4. Multi-Electron Atoms

Approximate each electron in spherically symmetric potential:

Effective potential includes:

• Nuclear attraction: $-Ze^2/(4\pi\epsilon_0 r)$
• Average repulsion from other electrons (screening)

Separable → product of single-electron wave functions

Inner electrons screen nuclear charge:

where $\sigma$ is screening constant

Slater's rules: Empirical estimates of $\sigma$

• Electrons in same shell: $\sigma = 0.35$ each
• Electrons in $n-1$ shell: $\sigma = 0.85$ each
• Electrons in $n-2$ or lower: $\sigma = 1.00$ each

Approximate energy for orbital $(n,\ell)$:

Key point: Energy depends on both $n$ and $\ell$

• Higher $\ell$ → less penetration → more screening → higher energy
• Ordering: $E_{ns} < E_{np} < E_{nd} < E_{nf}$ for same $n$

Format: $n\ell^x$ where $x$ is number of electrons

Examples:

• Hydrogen (Z=1): 1s¹
• Helium (Z=2): 1s²
• Carbon (Z=6): 1s² 2s² 2p²
• Neon (Z=10): 1s² 2s² 2p⁶
• Iron (Z=26): [Ar] 3d⁶ 4s²

[Ar] denotes filled argon core: 1s² 2s² 2p⁶ 3s² 3p⁶

Spectroscopic notation: $^{2S+1}L_J$

• $S$: Total spin angular momentum
• $L$: Total orbital angular momentum (S, P, D, F, ...)
• $J$: Total angular momentum $J = L + S$
• $2S+1$: Multiplicity (singlet, doublet, triplet, ...)

Example: Carbon ground state $^3P_0$

• $S = 1$ (two unpaired electrons, parallel spins)
• $L = 1$ (P state)
• $J = 0$
• Triplet (three spin states)

For light atoms, orbital and spin momenta couple separately:

Energy levels split by:

• Electrostatic repulsion (depends on $L, S$)
• Spin-orbit coupling (depends on $J$)

For heavy atoms, spin-orbit coupling dominates:

Individual electron $j$ values couple to total $J$

Self-consistent field approach:

1. Guess initial wave functions $\psi_i^{(0)}$
2. Calculate effective potential from all other electrons
3. Solve Schrödinger equation for each electron
4. Update wave functions $\psi_i^{(1)}$
5. Iterate until self-consistent (convergence)

Includes exchange effects automatically through antisymmetrization

Beyond Hartree-Fock: Mix multiple configurations

where $|\Psi_0\rangle$ is ground configuration, $|\Psi_i\rangle$ are excited configurations

Captures electron correlation effects

Configuration: 1s² 2s² 2p²

Ground state: Two 2p electrons

• Hund's rule: parallel spins in different orbitals
• $m_\ell = 1, 0$ (or any two different)
• $m_s = +1/2, +1/2$ (parallel)

Quantum numbers:

• $L = 1$ (P state)
• $S = 1$ (triplet)
• $J = 0, 1, 2$ → Ground state $J = 0$ (Hund's third rule)

Term symbol: $^3P_0$

Configuration: 1s² 2s² 2p⁴

Ground state: Four 2p electrons

• Fill three orbitals with parallel spins: ↑ ↑ ↑
• Fourth electron pairs with one: ↑↓ ↑ ↑
• $S = 1$, $L = 1$

Term symbol: $^3P_2$