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A Dyson orbital is defined as: $$ \textrm{Dy}(x) = \int dx_1 \ldots dx_{N-1} \, (\Psi^+(x_1,\ldots,x_{N-1}))^* \, \Psi^0(x_1,\ldots,x_{N-1}, x),\tag{1} $$ where $\Psi^0$ and $\Psi^+$ are the total wavefunctions of the neutral and cation, respectively. For $N$-electron molecule within Hartree-Fock picture, it's known that in most cases the Dyson orbital is similar to the HOMO of the neutral species and this is due to the fact that the MO's of the cation often look similar to the first $N-1$ MO's of the neutral. Let's take an example of the $\ce{CO_2}$ molecular orbital obtained by Hartree-Fock method where the HOMO has $\pi$ symmetry and is thus doubly degenerate. One may identify these HOMO as $p_x$ and $p_y$. Now the HOMO of $\ce{CO_2^+}$ obtained with the same method also has $\pi$ symmetry but is not degenerate, and I guess depending on the initial guess, it may collapse to having either $p_x$ or $p_y$. If say, it has $p_y$ symmetry and then you calculate the Dyson orbital, it will have $p_x$ symmetry.

What looks strange to me is that the Dyson orbital does not seem to respect the degeneracy of the neutral. Or is it that there is a special treatment for degeneracy in calculating Dyson orbital?

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  • $\begingroup$ IMHO this is a problem of Hartree-Fock, not a problem of the Dyson orbitals. If you use FCI wavefunctions in the calculation of Dyson orbitals, you should get perfectly degenerate HOMOs $\endgroup$
    – wzkchem5
    Commented Dec 7, 2021 at 8:05
  • $\begingroup$ @wzkchem5 do you mean when I use FCI to get the neutral and cation wavefunctions, I will get $n$ different Dyson orbitals where $n$ is the degree of degeneracy? But how can I have more than one different Dyson orbitals when there is only one formula to get it? $\endgroup$
    – nougako
    Commented Dec 7, 2021 at 12:07
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    $\begingroup$ In this case you have an infinite number of possible choices of $\Psi^+$, therefore the Dyson orbital is indeed non-unique, up to unitary rotations in an $n$-dimensional space $\endgroup$
    – wzkchem5
    Commented Dec 7, 2021 at 19:19
  • $\begingroup$ Isn't that true already in HF because there is an infinite possible pair of neutral's degenerate HOMO? You can rotate the HOMO manifold without affecting ground state energy. $\endgroup$
    – nougako
    Commented Dec 8, 2021 at 13:38
  • $\begingroup$ You're right. What I said only explains that Dyson orbitals from FCI respect spatial symmetry, while those from HF don't, and degeneracy is a different thing from symmetry. But then, how do you define the degeneracy of Dyson orbitals, when there is only one Dyson orbital for a given molecule? Or are you instead referring to the non-degeneracy of the Dyson orbital of the neutral molecule with that of the cation? $\endgroup$
    – wzkchem5
    Commented Dec 8, 2021 at 15:07

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