Problem Of Symmetry Breaking Research Articles

C 21H 9Z ( Z=−3 to +3): a theoretical study on the redox behaviour of C3 symmetric fragment of C 60

B3LYP/6-31G* calculations were done on seven C 21H 9 Z fragments, where the charge, Z, is varied from −3 to +3; doublet states of the open shell systems and both singlet and triplet states of closed shell species are considered. Among them, only the D 3 h and C 3 v forms of the singlet states of C 21H 9 3− and C 21H 9 1+, and triplet-C 21H 9 1− correspond to the bowl-to-bowl inversion transition states and minimum energy structures, respectively. 6-31+G* basis set, which employs a set of diffuse functions on the non-hydrogen atoms, was employed on anionic systems. The rest encountered symmetry breaking problem, which necessitated us to go for lower symmetry structures to obtain the local minima and the bowl-to-bowl inversion transition states. The curvature, ascertained by π-orbital axis vector angles (POAV), is lowest for the trianion and highest for the cations. The triplet states of the closed shell species are found to be competitive in energy with the singlet states when Z=−1 and +3, whereas for Z=−3 and +1, the triplet states lie above the corresponding singlet states over 30 kcal/mol. The inversion barriers vary from 12.2 to 40.2 kcal/mol with C 21H 9 3− and C 21H 9 2+ having the lowest and highest inversion barriers, respectively. The electron affinities and ionization energies were calculated to assess the electron receiving and donating ability of the C 3-fragment in comparison to corannulene. The curvature and redox behavior of the C 21H 9 fragment are compared to those of C 60.

Rhombic Periodicity Lattice in Bifurcational Symmetry Breaking Problems

AbstractGeneral theory of symmetry breaking problems in branching theory is presented in the works [1–3]. Such problems are invariant relative to Euclidean space motions group and their solution which is invariant relatively this group is the rest state or uniform linear motion. At the stability loss cell structure solutions arise, which are invariant relative to the group of the definite period shifts along the definite directions, passing mutually under discrete subgroup transformations that is defined by the symmetry of elementary cell of the periodicity. Thus the Euclidean space motions group is changed by the symmetry of a certain crystallographic group.In this article for the case of 4–dimensional degeneracy of the linearized Fredholm operator abstract bifurcational symmetry breaking problems both for stationary and Andronov‐Hopf bifurcation with planar rhombic periodicity lattice are considered. Applications to hydrodynamical problems are given.

Orbital optimization: Density matrix-based procedure versus energy minimization

An iterative method based on the density matrix block diagonalization has recently been proposed. It includes part of the dynamical correlation in the orbital optimization process and provides either localized or delocalized symmetry adapted molecular orbitals. The present work compares the so-obtained orbitals with the orbitals given by the traditional energy minimization process. Both analytical derivations and numerical calculations on model systems show that the self-consistent field symmetry-breaking problem is avoided when using the density matrix-based procedure in the case of states presenting a left–right resonance, such as mixed valence compounds or bichromophoric molecules. Accurate calculations of excitation energies in a family of magnetic systems, which generally require highly correlated treatments, can also be obtained at a low computational cost when using these appropriate orbitals.

III: PROPERTIES OF COMPLEX SYSTEMS

The molecular structures, vibrational frequencies, dissociation energies and electron affinities of the Br2/Br− 2, Br2O/Br2O−, Br2O2/Br2O− 2, Br2O3/Br2O− 3, and Br2O4/Br2O− 4 systems have been investigated using density functional theory (DFT) and hybrid Hartree-Fock/density functional theory. The methods used have been carefully calibrated against a comprehensive tabulation of experimental electron affinities, (Rienstra-Kiracofe, J. C., Tschumper, G. S., Shaefer, H. F., Nandi, S. and Ellison, G. B., 2002, Chem. Rev., 102, 231). Four different types of neutral/anion energetic difference are reported in this work: the adiabatic electron affinity (EAad), the zero-point corrected EAad (EAzero), the vertical electron affinity (EAvert), and the vertical detachment energy (VDE). The basis set used in this work is of double-zeta plus polarization quality with additional s- and p-type diffuse functions, and is denoted as DZP++. The BHLYP method does well in reproducing the very limited experimental energetics, while the B3LYP method does well for the few known vibrational frequencies. The final predicted electron affinities with the BHLYP method are 3.41 eV (Br, experiment 3.36 eV), 3.02 eV (Br2, experiment 2.6 ± 0.2 eV), 3.27 eV (Br2O), 2.93 eV (Br2O2), 3.07 eV (Br2O3), and 2.54 eV (Br2O4). The global minimum structures for several of the larger dibromine oxides and their anions are unusual. For neutral Br2O2 the peroxide structure (BrOOBr) lies lowest, but for the anion a loosely bound Cs symmetry BrO-BrO− structure lies lowest for the hybrid functionals, while a C2 symmetry peroxide BrOOBr− structure lies lowest for the pure functionals. Furthermore, the C2 structures are found to exhibit an inverse symmetry breaking problem, and should be interpreted with caution. For neutral Br2O3, a chain structure Br-O3-Br lies lowest, while the complex Br···O3···Br− lies lowest for the negative ion. For neutral Br2O4, a chain structure Br-O4-Br lies lowest, while the complex BrO−···BrO3 lies lowest for the negative ion.

Structure and stability of [C2H4N]+ singlet‐state cations: Comparison between DFT and high‐level ab initio calculations

AbstractThe stationary points of the [C2H4N]+ singlet potential energy surface have been characterized by means of B3LYP/6‐311+G(3df,2p) and G2(MP2) approaches. A comparison between both sets of results shows that, although both methods yield similar values for the relative stability of the local minima, they differ significantly regarding the stability of some transition states, in particular those corresponding to 1,3‐H shift processes. This implies that the dynamics of the corresponding reaction may exhibit substantial differences if based on B3LYP or high‐level ab initio potential surfaces. The attachment of N+ to the CC bond of ethylene is preceded by a charge transfer and, as a consequence, a symmetry breaking problem arises. Our results show that inclusion of high‐order correlation contributions through the participation of triple excitations may be crucial. The B3LYP method is not a good alternative to treat this problem due to the spin contamination of the unrestricted treatment. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

Ring structure of the NO dimer radical cation: A possible new assignment of the mysterious IR absorption at 1424 cm−1

The complete assignment of IR absorptions for the nitric oxide dimer radical cation has been a difficult task for some time. Although the 1619 cm−1 band was recently assigned to the antisymmetric N–O stretch mode for the trans and cis ONNO+ structures, the 1424 cm−1 band has remained a mystery. The ring or rectangular structure of the (NO)2+ cation was examined in this research with density functional theory (DFT) and high-level ab initio methods in the prospect that it might be an energetically low-lying isomer. In conjunction with the above methods, two sets of basis functions were utilized. One is double-ζ plus polarization, and another is triple-ζ plus double polarization with f functions. The ground state of the ring (NO)2+ cation is of Bu2 symmetry. The antisymmetric N–O stretch vibrational frequency is predicted to be nonphysically large with the self-consistent field method due to the symmetry breaking problem. This fundamental is predicted in the ∼1800 cm−1 region based upon DFT methods, but the result is also doubtful because (NO)2+ exhibits the inverse symmetry breaking problem. Since these problems may impair the theoretical vibrational frequencies, higher theoretical levels, namely Brueckner coupled-cluster methods, were ultimately applied, and the harmonic vibrational frequency of this challenging mode was eventually predicted to be about 1400 cm−1. This ring structure lies only ∼5 kcal/mol above the global minimum, so that it might be observable in the laboratory. Moreover, the ring structure is predicted to lie ∼10 kcal/mol below separated NO+NO+. Since no other low-lying isomers were found, it is plausible to assign the 1424 cm−1 band to the ring structure.

Spectroscopic constants of MH and M2 (M=Tl, E113, Bi, E115): Direct comparisons of four- and two-component approaches in the framework of relativistic density functional theory

The two-component DFT-ZORA (density functional theory, zeroth order regular approximation) method is implemented into the BDF (Beijing four-component density functional) program package so that systematic and direct comparisons between two- and four-component approaches are made possible for the first time. Different implementations of the ZORA method are also compared in this work. The calculated spectroscopic constants (bond lengths, binding energies, and force constants) for MH and M2 (M=Tl, E113, Bi, E115) by the two- and four-component approaches are very similar. The ionization and excitation energies for the metals obtained by these methods also agree very well with each other. Still, minor higher order relativistic effects beyond ZORA can be identified occasionally, but can be “safely” neglected. Therefore, the applicability of transformed (two-component) Hamiltonians to valence properties is well justified. However, the computational efficiency of four-component DFT compares favorably with that of two-component DFT. The problems of symmetry breaking and different treatments of open-shell systems are discussed by taking the Bi atom as an example.

Rowers coupled hydrodynamically. Modeling possible mechanisms for the cooperation of cilia

We introduce a model system of stochastic entities, called 'rowers' which include some essentialities of the behavior of real cilia. We introduce and discuss the problem of symmetry breaking for these objects and its connection with the onset of macroscopic, directed flow in the fluid. We perform a mean field-like calculation showing that hydrodynamic interaction may provide for the symmetry breaking mechanism and the onset of fluid flow. Finally, we discuss the problem of the metachronal wave in a stochastic context.

Supersymmetry and electroweak breaking with large and small extra dimensions

We consider the problem of supersymmetry and electroweak breaking in a 5d theory compactified on an S 1/ Z 2 orbifold, where the extra dimension may be large or small. We consider the case of a supersymmetry breaking 4d brane located at one of the orbifold fixed points with the Standard Model gauge sector, third family and Higgs fields in the 5d bulk, and the first two families on a parallel 4d matter brane located at the other fixed point. We compute the Kaluza–Klein mass spectrum in this theory using a matrix technique which allows us to interpolate between large and small extra dimensions. We also consider the problem of electroweak symmetry breaking in this theory and localize the Yukawa couplings on the 4d matter brane spatially separated from the brane where supersymmetry is broken. We calculate the 1-loop effective potential using a zeta-function regularization technique, and find that the dominant top and stop contributions are separately finite. Using this result we find consistent electroweak symmetry breaking for a compactification scale 1/ R≈830 GeV and a lightest Higgs boson mass m h ≈170 GeV.

Bose glass in a large-Ncommensurate dirty boson model

The large-$N$ commensurate dirty boson model, in both the weakly and strongly commensurate cases, is considered via a perturbative renormalization-group treatment. In the weakly commensurate case, there exists a fixed line under renormalization-group flow, with varying amounts of disorder along the line. Including $1/N$ corrections causes the system to flow to strong disorder, indicating that the model does not have a phase transition perturbatively connected to the Mott insulator-superfluid (MI-SF) transition. I discuss the qualitative effects of instantons on the low-energy density of excitations. In the strongly commensurate case, a fixed point found previously is considered and results are obtained for higher moments of the correlation functions. To lowest order, correlation functions have a log-normal distribution. Finally, I prove two interesting theorems for large-$N$ vector models with disorder, relevant to the problem of replica symmetry breaking and frustration in such systems.

Electroweak symmetry breaking by strong supersymmetric dynamics at the TeV scale

We construct models in which electroweak symmetry is spontaneously broken by supersymmetric strong dynamics at the TeV scale. The order parameter is a composite of scalars, and the longitudinal components of the W and Z are strongly-coupled bound states of scalars. The usual phenomenological problems of dynamical electroweak symmetry breaking are absent: the sign of the S parameter unconstrained in strongly interacting SUSY theories, and fermion masses are generated without flavor-changing neutral currents or large corrections to the rho parameter. The lightest neutral Higgs scalar can be heavier than M_Z without radiative corrections from standard-model fields. All the mass scales in the model can be naturally related in low-scale models of supersymmetry breaking. The mu problem can also be solved naturally, and the model can incorporate perturbative unification of standard-model gauge couplings with intermediate thresholds.

Highly symmetric cellular automata and their symmetry-breaking patterns

A highly symmetric cellular automaton (in one dimension) is defined by two properties: (i) its local rule is a completely symmetric function of the n cells of its surrounding and (ii) it is invariant with respect to a regular, Abelian group acting on its M-letter alphabet, which can then be identified with this group. The motivation for studying such systems is provided by the unsolved problem of partial symmetry breaking in spin models with such an Abelian symmetry. It is shown that the symmetries (i) and (ii) do not contradict each other if and only if n and M have no common factor. As examples, the four-letter alphabets, which may be identified either with the cyclic group C(4) or with the Klein group K(4)≅ S(2)⊗ S(2) are studied for the standard n=3 surrounding. It is shown that these automata show complicated patterns of broken symmetries. These give information on the corresponding spin models, the chiral 4-state clock model and the (symmetric and general) Ashkin–Teller models, respectively.

On nonperturbative calculations in quantum electrodynamics

A new approach to nonperturbative calculations in quantum electrodynamics is proposed. The approach is based on a regular iteration scheme for solution of Schwinger-Dyson equations for generating functional of Green functions. The approach allows one to take into account the gauge invariance conditions (Ward identities) and to perform the renormalization program. The iteration scheme can be realized in two versions. The first one (perturbative vacuum) corresponds to chain summation in the diagram language. In this version in four-dimensional theory the non-physical singularity (Landau pole) arises which leads to the triviality of the renormalized theory. The second version (nonperturbative vacuum) corresponds to ladder summation and permits one to make non-perturbative calculations of physical quantities in spite of the triviality problem. For chiral-symmetrical leading approximation two terms of the expansion of the first-step vertex function over photon momentum are calculated. A formula for anomalous magnetic moment is obtained. A problem of dynamical chiral symmetry breaking (DCSB) is considered, the calculations are performed for renormalized theory in Minkowsky space. In the strong coupling region DCSB-solutions arise. For the renormalized theory a DCSB-solution is also possible in the weak coupling region but with a subsidiary condition on the value of $\alpha$.

On-line learning in RBF neural networks: a stochastic approach

The on-line learning of Radial Basis Function neural networks (RBFNs) is analyzed. Our approach makes use of a master equation that describes the dynamics of the weight space probability density. An approximate solution of the master equation is obtained in the limit of a small learning rate. In this limit, the on line learning dynamics is analyzed and it is shown that, since fluctuations are small, dynamics can be well described in terms of evolution of the mean. This allows us to analyze the learning process of RBFNs in which the number of hidden nodes K is larger than the typically small number of input nodes N. The work represents a complementary analysis of on-line RBFNs, with respect to the previous works ( Phys. Rev. E 56 (1997a) 907; Neur. Comput. 9 (1997) 1601), in which RBFNs with N≫ K have been analyzed. The generalization error equation and the equations of motion of the weights are derived for generic RBF architectures, and numerically integrated in specific cases. Analytical results are then confirmed by numerical simulations. Unlike the case of large N> K we find that the dynamics in the case N< K is not affected by the problems of symmetric phases and subsequent symmetry breaking.

Some surprising failures of Brueckner coupled cluster theory

Brueckner coupled cluster (B–CC) methods have seen a considerable rise in popularity over the last decade thanks, in part, to their apparent propensity for avoiding artifactual symmetry-breaking problems that sometimes plague Hartree–Fock-based approaches. Recent B–CC applications to problematic systems such as the tetraoxygen cation have provided encouraging examples of the success of this theory. In the present work, we examine the performance of the Brueckner technique for a number of other well-known symmetry-breaking problems, including the formyloxyl radical, the first excited state of NO2 and the nitrate radical. In these cases, B–CC methods are found to fail dramatically, predicting broken-symmetry equilibrium geometries in conflict with experimental and/or higher-level theoretical results. A framework is developed which indicates that these errors can be attributed to artificially exaggerated second-order Jahn–Teller interactions with nearby electronic states. Hence, in spite of their initial successes, Brueckner methods cannot be considered a panacea for symmetry-breaking problems.

Finite supersymmetric grand unified theory reexamined

We analyze the finite supersymmetric SU(5) grand unified model taking into account the threshold corrections at the grand unified theory and supersymmetry breaking scales and the problem of electroweak symmetry breaking which is particularly important in the case of large $\mathrm{tan}\ensuremath{\beta}.$ We find that there are still parameter regions where the low-energy experimental values are consistently reproduced and the Higgs potential parameters actually satisfy both constraints for large $\mathrm{tan}\ensuremath{\beta}$ and radiative electroweak symmetry breaking, provided that a new free parameter is introduced in the boundary condition of the Higgs soft-mass parameter, while it preserves the finiteness requirements.

Ab initio investigations on the HOSO2+O2→SO3+HO2 reaction

HOSO 2 radical is the key intermediate for the oxidation SO2 to SO3 by OH radical in the atmosphere. The structural aspects and the energetics of the reaction HOSO2+O2→SO3+HO2 have been studied using Mo/ller–Plesset (MP2) and density functional (DFT) techniques with 6-31G** and triple-ζ, quadruple-ζ, and quintuple-ζ quality basis sets including diffuse basis functions. The detailed theoretical analyses have further revealed that this reaction could proceed through the formation of intermediate complexes and an intramolecular proton transfer like transition state. The energetics of these intermediate reactions has been studied in detail. The use of MP2 methods to study such radical mechanisms had some characteristic symmetry-breaking problem with larger basis sets. This unphysical situation with larger basis set MP2 calculations in this hypervalent system has been explained through the interpretation of the relevant energy surface.

On the electronic structure of Li2 (X1) and its changes with internuclear distance

A compact, yet accurate, and strictly virial-compliant ab initio electronic wavefunction for ground-state Li2 is exploited for a study of the molecule's electronic structure and electron density. Symmetry-breaking problems that emerge at the single-configuration level are solved in a multiconfigurational spin-coupled approach that enables simultaneous optimization of angularly correlated “resonating” configurations. Particular emphasis is placed on the accurate determination of the electron density's bifurcation points and of the quadrupole moment as a function of internuclear distance R. Tentative connections are drawn between the R dependence of the electron density's topological structure and quadrupole moment and that of the electronic wavefunction. Computation of the latter constitutes the first application to systems other than isolated atoms of the optimized basis set generalized multiconfiguration spin-coupled method, which entails use of nonorthogonal orbitals and Slater-type basis functions with variationally optimized exponential parameters. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 378–397, 2000

On the performance of density functional theory for symmetry-breaking problems

Density functional theory (DFT) has been applied to three open-shell molecules (NO 3, O 4 +, and O 2 +) for which the unrestricted Hartree–Fock (UHF) wavefunction breaks spatial symmetry. In contrast to Hartree–Fock, all of the standard DFT methods we employed yielded symmetric densities for each of the molecules considered. Symmetry-broken solutions were obtained with DFT only when we used hybrid functionals including unusually large fractions of Hartree–Fock exchange. The exchange functional seems more important than the correlation functional in determining whether symmetry is preserved or broken.

Symmetry-Breaking Convective Dynamos in Spherical Shells

The convective dynamo is the generation of a magnetic field by the convective motion of an electrically conducting fluid. We assume a spherical domain and spherically invariant basic equations and boundary conditions. The initial state of rest is then spherically symmetric. A first instability leads to purely convective flows, the pattern of which is selected according to the known classification of O(3) -symmetry-breaking bifurcation theory. A second instability can then lead to the dynamo effect. Computing this instability is now a purely numerical problem, because the convective flow is known only by its numerical approximation. However, since the convective flow can still possess a nontrivial symmetry group G0 , this is again a symmetry-breaking bifurcation problem. After having determined numerically the critical linear magnetic modes, we determine the action of G0 in the space of these critical modes. Applying methods of equivariant bifurcation theory, we can classify the pattern selection rules in the dynamo bifurcation. We consider various aspect ratios of the spherical fluid domain, corresponding to different convective patterns, and we are able to describe the symmetry and generic properties of the bifurcated magnetic fields.