Broken spatial symmetry

Unfortunately, if spatial symmetrization (in contrast to spin symmetrization considered above) is used, discontinuities will arise when spatial symmetrybreaking nuclear motion is considered. Thus broken spatial symmetry is best treated via the methods discussed in the next section that unfortunately do not yield states of pure spatial symmetry. (The alternative energy-projection method of Noodleman when self-consistently applied also becomes discontinuous during nuclear symmetry breaking.)... 

In the latter case, one has to be aware of solutions with broken spatial symmetry. This problem arises also in NCSDFT the (initial) symmetry of a system, as described by a scalar Hamiltonian, is destroyed by the vector field term proportional to as, which, similarly to an external magnetic field, reduces the spatial symmetry of the one-electron Hamiltonian. In spin-polarized calculations including SO interaction, the conventional collinear approach, where only one component of the spin-density s = Tr a p) is used in the definition of the xc energy functional, has the major drawback of breaking the spatial symmetry of the energy functional [18,64]. 

All the early applications, as well as the vast majority of many-body calculations performed to this day, are single-reference in character. They start from an appropriate single determinant, usually (but not necessarily) Hartree-Fock, and include correlation by finite-order perturbation or infinite-order summation of certain perturbation terms (the CC approach). The starting determinant may be closed-shell or open-shell the latter leads to contaminated spin states and occasionally to broken spatial symmetry [8], but acceptable results are obtained in most cases. States involving degeneracy or quasidegeneracy, where a single determinant... 

For our SCF calculation the way to obtain such a solution is to optimize the orbitals not for the energy of determinant 7.6, but for the average energy of both determinants. This is termed the imposition of symmetry and equivalence restrictions. It involves imposing a constraint on a variational calculation, and consequently the symmetry and equivalence restricted solution will have an energy no lower than the broken symmetry solution it will usually have a higher energy. We may note that in a UHF calculation we impose neither spin nor spatial symmetry and equivalence restrictions — the -terms restricted and unrestricted were first used in exactly this context of whether to impose symmetry constraints on the wave function. 

Prior to 1956, it was believed that all reactions jn nature obeyed the law of conservation of parity, so that there was no fundamental distinction between left and right in nature. However, Yang and Lee pointed out that in reactions involving the weak interaction between particles, parity was not conserved, and that experiments could be devised that would absolutely distinguish between right and left. This was the first example of a situation where a spatial symmetry was found to be broken by one of the fundamental interactions. 

There is a general statement [17] that spin-orbit interaction in ID systems with Aharonov-Bohm geometry produces additional reduction factors in the Fourier expansion of thermodynamic or transport quantities. This statement holds for spin-orbit Hamiltonians for which the transfer matrix is factorized into spin-orbit and spatial parts. In a pure ID case the spin-orbit interaction is represented by the Hamiltonian //= a so)pxaz, which is the product of spin-dependent and spatial operators, and thus it satisfies the above described requirements. However, as was shown by direct calculation in Ref. [4], spin-orbit interaction of electrons in ID quantum wires formed in 2DEG by an in-plane confinement potential can not be reduced to the Hamiltonian H s. Instead, a violation of left-right symmetry of ID electron transport, characterized by a dispersion asymmetry parameter Aa, appears. We show now that in quantum wires with broken chiral symmetry the spin-orbit interaction enhances persistent current. 

Perhaps the greatest need for Brueckner-orbital-based methods arises in systems suffering from artifactual symmetry-breaking orbital instabili-ties, " ° where the approximate wavefunction fails to maintain the selected spin and/or spatial symmetry characteristics of the exact wavefunction. Such instabilities arise in SCF-like wavefunctions as a result of a competition between valence-bond-like solutions to the Hartree-Fock equations these solutions typically allow for localization of an unpaired electron onto one of two or more symmetry-equivalent atoms in the molecule. In the ground Ilg state of O2, for example, a pair of symmetry-broken Hartree-Fock wavefunctions may be constructed with the unpaired electron localized onto one oxygen atom or the other. Though symmetry-broken wavefunctions have sometimes been exploited to produce providentially correct results in a few systems, they are often not beneficial or even acceptable, and the question of whether to relax constraints in the presence of an instability was originally described by Lowdin as the symmetry dilemma. ... 

Zi) < E(Zt). The remaining two states arise from homolytic dissociation of the bond and therefore are diradical in character. Both singlet and triplet states arise. If any residual interaction persists (i.e., if the bond broken was a tt bond or the products are held together in a solvent cage), then the triplet diradical state is lower than the singlet diradical state. Otherwise the two have the same energy. Since the electrons end up in different orbitals, the spatial symmetry of the diradical states is determined by the symmetry properties of and The diradical is symmetric (S) if and are both... 

The existence of an instability means that it is possible to find solutions of symmetry lower than the original one. One speaks of broken symmetry solutions. In Fukutome s classification system (Table II), the new solutions can belong to the same or to a different class. In the former case it is the spatial symmetry which has been lowered. If the new solution belongs to a different Fukutome class, the symmetry with respect to spin and/or time reversal has been lowered. 

However, there is a drawback. In addition to the fact that the spatial symmetry is broken, the spin contamination of this spin symmetry broken UHF reference is unusually large, ((25 -F 1) = 2.83). This is because of the presence of two closely lying triplet states (triplets arising from 6b2 and 7b2u orbitals) that can mix. We note that the (25 -F 1) of CCSD(T) is 2.12 compared to 2.83 of the reference UHF state. This observation is consistent with findings in the hterature [72, 78]. However, in this particular case, it is also important to note that the UHF reference (in C2V symmetry) has large Tj amplitudes (those Ti amplitudes are strictly zero by symmetry in D2h)- The Brueckner reference eliminates large Ti amplitudes present in the symmetry broken solution. As further discussed below, the... 

In the majority of calculations, the RHF electron density shows (at molecular geometry close to the equilibrium) spatial symmetry identical with the point symmetry group (the nuclear configuration) of the Hamiltonian. But the RHF method may also lead to broken symmetry solutions. For example, a system composed of the equidistant H atoms uniformly distributed on a circle shows bond alternation, i.e. symmetry breaking of the BOAS type. ... 

Eqs. (8.2) and (8.3) is that the effective interaction between two sites on the polymer depends on their instantaneous position, and only on the entire macromolecule conformation in an average (implicit) sense via the direct and collective pair correlations. This simplification does not preclude describing situations of broken conformational symmetry, such as polymer collapse, solvated electron localization, or spatially inhomogeneous conformational characteristics such as occur in star polymers (see Section IX)... 

In NaCl (18189), this principle would require all atoms to be identical. Clearly this symmetry is already broken by the constraint imposed by the chemical formula which requires half the atoms to be Na" " and half CP. However, all the Na" " ions are indistinguishable from each other, and the same is true for the CP ions. The bonds likewise, six for each formula unit, are also equivalent in the bond graph (Fig. 2.4). The crystal structure (Fig. 1.1) is then determined by applying the principle of maximum symmetry to the constraints imposed by three-dimensional space as described in Section 11.2.2.4. The crystal structure is thus uniquely determined by the principle of maximum symmetry and the chemical and spatial constraints. 

Further, by the definition of broken symmetry, the proven asymmetry of the source dipole in the vacuum flux must receive virtual energy and output observable energy. Since we see only the 3-spatial output, to us it appears that the source dipole somehow extracts from the vacuum some unobservable energy, transduces it, and then pours it out as the observable EM energy that we do observe. 

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