The bond order between the adsorbate and surface atoms represents the effective number of "bonds" formed between the two fragments. It can be calculated using the one-particle density matrices associated with the FRZ, POL, and FULL states, which are in turn computed from the corresponding occupied fragment molecular orbital coefficients. To ensure the idempotency of the FRZ and POL density matrices, one constructs the overlap matrix s s s oo = C † occ SC occ among the occupied orbitals on both the metal and adsorbate fragments and uses it to construct the density matrix: