Building the Periodic Table — Question 9

Question 9

An open p shell is related to its complementary hole count

A p subshell contains six spin orbitals. Compare configurations $p^2$ and $p^4$. Each set of occupied spin orbitals defines one determinant; an unoccupied spin orbital is a hole.

1. Count the determinants for each configuration.

2. Use Hund filling to find the maximum S in each and explain the electron-hole correspondence.

Question 9 - Solution

An open p shell is related to its complementary hole count

(a) Method and calculation

$$\boxed{d(p^2)=\binom62=15,\quad d(p^4)=\binom64=15}.$$$$\binom6k=\binom6{6-k}:\quad\text{choosing four occupied slots equals choosing two holes}.$$

(b) Interpretation and further calculation

$$p^2:\ (\uparrow)(\uparrow)(\ )\Rightarrow\boxed{S_{\max}=1}.$$$$p^4:\ (\uparrow\downarrow)(\uparrow)(\uparrow)\Rightarrow\boxed{S_{\max}=1}.$$

Physical check

Complementary occupations have equal determinant counts and related term structures. Their energies and detailed spin-orbit level ordering need not be identical.