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4 changes: 2 additions & 2 deletions appendix.tex
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Expand Up @@ -16,7 +16,7 @@ \chapter{List of Gregory-Newton Contact Graphs}
%\end{figure}

\footnotesize{\begin{longtable}{lll}
\caption{List of all GN polyhedral graphs. The ordinal numbers $\omega$ in the first column
\caption{List of all \acs{GN} polyhedral graphs. The ordinal numbers $\omega$ in the first column
can be used to identify the individual polyhedral graphs.
$|E|$ is the number of edges, and the pairs of numbers refer to edges deleted
incident vertices $(k,l)$ as defined in the icosahedral graph
Expand Down Expand Up @@ -773,7 +773,7 @@ \chapter{List of Gregory-Newton Contact Graphs}


\footnotesize{\begin{longtable}{l@{\hskip 4pt}r@{\hskip 4pt}r@{\hskip 4pt}r@{\hskip 4pt}r@{\hskip 4pt}r@{\hskip 4pt}r@{\hskip 4pt}r@{\hskip 4pt}r@{\hskip 4pt}r@{\hskip 4pt}r@{\hskip 4pt}r@{\hskip 4pt}r@{\hskip 4pt}r@{\hskip 4pt}r@{\hskip 8pt}r}
\caption{GN polyhedron grouped by vertex and face degrees. $|N_n|$ is the
\caption{\acs{GN} polyhedron grouped by vertex and face degrees. $|N_n|$ is the
number of vertices of degree $n$, $|F_n|$ the number of $n$-gonal faces.
The ordinal numbers $\omega$ in the last column identify the polyhedral
graphs shown in table~\ref{tab:icosubgraphs}.}\\
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2 changes: 1 addition & 1 deletion introduction.tex
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@@ -1,5 +1,5 @@
%Introduction

\acresetall
\chapter{Introduction}
\label{sec:introduction}

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6 changes: 4 additions & 2 deletions methods.tex
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@@ -1,5 +1,7 @@
%methods

\acresetall

\part{Methods}
\label{sec:methods}

Expand All @@ -12,7 +14,7 @@ \chapter{Program Package \textsc{Spheres}}
coordinates of cluster structures with respect to a two-body potential was
required to be created. The optimisation routine needed to be flexible in the
way that it would be easy to implement different two-body potentials like
\ac{LJ} or \ac{eLJ}. Furthermore, the program needed to be able to analyse the
\acf{LJ} or \acf{eLJ}. Furthermore, the program needed to be able to analyse the
results regarding structure, energy and the matrix of second derivatives. The
resulting program was written in \Cpp with standard library version 11 and was
tested to compile in a Linux environment with the \texttt{clang++} compiler
Expand Down Expand Up @@ -110,7 +112,7 @@ \subsection{Optimisation of Input Structures}
\textit{libconfig}.\autocite{Lindner_libconfiglibraryprocessing_2018} For
example, the optimisation can be controlled using the \texttt{opt} group. In
this group, the optimisation model can be chosen with the \texttt{name} setting,
which currently can be set to either \texttt{BFGS} for the \ac{BFGS} algorithm
which currently can be set to either \texttt{BFGS} for the \acf{BFGS} algorithm
or \texttt{CG} for the conjugate gradient method. The optimiser uses the machine
learning library dlib\autocite{King_DlibmlMachineLearning_2009} as a back-end,
which allows for the implementation of additional optimisation models with
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89 changes: 46 additions & 43 deletions results.tex
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@@ -1,5 +1,7 @@
%results

\acresetall

\part{Results}
\label{sec:results}

Expand Down Expand Up @@ -1113,8 +1115,7 @@ \section{Conclusion}
icosahedral \ce{Au12},
drastically.\autocite{Pyykko_IcosahedralWAu12Predicted_2002}



\acresetall

\chapter[From Sticky-Hard-Sphere to Lennard-Jones-Type Clusters]{From
Sticky-Hard-Sphere to Lennard-Jones-Type clusters\footnote{This chapter is
Expand Down Expand Up @@ -1199,7 +1200,7 @@ \section{Introduction}
%
Using the Gregory-Newton argument results in $f(N)=6$, however, a tighter upper
bound has been published more recently by
\citeauthor{Bezdek-2013}\autocite{Bezdek-2013}.
\citeauthor{Bezdek-2013}\autocite{Bezdek-2013}
%
\begin{equation}
f(N)=6N-3(18)^{1/3}\pi^{-2/3}N^{2/3}.
Expand Down Expand Up @@ -1247,10 +1248,11 @@ \section{Introduction}
%
\begin{figure}[htb]\centering
\includegraphics[width=0.8\columnwidth]{kslj/exampleLJ.pdf}
\caption{Lennard-Jones potentials for different exponents $(m,n)$ with
fixed $n=2m$. As the exponents grow larger, the well of attraction becomes narrower
and its shape approaches the \acs{SHS} potential. The dashed line
shows the extended LJ potential for the xenon dimer \autocite{Jerabek_relativisticcoupledclusterinteraction_2017}.}
\caption{Lennard-Jones potentials for different exponents $(m,n)$ with fixed
$n=2m$. As the exponents grow larger, the well of attraction becomes
narrower and its shape approaches the \acs{SHS} potential. The dashed line
shows the extended Lennard-Jones potential for the xenon dimer
\autocite{Jerabek_relativisticcoupledclusterinteraction_2017}.}
\label{fig:LJ}
\end{figure}
%
Expand Down Expand Up @@ -1758,8 +1760,8 @@ \section{(6,12)-Lennard-Jones Clusters from Basin-Hopping}
\begin{table}[htb]\centering
\begin{threeparttable}
\caption{Number of missing structures after optimisation belonging to the
same "seed" (figure~\ref{fig:seeds}). $N=8$ is excluded because all LJ minima were
found starting from the \acs{SHS} model.}
same "seed" (figure~\ref{fig:seeds}). $N=8$ is excluded because all \acs{LJ}
minima were found starting from the \acs{SHS} model.}
\label{tab:seeds}
\begin{tabular}{llllll}\toprule
seed & $N=9$ & $N=10$ & $N=11$ & $N=12$ & $N=13$ \\ \midrule
Expand Down Expand Up @@ -1796,8 +1798,8 @@ \section{(6,12)-Lennard-Jones Clusters from Basin-Hopping}
\subfloat[$N=13$]{\includegraphics[width=.5\textwidth]{kslj/lj13var.pdf}}
\caption{Histograms of the difference between the longest and shortest bond
distances $d_\Delta=d_\text{max}-d_\text{min}$ for the complete set of
distinct LJ minima $\mathcal{M}_\text{LJ}(N)$ for $N=\{11,12,13\}$. Orange
bars give the number of distinct structures not contained in
distinct \acs{LJ} minima $\mathcal{M}_\text{LJ}(N)$ for $N=\{11,12,13\}$.
Orange bars give the number of distinct structures not contained in
$\mathcal{M}_\mathrm{LJ}$ as obtained from the basin-hopping algorithm.}
\label{fig:bondlength-variance}
\end{figure}%
Expand All @@ -1812,10 +1814,13 @@ \section{(6,12)-Lennard-Jones Clusters from Basin-Hopping}
\ac{SHS} boundary conditions are not satisfied. The data in
table~\ref{tab:energies} shows that the unmatched (UM) structures for a specific
$N$ value have much higher energies compared to the one of the global minimum
(which is set to zero, i.e. $E_0=0$).
%
(which is set to zero, i.e. $E_0=0$). They are always positioned in the upper
half of the energy spectrum, making them energetically unfavourable. However, no
correlation between $d_\Delta$ and the energetic position of the \ac{LJ}
clusters was found.

\begin{table}[htb]\centering
\caption{Range $[E_0,E_\text{max}]$ of the energy spectrum of all LJ
\caption{Range $[E_0,E_\text{max}]$ of the energy spectrum of all \acs{LJ}
minima, position of the second lowest minimum structure $E_1$ and position
of the first unmatched (UM) structure $E_0^\text{UM}$ relative to the
respective global minimum (in reduced units and $E_0=0$).}
Expand All @@ -1830,10 +1835,6 @@ \section{(6,12)-Lennard-Jones Clusters from Basin-Hopping}
13 & 9.26 & 2.85 & 6.14 \\\bottomrule
\end{tabular}
\end{table}%
%
They are always positioned in the upper half of the energy spectrum, making them
energetically unfavourable. However, no correlation between $d_\Delta$ and the
energetic position of the \ac{LJ} clusters was found.

Last, the geometries of the missing structures were investigated in more detail.
As it turns out, almost all of the missing stable \ac{LJ} clusters can be
Expand Down Expand Up @@ -1930,6 +1931,7 @@ \section{Conclusion}
However, that would require a considerable amount of programming as the current
program was developed with only two-body forces in mind.

\acresetall

\chapter[The Gregory-Newton Clusters]{The Gregory-Newton Clusters\footnote{This
chapter is composed of sections previously published in the articles
Expand Down Expand Up @@ -2293,11 +2295,11 @@ \section{Rigid Gregory-Newton Clusters and Corresponding Graphs}
\centering
\subfloat[hcp, $|E|=24, \omega =1$.\label{subfig:hcpgraph}]{\includegraphics[width=0.43\columnwidth]{gregory-newton/hcp-new.pdf}\hspace{0.03\textwidth}\includegraphics[width=.37\columnwidth]{gregory-newton/hcp.png}}\\
\subfloat[fcc, $|E|=24, \omega =2$.\label{subfig:fccgraph}]{\includegraphics[width=0.43\columnwidth]{gregory-newton/fcc-new.pdf}\hspace{0.03\textwidth}\includegraphics[width=.37\columnwidth]{gregory-newton/fcc.png}}
\caption{GN hcp (triangular orthobicupola) and fcc (cuboctahedron) graphs
(central sphere removed) as sub-graphs of the icosahedral graph and
corresponding rigid \acs{GNC}s. Red lines indicate the edges that were removed to
create the GN graph. The ordinal numbers $\omega$ refer to
Table~\ref{tab:icosubgraphs} in the appendix.}
\caption{\acs{GN} \acs{hcp} (triangular orthobicupola) and \acs{fcc}
(cuboctahedron) graphs (central sphere removed) as sub-graphs of the
icosahedral graph and corresponding rigid \acsp{GNC}. Red lines indicate
the edges that were removed to create the \acs{GN} graph. The ordinal
numbers $\omega$ refer to Table~\ref{tab:icosubgraphs} in the appendix.}
\label{fig:GNshellgraphs}
\end{figure}

Expand Down Expand Up @@ -2334,9 +2336,9 @@ \section{Rigid Gregory-Newton Clusters and Corresponding Graphs}
\begin{figure}[htb]
\centering
\includegraphics[width=.8\columnwidth]{gregory-newton/plane.png}
\caption{Illustration of one zig-zag path (light blue spheres) that needs
to be deformed such that it aligns with the triangular plane (shown in
grey) of the fcc crystal.}
\caption{Illustration of one zig-zag path (light blue spheres) that needs to
be deformed such that it aligns with the triangular plane (shown in grey) of
the \acs{fcc} crystal.}
\label{fig:ico-fcc-trans}
\end{figure}
%
Expand All @@ -2363,11 +2365,12 @@ \section{Rigid Gregory-Newton Clusters and Corresponding Graphs}
\centering
\subfloat[icosahedral motif, $|E|=22, \omega =4$.\label{subfig:ico4graph}]{\includegraphics[width=0.47\columnwidth]{gregory-newton/4-22-new.pdf}\hspace{0.03\textwidth}\includegraphics[width=.3\columnwidth]{gregory-newton/GNico4-22}}\\
\subfloat[icosahedral motif, $|E|=22, \omega =7$.\label{subfig:ico7graph}]{\includegraphics[width=0.47\columnwidth]{gregory-newton/7-22-new.pdf}\hspace{0.03\textwidth}\includegraphics[width=.3\columnwidth]{gregory-newton/GNico7-22.png}}
\caption{Representative GN graphs (central sphere removed) with $|F_3|=10$
as sub-graphs of the icosahedral graph and corresponding rigid \acs{GNC}s. The
icosahedral motif in the 3D embedding is clearly visible. Red lines
indicate the edges that were removed to create the GN graph. The ordinal
numbers $\omega$ refer to Table~\ref{tab:icosubgraphs} in the appendix.}
\caption{Representative \acs{GN} graphs (central sphere removed) with
$|F_3|=10$ as sub-graphs of the icosahedral graph and corresponding rigid
\acsp{GNC}. The icosahedral motif in the 3D embedding is clearly visible.
Red lines indicate the edges that were removed to create the \acs{GN} graph.
The ordinal numbers $\omega$ refer to Table~\ref{tab:icosubgraphs} in the
appendix.}
\label{fig:GNicographs}
\end{figure}

Expand All @@ -2377,11 +2380,11 @@ \section{Rigid Gregory-Newton Clusters and Corresponding Graphs}
\begin{figure}[htb]
\centering
\subfloat[Distorted elongated pentagonal bipyramid (Johnson solid $J_{16}$), $|E|=23, \omega =3$.\label{subfig:johnsongraph}]{\includegraphics[width=0.47\columnwidth]{gregory-newton/johnson-new.pdf}\hspace{0.03\textwidth}\includegraphics[width=.3\columnwidth]{gregory-newton/johnson.png}}
\caption{GN graph (central sphere removed) as sub-graphs of the icosahedral
graph and corresponding GN Johnson-like solid (with edges removed). Red
lines indicate the edges that were removed from the icosahedral graph to
create the GN graph. The ordinal number $\omega$ refers to
Table~\ref{tab:icosubgraphs} in the appendix.}
\caption{\acs{GN} graph (central sphere removed) as sub-graphs of the
icosahedral graph and corresponding \acs{GN} Johnson-like solid (with edges
removed). Red lines indicate the edges that were removed from the
icosahedral graph to create the \acs{GN} graph. The ordinal number $\omega$
refers to Table~\ref{tab:icosubgraphs} in the appendix.}
\label{fig:GNJohnsongraph}
\end{figure}
%
Expand Down Expand Up @@ -2425,7 +2428,7 @@ \section{Symmetry-Broken Lennard-Jones Gregory-Newton Clusters}
\begin{figure}[htb]\centering
\includegraphics[width=.8\columnwidth]{gregory-newton/ico-2d.pdf}
\caption{Number of unique structures resulting from an optimisation with a
LJ$(a,b)$ potential. The lowest contour line shows the point where more
$(a,b)$-\acs{LJ} potential. The lowest contour line shows the point where more
than one structure results from the optimisation and the distance between
contour lines is 1.}
\label{fig:ico-2d}
Expand All @@ -2446,7 +2449,7 @@ \section{Symmetry-Broken Lennard-Jones Gregory-Newton Clusters}
\begin{figure}[htb]\centering
\includegraphics[width=.8\columnwidth]{gregory-newton/no-ico.pdf}
\caption{Different types of energy landscapes arising from combinations of
the LJ $(a,b)$ exponents. (1) One single (icosahedral) minimum, (2) more
the $(a,b)$-\acs{LJ} exponents. (1) One single (icosahedral) minimum, (2) more
than one minimum with the icosahedron as the global minimum, (3) more than
one minimum with the icosahedron becoming a local (and not global) minimum,
(4) the icosahedral motif disappears completely. The unshaded small area
Expand Down Expand Up @@ -2474,11 +2477,11 @@ \section{Symmetry-Broken Lennard-Jones Gregory-Newton Clusters}
%
\begin{figure}[htb]\centering
\includegraphics[width=.8\columnwidth]{gregory-newton/compareLJ.pdf}
\caption{Comparison of different shapes of LJ potentials at the phase
\caption{Comparison of different shapes of \acs{LJ} potentials at the phase
transition lines shown in figure~\ref{fig:no-ico} with the traditional
LJ(6,12) potential (black solid line). Dashed lines refer to potentials
with low $a$ values (left side of figure~\ref{fig:no-ico}), while solid lines
refer to potentials with high $a$ values (right side of
(6,12)-\acs{LJ} potential (black solid line). Dashed lines refer to
potentials with low $a$ values (left side of figure~\ref{fig:no-ico}), while
solid lines refer to potentials with high $a$ values (right side of
figure~\ref{fig:no-ico}).}
\label{fig:compareLJ}
\end{figure}
Expand All @@ -2501,7 +2504,7 @@ \section{Symmetry-Broken Lennard-Jones Gregory-Newton Clusters}
%
\begin{figure}[htb]\centering
\includegraphics[width=.8\columnwidth]{gregory-newton/sigma.pdf}
\caption{Hard-sphere radii $\sigma$ in reduced units for the LJ$(a,b)$
\caption{Hard-sphere radii $\sigma$ in reduced units for the $(a,b)$-\acs{LJ}
potentials on the transition lines shown in figure~\ref{fig:no-ico}.}
\label{fig:hardsphere}
\end{figure}
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13 changes: 9 additions & 4 deletions theory.tex
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Expand Up @@ -3,6 +3,8 @@
\part{Theoretical Background}
\label{sec:theory}

\acresetall

\chapter{Graph Theory}
\label{sec:graphtheory}

Expand Down Expand Up @@ -616,7 +618,7 @@ \section{Graph Matching}
version included in the \textit{boost graph
library}\autocite{Siek_BoostGraphLibrary_2002} utilised in this thesis.


\acresetall

\chapter{Quantum Chemistry}
\label{sec:basicsofQC}
Expand Down Expand Up @@ -1448,6 +1450,8 @@ \subsection{Projector-Augmented Wave Method}
which implies that the projector functions must be orthonormal to the pseudo
partial wave functions $\widetilde{\phi}_i$.

\acresetall

\chapter{Geometry Optimisation}
\label{sec:geometryoptimisation}

Expand Down Expand Up @@ -1965,7 +1969,7 @@ \section{Implementation for Two-Body Interaction Potentials}
three-dimensional space that interact via a given potential some modifications
have to be made to use the previously described methods. In the following
paragraphs the mathematical background for the implementation of potentials
that only depend on the distance between two objects like \ac{LJ} and \ac{eLJ}
that only depend on the distance between two objects like \acf{LJ} and \acf{eLJ}
for the program \textsc{Spheres} (chapter~\ref{sec:theprogramspheres}) is
explained. The physical objects in this case are called spheres and the
optimisation procedure tries to locate the minimiser corresponding to the
Expand Down Expand Up @@ -2258,6 +2262,7 @@ \subsection{Algorithms}
\ac{PES}. Other algorithms that can be used to solve the problem at hand are,
for example, the particle swarm algorithm or the ant colony optimisation method.

\acresetall

\chapter{Interaction Potentials}
\label{sec:energylandscapes}
Expand Down Expand Up @@ -2470,7 +2475,7 @@ \subsection{Equation of State from the Partition Function}
\section{Lennard-Jones Potential}
\label{sec:LennardJones}

One of the most widely used interaction potentials today is the \ac{LJ}
One of the most widely used interaction potentials today is the \acf{LJ}
potential. It was first introduced by
\citeauthor{Jones_DeterminationMolecularFields_1924} (later Lennard-Jones) on
April 22, 1924\autocite{Jones_DeterminationMolecularFields_1924}, however, the
Expand Down Expand Up @@ -2599,7 +2604,7 @@ \subsection{Lennard-Jones Clusters}
atoms due to dispersive forces, which are approximated well by the \ac{LJ}
potential.

The hypersurface upon which the particles move, also called a \ac{PES} or energy
The hypersurface upon which the particles move, also called a \acf{PES} or energy
landscape, has been explored extensively for the \ac{LJ} potential.
\autocite{Tsai_Useeigenmodemethod_1993,Ball_Dynamicsstatisticalsamples_1999,Doye_Saddlepointsdynamics_2002} \citeauthor{Doye_Saddlepointsdynamics_2002}\autocite{Doye_Saddlepointsdynamics_2002}
employed the eigenvector following method (section~\ref{sec:GOAlgorithms}) to
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