Atomic Orbitals, Electron Configurations, and Subshell Filling Rules

Atomic Orbital Diagrams and Quantum Principles

• Structure of Atomic Orbital Diagrams:

  • Atomic orbital diagrams visually organize orbitals into energy levels (shells) and subshells using boxes:
  • Shell $n = 1$ (bottom level): Contains $1$ box, corresponding to the $s$ subshell ($1s$).
  • Shell $n = 2$: Contains $1$ box for the $2s$ subshell and $3$ boxes for the $2p$ subshell.
  • Shell $n = 3$: Contains $1$ box for the $3s$ subshell, $3$ boxes for the $3p$ subshell, and $5$ boxes for the $3d$ subshell.
  • Shell $n = 4$ and above: Contains $s$ ($1$ box), $p$ ($3$ boxes), $d$ ($5$ boxes), and $f$ ($7$ boxes) subshells.

• Electron Addition and Spin:

  • Electrons fill orbitals starting at the lowest energy shell (at the bottom of the diagram).
  • The first electron enters the $1s$ orbital with its spin represented as an up arrow.
  • The second electron enters the same $1s$ orbital with an opposite spin, represented as a down arrow.
  • Electron spin quantum values are $+\frac{1}{2}$ and $-\frac{1}{2}$, corresponding to clockwise or counterclockwise spin orientations.
  • Two electrons residing in the same orbital box must have opposite spins (one pointing up, one pointing down).

• The Pauli Exclusion Principle:

  • The third electron enters the $2s$ subshell rather than pairing further in the $1s$ subshell.
  • Pauli Exclusion Principle: No two electrons in the same atom can possess the exact same set of four quantum numbers ($n$, $l$, $m_l$, $m_s$).
  • Theoretical Implication: Without the Pauli exclusion principle, every electron in an atom would collapse into the lowest energy subshell ($1s$). All electrons would have identical energies and behaviors, resulting in a universe that functions completely differently from the observed physical world.

Rules Governing Electron Configurations

• Electron Configuration Notation:

  • Electron configurations summarize orbital diagram information in a concise, written format.
  • Format rules:
  • Subshell letters ($s$, $p$, $d$, $f$) are written in lowercase.
  • The number of electrons in a subshell is denoted by a superscript (e.g., $1s^1$).
  • Square brackets are used when writing shorthand configurations.

• Hund's Rule of Maximum Multiplicity:

  • Orbitals within the same subshell that possess equal energy are termed degenerate orbitals (e.g., the three boxes of the $2p$ subshell: $2p_x$, $2p_y$, and $2p_z$).
  • Hund's Rule of Maximum Multiplicity: When electrons enter degenerate orbitals, the lowest energy state is achieved by maximizing the number of unpaired electrons with parallel spins before pairing electrons in the same orbital box.
  • Electrons avoid occupying the same orbital box when unoccupied degenerate orbitals are available, minimizing electron-electron electrostatic repulsion.
  • While electron configurations could explicitly specify individual degenerate orbitals (e.g., $1s^2 2s^2 2p_x^1 2p_y^1$), standard practice consolidates them by subshell (e.g., $1s^2 2s^2 2p^2$).

Electron Configurations for Elements 1 Through 18

• First Period Elements ($n = 1$):

  • Hydrogen ($Z = 1$): Contains $1$ electron $\rightarrow 1s^1$.
  • Helium ($Z = 2$): Contains $2$ electrons $\rightarrow 1s^2$.

• Second Period Elements ($n = 2$):

  • Lithium ($Z = 3$): Contains $3$ electrons $\rightarrow 1s^2 2s^1$.
  • Beryllium ($Z = 4$): Contains $4$ electrons $\rightarrow 1s^2 2s^2$.
  • Boron ($Z = 5$): Contains $5$ electrons $\rightarrow 1s^2 2s^2 2p^1$.
  • Carbon ($Z = 6$): Contains $6$ electrons; the second $2p$ electron enters a separate degenerate orbital $\rightarrow 1s^2 2s^2 2p^2$.
  • Nitrogen ($Z = 7$): Contains $7$ electrons; all three $2p$ orbitals are singly occupied $\rightarrow 1s^2 2s^2 2p^3$.
  • Oxygen ($Z = 8$): Contains $8$ electrons; pairing begins in the $2p$ subshell $\rightarrow 1s^2 2s^2 2p^4$.
  • Fluorine ($Z = 9$): Contains $9$ electrons $\rightarrow 1s^2 2s^2 2p^5$.
  • Neon ($Z = 10$): Contains $10$ electrons; the $2p$ subshell is fully filled $\rightarrow 1s^2 2s^2 2p^6$.

• Third Period Elements ($n = 3$):

  • Each orbital box holds a maximum of $2$ electrons; an entire $p$ subshell ($3$ boxes) holds a maximum of $6$ electrons.
  • Sodium ($Z = 11$): Begins filling the $3s$ subshell \rightarrow 1s^2 2s^2 2p^6 3s^1$.\n * Magnesium (Z = 12$): Fully fills the$3s$subshell$\rightarrow 1s^2 2s^2 2p^6 3s^2$.
  • Aluminum ($Z = 13$) through Argon ($Z = 18$): Progressively fills the $3p$ subshell from $3p^1$ to 3p^6$.\n\n# Periodic Trends and Noble Gas Shorthand Notation\n\n* **Periodic Patterns in Configurations:**\n * Group 1 (Alkali metals): Configuration ends in 1s^1, 2s^1, 3s^1, 4s^1, 5s^1, 6s^1.\n * Group 2 (Alkaline earth metals): Configuration ends in 1s^2, 2s^2, 3s^2, 4s^2, 5s^2, 6s^2.\n * Group 13-18 (p$-block): Configuration ends in$n p^1$through$n p^6$(e.g., Row 6 Group 15 ends in$6p^3$; Row 6 Group 1 ends in$6s^1$; Row 6 Group 2 ends in$6s^2).\n\n* **Noble Gas Shorthand Notation Rules:**\n * Writing full configurations for heavy elements becomes repetitive; noble gas shorthand replaces core electron configurations with the symbol of the noble gas from the end of the preceding period enclosed in square brackets.\n * Neon core (1s^2 2s^2 2p^6$) is represented as$[\text{Ne}]:\n * Silicon (Z = 14$):$[\text{Ne}] 3s^2 3p^2\n * Phosphorus (Z = 15$):$[\text{Ne}] 3s^2 3p^3\n * Sulfur (Z = 16$):$[\text{Ne}] 3s^2 3p^4\n * Chlorine (Z = 17$):$[\text{Ne}] 3s^2 3p^5\n * Argon (Z = 18$):$[\text{Ne}] 3s^2 3p^6\n * Potassium (Z = 19$): Located in Period 4. Its configuration must start with the noble gas at the end of Period 3 (Argon,$[\text{Ar}]$)$\rightarrow [\text{Ar}] 4s^1$.
  • Calcium ($Z = 20$): [\text{Ar}] 4s^2$.\n * Crucial Rule: Krypton ([\text{Kr}]$) cannot be used for Potassium ($Z = 19$) because Krypton represents$36 electrons, including electrons that Potassium does not possess. Shorthand must always use the noble gas from the previous row.\n\n# Energy Levels, Quantum Mechanics, and Subshell Filling Order\n\n* **Energy Ordering of the 4s$and$3d Subshells:**\n * Potassium (Z = 19$) fills$4s$before filling$3d$because the$4s$subshell is lower in energy than the$3d$subshell at$Z = 19$and$Z = 20.\n\n* **Schrödinger Equation and Atomic Number (Z):**\n * The atomic number Z is an explicit variable in the Schrödinger equation.\n * As atomic number Z$increases, the positive charge of the nucleus increases, causing greater attraction for electrons. Consequently, subshell energy levels generally decrease (move lower in potential energy) as$Z increases.\n * Subshell energy differences depend primarily on the azimuthal quantum number l$(and less on the magnetic quantum number$m_l).\n * Energy curve behaviors as Z increases:\n * s subshells drop rapidly in energy.\n * p subshells start with a milder negative slope before dropping off.\n * d subshells start flat, drop, exhibit a slight hump, and then drop again.\n * The 4s$curve drops faster than the$3d$curve, resulting in the$4s$orbital falling below$3d$in energy specifically at Potassium ($Z = 19$) and Calcium ($Z = 20).\n\n* **Principal Shell Spacing and Subshell Hierarchy:**\n * Moving from shell n = 1$to$n = 2$,$n = 3$, and$n = 4, overall energy increases, but the energy gap between successive principal shells decreases.\n * Within any given principal shell, subshell energies increase in the order s < p < d < f$.
  • Because the energy gap between shell $n = 3$ and shell $n = 4$ is extremely small, the elevated energy of the $3d$ subshell exceeds the energy of the $4s$ subshell prior to the transition elements.

• Diagonal Filling Rule (Zigzag Method):

  • Subshells are filled in order of increasing energy according to the diagonal rule diagram:
  • List subshells by principal quantum number:
    • Row 1: $1s$
    • Row 2: $2s, 2p$
    • Row 3: $3s, 3p, 3d$
    • Row 4: $4s, 4p, 4d, 4f$
  • Follow diagonal arrows top-right to bottom-left: $1s \rightarrow 2s \rightarrow 2p \rightarrow 3s \rightarrow 3p \rightarrow 4s \rightarrow 3d \rightarrow 4p \rightarrow 5s \dots$

Anomalous Electron Configurations in Transition Metals

• Standard Transition Element Filling:
  • After Potassium ($[\text{Ar}] 4s^1$) and Calcium ($[\text{Ar}] 4s^2$), the next ten elements fill the $3d$ subshell from $3d^1$ to $3d^{10}$, followed by the $4p$ subshell from $4p^1$ to 4p^6$.\n\n* **Stability of Half-Filled and Fully-Filled Subshells:**\n * Subshells that are exactly half-filled (d^5$) or completely filled ($d^{10}) exhibit special thermodynamic stability compared to other electron arrangements.\n\n* **Chromium Exception (Z = 24):**\n * Expected configuration: [\text{Ar}] 4s^2 3d^4\n * Actual configuration: [\text{Ar}] 4s^1 3d^5\n * Mechanism: The energy difference between 4s$and$3d$is very small (analogous to walking up$6$stairs instead of$3$or$4$full flights of stairs). Electrons experience unfavorable inter-electronic repulsion when sharing a single$4s$orbital box. An electron absorbs a tiny amount of energy to step up$6$stairs into an empty$3d$box, resulting in six half-filled orbitals ($4s^1 3d^5). The net stability gained from having half-filled subshells overcomes the small energy cost of promotion.\n\n* **Copper Exception (Z = 29):**\n * Expected configuration: [\text{Ar}] 4s^2 3d^9\n * Actual configuration: [\text{Ar}] 4s^1 3d^{10}\n * Mechanism: An electron is promoted from the 4s$subshell to the$3d$subshell. This converts a$90\%$filled$3d$subshell and$100\%$filled$4s$subshell into a$100\%$filled$3d^{10}$subshell and a$50\%$filled$4s^1$subshell. A system with one$100\%$filled subshell and one$50\%$filled subshell is energetically more stable than a system with a$90\%$filled subshell and a$100\% filled subshell.\n\n# Questions & Discussion\n\n* **Clarification on Electron Configurations (Lithium Example):**\n * *Question:* Why is the electron configuration for Lithium written as 1s^2 2s^1?\n * *Response:* The configuration represents the total count of electron arrows in each subshell box. Lithium has 2$electron arrows filling the$1s$orbital box and$1$electron arrow in the$2s$orbital box, yielding$1s^2 2s^1\n\n* **Meaning of Orbital Arrows and Spin:**\n * *Question:* What do the arrows in the orbital diagram represent?\n * *Response:* The arrows represent individual electrons, and their direction indicates spin state (+\frac{1}{2}$or$-\frac{1}{2}), which corresponds to clockwise or counterclockwise spin orientations. The exact direction assignment does not matter as long as two electrons sharing the same orbital box have opposite pointing arrows (one pointing up and one pointing down).\n\n* **Apparent Anomalies in Transition Metal Configurations:**\n * *Question:* Are configurations containing repeated electron counts (such as two d^5$and two$d^{10}$$ terms) typographical errors?
  • Response: These are not typos; they are intentional exceptions resulting from the enhanced thermodynamic stability of half-filled and fully-filled subshells.