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It seems that all wave functions studied in physical chemistry are orthogonal (e.g. particle in a box, hydrogen atomic orbitals). Does this come about because we purposefully make them orthogonal, or are they derived that way naturally? Can there be useful wave functions that are not orthogonal?

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    $\begingroup$ They are eigenfunctions of a hermitian operator. $\endgroup$ – user26143 Nov 22 '13 at 9:10
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In general, orthogonal wavefunctions are much easier to treat. In some cases they appear naturally, but usually, the orthogonality is imposed as a constrain while constructing the wavefunction.

For example, if you construct electronic wavefunction in the atomic orbital basis, you try to construct the orthogonal basis. This guarantees that the AOs are linearly independent. (Implication, not equivalence). Would you fail to fulfill this, the solution might still be possible, but much more difficult.

If you manage to solve the eigenvalue - eigenvector problem, the solutions are by definition orthogonal to each other. This is the case for the examples you provided.

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  • $\begingroup$ When AOs are linearly independent, does this mean that we are ignoring any interaction between the AOs? How does this relate to Huckel MO theory when neighboring AOs' interactions are taken into account? $\endgroup$ – halcyon Nov 23 '13 at 19:11
  • $\begingroup$ Sorry, I messed some things together. The atomic orbitals on one given atom are orthogonal (have zero overlap). Orbitals on two different atoms usually do have non-zero overlap (can be zero due to symmetry). $\endgroup$ – ssavec Nov 24 '13 at 19:28
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    $\begingroup$ Eigenvectors are not orthogonal "by definition". It's just that two eigenvectors of a Hermitian operator $A$, to different eigenvalues, are necessarily orthogonal (else e.g. decomposing in the middle of $A^2$ would lead to contradiction). They're however typically normalised by definition, thus orthonormal. $\endgroup$ – leftaroundabout Apr 29 '14 at 20:16
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    $\begingroup$ One should be more clear about if the topic is orthonormal BASISes or WAVEFUNCTIONs. The answer seems to talk about the first, the question about the second. $\endgroup$ – Greg Oct 17 '14 at 3:38

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