Molecular dynamics simulations are reported for a solute immersed in a monatomic solvent; systems modeled represent monatomic and diatomic solute species (whose atoms are larger and heavier than the solvent), with varying force constant and bond length for the diatomic. From these simulations, autocorrelation functions, diffusion coefficients (D), and friction coefficients (ξ) are determined; for the diatomic, these are found for both the center-of-mass and relative coordinates. These results are used to develop simple models for D and ξ, including (for the diatomic relative coordinate) their frequency dependence. The models enable D and ξ to be readily determined from properties such as bulk viscosity, potential parameters, etc. These D and ξ can be used to interpret and predict picosecond time scale data for solute dynamics using stochastic models (e.g., the Kramers or Langevin equations) at the molecular level; their theoretical basis is such that they should apply to many types of solute moieties (e.g., aromatic rings) as well as to the large atoms used in the simulations.