Broken symmetry structure

As observed in many theoretical studies of symmetric radicals (92), a HF wavefunction describing it ->- tt excited states preferentially localizes the hole on one moiety leading to a broken symmetry structure of the states with unequal bond lengths for the minimum energy structure. Similarly, for the n tt ... 

The weak correlation model Broken symmetry structure... 

Fig. 117. Band structure for YH3 for (i) the HoD3-type structure and (ii) broken symmetry structure (Kelly et al.,...

Anderson P Wand Stein D 1987 Broken symmetry, emergent properties, dissipative structures, life Self-Organizing Systems ed F E Yates (New York Plenum) pp 445-57... 

Minimal END has also been applied to a model system for intramolecular electron transfer. The small triatomic system LiHLi is bent C2v structure. But the linear structure presents an unrestricted Haiti ee-Fock (TJHF) broken symmetry solution with the two charge localized stmctures... 

Interpreted, as it is, within the standard model, Higgs theory has little meaning in the real world, failing, as it does to relate the broken symmetry of the field to the chirality of space, time and matter. Only vindication of the conjecture is expected to be the heralded observation of the field bosons at stupendous temperatures in monstrous particle accelerators of the future. However, the mathematical model, without cosmological baggage, identifies important structural characteristics of any material universe. The most obvious stipulation is to confirm that inertial matter cannot survive in high-symmetry euclidean space. 

All electron calculations were carried out with the DFT program suite Turbomole (152,153). The clusters were treated as open-shell systems in the unrestricted Kohn-Sham framework. For the calculations we used the Becke-Perdew exchange-correlation functional dubbed BP86 (154,155) and the hybrid B3LYP functional (156,157). For BP86 we invoked the resolution-of-the-iden-tity (RI) approximation as implemented in Turbomole. For all atoms included in our models we employed Ahlrichs valence triple-C TZVP basis set with polarization functions on all atoms (158). If not noted otherwise, initial guess orbitals were obtained by extended Hiickel theory. Local spin analyses were performed with our local Turbomole version, where either Lowdin (131) or Mulliken (132) pseudo-projection operators were employed. Broken-symmetry determinants were obtained with our restrained optimization tool (136). Pictures of molecular structures were created with Pymol (159). 

The question in the title can be reformulated by asking how much can be dug out of an analogy between broken symmetry in dissipative structures (such as the ripple marks generated by wind, i.e., an external perturbation, in an otherwise flat surface of sand) and broken symmetry defined as phenomena of condensed matter systems of the kind observed near the critical points. The value of Anderson s discussion is to be seen more in the deepening of the question itself than in the answer that cannot yet be final, and for the moment, according to the author, appears to be more on the negative side. 

The problem of symmetry breaking (SB) is well known and multiply discussed in literature. Briefly, we can formulate it as follows. The Hamiltonian of any system of particles forming the Universe is totally symmetric with respect to rotations and reflections in the isotropic space-time, as well as transmutations of identical and equivalent particles, whereas the real objects of the material world composed by these particles do not possess such symmetry. This is seen already from the examples that we live in a world of particles, not antiparticles, and in condensed matter, we have mostly low-symmetry structures. This circumstance can be expressed by the statement that the world is in a state of broken symmetry. An obvious explanation of the contradiction between the totally symmetric Hamiltonian and the broken symmetry of the real world is that the latter is not a solution of its Schrodinger equation. 

The picture of formation of states with broken symmetry is of significant interest. Approximately 15 billion years ago, immediately after the Big Bang at the temperature of T 1045 K the Universe system was in a quite symmetric state with respect to all its elementary particles. By expansion and cooling, a series of consequent spontaneous SB took place resulting in states with consequently decreasing symmetry. At T 104 — 103 K the first elements of condensed matter -atoms—were formed. Further cooling lead to the next consequent spontaneous SB resulting in new phases of condensed states with lower symmetry. However, the mechanism of formation of these SB states and its relation to the structure of matter is not fully explored. 

An interesting case of SB is presented by enantiomer formation. In recent papers [15] it was shown that enantiomers can be presented as the low symmetry, PJT distorted configurations of a hypothetical high-symmetry structure, and as such their interaction in the liquid phase via collisions under special conditions may lead to some kind of cooperativity and phase transition (SB) resulting in singleenantiomer broken symmetry configuration. 

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