A computer simulation methodology with which to study the single-particle dynamics in complex molecular liquids is presented. Molecular dynamics simulations of liquid water are performed in the temperature range of 238-473 K using the polarizable point charge (PPC) potential. The self part of the van Hove density-density correlation function is calculated. Using the Gaussian approximation of the van Hove function the mean self-diffusion coefficient for the PPC potential is calculated. The singularity temperature for supercooled PPC water, T-s=218 K, estimated from the self-diffusion data appears to agree well with most estimates for real water. In order to elucidate the spatial picture of the single-particle molecular density in this complex liquid and its time evolution, we explicitly resolve the self van Hove function in the local frame of water molecules. The self-diffusion tensor is introduced and numerically evaluated from this spatial (separation and direction dependent) self van Hove function. The fluctuations of the single-particle molecular density in liquid water appear to be spatially anisotropic (nonspherical). At low temperatures these dynamical heterogeneities in liquid water tend to increase.