Analytic structure and power-series expansion of the Jost function for the two-dimensional problem

For a two-dimensional quantum mechanical problem, we obtain a generalized power-series expansion of the S-matrix that can be done near an arbitrary point on the Riemann surface of the energy, similarly to the standard effective range expansion. In order to do this, we consider the Jost-function and analytically factorize its momentum dependence that causes the Jost function to be a multi-valued function. The remaining single-valued function of the energy is then expanded in the power-series near an arbitrary point in the complex energy plane. A systematic and accurate procedure has been developed for calculating the expansion coefficients. This makes it possible to obtain a semi-analytic expression for the Jost-function (and therefore for the S-matrix) near an arbitrary point on the Riemann surface and use it, for example, to locate the spectral points (bound and resonant states) as the S-matrix poles. The method is applied to a model simlar to those used in the theory of quantum dots.Comment: 42 pages, 9 figures, submitted to J.Phys.

LiFE Assessment Tool

As part of an ongoing study to construct a molecular Turing machine in which a polymer chain is encoded via allosteric information transfer between macrocyclic complexes, we describe the thermodynamic and kinetic characterization of a multicomponent self-assembled system based on a zinc porphyrin macrocyclic compound, a bidentate ligand (1,4-diazabicyclo[2.2.2]octane, DABCO), and a viologen-substituted polymer guest. Initial addition of DABCO to the porphyrin macrocycle in chloroform solution leads to the formation of a stable 2:1 (porphyrin:DABCO) dimeric complex, even under dilute conditions, by means of strong cooperative interactions involving hydrogen and metal-ligand bonds. Further titration of the porphyrin-DABCO mixtures with the polymer gives rise to a complex array of species in the solution. The system is analyzed in detail by a combination of spectroscopic measurements and computational modeling. Each association constant in the binding scheme and the fraction of each individual complex that is formed in solution are determined precisely using a mass-balance model. Kinetic studies revealed that the rates of the polymer threading and dethreading in and out of the dimeric system are remarkably slow, indicating that the polymer is locked inside the cavity of the stable 2:1 dimeric complex as a result of strong allosteric interactions

Supramolecular copolymers: structure and composition revealed by theoretical modeling

Supramolecular copolymers, non-covalent analogues of synthetic copolymers, constitute a new and promising class of polymers. In contrast to their covalent counterparts, the details of their mechanism of formation, as well as the factors determining their composition and length, are still poorly understood. Here, the supramolecular copolymerization between two slightly structurally different benzene-1,3,5-tricarboxamide (BTA) monomers functionalized with either oligodimethylsiloxane (oDMSi) or alkyl side chains is unraveled by combining experimental and theoretical approaches. By applying the “sergeant-and-soldiers” approach using circular dichroism (CD) experiments, we are able to obtain detailed insights into the structure and composition of these supramolecular copolymers. Moreover, we observe an unexpected chiral induction upon mixing two independently CD-silent solutions of the achiral (soldier) and chiral (sergeant) monomers. We find that the subtle differences in the chemical structure of the two monomers impact their homopolymerization mechanism: whereas alkyl-BTAs cooperatively self-assemble, oDMSi-BTAs self-assemble in an isodesmic manner. The effect of these mechanistic differences in the supramolecular copolymerization process is investigated as a function of the composition of the two monomers and explicitly rationalized by mathematical modeling. The results show that, at low fractions of oDMSi-BTA sergeants (25 mol%), the isodesmic assembly of the increasing amounts of sergeant becomes more dominant, and different species start to coexist in the copolymerization process. The analysis of the experimental data with a newly developed theoretical model allows us to quantify the thermodynamic parameters, the distribution of different species, and the compositions and stack lengths of the formed supramolecular copolymers existing at various feed ratios of the two monomers.This work was financially supported by The Netherlands Organization for Scientific Research (NWO-TOP PUNT Grant 10018944) and the Dutch Ministry of Education, Culture and Science (Gravity Program 024.001.035)

Selection for Replicases in Protocells

PMCID: PMC3649988This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited