In the general linear model consider the experimental design problem for the Gauß-Markov estimator or least squares estimator when the observations are correlated. We prove new formulas for the efficiency of an exact design with respect to theD-criterion. For models with intercept term, for example, these formulas are useful to derive better lower bounds for the efficiency than the bounds recently given for an arbitrary linear model. These bounds are applied in examples to symmetrical regular circulants as covariance matrices. A byproduct of the investigations is some insight as to what kinds of designs might retain their optimality or high efficiency (for the uncorrelated homoscedastic case) under correlated observations.