A $(2\ifmmode\times\else\texttimes\fi{}2)$ structure formed by thallium (Tl) atoms on the Si(001) surface at 0.5 monolayer coverage at room temperature has been studied by a tensor low-energy electron-diffraction (LEED) analysis. Twenty-four structural models were tested. The valley-bridge para dimer model, being produced also by other trivalent metals, was found to be the most optimal. It is concluded, therefore, that the so-called ``inert pair effect'' of Tl atoms does not play a significant role in the case of this $(2\ifmmode\times\else\texttimes\fi{}2)$ structure. The structural parameters obtained by first-principles total-energy calculations are in good agreement with those obtained by the LEED analysis.