  Understanding the structural underpinnings of magnetism is of great
fundamental and practical interest. Se_{1-x}Te_{x}CuO_{3} alloys are model
systems for the study of this question, as composition-induced structural
changes control their magnetic interactions. Our work reveals that this
structural tuning is associated with the position of the supposedly dummy atoms
Se and Te relative to the super-exchange (SE) Cu--O--Cu paths, and not with the
SE angles as previously thought. We use density functional theory,
tight-binding, and exact diagonalization methods to unveil the cause of this
surprising effect and hint at new ways of engineering magnetic interactions in
solids.
