%Document Setup.
\documentclass[a4paper,11pt]{article}
%Load useful packages.
\usepackage[a4paper,margin=2.0cm]{geometry}  %Set page size to A4.
\usepackage{graphicx}   %Allow import of images.
\usepackage{floatrow}   %Positioning of table & figure captions.
\usepackage{siunitx}    %SI Units formatting.
\usepackage[version=4]{mhchem}  %Chemical Equations.
\usepackage{amsmath}    %Mathematical alignments.
\usepackage{amssymb}    %Mathematical symbols e.g. therefore.
\usepackage[parfill]{parskip}   %Blank lines between paragraphs.
\usepackage{array}  %Table centring.
\usepackage[autostyle]{csquotes}   %Recommended for babel.
\usepackage[british]{babel} %British localisation.
\usepackage[backend=biber,style=chem-rsc]{biblatex} %Bibliography.
\usepackage[extreme,mathspacing=normal,leading=normal]{savetrees}  %Reduce whitespace.
\usepackage{multirow}   %Use fields over multiple rows in tables.

%Configure settings.
\numberwithin{equation}{subsection}    %Number equations based on their section
\bibliography{../../bibliographies/reference,../../bibliographies/coretextbooks,../../bibliographies/additionaltextbooks,specific}   %Use Bibliography of reference books.
\bibliography{specific}
\floatsetup[table]{style = plaintop}  %Put captions on top of tables.
\floatsetup[figure]{style = plain}    %Ensure figure captions are below figures.
\sisetup{range-phrase = --,range-units = single} %Use - for ranges and only have a single unit shown after.
\sisetup{separate-uncertainty = true,multi-part-units=single}   %Use \pm for uncertainties.
\setlength{\parindent}{0cm} % Don't indent paragraphs.

%Document Headings.
\begin{document}
\title{Experiment 4B: The rotational-vibrational spectrum of \ce{HCl}}
\author{Sam White (Author) and Emma Winful (Laboratory Partner)}
\date{30/11/2018}
\maketitle

\begin{abstract}
    The rotational constants, $\tilde{B_{\nu}}$, were determined for both common
    isotopologues of \ce{HCl} gas and hence the bond force constants, $k$. The
    rotational constants for \ce{H^{35}Cl} were in agreement with published
    values and the bond force constants found were
    \SI{477.383(1)}{\newton\per\meter} and \SI{480.698(6)}{\newton\per\meter}
    for \ce{H^{35}Cl} and \ce{H^{37}Cl} respectively. The bond force constants
    are not identical within reasonable errors, hence it is likely that the
    isotopic mass affects the value of $k$, however additional data should be
    collected to verify this.
\end{abstract}

\section{Results and Analysis}
\subsection{Collected Data}
\begin{table}[H]
    \caption{Rotational-vibrational absorbance wavelengths for the fundamental
    transition bands.}
    \label{tbl:fundamental-results}
    \centering
    \begin{tabular}{|c|c|c|c|c|c|}
        \hline
        & $J$ & $R(J)$ / \si{\per\centi\meter} & $P(J)$ / \si{\per\centi\meter} & $R(J) - P(J)$ /
        \si{\per\centi\meter} & $R(J - 1) - P(J + 1)$ / \si{\per\centi\meter} \\
        \hline
        \multirow{13}{*}{\ce{H^{35}Cl}} & 0 & 2906.4 & & & \\
        \cline{2-6}
        & 1 & 2926.03 & 2865.24 & 60.79 & 62.64 \\
        \cline{2-6}
        & 2 & 2945.05 & 2843.76 & 101.29 & 104.33 \\
        \cline{2-6}
        & 3 & 2963.42 & 2821.70 & 141.72 & 145.98 \\
        \cline{2-6}
        & 4 & 2981.14 & 2799.07 & 182.07 & 187.53 \\
        \cline{2-6}
        & 5 & 2998.19 & 2775.89 & 222.30 & 228.97 \\
        \cline{2-6}
        & 6 & 3014.57 & 2752.17 & 262.40 & 270.26 \\
        \cline{2-6}
        & 7 & 3030.27 & 2727.93 & 302.34 & 311.4 \\
        \cline{2-6}
        & 8 & 3045.28 & 2703.17 & 342.11 & 352.37 \\
        \cline{2-6}
        & 9 & 3059.52 & 2677.90 & 381.62 & 393.13 \\
        \cline{2-6}
        & 10 & 3073.10 & 2652.15 & 420.95 & 433.62 \\
        \cline{2-6}
        & 11 & 3085.88 & 2625.90 & 459.98 & 473.88 \\
        \cline{2-6}
        & 12 & & 2599.22 & & \\
        \hline
        \multirow{13}{*}{\ce{H^{37}Cl}} & 0 & 2904.29 & & & \\
        \cline{2-6}
        & 1 & 2923.91 & 2863.20 & 60.71 & 62.53 \\
        \cline{2-6}
        & 2 & 2942.90 & 2841.76 & 101.14 & 104.19 \\
        \cline{2-6}
        & 3 & 2961.24 & 2819.72 & 141.52 & 145.76 \\
        \cline{2-6}
        & 4 & 2978.93 & 2797.14 & 181.79 & 187.25 \\
        \cline{2-6}
        & 5 & 2995.96 & 2773.99 & 221.97 & 228.62 \\
        \cline{2-6}
        & 6 & 3012.32 & 2750.31 & 262.01 & 269.86 \\
        \cline{2-6}
        & 7 & 3027.98 & 2726.10 & 301.88 & 310.95 \\
        \cline{2-6}
        & 8 & 3042.97 & 2701.37 & 341.60 & 351.84 \\
        \cline{2-6}
        & 9 & 3057.22 & 2676.14 & 381.08 & 392.57 \\
        \cline{2-6}
        & 10 & 3070.73 & 2650.40 & 420.33 & 432.99 \\
        \cline{2-6}
        & 11 & 3083.51 & 2624.23 & 459.28 & 473.20 \\
        \cline{2-6}
        & 12 & & 2597.53 & & \\
        \cline{2-6}
        \hline
    \end{tabular}
\end{table}

\begin{table}[H]
    \caption{Rotational-vibrational absorbance wavelengths for the overtone
    transition bands.}
    \label{tbl:overtone-results}
    \centering
    \begin{tabular}{|c|c|c|c|c|c|}
        \hline
        & $J$ & $R(J)$ / \si{\per\centi\meter} & $P(J)$ / \si{\per\centi\meter} & $R(J) - P(J)$ /
        \si{\per\centi\meter} & $R(J - 1) - P(J + 1)$ / \si{\per\centi\meter} \\ 
        \hline
        \multirow{9}{*}{\ce{H^{35}Cl}} & 0 & 5688.16 & & & \\
        \cline{2-6}
        & 1 & 5706.59 & 5647.71 & 58.88 & 62.66 \\
        \cline{2-6}
        & 2 & 5723.78 & 5625.50 & 98.28 & 104.35 \\
        \cline{2-6}
        & 3 & 5739.75 & 5602.24 & 137.51 & 145.98 \\
        \cline{2-6}
        & 4 & 5754.53 & 5577.80 & 176.73 & 187.53 \\
        \cline{2-6}
        & 5 & 5767.89 & 5552.22 & 215.67 & 229.05 \\
        \cline{2-6}
        & 6 & 5780.07 & 5525.48 & 254.59 & 270.26 \\
        \cline{2-6}
        & 7 & 5790.92 & 5497.63 & 293.29 & 311.52 \\
        \cline{2-6}
        & 8 & & 5468.55 & & \\
        \hline
        \multirow{9}{*}{\ce{H^{37}Cl}} & 0 & 5684.12 & & & \\
        \cline{2-6}
        & 1 & 5703.13 & 5643.60 & 59.53 & 62.58 \\
        \cline{2-6}
        & 2 & 5719.67 & 5621.54 & 98.13 & 104.80 \\
        \cline{2-6}
        & 3 & 5735.61 & 5598.33 & 137.28 & 145.76 \\
        \cline{2-6}
        & 4 & 5750.30 & 5573.91 & 176.39 & 197.19 \\
        \cline{2-6}
        & 5 & 5763.27 & 5548.42 & 214.85 & 228.64 \\
        \cline{2-6}
        & 6 & 5775.12 & 5521.66 & 253.46 & 269.43 \\
        \cline{2-6}
        & 7 & 5786.65 & 5493.84 & 292.81 & 310.49 \\
        \cline{2-6}
        & 8 & & 5464.63 & & \\
        \hline
    \end{tabular}
\end{table}

\subsection{Determination of Rotational Constants and Bond Lengths}
The values of the rotational constants $\tilde{B_0}$, $\tilde{B_1}$ and
$\tilde{B_2}$ were calculated accounting for the centrifugal distortion of the
molecules by plotting graphs of $\frac{R(J) - P(J)}{J + \frac{1}{2}}$ and
$\frac{R(J - 1)- P(J + 1)}{J + \frac{1}{2}}$ against $(J + \frac{1}{2})^2$ and
performing linear regressions (as shown in figures \ref{fig:cl35-fu} and
\ref{fig:cl35-ou}). This was completed since if combination differences between
the $R(J)$ and $P(J)$ and then the $R(J-1)$ and $P(J+1)$ bands are taken and the
centrifugal distortion accounted for the resultant equations can be re-arranged
to give equations \ref{eq:comb-diff-upper} and \ref{eq:comb-diff-lower}
respectively, where $\tilde{D_{\nu}}$ are the centrifugal distortion
coefficients. These equations are in the form of the general equation of a
straight line, $y = m x + c$, hence the plotted data can be fitted by a function
of this form.

\begin{align}
    \label{eq:comb-diff-upper}
    \frac{R(J) - P(J)}{J + \frac{1}{2}} &= -8\tilde{D_1} \left(J + \frac{1}{2}\right)^2 + 4\tilde{B_1} -
    6\tilde{D_1} \\
    %
    \label{eq:comb-diff-lower}
    \frac{R(J - 1) - P(J + 1)}{J + \frac{1}{2}} &= -8\tilde{D_0} \left(J + \frac{1}{2}\right)^2 +
    4\tilde{B_0} - 6\tilde{D_0}
\end{align}

The values of  $\tilde{D_{\nu}}$ were then determined (see table
\ref{tbl:centrifugal-distortion-const} in the supplementary information) and
hence the rotational constants, $\tilde{B_{\nu}}$, shown in table
\ref{tbl:rot-const-bond-lengths} were calculated. The error propagation shown in
equation \ref{eq:b-err-prop} was then completed to estimate the uncertainties in
$\tilde{B_{\nu}}$, where $\alpha_{\tilde{B_{\nu}}}$, $\alpha_m$ and $\alpha_c$ are the
uncertainties in $\tilde{B_{\nu}}$, the gradient of the fitted line and the
intercept respectively.

\begin{figure}[h]
    \centering
    \includegraphics[width=0.9\textwidth]{figures/cl35fu.png}
    \caption{Graph showing the linear regression performed for the upper rotational (R) branch
    of the fundamental transition in \ce{H^{35}Cl}.}\label{fig:cl35-fu}
\end{figure}

\begin{figure}[h]
    \centering
    \includegraphics[width=0.9\textwidth]{figures/cl35ou.png}
    \caption{Graph showing the linear regression performed for the upper rotational (R)
    branch for the overtone transition in \ce{H^{35}Cl}.}\label{fig:cl35-ou}
\end{figure}

\begin{equation}
    \label{eq:b-err-prop}
    \alpha_{\tilde{B_{\nu}}} = \sqrt{\left( \frac{3}{16} \alpha_m \right)^2 + \left(
    \frac{1}{4} \alpha_c \right)^2}
\end{equation}

The literature values of $\tilde{B_{\nu}}$ for \ce{H^{35}Cl},
$\tilde{B}_{\nu_{Lit.}}$, in table \ref{tbl:rot-const-bond-lengths} were
determined using published equilibrium rotational constant, $\tilde{B_e}$, and
rotational constant parameter, $\alpha_e$, values\autocite{nist-hcl} with
equation \ref{eq:b-eq-nu}.

\begin{equation}\label{eq:b-eq-nu}
    \tilde{B}_{\nu_{Lit.}} = \tilde{B_e} - \alpha_e \left( \nu + \frac{1}{2} \right)
\end{equation}

The bond lengths in table \ref{tbl:rot-const-bond-lengths} were determined from
the $\tilde{B_{\nu}}$ values using equation \ref{eq:bond-length}. Furthermore
the calculus-based approximation\autocite{hughes-hase-uncertainties} was
utilised to propagate the uncertainties in $\tilde{B_{\nu}}$ for $r_{\nu}$
($\alpha_{r_{\nu}}$) as the $\alpha_{\tilde{B_{\nu}}}$ values are small hence producing equation
\ref{eq:r-err-prop}. The uncertainties in the values of the constants and
reduced mass used\autocite{crc-handbook} are insignificant compared to that of
$\alpha_{\tilde{B_{\nu}}}$, and hence they were excluded from the error propagation.

\begin{align} \label{eq:bond-length}
    r_{\nu} &= \sqrt{\frac{h}{8 \pi^2 c \mu \tilde{B_{\nu}}}} \\
    \label{eq:r-err-prop}
    \alpha_{r_{\nu}} &= \frac{1}{2} \sqrt{\frac{h}{8 \pi^2 c \mu \tilde{B_{\nu}}^3}} \alpha_{\tilde{B_{\nu}}}
\end{align}

\begin{table}[h]
    \caption{Rotational constants and bond lengths.}
    \label{tbl:rot-const-bond-lengths}
    \centering
    \begin{tabular}{|c|c|c|c|c|}
        \hline
        & $\nu$ & $\tilde{B_{\nu}}$ / \si{\per\centi\metre} &
        $\tilde{B}_{\nu_{Lit.}}$ / \si{\per\centi\metre} & $r_{\nu}$ / \si{\pico\meter} \\
        \hline
        \multirow{3}{*}{\ce{H^{35}Cl}} & 0 & \num{10.4408(4)} &  \num{10.43982} & \num{128.823(2)} \\
        \cline{2-5}
        & 1 & \num{10.1360(2)} & \num{10.13264} & \num{130.745(1)} \\
        \cline{2-5}
        & 2 & \num{9.829(4)} & \num{9.82546} & \num{132.77(2)} \\
        \hline
        \multirow{3}{*}{\ce{H^{37}Cl}} & 0 & \num{10.4248(3)} & & \num{128.385(2)} \\
        \cline{2-5}
        & 1 & \num{10.1214(1)} & & \num{130.2953(9)} \\
        \cline{2-5}
        & 2 & \num{9.86(2)} & & \num{132.0(1)} \\
        \hline
    \end{tabular}
\end{table}

\subsection{Determination of Vibrational Constants}

The harmonic constant, $\tilde{\nu_e}$, and the anharmonicity constant, $x_e$,
in table \ref{tbl:vib-const} were determined using equations \ref{eq:nu-e} and
\ref{eq:xe}. The derivation method of these equations is included in section
\ref{sec:vib-const-deriv} within the supplementary information.

\begin{align}
    \label{eq:nu-e}
    \tilde{\nu_e} &= R(0) - 3 \tilde{B_1} + \tilde{B_2} \\
    \label{eq:xe}
    x_e &= \frac{1}{2} \frac{\tilde{B_2} - \tilde{B_1}}{R(0) - 3 \tilde{B_1} + \tilde{B_2}}
\end{align}

The uncertainties in these values was estimated using equations
\ref{eq:nu-e-uncert} and \ref{eq:xe-uncert} with the uncertainty in $R(0)$
excluded from the propagation as it is negligible compared to that in
$\tilde{B_1}$ and $\tilde{B_2}$ since it is determined by reading the absorption
wavenumber directly from the spectrum.

\begin{align}
    \label{eq:nu-e-uncert}
    \alpha_{\tilde{\nu_e}} &= \sqrt{ \left( 3 \alpha_{\tilde{B_1}} \right)^2 + \left( \alpha_{\tilde{B_2}}
    \right)^2} \\
    \label{eq:xe-uncert}
    \alpha_{x_e} &= x_e \sqrt{ \frac{ \left( \alpha_{\tilde{B_1}} \right)^2 + \left(
    \alpha_{\tilde{B_2}} \right)^2 }{ \left( \tilde{B_2} - \tilde{B_1} \right)^2} + \left( \frac{\alpha_{\tilde{\nu_e}}}{\tilde{\nu_e}} \right)^2}
\end{align}

\begin{table}[h]
    \caption{Vibrational Coefficients.}
    \label{tbl:vib-const}
    \centering
    \begin{tabular}{|c|c|c|}
        \hline
        & \ce{H^{35}Cl} & \ce{H^{37}Cl} \\
        \hline
        $\tilde{\nu_e}$ / \si{\per\centi\metre} & \num{2885.821(4)} &
        \num{2883.78(2)} \\
        \hline
        $x_e$ / \num{e-5} & \num{-5.32(6)} & \num{-4.5(3)} \\
        \hline
    \end{tabular}
\end{table}


\subsection{Determination of Bond Force Constants}
The bond force constants, $k$, in table \ref{tbl:force-constants} were determined
using equation \ref{eq:force-constant} and the error in $k$, $\alpha_k$, was determined
using equation \ref{eq:force-constant-err} (derived using the calculus-based
approximation). The errors in $\mu$ and the constants were again not considered
as they are negligible compared to that of $\nu_e$.

\begin{align}
    \label{eq:force-constant}
    k &= 4 \pi^2 c^2 \mu \tilde{\nu_e}^2 \\
    %
    \label{eq:force-constant-err}
    \alpha_k &= 8 \pi^2 c^2 \mu \tilde{\nu_e} \alpha_{\tilde{\nu_e}} = 8 \pi^2 c^2 \mu \tilde{\nu_e} \sqrt{ \left( 3 \alpha_{\tilde{B_1}} \right)^2 + \left( \alpha_{\tilde{B_2}}
    \right)^2}
\end{align}

\begin{table}[H]
    \caption{Force Constants.}
    \label{tbl:force-constants}
    \centering
    \begin{tabular}{|c|c|c|}
        \hline
        & \ce{H^{35}Cl} & \ce{H^{37}Cl} \\
        \hline
        k / \si{\newton\per\metre} & \num{477.383(1)} & \num{480.698(6)} \\
        \hline
    \end{tabular}

\end{table}

\section{Discussion}
\subsection{Errors and Analysis Method}\label{sec:errors}
It is likely that the errors used for the data collected are underestimates
since they are derived from linear regressions performed with a limited number
of data points (less than 12). Furthermore while the linear regression was (in
general) good for the data obtained for fundamental transitions ($ \num{2e-5} <
\chi^2 < \num{2e-4}$) it was much poorer for data derived from the overtone
transition ($\num{4e-4} < \chi^2 < \num{0.2}$). This was due to the considerable
signal:noise ratio on the spectrum for the overtone transition resulting in the
data being significantly affected by random noise (as can be seen in figure
\ref{fig:cl35-ou}) and hence the true uncertainty in these values is likely to
be even greater.

The high signal:noise ratio for the overtone transition data is due to the low
overtone transition probability since the transition is forbidden by the $\Delta
\nu = \pm 1$ selection rule obtained for the purely harmonic potential and while
the oscillator is anharmonic it still has considerable harmonic character. The
signal:noise ratio could be reduced by recording additional spectra for just the
overtone region (\SIrange{6000}{5000}{\per\centi\meter}) with a greater
concentration of \ce{HCl} in the gas cell. The reduced wavenumber range should
be used since the increased concentration will also increase the intensity of
the fundamental transition peaks and will cause them to be 'cut' hence
preventing a distinct wavenumber from being recorded for the fundamental
transition absorption peak.  In addition to this further spectra for the
fundamental transition could be recorded as well to allow a better estimation of
the errors in these values by aiding in determining the reproducibility of the
results.

Corrections were made in the data analysis for centrifugal distortion since when
the effect of this was ignored the residuals for the resultant regression
followed a distinctive parabolic-like pattern when plotted opposed to being
randomly distributed, thus indicating the presence of a systematic error. When
the centrifugal distortion was considered the residuals instead appeared to be
much more randomly distributed.

\subsection{Data}

\subsubsection{Rotational Constants}\label{sec:rot-const}
The $\tilde{B_0}$ values tabulated in table \ref{tbl:rot-const-bond-lengths}
were derived using data from the fundamental transition only even though
they could have also been determined using data from the overtone transition.
This method was used due to the large random errors in the overtone transition
data (due to the high signal:noise ratio as discussed in section
\ref{sec:errors}), so any use of the data from this would be likely to greatly
increase the random errors associated with the $\tilde{B_0}$ value.

From equation \ref{eq:rot-const-b} it was expected that $\tilde{B_{\nu}}$
for \ce{H^{35}Cl} should be larger than the corresponding values for
\ce{H^{37}Cl} due to the larger reduced mass of \ce{H^{37}Cl}.

\begin{equation}\label{eq:rot-const-b}
    \tilde{B_{\nu}} = \frac{h}{8 \pi^2 c \mu r^2}
\end{equation}

The data in table \ref{tbl:rot-const-bond-lengths} agrees with this as almost
all values of $\tilde{B_{\nu}}$ stated for \ce{H^{35}Cl} are larger than the
corresponding values for \ce{H^{37}Cl} except for the $\tilde{B_2}$ value.
However using the uncertainty in the $\tilde{B_2}$ value for \ce{H^{37}Cl} there
is a probability of \SI{6.06}{\percent} that the true value actually satisfies
this condition (calculated by assuming the measurements are normally distributed
around the true value). In fact since the uncertainties estimated are likely
underestimates (see section \ref{sec:errors}) this probability will be higher in
reality, hence it is probable that the data agrees with the expectation
drawn from equation \ref{eq:rot-const-b}. To increase the certainty of this
statement more spectra should be obtained for the overtone transition with a
lower signal:noise ratio and hence smaller random error (as discussed in section
\ref{sec:errors}).

The values of $\tilde{B_{\nu}}$ in table \ref{tbl:rot-const-bond-lengths}
all agree with the literature values, $\tilde{B}_{\nu_{Lit.}}$, within
\SI{\pm0.004}{\per\centi\metre}. While this is one order of magnitude greater
than the estimated uncertainties in all values (except for $\tilde{B_{2}}$) it
is likely that the obtained data does agree with these literature values (as
discussed in section \ref{sec:errors}) as the uncertainties stated are likely
underestimates.

\subsubsection{Bond Lengths}
It was expected that the bond lengths, $r_{\nu}$, of both isotopologues would
increase as $\nu$ increased since the asymmetry of the anharmonic bond potential should
cause the mean bond length to increase with the vibrational energy level. All of
the values of $r_{\nu}$ obtained in table \ref{tbl:rot-const-bond-lengths} show
this trend.

In table \ref{tbl:rot-const-bond-lengths} it can also be seen that the bond
lengths of the \ce{H^{37}Cl} isotopologue in each vibrational state are greater
than those for the \ce{H^{35}Cl} isotopologue. This can be justified
theoretically using the one dimensional Sch\"odinger equation (equation
\ref{eq:s-eq}) where for a (stationary) state described by $\psi_{\nu}$ in a
constant potential $V(x)$ increasing the reduced mass, $\mu$, should reduce the
energy of the state, $E_{\nu}$, hence reducing the mean bond length due to the
anharmonicity of the potential.

\begin{align}
    E_{\nu} \psi_{\nu} &= \hat{H} \psi_{\nu} \nonumber \\ 
    \label{eq:s-eq}
        &= \left( \frac{\hat{p}^2}{2 \mu} + V(x) \right) \psi_{\nu}
\end{align}

\subsubsection{Bond Force Constants}
It was expected that the bond force constants, $k$, displayed in table
\ref{tbl:force-constants} should be equal for both isotopologues as it was
assumed that $k$ depends only on the chemical bonding and hence should be the
same.

Despite this the force constants differ by \SI{3.315}{\newton\per\meter} while
both values have uncertainties in the order of \num{e-3}. It is unlikely that
the underestimation of the uncertainties of $\tilde{B_1}$ and $\tilde{B_2}$
discussed in section \ref{sec:errors} can account for this because even if the
errors in these values are increased by a factor of ten (hence making the values
for \ce{H^{35}Cl} be in very good agreement with the published ones: see section
\ref{sec:rot-const}) and propagated the uncertainties in $k$ increase by only a
little over a factor of ten.  This gives the difference between the obtained
values as over 500 standard deviations hence showing that this is very
unlikely to be caused due to random errors.

Systematic errors are also unlikely to explain this difference as both values of
$k$ were obtained from data from the same spectrum and were calculated using
differences in wavenumber values thus any constant systematic errors in the
absorption wavenumber will have been removed and if any systematic errors were
introduced in the data analysis they should be the same for both $k$ values
hence still making them comparable.

It is possible that the isotopic mass affects the value of $k$ indirectly hence
leading to the observed differences as previously found by Biernacki and
Clerjaud for the \ce{SiH4} and \ce{SiD4} isotopologues.\autocite{biernacki-clerjaud}
However to more confidently confirm this more data with a lower random error in
the data derived from the overtone transition should be obtained (as discussed
in section \ref{sec:errors}). Additionally the rotational-vibrational
spectrum of deuterated \ce{HCl} gas could be recorded which should have an even
greater value of $k$ since the reduced mass is around a factor of two greater.

\printbibliography

\section{Supplementary Information}

\subsection{Centrifugal Distortion Coefficients}

The values of the centrifugal distortion coefficients, $\tilde{D_{\nu}}$, were
determined from the gradients determined in the linear regressions.

\begin{table}[h]
    \caption{Centrifugal distortion coefficients.}
    \label{tbl:centrifugal-distortion-const}
    \centering
    \begin{tabular}{|c|c|c|}
        \hline
        & $\nu$ & $\tilde{D_{\nu}}$ / \num{e-4} \si{\per\centi\metre}\\
        \hline
        \multirow{3}{*}{\ce{^{35}Cl}} & 0 & \num{5.25(4)} \\
        \cline{2-3}
        & 1 & \num{5.11(2)} \\
        \cline{2-3}
        & 2 & \num{4.3(8)} \\
        \hline
        \multirow{3}{*}{\ce{^{37}Cl}} & 0 & \num{5.20(3)} \\
        \cline{2-3}
        & 1 & \num{5.13(1)} \\
        \cline{2-3}
        & 2 & \num{11(4)}\\
        \hline
    \end{tabular}
\end{table}

\subsection{Derivation of Vibrational Constant Equations}\label{sec:vib-const-deriv}
The energies associated with the discrete vibrational energy levels in a molecule are given by equation
\ref{eq:vib-energy} which can be found through the application of perturbation
theory on the harmonic potential with the perturbation of the potential
including terms a of higher order than two from the Taylor expansion of the
potential energy.

\begin{equation}\label{eq:vib-energy}
    \tilde{E_{\nu}} = \tilde{\nu_e} \left( \nu + \frac{1}{2} \right) -
    \tilde{\nu_e} x_e \left( \nu + \frac{1}{2} \right)^2
\end{equation}

From equation \ref{eq:vib-energy} we can obtain the energy related to the pure vibrational transition
$\tilde{E}(\nu_f \leftarrow 0)$ (the $\nu = 0$ to $\nu = \nu_f$ transition) as equation \ref{eq:delta-e-coeff}.

\begin{align}
    \tilde{E}\left( \nu_f \leftarrow 0 \right) &= \tilde{E}_{\nu_f} - \tilde{E}_0 \nonumber \\
        &= \tilde{\nu_e}\left( \nu_f + \frac{1}{2} - \frac{1}{2} \right) - \tilde{\nu_e} x_e \left(
            \frac{1}{4} - \left( \nu_f + \frac{1}{2} \right)^2 \right) \nonumber \\
        \label{eq:delta-e-coeff}
        &= \tilde{\nu_e} \left( \nu_f - \left( \nu_f^2 + \nu_f \right) x_e \right)
\end{align}

The energy related to the R branch transitions can be determined to yield
equation \ref{eq:r-trans-energy} where $J$ is the rotational state adopted in
the lower vibrational state ($\nu = 0$).

\begin{equation}\label{eq:r-trans-energy}
    R(J) = \Delta \tilde{E}(\nu_f \leftarrow 0) + (\tilde{B_{\nu_f}} + \tilde{B_0}) (J + 1) + (\tilde{B_{\nu_f}} -
    \tilde{B_0}) ( J + 1)^2
\end{equation}

Setting $J = 0$ hence gives equation \ref{eq:r-trans-energy-j0}.

\begin{equation}\label{eq:r-trans-energy-j0}
    R(J) - 2 \tilde{B_{\nu_f}} =  \Delta \tilde{E}(\nu_f \leftarrow 0)
\end{equation}

Substituting equation \ref{eq:delta-e-coeff} into equation \ref{eq:r-trans-energy-j0} yields equation
\ref{eq:r-trans-energy-sub}.

\begin{equation}\label{eq:r-trans-energy-sub}
    R(0) - 2 \tilde{B_{\nu_f}} = \tilde{\nu_e} \left( \nu_f - \left( \nu_f^2 + \nu_f \right) x_e \right)
\end{equation}

Now from equation \ref{eq:r-trans-energy-sub} a system of linear equations can be
obtained by setting $\nu_f = 1$ and $\nu_f = 2$ (equations \ref{eq:coeff-1} and
\ref{eq:coeff-2}).

\begin{align}
    \label{eq:coeff-1}
    R(0) - 2 \tilde{B_1} &= \tilde{\nu_e} \left( 1 - 2 x_e \right) \\
    \label{eq:coeff-2}
    R(0) - 2 \tilde{B_2} &= \tilde{\nu_e} \left( 1 - 6 x_e \right)
\end{align}

Solving equations \ref{eq:coeff-1} and \ref{eq:coeff-2} for $\tilde{\nu_e}$ and
$x_e$ then gives equations \ref{eq:nu-e} and \ref{eq:xe}.

\end{document}
%Note: gq means : reformat the text included in the motion
% vim:tw=80
