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Based on the results published recently [J. Phys. A: Math. Theor. 50, 065201(2017)], the universal finite-size contributions to the free energy of thesquare lattice Ising model on the $L\times M$ rectangle, with open boundaryconditions in both directions, are calculated exactly in the finite-sizescaling limit $L,M\to\infty$, $T\to T_\mathrm{c}$, with fixed temperaturescaling variable $x\propto(T/T_\mathrm{c}-1)M$ and fixed aspect ratio$\rho\propto L/M$. We derive exponentially fast converging series for therelated Casimir potential and Casimir force scaling functions. At the criticalpoint $T=T_\mathrm{c}$ we confirm predictions from conformal field theory byCardy & Peschel [Nucl. Phys. B 300, 377 (1988)] and by Kleban & Vassileva [J.Phys. A: Math. Gen. 24, 3407 (1991)]. The presence of corners and the relatedcorner free energy has dramatic impact on the Casimir scaling functions andleads to a logarithmic divergence of the Casimir potential scaling function atcriticality.

The square lattice Ising model on the rectangle II: Finite-size scaling limit