A trial function for the ground state of the antiferromagnetic linear chain leads to an equation for an upper bound of the ground-state energy per spin. This equation is solved exactly.
van de Braak, H. P., Caspers, W. J., Gragert, P. K. H., & Willemse, M. W. M. (1979). An upper bound for the ground-state energy of an antiferromagnetic chain with next-nearest-neighbor interaction. Journal of statistical physics, 20(5), 487-497. https://doi.org/10.1007/BF01012896