Why only sigma bond determine hybridisation?
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VSEPR is used to predict highly idealized geometries corresponding to Platonic solids or combinations thereof (e.g., the trigonal bipyramidal geometry being essentially the result of fusing two faces of a tetrahedron). The basic rationale behind its use, and the reason for its success in many situations, is the recognition that electrons, being like-charged, repel one another electrostatically. Hence, as a first approximation, it's reasonable to assume that the most stable spatial configuration of any molecule would have the atoms spaced as uniformly and as distant from one another as possible. VSEPR achieves this by placing the central atom of a simple molecule (or some molecular sub-structure) at the center of a geometrically perfect polyhedron having as many vertices as non-central atoms and lone pairs, and then placing the non-central atoms and lone pairs at those vertices.
The reason that sigma bonds (and lone pairs) determine the geometry is that they form the basic skeleton of the molecule. Recall that sigma bonds are formed by head-on overlap of atomic orbitals, meaning that they are oriented along the imaginary axis connecting two atomic nuclei, and hence concentrate electron density in the region directly between the two nuclei. pi bonds, on the other hand, are essentially orthogonal to the sigma bond skeleton, and are substantially weaker. Moreover, pi bonds do not exist in isolation, meaning any pi bond between two given atoms is always formed secondarily to the sigma bond between said atoms. As such, pi bonds do not alter the basic idealized geometry of a molecule as dictated by sigma bonding, although in practice, because they do actually introduce additional electron density and require closer orbital overlap, bond multiplicity affects bond length and bond angles (as does, for that matter, the size of the atoms involved). The magnitude of those deviations from the idealized VSEPR geometry could, in principle, be quite large, although most often those deviations are slight enough that VSEPR remains useful as a first approximation
The reason that sigma bonds (and lone pairs) determine the geometry is that they form the basic skeleton of the molecule. Recall that sigma bonds are formed by head-on overlap of atomic orbitals, meaning that they are oriented along the imaginary axis connecting two atomic nuclei, and hence concentrate electron density in the region directly between the two nuclei. pi bonds, on the other hand, are essentially orthogonal to the sigma bond skeleton, and are substantially weaker. Moreover, pi bonds do not exist in isolation, meaning any pi bond between two given atoms is always formed secondarily to the sigma bond between said atoms. As such, pi bonds do not alter the basic idealized geometry of a molecule as dictated by sigma bonding, although in practice, because they do actually introduce additional electron density and require closer orbital overlap, bond multiplicity affects bond length and bond angles (as does, for that matter, the size of the atoms involved). The magnitude of those deviations from the idealized VSEPR geometry could, in principle, be quite large, although most often those deviations are slight enough that VSEPR remains useful as a first approximation
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