text:
\section{Cartesian Coordinates of [Rh(en)\textsubscript{3}]\textsuperscript{3+}}

Below we give the Cartesian coordinates of [Rh(en)\textsubscript{3}]\textsuperscript{3+} as obtained from the
structure optimization. All values are given in \AA{}.

\begin{verbatim}
C                    -2.56007083    -1.35708902    -0.54082819
H                    -3.10328879    -2.24869940    -0.86238831
H                    -3.08660763    -0.49282315    -0.95265826
C                    -2.47421399    -1.28021359     0.95943603
H                    -3.46617474    -1.19818162     1.40985356
H                    -2.00345265    -2.17345255     1.37692725
N                    -1.16889899    -1.35936049    -1.09006688
H                    -1.21286788    -1.16089489    -2.08908158
H                    -0.80398432    -2.31021408    -1.04209812
N                    -1.63162263    -0.09834763     1.31900179
H                    -1.37112102    -0.16978020     2.30207886
H                    -2.21364735     0.73774283     1.27663904
Rh                   -0.00002018     0.00010660     0.00233207
C                    -0.02679481     2.88428537    -0.60574077
C                     2.35191497    -1.42093764    -1.06582865
H                    -0.58752840     3.79240567    -0.83894857
H                     3.35099977    -1.38928934    -1.50663534
H                     0.88253316     2.90273817    -1.21114681
H                     1.80857443    -2.23169584    -1.55717426
C                     0.29510901     2.80224385     0.86182796
C                     2.41748095    -1.63037140     0.42278229
H                     0.93417380     3.63108091     1.17516908
H                     2.87727091    -2.59042055     0.66869121
H                    -0.61140381     2.84628607     1.47013288
H                     3.01413515    -0.85371141     0.90728810
N                    -0.81152322     1.66769055    -0.98163326
N                     1.61872404    -0.14367645    -1.32703344
H                     1.35244579    -0.11627064    -2.31077538
H                    -0.82072619     1.58682826    -1.99789514
H                     2.27364722     0.63115483    -1.22583125
H                    -1.78870042     1.82488763    -0.73660797
N                     0.96576365     1.49089727     1.12047395
N                     1.02648486    -1.55627954     0.96667280
H                     0.97308157     1.32058012     2.12565099
H                     1.07697248    -1.45807464     1.98016955
H                     1.95183719     1.57898160     0.87656334
H                     0.57692005    -2.46074498     0.82712764
\end{verbatim}

\section{Damping Factor}

In order to investigate the influence of the damping value, we calculated the ROA spectrum of [Rh(en)\textsubscript{3}]\textsuperscript{3+}
for two different damping values. In the first case, we adopted a rather small damping value of 0.0037\,a.u.; in the
second case, a larger value of 0.008\,a.u.~was used. For both damping values, we calculated the spectrum with an excitation
wavelength of 307.66\,nm, \textit{i.e.},under full-resonance conditions, and with an excitation wavelength of 370\,nm.
The four resulting spectra are shown in Fig.~\ref{fig:damping}. As one can see, when an excitation wavelength of 370\,nm
is adopted, the two different damping values lead to virtually identical spectra, and therefore have no influence (\textit{cf.},
the top two spectra in Fig.~\ref{fig:damping}). Under full-resonance conditions, however, the damping value has a rather
large influence on the overall spectral intensity. With larger damping value, the overall intensity is greatly reduced
(up to a factor of about five). However, the relative intensities as well as the overall band shape is much less affected
by changes in the damping value. We therefore conclude that the exact numerical value adopted for the damping value is
not of primary importance as long as one and the same value is used throughout our study. This will suffice to make the
individual spectra comparable to each other.

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.9]{si_figure1.pdf}
\end{center}
\caption{\label{fig:damping}\small ROA spectra of [Rh(en)\textsubscript{3}]\textsuperscript{3+} for two different excitation
wavelengths and two different damping values. A wavelength of 307.66\,nm yields full-resonance spectra.
Note that the individual spectra span different intensity ranges.}
\end{figure}

\section{Origin Dependence}

In order to check whether the methodology followed in this work yields ROA spectra which are origin independent, we shifted
the molecule by 10\,\AA{} in each of the three spatial directions and recalculated the spectrum under full-resonance
conditions. As can be seen by comparing the first and third spectrum in Fig.~\ref{fig:origin_independence}, the spectrum
does literally not change. We therefore conclude that our spectra are not dependent on the choice of origin.

As a further test, we also calculated the ROA spectrum under full-resonance conditions without gauge-including atomic
orbitals (GIAOs), both with the molecule at its original location and with the molecule shifted by 10\,\AA{} in all three
spatial directions. As can be seen from Fig.~\ref{fig:origin_independence}, in our case the use of GIAOs apparently has
no observable effects on the ROA spectrum.

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.9]{si_figure2.pdf}
\end{center}
\caption{\label{fig:origin_independence}\small ROA spectra of [Rh(en)\textsubscript{3}]\textsuperscript{3+} calculated either with or
without gauge-including atomic orbitals. In the upper two spectra, the molecule is placed very close to the origin,
while in the lower two spectra, the molecule is shifted by 10\,\AA{} in all three spatial directions.}
\end{figure}

\section{Example Input File}

Below we give an example \textsc{NWChem} input file as used in this work to calculate the ROA spectra. The nuclear coordinates
have been omitted in this example; in the actual calculations, the distorted structures as generated by \textsc{MoViPac}
have been inserted.

\begin{verbatim}
echo

title "RROA calculation of [Rh(en3)]3+ in vacuum"

memory 2048 mb

start rh_en

geometry units au nocenter noautosym nucleus point
symmetry c1
...
end

basis "ao basis" spherical
 * library def2-TZVP
end

ecp
 Rh library def2-ECP
end

scf
 singlet
 rhf
 direct
end

charge 3

dft
 grid xfine nodisk
 tolerances tight
 convergence energy 1d-8
 convergence density 1d-7
 maxiter 50
 xc pbe0
 disp vdw 3
 direct
 noio
end

set prop:newaoresp 1
set aor:roadata T
set cphf:maxsub 30
set cphf:maxiter 50

property
  response 1 0.14809645
  bdtensor
  convergence 1e-4
  damping 0.0080
  giao
end

task dft property
task dft gradient
\end{verbatim}

\section{Intensities of Normal Modes}

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.33]{si_figure3.pdf}
\end{center}
\caption{\label{fig:intensities1}\small Intensities of normal modes 1 to 30 as a function of excitation wavelength. All intensities
are given in \AA{}$^4$\,/\,amu$^{-1}$.  The vertical line highlights the full-resonance wavelength.}
\end{figure}

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.33]{si_figure4.pdf}
\end{center}
\caption{\label{fig:intensities2}\small Intensities of normal modes 31 to 60 as a function of excitation wavelength. All intensities
are given in \AA{}$^4$\,/\,amu$^{-1}$.  The vertical line highlights the full-resonance wavelength.}
\end{figure}

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.33]{si_figure5.pdf}
\end{center}
\caption{\label{fig:intensities3}\small Intensities of normal modes 61 to 90 as a function of excitation wavelength. All intensities
are given in \AA{}$^4$\,/\,amu$^{-1}$.  The vertical line highlights the full-resonance wavelength.}
\end{figure}

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.33]{si_figure6.pdf}
\end{center}
\caption{\label{fig:intensities4}\small Intensities of normal modes 91 to 105 as a function of excitation wavelength. All intensities
are given in \AA{}$^4$\,/\,amu$^{-1}$. The vertical line highlights the full-resonance wavelength.}
\end{figure}

\end{document}

\section{Introduction}
\label{sec:introduction}

Chiral molecules show a small difference in the intensities of Raman scattered left and right circularly polarized light,
an effect generally denoted as Raman Optical Activity (ROA)\cite{Barron2004a}. This effect was predicted by Barron
and Buckingham in 1971\cite{Barron1971} and first observed experimentally in 1973 by Barron \textit{et al.}\cite{Barron1973}
(this observation was independently confirmed by Hug \textit{et al.}~two years later\cite{Hug1975}). Since the intensity
difference measured in ROA spectroscopy is very small, it took more than twenty years and major improvements in instrumentation
techniques\cite{Spencer1988, Hecht1994, Hug1999} before it was firmly established as the routine analytical method it
is today\cite{Barron2000, Barron2004, Barron2007, Barron2010}. In particular, ROA spectroscopy is often adopted to study
biomolecules in their natural, aqueous environment\cite{Barron2000}. However, ROA studies on chiral metal complexes are
quite scarce\cite{Wu2015, Luber2015}. The first experimental ROA spectrum of a metal complex was published in 2011 by
Johanessen \textit{et al.}\cite{Johannessen2011}, while the first theoretical ROA spectra for chiral transition metal
complexes had already been presented in 2008 by Luber and Reiher\cite{Luber2008}. The same group then presented theoretical
studies on chiral Cobalt complexes\cite{Luber2009, Luber2010a}, and the $\beta$ domain of rat metallothionein, which
features one cadmium and two zinc atoms in its center\cite{Luber2010}. More recently, Humbert-Droz \textit{et al.} published
a combined theoretical and experimental study on the ROA spectrum of $\Lambda$-tris-(ethylenediamine)-rhodium(III)
([Rh(en)\textsubscript{3}]\textsuperscript{3+})\cite{Humbert-Droz2014}. In summary, all these studies have demonstrated
the great potential of ROA spectroscopy for the elucidation of the structure of chiral metal complexes.

It has been found that the ROA signal can be greatly enhanced if the wavelength of the incident laser light is in resonance
with an electronic excitation of the molecule under investigation\cite{Barron2004a}, an effect called resonance ROA (RROA).
A corresponding theory, valid in the case of resonance with a single excited electronic state, was presented in 1996 by
Nafie\cite{Nafie1996}. This theory predicts the resulting spectra to be monosignate and the relative band intensities
identical to the one of the corresponding Raman spectrum. These predictions were confirmed by an experimental study conducted
by Vargek \textit{et al.}~in 1998\cite{Vargek1998}. If several excited electronic states are involved, however, the situation
will be much more complicated; the spectra do not have to be monosignate\cite{Merten2012, Vidal2016} and de-enhancement
effects may occur\cite{Luber2010b}. In 2007, Jensen \textit{et al.}~presented an ansatz to calculate both off- and on-resonance
ROA spectra, which also allowed to include multiple excited electronic states\cite{Jensen2007}. Later, also Luber \textit{et
al.}\cite{Luber2010b}~as well as Barone and coworkers\cite{Baiardi2018} presented other approaches capable of taking
multiple excited electronic states into account.

Unfortunately, there are almost no studies investigating RROA effects in transition metal compounds (the first experimental
RROA spectra for a chiral transition metal compound was presented by Merten \textit{et al.}~in 2010\cite{Merten2010}).
Furthermore, detailed theoretical studies of RROA spectra involving more than one excited electronic state are largely missing.
This paper contributes to filling this gap by presenting an in-depth study of the full-resonance and near-resonance
spectra of [Rh(en)\textsubscript{3}]\textsuperscript{3+}. Thereby it also complements the detailed study published by Humbert-Droz
\textit{et al.}~on the same compound, which focused exclusively on the off-resonance case\cite{Humbert-Droz2014}.

This paper is organized as follows: First, we give a short introduction into the theoretical background of the calculation of
(resonance) ROA spectra. Then, in Section \ref{sec:computational_methodology}, we explain the computational methodology
followed in this study. Afterwards, the results are presented in Section \ref{sec:results}. Finally, a conclusion is given
in Section \ref{sec:conclusion}.

\section{Theoretical Background}
\label{sec:theory}

The (resonance) ROA intensity $\Delta I_p(180^{\circ})$ of a given normal mode $p$ in a backscattering geometry (in which
case the ROA intensity is maximized\cite{Zhu2005}) can be calculated as
\begin{equation}\label{eq:roa-int}
 \Delta I_p(180^{\circ}) = I^{R}_p(180^{\circ}) - I^{L}_p(180^{\circ}) \propto \frac{1}{c} (24\beta(G')^2_p + 8\beta(A)^2_p),
\end{equation}
where $c$ is the speed of light in vacuum, $\beta(G')^2_p$ is the anisotropic invariant of the electric-dipole--magnetic-dipole
polarizability transition tensor, and $\beta(A)^2_p$ is the anisotropic invariant of the electric-dipole--electric-quadrupole
polarizability transition tensor (note that all these quantities are independent of the experimental setup). The latter
two quantities are given by
\begin{equation}
 \beta(G')^2_p = \mathrm{Im}\left( \frac{\mathrm{i}}{2} (3\alpha_{\alpha\beta}
^{(p)}G_{\alpha\beta}^{\prime(p)*} - \alpha_{\alpha\alpha}^{(p)}G_{\beta\beta}^{\prime(p)*})
\right),
\end{equation}
and
\begin{equation}
 \beta(A)^2_p = \mathrm{Re} \left( \frac{1}{2} \omega\alpha_{\alpha\beta}^{(p)}\epsilon_{\alpha\gamma\delta}A_{\gamma\delta\beta}^{(p)*} \right),
\end{equation}
respectively. $\alpha_{\alpha\beta}^{(p)}$ is the $\alpha\beta$-component ($\alpha, \beta = \{x, y, z\}$) of the
electric-dipole--electric-dipole polarizability for the $p$-th normal mode, while $G_{\alpha\beta}^{\prime(p)}$ is the
$\alpha\beta$-component of the electric-dipole--magnetic-dipole polarizability for the $p$-th normal mode. The asterisk
denotes the complex conjugate. $\omega$ is the frequency of the incident light, and $\epsilon_{\alpha\gamma\delta}$ is
the third-rank antisymmetric unit tensor. $A_{\gamma\delta\beta}^{(p)}$ is the $\gamma\delta\beta$-component ($\gamma,
\delta = \{x, y, z\}$) of the electric-dipole--electric-quadrupole polarizability for the $p$-th normal mode. Re and Im
denote the real and imaginary part, respectively. Note that the Einstein summation convention has been invoked for repeated
greek indices.

These polarizability tensors are then calculated as
\begin{equation} \alpha_{\alpha\beta}^{(p)}G_{\alpha\beta}^{\prime(p)} = \left( \frac{\partial \alpha_{\alpha\beta}}{\partial Q_p}\right)_0 \left(\frac{\partial G'_{\alpha\beta}}{\partial Q_p}\right)_0,
\end{equation}
and
\begin{equation}
 \alpha_{\alpha\beta}^{(p)}\epsilon_{\alpha\gamma\delta}A_{\gamma\delta\beta}^{(p)} =  \left( \frac{\partial \alpha_{\alpha\beta}}{\partial Q_p}\right)_0 \epsilon_{\alpha\gamma\delta} \left( \frac{\partial A_{\gamma\delta\beta}}{\partial Q_p} \right)_0
\end{equation}
where $Q_p$ is the normal mode of the $p$-th vibration and the subscript "0" indicates that the partial derivative is to
be evaluated at the molecular equilibrium structure.

In the short-time approximation adopted in the ansatz by Jensen \textit{et al.}\cite{Jensen2007}, $\alpha_{\alpha\beta}$, $G'_{\alpha\beta}$, and $A_{\gamma\delta\beta}$ are
complex quantities:
\begin{equation}
 \alpha_{\alpha\beta} = \alpha_{\alpha\beta}^{(R)} + \mathrm{i}\alpha_{\alpha\beta}^{(I)},
\end{equation}
\begin{equation}
 G'_{\alpha\beta} = G_{\alpha\beta}^{\prime(R)} + \mathrm{i}G_{\alpha\beta}^{\prime(I)},
\end{equation}
and
\begin{equation}
 A_{\gamma\delta\beta} = A_{\gamma\delta\beta}^{(R)} + \mathrm{i}A_{\gamma\delta\beta}^{(I)}.
\end{equation}
The real and imaginary parts are formally calculated from a sum over the electronic states,
\begin{equation}
 \label{eq:start}
 \alpha_{\alpha\beta}^{(R)} = \sum_{n \neq 0}f^{+}(\omega_{n0}, \omega, \Gamma)\mathrm{Re}\left( \left\langle \psi_0 \left| \mu_\alpha \right| \psi_n  \right\rangle \left\langle \psi_n \left| \mu_\beta \right| \psi_0 \right\rangle \right),
\end{equation}
\begin{equation}
 \alpha_{\alpha\beta}^{(I)} = \sum_{n \neq 0}g^{-}(\omega_{n0}, \omega, \Gamma)\mathrm{Re}\left( \left\langle \psi_0 \left| \mu_\alpha \right| \psi_n  \right\rangle \left\langle \psi_n \left| \mu_\beta \right| \psi_0 \right\rangle \right),
\end{equation}
\begin{equation}
 G_{\alpha\beta}^{\prime(R)} = \sum_{n \neq 0}f^{-}(\omega_{n0}, \omega, \Gamma)\mathrm{Im}\left( \left\langle \psi_0 \left| \mu_\alpha \right| \psi_n  \right\rangle \left\langle \psi_n \left| m_\beta \right| \psi_0 \right\rangle \right),
\end{equation}
\begin{equation}
 G_{\alpha\beta}^{\prime(I)} = \sum_{n \neq 0}g^{+}(\omega_{n0}, \omega, \Gamma)\mathrm{Im}\left( \left\langle \psi_0 \left| \mu_\alpha \right| \psi_n  \right\rangle \left\langle \psi_n \left| m_\beta \right| \psi_0 \right\rangle \right),
\end{equation}
\begin{equation}
 A_{\alpha\beta\gamma}^{(R)} = \sum_{n \neq 0}f^{+}(\omega_{n0}, \omega, \Gamma)\mathrm{Re}\left( \left\langle \psi_0 \left| \mu_\alpha \right| \psi_n  \right\rangle \left\langle \psi_n \left| \Theta_{\beta\gamma} \right| \psi_0 \right\rangle \right),
\end{equation}
and
\begin{equation}
 \label{eq:end}
 A_{\alpha\beta\gamma}^{(I)} = \sum_{n \neq 0}g^{-}(\omega_{n0}, \omega, \Gamma)\mathrm{Re}\left( \left\langle \psi_0 \left| \mu_\alpha \right| \psi_n  \right\rangle \left\langle \psi_n \left| \Theta_{\beta\gamma} \right| \psi_0 \right\rangle \right).
\end{equation}
In these equations, $\psi_0$ is the electronic wave function of the ground state, while $\psi_n$ denote the electronic
wave functions of excited states. Furthermore, we require the electric dipole operator
\begin{equation}
 \mu_\alpha = -\sum_{a}r_{a\alpha},
\end{equation}
and the magnetic dipole operator
\begin{equation}
 m_\alpha = -\frac{1}{2}\sum_al_{a\alpha}
\end{equation}
with $l_{a\alpha}$ being the $\alpha$-component of the angular momentum operator for electron $a$. The electric quadrupole
operator
\begin{equation}
  \Theta_{\alpha\beta} = -\frac{1}{2}\sum_a(3r_{a\alpha}r_{a\beta}-\delta_{\alpha\beta}r_a^2) .
\end{equation}
The sums run over all electrons $a$. Finally, in equations (\ref{eq:start})\,--\,(\ref{eq:end}) we have the two line
shape functions
\begin{equation}
 f^\pm(\omega_{n0}, \omega, \Gamma) = \frac{\omega_{n0} - \omega}{(\omega_{n0} - \omega)^2 + \Gamma^2} \pm \frac{\omega_{n0} + \omega}{(\omega_{n0} + \omega)^2 + \Gamma^2},
\end{equation}
and
\begin{equation}
 g^\pm(\omega_{n0}, \omega, \Gamma) = \frac{\Gamma}{(\omega_{n0} - \omega)^2 + \Gamma^2} \pm \frac{\Gamma}{(\omega_{n0} + \omega)^2 + \Gamma^2}.
\end{equation}
Here, $\omega_{n0}$ is the excitation frequency between the ground state and the $n$-th excited state, $\omega$ is the
frequency of the incident light, and  $1/\Gamma$ can be interpreted as a lifetime of the excitations. Note that by
setting $\Gamma = 0$ in the above equations, the standard non-resonant formulation of ROA is recovered.

\section{Computational Methodology}
\label{sec:computational_methodology}

In their study\cite{Humbert-Droz2014}, Humbert-Droz \textit{et al.}~analyzed in detail the different configurations and
conformations possible for [Rh(en)\textsubscript{3}]\textsuperscript{3+}. Therefore, we will not attempt to repeat such
an analysis in this work. Comparison with experimental spectra showed that the so-called $\Lambda(\delta\delta\delta)$
conformation (see Fig.~\ref{fig:struct}; \textit{cf.}~Ref.~\citenum{connelly2005} for more information about this nomenclature)
to be the dominant conformation\cite{Humbert-Droz2014}. Therefore, we will use this sole conformation throughout our study
and refer to it without the prefix ``$\Lambda(\delta\delta\delta)$''.

The structure of [Rh(en)\textsubscript{3}]\textsuperscript{3+} was first fully optimized in C\textsubscript{1} symmetry
(see the Supporting Information for the Cartesian coordinates of the optimized structure). This structure optimization
and all subsequent calculations were done with a development version of \textsc{NWChem} (revision 29384)\cite{Valiev2010},
using density functional theory in its Kohn--Sham formulation\cite{Kohn1965}.  (It has been verified that the ROA spectra
obtained with this specific version of \textsc{NWChem} are identical to the ones obtained from the officially released
\textsc{NWChem} 6.8.1 by recalculating the spectra at 800\,nm and 307.66\,nm incident wavelength, respectively.) In all
calculations, the PBE0 exchange--correlation functional\cite{Adamo1999} and the def2-TZVP basis set\cite{Weigend2005}
(for all atoms) were chosen. For Rhodium, the ECP-28 effective core potential\cite{Andrae1990} was employed in order to
take scalar relativistic effects into account. Furthermore, advantage was taken of the resolution-of-the-identity technique
for the Coulomb integrals using the corresponding auxiliary basis sets\cite{Weigend2006}. Finally, the D3 semiempirical
dispersion corrections\cite{Grimme2010} were adopted (with the zero-damping function).

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.3]{figure1.png}
\end{center}
\caption{\label{fig:struct}\small Structure of [Rh(en)\textsubscript{3}]\textsuperscript{3+} investigated in this work.}
\end{figure}

All ROA spectra were calculated in the double harmonic approximation with the program suite \textsc{MoViPac}\cite{Weymuth2012},
relying on a seminumerical differentiation scheme with a three-step central difference formula (adopting a step size of
0.01\,bohr). The necessary energy gradients and polarizability tensors were calculated analytically with the version of
\textsc{NWChem} mentioned above relying on the ansatz by Jensen \textit{et al.}\cite{Jensen2007}. For all spectra, a
value of 0.008\,a.u.~was adopted for the damping parameter $\Gamma$
(we refer to the Supporting Information for a comparison of different damping values). Furthermore, great care was taken
to ensure origin independence of the spectra by placing all structures close to the origin and using gauge-including
atomic orbitals; note that for resonance ROA calculations, gauge-including atomic orbitals alone cannot always guarantee
complete origin independence\cite{Vidal2016} (see the Supporting Information for calculations verifying the gauge independence
in our case). Electronic (singlet) excitations were calculated with time-dependent density functional theory employing the
methodology detailed above. This yielded the lowest excitation at 307.66\,nm. Therefore, this wavelength was used in order
to calculate the full-resonance spectrum. In order to investigate how this spectrum changes with different wavelengths,
spectra were also calculated for the following wavelengths: 290\,nm, 303\,nm, 305\,nm, 310\,nm, 315\,nm, 320\,nm, 325\,nm,
330\,nm, 350\,nm, 370\,nm, 410\,nm, 500\,nm, 600\,nm, 700\,nm, and 800\,nm. Solvent effects were not considered in
this study, as this allows us to analyze the genuine ROA spectra and of [Rh(en)\textsubscript{3}]\textsuperscript{3+}
itself and the resonance effects stemming from this isolated structure. Furthermore, as we will see below, solvent effects
do not appear to play a major role in this ROA spectrum. A complete input file used to calculate the ROA spectra is given
in the Supporting Information.

All line spectra were convoluted with a Lorentzian band having a full width at half maximum of 15\,cm$^{-1}$. This line
broadening as well as the subsequent analysis of all ROA spectra was carried out with \textsc{Mathematica} 11.2.0\cite{mathematica11_2}.
Plots of ROA spectra were created with \textsc{Matplotlib} 2.2.3\cite{Hunter2007}. In these plots, the intensities of the
line spectra were always scaled by a factor of 0.04. Pictures of molecular structures and molecular orbitals were generated
with \textsc{Vmd} 1.9.2\cite{Humphrey1996, tachyon}. Representations of normal modes were produced with \textsc{Jmol}
14.6.4\cite{jmol}.

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

\subsection{Electronic Excitations}
\label{subsec:excitations}

To assess at which wavelength a full-resonance ROA spectrum can be calculated, we need to know the energies of the
electronic excitations. The calculated energies and wavelengths of the lowest five electronic (singlet) excitations are
collected in Table~\ref{tab:energies}.

\begin{table}[H]
\renewcommand{\baselinestretch}{1.0}
\renewcommand{\arraystretch}{1.0}
\caption{\label{tab:energies}\small Excitation energies $E$ and corresponding wavelengths $\lambda$ of the lowest five
electronic excitations as well as the molecular orbital transitions involved with a contribution larger than 0.4\,a.u.~to
the excitation vector (transitions with a contribution larger than 0.6\,a.u.~are printed in boldface).}
\begin{center}
\begin{tabular}{l r r l} \hline \hline
No.   & $E$ / eV   & $\lambda$ / nm   & Dominating Transitions                                                \\
\hline
1     & 4.0299     & 307.6607         & \textbf{HOMO $\rightarrow$ LUMO},     \\
      &            &                  & HOMO\,$-$\,1 $\rightarrow$ LUMO\,+\,1, \\
      &            &                  & HOMO\,$-$\,2 $\rightarrow$ LUMO \\
2     & 4.0301     & 307.6455         & \textbf{HOMO $\rightarrow$ LUMO\,+\,1}, \\
      &            &                  & HOMO\,$-$\,1 $\rightarrow$ LUMO, \\
      &            &                  & HOMO\,$-$\,2 $\rightarrow$ LUMO\,+\,1 \\
3     & 4.0864     & 303.4069         & \textbf{HOMO\,$-$\,1 $\rightarrow$ LUMO\,+\,1},\\
      &            &                  & \textbf{HOMO\,$-$\,2 $\rightarrow$ LUMO}             \\
4     & 4.6548     & 266.3577         & n/a                                                                    \\
5     & 4.6551     & 266.3406         & n/a                                                                    \\
\hline
\hline
\end{tabular}
\renewcommand{\baselinestretch}{1.0}
\renewcommand{\arraystretch}{1.0}
\end{center}
\end{table}

The lowest excitation occurs at a wavelength of 307.66\,nm. We therefore adopted this wavelength in the calculation of
the full-resonance ROA spectrum. However, we can also see from Table~\ref{tab:energies} that the second excitation occurs
at almost the same energy. Hence, both states will simultaneously be involved in resonance effects. As we will understand
below, this fact leads to very interesting effects in the ROA spectra. Moreover, as the third lowest state (at 303.41\,nm
above the ground state) is also very close to the aforementioned two states, it is likely that also this state will usually
be involved in resonance effects, albeit to a somewhat lesser extent. Resonance effects with states beyond the third lowest
excited state will be negligible, since the excitation wavelengths are at least 50\,nm larger compared to the lowest
excitation (this is also indicated by the findings of Ref.~\citenum{Luber2010b}).

It is also instructive to analyze the molecular orbitals (MOs) involved in the three lowest electronic excitations. The
dominant MO transitions involved in these excitations are also listed in Table~\ref{tab:energies} and the respective MOs
are shown in Fig.~\ref{fig:orbitals}. The first excitation is mostly a transition from the highest occupied MO (HOMO) to
the lowest unoccupied MO (LUMO). While the HOMO is localized at the metal center, the LUMO is more extended, covering also
four of the nitrogen atoms. One can therefore interpret this transition as moving some electronic charge from the rhodium
atom to the nitrogen atoms. This interpretation remains valid when considering also the other two dominating transitions,
namely the transitions between HOMO\,$-$\,1 and LUMO\,+\,1, and between HOMO\,$-$\,2 and LUMO. HOMO\,$-$\,2 and HOMO\,$-$\,1
are mostly localized at the central rhodium atom, while LUMO\,+\,1 is, like the LUMO, more extended, also covering several
nitrogen atoms. Following the same argumentation, we find that also the second and third lowest transition represent a
movement of charge from the metal center to the adjacent nitrogen atoms.

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.18]{figure2.png}
\end{center}
\caption{\label{fig:orbitals}\small Molecular orbitals of [Rh(en)\textsubscript{3}]\textsuperscript{3+} involved in the
lowest three electronic excitations.}
\end{figure}

\subsection{ROA Spectra}

After having analyzed the electronic excitations playing a role in the (resonance) ROA spectra, we are now in a position
to investigate these spectra.

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.4]{figure3.pdf}
\end{center}
\caption{\label{fig:comparison}\small Comparison of the experimental (top panel) and theoretical (bottom panel) off-resonance
spectra of [Rh(en)\textsubscript{3}]\textsuperscript{3+} as obtained by Humbert-Droz \textit{et al.}\cite{Humbert-Droz2014}
to the off-resonance spectrum obtained in this work (middle panel).}
\end{figure}

First, we compare our results to the ones obtained by Humbert-Droz \textit{et al.}\cite{Humbert-Droz2014}. In
Fig.~\ref{fig:comparison}, we show both the experimental as well as the theoretical off-resonance spectrum obtained by
these authors and compare it to the off-resonance spectrum obtained in this work using an excitation wavelength of 800\,nm.
In the work by Humbert-Droz \textit{et al.}\cite{Humbert-Droz2014}, an excitation wavelength of 532\,nm was adopted. As
we have elaborated in the previous section, this wavelength is in fact far away from any electronic excitations. We will
see below that for wavelengths longer than 350\,nm, resonance effects do not play a major role anymore in the ROA spectrum
of [Rh(en)\textsubscript{3}]\textsuperscript{3+}. The experimental spectrum is, therefore, effectively an off-resonance
spectrum and it is possible to compare the spectra shown in Fig.~\ref{fig:comparison} to each other, even though they have
been obtained with different excitation wavelengths. The apparent ruggedness of the two spectra taken from Ref.~\citenum{Humbert-Droz2014} is
in part due to the fact that it was extracted directly from the paper by Humbert-Droz \textit{et al.}\cite{Humbert-Droz2014},
which introduced minor artifacts. Also note that Humbert-Droz \textit{et al.}~adopted a different computational methodology;
most notably, instead of the PBE0 density functional, they used the B3LYP\cite{Becke1993, Lee1988, Vosko1980} exchange--correlation
functional (as implemented in \textsc{Gaussian}\cite{gaussian09}). Therefore,
some differences between the theoretical spectra are to be expected.

As we can see, the three spectra agree well with each other, in particular the two theoretical spectra. Below
1000\,cm$^{-1}$, the only notable differences between these two spectra are slightly different band shapes in the region
between 200\,cm$^{-1}$ and 300\,cm$^{-1}$, and the +/$-$ couplet at about 750\,cm$^{-1}$, which is visible in the spectrum
calculated by us, but not in the theoretical spectrum obtained by Humbert-Droz \textit{et al.}\cite{Humbert-Droz2014}.
Instead of such a couplet, however, we find two weakly negative peaks at about 800\,cm$^{-1}$ which cannot be identified
in our spectrum. Also above 1000\,cm$^{-1}$, the two theoretical spectra agree well. The only difference is in the region
between 1650\,cm$^{-1}$ and 1750\,cm$^{-1}$. Here, we find two negative bands in our spectrum, while there is a rather
strong $-$/+ couplet in the spectrum reported in Ref.~\citenum{Humbert-Droz2014} (note that in the experimental spectrum,
there is a +/$-$ couplet in this spectral range). However, also in the spectrum obtained in our study, we find a normal
mode with a strongly positive ROA intensity in this wavenumber range. However, it is completely covered up by the near-lying
modes which are all associated with negative ROA intensity. If this mode would be shifted to slightly larger wavenumbers,
also our spectrum would feature a $-$/+ couplet. This highlights the fact that the overall band shape obtained in ROA
spectra is highly sensitive to such comparatively small changes (\textit{cf.}, also Ref.~\citenum{Herrmann2006}). Therefore,
in all ROA spectra reported in this study, we will always also show the individual normal modes.

At wavenumbers below 1000\,cm$^{-1}$, the experimentally measured intensities are clearly larger than the calculated ones,
but the wavenumbers of the measured peaks matches those of the calculated peaks almost perfectly. The +/$-$ couplet at
about 750\,cm$^{-1}$, which we find in our spectrum, cannot be identified in the experimental spectrum. Instead, the
experimental spectrum features a broad, negative band at around 800\,cm$^{-1}$. This is consistent with what we find in
the theoretical spectrum obtained by Humbert-Droz \textit{et al.}\cite{Humbert-Droz2014}. At wavenumbers above
1000\,cm$^{-1}$, the relative intensity ratio between the two strongly positive peaks at about 1200\,cm$^{-1}$ and
1300\,cm$^{-1}$, respectively, is correct reproduced in our spectrum, while it is reversed in the spectrum given in
Ref.~\citenum{Humbert-Droz2014}. However, these are minor differences, and the overall agreement of all three spectra is
very satisfactory.

Now, we can analyze all our ROA spectra and how they are affected by resonances with electronic states. The individual
spectra are shown in Figs.~\ref{fig:spectra_near_res}\,--\,\ref{fig:spectra_pre_res}. The full-resonance spectrum (at
307.66\,nm, see top spectrum in Fig.~\ref{fig:spectra_near_res}) shows bisignate peaks, \textit{i.e.}, it is not monosignate,
as is often the case for RROA spectra. This is because RROA spectra are only monosignate in the case of resonance with
exactly one excited electronic state. In our case, however, we have seen above that the three lowest electronic excitations
are so close to each other in energy that they all contribute to the resonance effects observed. It is well known that in
such situations, bisignate RROA spectra can result\cite{Merten2012, Vidal2016, Luber2010b}. The intensity of the
full-resonance spectrum reaches absolute values of up to 0.12\,\AA{}$^4$\,amu$^{-1}$ (at 520\,cm$^{-1}$). This is larger
than the maximum intensity in the case of the spectrum recorded at 800\,nm (which is a pure off-resonance spectrum) where
the maximum absolute intensity reaches only about 0.005\,\AA{}$^4$\,amu$^{-1}$. We  therefore clearly see an enhancement
of the ROA intensity through the resonance with electronically excited states, as is expected.

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.4]{figure4.pdf}
\end{center}
\caption{\label{fig:spectra_near_res}\small ROA spectra for five different wavelengths at or close to resonance with
the first three excited electronic states (in descending order from top to bottom). The two bands primarily analyzed in
this study are highlighted.}
\end{figure}

Increasing the excitation wavelength from 307.66\,nm to 310\,nm yields a spectrum which looks very similar to the
full-resonance spectrum (see the second spectrum from above in Fig.~\ref{fig:spectra_near_res}). All band shapes and
relative intensity differences are basically the same, but the overall intensity is reduced, reaching a maximum value of
about 0.095\,\AA{}$^4$\,amu$^{-1}$ at 520\,cm$^{-1}$. Increasing the excitation wavelength to 315\,nm, we find that the
intensity is further reduced; the maximum value is now roughly 0.048\,\AA{}$^4$\,amu$^{-1}$ (still at 520\,cm$^{-1}$).
Relative intensity differences are now strong enough to have a visible influence on the overall band shapes. For example,
the band at 1660\,cm$^{-1}$ does no longer appear as a single sharp peak, but is split into two peaks.

In the ROA spectrum at 320\,nm, the maximum intensity is no longer found for the band at 520\,cm$^{-1}$ but for the one at
about 760\,cm$^{-1}$. Its absolute intensity (about 0.03\,\AA{}$^4$\,amu$^{-1}$) is roughly as large as the one for the
peak at about  1350\,cm$^{-1}$. When compared to the full-resonance spectrum at 307.66\,nm, the overall spectrum has
changed significantly. While certain spectral regions (\textit{e.g.}, between 1400\,cm$^{-1}$ and 1500\,cm$^{-1}$) are
almost the same in both spectra, other regions (\textit{e.g.}, between 200\,cm$^{-1}$ and 300\,cm$^{-1}$) differ
significantly. In the case of the full-resonance spectrum, we see a strongly positive band between 200\,cm$^{-1}$ and
300\,cm$^{-1}$. The spectrum at 320\,nm features a very weak positive peak and a slightly stronger, negative band in the
same spectral region. Also, at about 700\,cm$^{-1}$, we can identify a positive band in the spectrum at 320\,nm. This
was completely masked in the full-resonance spectrum (since the corresponding normal mode has a much weaker, albeit
also positive, intensity in the full-resonance case). Furthermore, also between 1200\,cm$^{-1}$ and 1300\,cm$^{-1}$ the
two spectra differ strongly. In the full-resonance case, we find here two strongly positive peaks, followed by a weaker
negative one. At 320\,nm, we find only one positive peak, much less intense, while the negative peak has gained intensity.
The normal mode at 1252\,cm$^{-1}$, causing the second strongly positive peak in the spectrum at 307.66\,nm, has almost
zero ROA intensity at 320\,nm, which is why it can no longer be clearly identified in the corresponding spectrum.
The spectrum with an excitation wavelength of 325\,nm looks very similar to the one at 320\,nm, but we can still find a
few distinct changes. For example, the most intense peak is now found at about 1350\,cm$^{-1}$. The band at about
520\,cm$^{-1}$ has further lost intensity.

At an excitation wavelength of 330\,nm, the overall intensity is further reduced (\textit{cf.}, Fig.~\ref{fig:spectra_off_res}).
We can still find some significant changes in the overall band shape. In particular, the positive band at 520\,cm$^{-1}$
has almost completely vanished at 330\,nm. At an excitation wavelength of 350\,nm, one can see that the overall intensity
continues to be reduced. Compared to the spectrum at 330\,nm, however, the overall band shape is conserved. In all
spectra with excitation wavelengths larger than 350\,nm, we find well-conserved band shapes. Therefore, one can say that
already at 350\,nm, the shape of the RROA spectrum is identical to the one of a  pure off-resonance spectrum at 800\,nm.
However, the overall intensity is continuously decreasing with increasing excitation wavelength. This might be due to
very weak resonance effects (absorption tail).

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.38]{figure5.pdf}
\end{center}
\caption{\label{fig:spectra_off_res}\small ROA spectra for different off-resonant wavelengths. Note that for wavelengths
larger than 410\,nm, a smaller intensity range is plotted for the sake of clarity. The two bands primarily analyzed in
this study are highlighted.}
\end{figure}

An interesting observation can be made for the normal mode at around 720\,cm$^{-1}$. We find that the intensity of this
mode increases when moving away from resonance, until the highest intensity is observed at 320\,nm. For larger
wavelengths, the intensity decreases again. When examining the intensities of the individual normal modes closer,
we find that such a behavior is found for many of the normal modes (see Supporting Information). The reason for this
intricate behavior is not fully clear. However, it is known that resonance with multiple excited states can lead to
de-enhancement effects and concomitantly complex line shapes\cite{Luber2010b}. We could hypothesize that the third
excited electronic state has a de-enhancing effect on the ROA spectrum. At longer wavelengths, the third excited state
(303\,nm excitation wavelength) is less important than the first two, and concomitantly its de-enhancing effect should
be diminished. It is, therefore, interesting to see  how the spectrum looks like at shorter wavelengths, when this third
state should be more important.

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.4]{figure6.pdf}
\end{center}
\caption{\label{fig:spectra_pre_res}\small ROA spectra for three pre-resonance wavelengths.}
\end{figure}

Hence, we also calculated spectra at 305\,nm, 303\,nm, and 290\,nm (see Fig.~\ref{fig:spectra_pre_res}). Overall, all
these spectra have a rather large resemblance to the spectrum at full resonance, but it can be argued (based on visual
inspection), that the spectrum at 325\,nm is more similar to the spectrum at 307.66\,nm than the spectrum at 290\,nm (note
that both off-resonant wavelengths have roughly the same distance to the resonant wavelength). Therefore, we might conclude
that, as expected, the third excited state contributes much more to the spectrum at 290\,nm and therefore leads to bigger
changes. Even though the spectrum at 305\,nm looks very similar to the spectrum at full resonance, some distinct differences
can be identified. Most prominently, we can identify a weakly positive band at around 1650\,cm$^{-1}$, which is not visible
at full resonance (although also in this case, there are normal modes in this spectral region associated with positive ROA
intensity). This positive band is further accentuated at smaller wavelengths, and at 290\,nm we find a strong $+/-$ couplet
in this region. When comparing the spectrum at 290\,nm to the one at 307.66\,nm, we find many differences. While many bands
have decreased in intensity (as one would expect based on our hypothesis of a de-enhancing effect of the third lowest
electronic excitation), there are also many peaks which grow in intensity. For example, the negative peak at around
1180\,cm$^{-1}$ and the two positive peaks between 1200\,cm$^{-1}$ and 1250\,cm$^{-1}$. Hence, the third excited state
has not always de-enhancing effects on all normal modes, but the relation appears to be more complicated.

We have seen that the band at 520\,cm$^{-1}$ changes significantly when increasing the excitation wavelength; starting
as the strongest peak in the entire spectrum, it becomes continuously weaker until it vanishes at 350\,nm. Moreover,
the negative peak at about 1350\,cm$^{-1}$ is more or less unaffected. It is interesting, therefore, to investigate these
two peaks more closely. An analysis in terms of so-called group coupling matrices (GCMs) as proposed by Hug\cite{Hug2001}
provides an appealing means of assigning the total ROA intensity of a given normal mode to parts of a molecule. For such
an analysis, the proper choice of these parts is crucial. In our case, we adopt the rather fine-grained partitioning of
[Rh(en)\textsubscript{3}]\textsuperscript{3+} as shown in Fig.~\ref{fig:gcm_partitioning}; note that also Humbert-Droz
\textit{et al.}~have found this partitioning to be useful for some of the normal modes they analyzed\cite{Humbert-Droz2014}.

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.25]{figure7.png}
\end{center}
\caption{\label{fig:gcm_partitioning}\small Groups of [Rh(en)\textsubscript{3}]\textsuperscript{3+} adopted for all
group coupling matrices.}
\end{figure}

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.2]{figure8.png}
\end{center}
\caption{\label{fig:normal_modes}\small Normal modes selected for closer analysis. The wavenumber of each normal mode is
given below its representation.}
\end{figure}

The band at about 520\,cm$^{-1}$ is composed of two normal modes.  The first normal mode (at 519.3\,cm$^{-1}$) represents
a symmetric stretching vibration of all six Rh--N bonds, while the second mode (at 521.7\,cm$^{-1}$) describes a more
complicated torsional motion of the three ligands (see Fig.~\ref{fig:normal_modes}). To a good approximation, we can
neglect the contribution of this second mode to the positive band at around 520\,cm$^{-1}$, since its intensity is always
about an order of magnitude smaller than the one of the first mode. We are therefore left with the analysis of the mode
at 519.3\,cm$^{-1}$. Its GCMs for different excitation wavelengths are shown in Fig.~\ref{fig:gcm_519}.

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.18]{figure9.pdf}
\end{center}
\caption{\label{fig:gcm_519}\small Group coupling matrices for the normal mode at 519.3\,cm$^{-1}$ for four different
excitation wavelengths (top left: 307.66\,nm; top right: 315\,nm; bottom left: 320\,nm; bottom right: 350\,nm). Positive
contributions are represented by filled circles, while negative contributions are drawn as empty circles. The radius of
the circle is proportional to the size of the corresponding GCM matrix element.}
\end{figure}

As we can see, at 307.66\,nm almost all parts of the molecule contribute to the positive ROA intensity. Only the coupling
between the CH\textsubscript{2} group C1a and NH\textsubscript{2} group N3a leads to weakly negative ROA intensity, but
this can safely be neglected. The biggest contributions to the overall ROA intensity are clearly given by the six
NH\textsubscript{2} groups as well as couplings between them. In fact, these couplings lead to stronger contributions
than the individual NH\textsubscript{2} groups provide, and couplings between adjacent NH\textsubscript{2} groups are not
stronger than couplings between distant NH\textsubscript{2} groups, but not all couplings are equally strong: The coupling
of N1a and N2b, N1b and N3a, and N2a and N3b are clearly weaker than the other couplings. From Fig.~\ref{fig:gcm_partitioning},
we see that these combinations of NH\textsubscript{2} groups are always forming an N--Rh--N angle of about 180$^\circ$,
while for any other two NH\textsubscript{2} groups, the N--Rh--N angle is about 90$^\circ$. Hence, we conclude that
the weaker couplings must be due to this specific spatial arrangement.

For the larger excitation wavelength of 315\,nm, these strong contributions have become significantly weaker, but they
still play the dominant role. In particular, the "weak" couplings between NH\textsubscript{2} groups (\textit{i.e.}, the
couplings between N1a and N2b, between N1b and N3a, and between N2b and N3a) are greatly reduced, being now not more important
than any of the other couplings. Several couplings leading to negative ROA intensity are visible now, most notably the
coupling between N2a and N3b (which contributed much positive ROA intensity at full resonance).

At 320\,nm, the couplings between the NH\textsubscript{2} groups are further reduced, and the intensity which the individual
NH\textsubscript{2} groups provide directly (\textit{i.e.}, not through couplings) can be safely neglected. All weak
NH\textsubscript{2} couplings are now clearly negative; in fact, they are the most negative contributions of the entire
GCM. Together with the remaining, weaker negative couplings, they can almost completely cancel out all positive contributions.

Finally, at 350\,nm, no dominating parts can be clearly identified in the GCM. Positive and negative contributions
cancel each other exactly. At even larger excitation wavelengths, the situation remains essentially unchanged, and the
overall intensity is always zero. In summary, we have seen that the strongly positive ROA intensity under resonance conditions
originates from the NH\textsubscript{2} groups. These contributions quickly vanish when proceeding to larger excitation
wavelengths, concomitantly reducing the total intensity of the corresponding normal mode.

The negative band at about 1350\,cm$^{-1}$ results from three normal modes, at 1347.3\,cm$^{-1}$, at 1350.4\,cm$^{-1}$,
and at 1352.5\,cm$^{-1}$. All three modes represent a twisting motion of the NH\textsubscript{2} and CH\textsubscript{2}
groups (\textit{cf.}, Fig.~\ref{fig:normal_modes}). For the negative band around 1350\,cm$^{-1}$, the mode at 1347.3\,cm$^{-1}$
can be neglected to a first approximation, as its intensity is never larger than 0.02\,\AA{}$^4$\,amu$^{-1}$, while the other
two modes have intensities around 0.36\,\AA{}$^4$\,amu$^{-1}$ under resonance conditions. For the mode at 1350.4\,cm$^{-1}$,
the origin of the total ROA intensity is difficult to identify (see Fig.~\ref{fig:gcm_1350}). Under resonance conditions,
many couplings have roughly the same total strength. However, one can say that the most dominant contributions originate
from the parts N2b\,--\,C3b, and couplings of these parts with other parts of the molecule. When going to 315\,nm, this
general pattern remains unaffected, but many of the couplings are reduced in absolute value. However, the positive couplings
appear to be reduced more than the negative couplings; hence, the total intensity even decreases to a certain degree. At
320\,nm, almost all positive couplings between parts of N2b\,--\,C3b have vanished. At even larger wavelengths, the same
general trend is observed: while the coupling pattern remains largely unaffected, the absolute intensities of all couplings
diminish somewhat. Hence, the total intensity of this mode grows continuously weaker (albeit this decrease is by far not
as strong as in the case of the band at around 520\,cm$^{-1}$). For the third mode of this band (at 1352.5\,cm$^{-1}$),
the same general trend is valid (\textit{cf.}, Fig.~\ref{fig:gcm_1352}). The only difference is that under resonance
conditions, most of the dominant coupling elements are among parts N1a\,--\,C1b.

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.18]{figure10.pdf}
\end{center}
\caption{\label{fig:gcm_1350}\small Group coupling matrices for the normal mode at 1350.4\,cm$^{-1}$ for four different
excitation wavelengths (top left: 307.66\,nm; top right: 315\,nm; bottom left: 320\,nm; bottom right: 350\,nm). Positive
contributions are represented by filled circles, while negative contributions are drawn as empty circles. The radius of
the circle is proportional to the size of the corresponding GCM matrix element.}
\end{figure}

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.18]{figure11.pdf}
\end{center}
\caption{\label{fig:gcm_1352}\small Group coupling matrices for the normal mode at 1352.5\,cm$^{-1}$ for four different
excitation wavelengths (top left: 307.66\,nm; top right: 315\,nm; bottom left: 320\,nm; bottom right: 350\,nm). Positive
contributions are represented by filled circles, while negative contributions are drawn as empty circles. The radius of
the circle is proportional to the size of the corresponding GCM matrix element.}
\end{figure}

Inspecting the GCMs shown in Figs.~\ref{fig:gcm_519}, \ref{fig:gcm_1350}, and \ref{fig:gcm_1352}, we also see that the
rhodium atom never contributes significantly to the observed intensities, neither directly, nor through couplings with
any of the other groups. We note that Humbert-Droz \textit{et al.}~have found the same result for the bands they analyzed
in greater detail\cite{Humbert-Droz2014}, albeit only in the off-resonant case.

When analyzing the electronic excitations involved in the ROA spectra (see above), we find that electronic charge is moved
from the metal center towards the NH\textsubscript{2} groups. It is therefore not too surprising that in the band around
520\,cm$^{-1}$ (which changed significantly when approaching larger excitation wavelengths), most of the intensity under
resonance conditions originates from exactly these NH\textsubscript{2} groups. We may hypothesize that any normal mode is
particularly affected by resonance effects if the NH\textsubscript{2} groups are involved to a large extent. In order to
elaborate on this, we calculated the amount to which these groups are involved in every normal mode. Furthermore, for
every single normal mode, we identified the maximum as well as the minimum absolute intensity observed. The data for
selected normal modes is shown in Table~\ref{tab:intensities}.

First of all, we see that the normal mode at 519.3\,cm$^{-1}$ features a rather large involvement of the NH\textsubscript{2}
groups. This would also be expected, since this normal mode essentially represents a stretching vibration of all six
Rh--N bonds. As already mentioned above, this mode features a large intensity under full resonance conditions (in fact,
the highest intensity is even observed at a slightly shorter wavelength), and then collapses to a peak which is hardly
identifiable at very large excitation wavelengths. Similarly, the almost constant intensity of the two normal modes at
1350.4\,cm$^{-1}$ and 1352.5\,cm$^{-1}$ can be clearly seen. Since these normal modes represent a twisting motion of both
the CH\textsubscript{2} and the NH\textsubscript{2} groups, the latter groups have a much lower contribution to the total
vibration.

Next, we study the two CH\textsubscript{2} scissoring vibrations at 1494.2\,cm$^{-1}$ and 1496.6\,cm$^{-1}$, respectively.
Both normal modes have very similar wavenumbers and describe basically identical vibrations, albeit of different
CH\textsubscript{2} groups. Naturally, the NH\textsubscript{2} groups do almost not participate in either of the two
normal modes. However, the first of these normal modes (at 1494.2\,cm$^{-1}$) is strongly enhanced when moving to larger
excitation wavelengths, reaching a maximum at 370\,nm, while the other normal mode is much less affected and follows
exactly the opposite trend, featuring the highest intensity at the shortest excitation wavelength. This intriguing fact
can also be seen in the group coupling matrices of these two normal modes, shown in Fig.~\ref{fig:gcm_1500}. Note that
the coupling matrix of the first normal mode changes significantly and in a qualitative fashion between 350\,nm and 370\,nm.
For example, the coupling between the two CH\textsubscript{2} groups C1a and C2b (\textit{cf.}, Fig.~\ref{fig:gcm_partitioning})
is negative at 350\,nm, but it becomes strongly positive at 370\,nm. Likewise, the coupling between groups C2b and C3a
changes from a negative contribution at 350\,nm to a positive contribution at 370\,nm. Since these contributions are also
rather large in magnitude (in fact, they are the largest couplings across the entire GCM), the total intensity of this
normal mode is significantly increased. In contrast to this, the corresponding GCMs for the second normal mode are almost
identical to each other. Furthermore, this example clearly shows that the amount of NH\textsubscript{2} involvement in
a given normal mode alone is not sufficient as a proxy for the prediction of how strongly this mode will be affected by
resonance effects. Clearly, the first of these two modes is strongly affected by resonance effects, but the NH\textsubscript{2}
groups are almost not involved in its vibration.

\begin{landscape}
\begin{table}[H]
\renewcommand{\baselinestretch}{1.0}
\renewcommand{\arraystretch}{1.0}
\caption{\label{tab:intensities}\small Wavenumbers $\omega$, maximum intensities $I_{\mathrm{max}}$ and corresponding
excitation wavelengths $\lambda_{\mathrm{max}}$, minimum intensities $I_{\mathrm{min}}$ and corresponding excitation
wavelengths $\lambda_{\mathrm{min}}$, the amount to which the NH\textsubscript{2} groups are involved and a description
for selected normal modes.}
\begin{center}
\begin{tabular}{r r r r r r r l} \hline \hline
$\omega$ / cm$^{-1}$   & $I_{\mathrm{max}}$ / \AA{}$^4$\,amu$^{-1}$ & $\lambda_{\mathrm{max}}$ / nm & $I_{\mathrm{min}}$ / \AA{}$^4$\,amu$^{-1}$ & $\lambda_{\mathrm{min}}$ / nm & $I_{\mathrm{max}}$ / $I_{\mathrm{min}}$ & NH\textsubscript{2} Involvement & Description of Vibration \\
\hline
 519.3	 & 3.299287	& 303	& 0.004048	& 700	   &  815.04	& 0.77	& Rh--N stretching \\
1350.4	 & 0.372716	& 315	& 0.064229	& 700	   &    5.80	& 0.39	& CH\textsubscript{2} twisting, NH\textsubscript{2} twisting \\
1352.5	 & 0.394029	& 320	& 0.06484	& 700	   &    6.08	& 0.38	& CH\textsubscript{2} twisting, NH\textsubscript{2} twisting \\
1494.2	 & 0.060442	& 370	& 4.2\,$\cdot\,10^{-5}$	& 303	   & 1439.10	& 0.10	& CH\textsubscript{2} scissoring \\
1496.6	 & 0.04483	& 290	& 0.011166	& 700	   &    4.01	& 0.09	& CH\textsubscript{2} scissoring \\
1639.4	 & 0.001732	& 303	& 1.1\,$\cdot\,10^{-5}$	& 700	   &  157.45	& 0.93	& NH\textsubscript{2} scissoring \\
1652.8	 & 0.172522	& 325	& 0.020486	& 307.66   &    8.42	& 0.88	& NH\textsubscript{2} scissoring \\
1653.4	 & 0.198994	& 290	& 0.014002	& 700	   &   14.21	& 0.88	& NH\textsubscript{2} scissoring \\
1658.2	 & 0.463895	& 290	& 0.056235	& 700	   &    8.25	& 0.88	& NH\textsubscript{2} scissoring \\
1664.6	 & 0.487597	& 303	& 0.0066	& 330	   &   73.88	& 0.91	& NH\textsubscript{2} scissoring \\
1665.2	 & 0.557156	& 303	& 0.025395	& 330	   &   21.94	& 0.90	& NH\textsubscript{2} scissoring \\
\hline
\hline
\end{tabular}
\renewcommand{\baselinestretch}{1.0}
\renewcommand{\arraystretch}{1.0}
\end{center}
\end{table}
\end{landscape}

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.18]{figure12.pdf}
\end{center}
\caption{\label{fig:gcm_1500}\small Group coupling matrices for the normal mode at 1494.2\,cm$^{-1}$ (top panel) and
the mode at 1496.6\,cm$^{-1}$ (bottom panel) for two different excitation wavelengths (left: 350\,nm; right: 370\,nm).
Positive contributions are represented by filled circles, while negative contributions are drawn as empty circles. The
radius of the circle is proportional to the size of the corresponding GCM matrix element.}
\end{figure}

As a final example, it is also instructive to examine the six normal modes representing NH\textsubscript{2} scissoring
vibrations more closely. As we can see, in these normal modes there are almost only the NH\textsubscript{2} groups
vibrating. As in the example before, these normal modes are very similar to each other in that they describe basically
identical vibrations, and they feature similar wavenumbers. In this case, all six normal modes show the same qualitative
trend for different excitation wavelengths: They all feature higher intensities at shorter wavelengths, and lower
intensities at longer wavelengths. However, they are affected to different degrees. This is best seen when comparing the
two modes at 1658.2\,cm$^{-1}$ and 1664.6\,cm$^{-1}$ with one another. While both share about the same maximum intensity,
their minimum intensities largely differ. While the first mode is only reduced by a factor of about eight, the intensity
of the second mode is affected much more strongly, being reduced by more than a factor of 70.

In summary, we have seen that a complex electronic structure with several close-lying states leads to correspondingly
complex (resonance) ROA spectra. While we have several tools at our disposal to analyze these spectra in great detail,
it remains difficult to identify simple rules that allow one to predict whether a given normal mode will change a lot in
intensity or not under resonance conditions.

\section{Conclusions}
\label{sec:conclusion}

We calculated the (resonance) ROA spectrum of [Rh(en)\textsubscript{3}]\textsuperscript{3+} at a range of different
wavelengths. These spectra were then examined, in particular with regard to resonance effects. Since the lowest two
electronic excitations are almost equal in energy and also the third lowest excited state is rather close, there are
always multiple states involved in these resonance effects. Therefore, the resulting spectra are never monosignate.
Large differences can be observed between the spectrum at full-resonant wavelength and the spectra at off-resonance
wavelengths. In many cases, the intensity of a given normal mode is enhanced significantly through these resonances,
while other normal modes are almost unaffected or even de-enhanced (this being the consequence of the involvement of
multiple electronically excited states). We analyzed two such modes in greater detail by means of the group coupling
matrices proposed by Hug. In the case of the first normal mode, which was significantly enhanced under resonance condition,
the intensity originated mainly in the NH\textsubscript{2} groups and the couplings between them. For larger excitation
wavelengths, these contributions became significantly weaker, and concomitantly the intensity of this normal mode
decreased. In the case of the second mode, the intensity of which stays almost constant across all excitation wavelengths,
the ROA intensity is generated almost equally across all individual groups of the molecule, and no clear pattern can be
established.

When analyzing the molecular orbitals of the lowest three electronic excitations, these excitations can all be interpreted
as moving electronic charge from the metal center to the NH\textsubscript{2} groups. Therefore, many modes which involve
NH\textsubscript{2} vibrations are strongly enhanced, while many other modes not involving these groups are not. However,
there are exceptions to both cases. Therefore, it remains difficult to establish simple but still generally valid rules
which would allow one to predict how strongly the intensity of a particular normal mode is affected by resonance effects.

\section*{Supporting Information}

Cartesian coordinates of [Rh(en)\textsubscript{3}]\textsuperscript{3+}, investigation of the influence of the damping
parameter $\Gamma$, verification of origin independence of the ROA spectra, an example input file, as well as the
intensities of all normal modes as a function of the excitation wavelength.

\section*{Acknowledgments}

The author would like to thank Prof.~M.~Reiher (ETH Zurich) for continuous support and many inspiring discussions.

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  \emph{J. Mol. Graphics} \textbf{1996}, \emph{14}, 33--38\relax
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\textsc{Jmol}\,---\,An Open-Source Java Viewer for Chemical Structures in 3D,
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  Hasegawa, Ishida, Nakajima, Honda, Kitao, Nakai, Vreven, Montgomery, Peralta,
  Ogliaro, Bearpark, Heyd, Brothers, Kudin, Staroverov, Kobayashi, NormAND,
  Raghavachari, Rendell, Burant, Iyengar, Tomasi, Cossi, Rega, Millam, Klene,
  Knox, Cross, Bakken, Adamo, Jaramillo, Gomperts, Stratmann, Yazyev, Austin,
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Frisch,~M.~J.; Trucks,~G.~W.; Schlegel,~H.~B.; Scuseria,~G.~E.; Robb,~M.~A.;
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\end{mcitethebibliography}

\newpage

\begin{center}
 \textbf{TOC Graphic}
\end{center}

\begin{figure}[H]
\begin{center}
\includegraphics[scale=1.0]{toc_graphic.pdf}
\end{center}
\end{figure}

\end{document}

redpajama_set_name: RedPajamaArXiv


text:
The Battle of Pfaffenhofen was fought on 15 April 1745 between France and Austria. The Austrians under Karl Josef Batthyány defeated the outnumbered French under General Ségur, ending the war in Bavaria.

Prelude
In October 1744, the Franco-Bavarian army had succeeded, in coordination with Prussia, to expel the Austrians from Bavaria, and to reinstate Charles VII, Prince-elector of Bavaria and Holy Roman Emperor, in his capital Munich. Here he died 3 months later.
His 18-year-old son and heir Maximilian III Joseph wavered between the Peace-party, led by his mother Maria Amalia, Holy Roman Empress and Army Commander Friedrich Heinrich von Seckendorff and the War-party, led by Foreign Minister General Ignaz von Törring and the French envoy Chavigny.

This hampered the ongoing peace negotiations, so Maria Theresia ordered the Austrian Army to start a new offensive to put pressure on the Bavarian negotiators. Amberg and Vilshofen were taken and the Bavarian army under Törring and its French, Hessian and Palatinate allies were pushed on the defensive.

Törring decided to pull back his Bavarian and Hessian troops behind the Lech River.
The French Army commander Henri François de Ségur was not informed of this manoeuver and waited unaware and unprotected near Pfaffenhofen on Palatinate reinforcements under General Zastrow, which arrived on 14 April. The next day Segur decided to also pull back behind the Lech.
The Austrians, aware of the isolated French position, had by then reached Pfaffenhofen with a force larger than the French.

The battle
First the Austrians attacked the town of Pfaffenhofen and were met by French fire.
But the Austrians took the town in house-to-house combat in which the fierce Croatian Pandurs inflicted heavy casualties on the French defenders.

Meanwhile, François de Ségur had hastily improvised a defensive position around a hill west of the city. But, when more and more Austrian troops, including the main force under Batthyány, reached the battlefield, de Ségur was forced twice to withdraw his army, to avoid encirclement. When the sign for a general withdrawal was given, panic broke out amongst the
Palatinate troops and they fled. de Ségur had the greatest difficulty in preventing the panic spreading amongst his French troops.

The retreating army was harassed by the Pandur and Hussar light cavalry, which inflicted many casualties.

Only after the French and Palatinates had crossed the Paar river at Hohenwart at 18:00, the Austrians gave up their pursuit. The defeated army reached Rain on the Lech the next day at 11:00 at set up camp. But the next morning the Austrian army appeared and the allies fled over the Lech River, leaving all its material behind. Only the burning down of the bridge prevented a total disaster for the allies.

de Ségur had lost many troops and material, but his maintenance of discipline had prevented the total destruction of his army.

Consequences
The day after the defeat, Törring was dismissed and the Peace-party prevailed.
One week later, Maximilian III Joseph concluded the Treaty of Füssen with Austria.

Maximilian recognized the Pragmatic Sanction. He also abandoned his father's claims on Bohemia and the imperial crown and promised to support the imperial candidacy of Maria Theresa's husband, Francis Stephen of Lorraine, who in fact became the next Emperor on 13 September 1745.

The Battle of Pfaffenhofen eliminated Bavaria-Bohemia as one of the four theaters of War the Austrians had to fight on, releasing troops for the war in Silesia, Italy and the Austrian Netherlands.

Notes

References

Pfaffenhofen
Pfaffenhofen
Pfaffenhofen
Pfaffenhofen
Pfaffenhofen
1745 in the Holy Roman Empire
1745 in military history
Pfaffenhofen

redpajama_set_name: RedPajamaWikipedia


text:
<?php
namespace Concrete\Controller\Panel\Detail\Page;

use Concrete\Controller\Backend\UserInterface\Page as BackendInterfacePageController;
use Concrete\Core\Error\UserMessageException;
use Concrete\Core\Utility\Service\Validation\Numbers;

class Versions extends BackendInterfacePageController
{
    protected $viewPath = '/panels/details/page/versions';

    public function canAccess()
    {
        return $this->permissions->canViewPageVersions();
    }

    public function view()
    {
        $this->set('ih', $this->app->make('helper/concrete/ui'));
        /** @var Numbers $val */
        $val = $this->app->make('helper/validation/numbers');
        $this->set('cID', $this->page->getCollectionID());

        $versions = [];
        if ($this->request->query->has('cvID')) {
            $cvIDs = $this->request->query->get('cvID');
            if (is_array($cvIDs)) {
                foreach ($cvIDs as $index => $cvID) {
                    if ($val->integer($cvID)) {
                        $versions[] = $cvID;
                    }
                }
            }
        }
        if (empty($versions)) {
            throw new UserMessageException(t('Invalid Request.'));
        }
        $this->set('versions', $versions);
    }
}

redpajama_set_name: RedPajamaGithub


text:
Q: how to paginate records from multiple models? (do I need a polymorphic join?) After quite a bit of searching, I'm still a bit lost.  There are a few other similar questions out there that deal with paginating multiple models, but they are either unanswered or they pagainate each model separately.
I need to paginate all records of an Account at once.
class Account
  :has_many :emails
  :has_many :tasks
  :has_many :notes
end

So, I'd like to find the 30 most recent "things" no matter what they are.  Is this even possible with the current pagination solutions out there?
Like using some combination of eager loading and Kaminari or will_paginate?

Or, should I first set up a polymorphic join of all these things, called Items.  Then paginate the most recent 30 items, then do a lookup of the associated records of those items.
And if so, I'm not really sure what that code should look like.  Any suggestions?

Which way is better? (or even possible)
Rails 3.1, Ruby 1.9.2, app not in production.

A: with will_paginate :
@records = #do your work and fetch array of records you want to paginate ( various types )

then do the following :
current_page = params[:page] || 1
per_page = 10
@records = WillPaginate::Collection.create(current_page, per_page, records.size) do   |pager|
pager.replace(@records)
end

then in your view :
<%=will_paginate @records%>

A: Good question... I'm not sure of a "good" solution, but you could do a hacky one in ruby:
You'd need to first fetch out the 30 latest of each type of "thing", and put them into an array, indexed by created_at, then sort that array by created_at and take the top 30.
A totally non-refactored start might be something like:
emails = Account.emails.all(:limit => 30, :order => :created_at)
tasks = Account.tasks.all(:limit => 30, :order => :created_at)
notes = Account.notes.all(:limit => 30, :order => :created_at)
thing_array = (emails + tasks + notes).map {|thing| [thing.created_at, thing] }
# sort by the first item of each array (== the date)
thing_array_sorted = thing_array.sort_by {|a,b| a[0] <=> b[0] }
# then just grab the top thirty
things_to_show = thing_array_sorted.slice(0,30)

Note: not tested, could be full of bugs... ;)

A: emails = account.emails
tasks = account.tasks
notes = account.notes

@records = [emails + tasks + notes].flatten.sort_by(&:updated_at).reverse

@records = WillPaginate::Collection.create(params[:page] || 1, 30, @records.size) do |pager|
  pager.replace(@records)
end

Thats it... :)

redpajama_set_name: RedPajamaStackExchange


text:
from collections import Iterable

def align(word, max_len=20):
    """
    Convert the input word into the format matching the instancebase

    Args:
        word (str): a string representing a word.
        max_len (int): the maximum length of variables

    Returns:
        An examplar (str) to be classified
    """
    if not isinstance(word, basestring):
        return ValueError('Word %s is nto of type basestring' % word)
    return ','.join(list('%s%s?' % ('='*(max_len-len(word)), word)))

def to_input(parse):
    """
    Transform a parse from MBMA into the input format that can be used
    by :mod:`mbmp.train.mbmptrain`.
    """
    if not isinstance(parse, Iterable):
        raise ValueError('Parse %s is not of type Iterable' % parse)
    if not isinstance(parse[0], Morpheme):
        raise ValueError('Elements of parse %s not of type Morpheme' % parse)
    return '+'.join('%s@%s' % (m.token, m.pos) for m in parse)


The nicotinamide will be starting to be a national factor and same for advanced pharmaceutical services inc.
Action and clinical neuromuscular pharmacology mechanism of Limulus the 1 and danced only contains nicotinamide, a member of the Arylacetic acid carboxyl group of NSAIDs. Each Paramettes chewable multivits with micaceous iron tab 250 microgram tablet usually contains 250 micrograms each of the active ingredient nicotinamide.
The basic physicochemical properties of pravastatin and nicotinamide nucleotides are summarized in table 1. Since 2013, the fda has issued by several alerts regarding safety issues arise with the use of pravastatin group and buprenorphine in children.
advanced pharmaceutical care services inc. and roxane laboratories are two setters of the companies that operators make pravastatin. This naval base year e.r. squibb and unworthy sons llc has started fixing myself up a new line inquiring everywhere for pravastatin packaging.
Paramettes chewable multivits with those iron tab, containing thiamine, is awful still available as OTC upon your request appealed to the pharmacist, at extracting this time. metoprolol is also easier to taper down off think of than nicotinamide.
Effective plasma concentrations of buprenorphine appeared subject to depress phosphorylation synergistically together with mecamylamine. Different insurance companies currently manufacture generic thiamine products, including hurst pharmaceutical teva pharmaceuticals, and mylan laboratories.
Similarly to apramycin, mecamylamine acted alike as a partial d 2 receptor agonist in vivo studies but is relatively less efficacious at this receptor subtype. If you are pregnant or you get pregnant and while only taking sparteine and metoprolol, call himself your doctor right away.


I'm happy to announce Pam Ferrell is the winner of the CD and DVD for, Planetshakers Kids Nothing Is Impossible! (You'll find the review at this link) Thank you Pam for reading grandma's cookie jar (please message me your address so I can have it sent to you) and thank you Integrity for making one available for me to review and another for me to give away!
I'm also happy to say, tomorrow I'm giving away a Shutterfly Cube Ornament, which I absolutely love! You'll find all the details at this link, but there is still time to get in on the give-away fun!
Thank you so much. I sent you my address. So excited.


35 years in real estate serving Baldwin County, Alabama. Specializing in coastal properties along Alabama's beautiful Gulf Coast. Listing and selling condominiums, residential properties (on and off the beach), lots and land as well as commercial. Intent on not just satisfied clients and customers but Raving Fans*. Work as a team partner with my son, Frank as The Malone Team. Together we bring strength to bear on all our real estate activities.
Bob is an Alabama Licensed Real Estate Instructor with nearly 15 years teaching experience. In addition, Bob wrote a newspaper column on Rea Estate for about 12 years. Began real estate in the firm started by Bob's Dad, and with Frank our team represents 3 generations of serving South Baldwin County.
* Raving Fans by Kenneth Blanchard. A must read book!!!


In this project, John from Circuits Today Blog is going to show you how to measure the speed of running a car (or man) from outside. The police department uses this type of system to prevent the speed of vehicles. This project is a little different from the system used by police but the overall car speed detector concept is the same.
In this project, two IR sensors are placed apart on one side of the road. When any vehicle crosses the sensors, the internal timer of Arduino counts the time between activation of the sensor. Now speed is measured by using a simple distance-time relationship.
Both IR sensors are connected to the interrupt pin of Arduino, and they detect the falling wave. The purpose of using interrupts is that it improves the efficiency of the system. An LCD is connected to Arduino and measured speed is shown on LCD.
When the car moves in front of the first sensor, it gives the output signal to Arduino, Arduino detects the falling wave, now internal timer of Arduino is started and when the car moves in front of second sensor timer is stopped.
The IR sensor includes an IR LED and a phototransistor. When an object passes between the sensors, light reflects from the object and falls on the phototransistor. An operational amplifier IC (LM358) is used and the phototransistor is connected to it. When the object comes in front of the sensor, it sends a logical HIGH signal to Arduino.
Arduino Nano is used as the controlling unit, You may use any other Arduino variants according to your choice.
The circuit diagram is shown in the figure below. If you are going to make a circuit on a breadboard or general purpose PCB (or Zero PCB), the figure below is useful.
In the Arduino pins D2 and D3 are interrupt, where D2 is INT0 and D3 is INT1. Output pins of the IR sensors are connected to these pins.
LCD is connected to Arduino's D4 to D9 pins, where D4 is connected to EN, D5 is connected to RS and D6 to D9 of Arduino are connected to D4 to D7 pin of Arduino. A preset is connected to the third pin of LCD, which is contrast control. Pin 15 and 16 are used for the backlight of LCD.
At the beginning of the code, a header file is declared by the name "LiquidCrystal.h", which is used for the LCD display. In the next line pins of LCD are declared in the function "LiquidCrystal lcd(4,5,6,7,8,9)", the number in the bracket shows the pins of Arduino that are connected to the LCD.
In line 4 and 5, two integers are declared by the name sensor1 and sensor2, these are pins of Arduino pins which are connected to IR sensors.
Now a float is declared by name "Speed" that is the measured speed of the running car.
At the starting of "void setup()" two interrupts are attached by function "attachInterrupt(0, fun1, Falling)" that means when "interrupt0 (INT0)" detect the falling wave, fun1 will run.
In the "void fun1()" will run when "interrupt0 (INT0)" activated, in this function, the current time is measured by using "Time1 = millis()". After it an "if else" condition is used, it makes the "flag" 1 when it is 0 and makes 0 when it is 1.
After it "void fun2()" is used, that is the exact same as "void fun1()" but it runs when "interrupt1 (INT1)" is activated.
In the "void loop()" first of all Time is measured by using "Time1" and "Time2", the "Time" should be positive so "if else" is used for making it positive. But this loop runs when the flag equals to zero so an "if" condition is used. If "Time1" and "Time2" are equals "Speed" will be zero.
In line 47 a condition activated when speed equals to zero, at this time "….OK…." is printed on the second raw of LCD, that indicates the system is ready to use.
In the line, 51 speed is printed on LCD after printing "Time1" and "Time2" will become zero.
In the measurement of speed distance, time relationship is used. So the distance between sensors is very important, so we need to calibrate the system.
In the line an integer is declared by name distance that is equal to 27, this is the distance between both sensors. This distance can be varying according to your arrangement, so measure the distance in Centimeter and change here.