⚛️ Atomic Structure & Quantum Mechanics

From Bohr's planetary model to the probabilistic Schrödinger wave mechanics, atomic physics and chemistry converge at the quantum level. Advanced competitive examinations focus heavily on radial/angular node topology, wavefunction boundary behaviors, Slater shielding calculations, and orbital exchange degeneracy.

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1. 🌌 The Quantum Transition

1.1 De Broglie Matter Waves & Heisenberg Uncertainty

For any particle of mass $m$ and velocity $v$:

$$\lambda =\frac{h}{p}=\frac{h}{mv}=\frac{h}{\sqrt{2mK}}=\frac{h}{\sqrt{2mqV}}$$

The Heisenberg Uncertainty Principle sets the fundamental limit of simultaneous measurement:

$$\mathrm{\Delta }x\cdot \mathrm{\Delta }{p}_x\ge \frac{h}{4\pi }=\frac{\hslash }{2}⟹\mathrm{\Delta }x\cdot \mathrm{\Delta }{v}_x\ge \frac{h}{4\pi m}$$$$\mathrm{\Delta }E\cdot \mathrm{\Delta }t\ge \frac{\hslash }{2}$$

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2. 🌊 The Schrödinger Wave Equation

For a single-electron hydrogenic atom ($Z$) in spherical polar coordinates $(r,\theta ,\varphi )$:

$$\hat{H}\psi =E\psi$$$${\mathrm{\nabla }}^2\psi +\frac{8{\pi }^{2}m}{h^2}(E-V)\psi =0$$

The total wavefunction factors into separable radial and angular components:

$${\psi }_{n,l,m_l}(r,\theta ,\varphi )=R_{n,l}(r)\cdot Y_{l,m_l}(\theta ,\varphi )=R_{n,l}(r)\cdot {\mathrm{\Theta }}_{l,m_l}(\theta )\cdot {\mathrm{\Phi }}_{m_l}(\varphi )$$

2.1 Nodes Topology Rules

Node Type	Geometric Nature	Mathematical Formula	Physical Meaning
Radial Nodes (Spherical)	Concentric spherical surfaces where $\psi =0$	$N_r=n-l-1$	Points where radial wavefunction $R(r)$ crosses zero
Angular Nodes (Planar / Nodal Cones)	Planes or conical surfaces passing through nucleus	$N_a=l$	Dependent purely on angular momentum quantum number $l$
Total Nodes	All nodal surfaces combined	$N_{\text{total}}=N_r+N_a=n-1$	Independent of orbital type, depends strictly on shell $n$

Examples:

• $1s$ orbital: $n=1,l=0⟹N_r=0,N_a=0,N_{\text{total}}=0$.
• $2p$ orbital: $n=2,l=1⟹N_r=0,N_a=1,N_{\text{total}}=1$.
• $3d$ orbital: $n=3,l=2⟹N_r=0,N_a=2,N_{\text{total}}=2$.
• $3s$ orbital: $n=3,l=0⟹N_r=2,N_a=0,N_{\text{total}}=2$.
• $d_{z^2}$ orbital: Has 2 nodal cones (cone angle $\theta ={54.74}^{\circ }$ where $3{\cos }^2\theta -1=0$).

2.2 Probability Density vs. Radial Probability Function

1. Probability Density at a point: $P(r)={\psi }^2$ or $R^2(r)$ (Probability per unit volume).
   • For all $s$-orbitals ($l=0$), $R(0)\ne 0$ (Maximum probability density is right at the nucleus!).
   • For all non-$s$ orbitals ($p,d,f$ where $l\ge 1$), $R(0)=0$ (Zero probability density at the nucleus).
2. Radial Probability Distribution Function (RDF):$$4\pi r^2{R}^2(r)dr$$
   • Probability of finding the electron in a spherical shell of thickness $dr$ at distance $r$ from nucleus.
   • For all orbitals without exception, $\text{RDF}(r=0)=0$ because $4\pi (0{)}^2=0$.
   • The radius of maximum probability for $1s$ hydrogen orbital is exactly the Bohr radius $a_0=0.529\text{AA}$.

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3. 🔢 Quantum Numbers & Electronic Selection Rules

1. Principal Quantum Number ($n=1,2,3,\dots$):
   • Determines the main energy shell and average orbital size: $r_n=\frac{a_0{n}^2}{Z}$, $E_n=-13.6\frac{Z^2}{n^2}\text{ eV}$.
2. Azimuthal / Orbital Angular Momentum Quantum Number ($l=0,1,\dots ,n-1$):
   • Determines orbital subshell shape and orbital angular momentum:$$L=\sqrt{l(l+1)}\frac{h}{2\pi }=\sqrt{l(l+1)}\hslash$$
   • For an electron in any $s$-orbital ($l=0$), orbital angular momentum is strictly zero.
3. Magnetic Quantum Number ($m_l=-l,\dots ,0,\dots ,+l$):
   • Specifies spatial orientation of the orbital in an external magnetic field ($2l+1$ degenerate orientations).
4. Spin Quantum Number ($s=1/2,m_s=\pm 1/2$):
   • Spin angular momentum: $S=\sqrt{s(s+1)}\hslash =\frac{\sqrt{3}}{2}\hslash$.
   • Magnetic spin moment: ${\mu }_s=\sqrt{n(n+2)}\text{ BM}$ (where $n$ is number of unpaired electrons).

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4. 🧮 Multi-Electron Energy Degeneracy & Exchange Energy

In single-electron hydrogenic atoms ($H,H{e}^{+},L{i}^{2+}$), energy depends only on $n$:

$$E_{3s}=E_{3p}=E_{3d}$$

In multi-electron atoms, electron-electron repulsion removes this degeneracy via shielding:

$$E_{(n+l)\text{ rule}}:1s<2s<2p<3s<3p<4s<3d<4p<5s<4d<5p<6s<4f<5d<6p$$

4.1 Exchange Energy & Anomalous Stability of $d^5,d^{10}$

Electrons with parallel spins in degenerate orbitals can exchange positions, releasing stabilizing Exchange Energy ($K$):

$$\text{Number of Exchanges }(E_{\text{ex}})=\sum \frac{n_i(n_i-1)}{2}$$

• For $Cr$ ($[Ar]3d^{5}4s^1$): $3d^5$ has 5 parallel spins $⟹\frac{5\times 4}{2}=10$ exchanges.
• If $Cr$ were $[Ar]3d^{4}4s^2$: $3d^4$ has only 4 parallel spins $⟹\frac{4\times 3}{2}=6$ exchanges.
• The extra 4 exchanges provide large stabilization overcoming the small pairing promotion energy.