How do you calculate the norm of a function describing a molecular orbital? I'm a bit confused by a past paper question I came across for one of my courses, shown below.


  The wavefunctions corresponding to the lowest, $\psi_{1}$, and second-lowest, $\psi_{2}$, energy molecular orbitals in 1,3-butadiene are:
  $$\psi_{1}=0.372\phi_{1} + 0.602\phi_{2}+0.602\phi_{3} +0.372\phi_{4}$$
  $$\psi_{2}=0.602\phi_{1} + 0.372\phi_{2}-0.372\phi_{3} -0.602\phi_{4}$$
  where $\phi_{i}$ is the $p_{z}$ orbital on atom $i$. Calculate the norm of $\psi_{1}$, and the overlap between $\psi_{1}$ and $\psi_{2}$. Is the result what you would expect? Explain.


Neither calculating norm nor the overlap between 2 orbitals in the way the question asks came up.
I assume by norm it means the normalisationg factor, but that makes no sense to me since it's already normalised. I assume the overlap should be 0 since they should be orthonormal, but don't know how to show that. I'd really appreciate some help!
Sadly the lecturer for the course has contracted Covid-19 and so of course is not available to answer questions.
I'm unsure if you're meant to solve this via the Hückel molecular orbital theory (see Chemistry LibreTexts), however, if you need to calculate the norm of a wave function $\Psi$, this is $|\Psi |$. Given what we know in your post about both $\Psi_{1, 2}$ then
\begin{align}
  |\Psi_1|^2 
    &amp;= \langle\Psi_1 | \Psi_1\rangle \\
    &amp;= 2\cdot(0.602^2)+2 \cdot (0.372^2) \\
    &amp;= 1.001576 \\
  \to |\Psi_1| &amp;= 1.000788 &gt; 1\\
\end{align}
I write that this is greater than one, as, on closer inspection,  is down to rounding from the coefficients of each $p_z$ orbital.

The repeat process can be found for the overlap part, which I am interpreting as $\langle\Psi_1 | \Psi_2\rangle$.