d- and f-Orbitals Cheatsheet

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1. Introduction to Orbitals Atomic Orbitals: Regions around the nucleus where electrons are most likely to be found. Defined by quantum numbers. Quantum Numbers: Principal Quantum Number ($n$): Energy level (shell). $n=1, 2, 3, ...$ Azimuthal/Angular Momentum Quantum Number ($l$): Shape of orbital (subshell). $l = 0, 1, 2, ..., n-1$. Magnetic Quantum Number ($m_l$): Orientation of orbital in space. $m_l = -l, ..., 0, ..., +l$. Spin Quantum Number ($m_s$): Electron spin. $m_s = +1/2$ or $-1/2$. Orbital Types: $l=0 \implies$ s-orbital (spherical) $l=1 \implies$ p-orbital (dumbbell) $l=2 \implies$ d-orbital (complex shapes) $l=3 \implies$ f-orbital (even more complex shapes) 2. d-Orbitals 2.1. Characteristics Exist for $n \ge 3$. (e.g., $3d, 4d, 5d, ...$) For $l=2$, $m_l$ can be $-2, -1, 0, +1, +2$. This means there are 5 degenerate d-orbitals in a free atom. Each d-orbital can hold 2 electrons, so a d-subshell can hold a maximum of 10 electrons. 2.2. Shapes of d-Orbitals Four "Cloverleaf" Shapes: $d_{xy}$: Lobes between x and y axes. $d_{yz}$: Lobes between y and z axes. $d_{xz}$: Lobes between x and z axes. $d_{x^2-y^2}$: Lobes along x and y axes. One "Dumbbell with Donut" Shape: $d_{z^2}$: Two lobes along the z-axis and a donut (torus) in the xy-plane. d-orbital shapes (schematic): d$_{x^2-y^2}$ d$_{xy}$ d$_{z^2}$ 2.3. Electronic Configuration Rules for d-Orbitals Aufbau Principle: Fill lower energy orbitals first. Pauli Exclusion Principle: Max 2 electrons per orbital, with opposite spins. Hund's Rule: For degenerate orbitals, fill each orbital with one electron of parallel spin before pairing. Stability of Half-filled and Fully-filled d-orbitals: $d^5$ (half-filled) and $d^{10}$ (fully-filled) configurations are exceptionally stable due to symmetry and exchange energy. This often leads to exceptions in electron configurations (e.g., Cr: $[Ar]3d^54s^1$ instead of $3d^44s^2$; Cu: $[Ar]3d^{10}4s^1$ instead of $3d^94s^2$). 2.4. Transition Metals (d-block elements) Elements with partially filled d-subshells in their common oxidation states. Key Properties: Variable oxidation states. Formation of colored compounds (due to d-d electronic transitions). Paramagnetism (due to unpaired d-electrons). Good catalysts (due to variable oxidation states and ability to form coordination complexes). Formation of coordination compounds (ligands donate lone pairs to empty d-orbitals). Crystal Field Theory (CFT) / Ligand Field Theory (LFT): Describes splitting of d-orbital degeneracy in coordination complexes. In an octahedral field, d-orbitals split into two sets: $t_{2g}$ (lower energy: $d_{xy}, d_{yz}, d_{xz}$) and $e_g$ (higher energy: $d_{x^2-y^2}, d_{z^2}$). The energy difference ($\Delta_o$) determines color and magnetic properties. 3. f-Orbitals 3.1. Characteristics Exist for $n \ge 4$. (e.g., $4f, 5f, ...$) For $l=3$, $m_l$ can be $-3, -2, -1, 0, +1, +2, +3$. This means there are 7 degenerate f-orbitals in a free atom. Each f-orbital can hold 2 electrons, so an f-subshell can hold a maximum of 14 electrons. Shapes are very complex and not typically memorized, often described as having multiple lobes. 3.2. Lanthanides (4f-block elements) Fill the $4f$ subshell. Located after Barium (Ba) and before Hafnium (Hf). Often have a stable $+3$ oxidation state. Key Properties: Lanthanide Contraction: Poor shielding by $4f$ electrons leads to a smaller than expected atomic/ionic radius across the series, affecting subsequent elements. Generally less reactive than Group 1 and 2 metals. Compounds are often colored and luminescent (used in lasers, phosphors). Paramagnetic properties. 3.3. Actinides (5f-block elements) Fill the $5f$ subshell. Located after Radium (Ra) and before Rutherfordium (Rf). All are radioactive. Exhibit a wider range of oxidation states than lanthanides (e.g., U: $+3, +4, +5, +6$). Key Properties: Most are synthetic (transuranic elements). Used in nuclear energy and weapons. Poor shielding by $5f$ electrons also contributes to actinide contraction, but less pronounced and more complex due to relativistic effects. 4. Summary Table Orbital Type $l$ value Min. $n$ Number of Orbitals Max Electrons Shape Complexity Associated Elements s 0 1 1 2 Spherical All elements p 1 2 3 6 Dumbbell p-block d 2 3 5 10 Cloverleaf, Dumbbell+Donut Transition Metals (d-block) f 3 4 7 14 Very Complex Lanthanides, Actinides (f-block)