# How to calculate the $d$ orbital energy from Atomic Spectra Database?

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?

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.

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 $$\varepsilon_{3d\beta}$$).
• @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. Nov 15, 2023 at 20:08