Using the beyond-Skyrme-Hartree-Fock approach with various Skyrme-type $NN$ effective interactions, we explore the large spectroscopic quadrupole moment $Q_s$ of the first excited $2_1^{+}$ state in ${}^{20}\mathrm{Ne}$. Our calculated $Q_s\left(2_1^{+}\right)\approx -0.20e\text{b}$ aligns closely with the experimental value of $-0.22\left(2\right)e\text{b}$. Consistent with other theoretical methods, our results confirm the presence of an $\alpha$ cluster structure in this state. We also investigate the robustness of the $\alpha$ cluster against tensor force, demonstrating that incorporating tensor components significantly enhances $Q_s\left(2_1^{+}\right)$ and strengthens the cluster structure along the symmetrical axis.

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• Nuclear density functional theory
• Nuclear forces
• Nuclear shapes and moments
• Nuclear structure & decays
• Nucleon-nucleon interactions
• 20 ≤ A ≤ 38