Nonclassical symmetry

The principal aim of the present chapter is twofold. First, we will review the already known ideas, methods, and results centered around the solution techniques that are based on the symmetry reduction method for the Yang-Mills equations (1) in Minkowski space. Second, we will describe the general reduction routine, developed by us in the 1990s, which enables the unified treatment of both the classical and nonclassical symmetry reduction approaches for an arbitrary relativistically invariant system of partial differential equations. As a byproduct, this approach yields exhaustive solution of the problem of... 

With all the wealth of exact solutions obtainable through Lie symmetries of the Yang-Mills equations, it is possible to construct solutions that cannot be derived by the symmetry reduction method. The source of these solutions is conditional or nonclassical symmetry of the Yang-Mills equations. 

The first paper devoted to nonclassical symmetry of partial differential equations was published by Bluman and Cole [57]. However, the real importance of these symmetries was understood much later after the explanations given in several papers [31,32,58-61] where the method of conditional symmetries had been used in order to construct new exact solutions of a number of nonlinear partial differential equations. 

The consideration of HO—LU interaction is useful also in the interpretation of the stability of "nonclassical carbonium ions. For instance, the 7-norbornenyl cation would be stabilized by the symmetry-allowed... 

The l3C NMR spectrum of the C4H7+ cation in superacid solution shows a single peak for the three methylene carbon atoms (72) This equivalence can be explained by a nonclassical single symmetric (three-fold) structure. However, studies on the solvolysis of labeled cyclopropylcarbinyl derivatives suggest a degenerate equilibrium among carbocations with lower symmetry, instead of the three-fold symmetrical species (13). A small temperature dependence of the l3C chemical shifts indicated the presence of two carbocations, one of them in small amounts but still in equilibrium with the major species (13). This conclusion was supported by isotope perturbation experiments performed by Saunders and Siehl (14). The classical cyclopropylcarbinyl cation and the nonclassical bicyclobutonium cation were considered as the most likely species participating in this equilibrium. 

Unusual cationic species, e.g. nonclassical bicyclobutonium ions, are discernible in the solvolytic rearrangements, depending on the structure of the substrate.3,4 Computational studies using ab initio at the STO 4-31G level of theory indicate that the molecular symmetry of the unsubstituted parent cation C4H, is a bisected form of the cyclopropylmethyl system.5... 

In the light of experimental difficulties associated with the identification of intermediates, a MINDO/3 quantum-mechanical study of the singlet state ( a2) of the cyclobutylidene to methylenecyclopropane rearrangement has been carried out. It has been proposed that the whole process is initiated by electrophilic attack from the C3 methylene group of cyclobutylidene at the empty p atomic orbital on the Cl carbene site, so that a shift of electron density towards Cl can take place to give the bicyclobutane-like nonclassical carbene intermediate 4. Finally, the bicyclobutane intermediate 4 undergoes a symmetry-allowed conrotatory bond-fission process to generate methylenecyclopropane. The activation enthalpy calculated for the two steps is 8 kcal mol-1.2... 

The present review is based mainly on our publications [33,35-39,49-53]. In Section II we give a detailed description of the general reduction routine for an arbitrary relativistically invariant systems of partial differential equations. The results of Section II are used in Section III to solve the problem of symmetry reduction of Yang-Mills equations (1) by subgroups of the Poincare group P 1,3) and to construct their exact (non-Abelian) solutions. In Section IV we review the techniques for nonclassical reductions of the STJ 2) Yang-Mills equations, which are based on their conditional symmetry. These techniques enable us to obtain the principally new classes of exact solutions of (1), which are not derivable within the framework of the standard symmetry reduction technique. In Section V we give an overview of the known invariant solutions of the Maxwell equations and construct multiparameter families of new ones. 

Olah and co-workers represented the nonclassical structure as a corner-protonated nor-tricyclane (52) the symmetry is better seen when the ion is drawn as in 53. Almost all the... 

From the discussion in the foregoing section, it should be clear that the structure of the product(s) from the cycloaddition reactions is dependent on the nature of the reactants as well as reaction conditions. In this section an attempt will be made to look at the general factors influencing the selectivity and reactivity of the nonclassical A,B-diheteropentalenes. In simple terms, cycloaddition or bond formation between terminal carbon atoms occurs when the topology and symmetry of the orbitals of the reacting ylide system and the dipolarophile allow parallel approach (Figure 2). 

Such cycloadditions involve the addition of a 2tt- electron system (alkene) to a 4ir- electron system (ylide) and have been predicted to occur in a concerted manner according to the Woodward-Hoffmann rules. The two most important factors involved in the cycloaddition reactions are (i) the energy and symmetry of the reacting orbitals and (ii) the thermodynamic stability of the cycloadduct. The reactivity of 1,3-dipolar systems has been successfully accounted for in terms of HOMO-LUMO interactions using frontier MO theory (71TL2717). This approach has been extended to explain the 1,3 reactivities of the nonclassical A,B-diheteropentalenes <77HC(30)317). 

Extensive theoretical studies have been carried out in order to rationalize the rearrangement pathways in the cationic system92. The characterization of the potential surface of C9H9+ cations93 at MP2-, MP3- and MP4(SDQ)/6-31G levels showed that the open 9-barbaralyl cation 83a is more stable than the completely charge delocalized (nonclassical) structure of Z)1h symmetry, and the bicyclo[3.2.2]nona-3,6,8-trien-2-yl cation 83b by 6.9... 

The observed fivefold symmetry in the 1H and 13C NMR spectra even at very low temperature (— 150°C) with no line broadening leaves only two alternatives for the structure of the dication the nonclassical fivefold symmetrical, static structure 437 or... 

Whereas the dipole moments of thieno[3,2-b]thiophene (3) and selenolo[3,2-6]selenophene (4) are 0.00 D, the mixed system selenolo[3,2-6]thiophene (29) exhibits a dipole moment of 0.30 D. Perturbation of the symmetry of (3) by the introduction of an ethyl group in the molecule at C-2 generates a dipole moment of 0.30 D. A study of the other classical thienothiophenes, selenoloselenophenes and selenolothiophenes shows that the [2,3-6]-annelated systems exhibit a slightly higher dipole moment compared to the [3,4-6]- or [2,3-c]-annelated systems (76AHC(19)123). In the case of the nonclassical thiophenes (Id X = Y = S), (Id X = S, Y=NH) and (Id X = S, Y = 0) the predicted dipole moments are 0.00, 0.15 and 3.21 D respectively (74JA1817). Experimental verification is not possible since none of these compounds are known although it should be of interest to determine the dipole moments of the tetraphenyl derivatives (6) and (12) and the pentaphenyl compound (13). 

The totally degenerate barbaralyl cation, a C,HJ ion, has been proposed to be a rapidly equilibrating 9-barbaralyl cation [2] or the rapidly equilibrating nonclassical [37] which has Djh symmetry. Ahiberg et al. (1981) have perturbed a i C-Iabelled barbaralyl cation by eight deuteriums P CH(CD)g] chemical shifts created by this perturbation show that the ion has structure [2]. 

Olah and co-workers represented the nonclassical structure as a corner-protonated nortricyclane (62) the symmetry is better seen when the ion is drawn as in 63. Almost all the positive charge resides on C-1 and C-2 and very little on the bridging carbon C-6. Other evidence for the nonclassical nature of the 2-norbomyl cation in stable solutions comes from heat of reaction measurements that show that the 2-norbomyl cation is more stable (by 6-10 kcal mol or 25 0 kJ mol than would be expected without the bridging.Studies of ir spectra of the 2-norbomyl cation in the gas phase also show the nonclassical stmcture. Ab initio calculations show that the nonclassical stracture corresponds to an energy minimum. ... 

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