  We have implemented and tested the method we have recently proposed [J. Phys.
Chem. Lett. 10, 1537 (2019)] to treat dispersion interactions, which is derived
from a supramolecular wavefunction constrained to leave the diagonal of the
many-body density matrix of each monomer unchanged. The corresponding
variational optimization leads to expressions for the dispersion coefficients
in terms of the ground-state pair densities of the isolated monomers only,
which provides a framework to build new approximations without the need for
polarizabilities or virtual orbitals. The question we want to answer here is
how accurate this ``fixed diagonal matrices'' (FDM) method can be for isotropic
and anisotropic $C_6$ dispersion coefficients when using monomer pair densities
from different levels of theory, namely Hartree-Fock, MP2 and CCSD. For
closed-shell systems, FDM with CCSD monomer pair densities yields the best
results, with a mean average percent error for isotropic $C_6$ dispersion
coefficients of about 7\% and a maximum absolute error within 18\%. The
accuracy for anisotropic dispersion coefficients with FDM on top of CCSD ground
states is found to be similar. The performance for open shell systems is less
satisfactory, with CCSD pair densities not always providing the best result. In
the present implementation, the computational cost on top of the monomer's
ground-state calculations is $\mathcal{O}(N^4)$.
