 Solution

${\Phi }_i^a$ is the normalized determinant obtained by replacing occupied spin-orbital $i$ in ${\Phi }_0$ with previously unoccupied orbital $a$, using a consistent antisymmetric determinant ordering. Thus $i=1,\dots ,N$ labels occupied orbitals and $a=N+1,\dots ,2M$ labels virtual orbitals. Occupied-to-occupied rotations only change the representation of the same occupied subspace; the occupied-virtual rotations describe distinct nearby determinants.