Clear[Hmp$, "*Hmp*", m, n, p, Nmax, "\[Psi]*"]
Nmax = 10.; (*want 80*)
Hmp = Table[0, {p, Nmax}, {m, Nmax}];

Chop[ReplacePart[
  Hmp, {m_, p_} :> (p^2 \[Pi]^2)/2 KroneckerDelta[m, p] + 
    2 Integrate[
      Sin[m \[Pi] \[Chi]] Sin[p \[Pi] \[Chi]] V[\[Chi]], {\[Chi], 0, 
       1}]]]