Many-Electron Atoms and the Periodic Table

Schrödinger’s equation has no exact solution for many-electron atoms; approximate solutions are typically used, often through the orbital approximation.
Quantum numbers ( $n$ , $l$ , $m_l$ ) are used to describe orbitals in many-electron atoms, similar to the hydrogen atom.

Orbital energies depend on both $n$ and $l$ for a given shell (constant $n$ ).
Energy increases with increasing $l$ (e.g., Es < Ep < Ed < Ef).
Shielding Effect: Decrease in nuclear attraction for an electron due to the presence of other electrons.
Effective Nuclear Charge ( $Z<em>{eff}$ ): The net positive charge experienced by an electron, calculated as actual nuclear charge minus shielding effects ( $Z</em>{eff} = Z - s$ ).
A greater $Z_{eff}$ corresponds to lower electron energy.
Orbital Penetration: s > p > d > f (s-orbitals penetrate closest to the nucleus).
Shielding effect trend within a given shell: s < p < d < f (s-orbitals shield most effectively).
Effective nuclear charge trend: s > p > d > f (s-electrons experience the highest $Z_{eff}$ ).
Energy trend: s < p < d < f (s-electrons have the lowest energy).
Electron spin ( $m_s = +1/2$ or $-1/2$ ) is crucial for many-electron atoms.

No two electrons in an atom can possess the same values for all four quantum numbers ( $n$ , $l$ , $m<em>l$ , $m</em>s$ ).
Equivalently: A maximum of two electrons can occupy one orbital, and they must have opposite spins.

Electrons will singly occupy orbitals within a subshell with parallel spins before any orbital is doubly occupied.
This rule applies to orbitals that have the same energy.

Ground-state configuration: The lowest energy arrangement of electrons in an atom.
Aufbau Principle: Orbitals are filled in order of increasing energy, adhering to the Pauli Exclusion Principle and Hund's Rule.
Orbital filling order: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < \text{…}.
Orbital diagrams: Visual representations showing electrons in orbitals using arrows (vertical for unpaired, opposing for paired).
Noble gas cores: An abbreviation method using the electron configuration of the preceding noble gas (e.g., $[Ar]4s^1$ for potassium).
Exceptions to filling order: Chromium (Cr) and Copper (Cu) achieve stability through half-filled ( $d^5$ ) or fully-filled ( $d^{10}$ ) d-orbitals.
- Cr: Exists as $[Ar]4s^13d^5$ instead of $[Ar]4s^23d^4$ .
- Cu: Exists as $[Ar]4s^13d^{10}$ instead of $[Ar]4s^23d^9$ .

Cations (positive ions): Electrons are removed sequentially from the highest $n$ value first. If $n$ is the same, electrons are removed from the highest $l$ value (e.g., $4s$ electrons are removed before $3d$ ).
- Example: Fe ( $[Ar]4s^23d^6$ ) becomes Fe $^{2+}$ ( $[Ar]3d^6$ ).
Anions (negative ions): The ground-state configuration is the same as that of a neutral atom with the identical total number of electrons, often resulting in a noble gas configuration. ‘