Symmetry Group Research Articles

Amorphous Bismuth-Tin Oxide Nanosheets with Optimized C-N Coupling for Efficient Urea Synthesis.

Closing the carbon and nitrogen cycles by electrochemical methods using renewable energy to convert abundant or harmful feedstocks into high-value C- or N-containing chemicals has the potential to transform the global energy landscape. However, efficient conversion avenues have to date been mostly realized for the independent reduction of CO2 or NO3-. The synthesis of more complex C-N compounds still suffers from low conversion efficiency due to the inability to find effective catalysts. To this end, here we present amorphous bismuth-tin oxide nanosheets, which effectively reduce the energy barrier of the catalytic reaction, facilitating efficient and highly selective urea production. With enhanced CO2 adsorption and activation on the catalyst, a C-N coupling pathway based on *CO2 rather than traditional *CO is realized. The optimized orbital symmetry of the C- (*CO2) and N-containing (*NO2) intermediates promotes a significant increase in the Faraday efficiency of urea production to an outstanding value of 78.36% at -0.4 V vs RHE. In parallel, the nitrogen and carbon selectivity for urea formation is also enhanced to 90.41% and 95.39%, respectively. The present results and insights provide a valuable reference for the further development of new catalysts for efficient synthesis of high-value C-N compounds from CO2.

New anomaly free supergravities in six dimensions

An extended search for anomaly free matter coupled N = (1, 0) supergravity in six dimension is carried out by two different methods which we refer to as the graphical and rank methods. In the graphical method the anomaly free models are built from single gauge group models, called nodes, which can only have gravitational anomalies. We search for anomaly free theories with gauge groups G1 × … × Gn with n = 1, 2, … (any number of factors) and G1 × … × Gn × U(1)R where n = 1, 2, 3 and U(1)R is the R-symmetry group. While we primarily consider models with the tensor multiplet number nT = 1, we also provide some results for nT ≠ 1 with an unconstrained number of charged hypermultiplets. We find a large number of ungauged anomaly free theories. However, in the case of R-symmetry gauged models with nT = 1, in addition to the three known anomaly free theories with G1 × G2 × U(1)R type symmetry, we find only six new remarkably anomaly free models with symmetry groups of the form G1 × G2 × G3 × U(1)R. In the case of nT = 1 and ungauged models, excluding low rank group factors and considering only low lying representations, we find all anomaly free theories. Remarkably, the number of group factors does not exceed four in this class. The proof of completeness in this case relies on a bound which we establish for a parameter characterizing the difference between the number of non-singlet hypermultiplets and the dimension of the gauge group.

Multi-entropy at low Renyi index in 2d CFTs

For a static time slice of AdS_33 we describe a particular class of minimal surfaces which form trivalent networks of geodesics. Through geometric arguments we provide evidence that these surfaces describe a measure of multipartite entanglement. By relating these surfaces to Ryu-Takayanagi surfaces it can be shown that this multipartite contribution is related to the angles of intersection of the bulk geodesics. A proposed boundary dual [Phys. Rev. D 106, 126001 (2022), J. High Energy Phys. 08, 202 (2023), J. High Energy Phys. 05, 008 (2023)], the multi-entropy, generalizes replica trick calculations involving twist operators by considering monodromies with finite group symmetry beyond the cyclic group used for the computation of entanglement entropy. We make progress by providing explicit calculations of Renyi multi-entropy in two dimensional CFTs and geometric descriptions of the replica surfaces for several cases with low genus. We also explore aspects of the free fermion and free scalar CFTs. For the free fermion CFT we examine subtleties in the definition of the twist operators used for the calculation of Renyi multi-entropy. In particular the standard bosonization procedure used for the calculation of the usual entanglement entropy fails and a different treatment is required.

OPTIMIZATION OF GEOMETRY OF PIEZORESISTIVE EFFECT ON THE EXAMPLE OF CUBIC CRYSTALS

On the example of semiconductor crystals Ge, Si, PbTe, PbS, InSb with different levels of doping and different types of conductivity, the geometry of the piezoresistive effect was optimized, namely, such directions of voltage measuring and uniaxial pressure applying were determined, which ensure the maximum achievable value of the effect. The optimization is based on an approach using the construction and analysis of extreme surfaces that represent all possible maxima of the objective function (the magnitude of the effect) under different spatial orientations of interacting factors. The optimization parameters were the angles that determine the directions of the unit vectors of the directions of current and uniaxial pressure applying. The directions of the radius vectors of the points on the extreme surface coincide with the ones in which the electric voltage is measured, and the length of this radius vector for each point was determined by setting such optimization parameters for which the magnitude of the effect for a given direction of voltage measuring would be maximal. It is shown that the optimal interaction geometry in most of the studied cases is longitudinal, and only for n-PbTe, p-InSb crystals it is transverse (although not identical), and the optimal directions for the studied crystals are <100>, <110> or <111> depending on the composition of the crystal and the type of doping. Despite the fact that all investigated crystals belong to the same point symmetry group (m3m), the shapes of the extreme surfaces for them are significantly different, which is caused by different ratios between the piezoresistive coefficients. Typical forms of extreme surfaces have been identified, and in order to explain the obtained results, an analysis of limiting cases that differ in the ratio of piezoresistive coefficients has been carried out. Based on this analysis, four main types of extreme surfaces were established. A scheme has been built that allows, in the case of cubic crystals, to estimate the type of extreme surface and the corresponding optimal directions of voltage measuring, current density (for cubic crystals, these directions coincide) and uniaxial pressure applying. On the basis of this scheme, the forms of extreme surfaces obtained for the investigated crystals are explained.

Crystal-Inspired Cellular Metamaterials and Triply Periodic Minimal Surfaces

Triply periodic minimal surfaces (TPMSs) are found in many natural objects including butterfly wings, sea urchins, and biological membranes. They simultaneously have zero mean curvature at every point and a crystallographic group symmetry. A metamaterial can be created from such periodic surfaces or used as a reinforcement of a composite material. While a TPMS as a mathematical object has been known since 1865, only novel additive manufacturing (AM) technology made it possible to fabricate cellular materials with complex TPMS shapes. Cellular TPMS-based metamaterials have remarkable properties related to wetting/liquid penetration, shock absorption, and the absence of stress concentrators. Recent studies showed that TPMSs are also found in natural crystals when electron surfaces are considered. Artificial crystal-inspired metamaterials mimic such crystals including zeolites and schwarzites. These metamaterials are used for shock, acoustic waves, and vibration absorption, and as structural materials, heat exchangers, and for other applications. The choice of the crystalline cell of a material, as well as its microstructure, plays a decisive role in its properties. The new area of crystal-inspired materials has many common features with traditional biomimetics with models being borrowed from nature and adjusted for engineering applications.

Spin textures of coherent photons with SU(4) symmetry

We show coherent photons with spin and orbital angular momentum have symmetry of special unitary group of degree four, SU(4), both theoretically and experimentally. We show the degree of polarisation could be a measure to characterise the quantum coherency of a spin state, which was significantly affected by the partial trace out of orbital degree of freedom. By changing the amplitudes of the classically entangled state, we observed a crossover from quantum Heisenberg spin to classical Ising spin. We also demonstrated projections of coherent states, which recover the coherency of the spin state, while the SU(4) state is projected to be a simple product state of SU(2)×SU(2). Based on the developed platform, we have explored spin texture of coherent photons, and found a unique spin profile reminiscent of the Bengal cat upon wrapping to a three-dimensional sphere. We have controlled the phase between singlet and triplet states and found the spin texture rotated upon increasing the phase.

Cosmological solutions in the Brans–Dicke theory via invariants of symmetry groups

We proceed to obtain an exact analytical solution of the Brans–Dicke (BD) equations for the spatially flat ([Formula: see text]) Friedmann–Lamaître–Robertson–Walker (FLRW) cosmological model in both cases of the absence and presence of the cosmological constant. The solution method that we use to solve the field equations of the BD equations is called the “invariants of symmetry groups method” (ISG method). This method is based on the extended Prelle–Singer (PS) method and it employs the Lie point symmetries, [Formula: see text]-symmetries, and Darboux polynomials (DPs). Indeed, the ISG method tries to provide two independent first-order invariants associated to the one-parameter Lie groups of transformations keeping the ordinary differential equations (ODEs) invariant, as solutions. It should be noted for integrable ODEs that the ISG method guarantees the extraction of these two invariants. In this work, for the BD equations in FLRW cosmological model, we find the Lie point symmetries, [Formula: see text]-symmetries, and DPs, and obtain the basic quantities of the extended PS method (which are the null forms and the integrating factors). By making use of the extended PS method we find two independent first-order invariants in such a way that appropriate cosmological solutions from solving these invariants as a system of algebraic equations are simultaneously obtained. These solutions are wealthy in that they include many known special solutions, such as the O’Hanlon–Tupper vacuum solutions, Nariai’s solutions, Brans–Dicke dust solutions, inflationary solutions, etc.

Asymptotic symmetries of SU(2) Yang-Mills-Higgs theory in Hamiltonian formulation

We investigate the asymptotic symmetry group of a SU(2)-Yang-Mills theory coupled to a Higgs field in the Hamiltonian formulation. This extends previous work on the asymptotic structure of pure electromagnetism by Henneaux and Troessaert, and on electromagnetism coupled to scalar fields and pure Yang-Mills fields by Tanzi and Giulini. We find that there are no obstructions to global electric and magnetic charges, though that is rather subtle in the magnetic case. Again it is the Hamiltionian implementation of boost symmetries that need a careful and technically subtle discussion of fall-off and parity conditions of all fields involved.

Impact of the terminal fluorination of bisthienylethene-based liquid crystals on the self-assembly and mechanochromic behaviors

Four novel bisthienylethene-based rod-like liquid crystals containing different numbers of terminal fluorine atoms were designed and the effect of the terminal fluorination on liquid crystalline self-assembly, optical characteristics in solid state and mechanochromic behaviors was explored. The transition from non-liquid crystal via monotropic liquid crystal with smectic A phase to enantiotropic liquid crystal with smectic A phase can be achieved and the temperature ranges of smectic A phase gradually increase by increasing the terminal fluorination, which might be due to the increase in intermolecular interactions by increasing the number of terminal fluorine atom. All the compounds exhibited AIE characteristic and distinct solid-state fluorescence with yellow or orange color. In addition, the non-fluorosubstituted and difluorosubstituted bisthienylethene-based liquid crystals exhibited reversible mechanochromism, whereas the monofluorosubstituted and trifluorosubstituted bisthienylethene-based liquid crystals exhibited extremely weak mechanochromism due to different intermolecular interaction, molecular symmetry and molecular conformation. These investigations implied that the incorporation of fluorine atom could result in interesting self-assemblies and photophysical properties in different states, which also provided the understanding of the relationship between chemical structures and properties.

Enhanced self-collimation effect by low rotational symmetry in hexagonal lattice photonic crystals

In this study, we present the design of a photonic crystal (PC) structure with a hexagonal lattice, where adjustments to the PC unit cell symmetry reveal an all-angle self-collimation (SC) effect. By optimizing opto-geometric parameters, such as the rotational angle of auxiliary rods and adjacent distances, we analyze the SC property in detail, leveraging group velocity dispersion (GVD) and third-order dispersion (TOD) characteristics. We also investigate the relationship between symmetry properties and their influence on dispersion characteristics. Through symmetry manipulation, we gain a comprehensive understanding of the underlying mechanisms governing light collimation and confinement in the proposed configurations. The PC structure with a C1 symmetry group exhibits all-angle SC effect within the range of a/λ = 0.652 and a/λ = 0.668 normalized frequencies, with a bandwidth of Δω/ω c =2.4% Further breaking the symmetry, transforming from C1 to C2 group symmetry enhances the SC bandwidth to Δω/ω c =6.5% and reveals the perfect linear equi-frequency contours (EFC) at two different frequency bands: all angle SC between a/λ = 0.616 and a/λ = 0.344 normalized frequencies in the 4th transverse magnetic (TM) band and between a/λ = 0.712 and a/λ = 0.760 in the 5th TM band. Here, GVD and TOD values of the TM 4th band vary between 7.3 (a/2πc2)–254.3 (a/2πc2) and 449.2 (a2/4π 2c3)–1.3×105 (a2/4π 2c3), respectively. Also, GVD and TOD values of the TM 5th band vary between 182.5 (a/2πc2)–71.3 (a/2πc2) and −24380(a2/4π 2c3)–−9619 (a2/4π 2c3) values, respectively. Additionally, we propose a composite/hybrid PC structure resembling C2 group symmetry, where two auxiliary rods are replaced by rectangular photonic wires with the same refractive index and width equal to the diameter of auxiliary rods. This hybrid structure exhibits an all-angle SC effect with an operating bandwidth of Δω/ω c =11.7%, which displays near-zero GVD and TOD performance and offers enhanced robustness against potential fabrication precision issues.

SOME CHARACTERISTICS AND CAUSES OF CHANGES IN THE RATE OF DISCOVERY OF NEW MINERALS SINCE 1800

ABSTRACT Construction of a discovery curve for Earth's minerals illustrates an abrupt inflection in rates of discovery and acceptance of new species that occurred in the middle of the 20th century. Prior to that time new mineral discovery is found to have accelerated at a modest pace of 0.3 species / year2. Since 1950 the rate of acceptance has accelerated at an average of 1.04 species / year2 such that the first two decades of the 21st century have been characterized by the acceptance of over 100 new minerals per year. Despite this rapid acceleration of discovery, the relative distribution among the 10 mineral classes of the Nickel-Strunz classification system of minerals identified prior to and after 1950 are nearly identical. Variation in the rates of attestation of the divisions, families, and mineral groups show distinct differences in that the maximum rates of discovery of minerals populating higher taxonomic levels occurred prior to 1950 while the attestation curve of newly recognized groups closely follows that defined by the identification of mineral species. The application of any hierarchical classification system to the mineral species is somewhat arbitrary and subject to biases or inconsistencies associated with the systematics of classification. Therefore, the discovery of new minerals is also considered within the context of the mathematically defined crystallographic space group symmetries. The space group attestation curve has a shape similar to those exhibited by the higher taxonomic levels within the Nickel-Strunz system. However, not all space groups are represented by naturally occurring minerals. Minerals discovered before and after the inflection in rates of new mineral discovery illustrate significantly different patterns of paragenesis. Nearly all of the species have been associated with one or more of 57 distinct paragenetic modes. Minerals that have long been known tend to have a higher number of paragenetic modes than those more recently discovered such that the average number of paragenetic modes is shown to have decreased linearly from 1950 to 2022. Further, the earliest known occurrence of over 80% of the currently accepted mineral species has been tabulated by the International Mineralogical Association. For species discovered prior to the inflection of 1950 a linear distribution of oldest ages is observed indicating that these minerals have earliest ages of occurrence that are uniformly distributed across the interval 0 to 4700 Ma. Conversely, approximately 70% of the species identified since 1950 have oldest known occurrences of less than 600 Ma and the age distribution of these post-1950 minerals exhibit an exponential distribution suggesting increasing efforts in new mineral discovery in tectonically active settings. Despite the differences in the pre- and post-1950 oldest age distributions, both populations exhibit similar temporal excursions in rates of new mineral creation that likely reflect substantial changes in Earth system processes responsible for new mineral formation.

Concentration effects on the local structures and electronic properties of ErxBaY2-xF8: A first-principles study.

Er3+ doped barium yttrium fluoride (BaY2F8) crystal has gained long-term attention due to its great potential in laser and medical device applications. However, the local structures of Er3+ doped BaY2F8 system (Er:BYF) remain uncertain, and the effect of doping concentration on structures and properties is unknown. Therefore, in this study, the first-principles study of the structural evolution of ErxBaY2-xF8 (x = 0.125, 0.25) crystals was carried out. By means of density functional theory and particle swarm optimization algorithm, the stable structures of Er:BYF crystals with two different concentrations are shown as standard monoclinic structures with P2 symmetry for the first time. The impurity Er3+ ions successfully enter the main lattice, replacing the Y3+ ions, and forming a [ErF8]5- polyhedron with C2 point group symmetry. By calculating the electronic properties, the band gap values of the two structures are significantly reduced compared with that of pure BaY2F8 crystal. However, the conduction band does not break through the Fermi level, and the crystals still maintain the insulation characteristic. According to the calculation of the electron local density function, we conclude that Er-F and Y-F in Er:BYF are connected by ionic bonds. These results fill a theoretical gap in the study of Er:BYF crystals and provide inspiration for structural evolution and material design at different doping concentrations.

Conserved spin operator of Dirac’s theory in spatially flat FLRW space-times

New conserved spin and orbital angular momentum operators of Dirac’s theory on spatially flat FLRW space-times are proposed generalizing thus the recent results concerning the role of Pryce’s spin operator in the flat case (Cotăescu in Eur Phys J C, 82, 1073, 2022). These operators split the conserved total angular momentum generating the new spin and orbital symmetries that form the rotations of the isometry groups. The new spin operator is defined and studied in active mode with the help of a suitable spectral representation giving its Fourier transform. Moreover, in the same manner is defined the operator of the fermion polarization. The orbital angular momentum is derived in passive mode using a new method, inspired by Wigner’s theory of induced representations, but working properly only for global rotations. In this approach the quantization is performed finding that the one-particle spin and orbital angular momentum operators have the same form in any FLRW spacetime regardless their concrete geometries given by various scale factors.

Effects of doping with selected rare-earth elements (Eu3+, Dy3+, Eu3+/Dy3+) on the structural characteristics as well as electrical and luminescent properties of K0.5Na0.5NbO3.

The effects of adding Eu3+, Dy3+, and Eu3+/Dy3+ rare-earth elements on the microstructural, optical and electrical properties of lead-free piezoelectric materials based on K0.5Na0.5NbO3 (KNN) were investigated. A series of fine powders were synthesized via a modified Pechini route utilizing citric acid. Relatively large amounts of rare-earth elements were successfully incorporated into the crystal lattice of KNN. A combination of X-ray diffraction, X-ray absorption spectroscopy, energy-dispersive X-ray and scanning electron microscopy were used to characterize the structure and chemical composition of the powders and sinters. XRD patterns indicated the formation of a single-phase orthorhombic structure in the co-doped sinter. X-ray absorption spectroscopy revealed the nominal 3+ valence states of Eu and Dy in KNN ceramic sinters. The Mott model and group symmetry theory were used to discuss the characteristics of Raman spectra obtained for the EuDy:KNN ceramics. The ceramics exhibited good dielectric permittivity (ε≃1300) and low dielectric loss (tan δ=0.03) in the temperature range of 170-240°C. The electrical properties of the ceramics showed that the basic mechanism of conduction and relaxation was thermal activation, and that small polaron hopping was responsible for charge carrying. The Eu3+ and Dy3+ ions displayed characteristic red and orange emissions in sinters when excited with light in the wide ultraviolet spectra range (380-420nm).

Ab-initio investigation of the structural, electronic and optical properties of CsPbBr3 perovskite under pressure using the DFT-1/2 method

Pressure-induced phase transitions in metal halide perovskites lead to significantly different material properties and offer great potential for diverse applications. The electronic, structural, and optical properties of Cesium lead bromide (CsPbBr3) crystals are investigated using density functional theory (DFT) to draw an integrated picture of orbital morphology, absorption spectra, and band gap values. Spin-orbit coupling and DFT-1/2 correction are used to gain a better insight into electronic properties and band splitting in the cubic phase. Calculations of pressure-induced phase transitions, electronic band gap engineering, and manipulation of optical absorption edges of CsPbBr3 are conducted. In the cubic and tetragonal crystal lattices, a band gap closure and opening occurs demonstrating a Dirac cone upon application and increase of pressure. Indeed, in the tetragonal lattice of CsPbBr3, a topological band occurs. Orthorhombic perovskite crystal lattice undergoes successive direct-indirect and indirect-direct band gap transitions, followed by a space group change at extreme pressures. However, under pressure, orthorhombic non-perovskite lattice experiences a sudden band gap breakdown. Phase and band gap transitions are explained by orbital hybridizations and structural symmetry evolution (e.g., bond length contraction/dilation and octahedral cage rotation/distortion). These results provide comprehensive guidelines for further experimental studies on pressure engineering, advanced photovoltaic and optoelectronic applications, discovering and understanding new topological band structures of perovskite materials.

Thermal Stability and Non-Linear Optical and Dielectric Properties of Lead-Free K0.5Bi0.5TiO3 Ceramics.

Lead-free K0.5Bi0.5TiO3 (KBT) ceramics with high density (~5.36 g/cm3, 90% of X-ray density) and compositional purity (up to 90%) were synthesized using a solid-state reaction method. Strongly condensed KBT ceramics revealed homogenous local microstructures. TG/DSC (Thermogravimetry-differential scanning calorimetry) techniques characterized the thermal and structural stability of KBT. High mass stability (>0.4%) has proven no KBT thermal decomposition or other phase precipitation up to 1000 °C except for the co-existing K2Ti6O13 impurity. A strong influence of crystallites size and sintering conditions on improved dielectric and non-linear optical properties was reported. A significant increase (more than twice) in dielectric permittivity (εR), substantial for potential applications, was found in the KBT-24h specimen with extensive milling time. Moreover, it was observed that the second harmonic generation (λSHG = 532 nm) was activated at remarkably low fundamental beam intensity. Finally, spectroscopic experiments (Fourier transform Raman and far-infrared spectroscopy (FT-IR)) were supported by DFT (Density functional theory) calculations with a 2 × 2 × 2 supercell (P42mc symmetry and C4v point group). Moreover, the energy band gap was calculated (Eg = 2.46 eV), and a strong hybridization of the O-2p and Ti-3d orbitals at Eg explained the nature of band-gap transition (Γ → Γ).

Improved thermodynamic properties of (Sc, V, Ti, Fe, Mn, Co, and Ni) doped NaBH4 for hydrogen storage: First-principal calculation

Sodium borohydride (NaBH4) shows promise as a hydrogen storage option, but its substantial thermodynamic stability limits its practical application. Enhancing NaBH4 by introducing transition metals is an effective approach to enhance its hydrogen-related characteristics. Nevertheless, even with the most successful additives, the temperatures needed for hydrogen release remain too elevated, exceeding 400 °C, which hinders its practical utility. Using first-principles density functional, we have investigated the potential of sodium borohydride as a hydrogen-based fuel source. Our focus has been on the stable α-phase NaBH4 structure with space group symmetry F-43m (216), aiming to understand the impact of 3d transition metal (Sc, V, Ti, Fe, Mn, Co, and Ni) doping on the structural, electronic, and dehydrogenation properties of the NaBH4 system. Furthermore, the heat of formation and desorption temperature values obtained align with the stability and volumetric capacity criteria set by the U.S. Department of Energy (DOE). Our findings indicate that doping transition metals at the Na site weakens the B–H bonds, thereby enhancing the dehydrogenation performance. Among the various systems studied, those doped with Sc and Ti exhibit the most favorable dehydrogenation properties.

Mobility of geometric constraint systems with extrusion symmetry

AbstractIf we take a (bar-joint) framework, prepare an identical copy of this framework, translate it by some vector $$\tau $$ τ , and finally join corresponding points of the two copies, then we obtain a framework with ‘extrusion’ symmetry in the direction of $$\tau $$ τ . This process may be repeated t times to obtain a framework whose underlying graph has $$\mathbb {Z}_2^t$$ Z 2 t as a subgroup of its automorphism group and which has ‘t-fold extrusion’ symmetry. Extruding a framework is a widely used technique in CAD for generating a 3D model from an initial 2D sketch, and hence it is important to understand the flexibility of extrusion-symmetric frameworks. Using group representation theory, we show that while t-fold extrusion symmetry is not a point-group symmetry, the rigidity matrix of a framework with t-fold extrusion symmetry can still be transformed into a block-decomposed form in the analogous way as for point-group symmetric frameworks. This allows us to establish Fowler-Guest-type character counts to analyse the mobility of such frameworks. We show that this entire theory also extends to the more general point-hyperplane frameworks with t-fold extrusion symmetry. Moreover, we show that under suitable regularity conditions the infinitesimal flexes we detect with our symmetry-adapted counts extend to finite (continuous) motions. Finally, we establish an algorithm that checks for finite motions via linearly displacing framework points along velocity vectors of infinitesimal motions.

Automatic Determination of the Non-Covalent Stable Conformations of the NO2-Pyrene Cluster in Full Dimensionality (81D) Using the vdW-TSSCDS Approach.

We present the detailed topographical characterisation (stationary points and minimum energy paths connecting them) of the full dimensional (81D) intermolecular potential energy surface associated with the non-covalent interactions between the NO2 radical and the pyrene (C16H10) molecule. The whole procedure is (quasi) fully automated. We have used our recent algorithm vdW-TSSCDS as implemented on the freely-available AutoMekin software package. To this end, a series of inexpensive classical trajectories using forces from a low-level (semi-empirical) theory are used to sample the configuration space of the system in the search for candidates to first order saddle points. These guess structures are determined by means of a graph-theory based algorithm using the concept of adjacency matrix. Low-level optimizations are followed by re-optimizations at a final high-level of theory (DFT and CCSD(T)-F12 in our case.). The resulting set of stationary points and paths connecting them constitutes the so-called reaction network. In the case of NO2-pyrene, this network exhibits four major basins which can be characterized by their point-group symmetry. A central one, of global C2 symmetry, comprises the global minimum (as well as all other permutationally related conformers) together with the corresponding C2v saddle points connecting them. This central basin is connected to three others of lower C1 symmetry. The latter can be distinguished by the projection of the position of the NO2 nitrogen atom on the pyrene plane in combination with the relative orientation of the oxygen pair pointing either inwards, outwards, upwards or downwards.

K-Positivity and Schmidt number under orthogonal group symmetries

In this paper, we present a new application of group theory to develop a systematical approach to efficiently compute the Schmidt numbers. The Schmidt number is a natural quantification of entanglement in quantum information theory, but computing its exact value is generally a challenging task even for very concrete examples. We exhibit a complete characterization of all orthogonally covariant k-positive maps. This result generalizes earlier results by Tomiyama (Linear Algebra Appl 69:169–177, 1985). Furthermore, we optimize duality relations between k-positivity and Schmidt numbers under group symmetries. This new approach enables us to transfer the results of k-positivity to the computation of the Schmidt numbers of all orthogonally invariant quantum states.