Page 58 - Mechanism and Theory in Organic Chemistry

Hybrid Orbitals 47
the only way to have four equivalent hybrids is to let the coefficient of s be
.\/$ in each; these are the familiar sp3 orbitals.
We can use the generalized expression for hybrids to find the relation be-
tween hybridization indices and angle between two hybrids X, and X, (Equations
A1.lO). Since the expressions in braces are equivalent to ordinary three-dimen-
sional vectors of unit length, the three-dimensional vector dot product, found by

+
0,
XI = J Jx cos 4, fix + sin 0, sin 4, p, + cos 8,
{sin
s
1 + m1 1 + m1
J&
1
XZ= JGs+ (sin 0, cos 4, p, + sin 0, sin 4, p, + cos 0, p,}
summing the products of coefficients appearing inside the braces, must be equal
to the cosine of the angle a between the vectors (Equation Al.1 l).b But the
hybrids are directed along these vectors, so the angle between the hybrids is also

cos a = sin 8, cos 4, sin 8, cos 4, + sin 8, sin 4, sin 0, sin 4, + cos 0, cos 0, (Al.1 1)
a. We now bring in the requirement A1.5 that the two hybrids be orthogonal.
This condition gives Equation A1.12, which can be immediately simplified be-

xlxz = Jz /z JE Jz cos " sin 8' cos 4,
81
isin
+
+ sin 0, sin 4, sin 0, sin 4, + cos 0, cos 0,) (A1.12)

cause the expression in braces is equal to cos a from Equation Al. 1 1. Equations
A1.13 through A1.16 then follow.

If the two orbitals are equivalent, m, = m, = my Equation A1.16 reduces to the
even simpler expression A 1.17. The angle between two equivalent hybrids completely
- 1
COS a = -
m

Proof may be found, for example, in G. B. Thomas, Jr., Calculus and Analytic Geometry, Addison-
Wesley, Reading, Mass., 1953, p. 447.