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Koopmans' theorem is a useful approach to calculate the global reactivity parameters from the HOMO-LUMO energies. My question is, does it apply to the open-shell systems where we get two sets of singly occupied (alpha and beta) HOMO-LUMO energies?

I am working on rare earth systems. Rare earth (RE) elements generally show +3 oxidation states which result in open shell configurations for $\ce{Ce^3+}$, $\ce{Eu^3+}$, $\ce{Gd^3+}$ etc. I am performing unrestricted open shell DFT calculations for these systems, as restricted open shell calculations for RE systems require larger computational time and cost. In unrestricted open shell calculations, we get two sets of alpha and beta HOMO and LUMO. My question is, can I apply Koopmans' theorem here to calculate the global reactivity parameters?

Could anyone help with some literature on the reactivity parameters for open shell systems? Thank you!

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    $\begingroup$ Regarding the literature, it might be worth having a look at the section Books about Computational Chemistry and Quantum Chemistry. I'm no specialist in this field, so probably someone else could help you with the precise textbook for you. $\endgroup$ – andselisk Sep 14 at 13:00
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    $\begingroup$ Could you walk us through an example you understand, and then give an example of an open-shell system that you are wondering about? That would make it easier to answer. $\endgroup$ – Karsten Theis Sep 18 at 1:56
  • $\begingroup$ Hello @KarstenTheis, I am working on rare earth systems. Rare earth (RE) elements generally show +3 oxidation states which result in open shell configurations for Ce3+, Eu3+, Gd3+ etc. I am performing unrestricted open shell DFT calculations for these systems, as restricted open shell calculations for RE systems require larger computational time and cost. In unrestricted open shell calculations, we get two sets of alpha and beta HOMO and LUMO. My question is, can I apply Koopmans' theorem here to calculate the global reactivity parameters? $\endgroup$ – Ranjini Sarkar Sep 19 at 6:53
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You (probably) cannot apply Koopmans' theorem. (Please also note that it is named after Tjalling Charles Koopmans, the s at the end is part of the name.)
Open shell systems are usually hard to describe and unrestricted density functional approximations might not even describe the system correct qualitatively. The problem usually comes down to the multireference character of the partially occupied orbitals. In these cases you need a multireference wave function to correctly describe the ground state, e.g CASSCF methods. In a first order approximation, unrestricted methods can probably yield reasonable results, because (in a practical sense) it is using more than one determinante. However, when you use these methods, there will be fractional occupancy of multiple orbitals, which make the approximations of Koopmans' theorem break down. I would suggest a thorough literature review on the best practises for such systems. In short, HOMO and LUMO are not defined in these systems.

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  • $\begingroup$ Thank you so much for the clarification. $\endgroup$ – Ranjini Sarkar Sep 24 at 5:48

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