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Sp hybrid orbitals orthonormality

  1. Nov 8, 2011 #1
    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
     
  2. jcsd
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