# Wave Function for sp, sp^{2} and sp^{3} Hybrid Orbitals

The wave function of hybrid orbitals are obtained by the linear combination of the angular wave functions of the appropriate atomic orbitals.

1. Ψ

2. Ψ

The values of the linear combination coefficients a

I. Ψ

II. Ψ

III. Ψ

From Condition-1: Ψ

From Condition-2: Ψ

Sine the s-orbital is sperically symmetrical and the two hybrid orbitals Ψ

i.e.

## sp

Wave functions for three hybrid orbitals may be written as-

Ψ

Ψ

Ψ

Since, s orbital is equally distributed among the three hubrid orbitals, so, we have-

Assume that Ψ

Since, Ψ

Since, Ψ

a

a

Hence,

Again, Since Ψ

Further, Ψ

Let c

i.e. a

Thus, If, c

Hence, the three wave functions are-

## sp

The wave function of four sp

Ψ

Ψ

Ψ

Ψ

Since the s orbital is equally distributed among the four hybrid orbitals. So, we have-

Assuming that Ψ

that means, c

Since, Ψ

Hence,

The orthogonality conditions for Ψ

a

or, b

or, b

a

b

or, b

a

b

or, b

Assuming that Ψ

That means,

c

The normalized condition for Ψ

The requirement of orthogonality conditions for Ψ

a

or, d

or, d

and a

or, d

or, d

The normalization required for Ψ

Similarly, normalization requirement for Ψ

c

Hence, the four sp

Ψ

Ψ

Ψ

Ψ

## sp Hybrid Orbital Wave Function

The combination of one s and one 2p orbital, giving two hybrid orbitals may be expressed as-1. Ψ

_{1}= a_{1}Ψ_{s}+ b_{1}Ψ_{2p}---Equation-12. Ψ

_{2}= a_{2}Ψ_{s}+ b_{2}Ψ_{2p}---Equation-2The values of the linear combination coefficients a

_{1}, b_{1}, a_{2}and b_{2}may be determined by the following considerations-I. Ψ

_{1}and Ψ_{2}are normalizedII. Ψ

_{1}and Ψ_{2}are orthogonalIII. Ψ

_{1}and Ψ_{2}are equivalentFrom Condition-1: Ψ

_{1}and Ψ_{2}are normalizedFrom Condition-2: Ψ

_{1}and Ψ_{2}are orthogonalSine the s-orbital is sperically symmetrical and the two hybrid orbitals Ψ

_{1}and Ψ_{2}are equivalent, the share of 's' function is equal in both Ψ_{1}and Ψ_{2}i.e.

## sp^{2} Hybrid Orbital Wave Function

Wave functions for three hybrid orbitals may be written as-Ψ

_{sp2}(1) = a_{1}Ψ_{2s}+ b_{1}Ψ_{2px}+ c_{1}Ψ_{2py}---Equation-1Ψ

_{sp2}(2) = a_{2}Ψ_{2s}+ b_{2}Ψ_{2px}+ c_{2}Ψ_{2py}---Equation-2Ψ

_{sp2}(3) = a_{3}Ψ_{2s}+ b_{3}Ψ_{2px}+ c_{3}Ψ_{2py}---Equation-3Since, s orbital is equally distributed among the three hubrid orbitals, so, we have-

Assume that Ψ

_{sp2}(1) points towards the x-axis. Hence, this hybrid orbital will have contribution from 2s and 2p_{x}orbitals only. The contribution of the 2p_{y}orbital will be zero. So, c_{1}= 0.Since, Ψ

_{sp2}(1) is normalized,Since, Ψ

_{sp2}(1) and Ψ_{sp2}(2) are orthogonal and Ψ_{sp2}(1) and Ψ_{sp2}(3) are also orthogonal. Hence-a

_{1}a_{2}+ b_{1}b_{2}= 0a

_{1}a_{3}+ b_{1}b_{3}= 0Hence,

Again, Since Ψ

_{sp2}(2) is normalized-Further, Ψ

_{sp2}(3) is also normalized-Let c

_{2}have a positive value, then the value of c_{3}will have to be negative as Ψ_{sp2}(2) and Ψ_{sp2}(3) are orthogonal.i.e. a

_{2}a_{3}+ b_{2}b_{3}+ c_{2}c_{3}= 0Thus, If, c

_{2}= +1/√2 and c_{3}= −1/√2Hence, the three wave functions are-

## sp^{3} Hybrid Orbital Wave Function

The wave function of four sp^{3}hybrid orbitals may be written as-Ψ

_{sp3}(1) = a_{1}Ψ_{2s}+ b_{1}Ψ_{2px}+ c_{1}Ψ_{2py}+ d_{1}Ψ_{2pz}---Equation-1Ψ

_{sp3}(2) = a_{2}Ψ_{2s}+ b_{2}Ψ_{2px}+ c_{2}Ψ_{2py}+ d_{2}Ψ_{2pz}---Equation-2Ψ

_{sp3}(3) = a_{1}Ψ_{2s}+ b_{3}Ψ_{2px}+ c_{3}Ψ_{2py}+ d_{3}Ψ_{2pz}---Equation-3Ψ

_{sp3}(4) = a_{4}Ψ_{2s}+ b_{4}Ψ_{2px}+ c_{4}Ψ_{2py}+ d_{4}Ψ_{2pz}---Equation-4Since the s orbital is equally distributed among the four hybrid orbitals. So, we have-

Assuming that Ψ

_{sp3}(1) is developed along x-axis, so the contribution from 2p_{y}and 2p_{z}orbitals to the hybrid orbitals is zero.that means, c

_{1}= d_{1}= 0Since, Ψ

_{sp3}(1) is normalized.Hence,

The orthogonality conditions for Ψ

_{sp3}(1) and Ψ_{sp3}(2), Ψ_{sp3}(1) and Ψ_{sp3}(3) and Ψ_{sp3}(1) and Ψ_{sp3}(4) are -a

_{1}a_{2}+ b_{1}b_{2}= 0or, b

_{2}= − a_{1}a_{2}/b_{1}or, b

_{2}= − 1/2√3a

_{1}a_{3}+ b_{1}b_{3}= 0b

_{3}= − a_{1}a_{3}/b_{1}or, b

_{3}= − 1/2√3a

_{1}a_{4}+ b_{1}b_{4}= 0b

_{4}= − a_{1}a_{4}/b_{1}or, b

_{4}= − 1/2√3Assuming that Ψ

_{sp3}(2) lies in the xz-plane, the hybrid orbital will have contributions from 2s, 2p_{x}and 2p_{z}orbitals and the contribution of 2p_{y}to Ψ_{sp3}(2) is zero.That means,

c

_{2}= 0The normalized condition for Ψ

_{sp3}(2) gives-The requirement of orthogonality conditions for Ψ

_{sp3}(2) and Ψ_{sp3}(3), and for Ψ_{sp3}(2) and Ψ_{sp3}(4) gives-a

_{2}a_{3}+ b_{2}b_{3}+ d_{2}d_{3}= 0or, d

_{3}= −(a_{2}a_{3}+ b_{2}b_{3})/d_{2}or, d

_{3}= − 1/√6and a

_{2}a_{4}+ b_{2}b_{4}+ d_{2}d_{4}= 0or, d

_{4}= −(a_{2}a_{4}+ b_{2}b_{4})/d_{2}or, d

_{4}= − 1/√6The normalization required for Ψ

_{sp3}(3) gives-Similarly, normalization requirement for Ψ

_{sp3}(4) and the orthogonality condition between Ψ_{sp3}(3) and Ψ_{sp3}(4) gives-c

_{4}= − 1/√4Hence, the four sp

^{3}hybrid orbital wave functions can be written as-Ψ

_{sp3}(1) = (1/2)Ψ_{2s}+ (√3/2)Ψ_{2p}Ψ

_{sp3}(2) = (1/2)Ψ_{2s}− (1/2)Ψ_{2px}+ (√2/√3)Ψ_{2py}Ψ

_{sp3}(3) = (1/2)Ψ_{2s}− (1/2√3)Ψ_{2px}+ (1/√2)Ψ_{2py}− (1/√6)Ψ_{2pz}Ψ

_{sp3}(4) = (1/2)Ψ_{2s}− (1/2√3)Ψ_{2px}+ (1/√2)Ψ_{2py}− (1/√6)Ψ_{2pz}**Share**