Symmetry-breaking phenomena

Another aspect of wave function instability concerns symmetry breaking, i.e. the wave function has a lower symmetry than the nuclear framework. It occurs for example for the allyl radical with an ROHF type wave function. The nuclear geometry has C21, symmetry, but the Cay symmetric wave function corresponds to a (first-order) saddle point. The lowest energy ROHF solution has only Cj symmetry, and corresponds to a localized double bond and a localized electron (radical). Relaxing the double occupancy constraint, and allowing the wave function to become UHF, re-establish the correct Cay symmetry. Such symmetry breaking phenomena usually indicate that the type of wave function used is not flexible enough for even a qualitatively correct description. 

The fact that states with definite parities are almost never observed experimentally cannot be explained alone by the persistence of mixed-parity states. Obviously, symmetry-breaking phenomena are in operation whether or not external perturbations are present. For species in condensed media, symmetry breaking is brought about by intermolecular interactions. In the extreme case of solids, these interactions are so strong that it is proper... 

The Lithium Superoxide Radical Symmetry Breaking Phenomena and Potential Energy Surfaces. 

For this equality to hold, we would have to imagine that atoms within the same molecule were uncorrelated. The different atoms are, however, bonded and the configurational constraints imposed by the bonding reduce the mixing entropy from that estimated with the right-hand side of (27). Further, for the physical reasons discussed in the preceding section, we require an accurate treatment of the entropy contribution to understand transformation of phases, spontaneous assemblies and many other symmetry breaking phenomena. 

See, for example W. D. Allen, D. A. Horner, R. L. DeKock, R. B. Remington, and H. F. Schaefer, Chem. Phys., 133, 11 (1989). The Lithium Superoxide Radical Symmetry Breaking Phenomena and Potential Energy Surfaces. 

Symmetry breaking is a universal phenomenon, from eosmology to the microscopic world, a perfectly familiar and daily experience whien should not generate the reluctance that it induces in some domains of Physics, and especially in Quantum Chemistry. In elassieal physics, the symmetry breaking of an a-priori symmetrical problem is sometimes refered to as the lack of symmetry of the initial conditions. But it may be a deeper phenomenon, the symmetry-broken solutions being more stable than the symmetrical one. 

One is purely formal, it concerns the departure from symmetry of an approximate solution of the Schrodinger equation for the electrons (ie within the Bom-Oppenheimer approximation). The most famous case is the symmetry-breaking of the solutions of the Hartree-Fock equations [1-4]. The other symmetry-breaking concerns the appearance of non symmetrical conformations of minimum potential energy. This phenomenon of deviation of the molecular structure from symmetry is so familiar, confirmed by a huge amount of physical evidences, of which chirality (i.e. the existence of optical isomers) was the oldest one, that it is well accepted. However, there are many problems where the Hartree-Fock symmetry breaking of the wave function for a symmetrical nuclear conformation and the deformation of the nuclear skeleton are internally related, obeying the same laws. And it is one purpose of the present review to stress on that internal link. 

The most famous case concerns the symmetry breaking in the Hartree-Fock approximation. The phenomenon appeared on elementary problems, such as H2, when the so-called unrestricted Hartree-Fock algorithms were tried. The unrestricted Hartree-Fock formalism, using different orbitals for a and p electrons, was first proposed by G. Berthier [5] in 1954 (and immediately after J.A. Pople [6] ) for problems where the number of a andp electrons were different. This formulation takes the freedom to deviate from the constraints of being an eigenfunction. 

Tamura R, Takahashi H, Fujimoto D, Ushio T (2007) Mechanism and Scope of Preferential Enrichment, a Symmetry-Breaking Enantiomeric Resolution Phenomenon. 269 53-82 Tanaka H, see Matile S (2007) 277 219-250... 

Crystallization and reactivity in two-dimensional (2D) and 3D crystals provide a simple route for mirror-symmetry breaking. Of particular importance are the processes of the self assembly of non-chiral molecules or a racemate that undergo fast racemization prior to crystallization, into a single crystal or small number of enantiomorphous crystals of the same handedness. Such spontaneous asymmetric transformation processes are particularly efficient in systems where the nucleation of the crystals is a slow event in comparison to the sequential step of crystal growth (Havinga, 1954 Penzien and Schmidt, 1969 Kirstein et al, 2000 Ribo et al 2001 Lauceri et al, 2002 De Feyter et al, 2001). The chiral crystals of quartz, which are composed from non-chiral Si02 molecules is an exemplary system that displays such phenomenon. 

The results of Soai and coworkers [35] as well as those of Singleton and Vo [36] suggest the occurrence of mirror-symmetry breaking in the Soai reaction, which is a rare phenomenon that is basically limited to a few crystallization processes [37]. Under certain kinetic conditions, chiral autocatalysis... 

Fluctuations are inherent to any experimental chemical system. Even if these fluctuations are infinitesimally small, they are sufficient to drive the system away from an unstable state. The optically active state is characterized by two equivalent options starting from an unstable racemic situation, the system can evolve into either an R configuration or into an S one. However, each individual experiment remains unpredictable as to which of the optically active states the system will move towards. For a large number of experiments an equal and random distribution between R and S dominance is expected if the initial conditions do not involve any preferences. Due to this unpredictability of the chiral configuration, the phenomenon of mirror-symmetry breaking introduces another element of stochastic behavior into chemical reactions different from that occurring in clock reactions [38,39]. 

The allyl radical is of particular theoretical interest as a small molecule which exhibits the phenomenon of doublet instability, or symmetry breaking. As a consequence, the restricted open-shell HF (ROHF) method fails to reproduce the C2v equilibrium structure predicted by experimental smdies [74]. One must, therefore, resort to unrestricted (UHF or UKS) or multiconfigurational (MCSCF) methods. The results obtained using different functionals are reported in table 4. As for the methyl radical, a good agreement is found between the... 

So the primordial strong inhomogeneities in the distribution of total, dark matter and baryon density in the Universe is the new important phenomenon of cosmological models, based on particle models with hierarchy of symmetry breaking. 

Had this process proceeded unchecked it would have by-passed several important stages in cosmic evolution between t = 10 and Is, such as baryogenesis, electroweak symmetry breaking, combination of free quarks to form hadrons and interconversion between protons and neutrons. Not to interrupt this orderly evolution it would therefore be useful to have the phase transition postponed for a while. Many phase transitions are indeed known to be delayed by the phenomenon of supercooling. Why not this one ... 

We say that the chirality transfer occurs from a chiral to an achiral molecule if, in presence of the chiral molecule, the achiral one gains a property observable by a chirality-sensitive-method (CSM). The origin of this phenomenon is a symmetry breaking incident induced by intermoleculecular interactions between the two individua which shifts the achiral molecule from the achiral point symmetry group to the chiral one. This means that achiral molecule possessing either a symmetry plane, or symmetry center, or 2/t-fold inversion axis looses one or more such symmetry elements as a result of intermolecular interactions, and thus it falls into a chiral point symmetry group. 

It should be emphasized that seven c,b, t, Vt, IT/Z, gluon) out of the 16 particles have been predicted by the SM before they have been observed experimentally. The SM is completed by the introduction of an additional particle called the Higgs boson. The Higgs boson plays an important role in the SM as it provides an explanation for the masses of the elementary particles and gives rise to the phenomenon of elec-troweak symmetry breaking. Despite the large effort, the experimental verification of the existence of the Higgs boson has not been crowned with success so far. 

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