The Role of Symmetry in Non-Hermitian Scattering1

We review recent work on asymmetric scattering by Non-Hermitian (NH) Hamiltonians. Quantum devices with an asymmetric scattering response to particles incident from right or left in effective ID waveguides will be important to develop quantum technologies. They act as microscopic equivalents of familiar macroscopic devices such as diodes, rectifiers, or valves. The symmetry of the underlying NH Hamiltonian leads to selection rules which restrict or allow asymmetric response. NH-symmetry operations may be organized into group structures that determine equivalences among operations once a symmetry is satisfied. The NH Hamiltonian posseses a particular symmetry if it is invariant with respect to the corresponding symmetry operation, which can be conveniently expressed by a unitary or antiunitary superoperator. A simple group is formed by eight symmetry operations, which include the ones for Parity-Time symmetry and Hermiticity as specific cases. The symmetries also determine the structure of poles and zeros of the S matrix. The ground-state potentials for two-level atoms crossing properly designed laser beams realize different NH symmetries to achieve transmission or reflection asymmetries.

Modeling and Structure Determination of Homo-Oligomeric Proteins: An Overview of Challenges and Current Approaches

Protein homo-oligomerization is a very common phenomenon, and approximately half of proteins form homo-oligomeric assemblies composed of identical subunits. The vast majority of such assemblies possess internal symmetry which can be either exploited to help or poses challenges during structure determination. Moreover, aspects of symmetry are critical in the modeling of protein homo-oligomers either by docking or by homology-based approaches. Here, we first provide a brief overview of the nature of protein homo-oligomerization. Next, we describe how the symmetry of homo-oligomers is addressed by crystallographic and non-crystallographic symmetry operations, and how biologically relevant intermolecular interactions can be deciphered from the ordered array of molecules within protein crystals. Additionally, we describe the most important aspects of protein homo-oligomerization in structure determination by NMR. Finally, we give an overview of approaches aimed at modeling homo-oligomers using computational methods that specifically address their internal symmetry and allow the incorporation of other experimental data as spatial restraints to achieve higher model reliability.

Molecular Vibrations Are Not Asymmetric

There is considerable confusion when naming vibrations in infrared and Raman spectra. One of the most common errors is the identification of some stretching and bending vibrations as “asymmetric”. There are no asymmetric vibrations as such vibrations incur rotations and translations. The correct term is antisymmetric and it is demonstrated, through molecular symmetry operations, why this is the correct term.

Core Modulation of Porphyrins for Chemical Sensing

The inner core system of metal-free (‘free base’) porphyrins has continually served as a ligand for various metal ions, but it was only recently studied in organocatalysis due its highly tunable basicity. Highly conjugated porphyrin systems offer spectrophotometric sensitivity toward geometrical and/or electronic changes and, thus, utilizing the porphyrin core for the selective detection of substrates in solution offers significant potential for a multitude of applications. However, solvation and dilution drastically affect weak interactions by dispersing the binding agent to its surroundings. Thus, the spectroscopic detection of N–H···X-type binding in porphyrin solutions is almost impossible without especially designing the binding pocket. Here, we present the first report on the spectroscopic detection of N–H···X-type interplay in porphyrins formed by weak interactions. Protonated 2,3,7,8,12,13,17,18-octaethyl-5,10,15,20-tetrakis(2-aminophenyl) porphyrin contains coordination sites for the selective binding of charge-bearing analytes, revealing characteristic spectroscopic responses. While electronic absorption spectroscopy proved to be a particularly useful tool for the detection of porphyrin–analyte interactions in the supramolecular complexes, X-ray crystallography helped to pinpoint the orientation, flexibility, and encapsulation of substrates in the corresponding atropisomers. This charge-assisted complexation of analytes in the anion-selective porphyrin inner core system is ideal for the study of atropisomers using high-resolution NMR, since it reduces the proton exchange rate, generating static proton signals. Therefore, we were able to characterize all four rotamers of the nonplanar 2,3,7,8,12,13,17,18-octaethyl-5,10,15,20-tetrakis(2-aminophenyl) porphyrin by performing 1D and 2D NMR spectroscopic analyses of host-guest systems consisting of benzenesulfonic acid (BSA) and each porphyrin atropisomer. Lastly, a detailed assignment of the symmetry operations that are unique to porphyrin atropisomers allowed us to accurately identify the rotamers using NMR techniques only. Overall, the N–H···X-type interplay in porphyrins formed by weak interactions that form restricted H-bonding complexes is shown to be the key to unravelling the atropisomeric enigma.

Structure Topology and Graphical Representation of Decorated and Undecorated Chains of Edge-Sharing Octahedra

ABSTRACT Infinite chains of edge-sharing octahedra occur as fundamental building blocks (FBBs) in the structures of several hundred mineral species. Such chains consist of a backbone of octahedra to which decorating polyhedra may be attached. The general, stoichiometric formula of such chains may be written as c[MATxФz] where M is any octahedrally coordinated cation, T is any cation coordinated by a decoration polyhedron (regardless of coordination geometry), Ф is any possible ligand [O2–, (OH)–, (H2O), Cl–, or F–], and c indicates the configuration of backbone octahedra. In the minerals in which they occur, these types of chains will commonly (though not exclusively) form part of the structural unit (i.e., the strongly bonded part) of a mineral. Hence, investigating the topology, configuration, and arrangement of such chains may yield fundamental insights into the stability of minerals in which they occur. A discussion of the topological variability of chains is presented here, along with the formulae necessary for their characterization. It is shown that many aspects of chain topology can be efficiently communicated by a pair of values with the form ([x], [Bopqrst]), where [x] summarizes the symmetry operations necessary to characterize the configuration of backbone octahedra, B indicates the length of the topological repeat, and o through t indicate the number of individual decorations (related to B). A methodology for developing finite graphical representations for infinite chains is presented in detail, showing that for any given chain, a single, irreducible finite graph exists that contains all topological information. Such a graph, however, can correspond to multiple chain topologies, highlighting the importance of geometrical isomerism. The utility of the graphical approach in facilitating the development of a hierarchy of chains and chain-bearing structures is also discussed.

Enhancement of properties in Mizar

A “property” in the Mizar proof-assistant is a construction that can be used to register chosen features of predicates (e.g., “reflexivity”, “symmetry”), operations (e.g., “involutiveness”, “commutativity”) and types (e.g., “sethoodness”) declared at the definition stage. The current implementation of Mizar allows using properties for notions with a specific number of visible arguments (e.g., reflexivity for a predicate with two visible arguments and involutiveness for an operation with just one visible argument). In this paper we investigate a more general approach to overcome these limitations. We propose an extension of the Mizar language and a corresponding enhancement of the Mizar proof-checker which allow declaring properties of notions of arbitrary arity with respect to explicitly indicated arguments. Moreover, we introduce a new property—the “fixedpoint-free” property of unary operations—meaning that the result of applying the operation to its argument always differs from the argument. Results of tests conducted on the Mizar Mathematical Library are presented.