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The d orbital allows those elements that are period 3 and higher to have more room to fit electrons and therefore, satisfy the octet rule. This lets the central atom have extra lone pairs on them and therefore, the Lewis structure will have high repulsion to the bonded electrons form the higher density lone electron pairs. This will affect the shape of the molecule.
It allows expanded octets to form when the central atom is in the third period and down the periodic table. It can accommodate more electrons, as atoms in the p-block of the third, fourth, and fifth period have empty d-orbitals, and the examples of this in VSEPR are trigonal bipyramidal and octahedral.
In terms of hybridization, this allows an atom to have up to 5 or 6 areas of surrounding electron density (whether it may be lone pairs or bonds) and the hybridization for those central atoms would be sp3d and sp3d2 respectively.
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