(Solved): 5. The \( \pi \) molecular orbitals \( \left(\Psi_{i}\right) \) of the allyl system \( \left(\mathr ...

5. The \( \pi \) molecular orbitals \( \left(\Psi_{i}\right) \) of the allyl system \( \left(\mathrm{C}_{3} \mathrm{H}_{5}\right) \) may be described in the \( \mathrm{LCAO} / \) Hückel approach as follows, where the \( \varphi i \) represent atomic \( p \)-orbitals: \[ \begin{array}{l} \psi 1=0.50 \varphi 1+0.707 \varphi 2+0.50 \varphi 3 \\ \psi 2=0.707 \varphi 1+0.707 \varphi 3 \\ \psi 3=0.50 \varphi 1-0.707 \varphi 2+0.50 \varphi 3 \end{array} \] Show with complete indexing that the wavefunctions \( \psi 1 \), \( \psi 2 \), and \( \psi 3 \) are normalised. [9 marks]