Is the C atom in methane sp3 hybridized because it’s tetrahedral or is it tetrahedral because it’s sp3 hybridized? It’s funny how many students think to this date that the correct answer is the latter; specially those working in inorganic chemistry. I ignore the reason for such trend. What is true is that most chemistry teachers seem to have lost links to certain historical facts that have shaped our scientific discipline; most of those lay in the realm of physics, maybe that’s why.
What Linus Pauling, in a very clever way, stated was that once you have a set of eigenvectors (orbitals) of the atomic Hamiltonian any combination of them will also be an eigenvector (which is normal since one of the properties of Hermitian operators is that they are linear); so why not making a symmetry adapted one? Let’s take the valence hydrogenoid orbitals (hydrogenoid being the keyword here) and construct a linear combination of them, in such a way that the new set transforms under the irreducible representations of a given point group. In the case of methane, the 2s and 2p orbitals comprise the valence set and their symmetry-adapted-linear-combination under the Td point group constitutes a set of new orbitals which now point into the vertexes of a tetrahedron. Funny things arise when we move to the next period of the table; it has been a controversy for a number of years the involvement of empty d orbitals in pentacoordinated P(V) compounds. Some claim that they lay too high in energy to be used in bond formation; while others claim that their involvement depends on the nature (electronegativity mainly) of the surrounding substituents.
In many peer reviewed papers authors are still making the mistake of actually assigning a type of hybridization to set of valence orbitals of an atom based on the bond angles around it. Furthermore, it is not uncommon to find claims of intermediate hybridizations when such angles have values in between those corresponding to the ideal polyhedron. Symmetry is real, orbitals are not; they are just a mathematical representation of the electron density distribution which allows us to construct mind images of a molecule.
Linus Pauling is one of my favourite scientific historical figures. Not only did he build a much needed at the time bridge between physics and chemistry but he also ventured into biochemistry (his model of an alpha-helix for the alanine olygopeptide became the foundation to Watson & Cricks later double helix DNA model), X-ray diffractometry, and humanities (his efforts in reducing/banning the proliferation of nuclear weapons got him the Nobel Peace Prize long after he had already received the Nobel Price in Chemistry). He was a strong believer of ortho-molecular nutrition, suggesting that most illnesses can be related to some sort of malnutrition. Linus Pauling and his book On the Chemical Bond will remain a beacon in our profession for the generations to come.
Disclaimer: The question above, with which I opened this post, was taken from an old lecture by Dr. Raymundo Cea-Olivares at UNAM back in the days when I was an undergraduate student.