1. What is being asked?

   We need to identify which aspect of the electron is responsible for the formation of covalent bonds.

2. Key idea: covalent bond formation

   A covalent bond is formed by overlapping of atomic orbitals and sharing of electrons between atoms.

3. Why does overlap occur?

   Atomic orbitals arise from the wave description of electrons. The electron in an atom is not treated as a tiny classical particle moving in a fixed path; instead, it is described by a wave function $\psi$.

   The covalent bond is explained by constructive overlap of these electron wave functions, leading to increased electron density between the nuclei.

4. Check the options

   • A: wave nature of electron

     Correct, because bonding and orbital overlap are consequences of the electron's wave nature.

   • B: particle nature of electron

     Incorrect. The particle picture alone does not explain orbital overlap and bond formation.

   • C: both wave and particle nature of electron

     Although electrons do possess both natures, the specific phenomenon of covalent bond formation is explained primarily by the wave nature.

   • D: none of these

     Incorrect.

5. Final answer

   Therefore, the formation of covalent bonds exhibits the wave nature of electron.

$\boxed{\text{A}}$