An approach is presented that allows NMR relaxation rates to be determined for a complex mixture, and it is applied to a dimethyl sulfoxide/water solution. This approach is novel for such systems, having only been used for simple systems such as atomic liquids or atomic ions in liquids until now. It involves use of a predetermined, quantum mechanical, multidimensional property surface in a simulation. The results are used in conjunction with the simulated rotational correlation time to calculate the deuteron quadrupole coupling constant (DQCC), in an analogous approach to the one used by experimentalists, and to examine the surprising experimental findings for the composition dependence of the DQCC in the dimethyl sulfoxide/water mixture. Experiments have suggested that the DQCC for a mixture of 5% dimethyl sulfoxide in water is close to the DQCC of ice, whereas its value increases to a value close to the gas value with further dilution.(1) The results are further critically analyzed using combinations of different experimental and theoretical results from the literature.