(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the other sp hybrid orbital given Ψsp1 = 3Ψ2s + 4Ψ2pz using the orthonormal relationships.

2. Relevant equations

I know that you are supposed to use the orthonormal relationships like stated in the problem, and when finding the the second sp hybrid orbital for the normalized

Ψ[itex]_{1}[/itex]sp1 = [itex]\frac{1}{\sqrt{2}}[/itex](Ψ2s + Ψ2pz)

Ψ[itex]_{2}[/itex]sp1 = [itex]\frac{1}{\sqrt{2}}[/itex](Ψ2s - Ψ2pz)

and the integral for orthogonality and normalization is the standard:

[itex]\int[/itex] Ψ[itex]_{1}[/itex]sp1* Ψ[itex]_{2}[/itex]sp1=0

[itex]\int[/itex] Ψ[itex]_{1}[/itex]sp1* Ψ[itex]_{1}[/itex]sp1=1

3. The attempt at a solution

This is what I get and I'm not very confident about it. If someone could help me finish it or point me in the right direction I would appreciate it.

0=[itex]\int[/itex](3Ψ2s*+4Ψ2pz*)(c[itex]_{1}[/itex]Ψ2s+c[itex]_{2}[/itex]Ψ2pz)

Which I get reduces to

0= 3c[itex]_{1}[/itex] + 4c[itex]_2{}[/itex]

c[itex]_{2}[/itex] = -3/4c[itex]_{1}[/itex]

I then plugged in and used the relationship:

1= [itex]\int[/itex](c[itex]_{1}[/itex]Ψ2s* - 3/4c[itex]_{1}[/itex]Ψ2pz*)(c[itex]_{1}[/itex]Ψ2s - 3/4c[itex]_{1}[/itex]Ψ2pz)

which I get reduces to

1= -3/2c[itex]_{1}[/itex][itex]^{2}[/itex]

so

c[itex]_{1}[/itex]= +/- [itex]\sqrt{2/3}[/itex] and

c[itex]_{2}[/itex]= +/- [itex]\sqrt{6}[/itex]/4

Can someone double check me or point out any errors?

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Sp hybrid orbitals orthonormality

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