section 5.20.3.2.2). 

The 27T-electrons of the carbon-nitrogen double bond of 1-azirines can participate in thermal symmetry-allowed [4 + 2] cycloadditions with a variety of substrates such as cyclo-pentadienones, isobenzofurans, triazines and tetrazines 71AHC(13)45). Cycloadditions also occur with heterocumulenes such as ketenes, ketenimines, isocyanates and carbon disulfide. It is also possible for the 27r-electrons of 1-azirines to participate in ene reactions 73HCA1351). 

Fig. 33. Three-dimensional instanton trajeetories of a partiele in a symmetrie double well, interaeting with symmetrieally and antisymmetrieally eoupled vibrations with eoordinates and frequeneies q, to, and ru, respeetively. The curves are 1, ru, ru, P ojq (MEP) 2. to, (u, < (Oq (sudden approximation) 3. ru, < cOq, oj, P ojo 4. to, > (Oq, < (Oq.

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