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text:
\section{Introduction}

The transport through molecules as small as benzene \cite{Reed_1997} or as long as DNA \cite{Slinker_2011}
or carbon nanotubes \cite{Avouris_2002}
falls into the realm of molecular electronics. This field of research has a long
history \cite{Ratner_2013}, dating back
to 1956 \cite{Hippel_1956}. As a consequence of this latter paper and of the immediate
perception of its importance
the USA Air Force decided, already by that time,
to fund this type of research \cite{CuevasScheer}.

According to Cuevas and Scheer \cite{CuevasScheer} molecular electronics is the field of research
devoted to the study of the electronic and thermal transport through individual molecules. The
objective of this type of research is to produce electronic devices where the building blocks are
single molecules, exploring the quantum mechanical properties that necessarily appear at this small
length scales. This type of devices include: switches, sensors, diodes, transistors, and other
\cite{Ratner_2003}. On the other hand, another advantage of this approach is that it opens the door to the
discovery of new quantum phenomena. In this sense, one should remember that, historically,
technological breakthroughs were often a natural consequence of the explorations of new areas, and
not its finality. From a technological point of view, there are several advantages in pursuing the study
of molecular electronics
when compared to the present silicon technology, namely \cite{CuevasScheer,ChoiMody}:
\begin{itemize}
\item Size. Molecules have typical sizes ranging from $1$ to $100$ nm, which allows the construction of
functional nano-structures with subsequent advantages in cost, efficiency and power dissipation.
\item Speed. Estimations show that a good molecular wire could, for example, reduce the transit
time of a traditional transistor, around $10^{-14}$ s, hence obtaining faster operations.
\item New functionalities. As an example, many molecules have several stable geometric structures,
or isomers, that may have different electronic or optic properties, what can be an advantage for the
development of some devices, such as switches or sensors.
\item Synthetic tailorability. By changing, for example, the composition or the geometry of a system,
one can obtain specific transport or optical properties suitable for a specific end.
\end{itemize}

\noindent However, we need not only to know which molecules are the best for a certain usage, but also how
to connect them, namely through unidimensional (1D) nanowires. We shall focus now on the composition of
these nanowires, rather than on how to create them (for that, see, for example [\cite{CuevasScheer}]).

Historically, the carbon-based molecules always played an important role in the field of molecular
electronics, due to its
high versatility. Carbon is the basis of a wide range of structures, since tridimensional, like diamonds
or graphite, to bidimensional, like graphene, (quasi)unidimensional, like the nanotubes, or even
(quasi)zero-dimensional, like the fullerenes. In particular, the carbon nanotubes can be used to
connect individual molecules. Also, some experiments on quantum transport through carbon nanotubes
have already been performed \cite{Kasumov_2003}.

Another very important class of systems, related to the previous one, are the hydrocarbons.
The importance of this class
is that such molecules can have different transport properties depending on the degree of hybridization of
the molecular orbitals of its carbon atoms. If the valence electrons are $sp^3$ hybridized, there is a
tetrahedral arrangement of the bonds in space. This is the case of the alkanes, with the sum formula
$C_nH_{2n+2}$ [the simplest alkane, the ethane, is presented in figure \ref{fig:hydrocarbons}(a)].
Since each valence electron is used for the formation a different (single) chemical bond,
they are basically
localized, and thus the alkanes are insulating. On the other hand, the valence electrons may also
be $sp^2$ or $sp$ hybridized, forming, respectively, alkenes, with double bonds, and alkynes, with
triple bonds [the simplest of each kind, the ethylene and the acethylene, are presented in figure
\ref{fig:hydrocarbons}(b) and (c)]. A particular case of these occurs when double or triple bonds are
alternated with single bonds. These molecules are very stable and give rise to delocalized wave functions,
which makes them good conductors.

\begin{figure}[hb]
\begin{minipage}{0.33\textwidth}
\begin{flushleft}
\centering
\includegraphics{figure1a}
\label{fig:ethane}
\end{flushleft}
\end{minipage}
\begin{minipage}{0.33\textwidth}
\begin{center}
\centering
\includegraphics{figure1b}
\label{fig:ethylene}
\end{center}
\end{minipage}
\begin{minipage}{0.33\textwidth}
\begin{flushright}
\centering
\includegraphics{figure1c}
\label{fig:acetylene}
\end{flushright}
\end{minipage}
\begin{minipage}{0.33\textwidth}
\centering \scriptsize (a) Ethane.
\end{minipage}
\begin{minipage}{0.33\textwidth}
\centering \scriptsize (b) Ethylene.
\end{minipage}
\begin{minipage}{0.33\textwidth}
\centering \scriptsize (c) Acetylene.
\end{minipage}
\caption{This figure shows schematic representations of the simplest of each type of hydrocarbon:
(a) the ethane, (b) the ethylene and (c) the acetylene. These are,
respectively, an alkane, an alkene and an alkyne.}
\label{fig:hydrocarbons}
\end{figure}

Another advantage of these organic molecules is that they can assume different geometries. For example,
in figure \ref{fig:polyenes} are presented two polyenes with six carbon atoms each. Due
to their atoms' $sp^2$ hybridization, they tend to make 120$^{\rm{o}}$ angles between bonds. When the
angles are built to alternate sides, the polyene has a zigzag shape
[hexatriene, figure \ref{fig:polyenes}(a)].
However, when all angles are built to the same side, the polyene has a cyclic shape [benzene,
figure \ref{fig:polyenes}(b)]. In the particular case of the benzene, there is a fully delocalized
double bond, which makes this molecule a good conductor. This is the motivation of our study
of the transport through a benzene ring.

\begin{figure}[htbp]
\begin{minipage}{0.6\textwidth}
\begin{flushleft}
\centering
\includegraphics{figure2a}
\label{fig:hexatriene}
\end{flushleft}
\end{minipage}
\begin{minipage}{0.4\textwidth}
\begin{flushright}
\centering
\includegraphics{figure2b}
\label{fig:benzene}
\end{flushright}
\end{minipage}
\begin{minipage}{0.6\textwidth}
\centering \scriptsize (a) Hexatriene.
\end{minipage}
\begin{minipage}{0.4\textwidth}
\centering \scriptsize (b) Benzene.
\end{minipage}
\caption{This figure shows the schematic representation of two different molecules with six carbon atoms each:
(a) a hexatriene, with zigzag structure, and (b) a benzene,
with a cyclic structure. The dashed lines on the benzene indicate fully delocalized double bonds.}
\label{fig:polyenes}
\end{figure}

The main goal of this paper is to provide a theoretical introduction to molecular electronics,
studying the electronic transport through a benzene ring, using the formalism of Green's
functions. The transport through quantum rings has been considered in the literature before
\cite{Dias_2013}, but the approach used cannot be easily generalized to more complex problems.
On the other hand, the Green's function approach is the method of choice when the complexity
of the problem increases. The use of the  Green's function method to describe impurity states in a
tight-binding models  has also been considered \cite{Adame2012}.
The effect of phonons on transport across a Benzene molecule has also been considered using Green's
function methods \cite{Knap}.

The paper is written with the graduate students in mind. The method of Green's
function is often considered a difficult one, casting a way many students from its study.
Here we show, by two examples
of moderate complexity, that this is not the case. These two problems are solved analytically, which,
to our best knowledge has not been solved in the literature before using the given method.
Indeed, although, there are
many books on quantum transport, none give a detailed discussion of problems of moderate complexity.
Typically,  they solve the one-dimension tight-binding model \cite{Datta1,Datta2}.
The same happens with review articles \cite{Pecchia}.
We thus consider that the solution of this problem is a valuable addition to the literature.

\section{Formalism and the resonant level system}

The resonant level system is the simplest example of molecular electronics displaying non-trivial
transport properties (this model also describes the transport through a single quantum level
\cite{Jauho}).
We introduce the formalism used in the next section for describing
electronic transport through a benzene molecule with a {\it hands-on} philosophy by solving the
resonant level system using Green's functions. This approach will benefit the students wanting
to have a working knowledge of the method.

To explain the basics of the Green's functions formalism,
let us consider a problem that can be described by a Hamiltonian
written in the form
\begin{equation}
 H=H_0 + V,
\end{equation}
\noindent such that
\begin{equation}
 H_0 \ket{\psi_0} = E \ket{\psi_0}
 \label{eq:schro1}
\end{equation}
and
\begin{equation}
 H \ket{\psi} = E \ket{\psi}.
 \label{eq:schro2}
\end{equation}

Let us also consider that $\ket{\psi_0}$ and $\ket{\psi}$ are such that they can be related by
\begin{equation}
 \ket{\psi} = \ket{\psi_0} + \ket{\psi_S},
\end{equation}
\noindent where $\ket{\psi_S}$ represents the scattered component of the wave. From the previous
equations, one can easily find the Lippmann-Schwinger equation, of the form
\begin{equation}
 \ket{\psi} = \ket{\psi_0} + G_0 \cdot V \ket{\psi},
 \label{eq:green3:lippmann1}
\end{equation}
\noindent where we define the free Green's function operator as \cite{Economou}
\begin{equation}
G_{0} \equiv \frac{1}{E-H_0+i\epsilon}.
\label{eq:green3:greenfunction1}
\end{equation}
\noindent This definition is obtained naturally except for the term $i \epsilon$, where $\epsilon$ is
a infinitesimal positive real number. This term is added to the definition to guarantee that this
operator describes a wave propagating from the left to the right.

Furthermore, one can replace the term $\ket{\psi}$ on
right hand side of the Lippmann-Schwinger equation (\ref{eq:green3:lippmann1})  with itself
obtaining by iteration
\begin{equation}
 \ket{\psi} = \ket{\psi_0} + G \cdot V \ket{\psi_0},
 \label{eq:green3:lippmann2}
\end{equation}
\noindent where $G$ is the full Green's function operator, that can be defined as
\begin{equation}
G \equiv \frac{1}{1-G_0 \cdot V} \cdot G_0.
\label{eq:green3:greenfunction2}
\end{equation}
The advantage of Eq. (\ref{eq:green3:lippmann2}) is that only the known wave function
$\ket{\psi_0}$ enters in it. Recalling the explicit definition of $G_0$,
given in \eqref{eq:green3:greenfunction1},
the previous equation can be simplified to
\begin{equation}
G \equiv \frac{1}{E-H_0-V+i\epsilon},
\label{eq:green3:greenfunction3}
\end{equation}
\noindent which results in a definition analogue to that of $G_0$ except that
now the full Hamiltonian enters in the definition, and with $\epsilon$ having the same
meaning as before.
However, neither of the two previous definitions of $G$ are very useful, since both of them involve
inversions of operators, which can be difficult. Fortunately, the Green's function
operator can yet be written in a
third form, through a Taylor expansion of the term
$\ccpar{1-G_0 \cdot V}^{-1}$ in \eqref{eq:green3:greenfunction2}.
In doing so, one reaches Dyson's equation
\begin{equation}
 G = G_0 + G_0 \cdot V \cdot G.
\end{equation}
\noindent Its usefulness will be proven further in the text.

The key aspect of this method is to use the previous relations in order to obtain, in the end, an expression of the form
\begin{numcases}{\braket{n}{\psi}=\label{eq:green3:sistema1}}
C \ccpar{ e^{i k a n} + r e^{-i k a n}}, & $n<0$,  \label{eq:green3:sistema1_r} \\
C \ccpar{ \tau e^{i k a n}},  & $n>0$. \label{eq:green3:sistema1_tau}
\end{numcases}

\noindent where $C$ is some constant and $\ket{n}$ is a position state (we note that we have in mind
a description of our systems by tight-binding Hamiltonians; therefore $\ket{n}$ is a Wannier state).
From an expression of this
form, it is possible to obtain $r$ and $\tau$, which are respectively the reflection and transmission
amplitudes, such that the reflectance and transmittance through some defect, quantum dot, or molecule
are given by
\begin{equation}
 {\cal R} = \abs{r}^2
\end{equation}
and
\begin{equation}
 {\cal T} = \abs{\tau}^2.
\end{equation}

We now proceed to the study of the resonant level system. This problem will be useful to
illustrate the previously described method in a situation where the solution is non-trivial
(in general, textbooks tend to describe only trivial examples which produce some discomfort on the
students). To start with,
let us define the system we ought to study. This is composed of two semi-finite 1D chains, with $N$ atoms
each, and a different atom in between them -- which we shall call the defect. The distance
between each atom and its nearest two neighbours is $a$. In both 1D chains, the hopping energy between
two first neighbours is $-t$, whereas the hopping energy between higher order neighbours is zero.
However, due to the presence of the defect, we shall consider that, on the one hand,
the hopping energies around the defect are different from $-t$ and are different in both sides of it --
being $-\alpha_{\mathrm{L}}$ to the left and $-\alpha_{\mathrm{R}}$ to the right --, and,
on the other hand, there is an
on-site energy on the position of the defect ($n=0$), with value $-\varepsilon_0$.
In figure \ref{fig:OneLevelResonant}
there is a schematic representation of the described system.

\begin{figure}[htbp]
\centering
\includegraphics{figure3}
\caption{Schematic representation of the one level resonant system considered. This system is composed
of two 1D chains with $N$ atoms each, represented by circles, and a defect in the middle of them,
represented by a cross. In this figure are also identified the hopping energies between different
atoms ($-t$, $-\alpha_{\mathrm{L}}$ or $-\alpha_{\mathrm{R}}$, accordingly), the site energy on the defect ($-\varepsilon_0$)
and the distance between two neighbours ($a$).}
\label{fig:OneLevelResonant}
\end{figure}

From the previous description, it becomes clear that the Hamiltonian which describes this system
can be written as
\begin{equation}
 H = H_{\mathrm{L}} + H_{\mathrm{R}} + H_{\mathrm{C}} + V \equiv H_0 + V,
\end{equation}
\noindent where we shall define
\begin{equation}
H_{\mathrm{L}} = -t \sum_{n=-N}^{-2} \Big( \ket{n}\bra{n+1} + \ket{n+1}\bra{n} \Big),
\label{eq:green3:HL}
\end{equation}
\begin{equation}
H_{\mathrm{C}} = -\varepsilon_0 \ket{0}\bra{0} ,
\label{eq:green3:HC}
\end{equation}
\begin{equation}
H_{\mathrm{R}} = -t \sum_{n=1}^{N-1} \Big( \ket{n}\bra{n+1} + \ket{n+1}\bra{n} \Big)
\label{eq:green3:HR}
\end{equation}
\noindent and
\begin{equation}
V = -\alpha_{\mathrm{L}} \Big( \ket{-1}\bra{0} + \ket{0}\bra{-1} \Big) -\alpha_{\mathrm{R}} \Big( \ket{0}\bra{1} + \ket{1}\bra{0} \Big).
 \label{eq:green3:V}
\end{equation}

We assume that in the beginning of times (say $t=-\infty$) the two semi-infinite chains
(often called  leads
in  molecular electronics terminology) are decoupled from the central atom.
In this case a flux of electrons incoming from the left of the defect ($n<0$) cannot
continue to the $n=0$ position being reflected in the end of the semi-infinite lead.
This means that at $t=-\infty$ the unperturbed wave function must read
\begin{numcases}{\ket{\psi_0}=\label{eq:green3:psi0}}
\ket{\psi_{\mathrm{L}}}, & $n<0$,  \label{eq:green3:psi0L} \\
0,  & $n>0$, \label{eq:green3:psi0R}
\end{numcases}
\noindent where $\ket{\psi_{\mathrm{L}}}$ is such that
\begin{equation}
 H_{\mathrm{L}} \ket{\psi_{\mathrm{L}}} = E \ket{\psi_{\mathrm{L}}},
 \label{eq:schropsiL}
\end{equation}
where $\ket{\psi_{\mathrm{L}}}$ plays the role of $\ket{\psi_0}$ in the formalism described above.
\noindent Hence, to apply the previously described method, we must find $\ket{\psi_{\mathrm{L}}}$.
It can be written, in general, in a basis composed by every position state available, that is,
\begin{equation}
 \ket{\psi_{\mathrm{L}}} = \sum_{m=-N}^{-1} c(m) \ket{m},
\end{equation}
\noindent where we do not know yet the explicit form of the coefficients $c(m)$. However, we
can use this result, as well as the definition of $H_{\mathrm{L}}$ given in \eqref{eq:green3:HL},
and replace them in \eqref{eq:schropsiL}. Multiplying  both sides of the
resulting equation with the position state $\bra{l}$ ($l=-N,\ldots,-1$), one gets that
(called the tight-binding equations)
\begin{numcases}{}
 -t c(-N+1) = E c(-N), & $l=-N$, \\
 -t \sqpar{c(l+1)+c(l-1)} = E c(l), & $-N<l<-1$, \label{eq:green3:schrodinger1}\\
 -t c(-2) = E c(-1), & $l=-1$.
\end{numcases}
\noindent It should be clear that \eqref{eq:green3:schrodinger1}
becomes general if we introduce the condition
\begin{equation}
 c(-N-1) = c(0) = 0,
\end{equation}
where it is assumed that $N$ will tend to infinite in the of the calculation.
On the other hand, we can consider that the flux must be composed by the sum of two plane waves,
one travelling forward and the other travelling backward, which means that the coefficients $c(m)$
must have the form
\begin{equation}
 c(m) = C e^{i kam} + D e^{-i kam},
\end{equation}
\noindent where $C$ and $D$ are constants and $k$ is the wave vector of the plane waves. From these
two conclusions, one finds that
\begin{equation}
 c(m) = C \sin \ccpar{k a m}
 \label{eq:c(m)}
\end{equation}
and, also,
\begin{equation}
 k \equiv k_n = \frac{n \pi}{a (N+1)},
 \label{eq:k1D}
\end{equation}
\noindent where $n$ is some integer value between $1$ and $N$. Through further normalization of
the wave function, one finally obtains
\begin{equation}
 \ket{\psi_{\mathrm{L}}} = \sqrt{\frac{2}{N+1}} \sum_{m=-N}^{-1} \sin \ccpar{k a m} \ket{m},
 \label{eq:green3:psiL}
\end{equation}
and also, from the \eqref{eq:green3:psi0},
\begin{equation}
 \braket{n}{\psi_0} = \braket{n}{\psi_{\mathrm{L}}} = \sqrt{\frac{2}{N+1}} \sin \ccpar{k a n},
 \label{eq:green3:npsi0}
\end{equation}
\noindent if $n<0$, and zero otherwise.
Moreover, it is also possible to find, from \eqref{eq:green3:schrodinger1} and \eqref{eq:c(m)},
that the allowed stationary energies of this problem are given by
\begin{equation}
 E(k)=-2t \cos\ccpar{ka}.
 \label{eq:green3:EL}
\end{equation}

In analogue way,
it is also possible to show that for the 1D chain to the right of the defect
the eigenstates of $H_{\mathrm{R}}$ are given by
\begin{equation}
 \ket{\psi_{\mathrm{R}}} = \sqrt{\frac{2}{N+1}} \sum_{m=1}^{N} \sin \ccpar{k a m} \ket{m},
 \label{eq:green3:psiR}
\end{equation}
and the allowed stationary energies are equally given by \eqref{eq:green3:psiL}. For this
case, $k$ is defined as before, see \eqref{eq:k1D}. This results will be useful later.

Now that  the coefficients $\braket{n}{\psi_0}$ have been found, we can use the Lippmann-Schwinger
equation, as given in \eqref{eq:green3:lippmann2}, and multiply this equation by the position
state $\bra{n}$ on both sides, obtaining [after recalling the definition of $V$ in \eqref{eq:green3:V}]
\begin{numcases}{\braket{n}{\psi}= \label{eq:green3:psi03}}
  \sqrt{\frac{2}{N+1}} \sin \ccpar{k a n} -\alpha_{\mathrm{L}} \braketop{n}{G}{0}\sqrt{\frac{2}{N+1}} \sin \ccpar{-k a}, & $n<0$, \\
  -\alpha_{\mathrm{L}} \braketop{n}{G}{0}\sqrt{\frac{2}{N+1}} \sin \ccpar{-k a}, & $n>0$.
\end{numcases}

In order to write an equation of the form of \eqref{eq:green3:sistema1}, we must now find
the matrix element $\braketop{n}{G}{0}$ of the operator $G$. Using the Dyson equation,
and considering that, if $n \neq 0$, $\braketop{n}{G_0}{m}$ is only non-zero if
$n$ and $m$ are both positive or both negative\cite{footnote1} one gets
\begin{numcases}{\braketop{n}{G}{0}= \label{eq:green3:G3}}
 -\alpha_{\mathrm{L}} \braketop{n}{G_0}{-1}\braketop{0}{G}{0},  & $n<0$, \\
 -\alpha_{\mathrm{R}} \braketop{n}{G_0}{1}\braketop{0}{G}{0},  & $n>0$.
\end{numcases}
\noindent This means it is now necessary to find the element $\braketop{0}{G}{0}$. This can be done
by solving the system
\begin{numcases}{}
  \braketop{0}{G}{0} = \braketop{0}{G_0}{0} \sqpar{1 -\alpha_{\mathrm{L}} \braketop{-1}{G}{0} - \alpha_{\mathrm{R}} \braketop{1}{G}{0}}, \\
  \braketop{-1}{G}{0} = -\alpha_{\mathrm{L}} \braketop{-1}{G_0}{-1}\braketop{0}{G}{0}, \\
  \braketop{1}{G}{0} = -\alpha_{\mathrm{R}} \braketop{1}{G_0}{1}\braketop{0}{G}{0},
\end{numcases}
\noindent whose equations were found using the Dyson's equation over again. By doing so, and defining
$G_0(n,m) \equiv \braketop{n}{G_0}{m}$ and also $G(n,m) \equiv \braketop{n}{G}{m}$, we get
\begin{equation}
 G(0,0) = \frac{G_0(0,0)}{1-G_0(0,0) \sqpar{\alpha_{\mathrm{L}}^2 G_0(-1,-1) + \alpha_{\mathrm{R}}^2 G_0(1,1)}}.
 \label{eq:green3:G00}
\end{equation}

To close the problem, it is now necessary to find the matrix elements of the operator $G_0$. That
calculation is presented in \ref{sec:anexoG03}, where it was found the result
\begin{numcases}{ G_0(n,m) =\label{eq:green3:G0nm}}
 \frac{1}{E+\varepsilon_0}, & $n=m=0$, \\
 \frac{i}{2t\sqrt{1-\ccpar{\frac{E}{2t}}^2}} \ccpar{e^{i k a \abs{n+m}} - e^{i k a \abs{n-m}}}, & $n \times m>0$, \\
 0, & otherwise.
\end{numcases}

Using this result, and defining now the dimensionless variables $Z \equiv {E}/{2t}$, $X \equiv {\varepsilon_0}/{t}$,
 $Y_{\mathrm{L}} \equiv {\alpha_{\mathrm{L}}}/{t}$ and $Y_{\mathrm{R}} \equiv {\alpha_{\mathrm{R}}}/{t}$,
\noindent \eqref{eq:green3:psi03} simplifies to
\begin{scriptsize}
\begin{numcases}{\braket{n}{\psi}= \label{eq:green3:psi06}}
  \sqrt{\frac{2}{N+1}} \sqpar{ \frac{e^{i k a n}}{2i} + \sqpar{\frac{i Y_{\mathrm{L}}^2 \sqrt{1-Z^2}}{Z+\frac{X}{2}+\ccpar{\frac{Y_{\mathrm{L}}^2+Y_{\mathrm{R}}^2}{2}}\ccpar{-Z+i\sqrt{1-Z^2}}}-1} \frac{e^{-i k a n}}{2i}}, & $n<0$, \\
  \sqrt{\frac{2}{N+1}} \sqpar{\frac{i Y_{\mathrm{L}} Y_{\mathrm{R}} \sqrt{1-Z^2}}{Z+\frac{X}{2}+\ccpar{\frac{Y_{\mathrm{L}}^2+Y_{\mathrm{R}}^2}{2}}\ccpar{-Z+i\sqrt{1-Z^2}}}} \frac{e^{i k a n}}{2 i}, & $n>0$.
\end{numcases}
\end{scriptsize}

\noindent where the sine functions were written in the Euler notation, $\sin x = \ccpar{e^{i x}-e^{-i x}}/\ccpar{2i}$.
Now, this result is in the wanted form, whereby comparing it to \eqref{eq:green3:sistema1},
one finds that
\begin{numcases}{\label{eq:green3:coefs}}
  r=\frac{i Y_{\mathrm{L}}^2 \sqrt{1-Z^2}}{Z+\frac{X}{2}+\ccpar{\frac{Y_{\mathrm{L}}^2+Y_{\mathrm{R}}^2}{2}}\ccpar{-Z+i\sqrt{1-Z^2}}}-1, \\
  \tau=\frac{i Y_{\mathrm{L}} Y_{\mathrm{R}} \sqrt{1-Z^2}}{Z+\frac{X}{2}+\ccpar{\frac{Y_{\mathrm{L}}^2+Y_{\mathrm{R}}^2}{2}}\ccpar{-Z+i\sqrt{1-Z^2}}},
\end{numcases}
\noindent and, also,
\begin{numcases}{\label{eq:green3:coefs2}}
  \mathcal{R}=\frac{ \ccpar{\frac{Y_{\mathrm{L}}^2-Y_{\mathrm{R}}^2}{2}}^2 + \sqpar{Z\ccpar{1-Y_{\mathrm{L}}^2} + \frac{X}{2}} \sqpar{Z\ccpar{1-Y_{\mathrm{R}}^2} + \frac{X}{2}}  }{\ccpar{\frac{Y_{\mathrm{L}}^2+Y_{\mathrm{R}}^2}{2}}^2 + \ccpar{Z+\frac{X}{2}}\sqpar{Z\ccpar{1-Y_{\mathrm{L}}^2-Y_{\mathrm{R}}^2} + \frac{X}{2}}}, \\
  \mathcal{T}=\frac{Y_{\mathrm{L}}^2 Y_{\mathrm{R}}^2 \ccpar{1-Z^2}}{\ccpar{\frac{Y_{\mathrm{L}}^2+Y_{\mathrm{R}}^2}{2}}^2 + \ccpar{Z+\frac{X}{2}}\sqpar{Z\ccpar{1-Y_{\mathrm{L}}^2-Y_{\mathrm{R}}^2} + \frac{X}{2}}}.
\end{numcases}

\noindent In particular, as expected,
\begin{equation}
 \mathcal{R} + \mathcal{T} = 1.
\end{equation}
It is worth stressing that the formalism allowed us to obtained analytical equations for
both $\mathcal{R}$ and $\mathcal{T}$.
In figure \ref{fig:TR_resonant_level}, we present some graphical representations of $\mathcal{R}$ and $\mathcal{T}$
as function of the energy $Z$, for several combinations of the parameters
$X$, $Y_{\mathrm{L}}$ and $Y_{\mathrm{R}}$. We leave the discussion of these results to the
final section of the paper (which can either be read immediately or afterwards).

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{figure4}
\caption{Graphic representations of the transmittance and reflectance for the resonant level system, as a
function of energy, for different values of the parameters $X$, $Y_{\mathrm{L}}$ and $Y_{\mathrm{R}}$.}
\label{fig:TR_resonant_level}
\end{figure}

\section{Transmittance through a benzene ring}

We now turn to the central part of this work, where using the formalism above we compute the
electronic transmittance through a benzene ring. Also the local density of states at the different
carbon atoms of the ring is given.

In order to do so, we start by defining the system we shall study. It is quite similar to the
level resonant system, but now the defect in not a different atom from the rest of the chain, but a benzene molecule connecting
the two 1D chains, as represented in figure \ref{fig:1DChainBenzene}. These chains are equivalent to
the ones presented before, and the hopping energies both on the chains and around the defect are equal
as well. However, now the {\it defect} is composed by a benzene molecule, with all the carbon atoms
placed
at a distance $a$ from its nearest neighbours, and the hopping energy between these atoms
being $-t_{\mathrm{b}}$.
We also consider on-site energies in the positions of every atom of the benzene molecule, all of them
with the value $-\varepsilon_0$. This makes the model more realistic, since there is a priori no particular
reason for the on-site energy in the leads be equal to that in the benzene.

\begin{figure}[htbp]
\centering
\includegraphics{figure5}
\caption{Schematic representation of the benzene system considered. This system is composed
of two 1D chains with $N$ atoms each, represented by circles, and a benzene molecule in the middle
of them. In this figure are also identified the hopping energies between different
atoms ($-t$, $-t_{\mathrm{b}}$, $-\alpha_{\mathrm{L}}$ or $-\alpha_{\mathrm{R}}$, accordingly), the site energy on the
atoms of the benzene ($-\varepsilon_0$) and the distance between two neighbours ($a$).}
\label{fig:1DChainBenzene}
\end{figure}

The Hamiltonian of the given system can be written as
\begin{eqnarray}
 H =H_{\mathrm{L}}+H_{\mathrm{C}}+H_{\mathrm{R}}+V \equiv H_0+V,
 \label{eq:green4:H}
\end{eqnarray}

\noindent with
\begin{equation}
H_{\mathrm{L}} = -t \sum_{n=-N}^{-2} \Big( \ket{n}\bra{n+1} + \ket{n+1}\bra{n} \Big),
\label{eq:green4:HL}
\end{equation}
\begin{eqnarray}
H_{\mathrm{C}} = \sum_{i=1}^{6} \Big[ -t_{\mathrm{b}} \Big( \ket{b_{i}}\bra{b_{i+1}} + \ket{b_{i+1}}\bra{b_{i}} \Big) -\varepsilon_0 \ket{b_i}\bra{b_i} \Big],
\label{eq:green4:HC}
\end{eqnarray}
\begin{equation}
H_{\mathrm{R}} = -t \sum_{n=1}^{N-1} \Big( \ket{n}\bra{n+1} + \ket{n+1}\bra{n} \Big)
\label{eq:green4:HR}
\end{equation}
\noindent and
\begin{equation}
V = -\alpha_{\mathrm{L}} \Big( \ket{-1}\bra{b_1} + \ket{b_1}\bra{-1} \Big) -\alpha_{\mathrm{R}} \Big( \ket{b_4}\bra{1} + \ket{1}\bra{b_4} \Big),
 \label{eq:green4:V}
\end{equation}

\noindent where, due to the cyclic structure of the benzene molecule, $\ket{b_{i+6}}=\ket{b_{i}}$.

Considering now, as before, a flux incoming from the left of the defect, then \eqref{eq:green3:psi0}
is still valid. On the other hand, the 1D chains surrounding the benzene molecule are similar as the ones
considered in the previous example, which means that \eqref{eq:k1D}, \eqref{eq:green3:psiL},
\eqref{eq:green3:EL} and \eqref{eq:green3:psiR} are also still valid. As a consequence, the same is true
for \eqref{eq:green3:npsi0}, for $n<0$, and zero otherwise. With this result, we can use the Lippmann-Schwinger
equation, as given in \eqref{eq:green3:lippmann2}, and proceed the same way as before,
obtaining
\begin{numcases}{\braket{n}{\psi}= \label{eq:green4:psi03}}
  \sqrt{\frac{2}{N+1}} \sin \ccpar{k a n} -\alpha_{\mathrm{L}} \braketop{n}{G}{b_1}\sqrt{\frac{2}{N+1}} \sin \ccpar{-k a}, & $n<0$, \\
  -\alpha_{\mathrm{L}} \braketop{n}{G}{b_1}\sqrt{\frac{2}{N+1}} \sin \ccpar{-k a}, & $n>0$.
\end{numcases}

We must now find the matrix element $\braketop{n}{G}{b_1}$ of the operator $G$. Using the Dyson equation,
and considering that, in the cases in which $n \neq b_{1,...,6}$, $\braketop{n}{G_0}{m}$ is only non-zero if
$n$ and $m$ are both positive or both negative, we get
\begin{numcases}{\braketop{n}{G}{b_1}= \label{eq:green4:G3}}
 -\alpha_{\mathrm{L}} \braketop{n}{G_0}{-1}\braketop{b_1}{G}{b_1},  & $n<0$, \\
 -\alpha_{\mathrm{R}} \braketop{n}{G_0}{1}\braketop{b_4}{G}{b_1},  & $n>0$,
\end{numcases}
\noindent We need now to calculate the matrix elements $\braketop{n}{G_0}{-1}$, $\braketop{n}{G_0}{1}$, $\braketop{b_1}{G}{b_1}$
and $\braketop{b_4}{G}{b_1}$. Due to the higher complexity of this case, we start now by the calculation of the matrix elements
of $G_0$ instead of the ones of $G$, as done in the previous example. To do so, we use the result obtained in \ref{sec:anexoG04},
\begin{numcases}{ G_0(n,m) =\label{eq:green4:G0nm}}
 \frac{i}{2t\sqrt{1-\ccpar{\frac{E}{2t}}^2}} \ccpar{e^{i  k a \abs{n+m}} - e^{i  k a \abs{n-m}}}, & $n \times m>0$, \\
 \frac{1}{6} \sum_{l=1}^6 \frac{e^{2 \pi i \frac{l (n-m)}{6}} }{E+ \varepsilon_0 + 2 t_{\mathrm{b}} \cos \ccpar{ \frac{2 \pi l}{6} }}, & $n,m=b_{i,j}$, \\
 0, & otherwise,
\end{numcases}
where $i,j=1,\dots,6$, from where we find
\begin{equation}
 \braketop{n}{G_0}{-1} = -\frac{e^{-i k a n}}{t}
\end{equation}
and
\begin{equation}
 \braketop{n}{G_0}{1} = -\frac{e^{i  k a n}}{t}.
\end{equation}
Through these results, we get
\begin{small}
\begin{numcases}{\braket{n}{\psi}= \label{eq:green4:psi05}}
  \sqrt{\frac{2}{N+1}} \sqpar{ \frac{e^{i  k a n}}{2 i} + \frac{e^{- i k a n}}{2 i} \ccpar{-1+ 2 i Y_{\mathrm{L}}^2 \sqrt{1-Z^2} t\braketop{b_1}{G}{b_1} }}, & $n<0$, \\
  \sqrt{\frac{2}{N+1}} \frac{e^{i  k a n}}{2 i} \sqpar{ 2 i Y_{\mathrm{L}} Y_{\mathrm{R}} \sqrt{1-Z^2} t \braketop{b_4}{G}{b_1} }, & $n>0$.
\end{numcases}
\end{small}

Finally, we only need the elements $\braketop{b_1}{G}{b_1}$ and $\braketop{b_4}{G}{b_1}$.
These elements can be
calculated through the system
\begin{numcases}{}
 \braketop{b_1}{G}{b_1} = \braketop{b_1}{G_0}{b_1} -\alpha_{\mathrm{L}} \braketop{b_1}{G_0}{b_1}\braketop{-1}{G}{b_1} -\alpha_{\mathrm{R}} \braketop{b_1}{G_0}{b_4}\braketop{1}{G}{b_1}, \\
 \braketop{b_4}{G}{b_1} = \braketop{b_4}{G_0}{b_1} -\alpha_{\mathrm{L}} \braketop{b_4}{G_0}{b_1}\braketop{-1}{G}{b_1} -\alpha_{\mathrm{R}} \braketop{b_4}{G_0}{b_4}\braketop{1}{G}{b_1}, \\
 \braketop{-1}{G}{b_1} = -\alpha_{\mathrm{L}} \braketop{-1}{G_0}{-1}\braketop{b_1}{G}{b_1} \\
 \braketop{1}{G}{b_1} = -\alpha_{\mathrm{R}} \braketop{1}{G_0}{1}\braketop{b_4}{G}{b_1},

\end{numcases}
\noindent where all equations were derived from the Dyson equation.
It follows that, defining $G_{n,m} \equiv \braketop{n}{G}{m}$
and $G^0_{n,m} \equiv \braketop{n}{G_0}{m}$, we get
\begin{footnotesize}
\begin{equation}
G_{b_1,b_1} = \frac{G^0_{b_1,b_1} + \alpha_{\mathrm{R}}^2  G^0_{1,1} \ccpar{G^0_{b_1,b_4}G^0_{b_4,b_1}-G^0_{b_1,b_1}G^0_{b_4,b_4}}}{1-\alpha_{\mathrm{L}}^2 G^0_{-1,-1}G^0_{b_1,b_1}-\alpha_{\mathrm{R}}^2 G^0_{1,1}G^0_{b_4,b_4}-\alpha_{\mathrm{L}}^2\alpha_{\mathrm{R}}^2 G^0_{-1,-1} G^0_{1,1} \ccpar{G^0_{b_1,b_4}G^0_{b_4,b_1}-G^0_{b_1,b_1}G^0_{b_4,b_4}}}
 \label{eq:green4:Gb1b1}
\end{equation}
\end{footnotesize}
and
\begin{footnotesize}
\begin{equation}
G_{b_4,b_1} = \frac{G^0_{b_4,b_1}}{1-\alpha_{\mathrm{L}}^2 G^0_{-1,-1}G^0_{b_1,b_1}-\alpha_{\mathrm{R}}^2 G^0_{1,1}G^0_{b_4,b_4}-\alpha_{\mathrm{L}}^2\alpha_{\mathrm{R}}^2 G^0_{-1,-1} G^0_{1,1} \ccpar{G^0_{b_1,b_4}G^0_{b_4,b_1}-G^0_{b_1,b_1}G^0_{b_4,b_4}}}.
 \label{eq:green4:Gb4b1}
\end{equation}
\end{footnotesize}

To close the problem, we only need some matrix elements of $G_0$, which can be found using again the result from equation \eqref{eq:green4:G0nm}.
Doing so, and defining $\Upsilon \equiv {t_{\mathrm{b}}}/{t}$, we get
\begin{equation}
 G^0_{-1,-1}=G^0_{1,1} = - \frac{e^{i  k a}}{t} = \frac{1}{t} \sqpar{Z - i\sqrt{1-Z^2}},
\end{equation}
\begin{equation}
 G^0_{b_1,b_1}=G^0_{b_4,b_4} = \frac{1}{t} \left\{\frac{-\frac{1}{2\Upsilon^3}\ccpar{Z+\frac{X}{2}}\sqpar{\frac{1}{\Upsilon^2}\ccpar{Z+\frac{X}{2}}^2 - 3}}{ \sqpar{\frac{1}{\Upsilon^2}\ccpar{ Z+\frac{X}{2} }^2-1}\sqpar{\frac{4}{\Upsilon^2}\ccpar{ Z+\frac{X}{2} }^2-1}}\right\},
\end{equation}
and
\begin{equation}
 G^0_{b_4,b_1}= G^0_{b_1,b_4} = \frac{1}{t} \left\{\frac{\frac{1}{2\Upsilon}}{ \sqpar{\frac{1}{\Upsilon^2}\ccpar{ Z+\frac{X}{2} }^2-1}\sqpar{\frac{4}{\Upsilon^2}\ccpar{ Z+\frac{X}{2} }^2-1}}\right\}.
\end{equation}

Note that all these elements are only numbers (real or complex), that are function of the energy and
several other parameters, but not the position $n$. As a consequence, the same is true for the elements
$G_{b_1,b_1}$ and $G_{b_4,b_1}$. This means that we find
\begin{numcases}{\label{eq:green4:coefs}}
  r= 2 i Y_{\mathrm{L}}^2 \sqrt{1-Z^2} \sqpar{ t G_{b_1,b_1} } -1, \\
  \tau= 2 i Y_{\mathrm{L}} Y_{\mathrm{R}} \sqrt{1-Z^2} \sqpar{ t G_{b_4,b_1} },
\end{numcases}

\noindent and, finally,
\begin{numcases}{\label{eq:green4:coefs2}}
  \mathcal{R}= 1 + 4 Y_{\mathrm{L}}^4 \ccpar{1-Z^2} t^2 \abs{ G_{b_1,b_1} }^2 + 4 Y_{\mathrm{L}}^2 \sqrt{1-Z^2} \mathrm{Im} \sqpar{ t G_{b_1,b_1} },  \\
  \mathcal{T}= 4 \ccpar{Y_{\mathrm{L}} Y_{\mathrm{R}}}^2 \ccpar{1-Z^2} t^2 \abs{ G_{b_4,b_1} }^2.
\end{numcases}

In figure \ref{fig:TR_benzene}, we present some graphical representations of $\mathcal{R}$ and $\mathcal{T}$
as function of the energy $Z$, for several combinations of the parameters
$X$, $\Upsilon$, $Y_{\mathrm{L}}$ and $Y_{\mathrm{R}}$. The discussion of the results is left for the
final section of the paper.

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{figure6}
\caption{Graphic representations of the transmittance and reflectance through a benzene ring, as a
function of energy, for different values of the parameters $X$, $\Upsilon$, $Y_{\mathrm{L}}$ and $Y_{\mathrm{R}}$.}
\label{fig:TR_benzene}
\end{figure}

\subsection{Local Density of States}
\label{sec:LDOS}

One of the main advantages of the Green's function approach is that the calculation of the Local Density
of States (LDOS) in the atom of the position $n$ is straightforward. It is given by \cite{Economou}
\begin{equation}
 \rho_n(Z) = - \frac{1}{\pi} \mathrm{Im} G_{n,n}(Z),
\end{equation}
\noindent where $G_{n,n}\equiv \braketop{n}{G}{n}$. For the benzene system, we are interested in the LDOS
in each of the atoms of the benzene ring. In general, it should be different for each atom except, due to
the symmetry of the system, $\rho_{b_2}(Z)=\rho_{b_6}(Z)$ and $\rho_{b_3}(Z)=\rho_{b_5}(Z)$. In the particular
case when $\alpha_{\mathrm{L}}=\alpha_{\mathrm{R}}$, there is a higher level of symmetry which imposes that $\rho_{b_2}(Z)=\rho_{b_3}(Z)=\rho_{b_5}(Z)=\rho_{b_6}(Z)$
and also $\rho_{b_1}(Z)=\rho_{b_4}(Z)$.

To calculate each of this different local densities of states, we need to find the matrix elements $\braketop{b_n}{G}{b_n}$,
with $n=1,...,6$. Through Dyson equation, we find
\begin{numcases}{}
 \braketop{b_n}{G}{b_n} = \braketop{b_n}{G_0}{b_n} - \alpha_{\mathrm{L}} \braketop{b_n}{G_0}{b_1}\braketop{-1}{G}{b_n} - \alpha_{\mathrm{R}} \braketop{b_n}{G_0}{b_4}\braketop{1}{G}{b_n}, \\
 \braketop{ -1}{G}{b_n} = - \alpha_{\mathrm{L}} \braketop{-1}{G_0}{-1}\braketop{b_1}{G}{b_n}, \\
 \braketop{  1}{G}{b_n} = - \alpha_{\mathrm{R}} \braketop{ 1}{G_0}{ 1}\braketop{b_4}{G}{b_n}, \\
 \braketop{b_1}{G}{b_n} = \braketop{b_1}{G_0}{b_n} - \alpha_{\mathrm{L}} \braketop{b_1}{G_0}{b_1}\braketop{-1}{G}{b_n} - \alpha_{\mathrm{R}} \braketop{b_1}{G_0}{b_4}\braketop{1}{G}{b_n}, \\
 \braketop{b_4}{G}{b_n} = \braketop{b_4}{G_0}{b_n} - \alpha_{\mathrm{L}} \braketop{b_4}{G_0}{b_1}\braketop{-1}{G}{b_n} - \alpha_{\mathrm{R}} \braketop{b_4}{G_0}{b_4}\braketop{1}{G}{b_n}.
\end{numcases}

\noindent These five equations make up a closed system from where it is possible to find the value of $\braketop{b_n}{G}{b_n}$.
However, the analytic solutions, although possible to find, are too complex to have any interest in being presented. As an
alternative, we present, in figure \ref{fig:LDOS_benzene}, graphical representations of the LDOS in each atom, as function
of the energy $Z$, for several combinations of the parameters
$X$, $\Upsilon$, $Y_{\mathrm{L}}$ and $Y_{\mathrm{R}}$. From the study of this figure it is clear that
the behaviour of the LDOS is rather complex, depending on the actual values of the parameters.

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{figure7}
\caption{Graphic representations of the local density of states in the atoms of a benzene ring, as a
function of energy, for different values of the parameters $X$, $\Upsilon$, $Y_{\mathrm{L}}$ and $Y_{\mathrm{R}}$. Note that,
in figures (a)--(d) and (f), the red and orange lines are superimposed, as well as the blue and green
ones.}
\label{fig:LDOS_benzene}
\end{figure}

\section{Discussion}

In conclusion, we have developed a formalism, based on Green's functions, for describing the
electronic transport through molecules represented by tight-binding models. We started by
explaining the general method, defining some useful operators. To illustrate its usage, we
then applied it to the case of the resonant level system, where we were able to find
its transmittance. To do so, we also found the wave functions and the allowed energies of electrons
propagating in an ideal unidimensional chain. From the expressions found for the transmittance
and reflectance of this system, and the subsequent presented plots, we draw a number of useful results:
\begin{itemize}
\item both $\mathcal{T}$ and $\mathcal{R}$ are symmetrical with respect to the line
$\mathcal{T} = \mathcal{R} = 0.5$. This is due to the condition $\mathcal{T}+\mathcal{R}=1$;

\item in most cases, at $Z=\pm 1$, we found that $\mathcal{T}=0$ and $\mathcal{R}=1$. The exception
was the plot presented in figure \ref{fig:TR_resonant_level}(h), in which $\mathcal{T}=1$ and
$\mathcal{R}=0$ at $Z=1$. Through an analysis of the limits
of \eqref{eq:green3:coefs2}, we found that this exceptions occur whenever one set of parameters
respects the condition $\abs{X/2}=\abs{(Y_{\mathrm{L}}^2+Y_{\mathrm{R}}^2)/2-1}$ [which is the case of the parameters in figure
\ref{fig:TR_resonant_level}(h)]. In these cases, the transmittance is total in at least one
of the extremes of the band;

\item when $X=0$, the these functions are also symmetrical with respect to $Z=0$. This means
that they must have a maximum or minimum at this energy. On the other hand, when $X$ was a
positive (negative) number, the peak tended to slide to the right (left), for the same parameters
$Y_{\mathrm{L}}$ and $Y_{\mathrm{R}}$. Analysing \eqref{eq:green3:coefs2}, we found that the extreme of these functions
depends on the other parameters through the expression
\begin{numcases}{Z_{\mathrm{extremo}}=}
\frac{\frac{X}{2}}{\frac{Y_{\mathrm{L}}^2+Y_{\mathrm{R}}^2}{2}-1}, & $\abs{\frac{X}{2}}\leq\abs{\frac{Y_{\mathrm{L}}^2+Y_{\mathrm{R}}^2}{2}-1}$, \\%[0.4cm]
\frac{\frac{Y_{\mathrm{L}}^2+Y_{\mathrm{R}}^2}{2}-1}{\frac{X}{2}}, & $\abs{\frac{X}{2}} >  \abs{\frac{Y_{\mathrm{L}}^2+Y_{\mathrm{R}}^2}{2}-1}$.
\end{numcases}
This result explains both the behaviour of the peak with a change in $X$ and also the exception presented
in the previous point;

\item through the two previous points, we conclude that the extreme found must be a maximum of the
transmittance and a minimum of the reflectance. Using \eqref{eq:green3:coefs2}, we obtain that
\cite{footnote2}
\begin{numcases}{\mathcal{T}_{\mathrm{max}}= \label{eq:maxtran1}}
\frac{4Y_{\mathrm{L}}^2Y_{\mathrm{R}}^2}{\ccpar{Y_{\mathrm{L}}^2+Y_{\mathrm{R}}^2}^2}, & $\abs{\frac{X}{2}}\leq\abs{\frac{Y_{\mathrm{L}}^2+Y_{\mathrm{R}}^2}{2}-1}$, \\%[0.4cm]
\frac{Y_{\mathrm{L}}^2Y_{\mathrm{R}}^2}{\ccpar{\frac{X}{2}}^2 + Y_{\mathrm{L}}^2+Y_{\mathrm{R}}^2 -1}, & $\abs{\frac{X}{2}} >  \abs{\frac{Y_{\mathrm{L}}^2+Y_{\mathrm{R}}^2}{2}-1}$.
\end{numcases}
In particular, when $Y_{\mathrm{L}}=Y_{\mathrm{R}} \equiv Y$, we get the important result
\begin{numcases}{\mathcal{T}_{\mathrm{max}}= \label{eq:maxtran2}}
1, & $\abs{\frac{X}{2}} \leq \abs{Y^2-1}$, \\%[0.4cm]
\frac{Y^4}{\ccpar{\frac{X}{2}}^2 + 2Y^2 -1}, & $\abs{\frac{X}{2}} > \abs{Y^2-1}$.
\end{numcases}
This result is important because, as we can see, when $Y_{\mathrm{L}}=Y_{\mathrm{R}}$, we can adjust the parameters $X$ and $Y$
so that, at some energy, we have maximum transmittance. This is not possible when $Y_{\mathrm{L}}\neq Y_{\mathrm{R}}$, nor in
the simplest case of point defect in which $Y_{\mathrm{L}}=Y_{\mathrm{R}}=1$.
This is why this problem is called the level resonant
system. In the diagram of figure \ref{fig:phasespace}, the region of the phase space $XY$ where
$\mathcal{T}_{\mathrm{max}}=1$ is presented in grey. On the other hand, in the white region of the same diagram,
the maximum transmittance is always lower than one, decreasing with the distance to the bold line
defined by $\abs{X/2} = \abs{Y^2-1}$. To understand the topology of this region,
some level lines are represented;

\begin{figure}[htbp]
\centering
\includegraphics{figure8}
\caption{This diagram shows: in a bold line, the points for which $\abs{X/2} = \abs{Y^2-1}$;
in a grey shadow, the region of the phase space $XY$ where $\mathcal{T}_{\mathrm{max}}=1$; in white,
the region where $\mathcal{T}_{\mathrm{max}}<1$; in thin lines, some level lines, for $\mathcal{T}_{\mathrm{max}}=0.1$,
$\mathcal{T}_{\mathrm{max}}=0.5$ and $\mathcal{T}_{\mathrm{max}}=0.9$; and a black dot which represents
the point in the phase space corresponding to the ideal system, that is, $Y=1$ and $X=0$. The grey shaded
region continues as $Y \rightarrow \pm \infty$. Note that
the diagram is symmetrical both with respect to the lines $X=0$ and $Y=0$.}
\label{fig:phasespace}
\end{figure}

\item in the general case, for the same $Y_{\mathrm{L}}$ and $Y_{\mathrm{R}}$, the transmittance increases with the decreasing of $X$,
as expected, but only when $\abs{X/2} >  \abs{(Y_{\mathrm{L}}^2+Y_{\mathrm{R}}^2)/2-1}$. Elsewhere, this value is independent of $X$.
On the other hand, for the same $X$, the transmittance tends to increase when $Y_{\mathrm{L}}=Y_{\mathrm{R}}$ and also when
$Y_{\mathrm{L}}$ and $Y_{\mathrm{R}}$, as represented in the phase space $XY_{\mathrm{L}}Y_{\mathrm{R}}$, get closer
to the surface defined by $\abs{X/2} = \abs{(Y_{\mathrm{L}}^2+Y_{\mathrm{R}}^2)/2-1}$, and decrease with the distance to
that surface. In particular, it is null when either $Y_{\mathrm{L}}$ or $Y_{\mathrm{R}}$ are null.

\end{itemize}

We then studied the case of the benzene system,
and presented  plots of its transmittance and reflectance, for different sets of
parameters. Here, the plots obtained for the transmittance
were much more complex, depending on four parameters,
which means that a study like the one presented for the resonant level system above is much more
difficult to elaborate. Indeed, It would require an extensive study of the limits of the transmittance and
reflectance for several values of the presented parameters, and also the production of much
more plots than the ones presented here, what
would worth of paper of itself. Our goal was rather to show that the Green's function method an
easily be applied to a problem of moderate difficulty.
The same can be said about the local density
of states in each atom of the benzene ring, which was also calculated in this paper.

Nevertheless, some qualitative aspects can be drawn from the results of Fig. \ref{fig:TR_benzene}.
Taking the single example of panel (h) of that figure we seen that the transmittance is strongly
suppressed for values of the energy such that $Z<-0.5$. If we now imagine that the energy of
the electrons can be tuned, such a system would correspond to an electronic filter, where
only the electrons with energies in the range $-0.5<Z<0$ would be transmitted. The same can be said
for the results in panel (g) and energies $Z>0$.

This method, as should now be clear, can be easily generalized to more complex problems, such
as a chain of atoms, like a polyacetylene, or two (or more) benzene rings in a row. However, we
can conclude that the analytical expressions of the found quantities, although possible to find
through this method, can get rather complex when the complexity of the problem increases,
due to the increase of the number of atoms involved in the calculation. In this case, the method
can be formulated using efficient numerical methods.

In synthesis, as should now be obvious, the Green's functions
method is ideal for tackling transport problems in molecular electronics, providing, in many cases,
analytical expressions for the quantum transmittance, which can further
be analysed in detail for the benefit of gaining insight  on the transport properties
of a given system.

\newpage
\setcounter{section}{1}

redpajama_set_name: RedPajamaArXiv


text:
Calvin and Radicalism in Geneva
Ozment, Steven E. The Age of Reform: 1250-1550 : An Intellectual and Religious History of Late Medieval and Reformation Europe. New Haven, Conn. ; London: Yale University Press, 1980. Kindle Electronic Edition.
Chapter 11, "Calvin and Calvinism" pp. 352-381. Part 2, "Political Revolt and Religious Reform in Geneva" pp. 358-362.
Between 1534 and 1536 Calvin, while travelling in France and Italy, wrote his Institutes of the Christian Religion. "This work grew through subsequent editions into a summa of Reformed Protestantism by the final edition of 1559" (Ozment 1980, 358). In 1536 Calvin settled in Geneva. "Whereas in Saxony a religious reform, undertaken at the initiative of theologians, led to a political revolt b princes against the emperor, in Geneva a political revolt against the House of Savoy, led by the city's magistrates, prepared the way for the introduction of Protestant religious reforms" (Ozment 1980, 358). Ozment describes the political background in some detail. Geneva became independent in 1528 then experienced a Protestant Reformation as "part of the consolidation of the political revolution of the 1520s, not of that revolution itself" Ozment 1980, 360). Disputes on religious and civil matters flared up around 1530 and again in 1535, at which time the city council suspended the Mass and gave Catholic clergy the choice of becoming Protestant or being exiled. Conflict continued for some years. Ozment makes an interesting statement about the resolution. "On May 25, 1536 the Genevan people accepted the Reformation, pledging 'to live according to the Law of the Gospel and the Word of God, and to abolish all Papal abuses.' Calvin arrived that summer, just as Farel and his co-workers were beginning to implement this new mandate" (Ozment 1980, 361). One must question the definition Ozment uses of "Gospel." Calvin quickly received a position of leadership. In 1537 he presented articles for church organization to magistrates. He also prepared a catechism. Churchmen were to be selected to govern and report legal failures to ministers for discipline. The populace did not receive these proposals willingly, as they seemed akin to the magisterial authority the people had opposed in the past. The Genevan council removed Calvin's right of excommunication and re-established traditional holiday observances. Calvin was exiled in 1538.

redpajama_set_name: RedPajamaCommonCrawl


Last year President Obama declared war on methane emissions from one of the most dangerous emitters in the world; cows.
While many of Obama's detractors were busy lambasting the President's plan by writing bovine fart jokes, some in the scientific community actually heeded his call for action by devising, in essence, Gas-X for cows.
Cows have a huge flatulence problem that is damaging the planet.
Last year, the Obama administration said that it would make a multi-pronged attack on cow flatulence in an attempt to cut emissions.
Now, one group of scientists believes it has come up with a solution; adding a special supplement to cows' feed to cut down on their methane production by 30 per cent.
Over the course of the 12-week study, cows consumed a diet supplemented by the methane inhibitor 3-nitrooxypropanol, or 3NOP.
The study was conducted at Penn State, released through the National Academy of Sciences, and was partially funded by DSM Nutritional Products.
So while some scientists are more than happy to take President Obama's war on bovine flatulence seriously, the rest of us, also known as the sane individuals, will continue to express the opinion that wasting time and the taxpayers money on a plan to fight cow farts, instead of spending those resources on more important issues, really…just…stinks.


In the March 1996 issue of 73 Magazine, Wayne Green asked readers to consider generating some articles on fellow hams who they felt deserved recognition before becoming a "Silent Key" and were recorded only as a memory. It's easy to recount an individual's achievements after he or she is gone, but if someone is doing something worthwhile, doesn't it deserve exposure while they're still around? I think so too.
When Uncle Wayne wrote about it in his editorial, I immediately thought of a friend of mine, someone who's overcome a great obstacle to become recognized as doing something worthwhile each and every week for the Amateur Radio community…Hap Holly, KC9RP, founder, guiding light and moderator of the weekly amateur radio audio feature magazine known as the RAIN(Radio Amateur Information Network) Report. From his home studio/ham shack in suburban Chicago, Hap produces his 10 to 15- minute weekly amateur radio program service, featuring timely interviews, occasional thought-provoking commentaries from other hams, excerpts from Hamvention Forums and other items of general interest to the ham radio community at large. Ham radio is traditionally an aural – as opposed to a visual – medium; we meet and recognize fellow hams primarily by voice, seldom seeing them in person. RAIN programming is also an aural medium, listened to by thousands of Internet users and hundreds of repeaters across the country at http://www.therainreport.com and via Twitter. Vern Jackson, WA0RCR, has carried the RAIN Report since its inception in the late 1980s on his Gateway 160 meter Radio Newsletter AM bulletin service from Wentzville, Missouri, on 1.860 Mhz, Saturday afternoon/evenings. Those aforementioned repeater groups and others replay the weekly RAIN program over their local repeater systems on their regular "net night" gatherings, permitting thousands of hams to hear Hap's offerings for that particular week. It's all done with volunteer help and – in the best of amateur traditions – without profit to anyone involved.
Hap Holly makes it happen; he's an uncommonly intelligent and outgoing individual. Hap is also very aurally oriented and insists on quality in every way in his RAIN programming. He conducts most of the interviews, edits and engineers all of the program material on his talking computer, writes the scripts for others to record for him as MP3 files, and archives/ catalogs the RAIN library. It's all accomplished by touch and by ear…Hap is one of a number of non-sighted radio amateurs within our ranks, but you'd never know it.
For Hap blindness happened literally overnight…when he was only 7. Problems with his vision began when he was 4, but then he awoke one morning totally blind. The condition is untreatable, at least within current medical terms, but to Hap, it hasn't been a handicap. In fact, in talking to him over the years, most of his friends tend to forget his unusual challenges, because he does so much of what we all do, without mention of his visual impairment. Both of Hap's parents were blind, his father's resulting from a football accident in high school and his mother's when she was around the age of 12. His father, nonetheless, was a successful building contractor and architect, formulating the plans he devised within his "mind's eye", then describing the details precisely to his secretary, who constructed balsa wood models. Then only by touch, Hap's dad would go over the "3-dimensional plans", incorporating additions or changes as needed.
Hap's father was also a long time columnist for the Christian Science Monitor and was known world-wide for his "Ask a Builder" column, which he wrote from 1965 until his passing in 1984. He was a touch typist – as is Hap – and was able to generate his column by that method alone (before the days of computers, word processors and voice synthesizers).
In 1988 a 420-page book was written about Hap's mother, depicting her struggle for independence from her New England industrialist father, her marriage to Hap's dad and the eventual challenges that two blind parents encountered raising four sighted children–then Hap too became blind. That book, entitled "What Love Sees" was made into a made-for-TV movie and was first aired on the CBS television network, September 22, 1996. It starred Richard Thomas of the original "Walton's" series and Anna Beth Gish, who played Pat Nixon in the widely acclaimed motion picture "Nixon", which also starred Anthony Hopkins. The TV movie, which is also entitled "What Love Sees", primarily took place in the small southern California ranching town of Ramona (where much of it was filmed) and followed the Holly household up until shortly after the time that Hap's sight was lost, and the family was forced to move to a larger city -Escondido – to seek specialized educational opportunities for their youngest son. Licensed to the Lifetime for Women cable network after CBS aired the movie a second time in June of 1999, "What Love Sees" was shown on Lifetime as many as three times a year beginning September, 1999 through June, 2002.
Hap's ham career began when he earned his Novice ticket in 1965 at the age of 14, receiving the call sign WN6UJH, while living in Escondido. He became a General – dropping the "N" in his call for a "B" – a year later in 1966 and served as a phone-patch station and net control for the Westcars traffic net until 1970. Hap then headed off to Principia College in Elsah, Illinois and from 1970 to graduation in 1974, ran phone patches and kept radio schedules for his fellow students. He graduated with a Bachelor of Arts degree in Sociology, and soon found himself in the Chicago area, where he sought out world-class jazz accordionist Leon Sash to pursue his other love, music. A year later another love, Stephanie, became the center of Hap's attention and the two were married in August of 1976 after meeting the previous summer at Roundup Ranch in Buena Vista, Colorado, where Hap was a bunkhouse counselor. Hap taught a class in non-visual perception to the high school campers at the ranch, all of whom were sighted, but who learned to "see" in yet another way …with Hap's patient guidance. Hap's wife, Stephanie – who is sighted – received her ham ticket in 1986 and the call KA9WKD, after realizing how much ham radio, and the good it was able to accomplish, meant to Hap. In 1977, Hap picked up the "9-land" call of WD9GJQ. A year later he and Stephanie moved into their small but comfortable home in Des Plaines, Illinois, where he resumed his phone-patching and DXing activities with his now permanent antenna and station set-up.
In 1981 Hap passed his Advanced class license exam and changed calls once more, this time to his current KC9RP call sign. I met Hap when he served as ham radio informational programming guru for the BEAR, the Broadcast Employees Amateur Repeater, in suburban Chicago from 1984 to 1989. Hap's weekly "net nights" became something of a legend in the Chicago area, sometimes attracting in excess of 100 check-ins, via simulcasts on five area repeaters. This stint led to Hap's forming RAIN – the Radio Amateur Information Network – and the weekly RAIN Report dial-up line, and the bimonthly RAIN Journal tape, which was produced during the 1990s especially for the blind amateur radio operator.
According to Hap, "My inspiration for producing weekly ham radio programming resulted from my 'need to know' . I faithfully listened to the weekly ham radio newscasts from the Amateur Radio Newsline (formerly WestLink Radio) in the early 1980's on a local repeater. I owe a great deal of gratitude to Newsline's founder and producer, Bill Pasternak, WA6ITF, for his encouragement and direction. RAIN is truly an offspring of Newsline. In recent years Hap has written for Spec-Com Journal, Radio Scan Magazine and occasionally has reported for Newsline but with a human interest focus, as opposed to a news-only approach.
Hap is a common sight at the yearly Dayton Hamvention, taping a number of the forums with the help and cooperation of the Hamvention organizers. These forums are then incorporated throughout the year in Hap's RAIN Reports, making the Hamvention come alive for those of us who may not be able to attend personally.
According to Hap, being named 2002 Hamvention amateur of the Year was a VERY special honor. I had hoped I would earn this prestigious award sometime down the road a bit. When I was notified via telephone by the Awards committee chair, I was flabbergasted and almost speechless – something that rarely happens to me.- Hamvention is the largest international ham radio convention; it is held in suburban Dayton, Ohio in May.
Hap is active on HF, VHF, and The Internet, using a Pc equipped with the JAWS speech screen reader program. Today he masters all RAIN audio digitally; encodes the weekly programs into downloadable MP3 files using Sound Forge; then uploads those files to an audio server for Internet distribution and now podcasting. Both the web site and audio bandwidth are subsidized by supporters of RAIN via PayPal and the U.S. Mail.
So what does Hap do when he's not producing the RAIN Report? As a professional keyboardist his diverse repertoire of American music from the past 60 years has made him a popular choice in the Chicago area. Since 1975 Hap has been an active and honored member of the Des Plaines Lions Club, receiving the distinguished international Melvin Jones Fellowship plaque for his "Service to Humanity" in 1994. He served as club President two terms, 2004 and 2005. As a member of the Des Plaines Toastmasters club since 1976, Hap has served as its president a number of times. Three times since 1994 Hap has served as Northwest 3 Area Governor. Hap and Stephanie were radio operators with Des Plaines EMA, Emergency Management Agency, in the early 2000's, passing federal exams IS100, IS200 and IS700 in early 2006.
Since May, 2000, Hap has been employed as an audio engineer and monitor full time for Horizons for the Blind, a 45-minute rail commute to the far-flung northwestern suburb of Crystal Lake. His experience in producing audio programming helped him in being hired. More importantly, however, like so many other folks with disabilities, Hap would not be employed today without the skills he developed from an intensive computer class he took in the fall of 1999 that taught him how to use Windows combined with speech screen reader technology. this technology has helped Hap immensely in his everyday activities, including serving as Chair for his 30th Principia college reunion in 2003.
Hap's charming wife, Stephanie, has been employed since 1994 as a billing clerk for Nelson Westerberg, an agent of Atlas World Group, also known as Atlas Van Lines. She completed her recovery from a serious, summer, 2003, bicycle mishap that laid her up in a rehab center for almost four months. They opted to not have children.
For Hap Holly, blindness has never been a disability, only a challenge that has served to "fine-tune" his other senses. If you have a friend or relative with a serious handicap or disability, and you've felt that Amateur Radio would be a worthwhile hobby for them to pursue, perhaps Hap's extra-ordinary story might be just the inspiration they would need to take the plunge. Hap would disagree with my choice of terms (extra-ordinary) since he considers his accomplishments no more than the expected effort required to achieve anything worthwhile…but then that's the sign of extra-ordinary people isn't it?
published in "73" magazine, August, 1996; Dave Miller, NZ9E penned his '73' Ham to Ham column in the '90's; he periodically voiced said column for the RAIN Report. Dave resided in Niles, Illinois, when he wrote this article. He and his long-time wife, Sue, retired to Sheboygan,WI in 2002; he became a silent key in 2008.
Hap updated the above article in 2014. His email address is: hap@therainreport.com; phone: 847-334-6352.
It was decades ago in a house far, far away in California; the town was Escondido, where my family moved in February, 1959 from the small chicken ranching town of Ramona. Both are located in north San Diego county some 60 miles north of the Mexican border.
My mother used to correspond with my grandmother using an old recording machine called a Soundscriber in the early 1960's. It used flexible discs about the size of floppy disks but a lot floppier. I would borrow that recorder and place the microphone in the open bedroom window to pick up whatever was going on outside. I loved the sounds of nature and with no airplane noise overhead or traffic roar nearby in those days, it was easy to capture the songs of feathered songsters like the meadowlarks and mockingbirds perched atop the telephone wire that ran along Avocado Avenue. � What a pity that none of those flexible disc recordings of Escondido, California survived the decades since.
Severely bitten by the "tape worm", In 7th grade I was given a portable, battery-operated Norelco tape recorder for Christmas in 1963 by my parents. It used three-inch reels. I was in heaven. Alas, my carelessness and the fragile design of the machine doomed it to an early demise, but the damage had been done and the archiving seeds permanently planted.
During my junior high school years I recorded what I called "rain music". Grabbing empty tin cans of all sizes, I would position each upside down under a stream of rain water dripping off the eaves just outside my bedroom window. Like a Caribbean steel drum band, the large fruit cocktail cans played the low notes and the smaller juice cans chimed the highest. The cans under constant streams gave me the basic rhythm line while simultaneously some short distance away the accent beats resounded from the intermittent drips. I learned some years later the racket drove my Mom and Dad crazy because the "music" was a bit too close to their bedroom window! What I would give to have saved any of those recordings.
In 1965 I began to record special family events, like Christmas. Many of those recordings were saved, thus started my archiving pastime. In 1967, as a sophomore at Escondido high school, I started keeping an audio journal. The open reel tape recorder became my confidante. I would talk to it often … about everything. My blind parents and I got along well overall but there were just some things I as a blind teen did not feel I wanted to discuss with them; pretty normal actually.
After attending Palomar Junior College in nearby San Marcos for a year, I spent the summer of 1970 at the Adventure Unlimited ranches in Buena Vista, Colorado, where I was a bunkhouse counselor and resident musician. My trusty portable cassette recorder was seemingly attached to my hip or hand during that summer. Sometimes I connected my cassette and open reel equipment to the sound system in Valerie Lodge at Roundup ranch; that began the archiving of my summers there from 1970 to 1976 at that camp. My summer camp sweetheart, Stephanie Eckman, whom I met in July, 1975, has been my forever-young wife since August 28, 1976. I still archive important events in my life, many of which I am preserving digitally, like the MP3 files on Volume 1 of the RAIN Collection .
In 1984 I began producing local ham radio programming as part of the newly organized BEAR Information Service, a weekly Chicago-based Amateur Radio program, sponsored by the Broadcast Employees Amateur Repeater. From 1987 till 1989 I produced ham radio programming under the name of the RP Report. Those archives will eventually be released as part of the RAIN Collection. In the early 1990's all my audio and ham radio production work became associated with the acronym RAIN, the Radio Amateur Information Network. I've been conducting interviews, scripting, editing, packaging and encoding RAIN Reports on the Internet since 1996. For ten years Mark Bohnhoff, WB9UOM. donated the web site (rainreport.com) and the necessary audio bandwidth for the RAIN Report at no charge to me.
I was honored to be named 2002 Radio Amateur of the Year by the Dayton Hamvention for my "years of service" to the U.S. Amateur Radio Service.
Today all RAIN audio is produced in the digital domain using Sound Forge 8.0, a suite of programs that enables the blind and visually impaired to do what I do. While the current RAIN files are produced digitally, all 100-plus RAIN highlights in the 1990 and 1991 folders on Volume 1 of the RAIN Collection CD were converted from audio cassette–an arduous task indeed. This first CD of RAIN programming allows folks, who are likewise fascinated with recording, archiving and listening to history, to own a piece of ham radio history for their personal use. Except for a reasonable fulfillment fee being collected by M. Bohnhoff Inc. the proceeds from the sales of this first RAIN CD will be used for the Dayton Hamvention presentation of RAIN.


text:
Q: How can I dynamically index through datalayer tags in GTM? I'm using the DuracellTomi datalayer plugin to push cart data from woocommerce to a GTM model to handle some tracking.
The DuracellTomi plugin pushes content to the transactionProducts[] array in the following format:
transactionProducts: Array[1]
0 : Object
category:""
currency:"USD"
id:8
name:"Test"
price:100
quantity:"1"
sku:8

I'd like to loop through this array and unstack it into three separate arrays, pricelist, skulist, and quantitylist. Currently I anticipate doing so as some variation on
  //Get Product Information
  if(stack = {{transactionProducts}}){
    for(i = 0; i < stack.length; i++) {
        if(stack.i.sku){
            skulisttemp.i = stack.i.sku;
            }
      if(stack.i.price){
            pricelisttemp.i = stack.i.price;
            }
      if(stack.i.sku){
            quantitylisttemp.i = stack.i.quantity;
            }
        }
    {{skulist}} = skulisttemp;
    {{pricelist}} = pricelisttemp;
    {{quantitylist}} = quantitylisttemp;

    }

Obviously this is not going to work because of how the tag referencing is set up, but I'm wondering if anyone has dealt with this and knows what the best way to index through these arrays might be. (For those who don't know, the square bracket array call doesn't work with GTM variables and instead the . format is used instead.)

A: You would need to create 3 variable type custom javascript function that picks your required value from dataLayer and returns it in an array.
Something like
function(){
var products =  {{transactionProducts}};
var skuArray = [];
for(i = 0; i < products.length; i++) {
    if(products[i].sku){
        skuArray.push(products[i].sku)
    }
}
return skuArray

}
hope this helped you :)

redpajama_set_name: RedPajamaStackExchange


text:
Wahlkreis 12 steht für folgende Wahlkreise:

 Bundestagswahlkreis 12
 Landtagswahlkreis 12
 Reichstagswahlkreis 12

redpajama_set_name: RedPajamaWikipedia


text:
Category Archives: Lincoln Red Imps FC
Posted in Champions League, Europa FC, Europa League, Football, Lincoln Red Imps FC, Men, St Josephs FC	| Leave a comment
Bardon: "We must continue to provide a platform for the youngsters to get better"
Part One of our three part series with Anthony Bardon as we talk about his time at Manchester 62, Lincoln Red Imps and much more.
FG – How would you look at your time at Man62?
Bardon – Man62 was the first club that took me in as a player in Gibraltar and I have to always thank them as they made me feel so welcome and one of their own from the very start. It was a great time in which I learned about Gibraltar football in the short 6 months we had together.
FG – How did it feel signing for Lincoln Red Imps?
Bardon – Signing for Lincoln was something I couldn't turn down. They showed great interest in me becoming a Lincoln player so I couldn't say no and it was the best decision I ever made playing there for the best part of 4-5 years.
FG – You won many trophies at Lincoln but what is your greatest achievement?
Bardon – My greatest achievement was when we beat Celtic. Not only that we beat them but the whole 11 months was an incredible season and that was the icing on the cake.
FG – How has your time at Lincoln benefitted you as a player?
Bardon – Lincoln benefitted me greatly because it provided me with an opportunity to train with quality players every single day, and it turned me into a much better overall player in everything.
FG – Since the day you arrived to when you left, what your thoughts on football in Gibraltar?
Bardon – From the day I came to the day I left the football in Gibraltar was so much different. The league when I left to when I started was so much different and the quality was on another level. Football in Gibraltar now is in such an important moment and we must continue to provide a platform for the youngsters to get better.
FG – Who is the best player you've played with?
Bardon – The best player overall would be Lee Casciaro. We fought every game and constantly got into each other but it's because we only wanted to win. He is an incredible athlete and player.
FG – Who is the toughest opponent you've come across in the league?
Bardon – Mustapha Yahaya. He was the most complete and most dominant midfielder I ever faced. He is a very good player.
FG – Who is the best manager you've worked with?
Bardon – Julio Ribas. He has been a football father figure to me. He is an inspiration and beyond smart.
FG – What local player/s do you think have the most potential to succeed as a professional?
Bardon – I think Tjay De Barr, Jayce Olivero and Louie Annesley are the players with the age and talent to make it. Two of them now are in pro clubs and have a real chance and Jayce can do the same. Another one is Ethan Britto.
FG – What is your view on the new rules being implemented by the Gibraltar FA in regard of HGP & 1 league?
Bardon – I don't like the 1 league because I think it kills the competition in the lower half towards the end of the season. Many problems could arise with that I feel, but we all trust the GFA to make the right decision. HGP rule is tricky because we have to get more locals playing, but not lose the quality of play because we need our best players playing at the highest level possible every week. Again, we all have to put our faith and trust in the GFA.
Posted in Football, Lincoln Red Imps FC, Manchester 62 FC, Men, Premier Division	| Leave a comment
Bernardo Lopes post-match interview against Europa
Post match interview with Bernardo Lopes after Lincoln beat Europa 2-0 to clinch the Premier Division shield.
Posted in Football, Lincoln Red Imps FC, Men, Premier Division	| Leave a comment
Lincoln and Europa battle for Premier Division title
Posted on May 9, 2019 by FootballGib
Gibraltar football's top two collide on Saturday night, with Lincoln hoping to retain the Premier Division shield – but rivals Europa stand in their way in what would most likely be a title decider.
Reigning champions Lincoln have a two point lead in the table heading into Saturday's game and have scored eleven more goals than their opponents Europa.
Even though both teams will be desperate for a win, drawing could be a risk that Lincoln may not want to take. Depending on results elsewhere, Mons Calpe, who face Lincoln on the final day, could still be in with a chance of qualifying for the Europa League, while Europa take on FC Boca Juniors.
Since taking over in mid-December, Victor Afonso has recorded eight straight wins, since a draw against Europa on the 16th February, and will be hoping to continue the run this Saturday.
Joseph Chipolina is one player which Victor will be hoping is fully fit. Lincoln's top goalscorer this season hasn't featured since scoring in their 3-0 win over Gibraltar Phoenix on the 8th March, while Lolo Soler will be expected to return between the sticks with Jaylan Hankins moving to the bench despite a very successful couple of months.
Their opponents Europa have also been on an excellent run of form; you need to date back to the 25th of November 2018 for their last defeat and it was against none other than Lincoln.
After losing five of arguably their best players and their manager in the betting scandal, Europa have still managed to turn their season into a positive one with a title decider on Saturday and facing Gibraltar United in the Gibtelecom Rock Cup final.
They needed players to step up and it would be their locals players which would steer the ship to Christmas. The club would go on to sign five players in January and announce a new coach in Rafael Escobar in February.
Escobar waltzed into Gibraltar football with a debut win over Lincoln in the Rock Cup and since then has gone undefeated with ten wins and two draws. Javi Munoz will be expected to return as Europa's number one with Coleing making his way to the bench, who just like Hankins, has had a very successful couple of months.
We, just like everyone else, hope that Saturday's game can live up the hype and provide the fans with a great spectacle. Entrance to the match is free, and if you can't make it to the Victoria Stadium, you can follow it live on our Twitter and Facebook accounts.
Posted in Europa FC, Football, Lincoln Red Imps FC, Men, Premier Division	| Leave a comment
Chipolina: "The team is feeling confident"
We spoke to Lincoln captain Roy Chipolina about his sides league campaign, Tjay De Barr and more.
Lincoln go into Saturday's game two points clear of their opponents Europa with two games remaining. Roy reflects on his sides campaign so far, "We have had our ups and downs this season. Starting in our European campaign, I felt we were unlucky not to progress in both our Champions league and Europa league ties but that's football. Investment was made in strengthening our squad but the chopping and changing of managers hasn't helped in settling and gelling what has been a new look Lincoln side. At Lincoln, standards and expectations are very high and though we have had some excellent results we have had some disappointing ones too. Since Victor took over just before Christmas I do feel the team has been progressing well and is looking the best it has been all season which is perfect timing as we will need to be at our very best if we are to come away with a victory on Saturday."
We asked the Lincoln captain how his side are feeling going into Saturday's title decider, "The team is feeling confident. I believe recently we have seen this on the field with some very good performances such as our recent wins versus St Joseph's and Gibraltar United. We know Europa is a fantastic team and there are very fine margins between us, as recent results suggest. We are ready and are very much looking forward to Saturday's battle with only one thing on our mind and that is to win and retain our title."
One player Roy will have to keep an eye out on is in-form Tjay De Barr who he knows very well from club and country. We asked Roy what he has made of the newly appointed Europa captain's season, "What can i say about Tjay that hasn't been said already? Apart from being a great lad, he is a fantastic player with so much potential. He has had a great season and even at his tender age has been mature enough to carry the responsibility of being such an important and reliable player for Europa and the national team. Nothing seems to faze him. With his explosive power and pace he will be a real danger."
Premier Division Player and Manager of the Month – April 2019
As Lincoln managed to grow a two point lead over Europa in the title race, the current league leaders have won both the Football Gibraltar Player of the Month and Manager of the Month awards for April
Photo (C) Rough and Ready Media
Player of the Month – Sergio Molina Rivero (Lincoln Red Imps)
Lincoln, as mentioned above, are now two points clear in the title race with just three games to go, and key to their 100% record last month was Sergio Molina really stepping up for his side and adding yet another goalscoring threat to Lincoln's potent attack force. The ex-Balona man scored twice, including a world class free kick vs Lynx, and has now made himself an integral part of the first eleven, having played the full 90 in five of his last six matches.
Closely behind him this month in second place was his teammate, Marcos Perez, who has arguably gone under the radar with his performances and his part in helping Lincoln towards a potential league title, while Europa's Tjay De Barr, who was instrumental in wins vs Lions and St Joseph's, but couldn't repeat the feat against Gib United, is third in this months award – all three will no doubt be very important in the title race.
Manager of the Month – Victor Manuel Afonso Mateo (Lincoln Red Imps)
The only manager with a 100% record last month, Afonso has kept his Lincoln side both fresh and consistent, no mean feat considering we are in the business end of the season, and he should be credited for that. The win vs Lions may have been expected, but Lincoln still had to navigate what were potentially tricky matches vs Gib United and Lynx – something they managed successfully to put themselves in pole position ahead of next weekend's potential title decider vs Europa.
Perhaps unlucky to finish second is Glacis manager Jesus Napolitano, who managed to steer his side to two wins and a creditable draw vs Phoenix in April – ending a run of six straight defeats from the start of February to the end of March. Napolitano has seemingly steadied the Glacis ship for now – Europa manager Rafael Escobar finishes third this month in this award.
Ferro: "We set our minds out to win this game and we did"
Posted on February 27, 2019 by FootballGib
After Lions Women's win over Lincoln Women, we spoke to Kayleigh Ferro about her Rock Cup triumph, scoring four goals in the final and much more.
Lions beat Lincoln 6-0 to clinch this season's Rock Cup, we asked Kayleigh how she felt the team played – "I think we all played well, we set our minds out to win this game and we did."
Not only did Kayleigh win the Rock Cup with her team, she was also announced as the Player's Player of the Season – we asked how it felt winning both accolades, "Its overwhelming to win the Rock Cup and Player's Player. I think i've worked hard for the award and my team have worked hard for the Cup."
A couple of weeks ago, Lincoln beat Lions on the final day of the season to win the Women's Football League. Kayleigh reflected on how she felt the campaign went – "I feel that my team worked very hard and deserved to win the league but football is unfair. Well done to Lincoln though for winning it."
At just 17 years old and plenty of years to improve – we asked what her goals for the future are, "My goals are to do well in future competitions with my team and the national squad. Also to win next years League & Cup and overall improve as a player."
Posted in Football, Lincoln Red Imps FC, Lions Gibraltar FC, Women	| Leave a comment
Chipolina brace secures Lincoln victory over Europa
Lincoln returned to the top of the table on goal difference after Joseph Chipolina's brace helped them beat Europa 3-2 today.
Lincoln appoint Manolo Nunez Sanchez as new manager
Lincoln have announced the appointment of former Gib United manager Manolo Nunez Sanchez as their new manager, after the departure of Diego Jose 'Yiy' Perez Jimenez yesterday.
Photo (C) Lincoln Red Imps


Four Y-shaped coat hooks move up and down on the cord of the wardrobe. When clothes are hung on, they instantly fix in position. The polyamide rope is connected to a ceiling hook or the wall mount and is hanging down, straightened by a solid metal weight.


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The Fruitmarket Gallery
We are making room for more Fruitmarket
Work has begun on our refurbishment and extension
The Fruitmarket Gallery Exhibition DVDs
VERAMUSEMENT 9-2-5
Thursday 22 February, 2018 at 11:00 am - 6:00 pm
An event every day that begins at 11:00 am, repeating until Sunday 25 February, 2018
« Exhibition Preview: VERAMUSEMENT 9-2-5
Upstage: Artists' Film Lounge »
Exhibition Preview Wed 21 Feb, 5–7pm
Thu 22 – Sun 25 Feb, 11am–6pm.
Joe Coghill is a multi-disciplinary artist, musician and experimental publisher based in Edinburgh. His practice incorporates sound, video, performance, social sculpture, facilitation and media distribution. Over the last six years he has been researching, archiving and creating new work exploring the life, influence and oeuvre of iconoclastic No-Wave outsider artist John Margotti.
For this installation, Coghill will be re-imagining Margotti's troublesome 1988 New York exhibition – VERAMUSEMENT 9-2-5 – by enacting a series of interventions and durational performances within the gallery. These will reflect on the concept of 'Veramusment', a term coined by Henry Flynt, meaning 'approximately pure recreation'.
Through a selection of artworks and ephemera, the installation will give glimpses into the life and character of John Margotti, who was apparently involved in a diverse selection of creative/disruptive projects and artist-led initiatives during the 1970s, '80s and early '90s. These ranged from the illegal bootlegging and distribution of cassette tapes with a loose collective of artists who went by the name 'Infinity-nineteen', to face-melting sonic performances in which Margotti and his band 'Invisible Maggot' used industrial machinery to create a noise so ferocious in amplitude it is said to have made noses bleed. The rich historic tapestry woven by this installation gradually unravels as we pull at its threads. Who is John Margotti?
This event forms part of Open Out – a week of performance, installation, film, dance and music at The Fruitmarket Gallery.
Thursday 22 February, 2018
Exhibitions, Open Out, Performance
Installation, Joe Coghill, Veramusement
Fresh Fruit Pop Up
Lower Level, 3/48 Waverley Mall, Waverley Bridge
Edinburgh, EH1 1BQ United Kingdom + Google Map
http://fruitmarket.co.uk
Night Walk for Edinburgh
Sunday 19 January, 2020 at 3:30 pm - 5:00 pm
An event every day that begins at 3:30 pm, repeating until Friday 31 January, 2020
Monday 20 January, 2020 at 3:30 pm - 5:00 pm
Tuesday 21 January, 2020 at 3:30 pm - 5:00 pm
The Fruitmarket Gallery programmes exhibitions with the best Scottish and International artists and enriches these with a new publication and a wide variety of cultural and educational events.
The Gallery is closed for redevelopment but we are working offsite and organising events and workshops during the closure. Check the calendar above for events or follow us on social media.
Bookmarket: our pop up bookshop and events space is now open in Top level of Waverley Mall, 3/1 Princes Street, Edinburgh, EH1 1BQ
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The Fruitmarket Gallery, 45 Market Street, Edinburgh, EH1 1DF. Tel +44 (0)131 225 2383


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Team, With the holidays upon us and 2018 drawing to a close, it's a great time to look back at this year. Together with you we have reached new heights….


The City of Busselton is located in Western Australia's south west region, and is the perfect base to explore the internationally renowned Margaret River region.
The Mediterranean style climate offers pleasant summer months and mild winter rainfall. The Region is bordered by the Indian and Southern oceans, with the coastline featuring some of the best surfing beaches in Australia, towering forests, sheltered bays and a well established food and wine scene.
Superb restaurants and a thriving arts community with numerous galleries showcasing the works of talented artisans are spread throughout the region. Other attractions include national parks, beaches, caves, wineries, micro-breweries and eco and cultural tourism sites that have credited the Margaret River Region as the most popular tourist destination outside Perth in Western Australia.
With a resident population of around 38,300 people (2017), the Busselton district has one of the fastest growing populations outside the Perth metropolitan area, experiencing an average annual growth rate of around 4% per annum. Based on the annual population growth achieved by the City over the past ten years, the City's population is expected to reach 53,080 in 2026.
Continued rollout of the NBN across the district, enabling high speed internet access supporting new and existing businesses and residents.
The City of Busselton is planning strategically, to meet the challenges of a fast growing community. A profile of the local economy and its demographic make-up can be downloaded by clicking the link below.
An online dashboard showing a summary of local economic and demographic conditions can be viewed here.
Investment Attraction and Marketing: Business attraction (or recruitment) is a means of contributing to the economic vitality of a local area. It is a process of understanding the local economy and identifying gaps for local provision of goods and services by businesses and institutions. It results in a marketing plan aimed at attracting companies and/or institutions that will diversify and build the local economy to match the community's assets and development goals.
Business Retention and Expansion: Business retention and expansion (BR&E) aims to support businesses that are already operating in the Busselton district, with the aim of growing the local economy from within. Its premise is that local entrepreneurs create the companies that bring new wealth and economic growth to a region in the form of jobs, increased revenues, and a vibrant local business sector.
Regional Workforce Development: The ability of a community to supply an appropriately skilled labour force can determine the success of existing businesses and influences the attractiveness of the Busselton district for new business and industry investment.
Industry Development: The Busselton district has a relatively diverse industrial base of small-medium enterprises and this has buffered its regional economy somewhat from a national trend of downturns in some regional economies. Resilience of the regional economy also needs to be maintained through support for new industries, and derivative enterprises from established industries.
Strategic Infrastructure: High quality economic infrastructure is a key enabler for industry growth, productivity and the attraction of investment. Access to appropriate infrastructure is also fundamental to ensuring the whole community can contribute to, and share in, the Region's wealth and quality of life.
Place Making and Activation: Vibrant activity centres are vital for sustainable urban living and are crucial in providing a memorable experience to visitors. Exciting, activated places foster a sense of community and develop a unique, individual character, image and style; evolve and grow with their community and reflect its character, heritage, and future aspirations; nurture small business and local employment; and, facilitate and encourage walking, cycling and public transport usage.
Council also established a new Economic Development Taskforce to help steer an implementation plan and facilitate collaboration between Government, industry and community groups that have a role in local economic development.


Caitlyn Bird, the recipient of the fourth annual Barbara Laronde Award, is an Anishinaabe artist from Noatkamegwanning First Nation.
Caitlyn Bird grew up in Thunder Bay, Ontario and is passionate about the traditional arts; beading. She utilizes traditional techniques and methods as well allows herself to explore in a contemporary setting through color, design, and methods. She obtained a lot of knowledge from her grandparents and Women within her community. In ensuring the survival of knowledge, she continues to share many teachings and hopes to inspire others to be creative and allow that creativeness to guide them through life. Her dreams are to explore the material culture of her people and showcase it where others learn and are encouraged to explore and learn about their history.
She currently goes back in forth to school from Thunder Bay, Ontario to Santa Fe, NM to attend the Institute of American Indian Arts (IAIA) in obtaining her BFA in Museum Studies. She hopes to become a curator and allow access to the community to come into the collections and learn through the elder objects (artifacts) by touching, feeling, smelling, and talking with the item. She hopes through this, Indigenous methodology will be applied within institutions such as museums, art galleries and allow the story of Indigenous people to told by the communities that these elder objects belong to. With attending she also is continuing to network and allow her dreams to become a reality. She shares a lot of her culture wherever she goes but more importantly through her art and the significance it holds to her people and home.