Dihedral symmetry of periodic chain: quantization and coherent states

Our previous work on quantum kinematics and coherent states over finite configuration spaces is extended: the configuration space is, as before, the cyclic group Z_n of arbitrary order n=2,3,..., but a larger group - the non-Abelian dihedral group D_n - is taken as its symmetry group. The corresponding group related coherent states are constructed and their overcompleteness proved. Our approach based on geometric symmetry can be used as a kinematic framework for matrix methods in quantum chemistry of ring molecules.Comment: 13 pages; minor changes of the tex

Classical Morphology of Plants as an Elementary Instance of Classical Invariant Theory

It has long been known that structural chemistry shows an intriguing correspondence with Classical Invariant Theory (CIT). Under this view, an algebraic binary form of the degree n corresponds to a chemical atom with valence n and each physical molecule or ion has an invariant-theoretic counterpart. This theory was developed using the Aronhold symbolical approach and the symbolical processes of convolution/transvection in CIT was characterized as a potential “accurate morphological method”. However, CIT has not been applied to the formal morphology of living organisms. Based on the morphological interpretation of binary form, as well as the process of convolution/transvection, the First and Second Fundamental Theorems of CIT and the Nullforms of CIT, we show how CIT can be applied to the structure of plants, especially when conceptualized as a series of plant metamers (phytomers). We also show that the weight of the covariant/invariant that describes a morphological structure is a criterion of simplicity and, therefore, we argue that this allows us to formulate a parsimonious method of formal morphology. We demonstrate that the “theory of axilar bud” is the simplest treatment of the grass seedling/embryo. Our interpretations also represent Troll's bauplan of the angiosperms, the principle of variable proportions, morphological misfits, the basic types of stem segmentation, and Goethe's principle of metamorphosis in terms of CIT. Binary forms of different degrees might describe any repeated module of plant organisms. As bacteria, invertebrates, and higher vertebrates are all generally shared a metameric morphology, wider implications of the proposed symmetry between CIT and formal morphology of plants are apparent

Mechanical Bonds and Topological Effects in Radical Dimer Stabilization

While mechanical bonding stabilizes tetrathiafulvalene (TTF) radical dimers, the question arises: what role does topology play in catenanes containing TTF units? Here, we report how topology, together with mechanical bonding, in isomeric [3]- and doubly interlocked [2]catenanes controls the formation of TTF radical dimers within their structural frameworks, including a ring-in-ring complex (formed between an organoplatinum square and a {2+2} macrocyclic polyether containing two 1,5-dioxynaphthalene (DNP) and two TTF units) that is topologically isomeric with the doubly interlocked [2]catenane. The separate TTF units in the two {1+1} macrocycles (each containing also one DNP unit) of the isomeric [3]catenane exhibit slightly different redox properties compared with those in the {2+2} macrocycle present in the [2]catenane, while comparison with its topological isomer reveals substantially different redox behavior. Although the stabilities of the mixed-valence (TTF2)^(•+) dimers are similar in the two catenanes, the radical cationic (TTF^(•+))_2 dimer in the [2]catenane occurs only fleetingly compared with its prominent existence in the [3]catenane, while both dimers are absent altogether in the ring-in-ring complex. The electrochemical behavior of these three radically configurable isomers demonstrates that a fundamental relationship exists between topology and redox properties

Computer Simulations of Hyperbranched Polymers: the Influence of the Wiener Index on the Intrinsic Viscosity and Radius of Gyration

The influence of the Wiener index on solution properties of trifunctional hyperbranched polymers has been investigated using Brownian dynamics simulations with excluded volume and hydrodynamic interactions. A range of degrees of polymerization (N) and degrees of branching (DB) were used. For each DB and N, several molecules with different Wiener indices (W) were simulated, where W depends on the arrangement of branch points. The intrinsic viscosity and the radius of gyration (Rg) of HPs were both observed to scale with W at a constant N via a power law relationship, as found in the literature. Through their relationships to W, an expression relating intrinsic viscosity to Rg was obtained. This relationship is found to fall centrally between the predictions of Flory and Fox for linear polymers and that of Zimm and Kilb for branched polymers. Molecular shape in solution is also found to depend on W and N, as observed through the W dependence of the ratio of Rg to the hydrodynamic radius, Rh