It is well known that for a closed shell (all electrons are paired up) molecule, the HOMO-LUMO gap is related to its stability. However, I often see that the same argument is used for open-shell systems (Unrestricted calculation). In such a case, the authors consider energies of both alpha and beta spin orbitals, and the orbital with the highest energy is considered as SOMO. Similarly, the LUMO is decided among both alpha and beta, and the respective gap is considered as the SOMO-LUMO gap. Next, the authors discuss the stability of the system based on that value.
However, to the best of my understanding, for an unrestricted calculation, three SOMO-LUMO gaps can be calculated (1) Gap between alpha spin HOMO and alpha spin LUMO (2) Gap between beta spin HOMO and beta spin LUMO, and (3). The method I mentioned above (considering both alpha and beta).
So, my questions are the following,
a) Is it technically correct to relate stability with the SOMO-LUMO gap calculated by method 3 for an open-shell system? Is it even possible to draw such a relation? How correct are such value and such correlation? If yes, then is there any reason why we are ignoring the other two gaps?
b) Are the gaps obtained by an unrestricted calculation have any practical significance at all (As argued here: https://joaquinbarroso.com/2018/09/27/the-homo-lumo-gap-in-open-shell-calculations-meaningful-or-meaningless/)
Any insightful response will be welcome. Thank you in advance.
I do not suggest using orbital gap to evaluate stability (especially for open-shell cases, as the definition has a degree of ambiguity), because there is a much more rigorous quantity defined in the framework of conceptual density functional theory to assess this, namely hardness, see J. Am. Chem. SOC., 105, 7512 (1983). For analogous systems, it is expected that the smaller the hardness, the lower the stability. The hardness is defined as IP - EA, where IP and EA are vertical ionization potential and vertical electron affinity, they can be easily and rigorously calculated for any chemical system including open-shell ones by popular quantum chemistry codes. An example of discussing hardness: Struct. Chem., 25, 1521–1533 (2014).
You are right. Method 3 is not always the correct way to estimate stability, methods 1 and 2 are sometimes more meaningful.
Regarding the stability of the open-shell systems in the course of a reaction depends on which spin-orbital (alpha or beta) an electron is entering (reduction) or from which spin-orbital (alpha or beta) an electron is going out (oxidation).
I do not suggest using orbital gap to evaluate stability (especially for open-shell cases, as the definition has a degree of ambiguity), because there is a much more rigorous quantity defined in the framework of conceptual density functional theory to assess this, namely hardness, see J. Am. Chem. SOC., 105, 7512 (1983). For analogous systems, it is expected that the smaller the hardness, the lower the stability. The hardness is defined as IP - EA, where IP and EA are vertical ionization potential and vertical electron affinity, they can be easily and rigorously calculated for any chemical system including open-shell ones by popular quantum chemistry codes. An example of discussing hardness: Struct. Chem., 25, 1521–1533 (2014).
Dear Prof. Lu,
Thank you so much for your answer. I also predicted that the conceptual DFT is the way to go, however, I was not so sure. Thank you for confirming that.
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Also, thanks Arup for your insightful comment.
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