The aufbau principle

1. Electrons are assigned to the various electronic states using the following principle of
   
   1. minimising the total energy (fill the lowest energy states first) and
      
   2. Pauli's Exclusion principle (not more than one electron per state).
2. In determining the relative energies of each state, we note the following :
   1. Each electron moves in a complicated field due to its many-body interactions with the other electrons and the nucleus. The net effective electric field is approximately that given by the nuclear charge $Ze$ decreased by the partial shielding of other electrons closer to the nucleus, considering the shape of the wave-functions of the electrons involved.
   2. Electrons with the same principal quantum number $n$ are the same average distance from the nucleus, and belong to the same shell. Electrons in the same shell have therefore very roughly the same energies $E_n = E_1/n^2$. Shells are denoted by the capital letters $K, L, M, N, \cdots$ for $n=1, 2, 3, 4, /cdots$. The electrons of lower principal quantum number $n$ are closer to the nucleus and consequently of lower energy. The states with lower principal quantum number $n$ will generally be filled first.
   3. The energy of an electron is also dependent on its wave-functions angular momentum $l$. The lower $l$ values have wave function shapes with probability densities closer to the nucleus, resulting in less shielding of the nuclear charge and therefore larger effective binding energies. Consequently the electrons with lower angular momenta $l$ have lower total energies. Electrons with the same $l$ value occupy the same sub shell, and have very similar energies, as the energy dependence on $m_l$ and $m_s$ is relatively minor.
3. The four quantum numbers

   
   

   Principal

   : $n=1,2,3, \cdots$,
   

   Orbital

   : $l = 0, 1, \cdots ,(n-1)$,
   

   Magnetic

   : $m_l = 0, \pm 1, \cdots ,\pm l$ and
   

   Spin Magnetic

   $m_s = \pm \frac12$ )

   therefore determine the energy and labelling of each state and the magnitude of $n$ and $l$ determine the capacity of the shells and sub shells.
4. An atomic shell or sub shell which contains its full quota of electrons is said to be closed. The net orbital and spin angular momentum of a closed sub shell is zero, and the net charge distribution is spherical (Unsöld's Theorem). Electrons in such a closed shell are especially well bound as the effective unshielded nuclear charge is relatively large. Atoms with filled shells therefore have strongly bound electrons with no net dipole moment. It is therefore much less reactive.
5. Hund's Rule maintains that in general, the electrons in a sub shell remain unpaired (parallel spins) whenever possible. The physical basis of this rule is that it achieves greater spatial separation of the electrons, and therefore a state of lower energy.

Exercise 11
Show that the capacity of the $l^{\mbox{th}}$ sub shell is $2l+1$ and the capacity of the $n^{\mbox{th}}$ shell is $2n^2$.

Exercise 12
Verify Unsöld's theorem for $l=1$ by showing $\Sigma^{+l}_{m_l=-l} = \vert\Theta\vert^2\vert\Phi\vert^2 = const$ for $l=0$. (See Tut 1, question 9)

The actual relative energies of each shell and sub shell, as found by experiment and illuminated by approximate many-body quantum mechanical calculations of the electron energies are shown in figure 9 below.

Figure 9: The sequence in energy of the various atomic states (schematically).

This diagram aids the selection of the correct order of filling of the available sub shells. The tables below traces the development of the electronic structure in detail, element by element indicating how the various sections of the periodic table (with similar properties) arise from a similar electronic configuration.

Figure 10: The aufbau sequence.

Figure 11: The aufbau sequence.

Exercise 13
Verify that you can explain the periodicities and trends in the diagrams of the variation of the ionisation energy (figure 12) and atomic radii (figure 13) of the elements of the periodic table using the systematic trends in the electronic structure of the elements

Figure 12: Periodicities in ionisation energy.

Figure 13: Periodicities in atomic radii.