What is the electron configuration for the f block?

I finally feel confident enough to post a table of the configurations, along with some detailed rationale for why the configurations are so riddled with 'Aufbau exceptions'.

For reference, the energy scales I will be using are small. For perspective, you can compare the numbers with the first ionization energy of #"N"# atom of #"14.53 eV"#, and the first ionization energy of #"H"# atom of #"13.61 eV"#.

The following graphs are from page #199 - 202# of this book by my advisor, as well as Michael Dolg and Kenneth Dyall. All energies here are in hartrees (#E_h#), where #1# #E_h = "27.2114 eV"#.

DISCLAIMER: LONG ANSWER!

LANTHANIDES

The order by atomic number is down the first column, and then down the second column. In #color(red)("red")# are the 'Aufbau exceptions'.

#color(white)([(color(red)(La),(color(red)([Xe] 6s^2 5d^1)),color(black)(Tb),(color(black)([Xe] 6s^2 4f^9))),(color(red)(Ce),(color(red)([Xe] 6s^2 4f^1 5d^1)),color(black)(Dy),(color(black)([Xe] 6s^2 4f^10))),(color(black)(Pr),(color(black)([Xe] 6s^2 4f^3)),color(black)(Ho),(color(black)([Xe] 6s^2 4f^11))),(color(black)(Nd),(color(black)([Xe] 6s^2 4f^4)),color(black)(Er),(color(black)([Xe] 6s^2 4f^12))),(color(black)(Pm),(color(black)([Xe] 6s^2 4f^5)),color(black)(Tm),(color(black)([Xe] 6s^2 4f^13))),(color(black)(Sm),(color(black)([Xe] 6s^2 4f^6)),color(black)(Yb),(color(black)([Xe] 6s^2 4f^14))),(color(black)(Eu),(color(black)([Xe] 6s^2 4f^7)),color(black)(Lu),(color(black)([Xe] 6s^2 4f^14 5d^1))),(color(red)(Gd),(color(red)([Xe] 6s^2 4f^7 5d^1)),"","")])#

The exceptions can be explained by looking at how the energies of the #6s#, #5d#, and #4f# orbitals vary for the lanthanides.

We can see that the #4f# orbitals decrease in energy as we go from left to right, but the #6s# and #5d# orbitals are consistently within #0.1# #E_h# (about #"2.7 eV"#) of each other.

The radii of the #(n-2)f# orbitals are also more contracted, particularly for the lanthanides, making them more core-like in size than even the #5s# and #5p# orbitals, in addition to the decreasing #4f# energies.

The exceptions occur mainly for the earlier lanthanides (#La, Ce#), where the #4f#'s are still fairly close in energy to the #5d# and #6s#.

• The radial compactness of the #4f# orbitals makes it more favorable to fill the #6s# and #5d# first for #La# and #Ce#, to minimize electron repulsion.

• For #Gd#, the repulsion that would be generated from pairing a #4f# electron would be enough to promote it to a #5d# orbital (about #0.6# #E_h# away, or about #"16 eV"#), so #Gd# takes on a #f^7 d^1# configuration instead of #f^8 d^0#.

ACTINIDES

The order by atomic number is down the first column, and then down the second column. In #color(red)("red")# are the 'Aufbau exceptions'.

#color(white)([(color(red)(Ac),(color(red)([Rn] 7s^2 6d^1)),color(black)(Bk),(color(black)([Rn] 7s^2 5f^9))),(color(red)(Th),(color(red)([Rn] 7s^2 6d^2)),color(black)(Cf),(color(black)([Rn] 7s^2 5f^10))),(color(red)(Pa),(color(red)([Rn] 7s^2 5f^2 6d^1)),color(black)(Es),(color(black)([Rn] 7s^2 5f^11))),(color(red)(U),(color(red)([Rn] 7s^2 5f^3 6d^1)),color(black)(Fm),(color(black)([Rn] 7s^2 5f^12))),(color(red)(Np),(color(red)([Rn] 7s^2 5f^4 6d^1)),color(black)(Md),(color(black)([Rn] 7s^2 5f^13))),(color(black)(Pu),(color(black)([Rn] 7s^2 5f^6)),color(black)(No),(color(black)([Rn] 7s^2 5f^14))),(color(black)(Am),(color(black)([Rn] 7s^2 5f^7)),color(black)(Lr),(color(black)([Rn] 7s^2 5f^14 6d^1))),(color(red)(Cm),(color(red)([Rn] 7s^2 5f^7 6d^1)),"","")])#

We can again examine the energies (pg. 199):

The energies of the #7s# and #6d# are likewise very close to each other (within #"2.7 eV"# as before), but the #5f# are at MOST #0.4# #E_h#, or about #11# #"eV"# away from the #7s# and #6d# orbitals.

That makes the energetic degeneracies of the #5f# with the #6d# and #7s# and the compactness of the #5f# orbitals even more significant in giving rise to #Ac-Np# as 'Aufbau exceptions'.

As before, the exceptions occur mainly for the earlier actinides (#Ac-Np#).

• For #Ac-Th#, since the #5f#'s and #6d#'s are very similar in energy, it is possible for #6d# occupation instead of the #5f#. I believe it is because the #5f# orbitals are barely bigger than the #6p#, #6s#, and #5d# orbitals for #Th# that the #5f# is about as core-like as them and thus not as accessible to fill... but this is difficult to explain.

You can see the radial extents here (pg. 202):

"Spinor" just means an electronic quantum state (in the Pauli Exclusion sense) with a specific spin (up/down). DHF stands for Dirac-Hartree-Fock.

• For #Pa - Np#, whose #6d-5f# gap is even smaller than the #5d-4f# gap of the lanthanides (but bigger than for #Ac-Th#), I believe that the repulsions generated from adding a second electron into the #6d# orbitals (even without pairing) are still enough that since the #5f# orbitals are lower in energy, it is preferable to proceed by filling them instead.

• And again for #Cm#, similar to #Gd#, the electron repulsion that occurs with pairing an #5f# electron would be enough to promote it to a #6d# orbital (about #0.15# #E_h# away, or about #"4 eV"#).