Orbital with maximum symmetry is — Atomic Structure Chemistry Question

💡 Solution & Explanation

**Step 1 - Concept of Orbital Symmetry** An orbital has maximum spatial symmetry if its electron probability density $|\psi|^2$ is identical in ALL directions from the nucleus — this is called spherical symmetry. Mathematically, this means the wave function depends only on $r$ (distance from nucleus), not on angles $\theta$ or $\phi$: $$\psi(r, \theta, \phi) = f(r) \quad \text{(angular-independent)}$$ --- **Step 2 - Analysis of Option (B): s-orbital** For the s-orbital, $l = 0$, $m_l = 0$. The angular wave function: $$Y_{0,0}(\theta, \phi) = \frac{1}{\sqrt{4\pi}} = \text{constant}$$ Since the angular part is a constant, $\psi$ depends only on $r$. The s-orbital is **perfectly spherically symmetric** — infinite rotational symmetry axes. This is the maximum possible symmetry. --- **Step 3 - Why Other Options Are Wrong** * **Option (A) p-orbital:** $l = 1$ → dumbbell shape, depends on $\theta$ and $\phi$. Oriented along a specific axis with one nodal plane. Only uniaxial/cylindrical symmetry. Incorrect. * **Option (C) $d_{xy}$-orbital:** $l = 2$ → clover-leaf shape in $xy$-plane with two nodal planes ($xz$ and $yz$). Low discrete rotational symmetry. Incorrect. * **Option (D) $d_{z^2}$-orbital:** $l = 2$ → two lobes along $z$-axis + torus in $xy$-plane. Cylindrical about $z$-axis but not spherically symmetric. Incorrect. $$\text{Correct Option: } \boxed{\text{B (s-orbital)}}$$