  The exchange-correlation potential formalism previously introduced and
applied to the one-dimensional Hubbard model has been extended to spin systems
and applied to the case of the one-dimensional antiferromagnetic
spin$-\frac{1}{2}$ Heisenberg model. Within the spin exchange-correlation
potential formulation, a new sum rule for spin-systems is derived. The
exchange-correlation potential for the Heisenberg model is extrapolated from
exact diagonalization results of small antiferromagnetic Heisenberg clusters.
This procedure is also employed to revisit and computationally improve the
previous investigation of the exchange-correlation potential of the half-filled
Hubbard model, which was based on the exchange-correlation potential of the
dimer. Numerical comparisons with exact benchmark calculations for both the
Heisenberg and the Hubbard models indicate that, starting from the
exchange-correlation potential of a finite cluster, the extrapolation procedure
yields a one-particle spectral function with favorable accuracy at a relatively
low computational cost. In addition, a comparison between the ground state
energies for the one-dimensional Hubbard and Heisenberg models displays how the
well known similarity in behavior of the two models at large interactions
manifests within the exchange-correlation potential formalism.
