Construction of Hybrid Orbitals

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!

  =  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 + a1 b1 + a1 b1 + b1² ↓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:
   =  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 − |x> − |y> + |z>) |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.

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