Hund’s rules

Hund’s rules, which are created by Friedrich Hund in 1927, consist of three rules that are used to identify the lowest energy term symbol of a given configuration of an atom. They are:

The term symbol with the highest multiplicity represents state(s) with the lowest energy.

This is due to the spin correlation between spin angular momenta of electrons, where the expectation value of the distance between a pair of electrons in the triplet state is greater than the expectation value of the distance between a pair of electrons in the singlet state. Consequently, there is less electron-electron repulsion for the triplet state (higher multiplicity) than for the singlet state.

For a particular multiplicity, the term symbol with the highest $L$ is lowest in energy.

The is a result of the Coulomb interaction between orbital angular momentum of a pair of electrons, where the electrons orbiting in the same direction meet less often than when they are orbiting in opposite directions. This implies that $\vert\boldsymbol{\mathit{r}}_i-\boldsymbol{\mathit{r}}_j\vert_{ave}(same)>\vert\boldsymbol{\mathit{r}}_i-\boldsymbol{\mathit{r}}_j\vert_{ave}(opposite)$ . Since the individual angular momentum of each electron orbiting in the same direction adds vectorially to give a higher total angular momentum (and hence a higher value of the quantum number $L$ ) than electrons orbiting in opposite directions, the term symbol with the highest $L$ is lowest in energy.

Generally, for a particular term describing an atom with a less than half-filled outermost subshell, the term symbol with the lowest $J$ is lowest in energy. If the subshell is more than half-filled, the term symbol with the highest $J$ is lowest in energy.

This is because the value of $A$ in eq289, which is empirically evaluated, generally changes from positive to negative when the subshell is more than half-filled.

Question

What about a half-filled subshell?

Answer

Examples of electron configuration of half-filled subshells are p³ and d⁵, with terms ⁴S and ⁶S respectively and corresponding levels of ⁴S_3/2 and ⁶S_5/2 respectively. Since the level of a half-filled subshell is described by only one term symbol, which always has the highest multiplicity of all microstates of the associated electron configuration, it is always lowest in energy. In other words, no rules are needed to determine the lowest energy term symbol of electron configurations like p³ and d⁵.

Since term symbols are based on Russell-Saunders coupling, where we assume that spin-spin coupling > orbit-orbit coupling > spin-orbit coupling, the three rules are applied in the above order (from rule 1 to rule 3) when we determine the lowest energy term symbol of a given configuration of an atom. The ground states of carbon and nitrogen are therefore ³P₀and ⁴S_3/2 respectively.

Hund’s rules are often stated as a single rule at introductory courses, with the rule being: electrons occupy the orbitals of a subshell singly and in parallel before pairing up. This statement is another way to state the first rule, which is the most dominant. Using the example of the $n=2$ shell of the ground state of carbon, the two electrons in the 2p orbitals are parallel because of the first rule.

Question

Why are the two electrons in the 2s orbital anti-parallel?

Answer

This is also a consequence of the first rule, where the spatial wavefunction associated with a triplet state is $\psi=\frac{1}{\sqrt{2}}[\psi_a(r_1)\psi_b(r_2)-\psi_b(r_1)\psi_a(r_2)]$ . If $r_1=r_2$ , and $\psi=0$ and $\int\vert\psi\vert^2d\tau=0$ , which implies that there is zero probability of finding the two parallel electrons at the same point.