Broken symmetries symmetry breaking

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. 

The major drawback of symmetry-broken solutions is the difficulty to exploit them at a higher level of accuracy. There are three possible attitudes (besides simply refusing symmetry -breaking). 

The dominant practice in Quantum chemistry is optimization. If the geometry optimization, for instance through analytic gradients, leads to symmetry-broken conformations, we publish and do not examine the departure from symmetry, the way it goes. This is a pity since symmetry breaking is a catastrophe (in the sense of Thom s theory) and the critical region deserves attention. There are trivial problems (the planar three-fold symmetry conformation of NH3 is a saddle point between the two pyramidal equilibrium conformations). Other processes appear as bifurcations for instance in the electron transfer... 

Again, the Pi are the equivalent densities obtained from symmetry breaking. Let us clarify this concept by using the examples given above. In the B2 case, the two equivalent symmetry broken Kohn-Sham Slater determinants are... 

The finite-temperature field theory has been the most popular approach to equilibrium phase transitions (L. Dolan et.al., 1974). The effective potential of quantum fluctuations around a classical background provides a convenient tool to describe phase transitions. The symmetry breaking or restoration mechanism can be illustrated by a scalar field model with broken symmetry... 

Therefore, the system restores the broken symmetry when T >TC but breaks the symmetry when T [Pg.277]

The high symmetry of the chains can be broken in the limit-ordered crystalline lattice of polymers (symmetry breaking). 

Chiral molecules have a nonsuperposable mirror image (see Fig. 11.1) and so possess intrinsic handedness in three-dimensional space. In a perfectly symmetric, chi-rally unbiased world, each handed version (enantiomer) must exist with an equal probability. The observed preference of one enantiomer over the other in biomolecules implies that this symmetry has been broken. The initial induction of a symmetry-breaking chiral excess is the sine qua non of eventual chiral dominance. Several mechanisms have been proposed to bring about chiral symmetry breaking. 

The second Higgs field acts in such a way that if the vacuum expectation value is zero, ( ) = 0, then the symmetry breaking mechanism effectively collapses to the Higgs mechanism of the standard SU(2) x U(l) electroweak theory. The result is a vector electromagnetic gauge theory 0(3)/> and a broken chiral SU(2) weak interaction theory. The mass of the vector boson sector is in the A(3) boson plus the W and Z° particles. 

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. 

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. 

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