QB_PH_SEC6_018: Atomic Spectra & Multi-Electron Atoms

Which of the following spectroscopic terms is/are Pauli-allowed for the equivalent electron configuration $n p^2$ in $L-S$ coupling?

Correct Answer

### Step-by-Step Multi-Select Evaluation: 1. **Total Quantum Microstates**: The $p$-subshell has $2(2l+1) = 6$ individual quantum spin-orbitals. Distributing 2 indistinguishable electrons yields: $N_{\text{microstates}} = \binom{6}{2} = \frac{6 \times 5}{2} = 15$ 2. **Symmetry Requirements**: - Spatial wave function parity under electron exchange is $(-1)^L$. - Spin wave function symmetry is $+1$ for triplet ($S=1$) and $-1$ for singlet ($S=0$). - Overall Pauli principle requires: $(-1)^L \times (-1)^{S+1} = -1 \implies L + S = \text{even}$. 3. **Allowed $(L, S)$ Combinations**: - $L = 2$ ($D$), $S = 0$ ($^1D$): $2 + 0 = 2$ (even, ALLOWED) $\implies ^1D_2$ (5 states). - $L = 1$ ($P$), $S = 1$ ($^3P$): $1 + 1 = 2$ (even, ALLOWED) $\implies ^3P_0, ^3P_1, ^3P_2$ (9 states). - $L = 0$ ($S$), $S = 0$ ($^1S$): $0 + 0 = 0$ (even, ALLOWED) $\implies ^1S_0$ (1 state). - Total allowed states: $5 + 9 + 1 = 15$. 4. **Forbidden Terms**: $^3D$ has $L=2, S=1 \implies 2+1=3$ (odd, FORBIDDEN by Pauli exclusion). 5. **Conclusion**: Options A, B, and C are correct.