Establishing the physical mechanism governing exchange interactions is fundamental for exploring exotic phases such as the quantum spin liquids (QSLs) in real materials. In this work, we address exchange interactions in Sr2CuTe$_{1-x}$W$_{x}$O, a series of double perovskites that realize the spin-1/2 square lattice and were suggested to harbor a QSL ground state arising from random distribution of non-magnetic ions. Our {it ab initio} multi-reference configuration interaction calculations show that replacing Te atoms with W atoms changes the dominant couplings from nearest to next-nearest neighbor explained by the crucial role of unoccupied states of non-magnetic ions in the super-superexchange mechanism. Combined with spin-wave theory simulations, our calculated exchange couplings provide an excellent description of the inelastic neutron scattering spectra of the end compounds, as well as explain the magnetic excitations in Sr2CuTe$_{0.5}$W$_{0.5}$O as emerging from the bond-disordered exchange couplings. Our results provide crucial understanding of the role of non-magnetic cations in exchange interactions paving the way to further exploration of QSL phases in bond-disordered materials.
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