QB_PH_SEC6_002: Atomic Spectra & Multi-Electron Atoms

Using Hund's rules, what is the ground-state spectroscopic term symbol $^{2S+1}L_J$ for a carbon atom with electron configuration $1s^2 2s^2 2p^2$?

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Rule 1 (Maximum $S$)**: For two equivalent $p$-electrons, the spins align parallel to minimize Coulomb repulsion: $s_1 = 1/2, s_2 = 1/2 \implies S = 1$, multiplicity $2S+1 = 3$. 2. **Rule 2 (Maximum $L$)**: With $m_{s1} = +1/2, m_{s2} = +1/2$, the spatial wave function must be antisymmetric. The maximum allowed $M_L = m_{l1} + m_{l2} = 1 + 0 = 1 \implies L = 1$ ($P$ term). 3. **Rule 3 ($J$ coupling)**: For a shell less than half-filled ($n_e = 2 < 3$ for $p$-shell), the state with minimum $J$ lies lowest: $J = |L - S| = |1 - 1| = 0$. 4. **Ground State Term**: Combining parameters yields $^{3}P_0$. Option (A) is correct.