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Chapter 20

Q. 20.9

Using the fact that the 2s  and  2p_{z} orbitals are normalized and orthogonal to each other, show that c_{1}  and  c_{2} both equal \sqrt{1/2} to normalize the hybrid orbitals.

Step-by-Step

Verified Solution

\int{ψ^{∗}_{2sp(1)}ψ_{2sp(1)}dq }=1

 

c^{2}_{1} \int{[−ψ_{2s}+ ψ_{2pz}][−ψ_{2s}+ ψ_{2pz}]dq } =1

 

c^{2}_{1} \int{ψ^{2}_{2s}dq} −2c^{2}_{1}\int{ψ_{2s}ψ_{2pz}dq} +c^{2}_{1} \int{ψ^{2}_{2pz}dq} =c^{2}_{1} (1-2×0+1)=2c^{2}_{1}=1

 

c_{1}=\frac{1}{\sqrt{2} }

 

\int{ψ^{∗}_{2sp(2)}ψ_{2sp(2)}dq }=1

 

c^{2}_{2} \int{[−ψ_{2s}− ψ_{2pz}]^{2}dq}=1

 

c^{2}_{2} \int{ψ^{2}_{2s} dq }+2c^{2}_{2}\int{ψ_{2pz}ψ_{2s}dq} +c^{2}_{2}\int{ψ^{2}_{2pz}dq}  =c^{2}_{2}(1+0+1) =2c^{2}_{2} =1

 

c_{2}=\frac{1}{\sqrt{2} }