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I have this code:

begin{center}

$left.% Use `left.` if don't want arrow on this side.

begin{tabular}{r r r }

toprule

 strength & region & comments \

midrule

 $H_B ll H_0$ & Landau levels \

$ H_B lessapprox H_0 $& $n$-mixing & $n$ not good\

  & $ell$-lmixing & $ell$ not good \

 $H_B ll B_0$ & Paschen-Back regime & $J$ noot good, $L$ and $S$ good \

 & Intermediate regime & $J$ and $I$ good, $F$ not good \

 & Zeeman effect & $F$ good \

 bottomrule

end{tabular}

quad

rightupdownarrow%  Use `right.` if don't want arrow on this side.

rotatebox[origin=100]{0}{Landau limit}$

caption{HAILO}

end{center}

that generates:

What I would like:

• how can I make the entries in the first column (strengh, region, comments) centred with the columns contents?




• How can I have the Landau limit text at the top right corner of the table, like I drew with the red arrow?

Thanks.

How about this, based on bigdelim and makecell? The margin for the caption was obtained by trial and error – it depends on the text of the last column.

documentclass{article}

usepackage{geometry} 

usepackage{amssymb}

usepackage{caption, booktabs, bigdelim, makecell}%

renewcommand{theadfont}{normalsizebfseries}



begin{document}



begin{table}[!htb]

centering

captionsetup{margin={0pt,28mm},skip=6pt}

caption{HAILO}

begin{tabular}{r r r c@{}l}

cmidrule[heavyrulewidth]{1-3}

 thead{strength} & thead{region} & thead{comments} & rdelimupdownarrow{7}{*} & Landau limit \

cmidrule{1-3}

 $H_B ll H_0$ & Landau levels \

$ H_B lessapprox H_0 $& $n$-mixing & $n$ not good\

  & $ell$-lmixing & $ell$ not good \

 $H_B ll B_0$ & Paschen-Back regime & $J$ noot good, $L$ and $S$ good \

 & Intermediate regime & $J$ and $I$ good, $F$ not good \

 & Zeeman effect & $F$ good \

cmidrule[heavyrulewidth]{1-3}

end{tabular}

end{table}



end{document} 

Without use of booktabs for the table (but clines instead) and with use of tikz with a n arrow that its height is found with tests:

documentclass{article}

usepackage{amssymb}

usepackage{caption}

usepackage{tikz}

usepackage{multirow}

begin{document}

begin{center}

$% Use `left.` if don't want arrow on this side.

begin{tabular}[t]{r r r c l}cline{1-3}

 multicolumn{1}{c}{strength} & multicolumn{1}{c}{region} & multicolumn{1}{c}{comments} &multirow{6}{*}{begin{tikzpicture}draw[<->](0,0)--(0,2.9cm);end{tikzpicture}}&Landau limit\

cline{1-3}

 $H_B ll H_0$ & Landau levels &&\

$ H_B lessapprox H_0 $& $n$-mixing & $n$ not good&&\

  & $ell$-lmixing & $ell$ not good &&\

 $H_B ll B_0$ & Paschen-Back regime & $J$ noot good, $L$ and $S$ good &&\

 & Intermediate regime & $J$ and $I$ good, $F$ not good &&\

 & Zeeman effect & $F$ good &&\

 cline{1-3}

end{tabular}

rotatebox[origin=100]{0}{}$

captionof{table}{HAILO}

end{center}

end{document}

Output:

PS: Just as a start...

(Needed packages for OP's non working example:

usepackage{booktabs}

usepackage{amssymb}

usepackage{rotating}

and

usepackage{caption}

for the fix with captionof{table} instead of caption{}

)