The title compounds, [AnO(2)(H2O)5](n+), n = 1 or 2 and An = U, Np, and Pu, are studied using relativistic density functional theory (DFT). Three rather different relativistic methods are used, small-core effective core potentials (SC-ECP), a scalar four-component all-electron relativistic method, and the zeroeth-order regular approximation. The methods provide similar results for a variety of properties, giving confidence in their accuracy. Spin-orbit and multiplet corrections to the An(VI)/An(V) reduction potential are added in an approximate fashion but are found to be essential. Bulk solvation effects are modeled with continuum solvation models (CPCM, COSMO). These models are tested by comparing explicit (cluster), continuum, and mixed cluster/ continuum solvation models as applied to various properties. The continuum solvation models are shown to accurately account for the effects of the solvent, provided that at least the first coordination sphere is included. Reoptimizing the structures in the presence of the bulk solvent is seen to be important for the equatorial bond lengths but less relevant for energetics. Explicit inclusion of waters in the second coordination sphere has a modest influence on the energetics. For the first time, free energies of solvation are calculated for all six [AnO(2)(H2O)(5)](n+) species. The calculated numbers are within the experimental error margins, and the experimental trend is reproduced correctly. By comparison of different relativistic methods, it is shown that an accurate relativistic description leads to marked improvements over the older large-core ECP (LC-ECP) method for bond lengths, vibrational frequencies, and, in particular, the An(VI)/An(V) reduction potential. Two approximate DFT methods are compared, B3LYP, a hybrid DFT method, and PBE, a generalized gradient approximation. Either method yields An(VI)/An(V) reduction potentials of comparable quality. Overall, the experimental reduction potentials are accurately reproduced by the calculations.