Symmetry Restrictions Research Articles

“There is no such animal (אין חיה כזו)”—Lessons of a discovery

The discovery of quasicrystals by Dan Shechtman in the early 1980s was a conspicuous event in materials science not only because it led to the production of a plethora of new materials but also because it signified the demise of a dogma in the science of condensed phase materials concerning symmetry restrictions. Having the discovery recognized was not easy and it required stamina on Shechtman’s part. The story of the quasicrystal discovery offers a set of lessons that might be useful to remember in similar situations.

Precise ab initio calculations of the 3d transition-metal clusters: Sc2

The ground ${}^5\Sigma _u^ -$5Σu− state of Sc2 was studied by the valence multireference configuration interaction method with single and double excitations plus Davidson correction (MRCISD(+Q)) at the complete basis set limit. The calculations were made under C2v symmetry restrictions, which allowed us to obtain at the dissociation limit the Sc atoms in different states (in all previous studies of Sc2 the D2h symmetry group was employed). From the Mulliken population analysis and energy calculations follows that in the ground state Sc2 dissociates in one Sc in the ground state and the other in the second excited quartet state, 4Fu. The corrected parameters of the ground potential curve are the following: Re = 5.2 bohr, De = 50.37 kcal/mol, and ωe = 234.5 cm-1. The dissociation energy in respect to the dissociation on two Sc in the ground states was estimated as De = 9.98 kcal/mol.

A CASPT2 Description of the Electronic Structures of FeO3(-/0) in Relevance to the Anion Photoelectron Spectrum.

DFT and multireference methods were used to investigate the electronic structure of FeO3 and FeO3(-) clusters. Geometries of different spin multiplicities and conformations were optimized without any symmetry restrictions at the BP/QZVP level and further refined with the CASPT2 method. Although the latter type of calculations were performed by using the C2v point group, all low-lying states relevant to the photoelectron spectrum were found to correspond to or to resemble closely a planar D3h iron trioxide with no bonds between the oxygen atoms. Depending on the computational method used, the ground state of the FeO3(-) anion can be either (2)E'' or (4)A1'. The two lowest binding energy bands of the photoelectron spectrum of FeO3(-) can only be ascribed to electron detachments from the (2)E'' state. The first band is the result of a transition to the (1)A1' ground state of FeO3, whereas the second band originates from the first excited (3)E'' state. A harmonic vibrational analysis of the symmetric stretch shows that the observed vibrational progressions of these two bands in the photoelectron spectrum of FeO3(-) are also in line with the assignment. A molecular orbital analysis led to the conclusion that the electronic structures of the anionic and neutral clusters can formally be described by an oxidation state of iron of +5 and +6, respectively. A population analysis, on the contrary, points to an ionization that takes place on the oxygen atoms.

US import demand for apple: source differentiated almost ideal demand system approach

We adopted a restricted version of Source Differentiated Almost Ideal Demand System (RSDAIDS) to analyse the US import demand for fresh apples, apple juice and other processed apples after differentiating each type of apple by import origins. Seasonality and time trend variables were included in the RSDAIDS model and the general demand restrictions of adding-up, homogeneity and slutsky symmetry were imposed. The system of equations is then estimated using Seemingly Unrelated Regression (SUR) and the results of the estimation are presented in the form of parameter estimates, as well as price and expenditure elasticities.

Dynamic Shortest Path Algorithm in Stochastic Traffic Networks Using PSO Based on Fluid Neural Network

The shortest path planning issure is critical for dynamic traffic assignment and route guidance in intelligent transportation systems. In this paper, a Particle Swarm Optimization (PSO) algorithm with priority-based encoding scheme based on fluid neural network (FNN) to search for the shortest path in stochastic traffic networks is introduced. The proposed algorithm overcomes the weight coefficient symmetry restrictions of the traditional FNN and disadvantage of easily getting into a local optimum for PSO. Simulation experiments have been carried out on different traffic network topologies consisting of 15-65 nodes and the results showed that the proposed approach can find the optimal path and closer sub-optimal paths with good success ratio. At the same time, the algorithms greatly improve the convergence efficiency of fluid neuron network.

Chiral non-periodic surface-confined molecular nanopatterns revealed by scanning tunnelling microscopy

Enantiomers of a 2,6-napthyl derivative—bearing a long alkyl chain at one position and a lactate group at the other—were synthesised and the formation of self-assembled physisorbed monolayers of the molecules on highly oriented pyrolytic graphite was studied by scanning tunnelling microscopy. The results of imaging this monolayer reveal the rare formation of an irregular packing which has been analysed in detail. The enantiomers appear as chiral dimers on the surface. The alkyl chains direct the assembly to a large degree, since they run along the graphite axes, but the chiral lactate group coupled with the symmetry restrictions of the naphthyl ring apparently causes a frustration in the packing. The dimers pack into rows but their relative orientation along the long row axis is not fixed, inducing a degree of disorder in this direction. However, the molecular dimers are perfectly ordered in a direction nearly perpendicular to it, in an effective one-dimensional crystal. In contrast the racemic compound shows a less densely packed and defective monolayer, where relatively few nanoscopic domains of apparently enantiopure compound are interspersed in the disordered array. These results shine light on the self-assembly mechanism and are relevant for understanding packing in three dimensions, and especially for chiral compounds which do not form ordered crystals.

Necessary and sufficient condition for generalized passivity, passification and application to multivariable adaptive systems

AbstractRecently, a generalized passivity concept for linear multivariable systems was obtained which allows circumventing the restrictiveness of the usual passivity concept. The latter is associated with the classical SPR (Strictly Positive Real) condition whereas the new concept of passivity is associated with the so called WSPR condition and its advantage in multivariable systems is that it does not require a restrictive symmetry condition of SPR systems. As a result, it allows the design of multivariable adaptive control that, unlike some existing factorization approaches, does not imply in additional overparameterization of the adaptive controller. In this paper, we complete a previously established WSPR sufficient condition and prove that it is also necessary. We also propose some methods of passification by either premultiplying the system output tracking error vector or the system input vector by an adequate passifying matrix multiplier, so that the resulting input/output transfer function becomes WSPR. The efficiency of our proposals are illustrated by simulation utilizing a well known robotics adaptive visual servoing problem.

The eigenfrequency spectrum of linear magnetohydrodynamic perturbations in stationary equilibria: A variational principle

The frequencies of the normal modes of oscillation of linear magnetohydrodynamic perturbations of a stationary equilibrium are related to the stationary points of a quadratic functional over the Hilbert space of Lagrangian displacement vectors, which is subject to a constraint. In the absence of a background flow (or of a uniform flow), the relation reduces to the well-known Rayleigh–Ritz variational principle. In contrast to the existing variational principles for perturbations of stationary equilibria, the present treatment does neither impose additional symmetry restrictions on the equilibrium, nor does it involve the generalization to bilinear functionals instead of quadratic forms. This allows a more natural interpretation of the quadratic forms as energy functionals.

Dynamical consequences of symmetry breaking in benzene and difluorobenzene

The systems benzene/benzene-d(1) and o-/m-/p-difluorobenzene were studied in the dense gas phase with ultrafast transient absorption spectroscopy to investigate the effect of symmetry reduction through monodeuteration and constitutional isomerism on the timescales of intramolecular vibrational energy redistribution (IVR). In both systems IVR proceeds faster in the molecules of lower symmetry. In addition the dynamics were simulated in vibrational quantum number space using a simple model based on scaling state-to-state interactions by coupling order and the energy gap law. These simulations (semi-) quantitatively reproduce the experimental data for benzene and benzene-d(1) without incorporating further molecular symmetry restrictions. The relative impact of molecular symmetry and vibrational state space structure on IVR is discussed.

Crystal chemistry of synthetic lawsonite solid-solution series CaAl2[(OH)2/Si2O7]{middle dot}H2O-SrAl2[(OH)2/Si2O7]{middle dot}H2O and the Cmcm-P21/m phase transition

Crystals of the solid-solution series of (Ca,Sr)-lawsonite were synthesized hydrothermally at 4 GPa and 600 and 800 °C in piston-cylinder experiments. Synthesis products were analyzed with SEM, EMP, and powder-XRD. Lawsonite was observed in both the orthorhombic space group Cmcm and in the monoclinic space group P 2 1 / m . It is exclusively orthorhombic at low x Sr bulk but monoclinic at high x Sr bulk ; in the range x Sr bulk = 0.18 to 0.4 both polymorphs coexist and the data suggest a two-phase field between x Sr ortho ~0.1–0.2 and x Sr mono ~0.3–0.4 at 4 GPa/600 °C. Linear regression to the refined lattice parameters yields a = 0.017· x Sr + 5.841 (A), b = 0.197· x Sr + 8.787 (A), c = 0.263· x Sr + 13.130 (A), and v = 4.62· x Sr + 101.46 (cm 3 /mol) for orthorhombic lawsonite and a = 0.119· x Sr + 5.306 (A), b = 0.118· x Sr + 13.160 (A), c = 0.025· x Sr + 5.833 (A), β = 0.38· x Sr + 124.07 (°), and v = 3.20· x Sr + 101.59 (cm 3 /mol) for monoclinic lawsonite. The data suggest an increasingly negative Δ v ortho-mono with increasing x Sr . In monoclinic lawsonite, structural expansion due to the incorporation of Sr is primarily accomplished by tilting and rotation within the Si 2 O 7 -group, whereas in orthorhombic lawsonite this tilting and rotation is prohibited by symmetry restrictions and expansion is mostly accomplished by an increase in lattice parameters. Combining the extrapolated Ca end-member volume for monoclinic lawsonite with published high- P data yields K 0 mono = 137(3) GPa ( K ′ = 4.4). Contrary to the Ca end-member system, the Cmcm–P 2 1 / m phase transition is quenchable within the Sr-bearing system. A tentative phase diagram for (Ca,Sr)-lawsonite at 600 °C indicates a narrow orthorhombic-monoclinic two-phase field that shifts significantly to lower pressure with increasing x Sr . The Cmcm–P 2 1 / m phase transition in the Sr end-member system is located at ≤1 GPa at ~400 to 600 °C, 6 to 9 GPa below the transition in the Ca-system, and has a negative P - T slope.

Symmetry, incommensurate magnetism and ferroelectricity: The case of the rare-earth manganites RMnO3

The complete irreducible co-representations of the paramagnetic space group provide a simple and direct path to explore the symmetry restrictions of magnetically driven ferroelectricity. The method consists of a straightforward generalization of the method commonly used in the case of displacive modulated systems and allows us to determine, in a simple manner, the full magnetic symmetry of a given phase originated from a given magnetic order parameter. The potential ferroic and magneto-electric properties of that phase can then be established and the exact Landau free energy expansions can be derived from general symmetry considerations.In this work, this method is applied to the case of the orthorhombic rare-earth manganites RMnO3. This example will allow us to stress some specific points, such as the differences between commensurate or incommensurate magnetic phases regarding the ferroic and magnetoelectric properties, the possible stabilization of ferroelectricity by a single irreducible order parameter or the possible onset of a polarization oriented parallel to the magnetic modulation. The specific example of TbMnO3 will be considered in more detail, in order to characterize the role played by the magneto-electric effect in the mechanism for the polarization rotation induced by an external magnetic field.

Polymerization Mechanism of α-Linear Olefin

The density functional theory on the level of B3LYP/6–31G was empolyed to study the chain growth mechanism in polymerization process of α-linear olefin in TiCl3/AlEt2Cl catalytic system to synthesize drag reduction agent. Full parameter optimization without symmetry restrictions for reactants, products, the possible transition states, and intermediates was calculated. Vibration frequency was analyzed for all of stagnation points on the potential energy surface at the same theoretical level. The internal reaction coordinate was calculated from the transition states to reactants and products respectively. The results showed as flloes: (i) Coordination compounds were formed on the optimum configuration of TiCl3/AlEt2Cl. (ii) The transition states were formed. The energy difference between transition states and the coordination compounds was 40.687 kJ/mol. (iii) Double bond opened and Ti–C(4) bond fractured, and the polymerization was completed. The calculation results also showed that the chain growth mechanism did not essentially change with the increase of carbon atom number of α-linear olefin. From the relationship between polymerization activation energy and carbon atom number of the α-linear olefin, it can be seen that the α-linear olefin monomers with 6–10 carbon atoms had low activation energy and wide range. It was optimum to synthesize drag reduction agent by polymerization.

The self-assembly and photophysical characterization of tri(cyclopenta[def]phenanthrene)-derived nanoparticles: a template free synthesis of hollow colloidosomes.

A covalently linked trimer of 4,4-dimethyl-4H-cyclopenta[def]phenanthrene (MCPP), referred to as tri-MCPP, undergoes spontaneous self-assembly to form highly ordered colloidosomes in the absence of a template when subjected to reprecipitation from THF–water. The colloidosomes, and the oligomers in neat tetrahydrofuran have also been analyzed using optical methods. While the monomer, MCPP, was found to display a low fluorescence quantum yield (ΦF ∼ 0.03) and low molar absorptivity (ε ∼ 900 M−1 cm−1) for the S1 ← S0 transition due to symmetry restrictions, short oligomers formed via connection at the para phenylene positions displayed corresponding optical features that were found to increase gradually as a function of oligomer length. In the case of the largest species studied, the pentamer (penta-MCPP), the molar absorptivity was ca. 140 times larger than that of MCPP (ε = 126 500) for the S1 ← S0 transition. The ΦF and radiative rates for the MCPP oligomers were found to be enhanced by a factor of ∼30 and ∼180, respectively, while going from monomer to pentamer, presumably because of the larger radiative size of the oligomers. Further, the colloidosomes derived from tri-MCPP displayed a fluorescence quantum yield (ca. 0.35 to ∼0.20) that was reduced by approximately 1.8 times relative to tri-MCPP. The fluorescence lifetime was likewise decreased by ∼10 times compared to tri-MCPP; presumably, this is the result of intermolecular interactions between the very closely organized oligomers. The fact that highly ordered structures are obtained in the case of tri-MCPP, but not the other species included in this study (monomer; other oligomers), is thought to reflect the mutually orthogonal orientation between the constituent monomeric subunits in the trimer, as well as an ability to stabilize favorable hydrogen bonding interactions with residual solvent entrapped under the conditions of THF–water induced precipitation. Specifically, it is proposed that, in the case of tri-MCPP, these interactions aid in the formation and subsequent stabilization of precise colloidosome superstructures. An analogous stabilization of a colloidosome superstructure was not found for the corresponding tetramer (tetra-MCPP) or pentamer (penta-MCPP), in spite of the fact that these species contain a greater number of monomeric MCPP subunits and would be expected to give rise to ostensibly similar structures. This leads us to suggest that specific structural requirements are required for MCPP-derived precursors to form self-assembled colloidosomes.

On the origin of the hemidirected geometry of tetracoordinated lead(II) compounds

Abstract In order to explain the difference between high-symmetric holodirected and less-symmetric hemidirected Pb(II) compounds, the structures of [PbF 4 ] 2− , [PbCl 4 ] 2− and [Pb(OH) 4 ] 2− have been optimized without any symmetry restrictions and/or within various symmetry groups using all-electron relativistic B3LYP treatment and alternatively using B3LYP and MP2 treatments with effective core potential for Pb atoms. Optimal geometries of both [PbF 4 ] 2− and [PbCl 4 ] 2− are of T d symmetry (holodirected structures). The [Pb(OH) 4 ] 2− optimal geometry of C 2 symmetry cannot be explained by pseudo-Jahn–Teller effect due to energy and symmetry reasons. The less-symmetric geometry of tetracoordinated Pb(II) complexes with non-halogen ligands may be explained by the mutual interactions of their less-symmetric ligands. This symmetry decrease is subsequently supported by the increasing p electrons content in the Pb(II) lone electron pair via influencing the hybridization scheme of the Pb–ligands bond orbitals (i.e. not due to the repulsion between the ligands and the Pb(II) lone electron pair).

Frequency distribution and selection of space groups in inorganic crystal chemistry

Statistical analysis of modern structural databases on inorganic crystals and minerals is carried out to elucidate the classification of crystal structure distributions over space groups. The known fact about extremely uneven occupation of space groups is confirmed; the relative abundance of all and, in particular, 24 most frequent groups (>1%) is derived, and the latter are noted to belong to holohedral and centrosymmetric classes. The data on occupation of extremely rare and empty space groups are analyzed and refined separately, which allows the reduction of a number of unreliable structural data. The theoretical analysis of empirical regularities is performed from the standpoint of filling rules for special structural positions (Wyckoff positions) and the symmetry of lattice complexes. It is shown that only combined consideration of different symmetry rules and restrictions provides an explanation for most regularities observed in occupation of symmetry point groups.

Ab Initio calculation of potential energy curves for the ion-molecule reaction O+ + N2 → NO+ + N

Configuration interaction calculations verified an earlier hypothesis that the experimentally observed apparently anomalous behavior of the O++N2 → NO+ + N reaction is due to the symmetry and spin restrictions which prohibit the ground state reactants or products from correlating directly with the ground state intermediate. Complex symmetry adapted functions were used, and all single and double excitations were included for the ground and first excited 2Πi states, the lowest and second 2Σ+ states, the 4Π state and the 4Σ‐ state of the intermediate NNO+ for two cuts through the potential surface close to the reaction path. The number of configurations included were: 2Πi (123), 2Σ+ (177), 4Π (1349) and 4Σ‐ (1310). Analyses of the first order CI density matrices at large distances in terms of their ``fragmental'' MO's (or AO's) verified that the 4Σ‐ NNO+ state was formed from the ground state reactants O+(4Su) + N2(1Σg+) and that it separated into ground state products NO+(1Σ+) + N(4Su). The positions of oth...

Testing and imposing Slutsky symmetry in nonparametric demand systems

Maximization of utility implies that consumer demand systems have a Slutsky matrix which is everywhere symmetric. However, previous non- and semi-parametric approaches to the estimation of consumer demand systems do not give estimators that are restricted to satisfy this condition, nor do they offer powerful tests of this restriction. We use nonparametric modeling to test and impose Slutsky symmetry in a system of expenditure share equations over prices and expenditure. In this context, Slutsky symmetry is a set of nonlinear cross-equation restrictions on levels and derivatives of consumer demand equations. The key insight is that due to the differing convergence rates of levels and derivatives and due to the fact that the symmetry restrictions are linear in derivatives, both the test and the symmetry restricted estimator behave asymptotically as if these restrictions were (locally) linear. We establish large and finite sample properties of our methods, and show that our test has advantages over the only other comparable test. All methods we propose are implemented with Canadian micro-data. We find that our nonparametric analysis yields statistically significantly and qualitatively different results from traditional parametric estimators and tests.

Determination of Experimental Charge Density in Model Nickel Macrocycle: [3,11-Bis(methoxycarbonyl)-1,5,9,13-tetraazacyclohexadeca-1,3,9,11-tetraenato-(2-)-κ4N]nickel(II)

Experimental and theoretical atomic charges and d-orbital populations were obtained for [3,11-bis(methoxycarbonyl)-1,5,9,13-tetraazacyclohexadeca-1,3,9,11-tetraenato(2-)-kappa(4)N]nickel(II) monocrystal (16Ni) using the Hansen and Coppens formalism. Several models of this structure were tested as a function of quality of the fit, convergence of the refinement, value of residual peaks and holes, and Hirshfeld's rigid-bond test. The models with Ni-atomic scattering factors applied for the metal center are significantly better than those with the Ni-ionic scattering factors. The properties of the final electron density distributions are very consistent and similar for all of the models tested. The values of the d-orbital populations for Ni roughly agree with the occupation order of the d-electron levels for square-planar complexes derived from crystal field theory and are comparable with those obtained by the natural population analysis method. Experimental atoms-in-molecules charges calculated for four different models agree quite well for models with different symmetry restrictions and the same scattering factor for the Ni. Both experimental and theoretical methods predict relatively high negative values of charges for the nitrogen and oxygen atoms and also a quite high positive charge for the C(7) atom. The values of rho(r(BCP)) for the Ni-N bonds are in the range from 0.60 up to 0.75e A(-3), with positive laplacian values indicating noncovalent bonding. For the laplacian of the electron density evaluated in the plane of the macrocyclic ring, a typical map for a square-planar complex was obtained with four charge concentrations--3d(xy) orbitals--pointing toward the regions between the M-N bonds and charge depletions directed toward the (3,-1) critical points of the negative laplacian for the nitrogen atoms.

Structural changes underlying the diffuse dielectric response inAgNbO3

Structural differences in the so-called M polymorphs of ${\text{AgNbO}}_{3}$ were analyzed using combined high-resolution x-ray diffraction, neutron total scattering, electron diffraction, and x-ray absorption fine-structure measurements. These polymorphs all crystallize with $Pbcm$ symmetry and lattice parameters $\ensuremath{\surd}2{a}_{c}\ifmmode\times\else\texttimes\fi{}\ensuremath{\surd}2{a}_{c}\ifmmode\times\else\texttimes\fi{}4{a}_{c}$ (where ${a}_{c}\ensuremath{\approx}4\text{ }\text{\AA{}}$ corresponds to the lattice parameter of an ideal cubic perovskite) which are determined by a complex octahedral tilt system $({a}^{\ensuremath{-}}{b}^{\ensuremath{-}}{c}^{\ensuremath{-}})/({a}^{\ensuremath{-}}{b}^{\ensuremath{-}}{c}^{+})$ involving a sequence of two in-phase and two antiphase rotations around the $c$ axis. Our results revealed that, similar to ${\text{KNbO}}_{3}$, the Nb cations in ${\text{AgNbO}}_{3}$ exhibit local off-center displacements correlated along Nb-Nb-Nb chains. The displacements appear to be present even in the high-temperature ${\text{AgNbO}}_{3}$ polymorphs where the Nb cations, on average, reside on the ideal fixed-coordinate sites. The onset of the $({a}^{\ensuremath{-}}{b}^{\ensuremath{-}}{c}^{\ensuremath{-}})/({a}^{\ensuremath{-}}{b}^{\ensuremath{-}}{c}^{+})$ tilting in the M polymorphs lifts the symmetry restrictions on the Nb positions and promotes ordering of the local Nb displacements into a long-range antipolarlike array. This ordering preserves the average $Pbcm$ symmetry but is manifested in electron diffuse scattering and corroborated by other local-structure sensitive techniques. Structural states previously identified as the ${\text{M}}_{3}$ and ${\text{M}}_{2}$ phases represent different stages of displacive ordering rather than distinct thermodynamic phases. Rietveld refinements indicated intimate coupling between the displacive behavior on the oxygen, Nb, and Ag sublattices. The $Pbcm$ symmetry of the octahedral framework precludes a complete ordering of Nb displacements so that some positional disorder is retained. This partial disorder likely gives a source to the dielectric relaxation which, according to previous spectroscopic studies, is the origin of the diffuse dielectric response exhibited by M-type ${\text{AgNbO}}_{3}$ at $\ensuremath{\approx}250\text{ }\ifmmode^\circ\else\textdegree\fi{}\text{C}$.

Spontaneous symmetry breaking approach to La 2CuO 4 properties: Hints for matching the Mott and Slater pictures

Special solutions of the Hartree–Fock (HF) problem for Coulomb interacting electrons described by a simple model of the Cu–O planes in La 2CuO 4 are presented. One of the mean field states obtained, is able to predict some of the most interesting properties of this material, such as its insulator character and the antiferromagnetic order. The natural appearance of pseudogaps in some states of this material is also indicated by another of the HF states obtained. These surprising results follow after eliminating spin and crystal symmetry restrictions usually imposed on the single particle HF orbitals, by employing the rotational invariant formulation of the HF scheme originally introduced by Dirac. Therefore, it is exemplified here, how up to know considered strong correlation effects, can be described by improving the HF solution of the considered system. In other words, it has been argued, that defining correlation effects as the ones shown by the system and not predicted by the HF best (lowest energy) solution, allows to explain important, up to know considered as strong correlation properties, as simple mean field ones. The discussion also helps to clarify the role of the antiferromagnetism and pseudogaps in the physical properties of the HTSC materials and indicates a promising way to start conciliating the Mott and Slater pictures in the physics of the transition metal oxides and other strongly correlated electron systems.