In multi-electron atoms, the Schrödinger equation cannot be solved exactly, so electron behavior is approximated using one-electron (Hartree) orbitals whose product represents the total wavefunction, giving rise to electron configuration notation. Orbital energies in these atoms depend on both the principal quantum number n and the angular momentum quantum number l, because electrons experience an effective nuclear charge (Z_effective) reduced from the true nuclear charge Z by shielding — the partial cancellation of nuclear attraction by repulsion from other electrons. Electron configurations are built via the Aufbau principle, constrained by the Pauli exclusion principle and Hund's rule, with orbital filling order and exceptions (e.g., stability of half-filled and filled d subshells) explained by penetration and shielding differences among subshells.
In multi-electron atoms, the Schrödinger equation cannot be solved exactly, so electron behavior is approximated using one-electron (Hartree) orbitals whose product represents the total wavefunction,…