For OE62 molecule MAJHUB, using the attached geometry and the official PBE0(vacuum) and G0W0@PBE0(vacuum, complete-basis-set) orbital-energy lists, identify the HOMO and LUMO at each level and compute both HOMO-LUMO gaps. Report both gaps in eV to four decimal places, state the signed difference (GW gap minus PBE0 gap), and explain whether this single molecule alone establishes a general relationship between the two levels of theory.

The attachment is the MAJHUB XYZ geometry (S0002), 41 atoms. Coordinates are in angstrom (1 nm = 10 angstrom). Atom rows are numbered starting at 1, immediately after the two header lines. Do not renumber or reorder atoms; use the row numbers exactly as given. This is a single relaxed molecular geometry, not a periodic cell: apply no periodic images or symmetry expansion. Occupied orbital energies are listed in ascending order; the HOMO is the last (highest) value in the occupied list. Unoccupied orbital energies are also listed in ascending order; the LUMO is the first (lowest) value in the unoccupied list. The HOMO-LUMO gap is LUMO minus HOMO, in eV. Only virtual levels below the vacuum level are listed for the unoccupied set.

PBE0(vacuum) occupied energies (eV): -6.45361 is the HOMO (last value of the full list). PBE0(vacuum) unoccupied energies (eV): -2.5701400000000003 is the LUMO (first value of the full list). G0W0@PBE0(vacuum, CBS) occupied energies (eV): -8.31924 is the HOMO (last value). G0W0@PBE0(vacuum, CBS) unoccupied energies (eV): -1.4863300000000002 is the LUMO (first value).