Page 54 - Mechanism and Theory in Organic Chemistry

Appendix 1

HYBRID

In order to obtain correct expressions for hybrid orbitals, we first need to describe
more precisely than has been done in Section 1.2 some properties of orbitals.
Recall that v2, the square of an orbital function, gives the probability of finding
the electron in each region of space. If we were to add up the values of this func-
tion over all points, we would have the total probability of finding the electron,
which should equal unity. Orbitals are ordinarily constructed so as to satisfjr this
requirement; when they are, they are said to be normalized. The normalization
condition is Equation Al.l, where d~ signifies integration over all coordinates.

Normalization: 1 T2 d~ = 1 (Al.1)
A second condition, which does not apply generally to orbitals but which
does apply to different atomic orbitals on the same atom is that they do not overlap
with each other. The correct terminology for orbitals that have zero overlap is
that they are orthogonal. We have seen that the overlap of two orbitals is found by
integrating over space the product v,v2. Since our s and p orbitals, and also the
resulting hybrids, are on the same atom, we require for any pair in the s, p set
and also for any pair in the hybrid set that Equation A1.2 be satisfied :

Orthogonality: 1 cpp, d~ = 0 (A 1.2)
We now write a general formulation for our set of hybrid orbitals. Each
hybrid, xi, is going to be written as a sum of contributions from the s, p,, p,, p,
atomic orbitals, each with a coefficient that tells the extent of its contribution.

"The discussion partly follows C. Hsu and M. Orchin, J. Chem. Educ., 50, 114 (1973).