Diagonal superexchange in a simple square <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>CuO</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> lattice

Many microscopic models with the interaction between the next-nearest neighbors as a key parameter for cuprate physics have inspired us to study the diagonal superexchange interaction in a ${\mathrm{CuO}}_{2}$ layer. Our investigation shows that models with extended hopping provide a correct representation of magnetic interactions only in a hypothetical square ${\mathrm{CuO}}_{2}$ layer, where the diagonal superexchange interaction with the next-nearest neighbors always has an antiferromagnetic (AFM) nature. The conclusions are based on the symmetry prohibition on the ferromagnetic contribution to the diagonal superexchange between the next-nearest neighbors for a simple square ${\mathrm{CuO}}_{2}$ layer rather than for a real ${\mathrm{CuO}}_{2}$ layer, where diagonal AFM superexchange may be overestimated. We also discuss the reasons for magnetic frustration effects and high sensitivity of spin nanoinhomogeneity to square symmetry breaking.