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The paper highlights the impact of channel correlations on the capacity of MIMO communications, with a focus on the so called diagonal correlations. Based on this analysis, the limitations of simplified mathematical representations, such as the Kronecker or the diagonal-decorrelation models, are pointed out. Finally, the correlation properties of characteristic geometrical scattering models are studied in order to analyze whether these models can be adequately represented by the simplified formalisms, as well as to quantify the errors introduced by these simplifications. With respect to channel correlations, neither the Kronecker model nor the diagonal-decorrelation assumptions are good representations of the investigated geometrical scattering models. When comparing capacity values, it is found that the diagonal-decorrelation model may yield significant errors. In particular, relative errors on ergodic or outage capacity may be significant for the Kronecker model.