Aufbau principle, Pauli exclusion principle and Hund's rule Knowledge-Based Study

Study notes

1. Arranging Electrons in Atoms

   In atomic structure, the chemical behavior of an element is largely dictated by how its electrons are distributed around the nucleus. The arrangement of electrons among the available energy levels, sublevels, and atomic orbitals is called an electron configuration.

   To determine the most stable, lowest-energy arrangement—known as the ground state—chemists apply three foundational guidelines:

   • The Aufbau Principle: Dictates the sequence in which subshells are filled based on increasing energy.
   • The Pauli Exclusion Principle: Limits the capacity of any individual orbital and defines the spin characteristics of paired electrons.
   • Hund's Rule: Governs the placement of electrons within orbitals of equal energy (degenerate orbitals).

   Building Up Electron Subshells

   The term Aufbau comes from the German word meaning "building up". The Aufbau principle states that electrons occupy the lowest-energy atomic orbitals available before filling higher-energy states.

   The $n + l$ Rule

   The relative energy of a subshell is governed primarily by the sum of its principal quantum number ($n$) and angular momentum quantum number ($l$):

   • Orbitals with a smaller value of $n + l$ possess lower energy and are filled first.
   • If two subshells share the same value of $n + l$, the subshell with the smaller $n$ lies lower in energy and is occupied first.

   For example, comparing the 4s and 3d subshells:

   • For 4s: $n = 4$ and $l = 0$, so $n + l = 4$.
   • For 3d: $n = 3$ and $l = 2$, so $n + l = 5$.

   Because $4 < 5$, the 4s subshell is lower in energy than 3d and fills first in multi-electron atoms.

   Fundamental Limits on Orbital Occupancy

   Proposed by Wolfgang Pauli in 1925, the Pauli exclusion principle states that no two electrons in an atom can have identical sets of all four quantum numbers ($n, l, m_l, m_s$).

   Consequences for Orbitals

   Every individual orbital is uniquely specified by three quantum numbers: the principal quantum number ($n$), angular momentum quantum number ($l$), and magnetic quantum number ($m_l$). Therefore, within any single orbital:

   • Electrons already share identical values of $n$, $l$, and $m_l$.
   • To satisfy the exclusion principle, they must differ in their fourth quantum number: the electron spin quantum number, $m_s$.
   • Because $m_s$ can only take values of $+\frac{1}{2}$ or $-\frac{1}{2}$, an individual orbital can hold a maximum of two electrons, and they must have opposite (antiparallel) spins.

   Representing spins with arrows, an orbital containing two electrons must display them as one pointing up and one pointing down (paired spins).

   Distributing Electrons in Degenerate Orbitals

   Orbitals that share the exact same energy level within a given subshell are called degenerate orbitals. For instance, the three 2p orbitals ($2p_x, 2p_y, 2p_z$) are degenerate.

   Hund's rule of maximum multiplicity states that when filling degenerate orbitals, electrons occupy them singly with parallel spins before any orbital is doubly occupied.

   Minimizing Electron Repulsion

   Electrons carry negative electric charges and naturally repel one another. Distributing them across separate spatial regions minimizes electrostatic repulsion and lowers the atom's overall energy.

   Consider a neutral carbon atom (atomic number 6), which has two electrons in its 2p subshell:

   • Correct (ground state): One electron enters $2p_x$ and one enters $2p_y$, both with parallel spins (e.g., both spin-up).
   • Incorrect (excited state or non-compliant): Pairing both electrons into $2p_x$ while leaving $2p_y$ and $2p_z$ empty creates unnecessary electron-electron repulsion.

   Applying the Principles to Ground-State Configurations

   Determining a ground-state orbital diagram requires applying all three rules simultaneously:

   1. Identify total electrons from the atomic number of the neutral atom or ionic charge.
   2. Apply Aufbau: Sequence subshells in increasing energy order (1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, ...).
   3. Apply Pauli: Restrict every orbital to no more than 2 electrons with opposing spins.
   4. Apply Hund: In any partially filled p, d, or f subshell, place single electrons in each orbital with matching spins before pairing begins.

   Illustrative Examples

   • Nitrogen ($Z = 7$): The configuration is $1s^2\, 2s^2\, 2p^3$. The three 2p orbitals each contain one single electron with parallel spins, illustrating Hund's rule.
   • Oxygen ($Z = 8$): The configuration is $1s^2\, 2s^2\, 2p^4$. After all three 2p orbitals receive one electron each, the fourth 2p electron must pair up with an opposite spin in one orbital, demonstrating the transition from Hund's rule to Pauli-governed pairing.