Chapter 16 Key Takeaways: Multi-Electron Atoms and the Periodic Table

The Big Picture

Multi-electron atoms cannot be solved exactly because the electron--electron repulsion terms couple all electrons together. The central field approximation --- replacing the full many-body potential with an effective spherically symmetric potential for each electron --- transforms the problem into manageable single-particle equations. This approximation, combined with the Pauli exclusion principle and Hund's rules, explains the entire structure of the periodic table. The periodic table is not an empirical curiosity; it is a deductive consequence of the Schrodinger equation, the Coulomb potential, electron spin, and fermion antisymmetry.

Key Equations

Multi-Electron Hamiltonian

$$\hat{H} = \sum_{i=1}^{N}\left[-\frac{\hbar^2}{2m_e}\nabla_i^2 - \frac{Ze^2}{4\pi\epsilon_0 r_i}\right] + \sum_{i

Central Field Single-Particle Equation

$$\left[-\frac{\hbar^2}{2m_e}\nabla^2 + V_{\text{eff}}(r)\right]\psi_{nlm}(\mathbf{r}) = \varepsilon_{nl}\,\psi_{nlm}(\mathbf{r})$$

Term Symbol

$${}^{2S+1}L_J$$

where $S$ = total spin, $L$ = total orbital angular momentum ($S, P, D, F, G, \ldots$ for $L = 0, 1, 2, 3, 4, \ldots$), $J$ = total angular momentum ($|L-S| \leq J \leq L+S$), and $2S+1$ = spin multiplicity.

Effective Nuclear Charge (Slater)

$$Z_{\text{eff}} = Z - \sigma$$

Approximate Orbital Energy (Slater)

$$\varepsilon_{nl} \approx -13.6\;\text{eV}\times\frac{Z_{\text{eff}}^2}{(n^*)^2}$$

Subshell Capacity

$$\text{Max electrons in subshell } l = 2(2l+1)$$

Shell Capacity

$$\text{Max electrons in shell } n = 2n^2$$

Hund's Rules (Ground State Selection)

Filling Order (Madelung Rule)

$$1s \to 2s \to 2p \to 3s \to 3p \to 4s \to 3d \to 4p \to 5s \to 4d \to 5p \to 6s \to 4f \to 5d \to 6p \to 7s \to 5f \to 6d \to 7p$$

Rule: Fill in order of increasing $n + l$; for equal $n + l$, lower $n$ first.

Notable exceptions: Cr ($3d^5\,4s^1$), Cu ($3d^{10}\,4s^1$), and analogues in the $4d$ and $5d$ series.

Period Structure of the Periodic Table

Screening and Penetration Summary

Term Symbols for Common Configurations

Five Things to Remember

1. Multi-electron atoms break the $l$-degeneracy: Unlike hydrogen (where energy depends only on $n$), multi-electron atoms have $\varepsilon_{ns} < \varepsilon_{np} < \varepsilon_{nd} < \varepsilon_{nf}$ due to screening. This single fact determines the filling order and the structure of the periodic table.

2. Filled subshells always give ${}^1S_0$: When determining term symbols, only the electrons in partially filled subshells contribute. Filled subshells add $L = 0$, $S = 0$.

3. Hund's first rule is the most important: Maximize spin first. The exchange interaction (parallel spins reduce repulsion via the Fermi hole) is the dominant effect for ground state selection.

4. The Madelung rule has exceptions: Half-filled ($d^5$) and fully-filled ($d^{10}$) subshells gain extra exchange stabilization that can override the $(n+l)$ ordering. Always check Cr, Cu, and their analogues.

5. Hartree--Fock is the starting point, not the final answer: HF captures ~99% of the total energy and correctly predicts shell structure, but misses the correlation energy. For chemical accuracy, post-Hartree--Fock methods (CI, coupled cluster, DFT) are needed.

Connections to Other Chapters

Common Mistakes to Avoid

1. Applying hydrogen-like energy formulas to multi-electron atoms: $E_n = -13.6/n^2$ eV is for hydrogen only. Multi-electron atoms have $l$-dependent energies.

2. Assuming $4s$ is always below $3d$ in energy: This is true for K and Ca but false for transition metal ions. Fe$^{2+}$ is $[\text{Ar}]\,3d^6$, not $[\text{Ar}]\,3d^4\,4s^2$.

3. Forgetting the Pauli constraint on equivalent electrons: For two equivalent $p$-electrons ($p^2$), $L + S$ must be even. Not all $(L, S)$ combinations are allowed.

4. Confusing spin multiplicity with the number of electrons: The superscript in ${}^3P$ is $2S + 1 = 3$, meaning $S = 1$ (triplet). It is not the number of electrons.

5. Applying Hund's rules to excited states: Hund's rules reliably predict only the ground state. The ordering of excited terms is more complex and requires explicit calculation.

6. Ignoring the half-filling rule for $J$: Carbon ($p^2$, less than half-filled) has $J = |L-S| = 0$; oxygen ($p^4$, more than half-filled) has $J = L + S = 2$. Getting this backwards is a common error.