Figure 7.

Scaling of $\overline{\alpha }$ as a function of the number of carbon atoms N_C for the QE, AACT, and SQE methods for the trans N-alkane series. In all cases, we have used (without optimization) the exponents of 0.70 and 0.78 a.u. for C and H taken from Chelli et al. (Ref. ⁵) and the full Coulomb J matrix with diagonal hardness elements given by the idempotential J_ii values. For the AACT and SQE cases, bond hardness parameters are applied so that the diagonal of the resulting G matrix is equal to twice the diagonal that would be obtained from the pure QE method. A value of λ=1 is employed in the SQE case. For the AACT^* case (stars), we have artificially reduced the diagonal bond hardness terms by a factor of 4 to illustrate the AACT instability problem when these terms become too small.