Sometimes the obvious is the most difficult to see. I was dealing with the topic of hybridization using my PowerPoints that you have seen from the Counting Orbitals II: Hybrids blogpost. N.B: Our book uses steric number to describe the number of Valence Shell Electron Pairs (VSEP). See Table 1 below.
In re-emphasizing the VSEPR geometry I was restating "Linear-Trigonal Planar-Tetrahedral sp, sp2, sp3 ". A student asks, "So, they follow the numbers 1, 2, 3 for the dimensions." That is sp1 (explicitly showing the number of p-orbitals used in the hybridization, 1) is 1-dimensional (= linear), sp2 is 2-dimensional (= trigonal planar), and sp3 is 3-dimensional (= tetrahedral). Not recalling the exact content I had in high school, I will discount that, and say I have been at this for 45 years, and this never occurred to me. Perhaps you are as good as my student and already teach it this way, but I was floored.
Table 2 below is a revision of Table 1:
It has to be this way, doesn't it? The rules of wavefunctions are:
1) # atomic orbitals in = # hybrid orbitals out, and
2) the hybrids are linear combinations of wavefunctions,
...so their character, especially from a spatial/directional point of view, is maintained. The hybrid, so to speak, "remembers" the directionality of the component p-orbitals. Forgive me for dispensing with the Ψ's and various subscripts and superscripts and constants. So, I am just giving you the Linear Combination of Atomic Orbitals (LCAO's) using the labels of the orbitals to represent the hybrids.
- sp1 + sp1 = Some Normalization Constant * [ (s + pz) + (s – pz) ] Linear—one-dimensional in the pz direction
- sp2 + sp2 + sp2 = Some Other Normalization Constants * [(s + pz) +(s + pz – px) + (s – pz – px)] Trigonal Planar—two-dimensional in the pz – px plane
- sp3 + sp3 + sp3 + sp3 = Yet Another Normalization Constant * [(s + px + py + pz) + (s – px – py + pz) + (s + px – py – pz) +(s – px + py – pz)] Tetrahedral—three dimensional
Now, I had to go to a Physical Chemistry text to get the conventional signs on the wave-functions for the sp2 and sp3 hybrids, so I definitely do not expect you to go into this much detail with General Chemistry students. You do not have to. The beauty of this is that you remember, just as the hybrid orbitals "remember", follow the numbers 1, 2, 3 for the dimensions.