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I'm beginning to study quantum chemistry and I became pretty confused about the total state of the molecule.

When we describe the single orbitals using irreducible representations like in the picture below, then we can get the total quantum state by direct product $$(2a_1)^2(1b_2)^2(3a_1)^2(1b_1)^2=(A_1)^2\otimes (B_2)^2\otimes (A_1)^2\otimes (B_1)^2 =A_1,$$

as it is described in this answer. But when I read about multi-particle wavefunctions, I realized, that the wavefunction of the whole system should be antisymmetric, so I should assemble Slater determinant to get the multi-particle wavefunction with the enforced antisymmetry property.

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Questions

1) How is Slater determinant connected with the direct product above?

2) Does $(A_1)^2$ mean $A_1 \otimes A_1$ or is it purely the notation for 2 electrons in the orbital?

Eenoku
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  • I haven't read through it in detail, but I think this should help: https://www.google.com/url?sa=t&source=web&rct=j&url=http://www.uwyo.edu/kubelka-chem/mm_notes_10.pdf&ved=2ahUKEwia1cyHqszZAhWE14MKHWdoBZoQFjAFegQIABAB&usg=AOvVaw0LHVQaMyEpqFSff8yvl-u8 – Tyberius Mar 01 '18 at 23:59
  • (1) It isn't. (2) Yes. – Ivan Neretin Mar 02 '18 at 04:40
  • @IvanNeretin Would you mind to respond in more detail? So, why can we find the total irreducible representation using direct product? And by "yes" you mean, that $()^2$ is purely the notation and not power? – Eenoku Mar 02 '18 at 09:32
  • The Slater determinant is just a linear combination of these direct products. The symmetry will still be the same. – orthocresol Mar 02 '18 at 10:14
  • Two electrons in the orbitals leads to $(A_1)^2 = A_1 \times A_1 = A_1$. So it isn't just notation. – pentavalentcarbon Mar 02 '18 at 13:12
  • @pentavalentcarbon So, that means that every IR, e.g. $(B_2)^2$ equals $A_1$? – Eenoku Mar 02 '18 at 13:24
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    Yes (extra text to satisfy comment length) – pentavalentcarbon Mar 02 '18 at 13:27
  • The electronic wavefunction must be anti-symmetric with respect to the exchange of any two electrons, and Slater determinants provide a practical way to do achieve that. – tobiuchiha Feb 12 '20 at 00:05

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