Atomic Shell Model

The Hamiltonian for an atom with \bgroup\color{black}$Z$\egroup electrons and protons is

\begin{displaymath}\bgroup\color{black}\left[ \sum\limits^Z_{i=1} \left({p^2_i\o...
...\vert\vec{r_i}-\vec{r_j}\right\vert}} \right]\psi=E\psi.\egroup\end{displaymath}

We have seen that the coulomb repulsion between electrons is a very large correction in Helium and that the three body problem in quantum mechanics is only solved by approximation. The states we have from hydrogen are modified significantly. What hope do we have to understand even more complicated atoms?

The physics of closed shells and angular momentum enable us to make sense of even the most complex atoms. Because of the Pauli principle, we can put only one electron into each state. When we have enough electrons to fill a shell, say the 1s or 2p, The resulting electron distribution is spherically symmetric because

\begin{displaymath}\bgroup\color{black}\sum^\ell_{m=-\ell}\left\vert Y_{\ell m}\...
...a , \phi\right)\right\vert^2
= {2\ell + 1\over{4\pi}}. \egroup\end{displaymath}

With all the states filled and the relative phases determined by the antisymmetry required by Pauli, the quantum numbers of the closed shell are determined. There is only one possible state representing a closed shell.

As in Helium, the two electrons in the same spatial state, \bgroup\color{black}$\phi_{n\ell m}$\egroup, must by symmetric in space and hence antisymmetric in spin. This implies each pair of electrons has a total spin of 0. Adding these together gives a total spin state with \bgroup\color{black}$s=0$\egroup, which is antisymmetric under interchange. The spatial state must be totally symmetric under interchange and, since all the states in the shell have the same \bgroup\color{black}$n$\egroup and \bgroup\color{black}$\ell$\egroup, it is the different \bgroup\color{black}$m$\egroup states which are symmetrized. This can be shown to give us a total \bgroup\color{black}$\ell=0$\egroup state.

So the closed shell contributes a spherically symmetric charge and spin distribution with the quantum numbers

& s=0 \\
& \ell=0 \\
& j=0

The closed shell screens the nuclear charge. Because of the screening, the potential no longer has a pure \bgroup\color{black}${1\over r}$\egroup behavior. Electrons which are far away from the nucleus see less of the nuclear charge and shift up in energy. This is a large effect and single electron states with larger \bgroup\color{black}$\ell$\egroup have larger energy. From lowest to highest energy, the atomic shells have the order

\begin{displaymath}\bgroup\color{black} 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p. \egroup\end{displaymath}

The effect of screening not only breaks the degeneracy between states with the same \bgroup\color{black}$n$\egroup but different \bgroup\color{black}$\ell$\egroup, it even moves the 6s state, for example, to have lower energy than the 4f or 5d states. The 4s and 3d states have about the same energy in atoms because of screening.

Jim Branson 2013-04-22