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I am struggling to understand how to calculate the $d$ orbital energy of the Fe atom. I tried to search the $\text{Fe I}$ in the Atomic Spectra Database, so I thought I could use the energy difference between $3d^64s^2$ and $3d^54s^2$, but there are many configurations for $3d^64s^2$ and $3d^54s^2$. Which one should I choose to represent the two configurations?

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Note that identifying ionization energies with orbital energies is an approximation based on the Koopmans theorem $\varepsilon\approx -E_{\text{IP}}=E_{N}-E_{N-1}$, and should not be taken too literally. The Koopmans theorem is based on the Hartree-Fock model, and gives acceptable results only because of a fortuitous error cancellation between the lack of electron correlation and the lack of orbital relaxation (the latter meant for the $N-1$ electron system).

I guess you are trying to get something like the $-11.66 \, \text{eV}$ value here for the ground state of Fe. This should be obtained from the lowest energy $3d^64s^2\rightarrow3d^54s^2$ ionization process.

On NIST, you can find the ground state ionization energy $$ E(3d^64s^1;{^6\hspace{-0.07cm}D}_{9/2})-E(3d^64s^2;{^5\hspace{-0.07cm}D}_{4})\approx 7.9025 \, \text{eV} \ , $$ and the excitation energy of the lowest lying $3d^54s^2$ state $$ E(3d^54s^2;{^6\hspace{-0.07cm}S}_{5/2})-E(3d^64s^1;{^6\hspace{-0.07cm}D}_{9/2})\approx2.8910 \, \text{eV} \ . $$ I used the lowest lying states of the fine structure corresponding to the highest $J$ value. From these, we get $$ E(3d^54s^2;{^6\hspace{-0.07cm}S}_{5/2})-E(3d^64s^2;{^5\hspace{-0.07cm}D}_{4})\approx 10.793 \, \text{eV} \ , $$ which (assuming the validity of the Koopmans theorem) translates to $\varepsilon_{3d}\approx-10.793 \, \text{eV}$. This is only in a rough agreement with the $-11.66 \, \text{eV}$ value, but I would rather trust the NIST results, especially since no reference is given for the other energy.

In case you are interested in the actual $3d$ orbital energies (not the ionization energy), then a HF calculation must be performed with the usual warning that orbital energies are not really physical quantities. Beyond the approximate identification with $-E_{\text{IP}}$, they do not have too much meaning.

Addendum

It should be noted that obtaining meaningful HF orbital energies in an open shell system is not that straightforward, since these energies are usually not unique: different implementations of the restricted open-shell Hartree-Fock (ROHF) model can yield vastly different orbital energies, while giving the same total HF energy. This poses a difficulty compared to closed shell systems, whose orbital energies are uniquely determined. One must impose extra conditions on the orbitals to bring them into a so-called canonical form, and only in this case the orbital energies can be compared with ionization energies; see J. Phys. Chem. A 113 45 12386 (2009) for details. Table 2 of the paper compares the computed orbital energies of Mn with experimental ionization energies that can be obtained from NIST just like above (see here and here), and shows a qualitative agreement.

I could not find any recent computation on Fe ROHF orbital energies. The only results I found are from a rather old paper (Phys. Rev. 119 1934 (1960)), where the $\varepsilon_{3d}\approx-0.6359 \, E_h\approx-17.3 \, \text{eV}$ value is reported. But, given the above mentioned ambiguity of open shell orbital energies, this should also be taken with a grain of salt.

Another approach to obtain unique, well-defined open shell orbital energies is to use unrestricted HF (UHF). This comes at the price of breaking the $\hat{S}^2$ (and possibly spatial) symmetry of the solutions though, and leads to different orbital energies for different spins. An even older paper (Phys. Rev. 107 995 (1957)) states the UHF result $\varepsilon_{3d\alpha}\approx-0.56115 \, E_h\approx-15.3 \, \text{eV}$ for a high-spin Fe atom (and a much smaller value for $\varepsilon_{3d\beta}$).

The main message is that orbital energies and ionization energies are in general not as closely connected as one would expect from naively looking at the Koopmans theorem (especially not for open shell systems). At least, one should not anticipate too accurate results when trying to use one to calculate the other.

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  • $\begingroup$ There are other results of 3d54s2 except 6S5/2, and some of them could show a way higher energy, which one should I choose? Or should I use the average energy value? $\endgroup$
    – Jack
    Nov 13, 2023 at 3:27
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    $\begingroup$ @Jack What exactly is your goal? The experimental energies of NIST should not be averaged; after all, these are peaks in your spectrum corresponding to transitions from the ground state of Fe to some state of Fe$^+$. Trying to match orbital energies with the lowest transition makes the most sense (following from the variational property of the Koopmans theorem), but in light of the above numbers, I wouldn't expect good results from this. I have a feeling that you are expecting too much from the orbital concept and the Koopmans theorem. If you could clarify a bit, maybe I can improve my answer. $\endgroup$ Nov 15, 2023 at 20:08

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