13 research outputs found
Two essays in computational optimization: computing the clar number in fullerene graphs and distributing the errors in iterative interior point methods
Fullerene are cage-like hollow carbon molecules graph of pseudospherical sym-
metry consisting of only pentagons and hexagons faces. It has been the object
of interest for chemists and mathematicians due to its widespread application
in various fields, namely including electronic and optic engineering, medical sci-
ence and biotechnology. A Fullerene molecular, Γ n of n atoms has a multiplicity
of isomers which increases as N iso ∼ O(n 9 ). For instance, Γ 180 has 79,538,751
isomers. The Fries and Clar numbers are stability predictors of a Fullerene
molecule. These number can be computed by solving a (possibly N P -hard)
combinatorial optimization problem. We propose several ILP formulation of
such a problem each yielding a solution algorithm that provides the exact value
of the Fries and Clar numbers. We compare the performances of the algorithm
derived from the proposed ILP formulations. One of this algorithm is used to
find the Clar isomers, i.e., those for which the Clar number is maximum among
all isomers having a given size. We repeated this computational experiment for
all sizes up to 204 atoms. In the course of the study a total of 2 649 413 774
isomers were analyzed.The second essay concerns developing an iterative primal dual infeasible path
following (PDIPF) interior point (IP) algorithm for separable convex quadratic
minimum cost flow network problem. In each iteration of PDIPF algorithm, the
main computational effort is solving the underlying Newton search direction
system. We concentrated on finding the solution of the corresponding linear
system iteratively and inexactly. We assumed that all the involved inequalities
can be solved inexactly and to this purpose, we focused on different approaches
for distributing the error generated by iterative linear solvers such that the
convergences of the PDIPF algorithm are guaranteed. As a result, we achieved
theoretical bases that open the path to further interesting practical investiga-
tion
The Structure of 4-Clusters in Fullerenes
Fullerenes can be considered to be either molecules of pure carbon or the trivalent plane graphs with all hexagonal and (exactly 12) pentagonal faces that models these molecules. Since carbon atoms have valence 4 and our models have valence 3, the edges of a perfect matching are doubled to bring the valence up to 4 at each vertex. The edges in this perfect matching are called a Kekule structure and the hexagonal faces bounded by three Kekule edges are called benzene rings. A maximal independent (disjoint) set of benzene rings for a given Kekule structure is called a Clar set, and the maximum possible size of a Clar set over all Kekule structures is the Clar number of the fullerene. For any patch of hexagonal faces in the fullerene away from all pentagonal faces, there is a perfect Kekule structure: a Kekule structure for which the faces of an independent set of benzene rings are packed together as tightly as possible. Starting with such a patch and extending it as far as possible results in a perfect Kekule structure except for isolated regions, called clusters, containing the pentagonal faces. It has been shown that clusters must contain even numbers of pentagonal faces. It has also been shown that the Kekule structure of the patch can be extended into each of these clusters to give a full Kekule structure. However, these Kekule extensions will not admit as tightly packed benzene rings as in the patch external to the clusters. A basic problem in computing the Clar number of a fullerene is to make these extensions in a way that maximizes the number of benzene rings in each cluster. The simplest case, that of 2-clusters, has been completely solved. This thesis is devoted to developing a complete understanding of the Clar structures of 4-clusters
Molecular dynamics simulations of conjugated semiconducting molecules
In this thesis, we present a study of conformational disorder in conjugated
molecules focussed primarily on molecular dynamics (MD) simulation methods.
Along with quantum chemical approaches, we develop and utilise MD simulation
methods to study the conformational dynamics of polyfluorenes and polythiophenes
and the role of conformational disorder on the optical absorption behaviour observed
in these molecules. We first report a classical force-field parameterisation scheme for
conjugated molecules which defines a density functional theory method of accuracy
comparable to high-order ab-initio calculations. In doing so, we illustrate the role of
increasing conjugated backbone and alkyl side-chain length on inter-monomer dihedral
angle potentials and atomic partial charge distributions. The scheme we develop
forms a minimal route to conjugated force-field parameterisation without substantial
loss of accuracy. We then present a validation of our force-field parameterisation
scheme based on self-consistent measures, such as dihedral angle distributions, and
experimental measures, such as persistence lengths, obtained from MD simulations.
We have subsequently utilised MD simulations to investigate the interplay of solvent
and increasing side-chain lengths, the emergence of conjugation breaks, and
the wormlike chain nature of conjugated oligomers. By utilising MD simulation geometries
as input for quantum chemical calculations, we have investigated the role
of conformational disorder on absorption spectral broadening and the formation of
localised excitations. We conclude that conformational broadening is effectively independent
of backbone length due to a reduction in the effect of individual dihedral
angles with increasing length and also show that excitation localisation occurs as a
result of large dihedral angles and molecular asymmetry
Sharp Upper Bounds on the Clar Number of Fullerene Graphs
The Clar number of a fullerene graph with n vertices is bounded above by ⌊n/6⌋ − 2 and this bound has been improved to ⌊n/6⌋ − 3 when n is congruent to 2 modulo 6. We can construct at least one fullerene graph attaining the upper bounds for every even number of vertices n ≥ 20 except n = 22 and n = 30