The sigma bonding orbitals of methane (CH₄) are derived using the Correspondence Theorem on basis functions, also known as the "Kim Method". The derived molecular orbitals are visualized with line sketches.
Even though the tetrahedral group (T𝒹) has 24 symmetry operations, we need not consider them individually, as we need to do with the traditional projection operator method. We also have no need for the characters of the irreducible representations of the group, only their basis functions.
This method makes it easy to determine which atomic orbitals of carbon combine with which molecular orbital - the A₁ representation has basis function "1", which signals an "s" type orbital, which we know (from energy considerations must be 2s), and the T₂ representation has basis functions x (so 2pₓ), y (so 2pᵧ) and z (2p𝓏 orbital of carbon)
Note an error in the video. In several places, the T₂ representation is written as "T₂𝓰". The "g" denotes "gerade"; the "gerade" and "ungerade" ("u") suffixes applies only if the molecule belongs to a centrosymmetric point group, and T𝒹 is not a centrosymmetric point group.
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