Exact diagonalization serves as the most intuitive numerical approach for solving quantum problems and is widely applied in both few-body and many-body systems, making it a core component of computational physics curricula. Guided by a scaffolded teaching philosophy that progresses from fundamental to advanced concepts, this paper systematically outlines the instruction of exact diagonalization in undergraduate computational physics courses—beginning with single-particle systems and gradually advancing to quantum many-body systems. Through a series of carefully designed pedagogical examples, students not only master key procedural steps such as basis selection, symmetry utilization, matrix construction, and diagonalization, but also develop a deeper comprehension of the method's strengths and limitations. This instructional framework effectively bridges the formalism of quantum mechanics with cutting-edge many-body numerical techniques, laying a solid groundwork for students' future exploration of advanced many-body computational methods such as the density matrix renormalization group and tensor networks.

Vanadium dichalcogenides have attracted increasing interests for the charge density wave phenomena and possible ferromagnetism. Here, we report on the multiphase behavior and gap opening in monolayer VTe₂ grown by molecular beam epitaxy. Scanning tunneling microscopy (STM) and spectroscopy study revealed the (4×4) metallic and gapped (2 $\sqrt{3}$ ×2 $\sqrt{3}$ ) charge-density wave (CDW) phases with an energy gap of ~ 40 meV. Through the in-plane condensation of vanadium atoms, the typical star-of-David clusters and truncated triangle-shaped clusters are formed in the (4×4) and (2 $\sqrt{3}$ ×2 $\sqrt{3}$ ) phases respectively, resulting in different surface morphologies and electronic structures as confirmed by density functional theory (DFT) calculations with on-site Coulomb repulsion. The CDW-driven reorganization of the atomic structure weakens the ferromagnetic superexchange coupling and strengthens the antiferromagnetic exchange coupling on the contrary, suppressing the long-range magnetic order in monolayer VTe₂. The electron correlation is found to be important to explain the gap opening in the (2 $\sqrt{3}$ ×2 $\sqrt{3}$ ) phase.