Advanced Quantum Chemistry and Electronic Configuration

The Quantum Mechanical Model and Principles of Quantum Numbers

• The quantum mechanical model approaches the atomic model by associating each electron with a wave equation.
• Similar to mathematical equations having specific solutions, electronic wave equations yield solutions known as quantum numbers.
• Quantum numbers serve as the "address" for an electron within an atom, providing a unique description of its location, energy, and properties.
• There are four distinct quantum numbers: the principal quantum number ($n$), the angular momentum quantum number ($l$), the magnetic quantum number ($m_l$), and the spin quantum number ($m_s$).

Principal Quantum Number ($n$)

• The principal quantum number, symbolized by $n$, identifies the shell or the main energy level where an electron is located.
• Allowed values: Positive integers ranging from $1, 2, 3$ upward.
• Energy relationship: As the value of $n$ increases, the energy of the orbital increases, and the electron is generally farther from the nucleus.

Angular Momentum Quantum Number ($l$)

• The angular momentum quantum number, symbolized by $l$, determines the shape of the orbital.
• Allowed values: Integers from $0$ to $n - 1$.
• Subshell designations:
  • If $l = 0$: It is an $s$ orbital (spherical shape).
  • If $l = 1$: It is a $p$ orbital (dumbbell shape).
  • If $l = 2$: It is a $d$ orbital (complex shape).
  • If $l = 3$: It is an $f$ orbital (complex shape).
• Radial Nodes: A node is a region where the probability of finding an electron is exactly zero. The formula for calculating radial nodes is:
  • $\text{Radial Nodes} = n - l - 1$

Magnetic Quantum Number ($m_l$)

• The magnetic quantum number, symbolized by $m_l$, specifies the orientation of the orbital in space relative to Cartesian axes ($x, y, z$).
• Allowed values: Integers ranging from $-l$ through $0$ to $+l$.
• Characteristics:
  • For $l = 0$ ($s$ orbital), only one orientation exists ($m_l = 0$) because a sphere is uniform in all directions.
  • For higher $l$ values, multiple orientations exist, calculated by the formula $2l + 1$.
  • Example: If $l = 1$ ($p$ orbital), there are $2(1) + 1 = 3$ orbitals: $m_l = -1, 0, +1$. These correspond to the $p_x, p_y,$ and $p_z$ orbitals.

Spin Quantum Number ($m_s$)

• The spin quantum number, symbolized by $m_s$, describes the intrinsic rotation or spin state of the electron.
• Allowed values: Plus one-half ($+1/2$) or negative one-half ($-1/2$), often referred to as "up spin" and "down spin."
• Significance: Each individual orbital can accommodate a maximum of two electrons, provided they have opposite spins.

Determining Orbital Energy: The ($n+l$) Rule

• While the principal quantum number $n$ primarily dictates energy, the subshell shape ($l$) also contributes to the overall energy of the orbital.
• The energy level of an orbital can be predicted using the $n + l$ value:
  • Higher $n + l$ = higher energy.
  • Lower $n + l$ = lower energy.
• Examples:
  • $2s$ orbital: $n = 2, l = 0$. Sum = $2$.
  • $2p$ orbital: $n = 2, l = 1$. Sum = $3$. (Therefore, $2p$ has higher energy than $2s$).
  • $3d$ orbital: $n = 3, l = 2$. Sum = $5$.
  • $4s$ orbital: $n = 4, l = 0$. Sum = $4$. (Therefore, $4s$ is lower in energy than $3d$).
• Degenerate Orbitals: Orbitals that possess the same energy are called degenerate orbitals (e.g., the three $p$ orbitals in a given shell).

Rules for Electronic Configuration

• Pauli's Exclusion Principle: No two electrons in the same atom can have the same set of all four quantum numbers. If $n, l,$ and $m_l$ are the same, the electrons must differ in their spin ($m_s$).
• Aufbau Principle: Electrons occupy the lowest energy orbitals available first before moving to higher energy levels.
• Hund's Maximum Multiplicity Rule: When filling degenerate orbitals, electrons enter each orbital singly with parallel spins before they begin to pair up. This minimizes repulsion and maximizes the overall spin.

Notation and Orbital Diagrams

• Electronic configuration notation consists of three parts:
  • The coefficient: Principal quantum number ($n$).
  • The letter: Orbital type/shape designation ($l$).
  • The superscript: The number of electrons in that subshell (e.g., $1s^2$).
• Increasing Energy Order: $1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, ...$
• Orbital Diagrams: Pictorial representations showing orbitals as boxes or lines and electrons as arrows (up or down).

Valence and Core Electrons

• Valence Electrons: Electrons occupying the outermost shell (highest $n$ value). These are primarily responsible for the chemical reactivity of an atom.
• Core Electrons: Electrons occupying the inner shells. They correspond to the electron configuration of the preceding noble gas.
• Abbreviated (Noble Gas) Notation: Using the symbol of the previous noble gas in brackets to represent core electrons (e.g., Sodium ($Na$) is written as $[Ne] 3s^1$).

Periods, Groups, and Blocks

• Elements in the same group of the periodic table share similar valence electron configurations, which explains their similar chemical properties.
• Group 1 elements (Alkali metals such as Lithium ($Li$), Sodium ($Na$), Potassium ($K$), Rubidium ($Rb$), Cesium ($Cs$), and Francium ($Fr$)) all end in $ns^1$ orientations.
• Blocks define where the valence electrons reside:
  • $s$-block: Groups 1 and 2.
  • $p$-block: Groups 13 through 18.
  • $d$-block: Transition metals (subshells accommodate up to $10$ electrons).
  • $f$-block: Inner transition metals (Lanthanides and Actinides; subshells accommodate up to $14$ electrons).

Anomalous Electronic Configurations

• Some elements deviate from the expected Aufbau filling order because half-filled or completely filled subshells provide enhanced stability.
• Chromium ($Cr, Z=24$): Expected $[Ar] 4s^2 3d^4$; Actual $[Ar] 4s^1 3d^5$ (half-filled $s$ and $d$).
• Copper ($Cu, Z=29$): Expected $[Ar] 4s^2 3d^9$; Actual $[Ar] 4s^1 3d^{10}$ (half-filled $s$ and full $d$).
• Other examples include Molybdenum ($Mo$) and Ruthenium ($Ru$).

Electronic Configuration of Ions

• The process involves determining the total electron count after considering the charge.
• Cations ($+$ charge): Formed by losing electrons (e.g., $Na \to Na^+$ loses 1 electron, leaving $10$ electrons, identical to Neon ($Ne$)).
• Anions ($-$ charge): Formed by gaining electrons (e.g., Phosphorus ($P$) with $15$ electrons becomes $P^{3-}$ with $18$ electrons, identical to Argon ($Ar$)).
• Calculation Strategy: Identify the atomic number of the neutral atom and subtract/add electrons based on the charge value.

Examples and Exercises

• Question: Is the set $n=2, l=2, m_l=2$ allowed?
  • Answer: No. If $n=2$, the maximum possible value for $l$ is $n-1 = 1$. Therefore, $l=2$ is invalid.
• Question: Electronic configuration for Magnesium ($Mg^{2+}$)?
  • Atomic number: $12$.
  • Charge: $+2$ means losing $2$ electrons.
  • Total electrons: $10$.
  • Configuration: $1s^2 2s^2 2p^6$ (Isoelectronic with Neon).
• Question: Identifying Technetium ($TC$)?
  • Ground state configuration given: $[Kr] 5s^2 4d^5$.
  • Krypton ($Z=36$) + $2$ ($s$ orbital) + $5$ ($d$ orbital) = $43$ electrons. Atomic number $43$ is Technetium.