Coefficients of wave function of a hybrid orbital

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SUMMARY

The discussion focuses on determining the coefficients of the hybrid orbital Ψ(sp3) expressed as Ψ(sp3) = aΨ(2s) + aΨ(2px) + aΨ(2py) + aΨ(2pz). The participants confirm that the coefficients can be derived by imposing orthonormality conditions on the wavefunctions. It is established that if the other three hybrid orbitals have negative coefficients, the coefficient for this specific hybrid orbital is +1/2, leading to the normalization of the wavefunction with a fixed value of a = 1/2.

PREREQUISITES
  • Understanding of hybridization in chemistry, specifically sp3 hybridization.
  • Knowledge of wavefunction normalization and orthonormality principles.
  • Familiarity with quantum mechanics terminology related to atomic orbitals.
  • Basic mathematical skills for solving equations involving coefficients.
NEXT STEPS
  • Study the principles of wavefunction normalization in quantum mechanics.
  • Learn about the mathematical derivation of hybrid orbitals in molecular geometry.
  • Explore the concept of orthonormality in quantum states and its applications.
  • Investigate the characteristics and implications of different hybridization types (e.g., sp, sp2, sp3).
USEFUL FOR

Chemistry students, quantum mechanics enthusiasts, and educators looking to deepen their understanding of hybrid orbitals and wavefunction properties.

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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