MO Theory, sp Hybridization question

[sgoodrow]

Recently I begun learning about molecular orbitals and the geometry of an sp orbital. It is my understanding that in the creation of an sp orbital with the combination of an s orbital and p orbital, one gets something that resembles half of a p-orbital (although larger). My professor noted that the other lobe exists infinitesimally small, in opposite phase. She said it was beyond the scope of our course to discuss how this lobe would affect the geometry of MOs, but I was curious regardless. When I asked her she said she would get back to me, and hasn't yet, so I thought I'd ask around on my own!

Why/when do these infinitesimally small lobes affect the geometry of an MO? Probabilistically speaking, what is the math behind it? Basically, if anyone knows anything about the relationship/importance of maintaining the infinitesimally small lobe of an sp-orbital, do tell!

Thanks, I look forward to hearing from you!

 

[JPRitchie]

The out-of-phase lobe of an sp hybrid is smaller than the in-phase lobe, but not infinitesimally small. It's needed to maintain orthogonality between different hybrids on the same atom. That is, the properly normalized sum and difference of an s and p orbital gives two sp hybrids that are orthogonal to one another. If the hybrids were somehow positive everywhere, the overlap integral couldn't be zero. Maintaining orthogonality ensures the orbitals span the orbital space most effectively.
-Jim