# Sp hybrid orbitals orthonormality

1. Nov 8, 2011

### josecuervo

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$_{1}$sp1 = $\frac{1}{\sqrt{2}}$(Ψ2s + Ψ2pz)

Ψ$_{2}$sp1 = $\frac{1}{\sqrt{2}}$(Ψ2s - Ψ2pz)

and the integral for orthogonality and normalization is the standard:

$\int$ Ψ$_{1}$sp1* Ψ$_{2}$sp1=0

$\int$ Ψ$_{1}$sp1* Ψ$_{1}$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=$\int$(3Ψ2s*+4Ψ2pz*)(c$_{1}$Ψ2s+c$_{2}$Ψ2pz)

Which I get reduces to

0= 3c$_{1}$ + 4c$_2{}$

c$_{2}$ = -3/4c$_{1}$

I then plugged in and used the relationship:

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

which I get reduces to

1= -3/2c$_{1}$$^{2}$

so

c$_{1}$= +/- $\sqrt{2/3}$ and
c$_{2}$= +/- $\sqrt{6}$/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