Resonances symmetry breaking

SYMMETRY BREAKING IN CASE OF WEAK RESONANCE BETWEEN POLARIZED FORMS... 

The symmetry-breaking of the HF function occurs when the resonance between the two localized VB form A+...A and A...A+ is weaker than the electronic relaxation which one obtains by optimizing the core function in a strong static field instead of keeping it in a weak symmetrical field. If one considers for instance binding MOs between A and A they do not feel any field in the SA case and a strong one in the SB solution. The orbitals around A concentrate, those around A become more diffuse than the compromise orbitals of A+ 2 and these optimisations lower the energy of the A. A form. As a... 

CAS F calculations are not a universal solution to symmetry-breaking of the wave functions, and for such weak resonance problems it is far more reliable to start from state average solutions which treat on an equal footing the two configurations which interact weakly. 

Of course the conformational symmetry breaking may appear or disappear depending on the level of sophistieation of the eomputation, which may unduly favor one term of the crucial (relaxation/resonance) ratio. As an example we would like to mention an open... 

MSN. 155. T. Petrosky and I. Prigogine, Poincare resonances and the extension of classical dynamics, in Special issue Time symmetry breaking in classical and quantum mechanics, Chaos, Solitons and Fractals 7, 441 97 (1996). 

PoIIicott-Ruelle Resonances and Time-Reversal Symmetry Breaking Microscopic Construction of the Diffusive Modes... 

IV. POLLICOTT-RUELLE RESONANCES AND TIME-REVERSAL SYMMETRY BREAKING... 

As well as providing a means of measuring 1 H/2H-exchange in proteins, NMR is a most powerful technique for studying the mobility of individual amino acids. For example, the rotational freedom of the aromatic side chains of tyrosine and phenylalanine about the C 3—Cy bond is readily studied by various NMR methods. ]H NMR can detect whether or not the aromatic ring is constrained in an anisotropic environment. In an isotropic environment or where there is rapid rotation on the NMR time scale, the 3 and 5 protons of phenylalanine and tyrosine are symmetrically related, as are the 2 and 6 (structures 1.12). The resultant spectrum is of the AA BB type, containing two pairs of closely separated doublets. But if there is slow rotation in an anisotropic environment, the symmetry breaks down to give four separate resonances (an ABCD spectrum), since the 5 and 6 protons are in different states from the 2 and 3. At an intermediate time... 

A curious effect, prone to appear in near degeneracy situations, is the artifactual symmetry breaking of the electronic wave function [27]. This effect happens when the electronic wave function is unable to reflect the nuclear framework symmetry of the molecule. In principle, an approximate electronic wave function will break symmetry due to the lack of some kind of non-dynamical correlation. A typical example of this case is the allyl radical, which has C2v point group symmetry. If one removes the spatial and spin constraints of its ROHF wave function, a lower energy symmetry broken (Cs) solution is obtained. However, if one performs a simple CASSCF or a SCVB [28] calculation in the valence pi space, the symmetry breaking disappears. On the other hand, from the classical VB point of view, the bonding of the allyl radical is represented as a superposition of two resonant structures. 

However, as we will show below, a molecule possessing C2v symmetry cannot possess two degenerate resonance hybrids. Thus, the allyl radical has a three electron pi bond, which cannot be described properly at the MO monoconfigurational level. This type of symmetry breaking is called artifactual . 

The subscript C/ in (4.1.45) stands for unitary since the system Hamiltonian is invariant under general unitary transformations in case all antiunitary symmetries are broken. The effects of time reversal symmetry breaking on the level statistics were recently investigated experimentally by So et al. (1995) and Stoffregen et al. (1995) by measuring the resonance spectrum of quasi-two-dimensional microwave cavities in the presence of time reversal breaking elements such as ferrites and directional couplers. These experiments are of considerable relevance for quantum mechanics since quasi-two-dimensional microwave cavities are generally considered to be excellent models for two-dimensional quantum billiards. 

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