Multiple Electrons in Rectangular Traps — Question 9

Question 9

Partial shell filling creates many degenerate ground determinants

In a square box, the lowest spatial orbital has energy $2\epsilon$, followed by two degenerate spatial orbitals at $5\epsilon$. Four noninteracting electrons are present, with no magnetic field and no interaction splitting.

1. Find the ground energy and number of degenerate ground determinants.

2. Group the open-shell determinants by total spin projection $M_S$ in units of hbar.

Question 9 - Solution

Partial shell filling creates many degenerate ground determinants

(a) Method and calculation

$$\text{Core: two electrons at }2\epsilon;\quad\text{open shell: two at }5\epsilon.$$$$\boxed{E_g=14\epsilon,\quad d=\binom42=6}.$$

(b) Interpretation and further calculation

$$M_S=+1:\binom22=1;\qquad M_S=-1:\binom22=1.$$$$M_S=0:\quad2\times2=4\Rightarrow\boxed{d_{+1},d_0,d_{-1}=1,4,1}.$$

Physical check

These are determinants classified by spin projection. Some linear combinations have definite total spin; interactions can lift the degeneracy, so Hund ordering cannot be inferred from this noninteracting model alone.