Symmetry-breaking

Symmetry breaking has also been found in the structure of the a form of sPS. Various models of packing 

5 PACKING EFFECTS ON THE CONFORMATION OF POFYMER CHAINS IN THE CRYSTAFS THE CASE OF ALIPHATIC POLYAMIDES 

Regardless of the kind of nylon, the main structural features of the polymer chains in the crystals are (1) the tendency of amide groups of adjacent chains to form 100% hydrogen bonds [118] (2) the frans-planar conformation of the aliphatic portions of chains [118] (3) the planar conformation of -CH2-CO-NH amide groups [119] and (4) a nearly linear geometry for hydrogen 

H 0=C interactions [118]. Furthermore, nylons normally crystallize with chains in nearly extended conformations small contractions of chain periodicity from the value of the fuUy extended chain may be allowed in order to accomplish the stringent requirement that 100% hydrogen bonds are always formed. As suggested by Natta and Corradini [119a] and Corradini et al. [119b], such deviations are in all cases associated with 

It is worth noting that in order to preserve the extended and straight conformation of the chains, for each couple of torsion angles and O adjacent to the same amide bond, deviations 5 from 180° should be nearly identical but of opposite sign [120], The values of and O adjacent to the same amide bond are, therefore. 

The Lagrangian formulation is equally effective for discussing the idea of spontaneous breakdown of symmetry and the effect thereof can be demonstrated in terms of a simple example [18, 19]. 

Consider a system consisting of scalar particles only and described by a Lagrangian density as a function of the field f . Let 

The lower curve corresponds to p 0 and appears to have negative mass for the field j . The potential has two minima which satisfy 

The extremum j = 0 does not correspond to the energy minimum, which occurs at f = u. If r (x) represents fluctuations around the minimum at v, one can write 

If this form of the field, translated to j + v, is now substituted into the Lagrangian it becomes 

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Prigogine I and Lefever R 1968 Symmetry breaking instabilities in dissipative systems J. Chem. Phys. 48 1695-700... 

There are a number of other technical details associated with HF and other ah initio methods that are discussed in other chapters. Basis sets and basis set superposition error are discussed in more detail in Chapters 10 and 28. For open-shell systems, additional issues exist spin polarization, symmetry breaking, and spin contamination. These are discussed in Chapter 27. Size-consistency and size-extensivity are discussed in Chapter 26. 

Coupling to these low-frequency modes (at n < 1) results in localization of the particle in one of the wells (symmetry breaking) at T = 0. This case, requiring special care, is of little importance for chemical systems. In the superohmic case at T = 0 the system reveals weakly damped coherent oscillations characterised by the damping coefficient tls (2-42) but with Aq replaced by A ft-If 1 < n < 2, then there is a cross-over from oscillations to exponential decay, in accordance with our weak-coupling predictions. In the subohmic case the system is completely localized in one of the wells at T = 0 and it exhibits exponential relaxation with the rate In k oc - hcoJksTY ". 

Looking back at the frequency output once again, we note that its symmetry is A", indicating that this is a symmetry-breaking mode. The molecular structure has C, symmetry, indicating that there is a single plane of symmetry (in this case, the plane of the carbon atoms). The structure wants to move down the PES to a lower-energy structure of equal or lower symmetry. 

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 optimum value of c is determined by the variational principle. If c = 1, the UHF wave function is identical to RHF. This will normally be the case near the equilibrium distance. As the bond is stretched, the UHF wave function allows each of the electrons to localize on a nucleus c goes towards 0. The point where the RHF and UHF descriptions start to differ is often referred to as the RHF/UHF instability point. This is an example of symmetry breaking, as discussed in Section 3.8.3. The UHF wave function correctly dissociates into two hydrogen atoms, however, the symmetry breaking of the MOs has two other, closely connected, consequences introduction of electron correlation and spin contamination. To illustrate these concepts, we need to look at the 4 o UHF determinant, and the six RHF determinants in eqs. (4.15) and (4.16) in more detail. We will again ignore all normalization constants. 

The induced absorption band at 3 eV does not have any corresponding spectral feature in a(co), indicating that it is most probably due to an even parity state. Such a state would not show up in a(co) since the optical transition IAK - mAg is dipole forbidden. We relate the induced absorption bands to transfer of oscillator strength from the allowed 1AS-+1 (absorption band 1) to the forbidden 1 Ak - mAg transition, caused by the symmetry-breaking external electric field. A similar, smaller band is seen in EA at 3.5 eV, which is attributed to the kAg state. The kAg state has a weaker polarizability than the mAg, related to a weaker coupling to the lower 1 Bu state. 

Recent applications of this method have shown that isolated benzene retains perfect six-fold symmetry to within numerical accuracy of the calculation [69] whereas in 5CB subtle symmetry breaking in the constituent phenyl rings occurs by contraction of the C—C bonds along the long axis of the molecule [69]. 

Symmetry breaking associated with chiral phenomena is a theme that recurs across the sciences—from the intricacies of the electroweak interaction and nuclear decay [1-3] to the environmentally influenced dimorphic chiral structures of microscopic planktonic foraminifera [4, 5], and the genetically controlled preferential coiling direction seen in the shells of snail populations [6, 7]. 

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. 

Quantum chemistry experienees two types of symmetry breakings. 

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. 

For Sz=0 problems, where the ground state is a singlet state, the use of such a wave function appeared to give significantly lower energies than the orthodox symmetry-adapted solution in many problems, as illustrated below. Later on other types of symmetry breaking have been discovered and Fukutome [7] has given a systematics of the various HF instabilities in a fundamental paper. 

Another well-known atomic HF symmetry breaking is the O problem but it is more... 

For 4t (Uthe relevant perturbation consists in perturbing the covalent (or neutral) VB structures by their interaction with the ionic ones this is the strongly correlated or magnetic domain. So that the Hartree Fock symmetry breaking occurs in a zone which covers the whole magnetic domain and a significant part of the "weakly" correlated domain... 

For more complex problems such as multiple bonds (N2for instance [13-14] and Metal-Metal bonds [15-17]) or extended systems (the K system of cyclic polyenes, among others), the symmetry-breakings may take several forms since one may leave different space-and spin-symmetry constraints independently or simultaneously. For C2for... 

The multiplicity of symmetry breakings have been explored in details in N2 where they occur near the equilibrium interatomic distance [12]. 

The fact that symmetry breaking occurs at shorter interatomic distances for multiple bonds than for single bonds may be understood within two different languages. One refers to the instability conditions of the symmetry-adapted solution. In multiple bonds some bonding... 

Symmetry breakings have been studied for systems with one electron percenter such as the... 

When the symmetry breaking of the wave function represents a biased procedure to decrease the weights of high energy VB stmctures which were fixed to umealistic values the tymmetry and single determinant constraints, one may expect that the valence CASSCF wave function will be symmetry-adapted, since this function optimizes the coefficients of all VB forms (the valence CASSCF is variational determination of the best valence space and of the best valence function, i.e. an optimal valence VB picture). In most problems the symmetry breaking should disappear when going to the appropriate MC SCF level. This is not always the case, as shown below. 

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... 

The HF symmetry-breaking also occurs in the valence shell for A systems when the... 

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