# Triplet states and the Hund's rule in identifying the ground state configuration (open shell)

I referred to some of the questions about Hund's rule on StackExchange, such as this for example, but still wasn't able to have my question resolved.

(the wikiepdia page has $E_{ex} = C - \frac{1}{2}J_{ex} - 2J_{ex }<\vec{s_a} \cdot \vec{s_b}>$ and I believe our goal is to pick spins of the two electrons such that $E_{ex}$ becomes minimum.)

We have two competing effects, Coulomb energy and exchange energy. A lot of sources on this topic have energy calculations which shows that the expectation value of the energy for the singlet state is higher.

Hund's rule states that the lowest energy state has parallel spins to maximize the exchange energy. However, there's one triplet state which has $\frac{1}{\sqrt{2}}{|\uparrow \downarrow>} +{|\downarrow \uparrow>}$.

So, it seems the Hund's rule is telling me the ground state is either of the triplet states where you have both spins up or both spins down, since you would want to have both spins pointing in the same direction to minimize $E_{ex}$ term I have above. (I guess whether they are both up or down has to do with the g factor?).

However, the sources I refer to don't seem to draw a distinction among different triplet states. I feel that if we go by the equation we have above, since all three triplet states give you the same energy, as $<\vec{s_a} \cdot \vec{s_b}> = \frac{1}{4}$ for all three triplet states, it doesn't necessarily rule out $\frac{1}{\sqrt{2}}{|\uparrow \downarrow>} +{|\downarrow \uparrow>}$ as your ground state, which seems different from what the Hund's rule says.

• My understanding is that since the core of the second part of Hund's rule is the minimization of energy, your $|S_{tot}=1,S_{tot,z}= 0>$ example is a valid choice. Maybe you took the "parallel spin" explanation (which seems suitable for high school textbook) literally. – wcc Jul 30 '18 at 4:04
• @IamAStudent I think Hund's first rule,which says you want to place as many into orbitals singly with aligned spin, is quite different from the exchange energy calculation showing any of the triplet states gives you energy lower than the singlet state by the same amount, I feel that there's a leap in logic. – Blackwidow Jul 30 '18 at 13:21
• I am not sure if I understood your point, but for any of the triplet states, each spin (spin 1 and 2) occupies a distinct orbital (e.g. px and py), so it is consistent with Hund's first rule. – wcc Jul 30 '18 at 17:31
• @IamAStudent but for spin $\frac{1}{2}(|\uparrow \downarrow> \ + |\uparrow \downarrow>)$ triplet, the spins don't seem to be aligned anymore – Blackwidow Jul 30 '18 at 17:44
• it seems the textbook definitions for Hund's rules are given as requirements on total $S$, total $L$ and total $J$, not what their polarization is ($S_z$, $L_z$, $J_z$). This language of "spins need to be aligned", for me, just seems to be a gross simplification of how the exchange interaction actually works (symmetry of the wavefunction). – wcc Jul 30 '18 at 20:49