text:
\section{Introduction} \label{sec:intro}

  The variability of the (equilibrium) carbon-carbon (C--C) bond lengths in $\pi$-electron hydrocarbons and all-carbon molecules (fullerenes, nanotubes, and graphene) is a consequence of the coupling between the nuclear framework and the system of \emph{mobile} electrons that occupy the molecular orbitals of the $\pi$-symmetry. This phenomenon poses a challenge to both experiment and theory.

  On the experimental side one faces difficulties in obtaining good-quality monocrystals. Moreover, the usual diffraction techniques (by X-rays or neutrons) suffer from the limitations of the theoretical model which translates the diffraction data into the molecular geometry. This model, in which spherical atoms are subject to uncorrelated thermal motions, neglects the changes of the electronic density due to chemical bonding (X-ray diffraction), as well as the rigid-body motions of the whole molecule. The latter effects, affecting both the X-ray and neutron studies, are especially important for large and rigid structures of $\pi$-electron hydrocarbons.

  On the theory side there are convergence problems of quantum-mechanical calculations with respect to the orbital basis sets and the electronic-correlation contributions. These problems become especially acute with the growing size of the studied molecules. As it has been already in the past, a carefully crafted semi-empirical quantum approach may come to the rescue.

  In the two recent papers by Stolarczyk and Krygowski~\cite{Stol2017a,Stol2017b} a very simple Augmented H\"{u}ckel Molecular Orbital (AugHMO) model has been designed for this purpose. It allows for a qualitative, as well as quantitative, study of the bond-length variation in general $\pi$-electron molecules (i.e., those which are planar or locally planar). The simplest realization of the AugHMO model, the the H\"{u}ckel-Su-Schrieffer-Heeger (HSSH) method, was parametrized~\cite{Stol2017b} for the $\pi$-electron hydrocarbons and all-carbon molecules. It was demonstrated~\cite{Stol2017b} that the HSSH method is capable of fairly accurate description of the C--C bond lengths in a variety of $\pi$-electron hydrocarbons and carbon systems, including polyacetylene, fullerene C$_{60}$, and graphene.

  In the present paper we employ the HSSH method to calculate the equilibrium C--C bond lengths for a set of the polycyclic aromatic hydrocarbons (PAHs). These are precisely the molecules whose spectroscopic properties were summarized and analysed in a review paper by Riegel and M\"{u}llen~\cite{Riegel2010}. These PAHs have been obtained by chemical synthesis (many of them by M\"{u}llen and his coworkers), but in most cases no diffraction data of sufficient quality is available to provide us with their geometries. On the other hand, these PAHs are rather big (up to 96 C atoms), and \textit{ab initio} methods of quantum chemistry are too expensive for a reliable geometry optimization.

  In our paper we keep the classification of PAHs, as well as the numbering of the molecules, employed in Ref.~\cite{Riegel2010}:
\begin{itemize}
\item Clarenes (i.e., all-benzenoid PAHs): molecules \textbf{1} -- \textbf{14}, see Sec.~\ref{sec:clar}, Figs.~\ref{fig:clar1} and \ref{fig:clar2}.
\item K-region PAHs: molecules \textbf{15} -- \textbf{20} (plus ovalene), see Sec.~\ref{sec:K-region}, Figs.~\ref{fig:PCO} and \ref{fig:KPAHs2}.
\item Phenacenes: molecules \textbf{21} -- \textbf{25}, see Sec.~\ref{sec:phen}, Fig.~\ref{fig:phen}.
\item Rylenes: molecules \textbf{26} -- \textbf{29}, see Sec.~\ref{sec:ryl}, Fig.~\ref{fig:ryl}.
\item Acenes: molecules \textbf{30} -- \textbf{34} (plus naphthalene), see Sec.~\ref{sec:ace}, Fig.~\ref{fig:ace}.
\end{itemize}

\section{HSSH calculations} \label{sec:calcs}

  The HSSH bond-length optimization procedure ia a (simplified) analog of the molecular-geometry optimization techniques that employ the analytical gradients and hessians of the total molecular energy~\cite{Yama1994}. In the HSSH method the hydrocarbon under study is fully characterized by its topology of C--C bonds (which is coded in the form of the topological matrix~\cite{Stol2017a,Stol2017b}).

  At the start of the calculations all the C--C bonds are put equal to the value for benzene, $R_{\rm benz}^{\rm e} = 1.397$ \AA. Then the usual H\"{u}ckel calculations follow, yielding the molecular orbitals (MOs) and the $\pi$-electron bond orders $p_{\rm bond}$. This is the 0th iteraction, fully equivalent to the standard H\"{u}ckel approach. In each subsequent step (iteration) the condition of the vanishing gradient of the total HSSH energy is enforced: this amounts to iterating the linear bond-order bond-length (LBOBL) relationship:
\begin{equation} \label{eq:LBOBL}
R_{\rm bond}^{\rm e} = R^{\rm o} - x \, p_{\rm bond}^{\rm e} \,,
\end{equation}
where $R_{\rm bond}^{\rm e}$ and $p_{\rm bond}^{\rm e}$, respectively, are the equilibrium bond length and the $\pi$-electron bond order for a given C--C bond. $R^{\rm o} = 1.523$ \AA{} and $x = 0.189$ \AA{} are two geometrical parameters of the AugHMO model of $\pi$-electron hydrocarbons~\cite{Stol2017a,Stol2017b}. In the HSSH method for hydrocarbons~\cite{Stol2017b} there is also a third geometrical parameter $y = 0.2756$ \AA{} which determines the slope of a linear function $\beta (R)$ describing the dependence of the ,,resonance integral'' $\beta$ on the C--C bond length $R$. In order to improve the rate of convergence of the iterative procedure, the hessian of the HSSH total energy is calculated and applied~\cite{Stol2017a}. In our calculations for PAHs we assumed that the iterations stopped when the subsequent values of bond-length differences dropped below $5 \cdot 10^{-6}$ \AA. To this end five ,,direct iterations'' using Eq.~(\ref{eq:LBOBL}) followed by two to three iterations involving the hessian were always sufficient.

  Four points concerning the HSSH method~\cite{Stol2017a,Stol2017b} should be made clear. First:  this method optimizes \emph{only} the bond lengths, and it is assumed that the optimal valence angles (the C--C--C ones in the case of PAHs) are consistent with the optimized C--C bond lengths, and as close to $120^{\rm o}$ as possible. By reducing the molecular geometry to bond lengths only seems justified for such molecules as PAHs and contribute to the computational effectiveness of the HSSH method. Second: while Eq.~(\ref{eq:LBOBL}) looks like a relationship of \emph{local character}, it should be remembered that the $\pi$-electron bond orders $p_{\rm bond}$ are derived from the set of completely delocalized occupied $\pi$ molecular orbitals~\cite{Stol2017a}. And in the HMO model (as well as in the AugHMO one) these bond orders appear to be sensitive probes of molecular topology. Third: PAHs belong to \emph{the class of alternant $\pi$-electron hydrocarbons}~\cite{Stol2017b}, and within the HMO (AugHMO) model the Coulson-Rushbrooke theorem~\cite{Couls1940} holds: (i) the $\pi$-electron orbital energies for the occupied and unoccupied states are placed symmetrically with respect to the value of the ,,Coulomb integral'' $\alpha$, (ii)  the net $\pi$-electron charges~\cite{Stol2017a} at the C atoms are equal to zero. Four: in the HSSH calculations for hydrocarbons we employ the H\"{u}ckel energy units:
\begin{equation} \label{eq:Huckel_units}
\alpha = 0 \,, \quad |\beta| \equiv |\beta(R_{\rm benz}^{\rm e})| = 1 \,.
\end{equation}
Thus, the only empirical parameters of the HSSH model are the above-mentioned three geometrical parameters: $R^{\rm o}$, $x$, and $y$.

  We used our HSSH optimized bond lengths to calculate the values of the aromaticity index HOMA of Krygowski and coworkers~\cite{Krusz1972, Kryg1993, Kryg1996, Cyra1999}. HOMA (Harmonic-Oscillator Measure of Aromaticity) is currently considered the most important indicator of the aromatic character of $\pi$-electron molecules (or their fragments), based solely on the values of bond lengths. For a $\pi$-electron hydrocarbon the definition of HOMA reads as~\cite{Krusz1972, Kryg1993}
\begin{equation} \label{eq:HOMA1}
{\rm HOMA} = 1 - \frac{\alpha}{N} \sum_{\rm bond} (R_{\rm bond}^{\rm e} - R_{\rm opt})^2  \,,
\end{equation}
where the summation goes over all $N$ carbon-carbon bonds in the $\pi$-electron hydrocarbon (or its fragment), $\alpha = 257.7$ \AA$^{-2}$ is a normalization constant, while $R_{\rm opt} = 1.388$ \AA{} represents the optimal value of the C--C bond length in the HOMA model. Although of energetic provenience~\cite{Krusz1972}, HOMA of Eq.~(\ref{eq:HOMA1}) is dimensionless and fulfils condition $0 \leq {\rm HOMA} \leq 1$ (smaller HOMA indicates lower aromatic character). It was found~\cite{Kryg1996, Cyra1999} that definition~(\ref{eq:HOMA1}) can be rewritten in a more revealing form:
\begin{equation} \label{eq:HOMA2}
{\rm HOMA} = 1 - {\rm EN} - {\rm GEO} = 1 - \alpha (R_{\rm av} - R_{\rm opt})^2  - \frac{\alpha}{N} \sum_{\rm bond} (R_{\rm bond}^{\rm e} - R_{\rm av})^2  \,,
\end{equation}
where
\begin{equation} \label{eq:Rav}
R_{\rm av} = \frac{1}{N} \sum_{\rm bond} R_{\rm bond}^{\rm e}
\end{equation}
is the average C--C bond length in the $\pi$-electron hydrocarbon (or its fragment). The components EN and GEO may be interpteted as some ,,dearomatization'' contributions originating from the bond-length elongation and bond alternation, respectively~\cite{Cyra1999}. Let us remark that the EN component is simply a quadratic function of $R_{\rm av}$, and that the EN and GEO components (and thus HOMA) are sensitive to the rounding errors in the values of $R_{\rm bond}^{\rm e}$.

\section{Modular architecture of PAHs} \label{sec:modarch}

  The carbon skeletons of PAHs are finite jigsaw cuts from the honeycomb lattice of graphene. At the perimeter of a PAH there is a number of methine groups ($>$CH) corresponding to the tertiary carbon atoms (each with two C neighbors); the remaining C atoms are of the quaternary kind (with three C neighbors). The varied shapes of PAHs determine the corresponding topologies of the C--C bonds which, in the AugHMO model~\cite{Stol2017a,Stol2017b}, directly translate into the bond-length pattern. Our HSSH calculations for PAHs reveal a considerable spread ($~ 0.1$ \AA) of the equilibrium C--C bond lengths: from $1.363$ \AA{}  [bond a in pyrene (\textbf{15}), see Fig.~\ref{fig:PCO}], to $1.461$ \AA{} [bond m' in perylene (\textbf{26}), see Fig.~\ref{fig:ryl}]. These values are to be compared with the HSSH result for graphene~\cite{Stol2017b} equal to $1.424$ \AA.

  Our HSSH calculations indicate that some classes of PAHs (clarenes, rylenes, and a subclass of the K-region PAHs) are well represented by a \emph{modular architecture}. The ground-state geometry of such a PAH can be assemblied by using certain standard molecular fragments (modules) of fixed geometries, connected by C--C bonds (linkers) of fixed lengths. The set of modules employed in the present study is depicted in Fig.~\ref{fig:ABCmodules}. These modules are assumed to have ,,natural'' symmetries, independent of the particular molecular environment. More details will be provided in sections devoted to clarenes (Sec.~\ref{sec:clar}), K-region PAHs (2) (Subsec.~\ref{ssec:KPAHs2}), and rylenes (Sec.~\ref{sec:ryl}).

\FloatBarrier
\begin{figure}[!h]
\begin{minipage}{\linewidth}
\centering
\includegraphics[scale=0.15]{Fig_01_Modules.png}
\caption{A-, B-, and C-modules. C--C bond topologies, with symbols of symmetry non-equivalent bonds; directions of linkers are indicated.}
\label{fig:ABCmodules}
\end{minipage}
\end{figure}
\FloatBarrier

  How the bond lengths corresponding to the modules and linkers were computed? Take the example of bond a in module A3: the length $\overline{R^{\rm e}_{\rm a}}$ was calculated by averaging over all the 28 symmetry-unrelated occurences of this type of bond in the molecules containig the A3-module (see Figs.~\ref{fig:clar1}, \ref{fig:clar2}, and \ref{fig:KPAHs2}). As a measure of the spread of the actual values of this bond length we used the absolute value of maximal difference,
\begin{equation} \label{eq:Delta}
\Delta_a = \textrm{max} | R_{\rm a}^{\rm e} - \overline{R_{\rm a}^{\rm e}} | \,,
\end{equation}
corresponding to the above-mentioned set of 28 values of $R_{\rm a}^{\rm e}$.

\section{Clarenes} \label{sec:clar}

  Eric Clar proposed~\cite{Clar1972, Sola2013} that the all-benzenoid PAHs are special because of being ideal superpositions of six-electron units -- the ,,Clar sextets.'' Thus, it seems to be quite proper to call this class of PAHs ,,clarenes.''  The family of clarenes, as taken from the paper by Riegel and M\"{u}llen\cite{Riegel2010}, consists of 14 molecules, form convenience partitioned into two subsets shown in Figs.~\ref{fig:clar1} and \ref{fig:clar2}.

\FloatBarrier
\begin{figure}[!h]
\centering
\includegraphics[scale=0.15]{Fig_02_Clarenes_1_.png}
\caption{Clarenes (1). C--C bond topologies, with symbols of symmetry non-equivalent A-modules and AA-linkers.}
\label{fig:clar1}
\end{figure}

\begin{figure}[!h]
\centering
\includegraphics[scale=0.15]{Fig_03_Clarenes_2_.png}
\caption{Clarenes (2). C--C bond topologies, with symbols of symmetry non-equivalent A-modules and AA-linkers.}
\label{fig:clar2}
\end{figure}
\FloatBarrier

  Every molecule of clarene contain $6n$ carbon atoms and can be partitioned into six-carbon-atom benzene-like fragments, hereafter called the A-modules. Each A-module carries six $\pi$ electrons -- this is the above-mentioned Clar sextet. Within a given molecule the A-modules (depicted in Fig.~\ref{fig:ABCmodules}) are connected by C--C bonds, hereafter called the AA-linkers. The An-module (n = 2, 3, 4, 5 ,6) uses n linkers to connect with other modules; benzene may be considered a ,,honorary'' A0 module, while the phenyl group (not appearing in PAHs) is the A1-module. Modules A4 and A4' correspond to two diffrent arrangements of four linkers. Moreover, linkers connecting the same pair of modules (eg., A3-A5) may correspond to different linking topologies: see, e.g., linker u in molecule \textbf{12}, and linkers u' and u'' in molecule \textbf{6}.

  Our HSSH calculations demonstrate that, with sufficient accuracy, each clarene can be additively assemblied from standard A-modules, connected via some standard AA-linkers. The results in Tabl.~\ref{tab:Re_A} and \ref{tab:Re_AAlink} are quite convincing: the An-modules (n = 2 -- 5) have well-defined geometries (with characteristic bond lengths), and the AA-linkers of a given type have nearly constant lengths. It is seen that within the A-modules the bond lengths are distictly shorter than those corresponding to the AA-linkers. Clearly, this is an indication that
the Clar sextets are to some degree \emph{localized} within the clarene molecule.

\FloatBarrier
\begin{table}[!h]
\centering
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|}
\hline & & \multicolumn{10}{c|}{Module} \\
\cline{3-12}
     & \textbf{Benz.} & \multicolumn{2}{c|}{\textbf{A2}} & \multicolumn{2}{c|}{\textbf{A3}} & \multicolumn{2}{c|}{\textbf{A4}} & \multicolumn{2}{c|}{\textbf{A4'}} & \multicolumn{2}{c|}{\textbf{A5}} \\
  \hline & & & & & & & & & & & \\[-2.0ex]
Bond & $R^{\rm e}$ & $\overline{R^{\rm e}}$ & $\Delta$ & $\overline{R^{\rm e}}$ & $\Delta$ & $\overline{R^{\rm e}}$ & $\Delta$ & $\overline{R^{\rm e}}$ & $\Delta$ & $\overline{R^{\rm e}}$ & $\Delta$ \\
\hline
\hline a  & 1.397 & 1.406 & 0.002 & 1.397 & 0.003 & 1.388 & 0.002 & 1.402 & 0.000 & 1.403 & 0.000 \\
\hline b  &  ---  & 1.389 & 0.002 & 1.404 & 0.004 & 1.412 & 0.003 & 1.416 & 0.000 & 1.417 & 0.001 \\
\hline c  &  ---  & 1.409 & 0.002 & 1.415 & 0.002 & 1.410 & 0.001 &  ---  &  ---  & 1.416 & 0.002 \\
\hline d  &  ---  & 1.409 & 0.001 &  ---  &  ---  & 1.420 & 0.002 &  ---  &  ---  &  ---  &  ---  \\
\hline
\end{tabular}
\caption{A-modules (up to A5). HSSH equilibrium C--C bond lengths (in \AA).}
\label{tab:Re_A}
\end{table}

\begin{table}[!h]
\begin{minipage}{\linewidth}
\centering
\begin{tabular}{|c|c|c|c||c|c|c|c|}
\hline \multicolumn{2}{c|}{} & & & \multicolumn{2}{c||}{} & & \\[-2.0ex]
 \multicolumn{2}{c|}{Linker} & $\overline{R^{\rm e}}$ & $\Delta$ & \multicolumn{2}{c||}{Linker} & $\overline{R^{\rm e}}$ & $\Delta$ \\
\hline \multicolumn{2}{l|}{\mbox{m}  (\textbf{A2-A2})} & 1.456 & 0.001 & \multicolumn{2}{l||}{\mbox{w } (\textbf{A4-A4})} & 1.447 &  ---  \\
\hline \multicolumn{2}{l|}{\mbox{n } (\textbf{A2-A3})} & 1.455 & 0.001 & \multicolumn{2}{l||}{\mbox{w'} (\textbf{A4-A4})} & 1.442 & 0.001 \\
\hline \multicolumn{2}{l|}{\mbox{o } (\textbf{A2-A4})} & 1.452 & 0.002 & \multicolumn{2}{l||}{\mbox{x } (\textbf{A4-A6})} & 1.439 &  ---  \\
\hline \multicolumn{2}{l|}{\mbox{p } (\textbf{A2-A4'})}& 1.456 & 0.000 & \multicolumn{4}{c|}{} \\
\hline \multicolumn{2}{l|}{\mbox{q } (\textbf{A2-A5})} & 1.454 & 0.001 & \multicolumn{4}{c|}{} \\
\hline
\hline \multicolumn{2}{l|}{\mbox{r } (\textbf{A3-A3})} & 1.455 & 0.001 & \multicolumn{2}{l||}{\mbox{y } (\textbf{A5-A5})} & 1.438 &  ---  \\
\hline \multicolumn{2}{l|}{\mbox{r'} (\textbf{A3-A3})} & 1.444 & 0.000 & \multicolumn{2}{l||}{\mbox{y'} (\textbf{A5-A5})} & 1.447 &  ---  \\
\hline \multicolumn{2}{l|}{\mbox{s } (\textbf{A3-A4})} & 1.452 & 0.001 & \multicolumn{2}{l||}{\mbox{z } (\textbf{A5-A6})} & 1.437 & 0.001 \\
\hline \multicolumn{2}{l|}{\mbox{s'} (\textbf{A3-A4})} & 1.443 & 0.001 & \multicolumn{2}{l||}{\mbox{z'} (\textbf{A5-A6})} & 1.436 & 0.000 \\
\hline \multicolumn{2}{l|}{\mbox{t } (\textbf{A3-A4'})}& 1.456 &  ---  & \multicolumn{4}{c|}{} \\
\hline \multicolumn{2}{l|}{\mbox{u } (\textbf{A3-A5})} & 1.453 & 0.000 & \multicolumn{4}{c|}{} \\
\hline \multicolumn{2}{l|}{\mbox{u'} (\textbf{A3-A5})} & 1.442 &  ---  & \multicolumn{4}{c|}{} \\
\hline \multicolumn{2}{l|}{\mbox{u"} (\textbf{A3-A5})} & 1.443 &  ---  & \multicolumn{4}{c|}{} \\
\hline \multicolumn{2}{l|}{\mbox{v } (\textbf{A3-A6})} & 1.440 & 0.003 & \multicolumn{2}{l||}{\mbox{z"} (\textbf{A6-A6})} & 1.436 & 0.002 \\
\hline
\end{tabular}
\caption{AA-linkers. HSSH equilibrium C--C bond lengths (in \AA).}
\label{tab:Re_AAlink}
\end{minipage}
\end{table}
\FloatBarrier

  The An-modules (n = 2 -- 5) necessarily have to be located at the perimeter of a given clarene. The interior of the molecule, if sufficiently big, is filled with the A6-modules. Quite surprisingly, A6 cannot be considered a rigid building block: in a less symmetric surrounding its perfect $6mm$ (C$_{6v}$) symmetry\footnote{We use here the symbols corresponding to the planar point groups} becomes visibly perturbed. The most important cases of A6 deformations are presented in Tabl.~\ref{tab:Re_A6}. In molecules \textbf{11} and \textbf{20} (see Fig.~\ref{fig:KPAHs2}) the molecular symmetry group $3m$ (C$_{3v}$) induces some \emph{bond alternation} in the central A6-module; however, this effect is absent in molecule \textbf{14} due to a different orientation of the molecular-symmetry elements with respect to the central A6 hexagon. A similar pattern of deformation is found in molecule \textbf{19}, despite its lower symmetry corresponding to the $m$ (C$_{\rm s}$) group. It is apparent that bond alternation is the softest mode of deformation for the A6-module.

\FloatBarrier
\begin{table}[!h]
\centering
\begin{tabular}{|c||c|c|c|c|}
\hline Molecule &$R_{\rm a'}^{\rm e}$ (\AA) &$R_{\rm a}^{\rm e}$ (\AA) &$\frac{1}{2}(R_{\rm a'}^{\rm e} + R_{\rm a}^{\rm e})$& $R_{\rm a'}^{\rm e} - R_{\rm a}^{\rm e}$ \\
\hline
\hline \textbf{11} & 1.420  & 1.414 & 1.417 & 0.006 \\
\hline \textbf{19} & 1.421  & 1.414 & 1.417 & 0.007 \\
\hline \textbf{20} & 1.424  & 1.413 & 1.419 & 0.010 \\
\hline
\end{tabular}
\caption{A6-modules, HSSH equilibrium C--C bond lengths (in \AA). In molecules \textbf{11}, \textbf{19}, and \textbf{20} a pattern of \emph{alternating} bonds (a' a)$_3$ is seen. Full averaging over the remaining molecules containing A6-modules gives $R_{\rm a'}^{\rm e} = R_{\rm a}^{\rm e} = 1.417$, $\Delta = 0.003$.}
\label{tab:Re_A6}
\end{table}
\FloatBarrier

  While it seems proven that the perimeter of a clarene assumes a definite geometric structure corresponding to some arrangement od the An-modules (n = 2 -- 5), its interior is expected to approach the structure of graphene (however, see the discussion in the last Section). To this end the equilibrium bond lengths corresponding to bonds a and a' in the A6 module (see Fig.~\ref{fig:ABCmodules}), and the equilibrium bond length corresponding to the A6-A6 linker (z'', see molecule \textbf{14} in  Fig.~\ref{fig:clar2}) should approach the HSSH value for graphene ($1.424$ \AA). For the largest PAH in this study, molecule \textbf{14} (96 C atoms), we found $R_{\rm a'}^{\rm e} = R_{\rm a}^{\rm e} = 1.419$ \AA{}, $R_{\rm z''}^{\rm e} = 1.435$ \AA{}, quite far from the graphene limit. However, when one averages over three bond lengths adjacent to any C atom in any A6 module, the result is very close to $1.424$ \AA. Thus, even in relatively small clarenes their interiors ,,prepare'' for becoming graphenelike.  It may be said with some exaggeration that a big clarene molecule has a hard shell made of some An-modules (n = 2 -- 5), protecting an interior consisting of softer A6-modules, which, with the growing size of the molecule, loose their identity and melt into the homogenous honeycomb lattice of graphene.

  The inspection of the HOMA values and its components in Tabl.~\ref{tab:homa_A} indicates that the An-modules retain, to some degree, the aromaticity of the benzene molecule. It is seen that it is EN (and thus $\mathrm{R_{\rm av}}$) which is mostly responsible for diminishing of HOMA with the increase of n (the number of linkers). The result for graphene suggests its low aromaticity, despite the fact that graphene is the most stable of all carbon and $\pi$-electron hydrocarbon molecules (as the molecular enthalpies of formation attest).

\FloatBarrier
\begin{table}[!h]
\centering
\begin{tabular}{|l||c|c|c|c|}
\hline Module & $\mathrm{R_{\rm av}}$ (\AA) & EN & GEO & HOMA \\
\hline
\hline \textbf{Benzene} & 1.397 & 0.021 & 0.000 & 0.979 \\
\hline \textbf{A2}      & 1.402 & 0.051 & 0.021 & 0.928 \\
\hline \textbf{A3}      & 1.405 & 0.076 & 0.013 & 0.911 \\
\hline \textbf{A4}      & 1.409 & 0.110 & 0.043 & 0.847 \\
\hline \textbf{A4'}     & 1.406 & 0.086 & 0.012 & 0.902 \\
\hline \textbf{A5}      & 1.412 & 0.150 & 0.011 & 0.839 \\
\hline \textbf{A6}      & 1.417 & 0.217 & 0.000 & 0.783 \\
\hline \textbf{Graphene}& 1.424 & 0.334 & 0.000 & 0.666 \\
\hline
\end{tabular}
\caption{A-modules. $\mathrm{R_{\rm av}}$, EN, GEO, and HOMA values.}
\label{tab:homa_A}
\end{table}

  The molecular values of HOMA and its components collected in Tabl.~\ref{tab:homa_clar} show that the aromatic character of clarenes is lowering with their size. Both the EN and GEO components contribute to this effect. As clarenes grow larger, one finds that (i) the value of $\mathrm{R_{\rm av}}$ increases and thus departs from the HOMA value of $\mathrm{R_{\rm opt}}$, and (ii) the HOMO-LUMO gap (as calculated with the HSSH model) diminishes. Even if big clarenes are still far from the graphene limit, one may speculate that the metallic character and aromaticity are in conflict. It should be also remembered that in a PAH all the quaternary C atoms are completely inert~\cite{scott2015}, and thus all the chemistry takes place at the perimeter of the molecule, involving the methine groups. Therefore, the interior of PAHs (and clarenes in particular) may be of little (or no at all) relevance to their chemical reactivity.

\FloatBarrier
\begin{table}[!h]
\centering
\begin{tabular}{|c||c|c|c|c|}
\hline Molecule & $\mathrm{R_{\rm av}}$ (\AA) & EN & GEO & HOMA \\
\hline
\hline \textbf{1}  & 1.409  & 0.119 & 0.106 & 0.775 \\
\hline \textbf{2}  & 1.412  & 0.147 & 0.105 & 0.748 \\
\hline \textbf{3}  & 1.412  & 0.143 & 0.115 & 0.742 \\
\hline \textbf{4}  & 1.413  & 0.164 & 0.103 & 0.733 \\
\hline \textbf{5}  & 1.414  & 0.175 & 0.099 & 0.727 \\
\hline \textbf{6}  & 1.414  & 0.176 & 0.102 & 0.722 \\
\hline \textbf{7}  & 1.414  & 0.175 & 0.100 & 0.725 \\
\hline \textbf{8}  & 1.413  & 0.158 & 0.112 & 0.730 \\
\hline \textbf{9}  & 1.416  & 0.199 & 0.090 & 0.711 \\
\hline \textbf{10} & 1.417  & 0.219 & 0.080 & 0.701 \\
\hline \textbf{11} & 1.416  & 0.208 & 0.086 & 0.706 \\
\hline \textbf{12} & 1.417  & 0.222 & 0.082 & 0.696 \\
\hline \textbf{13} & 1.417  & 0.214 & 0.086 & 0.700 \\
\hline \textbf{14} & 1.418  & 0.238 & 0.073 & 0.689 \\
\hline
\end{tabular}
\caption{Clarenes. Molecular values of $\mathrm{R_{\rm av}}$ (in \AA), EN, GEO, and HOMA.}
\label{tab:homa_clar}
\end{table}

\section{K-region PAHs} \label{sec:K-region}

 Riegel and M\"{u}llen~\cite{Riegel2010} coined the name ,,K-region PAH'' to denote the PAH which can be derived from a clarene having one or more ,,bays'' within its perimeter. The ,,bay'' consist of four C atoms arranged as in the s-cis butadiene molecule (at the ,,mouth'' of the bay there are two methine groups, causing some steric tension). Examples of such bays are seen in clarenes \textbf{1} - \textbf{12} and \textbf{14}. The K-region emerges when the mouth of the bay is closed with a two-carbon insertion of the formula --CH--CH-- ; we shall call this insertion the ,,f-handle'', alluding to the bond f in phenanthrene (\textbf{21}), see Fig.~\ref{fig:phen}. Obviously, the new molecule do not fit the clarene paradigm, and the K-region (corresponding to the f-handle) introduces some olefinic character to the molecule (for a detailed discussion on the chemical and spectral properties of the K-region PAHs, see Ref.~\cite{Riegel2010})

 The K-region PAHs of Ref.~\cite{Riegel2010} may be conveniently divided into two groups: one containing pyrene and coronene (and ovalene, added here), and second composed of molecules which may be derived from ,,superbenzene'' (\textbf{9}), see Fig.~\ref{fig:clar2}.

\subsection{K-region PAHs (1): pyrene, coronene, and ovalene}

  Pyrene, coronene, and ovalene, depicted in Fig.~\ref{fig:PCO}, are well-known PAHs. Their C--C bond lengths are listed in Tabl.~\ref{tab:Re_PCO}: a closer inspection of these data indicates that no obvious transferable fragments (modules) can be identified. This is a striking contrast with the clarenes of the previous Section. One can identify the ,,olefinic'' f-handles as a-bonds in all three molecules, and f-bond in ovalene.

\FloatBarrier
\begin{figure}[!h]
\centering
\includegraphics[scale=0.15]{Fig_04_Pyr_Cor_Ova.png}
\caption{K-region PAHs (1): pyrene, coronene, and ovalene. C--C bond topologies, with symbols of symmetry non-equivalent bonds.}
\label{fig:PCO}
\end{figure}
\FloatBarrier

\begin{table}[!h]
\centering
\begin{tabular}{|c||c|c|c|}
\hline      & \multicolumn{3}{c|}{Molecule} \\
\hline
\hline Bond & \textbf{Pyrene} & \textbf{Coronene} & \textbf{Ovalene} \\
\hline a & 1.363 & 1.374 & 1.382 \\
\hline b & 1.444 & 1.429 & 1.419 \\
\hline c & 1.418 & 1.411 & 1.412 \\
\hline d & 1.432 & 1.430 & 1.425 \\
\hline e & 1.406 &  ---  & 1.438 \\
\hline f & 1.397 &  ---  & 1.367 \\
\hline g &  ---  &  ---  & 1.440 \\
\hline h &  ---  &  ---  & 1.405 \\
\hline i &  ---  &  ---  & 1.422 \\
\hline j &  ---  &  ---  & 1.432 \\
\hline k &  ---  &  ---  & 1.417 \\
\hline l &  ---  &  ---  & 1.438 \\
\hline
\end{tabular}
\caption{Pyrene, coronene, and ovalene. HSSH equilibrium C--C bond lengths (in \AA).}
\label{tab:Re_PCO}
\end{table}
\FloatBarrier

 The molecular values of HOMA and its components are collected in Tabl.~\ref{tab:homa_PCO}. Although the general trend is similar to that for clarenes, Tabl.~\ref{tab:homa_clar}, this time also the GEO contribution is important. Let us note that HOMA for coronene is much lower than for benzene, and not very different from that of graphene, see Tabl.~\ref{tab:homa_A}.

\begin{table}[!h]
\centering
\begin{tabular}{|c||c|c|c|c|}
\hline Molecule &$\mathrm{R_{\rm av}}$ (\AA) & EN & GEO & HOMA \\
\hline
\hline \textbf{Pyrene}   & 1.411  & 0.141 & 0.142 & 0.717 \\
\hline \textbf{Coronene} & 1.414  & 0.179 & 0.118 & 0.703 \\
\hline \textbf{Ovalene}  & 1.416  & 0.203 & 0.118 & 0.679 \\
\hline
\end{tabular}
\caption{Pyrene, coronene, and ovalene. Molecular values of $\mathrm{R_{\rm av}}$, EN, GEO, and HOMA.}
\label{tab:homa_PCO}
\end{table}
\FloatBarrier

\subsection{K-region PAHs (2): ,,superbenzene'' derivatives}\label{ssec:KPAHs2}

  The second group of the K-region PAHs of Ref.~\cite{Riegel2010} can be derived by inserting the f-handles into the bay regions of clarene \textbf{9}, called~\cite{Iyer1997} ,,superbenzene'' (according to that nomenclature, clarene \textbf{12} is ,,supernaphthalene'', and clarene \textbf{14} is ,,superphenalene''). One insertion gives molecule \textbf{17}, two (different) insertions -- molecules \textbf{18} and \textbf{19}, and three insertions -- molecule \textbf{17}, see Fig.~\ref{fig:KPAHs2}.

\FloatBarrier
\begin{figure}[!h]
\centering
\includegraphics[scale=0.15]{Fig_05_K-regionPAHs_2_.png}
\caption{K-region PAHs (2). C--C bond topologies, with symbols of symmetry non-equivalent A- and C-modules, and linkers.}
\label{fig:KPAHs2}
\end{figure}
\FloatBarrier

  In molecules (\textbf{17} - \textbf{20}) one may identify the A3 and A6 modules corresponding to clarene \textbf{9}, with practically ,,frozen'' structures, see Tabl.~\ref{tab:Re_AAlink} (also linkers r and v are unchanged). But a new module can be identified: this is the C4-module (see Fig.~\ref{fig:ABCmodules}), involving fourteen C atoms (and fourteen $\pi$ electrons), which resembles the phenanthrene molecule (\textbf{9}), the ,,C0-module''. As seen in Tabl.~\ref{tab:Re_C}, the (averaged) equilibrium C--C bond lengths of the C4-module are pretty well transferable, and so are the corresponding linkers (linker k being here a bit of exception).

\FloatBarrier
\begin{table}[!h]
\begin{minipage}{\linewidth}
\centering
\begin{tabular}{|c||c|c|c||c|c|c|c|}
\hline & \textbf{Phenanth.} & \multicolumn{2}{c||}{\textbf{C4}} & \multicolumn{4}{c|}{} \\
\hline & & & & \multicolumn{2}{c||}{} & & \\[-2.0ex]
      Bond & $R^{\rm e}$ & $\overline{R^{\rm e}}$ & $\Delta$ & \multicolumn{2}{c||}{Linker} & $\overline{R^{\rm e}}$ & $\Delta$ \\
\hline
\hline a  & 1.411 & 1.419 & 0.003  & \multicolumn{2}{l||}{\mbox{j } (\textbf{A3-C4})} & 1.449 & 0.001 \\
\hline b  & 1.385 & 1.381 & 0.002  & \multicolumn{2}{l||}{\mbox{k}  (\textbf{A6-C4})} & 1.436 & 0.005 \\
\hline b' & 1.385 & 1.408 & 0.001  & \multicolumn{2}{l||}{\mbox{l } (\textbf{C4-C4})} & 1.442 & 0.001 \\
\hline c  & 1.416 & 1.421 & 0.003  & \multicolumn{4}{c|}{} \\
\hline c' & 1.414 & 1.425 & 0.001  & \multicolumn{4}{c|}{} \\
\hline d  & 1.410 & 1.412 & 0.001  & \multicolumn{4}{c|}{} \\
\hline e  & 1.442 & 1.434 & 0.001  & \multicolumn{4}{c|}{} \\
\hline e' & 1.446 & 1.432 & 0.001  & \multicolumn{4}{c|}{} \\
\hline f  & 1.365 & 1.370 & 0.001  & \multicolumn{4}{c|}{} \\
\hline
\end{tabular}
\caption{C-modules and AC- and CC-linkers. HSSH equilibrium C--C bond lengths (in \AA).}
\label{tab:Re_C}
\end{minipage}
\end{table}
\FloatBarrier

The values of HOMA and its components for phenanthrene and the C4-module are collected in Tabl.~\ref{tab:homa_C}. The molecular values of HOMA and its components presented in Tabl.~\ref{tab:homa_KPAHs} show that each insertion of the f-handle into clarene lowers its HOMA and thus diminishes its aromatic character.

\FloatBarrier
\begin{table}[!h]
\centering
\begin{tabular}{|l||c|c|c|c|}
\hline Module & $\mathrm{R_{\rm av}}$ (\AA) & EN & GEO & HOMA \\
\hline
\hline \textbf{Phenanth.} & 1.408  & 0.107 & 0.129 & 0.764 \\
\hline \textbf{C4}        & 1.413  & 0.158 & 0.093 & 0.749 \\
\hline
\end{tabular}
\caption{C-modules. $\mathrm{R_{\rm av}}$, EN, GEO, and HOMA values.}
 \label{tab:homa_C}
\end{table}

\begin{table}[!h]
\centering
\begin{tabular}{|c||c|c|c|c|}
\hline Molecule &$\mathrm{R_{av}}$ (\AA) & EN & GEO & HOMA \\
\hline
\hline \textbf{9}  & 1.416  & 0.199 & 0.090 & 0.711 \\
\hline \textbf{17} & 1.416  & 0.206 & 0.091 & 0.703 \\
\hline \textbf{18} & 1.417  & 0.213 & 0.091 & 0.696 \\
\hline \textbf{19} & 1.417  & 0.213 & 0.092 & 0.695 \\
\hline \textbf{20} & 1.418  & 0.225 & 0.113 & 0.662 \\
\hline
\end{tabular}
\caption{``Superbenzene'' (\textbf{9}) and K-region PAHs (2). Molecular values of $\mathrm{R_{\rm av}}$, EN, GEO, and HOMA.}
\label{tab:homa_KPAHs}
\end{table}
\FloatBarrier

\section{Phenacenes} \label{sec:phen}

  The first five members of the family of phenacenes are shown in Fig.~\ref{fig:phen}. With the growing size they approach a limit of a $\pi$-electron polymer (not the 2D structure of graphene). The bond lengths of phenacenes are presented in Tabl.~\ref{tab:Re_phen}: here also we see no clue to identify transferable modules, although some regularities in bond lengths are clearly visible.

\FloatBarrier
\begin{figure}[!h]
\centering
\includegraphics[scale=0.15]{Fig_06_Phenacenes.png}
\caption{Phenacenes. C--C bond topologies, with symbols of symmetry non-equivalent bonds.}
\label{fig:phen}
\end{figure}
\FloatBarrier

\begin{table}[!h]
\centering
\begin{tabular}{|c||c|c|c|c|c|}
\hline      & \multicolumn{5}{c|}{Molecule} \\
\hline Bond &\textbf{21} &\textbf{22} &\textbf{23} &\textbf{24} &\textbf{25}  \\
\hline
\hline a & 1.411 & 1.415 & 1.413 & 1.414 & 1.414 \\
\hline b & 1.385 & 1.382 & 1.383 & 1.382 & 1.383 \\
\hline b'& 1.385 & 1.382 & 1.383 & 1.383 & 1.383 \\
\hline c & 1.416 & 1.419 & 1.418 & 1.418 & 1.418 \\
\hline c'& 1.414 & 1.418 & 1.416 & 1.417 & 1.417 \\
\hline d & 1.410 & 1.411 & 1.411 & 1.411 & 1.411 \\
\hline e & 1.442 & 1.435 & 1.438 & 1.437 & 1.437 \\
\hline e'& 1.446 & 1.439 & 1.442 & 1.441 & 1.441 \\
\hline f & 1.365 & 1.370 & 1.368 & 1.369 & 1.368 \\
\hline f'&  ---  &  ---  & 1.376 & 1.374 & 1.375 \\
\hline g &  ---  & 1.433 & 1.436 & 1.435 & 1.435 \\
\hline g'&  ---  &  ---  & 1.426 & 1.428 & 1.427 \\
\hline h &  ---  & 1.400 & 1.404 & 1.403 & 1.403 \\
\hline i &  ---  &  ---  & 1.431 & 1.434 & 1.433 \\
\hline i'&  ---  &  ---  &  ---  & 1.429 & 1.428 \\
\hline j &  ---  &  ---  &  ---  & 1.407 & 1.406 \\
\hline k &  ---  &  ---  &  ---  &  ---  & 1.431 \\
\hline k'&  ---  &  ---  &  ---  &  ---  & 1.437 \\
\hline l &  ---  &  ---  &  ---  &  ---  & 1.372 \\
\hline
\end{tabular}
\caption{Phenacenes. HSSH equilibrium C--C bond lengths (in \AA).}
\label{tab:Re_phen}
\end{table}

 The molecular values of HOMA and its components for phenacenes are presented in Tabl.~\ref{tab:homa_phen}. It seems that these quantities may converge to some limiting values in the case of the polyphenacene polymer.

\begin{table}[!h]
\centering
\begin{tabular}{|c||c|c|c|c|}
\hline Molecule & $\mathrm{R_{\rm av}}$ (\AA) & EN & GEO & HOMA \\
\hline
\hline \textbf{21}  & 1.408  & 0.107 & 0.129 & 0.764 \\
\hline \textbf{22}  & 1.410  & 0.124 & 0.133 & 0.743 \\
\hline \textbf{23}  & 1.411  & 0.135 & 0.136 & 0.729 \\
\hline \textbf{24}  & 1.411  & 0.142 & 0.136 & 0.722 \\
\hline \textbf{25}  & 1.412  & 0.147 & 0.137 & 0.716 \\
\hline
\end{tabular}
\caption{Phenacenes. Molecular values of $\mathrm{R_{\rm av}}$, EN, GEO, and HOMA.}
\label{tab:homa_phen}
\end{table}

\newpage

\section{Rylenes} \label{sec:ryl}

  In Fig.~\ref{fig:ryl} we present the first four members of the family of rylenes. As in the case of the family of phenacenes, with the growing size rylenes approach a limit of a $\pi$-electron polymer. However, in  difference to phenacenes, rylenes display modular architecture, based on the ten C atom, ten $\pi$-electron B2 and B4 modules, see Fig.~\ref{fig:ABCmodules}. They may be considered certain analogs of naphthalene (the ,,B0'' module, see Fig.~\ref{fig:ace}). The structures of the B2 and B4 modules are presented in Tabl.~\ref{tab:Re_B}, together with the lengths of the corresponding BB-linkers.

\FloatBarrier
\begin{figure}[!h]
\centering
\includegraphics[scale=0.15]{Fig_07_Rylenes.png}
\caption{Rylenes. C--C bond topologies, with symbols of symmetry non-equivalent B-modules and BB-linkers.}
\label{fig:ryl}
\end{figure}
\FloatBarrier

  The values of HOMA and its components for the B-modules are presented in Tabl.~\ref{tab:Re_B}, and the molecular values of these parameters calculated for B-modules appear in Tabl.~\ref{tab:homa_B}, while for rylenes are in Tabl.~\ref{tab:homa_ryl}.

\begin{table}[!h]
\centering
\begin{tabular}{|c||c|c|c|c|c||c|c|c|c|}
\hline & \textbf{Naphth.} & \multicolumn{2}{c|}{\textbf{B2}} & \multicolumn{2}{c||}{\textbf{B4}} & \multicolumn{4}{c|}{} \\
\hline & & & & & & \multicolumn{2}{c||}{} & & \\[-2.0ex]
  Bond & $R^{\rm e}$ & $\overline{R^{\rm e}}$ & $\Delta$ & $\overline{R^{\rm e}}$ & $\Delta$  & \multicolumn{2}{c||}{Linker} & $\overline{R^{\rm e}}$ & $\Delta$ \\
\hline
\hline a  & 1.421 & 1.414 & 0.001 & 1.405 & 0.002 & \multicolumn{2}{l||}{\mbox{m'}  (\textbf{B2-B2})} & 1.461 &  ---  \\
\hline b  & 1.377 & 1.381 & 0.001 &  ---  &  ---  & \multicolumn{2}{l||}{\mbox{m }  (\textbf{B2-B4})} & 1.458 & 0.001 \\
\hline b' &  ---  & 1.390 & 0.002 & 1.395 & 0.002 & \multicolumn{2}{l||}{\mbox{n    } (\textbf{B4-B4})} & 1.454 & 0.001 \\
\hline c  & 1.425 & 1.424 & 0.000 &  ---  &  ---  & \multicolumn{4}{c|}{} \\
\hline c' &  ---  & 1.427 & 0.000 & 1.426 & 0.000 & \multicolumn{4}{c|}{} \\
\hline d  & 1.414 & 1.415 & 0.000 & 1.417 & 0.001 & \multicolumn{4}{c|}{} \\
\hline
\end{tabular}
\caption{B-modules and BB-linkers. HSSH equilibrium C--C bond lengths (in \AA).}
\label{tab:Re_B}
\end{table}

\begin{table}[!h]
\centering
\begin{tabular}{|l||c|c|c|c|}
\hline Module & $\mathrm{R_{\rm av}}$ (\AA) & EN & GEO & HOMA \\
\hline
\hline \textbf{Naphth.}  & 1.406  & 0.081 & 0.125 & 0.794 \\
\hline \textbf{B2}       & 1.408  & 0.102 & 0.081 & 0.817 \\
\hline \textbf{B4}       & 1.410  & 0.126 & 0.045 & 0.829 \\
\hline
\end{tabular}
\caption{B-modules. $\mathrm{R_{\rm av}}$, EN, GEO, and HOMA values.}
 \label{tab:homa_B}
\end{table}

\begin{table}[!h]
\centering
\begin{tabular}{|c||c|c|c|c|}
\hline Molecule & $\mathrm{R_{\rm av}}$ (\AA) & EN & GEO & HOMA \\
\hline
\hline \textbf{26}  & 1.408  & 0.107 & 0.129 & 0.764 \\
\hline \textbf{27}  & 1.410  & 0.124 & 0.133 & 0.743 \\
\hline \textbf{28}  & 1.411  & 0.135 & 0.136 & 0.729 \\
\hline \textbf{29}  & 1.411  & 0.142 & 0.136 & 0.722 \\
\hline
\end{tabular}
\caption{Rylenes. Molecular values of $\mathrm{R_{\rm av}}$, EN, GEO, and HOMA.}
\label{tab:homa_ryl}
\end{table}

\newpage

\section{Acenes} \label{sec:ace}

  The first six members of the family of acenes are shown in Fig.~\ref{fig:ace}. Acenes are similar to phenacenes of Sec.~\ref{sec:phen}. They also, with the growing size, approach a limit of a $\pi$-electron polymer. The bond lengths of acenes are presented in Tabl.~\ref{tab:Re_ace}: here also we see no clue to identify transferable modules despite some regularities in bond lengths.

\begin{figure}[!h]
\centering
\includegraphics[scale=0.15]{Fig_08_Acenes.png}
\caption{Acenes. C--C bond topologies, with symbols of symmetry non-equivalent bonds.}
 \label{fig:ace}
\end{figure}

\begin{table}[!h]
\centering
\begin{tabular}{|c||c|c|c|c|c|c|c|}
\hline      & \multicolumn{7}{c|}{Molecule} \\
\hline Bond & \textbf{Benzene} & \textbf{Naphth.} & \textbf{30} &\textbf{31} &\textbf{32} &\textbf{33} &\textbf{34}  \\
\hline
\hline a & 1.397 & 1.421 & 1.431 & 1.435 & 1.438 & 1.439 & 1.440 \\
\hline b &  ---  & 1.377 & 1.370 & 1.367 & 1.366 & 1.365 & 1.364 \\
\hline c &  ---  & 1.425 & 1.435 & 1.440 & 1.442 & 1.443 & 1.444 \\
\hline d &  ---  & 1.414 & 1.426 & 1.433 & 1.437 & 1.439 & 1.441 \\
\hline e &  ---  &  ---  & 1.404 & 1.395 & 1.390 & 1.388 & 1.387 \\
\hline f &  ---  &  ---  &  ---  & 1.416 & 1.422 & 1.425 & 1.427 \\
\hline g &  ---  &  ---  &  ---  & 1.432 & 1.437 & 1.439 & 1.441 \\
\hline h &  ---  &  ---  &  ---  &  ---  & 1.406 & 1.401 & 1.398 \\
\hline i &  ---  &  ---  &  ---  &  ---  &  ---  & 1.413 & 1.416 \\
\hline j &  ---  &  ---  &  ---  &  ---  &  ---  & 1.439 & 1.441 \\
\hline k &  ---  &  ---  &  ---  &  ---  &  ---  &  ---  & 1.408 \\
\hline
\end{tabular}
\caption{Acenes. HSSH equilibrium C--C bond lengths (in \AA).}
\label{tab:Re_ace}
\end{table}

  The molecular values of HOMA and its components for acenes are presented in Tabl.~\ref{tab:homa_ace}. As seen, for long acenes HOMA quickly deteriorates, which is consistent with a high chemical reactivity of these molecules.

\begin{table}[!h]
\centering
\begin{tabular}{|c||c|c|c|c|}
\hline Molecule & $\mathrm{R_{\rm av}}$ (\AA) & EN & GEO & HOMA\\
\hline
\hline \textbf{Benzene} & 1.397  & 0.021 & 0.000 & 0.979 \\
\hline \textbf{Naphth.} & 1.406  & 0.081 & 0.125 & 0.794 \\
\hline \textbf{30}      & 1.409  & 0.118 & 0.168 & 0.714 \\
\hline \textbf{31}      & 1.411  & 0.141 & 0.181 & 0.678 \\
\hline \textbf{32}      & 1.413  & 0.156 & 0.183 & 0.661 \\
\hline \textbf{33}      & 1.413  & 0.166 & 0.179 & 0.655 \\
\hline \textbf{34}      & 1.414  & 0.174 & 0.173 & 0.653 \\
\hline
\end{tabular}
\caption{Acenes. Molecular values of $\mathrm{R_{\rm av}}$, EN, GEO, and HOMA.}
 \label{tab:homa_ace}
\end{table}

\newpage

\section{Concluding remarks} \label{sec:concl}

  In a series of papers Tyutyulkov, M\"{u}llen, and their collaborators~\cite{Tyut1998, Tyut1999, Dietz2000, Tyut2003} posed a question: ,,is graphene an ultimate large hydrocarbon''? In order to find the answer, they analyzed various aspects: structural (topology), energetic (the energy spectra), and the influence of defects (including these with unpaired spins) and different edge structures. Their findings suggest~\cite{Tyut1998} that the electronic-correlation effects should lead to a non-zero gap in the energy spectrum of PAHs, even in the limit of an infinite molecule (thus, strictly speaking, the metallic limit is not reached). However, the experimental energy gap in the clarene family (\textbf{1} - \textbf{14}), as presented in Fig.~7 of Ref.~\cite{Riegel2010}, indicates that the band gap diminishes quickly with the size of clarene.

  Our HSSH study of the equilibrium C--C bond lengths for the PAHs considered in the review paper by Riegel and M\"{u}llen~\cite{Riegel2010} is too limited to attempt an answer to the question of Refs.~\cite{Tyut1998, Tyut1999, Dietz2000, Tyut2003}. However,  our elucidation of the modular architecture of some classes of PAHs (clarenes, rylenes, certain subclasses of the K-region PAHs) provides a suggestive indication that the properties of big PAHs may be understood on the basis of the calculations for smaller systems.

redpajama_set_name: RedPajamaArXiv


text:
Louisiana Highway 35 (LA 35) is a -long north-south state highway in Louisiana that serves Vermilion, Lafayette, Acadia, and Saint Landry parishes, extending from Louisiana Highway 82, intersecting with exit 87 of Interstate 10, ending at US Route 190.

Route description
From the south, LA 35 begins at a junction with LA 82 located just north of the Gulf Intracoastal Waterway and an area known as Forked Island.  The route heads northward and crosses LA 14 in the city of Kaplan.  North of Kaplan, LA 35 briefly overlaps LA 92 to the west of Indian Bayou.  About  later, LA 35 passes through the city of Rayne, where it runs concurrent with US 90 through the center of town.  After US 90 departs to the east, LA 35 passes through a diamond interchange with I-10 (exit 87) at the north end of town.  This interchange is followed immediately by a junction with LA 98.  Continuing northward, LA 35 passes through the rural community of Branch en route to the town of Church Point.  The route zigzags through the center of town, overlapping the route of LA 95.  North of Church Point, LA 35 meanders into St. Landry Parish and reaches its northern terminus at US 190 in Lawtell.

History
Prior to the 1955 Louisiana Highway renumbering, LA 35 was a road from Amite to Bogalusa via Franklinton. In 1955 the Amite-to-Franklinton segment became part of LA 16, and the Franklinton-to-Bogalusa segment became part of LA 10.

Major intersections

References

0035
Transportation in Acadia Parish, Louisiana
Transportation in Lafayette Parish, Louisiana
Transportation in St. Landry Parish, Louisiana
Transportation in Vermilion Parish, Louisiana

redpajama_set_name: RedPajamaWikipedia


This large plaza includes many historical buildings and the beauty of this location is unforgettable.
Not the most imposing square I have been to in Spain but nonetheless still a sight to see, surrounded by a few bars and eating places you can sit and relax and watch the world go by.
A typical Spanish Plaza De Mayor. A big surrounded by some official buildings. Several cafes and restaurants. Crowded during weekends specially when there are some activities .
No doubt one of the best Squares in Spain. Really welcoming, tourist info there,mpacked with small bars where to eat something. Also quite close to Virgen Blanca is a good starting point to visit themold city.
Get quick answers from Plaza Espana Plaza Nueva staff and past visitors.


text:
Single Dad Becomes Social Media Star After Asking Daughter For Fashion Advice.
Parenting | July 18, 2019
Jeff Saville couldn't stop talking to 18-year-old Carli about his upcoming date night, but she worried.
Being a single dad isn't easy, especially when you're looking to start dating again. If anyone can relate to that it's Jeff Saville from Texas. He had been married for 22 years and single for three. But the 56-year-old was still hoping it wasn't too late to start dating again. But he needed some serious fashion advice, and he didn't turn to his close friends or netizens. He went straight to the one person who would be open and honest—his daughter, Carli.
Her dad hadn't dated in forever and was afraid that he didn't know how to dress appropriately for his date. So, she asked him to send her some photos of himself wearing several different types of clothes.
Carli found it adorable that her dad was relying on her for fashion advice before his big date.
With her guidance, he was able to figure out which shirt looked best and whether or not he needed to tuck it in or let it hang. She even suggested he try using a belt to see if it would improve his overall look.
Jeff continued sending Carli different mirror selfies, and each time he had on something new.
She found the whole thing so cute that she decided to share the back and forth conversation on Twitter using the caption: "Life with a single dad, asking for advice on date outfits. My heart." But then, something unexpected happened.
When Carli shared the screencaps of her convo with her dad, it was intended for her friends.
They would undoubtedly have a good laugh. But as it turns out, other netizens caught wind of her exchange with her dad and fell head over heels in love with Jeff. Soon, everyone was posting supportive comments for Twitter's single dad.
Carli is obviously one awesome daughter, but it turns out that other daughters helped their dads too.
One person shared a shot of her dad asking for advice on their wardrobe. Clearly having daughters is awesome because they definitely have the 4-1-1 on what a single dad needs to wear before jumping into the dating pool.
As the post got more attention, one restaurant offered to provide Jeff with a free meal.
Although Kobe restaurant's offer was beyond generous, there was one requirement. Jeff needed to bring a date. Unfortunately, Carli tweeted back to the restaurant that her dad's date had stood him up.
Although his date was a no-show, the kindness Twitter users showed him lifted his spirits.
Carli made sure to let the Twitterverse know that everyone's words of encouragement have touched her dad and motivated him to continue working on his love life. And judging from this photo, we doubt he'll be single much longer.
Camila Villafañe

redpajama_set_name: RedPajamaCommonCrawl


As an architect, keeping your education current is essential. As zoning and environmental laws change, and more emphasis is placed on sustainable design, smart professionals can use training as a way to grow their business and earn new clients. It is for this reason that state architecture boards and other professional associations, such as the American Institute of Architects (AIA), mandate that all members undergo a minimal amount of training each year.
For busy professionals, however, taking time off to attend seminars and workshops can put a serious drain on their practice. Fortunately, Lorman Education Services offers AIA-approved continuing education training online, through onDemand content and other innovative course delivery methods.
Each state has different continuing education requirements for architects and engineers. The AIA is responsible for vetting educational institutions and providing other support to its more than 300 state and local chapters. On the AIA website, you can find a helpful interactive map that displays specific continuing education requirements in your area.
Lorman Education Services is an authorized provider of AIA training online. We help busy professionals meet their continuing education requirements through classes that deliver genuine value for participants. Courses cover everything from local building codes to environmental site design. Experienced professionals design and deliver all courses and other training based on an up-to-date understanding of current laws and best practices in your field.
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To me they look like M/T drag radials going by his #88 post on his thread. If their 19s 325 is widest available I've seen on their site.if 18s,345. But this is jist an educated guess untill he responds.
Old Z28 #158 made it the LSX Magazine.


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When David and Linda Boyden started Boyden Valley Winery in Cambridge in 1996, they had zero experience in viticulture or oenology, save for a class that David had taken at Cornell University. Initially, they planned to grow fruit other than grapes to make wine, because like most farms in northern Vermont, their farm is in a zone 4 region, which means average low temps of -25. Not exactly chardonnay-friendly temperatures.
But around that time, David's brother, Mark, was traveling up to Québec to buy tractor parts for his farm and told David he should join him to meet a local vintner who was successfully growing grapes in a climate even colder than in Cambridge. David accompanied his brother on that trip and started a relationship that was to result in one of Vermont's pioneering grape wineries.
Robert offered to mentor David and the two became good friends, sharing a passion for making fine wines in an unfriendly climate. Robert consulted for Boyden Farm and offered advice for varietals and techniques. As David recalls, "Having Robert helping us in the beginning was really instrumental. He gave us great insight into cold-climate viticulture and enology." One technique was to hill the vines in late fall to protect them from icy winds in winter. David has since changed to varietals that don't require hilling, but those early lessons resulted in very good yields from his first few plantings.
Québec vintners also collaborate with Cornell University and the University of Minnesota, which are the leading research facilities for grape growing in cold climates. With advice from Robert, Minnesota, and Cornell, David and Linda broke ground on a section of the family farm (which had previously been a dairy farm) and started Boyden Valley Winery.
Since those first grapes were planted, the Boydens have developed their own techniques and planted even more varietals, creating wines that have won six silver and eight gold medals in national and international competitions, plus numerous other awards. They are especially known for their exceptional ice wines, experimenting with both white and red varietals to develop complex profiles that are reminiscent of fine port wines. Twenty years later, they remain grateful to Québec growers like Robert LeRoyer who were so forthright with advice. Although they admit they do not see their old friend as much as they used to, David and Linda will always acknowledge their debt to their fellow oenophiles up north.


Many domestic and international flights used to fly in and out of this airport. The majority of its routes were transferred to Chubu Centrair International Airport when the newer, larger airport opened, and Nagoya Airport now serves as a hub for smaller aircraft such as commuter planes and business jets. The Japan Air Self-Defense Force's Komaki air base is right next door and shares a runway with the airport, so you can watch their planes take off and land from the observation deck.
Access From "Nagoya Station" of JR Tokaido Shinkansen, walk a short distance and take the bus with direct service to Nagoya Airport for approx. 20 minutes. Alight at "Ken'ei Nagoya Kuko" (Nagoya Airport).
* There are buses that go to "Ken'ei Nagoya Kuko" (Nagoya Airport) and/or "Aichi Koku Museum" (Aichi Museum of Flight) departing from Nagoya Station, Sakae Station, Kachigawa Station, and Nishiharu Station (approx. 20 to 40 minutes' journey time).


text:
Ochthebius sichuanensis är en skalbaggsart som beskrevs av Manfred A. Jäch 2003. Ochthebius sichuanensis ingår i släktet Ochthebius och familjen vattenbrynsbaggar. Inga underarter finns listade i Catalogue of Life.

Källor

Vattenbrynsbaggar
sichuanensis


Jerry Rogers & Sons Home Recordings is a compilation of recordings over the last several years by Jerry, his sons Andrew and Bobby, and different friends throughout. This CD is about as authentic as it gets when it comes to the heart of folk music. Some songs were recorded informally at a local radio station (after hours while doing "maintenance") while others were recorded under an old oak tree with a plain old cassette recorder thrown on the ground between the fiddle and the banjo. These recordings are raw, and give a sense of being in the middle of an impromptu jam session on a hot summer night in central Alabama.
I hope you enjoy this recording and catch a bit of the spirit of true folk music while listening.
This CD can be purchased at cdbaby.com or by sending a $10.00 check / money order to POB 621 Bynum, AL 36253.


Two VLCCs, the Atlantic Empress and the Aegean Captain collided during a tropical storm in the Caribbean Sea. The two vessels burst into flame on impact, claiming the lives of 26 seamen. The fire aboard the Aegean Captain was tamed by her crew, but the blaze on the Atlantic Empress raged uncontrolled.
She was towed away from the original collision site, still haemorrhaging crude oil until a series of explosions downed the vessel on August 3rd. Although massive in scale, the Atlantic Empress / Aegean Oil spill was only the second largest such incident of that calendar year.


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text:
/*
 * DeliveryHub
 *
 * DeliveryHub API
 *
 * OpenAPI spec version: 1.0.0
 *
 * Generated by: https://github.com/swagger-api/swagger-codegen.git
 *
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 *      http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */

using NUnit.Framework;

using System;
using System.Linq;
using System.IO;
using System.Collections.Generic;
using IO.Swagger.Api;
using IO.Swagger.Model;
using IO.Swagger.Client;
using System.Reflection;

namespace IO.Swagger.Test
{
    /// <summary>
    ///  Class for testing AccountStatus
    /// </summary>
    /// <remarks>
    /// This file is automatically generated by Swagger Codegen.
    /// Please update the test case below to test the model.
    /// </remarks>
    [TestFixture]
    public class AccountStatusTests
    {
        // TODO uncomment below to declare an instance variable for AccountStatus
        //private AccountStatus instance;

        /// <summary>
        /// Setup before each test
        /// </summary>
        [SetUp]
        public void Init()
        {
            // TODO uncomment below to create an instance of AccountStatus
            //instance = new AccountStatus();
        }

        /// <summary>
        /// Clean up after each test
        /// </summary>
        [TearDown]
        public void Cleanup()
        {

        }

        /// <summary>
        /// Test an instance of AccountStatus
        /// </summary>
        [Test]
        public void AccountStatusInstanceTest()
        {
            // TODO uncomment below to test "IsInstanceOfType" AccountStatus
            //Assert.IsInstanceOfType<AccountStatus> (instance, "variable 'instance' is a AccountStatus");
        }

    }

}

redpajama_set_name: RedPajamaGithub


19th Annual Jackrabbit Classic ONLINE REGISTRATION IS CLOSED. REGISTER IN PERSON AT PACKET PICK UP.
for runners only, proceed 2 mile to extreme north end of the park to post #44.
of getting everybody into the park. Please be courteous of others.


Summit Charter School took the Tri-State Conference Tournament Championship for the third year in a row on Saturday in Highlands against Tallualah Falls with a final of 34-19. The undefeated Lady Bears came out determined to maintain the title and Head Coach Kaylee O'Dwyer said they worked hard for the victory.
Summit Charter School girls basketball team three-peats the Tri-State Conference Tournament Championship. Front row from left are Taylor Robinson, Adisyn Westendorf, Ezra Branham, Beatrice Bryson, Anna Johnson, Chloe Stiwinter, and Riley Logan. Back row from left are Assistant Coach Jeremy Robinson, Caroline Woods, Lily Kate Bryson, Riley Conner, Chloe Crawford, Abby Doerter, and Head Coach Kaylee O'Dwyer.
Assistant Coach Jeremy Robinson said Summit has maintained their intensity throughout the season.
"They set the standard, and now they're going to work hard to match that," said Robinson.
He added that Summit defeated Highlands in the semifinal, but it was no easy feat.
Summit Adisyn Westnedorf feels out the perimeter against Tallulah Falls on Saturday at the Tri-State Conference Tournament Championships in Highlands. Summit won 34-19.
Summit Adisyn Westendorf said Tallulah played a good game and Summit had to step it up.
Summit Ezra Branham said taking the championship for a third year is a great accomplishment but said they had too many fouls, needed to move their feet more, and run the press better.
Highlands School hosted the tournament because Summit's new gym is still undergoing construction and is expected to open within a couple of weeks.
Summit Lily Kate Bryson powers through to the hoop for the lay up against Tallulah Falls.
Summit Riley Connor feels some pressure bringing the ball up against Tallualah Falls in Highlands on Saturday.
Summit Abby Doerter snags a rebound during the Tri-State Championship on Saturday in Highlands.
Summit Beatrice Bryson makes the play happen and drives to the hoop at the Tri-State Championships against Tallulah Falls.
Summit Charter School girls basketball team takes a moment to celebrate winning the Tri-State Conference Tournament Championship on Saturday at highlands School.
This entry was posted in News, Sports and tagged Basketball, February 2019, Girls, Middle, Summit Charter School, Three-peat, Tri-State. Bookmark the permalink.


WILDD's mission is to unleash the gifts of dyslexic children and adults so they may achieve the full potential they deserve, and of which they are capable. We do this through identification, remediation, therapy, support, outreach, community education, and advocacy.
Reading is an essential skill set needed to thrive in our society. WILDD creates learning and living independence that allows persons with dyslexia/SLD to achieve success and compete with their peers in school, higher education, the workforce, and their community.


Please submit a 500-word panel proposal, or a 200 word abstract for an individual paper to cern-admin@uow.edu.au by Friday 1 July 2016. For more information, and for updates about keynote speakers and other conference related information, please visit the CERN events page.
Convened by: Roger Patulny and Sukhmani Khorana (UOW CERN), Andrew Lynch (ARC CHE) and Rebecca Olson and Jordan McKenzie (TASA SEA).
'Silence' is trending. Ever more people embrace silence as a way of escaping the pressures of daily life. Yet silence itself is not new; it has a long and sometimes painful history. This book tries to capture the many faces of silence as a historical phenomenon. Dealing with topics such as art, trauma, migration, politics and education it addresses the history, cultural diversity and cultural force of silence.
Pieter Verstraete is a lecturer in Historical Pedagogy in the department of Education, Culture and Society of the University of Louvain. He has authored many books, including In the Shadow of Disability and Verminkte stilte. He was awarded with the Disability History Association Outstanding Book Award for his book The Imperfect Historian.
Josephine Hoegaerts is a postdoctoral researcher in the research group 'Modernity and Society 1750-2000' at the University of Louvain and the Helsinki Collegium for Advanced Studies. Her research focuses on the social history of the human voice in Western-Europe.


We own and operate a 200 acre Heritage Farm in Prince Edward County, acquired by my U.E.L. ancestors six generations ago in 1834. We respectfully demand to know why your government obviously has different rules for farmers than it has for multinational corporations, governing the siting of wind turbines. We would like you to explain why certain M.D.S. distances apply to us and not to wpd Canada.
Premier Kathleen Wynne, Minister of Agriculture and Food, and Hon. Jim Bradley, Minister of the Environment need to explain why MOE has deemed "complete" this proposal from wpd Canada for its White Pines Wind Farm project, even though the project clearly contravenes the OMAF web site instructions to farmers, to "make sure turbines are sited away from migratory bird corridors".
The White Pines Wind Farm project proposes to site dozens of huge Industrial Wind Turbines in the Southeastern tip of Prince Edward County, which is in Ontario's largest migratory bird corridor. This is an internationally-recognized "Important Bird Area", in a heavily flown migration flyway extending from Point Petre to Wolfe Island.
Each turbine will be taller than the Toronto Star building at 1 Yonge Street in Toronto.
For common sense advice from OMAF about how farmers should participate in the generation of wind energy with smaller turbines, go to http://www.omafra.gov.on.ca/english/engineer/ge_bib/wind.htm and scroll down to Benefits, Drawbacks and Potential Pitfalls.
Birds and bats are ravenous consumers of harmful insects and thus are natural aids to the farmer in his growing of good crops with minimal use of chemicals. This is also a plus for the alarmingly diminishing bee population. Wind turbines kill bats and birds and, as already demonstrated on Wolfe Island, the rate of avian slaughter by industrial wind turbines is exceptionally high in bird migration corridors.
We urge you, Ms. Afante, to remove this project from the EBR listing of projects awaiting technical approval. There are many reasons to not waste limited MOE resources on a detailed technical review of this project, and one very important reason for both farmers and consumers is that the White Pines Wind Farm proposal clearly contravenes OMAF's prudent advice cautioning that wind turbines kill the migrating bats and birds that reduce farmers' dependence on chemicals.
There are four major migratory flyways in North America. The cumulative effect of several proposed turbine projects surrounding eastern Lake Ontario creates a barrier across the Eastern Flyway that welcomes many millions of migrating birds from May to October. If approved, these projects will displace wildlife from our shorelines, the most important staging and resting habitats for many species. As appropriately advocated by the Ontario Ministry of Agriculture and Food, wind turbines must be sited away from migratory bird corridors.
We reside at and farm our Murdale Farms property as full-time farmers. To the best of our knowledge, none of the neighbouring properties on which wpd Canada proposes to erect its 29 "White Pines" turbines are owned by people who will be living there, and farming there, on a full-time basis. Most are absentee turbine landlords who will not have to live with the devastating ongoing consequences of this project. In a separate letter, we will explain how wpd Canada, a multinational corporation, plans to site its noisy, flickering turbines much too close to the place where our family has lived and farmed in peace and tranquillity for the past 180 years.


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Attac (celým názvem Association pour la Taxation des Transactions financières et pour l'Action Citoyenne, Sdružení pro zdanění finančních transakcí a pro občanskou akci) je mezinárodní alterglobalizační nevládní organizace, která požaduje, aby na spekulace se zahraničními měnami byla uvalena Tobinova daň. Sám James Tobin se však od tohoto hnutí distancoval.

Organizace byla založena v Paříži 3. června 1998. Její vznik inspiroval Ignacio Ramonet, který v měsíčníku Le Monde diplomatique uveřejnil článek o tom, že forex generuje značné příjmy, které nepodléhají žádné daňové povinnosti. Attac také odmítá existenci daňových rájů a vyzývá vlády, aby přijaly opatření proti daňovým únikům, která by umožnila financování řady sociálních programů. Má regionální pobočky ve 38 zemích světa. V jeho logu je symbol procenta, odkazující na snahu zavést Tobinovu daň ve výši alespoň jednoho procenta. Vystupuje proti politice Světová obchodní organizace a požaduje oddlužení rozvojových zemí. Také kritizoval postup Evropské centrální banky při sanaci bank během světové finanční krize 2008.

V roce 2013 byly firma Nestlé a bezpečnostní agentura Securitas AG usvědčeny, že sbíraly kompromitující materiály na aktivisty hnutí Attac. Švýcarský soud přiznal postiženým finanční kompenzaci.

V dubnu 2015 hnutí svolalo rozsáhlé mezinárodní demonstrace proti přijetí smlouvy TTIP.

Reference

Externí odkazy

 https://www.attac.org/

Sociální hnutí
Vzniklo 1998