Periodic Table

The ground states of atoms with $N=Z$ electrons in the period table can now be understood by forming Slater determinants (`configurations') with $N$ spin-orbitals $\vert\nu_i\rangle = \vert n_il_im_i\sigma_i\rangle$. The atoms are thus `built up' from these solutions. This is denoted as

$$\begin{array}{lll} {\rm H} & 1s & ^2S_{1/2}\\ {\rm He} & (1s... ... C} & ({\rm He})(2s)^2(2p)^2& ^3P_{0}\\ ...&...\\ \end{array}$$ (1.9)

These are built up by `filling up the levels' with electrons. For a given $(n,l)$ there are $2(2l+1)$ orbitals (2 spin states for each given $m$-value).

The spectroscopic description is given by the quantum numbers $S$, $L$, $J$ (total spin, orbital, angular momentum) in the form

$$^{2S+1}L_J.$$ (1.10)

Carbon is the first case where Hund's Rules kick in. These `rules' are rules and no strict theorems, but they seem to work well for the understanding of atoms. Here I cite them after Gasiorowicz (web-supplement)

1. The state with largest $S$ lies lowest: spin-symmetric WFs have anti-symmetric orbital WFs and therefore reduced electron-electron interaction.
2. For a given value of $S$, the state with maximum $L$ lies lowest: the higher $L$, the more lobes (and thereby mutual `escape routes' for interacting electrons) there are in the $Y_{lm}$s.
3. $L$, $S$ given. (i) not more than half-filled incomplete shell: $J=\vert L-S\vert$; (ii) more than half-filled shell: $J=L+S$: due to spin-orbit interaction.