# What is the electron configuration for the f block?

Dec 9, 2016

The f block has the lower level filled, then for valance electrons has 2 s electrons 1 d electron and then up to 14 f electrons filling the 7 f orbitals.

Jul 14, 2017

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 $\text{N}$ atom of $\text{14.53 eV}$, and the first ionization energy of $\text{H}$ atom of $\text{13.61 eV}$.

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 $1$ ${E}_{h} = \text{27.2114 eV}$.

LANTHANIDES

The order by atomic number is down the first column, and then down the second column. In $\textcolor{red}{\text{red}}$ are the 'Aufbau exceptions'.

$\textcolor{w h i t e}{\left[\begin{matrix}\textcolor{red}{L a} & \left(\textcolor{red}{\left[X e\right] 6 {s}^{2} 5 {d}^{1}}\right) & \textcolor{b l a c k}{T b} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{9}}\right) \\ \textcolor{red}{C e} & \left(\textcolor{red}{\left[X e\right] 6 {s}^{2} 4 {f}^{1} 5 {d}^{1}}\right) & \textcolor{b l a c k}{D y} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{10}}\right) \\ \textcolor{b l a c k}{P r} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{3}}\right) & \textcolor{b l a c k}{H o} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{11}}\right) \\ \textcolor{b l a c k}{N d} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{4}}\right) & \textcolor{b l a c k}{E r} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{12}}\right) \\ \textcolor{b l a c k}{P m} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{5}}\right) & \textcolor{b l a c k}{T m} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{13}}\right) \\ \textcolor{b l a c k}{S m} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{6}}\right) & \textcolor{b l a c k}{Y b} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{14}}\right) \\ \textcolor{b l a c k}{E u} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{7}}\right) & \textcolor{b l a c k}{L u} & \left(\textcolor{b l a c k}{\left[X e\right] 6 {s}^{2} 4 {f}^{14} 5 {d}^{1}}\right) \\ \textcolor{red}{G d} & \left(\textcolor{red}{\left[X e\right] 6 {s}^{2} 4 {f}^{7} 5 {d}^{1}}\right) & \text{ & }\end{matrix}\right]}$

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

We can see that the $4 f$ orbitals decrease in energy as we go from left to right, but the $6 s$ and $5 d$ orbitals are consistently within $0.1$ ${E}_{h}$ (about $\text{2.7 eV}$) of each other.

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

The exceptions occur mainly for the earlier lanthanides ($L a , C e$), where the $4 f$'s are still fairly close in energy to the $5 d$ and $6 s$.

• The radial compactness of the $4 f$ orbitals makes it more favorable to fill the $6 s$ and $5 d$ first for $L a$ and $C e$, to minimize electron repulsion.

• For $G d$, the repulsion that would be generated from pairing a $4 f$ electron would be enough to promote it to a $5 d$ orbital (about $0.6$ ${E}_{h}$ away, or about $\text{16 eV}$), so $G d$ 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 $\textcolor{red}{\text{red}}$ are the 'Aufbau exceptions'.

$\textcolor{w h i t e}{\left[\begin{matrix}\textcolor{red}{A c} & \left(\textcolor{red}{\left[R n\right] 7 {s}^{2} 6 {d}^{1}}\right) & \textcolor{b l a c k}{B k} & \left(\textcolor{b l a c k}{\left[R n\right] 7 {s}^{2} 5 {f}^{9}}\right) \\ \textcolor{red}{T h} & \left(\textcolor{red}{\left[R n\right] 7 {s}^{2} 6 {d}^{2}}\right) & \textcolor{b l a c k}{C f} & \left(\textcolor{b l a c k}{\left[R n\right] 7 {s}^{2} 5 {f}^{10}}\right) \\ \textcolor{red}{P a} & \left(\textcolor{red}{\left[R n\right] 7 {s}^{2} 5 {f}^{2} 6 {d}^{1}}\right) & \textcolor{b l a c k}{E s} & \left(\textcolor{b l a c k}{\left[R n\right] 7 {s}^{2} 5 {f}^{11}}\right) \\ \textcolor{red}{U} & \left(\textcolor{red}{\left[R n\right] 7 {s}^{2} 5 {f}^{3} 6 {d}^{1}}\right) & \textcolor{b l a c k}{F m} & \left(\textcolor{b l a c k}{\left[R n\right] 7 {s}^{2} 5 {f}^{12}}\right) \\ \textcolor{red}{N p} & \left(\textcolor{red}{\left[R n\right] 7 {s}^{2} 5 {f}^{4} 6 {d}^{1}}\right) & \textcolor{b l a c k}{M d} & \left(\textcolor{b l a c k}{\left[R n\right] 7 {s}^{2} 5 {f}^{13}}\right) \\ \textcolor{b l a c k}{P u} & \left(\textcolor{b l a c k}{\left[R n\right] 7 {s}^{2} 5 {f}^{6}}\right) & \textcolor{b l a c k}{N o} & \left(\textcolor{b l a c k}{\left[R n\right] 7 {s}^{2} 5 {f}^{14}}\right) \\ \textcolor{b l a c k}{A m} & \left(\textcolor{b l a c k}{\left[R n\right] 7 {s}^{2} 5 {f}^{7}}\right) & \textcolor{b l a c k}{L r} & \left(\textcolor{b l a c k}{\left[R n\right] 7 {s}^{2} 5 {f}^{14} 6 {d}^{1}}\right) \\ \textcolor{red}{C m} & \left(\textcolor{red}{\left[R n\right] 7 {s}^{2} 5 {f}^{7} 6 {d}^{1}}\right) & \text{ & }\end{matrix}\right]}$

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

The energies of the $7 s$ and $6 d$ are likewise very close to each other (within $\text{2.7 eV}$ as before), but the $5 f$ are at MOST $0.4$ ${E}_{h}$, or about $11$ $\text{eV}$ away from the $7 s$ and $6 d$ orbitals.

That makes the energetic degeneracies of the $5 f$ with the $6 d$ and $7 s$ and the compactness of the $5 f$ orbitals even more significant in giving rise to $A c - N p$ as 'Aufbau exceptions'.

As before, the exceptions occur mainly for the earlier actinides ($A c - N p$).

• For $A c - T h$, since the $5 f$'s and $6 d$'s are very similar in energy, it is possible for $6 d$ occupation instead of the $5 f$. I believe it is because the $5 f$ orbitals are barely bigger than the $6 p$, $6 s$, and $5 d$ orbitals for $T h$ that the $5 f$ 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 $P a - N p$, whose $6 d - 5 f$ gap is even smaller than the $5 d - 4 f$ gap of the lanthanides (but bigger than for $A c - T h$), I believe that the repulsions generated from adding a second electron into the $6 d$ orbitals (even without pairing) are still enough that since the $5 f$ orbitals are lower in energy, it is preferable to proceed by filling them instead.

• And again for $C m$, similar to $G d$, the electron repulsion that occurs with pairing an $5 f$ electron would be enough to promote it to a $6 d$ orbital (about $0.15$ ${E}_{h}$ away, or about $\text{4 eV}$).