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Mobile single interstitials can grow into extended interstitial defect structures during thermal anneals following ion implantation. The silicon tetra-interstitials present an important intermediate structure that can either provide a chain-like nucleation site for extended structures or form a highly stable compact interstitial cluster preventing further growth. In this paper, dimer searches using the tight-binding (TB) model by Lenosky et al. and density functional calculations show that the compact ground-state I^α ₄ and the I₄-chain are surrounded by high-lying neighboring local minima. To furthermore explore the phase space of tetra-interstitial structures an empirical potential is optimized to a database of silicon defect structures. The minima hopping method combined with this potential extensively searches the energy landscape of tetra-interstitials and discovers several new low-energy I₄ structures. The second lowest-energy I₄ structure turns out to be a distorted ground-state tri-interstitial bound with a single interstitial, which confirms that the ground-state tri-interstitial may serve as a nucleation center for the extended defects in silicon.