sp-Hybrids
Two new wave functions as linear combination of the functions for 2s and 2pz:
Ψ1 = a1Ψ2s
+ b1 Ψ2pz
Ψ2 = a2Ψ2s
+ b2 Ψ2pz
abbreviated to:
|i> = ai|s> + bi|z> (i = 1, 2)
The resulting wave functions shall be orthogonal!
<i|k> = δik |
We recieve three equations: <1|1> = 1; <1|2> = <2|1> = 0; <2|2> = 1. In addition, we shall get bonds of the same strength, i.e. the contribution of s is the same for both hybrid orbitals:
ai = a |
From this, we deduce (a1 = a2).
Thus, we obtain
<1|1> = a1² | <s|s> | + a1 b1 | <s|z> | + a1 b1 | <z|s> | + b1² | <z|z> |
↓ 1 |
↓ 0 |
↓ 0 |
↓ 1 |
<1|1> = a1² + b1² = 1
<2|2> = a2² + b2² = 1
<1|2> = a1a2 + b1b2 = 0
and (ai = a)
a1 = a2 |
a1² + b1² = 1
a1² + b2² = 1 |
} | b1² = b2² |
b1 = −b2
For b1 = +b2 is |1> ≡ |2>
a1² + b1b2 = 0 ⇒
a1 = b1
Inserted in a1² + b1² = 1 ⇒ 2a1² = 1 ⇒ a1 = 1/√2
|1> = 1/√2(|s> + |z>)
|2> = 1/√2(|s> − |z>) |
sp²-Hybrids
As shown above, it is possible to deduce a description for a sp2 hybrid:
|i> = ai |s> + bi |z> + ci |x> i = 1, 2, 3
In total, 9 coefficients have to be determined:
<i|k> = δik | 3 + 3 = 6 equations (3 related to normalization, 3 to orthogonality) |
ai = a for i = 1, 2, 3 | 2 equations |
c1 = 0 | 1 equation (one hybrid orbital is assumed to point in direction z) |
|1> = 1/√3(|s>
+ √2 |z>)
|2> = 1/√3(|s>−1/√2 |z> + √(3/2) |x>) |3> = 1/√3(|s>−1/√2 |z> − √(3/2) |x>) |
sp³-Hybrids
To describe sp³ hybridization, one hybrid orbital is assumed to have z-orientation, in consequence, the coefficients for |x>, an |y> are zero. The axis of one second hybrid orbital shall (without any limitation of generality) within the x-z-plane. Therefore, the respective coefficient for |y> is zero.
16 unknowns; 4 equations related to normalization , 6 to orthogonality,
3 following ai = a and 3 following the assumptions made with respect to orientation in the coordinate system
(c1 = d1 = d2 = 0)
|1> = ½ |s> + √(3/4) |z> |
|2> = ½ |s >−√(1/12) |z> + √(2/3) |x> |
|3> = ½ |s>−√(1/12) |z> − √(1/6) |x> + 1/√2 |y> |
|4> = ½ |s>−√(1/12) |z> − √(1/6) |x> − 1/√2 |y> |
A symmetric notation for the four hybrid orbitals is
|1> = ½ (|s> + |x> + |y> + |z>) |
|2> = ½ (|s |
|3> = ½ (|s> + |x> − |y> − |z>) |
|4> = ½ (|s>− |x> + |y> − |z>) |
These hybrid orbitals are as well normalized and orthogonal to each other which can be proved easily.