\documentclass{article}
\begin{document}

\title{Supplementary material (ESI) for \textit{Phys. Chem. Chem. Phys.} \\ This journal is copyright 2005 The Owner Societies}
\author{Andrew J. Alexander}

\begin{table*}
\begin{tabular}{cc@{\hspace{0.5cm}}c}\hline
Configuration & Term & $C_2(\gamma L)$ \\\hline
$np$   & ${^2P}$ & $-(2/5)^{1/2}$ \\
$np^2$ & ${^3P}$ & $(2/5)^{1/2}$ \\
       & ${^1D}$ & $-(28/50)^{1/2}$ \\
$nd$   & ${^2D}$ & $-(2/7)^{1/2}$ \\
$nd^2$ & ${^3P}$ & $(2/5)^{1/2}$ \\
       & ${^1G}$ & $-(220/343)^{1/2}$ \\
       & ${^1D}$ & $(18/343)^{1/2}$ \\
       & ${^3F}$ & $-(12/245)^{1/2}$ \\
$nd^3$ & ${^4F}$ & $(12/245)^{1/2}$ \\
$nd^4$ & ${^5D}$ & $(2/7)^{1/2}$ \\
\hline
\end{tabular}
\caption{Coefficients $C_2(\gamma L)$ for ground and excited terms
($^{2S+1}L$) in configurations with outer shell electrons listed
in the first column. Not all possible excited terms are listed.
For shells that are complete except for missing electrons, the
$C_2(\gamma L)$ have opposite sign to those tabulated. For
example, for an atom having a $p$ shell with $np^4$ (a full $p$
shell with 2 missing electrons) we look up the value for $np^2$ (2
electrons) and change the sign, giving $-(2/5)^{1/2}$ for $^3P$.
\label{tab-c2}}
\end{table*}

\begin{table*}
\begin{footnotesize}
\begin{tabular}{cr@{\hspace{0.5cm}}l@{\hspace{0.5cm}}rr}\hline
$\Omega$ & $\alpha(11\Omega)$ & Eigenvector & $\rak{a}^{(1)}_{0}$
& $\rak{a}^{(2)}_{0}$ \\\hline
0&$3.6ga$&\ket{+0.57735}{1}{-1}$_g$\ket{+0.81650}{0}{0}&$0.000$&$-0.500$\\
 &$0.0ga$&\ket{+0.81650}{1}{-1}$_g$\ket{-0.57735}{0}{0}&$0.000$&$0.000$\\
 &$0.0ub$&\ket{+1.00000}{1}{-1}$_u$&$0.000$&$0.500$\\
1&$-2.4g$&\ket{+1.00000}{1}{0}$_g$&$0.354$&$-0.250$\\
 &$0.0u$&\ket{+1.00000}{1}{0}$_u$&$0.354$&$-0.250$\\
2&$0.6g$&\ket{+1.00000}{1}{1}&$0.707$&$0.500$\\\hline
\end{tabular}
\end{footnotesize}
\caption{Eigenvalues $\alpha$ and eigenfuctions of the
quadrupole-quadrupole interaction matrix, for atoms $J_c=1$ and
$J_d=1$. The key to the column headed $\alpha$ is as follows: for
like atoms, the label $g$ means gerade, and $u$ means ungerade,
the label $a$ means that the state is $0^+$ if the parity
parameter (Eq.(\ref{eqn-parity})) $\delta=+1$, and $0^-$ if
$\delta=-1$; the label $b$ denotes a state that is $0^+$ if
$\delta=-1$ and $0^-$ if $\delta=+1$. For the eigenvectors, a
shorthand $|M_cM_d\rangle_{g,u}$ ket notation is used: see
Eqs.(\ref{eqn-ket}) and (\ref{eqn-gu}) in the text. The columns to
the far right of the table list adiabatic molecule-frame
polarization parameters $\rak{a}^{(K)}_0$ ($K=1,2$) for atoms
$J=1$. Conversion factors for $J=1$ are
$\rho^{(1)}_0=\rak{a}^{(1)}_{0}$ and
$\rho^{(2)}_0=\sqrt{\frac{2}{3}}\rak{a}^{(2)}_{0}$.\label{tab-11}}
\end{table*}

\begin{table*}
\begin{footnotesize}
\begin{tabular}{cr@{\hspace{0.4cm}}l@{\hspace{0.2cm}}rrrr}\hline
&&& \multicolumn{2}{c}{$J=\frac{3}{2}$} & \multicolumn{2}{c}{$J=1$}\\
$\Omega$ & $\alpha(\frac{3}{2}1\Omega)$ & Eigenvector &
$\rak{a}^{(1)}_{0}$ & $\rak{a}^{(2)}_{0}$&$\rak{a}^{(1)}_{0}$ &
$\rak{a}^{(2)}_{0}$\\\hline
0.5&$2.74589$&\ket{+0.60150}{\frac{3}{2}}{-1}\ket{+0.79465}{\frac{1}{2}}{0}\ket{+0.08198}{-\frac{1}{2}}{1}&$0.442$&$-0.221$&$-0.198$&$-0.410$\\
 &$0.00000$&\ket{+0.70711}{\frac{3}{2}}{-1}\ket{-0.57735}{\frac{1}{2}}{0}\ket{+0.40825}{-\frac{1}{2}}{1}&$0.430$&$0.000$&$-0.065$&$0.121$\\
 &$-1.04884$&\ket{+0.37175}{\frac{3}{2}}{-1}\ket{-0.18759}{\frac{1}{2}}{0}\ket{-0.90918}{-\frac{1}{2}}{1}&$-0.097$&$-0.579$&$0.545$&$0.488$\\
1.5&$0.17635$&\ket{+0.59466}{\frac{3}{2}}{0}\ket{-0.80398}{\frac{1}{2}}{1}&$0.441$&$-0.234$&$0.457$&$-0.030$\\
 &$-2.72193$&\ket{+0.80398}{\frac{3}{2}}{0}\ket{+0.59466}{\frac{1}{2}}{1}&$0.592$&$0.234$&$0.250$&$-0.470$\\
2.5&$0.84853$&\ket{+1.00000}{\frac{3}{2}}{1}&$0.775$&$0.800$&$0.707$&$0.500$\\
 \hline
\end{tabular}\end{footnotesize}
 \caption{As for Table \ref{tab-11}, but for for atoms $J_c=3/2$ and
$J_d=1$. Conversion factors for $J=3/2$ are
$\rho^{(1)}_0=\frac{\sqrt{3}}{4}\rak{a}^{(1)}_{0}$ and
$\rho^{(2)}_0=\frac{5}{8}\rak{a}^{(2)}_{0}$. Conversion factors
for $J=1$ are given in Table \ref{tab-11}.\label{tab-3h1}}
\end{table*}

\begin{table*}
\begin{footnotesize}
\begin{tabular}{cr@{\hspace{0.4cm}}l@{\hspace{0.2cm}}rrrr}\hline
&&& \multicolumn{2}{c}{$J=2$} & \multicolumn{2}{c}{$J=1$}\\
$\Omega$ & $\alpha(21\Omega)$ & Eigenvector & $\rak{a}^{(1)}_{0}$
& $\rak{a}^{(2)}_{0}$&$\rak{a}^{(1)}_{0}$ &
$\rak{a}^{(2)}_{0}$\\\hline
0&$2.32349a$&\ket{+0.23738}{1}{-1}\ket{+0.23738}{-1}{1}\ket{+0.94197}{0}{0}&$0.000$&$-0.944$&$0.000$&$-0.831$\\
 &$-0.29512a$&\ket{+0.66607}{1}{-1}\ket{+0.66607}{-1}{1}\ket{-0.33571}{0}{0}&$0.000$&$-0.556$&$0.000$&$0.331$\\
 &$-1.01419b$&\ket{+0.70711}{1}{-1}\ket{-0.70711}{-1}{1}&$0.000$&$-0.250$&$0.000$&$0.250$\\
1&$2.45293$&\ket{+0.70020}{2}{-1}\ket{+0.71300}{1}{0}\ket{-0.03680}{0}{1}&$0.608$&$0.235$&$-0.346$&$-0.263$\\
 &$0.04768$&\ket{+0.60914}{2}{-1}\ket{-0.56973}{1}{0}\ket{+0.55168}{0}{1}&$0.435$&$-0.096$&$-0.047$&$0.013$\\
 &$-1.48642$&\ket{+0.37238}{2}{-1}\ket{-0.40870}{1}{0}\ket{-0.83325}{0}{1}&$0.181$&$-0.639$&$0.393$&$0.249$\\
2&$0.35576$&\ket{+0.85689}{2}{0}\ket{+0.51550}{1}{1}&$0.708$&$0.601$&$0.188$&$-0.601$\\
 &$-2.89122$&\ket{+0.51550}{2}{0}\ket{-0.85689}{1}{1}&$0.517$&$-0.101$&$0.519$&$0.101$\\
3&$1.01419$&\ket{+1.00000}{2}{1}&$0.816$&$1.000$&$0.707$&$0.500$\\\hline
\end{tabular}
\end{footnotesize}
\caption{As for Table \ref{tab-11}, but for for atoms $J_c=2$ and
$J_d=1$. Conversion factors for $J=2$ are
$\rho^{(1)}_0=\sqrt{\frac{3}{5}}\rak{a}^{(1)}_{0}$ and
$\rho^{(2)}_0=\sqrt{\frac{2}{7}}\rak{a}^{(2)}_{0}$. Conversion
factors for $J=1$ are given in Table \ref{tab-11}.\label{tab-21}}
\end{table*}

\begin{table*}
\begin{footnotesize}
\begin{tabular}{cr@{\hspace{0.5cm}}l@{\hspace{0.5cm}}rr}\hline
$\Omega$ & $\alpha(\frac{3}{2}\frac{3}{2}\Omega)$ & Eigenvector &
$\rak{a}^{(1)}_{0}$ & $\rak{a}^{(2)}_{0}$ \\\hline
0&$3.2ua$&\ket{+0.70711}{\frac{3}{2}}{-\frac{3}{2}}$_g$\ket{+0.70711}{\frac{1}{2}}{-\frac{1}{2}}$_g$&$0.000$&$0.000$\\
 &$-0.8ua$&\ket{+0.70711}{\frac{3}{2}}{-\frac{3}{2}}$_g$\ket{-0.70711}{\frac{1}{2}}{-\frac{1}{2}}$_g$&$0.000$&$0.000$\\
 &$2.4gb$&\ket{+0.70711}{\frac{3}{2}}{-\frac{3}{2}}$_u$\ket{+0.70711}{\frac{1}{2}}{-\frac{1}{2}}$_u$&$0.000$&$0.000$\\
 &$0.0gb$&\ket{+0.70711}{\frac{3}{2}}{-\frac{3}{2}}$_u$\ket{-0.70711}{\frac{1}{2}}{-\frac{1}{2}}$_u$&$0.000$&$0.000$\\
1&$1.2u$&\ket{+1.00000}{\frac{1}{2}}{\frac{1}{2}}&$0.258$&$-0.800$\\
 &$-0.8u$&\ket{+1.00000}{\frac{3}{2}}{-\frac{1}{2}}$_g$&$0.258$&$0.000$\\
 &$-1.6g$&\ket{+1.00000}{\frac{3}{2}}{-\frac{1}{2}}$_u$&$0.258$&$0.000$\\
2&$-2.8u$&\ket{+1.00000}{\frac{3}{2}}{\frac{1}{2}}$_g$&$0.516$&$0.000$\\
 &$0.4g$&\ket{+1.00000}{\frac{3}{2}}{\frac{1}{2}}$_u$&$0.516$&$0.000$\\
3&$1.2u$&\ket{+1.00000}{\frac{3}{2}}{\frac{3}{2}}&$0.775$&$0.800$\\\hline
\end{tabular}
\end{footnotesize}
\caption{As for Table \ref{tab-11}, but for for atoms $J_c=3/2$
and $J_d=3/2$. Conversion factors to $\rho^{(K)}_0$ are given in
Table \ref{tab-3h1}.\label{tab-3h3h}}
\end{table*}

\begin{table*}
\begin{footnotesize}
\begin{tabular}{cr@{\hspace{0.4cm}}l@{\hspace{0.2cm}}rrrr}\hline
&&& \multicolumn{2}{c}{$J=2$} & \multicolumn{2}{c}{$J=\frac{3}{2}$}\\
$\Omega$ & $\alpha(2\frac{3}{2}\Omega)$ & Eigenvector &
$\rak{a}^{(1)}_{0}$ & $\rak{a}^{(2)}_{0}$&$\rak{a}^{(1)}_{0}$ &
$\rak{a}^{(2)}_{0}$\\\hline
0.5&$2.86401$&\ket{+0.75418}{2}{-\frac{3}{2}}\ket{+0.60370}{1}{-\frac{1}{2}}\ket{+0.23205}{0}{\frac{1}{2}}\ket{+0.11361}{-1}{\frac{3}{2}}&$0.608$&$0.326$&$-0.511$&$0.131$\\
 &$1.55051$&\ket{+0.16224}{2}{-\frac{3}{2}}\ket{+0.20225}{1}{-\frac{1}{2}}\ket{-0.93503}{0}{\frac{1}{2}}\ket{-0.24186}{-1}{\frac{3}{2}}&$0.014$&$-0.898$&$0.240$&$-0.664$\\
 &$-0.37762$&\ket{+0.54806}{2}{-\frac{3}{2}}\ket{-0.62070}{1}{-\frac{1}{2}}\ket{+0.10339}{0}{\frac{1}{2}}\ket{-0.55107}{-1}{\frac{3}{2}}&$0.279$&$-0.055$&$-0.094$&$0.166$\\
 &$-1.16835$&\ket{+0.32330}{2}{-\frac{3}{2}}\ket{-0.45757}{1}{-\frac{1}{2}}\ket{-0.24736}{0}{\frac{1}{2}}\ket{+0.79052}{-1}{\frac{3}{2}}&$-0.084$&$-0.374$&$0.365$&$0.367$\\
1.5&$0.91577$&\ket{+0.59863}{2}{-\frac{1}{2}}\ket{-0.20240}{1}{\frac{1}{2}}\ket{-0.77503}{0}{\frac{3}{2}}&$0.309$&$-0.263$&$0.383$&$0.161$\\
 &$-1.19523$&\ket{+0.80000}{2}{-\frac{1}{2}}\ket{+0.20000}{1}{\frac{1}{2}}\ket{+0.56569}{0}{\frac{3}{2}}&$0.539$&$0.300$&$0.093$&$-0.288$\\
 &$-1.87196$&\ket{+0.04052}{2}{-\frac{1}{2}}\ket{-0.95867}{1}{\frac{1}{2}}\ket{+0.28164}{0}{\frac{3}{2}}&$0.377$&$-0.537$&$0.298$&$-0.673$\\
2.5&$0.61882$&\ket{+0.62785}{2}{\frac{1}{2}}\ket{-0.77833}{1}{\frac{3}{2}}&$0.569$&$0.091$&$0.705$&$0.307$\\
 &$-2.77023$&\ket{+0.77833}{2}{\frac{1}{2}}\ket{+0.62785}{1}{\frac{3}{2}}&$0.656$&$0.409$&$0.643$&$0.018$\\
3.5&$1.43427$&\ket{+1.00000}{2}{\frac{3}{2}}&$0.816$&$1.000$&$0.775$&$0.800$\\
 \hline
\end{tabular}
\end{footnotesize}
\caption{As for Table \ref{tab-11}, but for for atoms $J_c=2$ and
$J_d=3/2$. Conversion factors to $\rho^{(K)}_0$ are given in
Tables \ref{tab-3h1} and \ref{tab-21}.\label{tab-23h}}
\end{table*}


\begin{table*}
\begin{footnotesize}
\begin{tabular}{cr@{\hspace{0.5cm}}l@{\hspace{0.5cm}}rr}\hline
$\Omega$ & $\alpha(22\Omega)$ & Eigenvector & $\rak{a}^{(1)}_{0}$
& $\rak{a}^{(2)}_{0}$ \\\hline
0&$3.26091ga$&\ket{+0.73597}{2}{-2}$_g$\ket{+0.58279}{1}{-1}$_g$\ket{+0.34453}{0}{0}&$0.000$&$0.253$\\
 &$1.50840ga$&\ket{+0.29184}{2}{-2}$_g$\ket{+0.18609}{1}{-1}$_g$\ket{-0.93819}{0}{0}&$0.000$&$-0.812$\\
 &$-0.48365ga$&\ket{+0.61088}{2}{-2}$_g$\ket{-0.79103}{1}{-1}$_g$\ket{+0.03312}{0}{0}&$0.000$&$0.059$\\
 &$2.77377ub$&\ket{+0.85065}{2}{-2}$_u$\ket{+0.52573}{1}{-1}$_u$&$0.000$&$0.585$\\
 &$-1.05949ub$&\ket{+0.52573}{2}{-2}$_u$\ket{-0.85065}{1}{-1}$_u$&$0.000$&$-0.085$\\
1&$1.12688g$&\ket{+0.46768}{2}{-1}$_g$\ket{+0.88390}{1}{0}$_g$&$0.204$&$-0.531$\\
 &$-1.41260g$&\ket{+0.88390}{2}{-1}$_g$\ket{-0.46768}{1}{0}$_g$&$0.204$&$0.031$\\
 &$1.20231u$&\ket{+0.16751}{2}{-1}$_u$\ket{+0.98587}{1}{0}$_u$&$0.204$&$-0.722$\\
 &$-0.91660u$&\ket{+0.98587}{2}{-1}$_u$\ket{-0.16751}{1}{0}$_u$&$0.204$&$0.222$\\
2&$0.85714g$&\ket{+0.39736}{2}{0}$_g$\ket{-0.91766}{1}{1}&$0.408$&$-0.421$\\
 &$-1.85714g$&\ket{+0.91766}{2}{0}$_g$\ket{+0.39736}{1}{1}&$0.408$&$-0.079$\\
 &$-2.00000u$&\ket{+1.00000}{2}{0}$_u$&$0.408$&$0.000$\\
3&$-2.57142g$&\ket{+1.00000}{2}{1}$_g$&$0.612$&$0.250$\\
 &$0.85714u$&\ket{+1.00000}{2}{1}$_u$&$0.612$&$0.250$\\
4&$1.71429g$&\ket{+1.00000}{2}{2}&$0.816$&$1.000$\\\hline
\end{tabular}
\end{footnotesize}
\caption{As for Table \ref{tab-11}, but for for atoms $J_c=2$ and
$J_d=2$. Conversion factors to $\rho^{(K)}_0$ are given in Table
\ref{tab-21}.\label{tab-22}}
\end{table*}

% ##############Special cases ###############

\begin{table*}
\begin{footnotesize}
\begin{tabular}{cr@{\hspace{0.4cm}}l@{\hspace{0.2cm}}rrr}\hline
&&& \multicolumn{2}{c}{$J=\frac{3}{2}$} & $J=\frac{1}{2}$\\
$\Omega$ & $\alpha'(\frac{3}{2}\frac{1}{2}\Omega)$ & Eigenvector &
$\rak{a}^{(1)}_{0}$ & $\rak{a}^{(2)}_{0}$&$\rak{a}^{(1)}_{0}$
\\\hline
0&$0.96ga$&\ket{+0.70711}{\frac{1}{2}}{-\frac{1}{2}}$_u$\ket{+0.70711}{-\frac{1}{2}}{\frac{1}{2}}$_u$&$0.000$&$-0.800$&$0.000$\\
 &$0.00gb$&\ket{+0.70711}{\frac{1}{2}}{-\frac{1}{2}}$_u$\ket{-0.70711}{-\frac{1}{2}}{\frac{1}{2}}$_u$&$0.000$&$-0.800$&$0.000$\\
 &$-0.96ua$&\ket{+0.70711}{\frac{1}{2}}{-\frac{1}{2}}$_g$\ket{+0.70711}{-\frac{1}{2}}{\frac{1}{2}}$_g$&$0.000$&$-0.800$&$0.000$\\
 &$0.00ub$&\ket{+0.70711}{\frac{1}{2}}{-\frac{1}{2}}$_g$\ket{-0.70711}{-\frac{1}{2}}{\frac{1}{2}}$_g$&$0.000$&$-0.800$&$0.000$\\
1&$0.00g$&\ket{+0.86603}{\frac{3}{2}}{-\frac{1}{2}}$_u$\ket{-0.50000}{\frac{1}{2}}{\frac{1}{2}}$_u$&$0.646$&$0.400$&$-0.289$\\
 &$-0.64g$&\ket{+0.50000}{\frac{3}{2}}{-\frac{1}{2}}$_u$\ket{+0.86603}{\frac{1}{2}}{\frac{1}{2}}$_u$&$0.387$&$-0.400$&$0.289$\\
 &$0.64u$&\ket{+0.50000}{\frac{3}{2}}{-\frac{1}{2}}$_g$\ket{+0.86603}{\frac{1}{2}}{\frac{1}{2}}$_g$&$0.387$&$-0.400$&$0.289$\\
 &$0.00u$&\ket{+0.86603}{\frac{3}{2}}{-\frac{1}{2}}$_g$\ket{-0.50000}{\frac{1}{2}}{\frac{1}{2}}$_g$&$0.646$&$0.400$&$-0.289$\\
2&$0.16g$&\ket{+1.00000}{\frac{3}{2}}{\frac{1}{2}}$_u$&$0.775$&$0.800$&$0.577$\\
 &$-0.16u$&\ket{+1.00000}{\frac{3}{2}}{\frac{1}{2}}$_g$&$0.775$&$0.800$&$0.577$\\
\hline
\end{tabular}
\end{footnotesize} \caption{Special case eigenvalues $\alpha'$ valid
for $np$ or $np^5$ configurations and ${^2P}$ term. Atoms are
$J_c=3/2$ and $J_d=1/2$. The label $g$ denotes a gerade state, and
$u$ denotes ungerade. States that are $0^+$ are labelled $a$ and
$0^-$ is labelled $b$. Conversion factor for $J=1/2$ is
$\rho^{(1)}_0=\sqrt{\frac{3}{2}}\rak{a}^{(1)}_{0}$. Conversion
factors for $J=3/2$ are given in Table
\ref{tab-3h1}.\label{tab-s3h1h}}
\end{table*}

\begin{table*}
\begin{footnotesize}
\begin{tabular}{cr@{\hspace{0.5cm}}l@{\hspace{0.5cm}}rr}\hline
$\Omega$ & $\alpha'(20\Omega)$ & Eigenvector & $\rak{a}^{(1)}_{0}$
& $\rak{a}^{(2)}_{0}$ \\\hline
0&$0.48ga$&\ket{+}{0}{0}$_g$&$0.000$&$-1.000$\\
 &$-0.48ua$&\ket{+}{0}{0}$_u$&$0.000$&$-1.000$\\
1&$-0.32g$&\ket{+}{1}{0}$_g$&$0.408$&$-0.500$\\
 &$0.32u$&\ket{+}{1}{0}$_u$&$0.408$&$-0.500$\\
2&$0.08g$&\ket{+}{2}{0}$_g$&$0.816$&$1.000$\\
 &$-0.08u$&\ket{+}{2}{0}$_u$&$0.816$&$1.000$\\
\hline
\end{tabular}
\end{footnotesize}
\caption{As for Table \ref{tab-s3h1h}, for atoms $J_c=2$, $J_d=0$
and valid for $np^2$ or $np^4$ (${^3P}$) configuration. The
polarization parameters for $J=2$ are shown: $J=0$ can not be
polarized. Conversion factors to $\rho^{(K)}_0$ are given in Table
\ref{tab-21}.\label{tab-s20}}
\end{table*}


\begin{table*}
\begin{footnotesize}
\begin{tabular}{cr@{\hspace{0.4cm}}l@{\hspace{0.2cm}}rrrr}\hline
&&& \multicolumn{2}{c}{$J=2$} & \multicolumn{2}{c}{$J=1$}\\
$\Omega$ & $\alpha'(21\Omega)$ & Eigenvector & $\rak{a}^{(1)}_{0}$
& $\rak{a}^{(2)}_{0}$&$\rak{a}^{(1)}_{0}$ &
$\rak{a}^{(2)}_{0}$\\\hline
0&$0.72ga$&\ket{+0.70711}{1}{-1}$_g$\ket{-0.70711}{-1}{1}$_g$&$0.000$&$-0.500$&$0.000$&$0.500$\\
 &$0.04988gb$&\ket{+0.42049}{1}{-1}$_g$\ket{+0.42049}{-1}{1}$_g$\ket{-0.80398}{0}{0}$_g$&$0.000$&$-0.823$&$0.000$&$-0.470$\\
 &$-0.76988gb$&\ket{+0.56850}{1}{-1}$_g$\ket{+0.56850}{-1}{1}$_g$\ket{+0.59466}{0}{0}$_g$&$0.000$&$-0.677$&$0.000$&$-0.030$\\
 &$0.52988ub$&\ket{+0.68526}{1}{-1}$_u$\ket{+0.68526}{-1}{1}$_u$\ket{-0.24667}{0}{0}$_u$&$0.000$&$-0.530$&$0.000$&$0.409$\\
 &$-0.28988ub$&\ket{+0.17442}{1}{-1}$_u$\ket{+0.17442}{-1}{1}$_u$\ket{+0.96910}{0}{0}$_u$&$0.000$&$-0.970$&$0.000$&$-0.909$\\
 &$-0.48ua$&\ket{+0.70711}{1}{-1}$_u$\ket{-0.70711}{-1}{1}$_u$&$0.000$&$-0.500$&$0.000$&$0.500$\\
1&$0.07416g$&\ket{+0.18759}{2}{-1}$_g$\ket{-0.90918}{1}{0}$_g$\ket{-0.37175}{0}{1}$_g$&$0.366$&$-0.516$&$0.073$&$-0.740$\\
 &$-0.19416g$&\ket{+0.79465}{2}{-1}$_g$\ket{-0.08198}{1}{0}$_g$\ket{+0.60150}{0}{1}$_g$&$0.518$&$0.266$&$-0.191$&$0.490$\\
 &$-0.48g$&\ket{+0.57735}{2}{-1}$_g$\ket{+0.40825}{1}{0}$_g$\ket{-0.70711}{0}{1}$_g$&$0.340$&$-0.250$&$0.118$&$0.250$\\
 &$0.68010u$&\ket{+0.44834}{2}{-1}$_u$\ket{-0.25509}{1}{0}$_u$\ket{-0.85669}{0}{1}$_u$&$0.191$&$-0.565$&$0.377$&$0.402$\\
 &$0.06955u$&\ket{+0.34489}{2}{-1}$_u$\ket{-0.83484}{1}{0}$_u$\ket{+0.42907}{0}{1}$_u$&$0.382$&$-0.414$&$0.046$&$-0.545$\\
 &$-0.38965u$&\ket{+0.82465}{2}{-1}$_u$\ket{+0.48783}{1}{0}$_u$\ket{+0.28632}{0}{1}$_u$&$0.652$&$0.479$&$-0.423$&$0.143$\\
2&$0.84g$&\ket{+0.57735}{2}{0}$_g$\ket{+0.81650}{1}{1}$_g$&$0.544$&$0.000$&$0.471$&$0.000$\\
 &$0.12g$&\ket{+0.81650}{2}{0}$_g$\ket{-0.57735}{1}{1}$_g$&$0.680$&$0.500$&$0.236$&$-0.500$\\
 &$0.12u$&\ket{+1.00000}{2}{0}$_u$&$0.816$&$1.000$&$0.000$&$-1.000$\\
 &$-0.48u$&\ket{+1.00000}{1}{1}$_u$&$0.408$&$-0.500$&$0.707$&$0.500$\\
3&$-0.36g$&\ket{+1.00000}{2}{1}$_g$&$0.816$&$1.000$&$0.707$&$0.500$\\
&$0.12u$&\ket{+1.00000}{2}{1}$_u$&$0.816$&$1.000$&$0.707$&$0.500$\\
\hline
\end{tabular}
\end{footnotesize}
\caption{As for Table \ref{tab-s3h1h}, for atoms $J_c=2$, $J_d=0$
and valid for $np^2$ or $np^4$ (${^3P}$) configuration. Conversion
factors to $\rho^{(K)}_0$ are given in Tables \ref{tab-11} and
\ref{tab-21}.\label{tab-s21}}
\end{table*}

%**********************
\begin{table*}
\begin{tabular}{lrr@{\hspace{0.5cm}}rr@{\hspace{0.5cm}}rr}\hline
States $(n\Omega,n'\Omega')$ & $M_a$ & $M_a'$ & ${\mathcal
T}^{n\Omega}$ & ${\mathcal T}^{n'\Omega'}$ & ${\mathcal
T}^{n\Omega}{\mathbf T_{n\Omega}}$ & ${\mathcal
T}^{n'\Omega'}{\mathbf T^{*}_{n'\Omega'}}$
\\\hline
%
$A1,B0$ & $\frac{3}{2}$ & $\frac{1}{2}$ & $\frac{1}{\sqrt{2}}$ &
$\frac{1}{2}$ & $\frac{1}{\sqrt{2}}r_A{\rm e}^{i(\phi_A+\psi_A)}$
& $\frac{1}{2}r_B{\rm e}^{-i(\phi_B+\psi_B)}$\\
%
& ${-\frac{1}{2}}$ & $-\frac{3}{2}$ & $-\frac{1}{\sqrt{2}}$ &
${-\frac{1}{2}}$ & $-\frac{1}{\sqrt{2}}r_A{\rm
e}^{i(\phi_A+\psi_A)}$ & $-\frac{1}{2}r_B{\rm
e}^{-i(\phi_B+\psi_B)}$\\
%
$a1,B0$ & $\frac{3}{2}$ & $\frac{1}{2}$ & $\frac{1}{\sqrt{2}}$ &
$\frac{1}{2}$ & $\frac{1}{\sqrt{2}}r_a{\rm e}^{i(\phi_a+\psi_a)}$
& $\frac{1}{2}r_B{\rm e}^{-i(\phi_B+\psi_B)}$\\
%
& ${-\frac{1}{2}}$ & $-\frac{3}{2}$ & $\frac{1}{\sqrt{2}}$ &
${-\frac{1}{2}}$ & $\frac{1}{\sqrt{2}}r_a{\rm
e}^{i(\phi_a+\psi_a)}$ & $-\frac{1}{2}r_B{\rm
e}^{-i(\phi_B+\psi_B)}$\\
%
$C1,(C,-1)$ & $1$ & $-1$ & $\frac{1}{\sqrt{2}}$ &
$\frac{1}{\sqrt{2}}$ & $\frac{1}{\sqrt{2}}r_C{\rm
e}^{i(\phi_C+\psi_C)}$
& $-\frac{1}{\sqrt{2}}r_C{\rm e}^{-i(\phi_C+\psi_C)}$\\
%
$A1,(z,-1)$ & $\frac{3}{2}$ & $-\frac{1}{2}$ &
$\frac{1}{\sqrt{2}}$ & $1$ & $\frac{1}{\sqrt{2}}r_A{\rm
e}^{i(\phi_A+\psi_A)}$
& $-r_z{\rm e}^{-i(\phi_z+\psi_z)}$\\
\hline
\end{tabular}
\caption{Fragment terms (columns 6 and 7) in the summation
(Eq.\ref{eqn-fk}) for the calculations of $f_K(q,q')$ outlined in
the text. Pairs of states are listed on the left in column 1:
unprimed variables refer to the first state, primed variables
refer to the second. Columns 2 and 3 show the $M_J$ states
involved, columns 4 and 5 show the expansion coefficients
$\mathcal{T}^{n\Omega}_{J_aM_aJ_bM_b}$ taken from the tabulated
eigenvectors.\label{tab-f1}}
\end{table*}

\end{document}