Coefficients of wave function of a hybrid orbital

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The discussion focuses on determining the coefficients for a hybrid orbital expressed as Ψ(sp3) = aΨ(2s) + aΨ(2px) + aΨ(2py) + aΨ(2pz), with the assumption that the 2s and 2p wavefunctions are normalized. Participants clarify that the coefficients must be fixed by normalizing the wavefunction and ensuring orthonormality among the four hybrid orbitals. It is concluded that if the other three hybrid orbitals have negative coefficients, the coefficient for this specific hybrid orbital should be +1/2. The normalization process confirms that the value of a is indeed 1/2. The conversation emphasizes the importance of orthonormality in determining the coefficients of hybrid orbitals.
Samuelriesterer
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Assuming the 2s and 2p wavefunctions are normalized, determine the coefficients in the hybrid orbital:

Ψ(sp3) = aΨ(2s) + aΨ(2px) + aΨ(2py) + aΨ(2pz) (the other 3 hybrids have – signs for some of the coefficients.

I have no clue where to start. I know this is a tetrahedral hybrid orbital but can't seem to grasp this question.
 
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Are the other 3 hybrids completely specified for you? If so, you should impose orthonormality to determine the coefficients for this wavefunction.
 
No this is the whole problem. I keep thinking it must be entirely simple but I guess it's not. (My teacher has a way of omission to the point of ridiculous)
 
Well, you've written down a fairly specific expression with the same coefficient ##a## in front of each term. Assuming that's given, then ##a## should be fixed by normalizing the wavefunction. You should verify that the 4 hybrid orbitals are orthonormal to complete the exercise.
 
If I assume orthonormality and the other 3 hybrids have some coefficients as negative, then am I right to conclude that the coefficients for this one are all +1/2?
 
Yes, normalizing this requires ##a=1/2##.
 

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