To differentiate between the site basis and the delocalized basis, {m, n, • • • } will be used to index the site basis while {j, k, • • • } to index the delocalized basis. Specifically, j = 0 corresponds to an all-symmetric entity (i.e., the bright state), while j = 1, 2 • • • N -1 are asymmetrical (i.e., dark states). Furthermore, the {j, k, • • • } ∈ [0, N -1] are defined explicitly in the reciprocal space, such that -j ≡ N -j, and j -k ≡ N + j -k for j < k.