Two new anharmonic forms for the Debye-Waller factor, aimed at modelling curvilinear and asymmetric motion, have been introduced. These forms permit the refinement of structures with these types of anharmonic motion using a small number of additional parameters. Molecular-dynamics-derived numerical probability density functions (PDFs) have been used to assess the merit of these new functions in real space. The comparison is favourable particularly for the curvilinear PDF based on a parabolic coordinate system change of a trivariate Gaussian distribution. The initial results also suggest that high-order even terms from the Gram-Charlier series may be important for modelling methyl-group libration. The molecular-dynamics data sets provide useful insights into the nature of anharmonic thermal motion. Addressing the problem in real space allows intuitive PDFs to be developed but numerical methods may be necessary for these methods to be implemented in refinement programs as an analytical Debye-Waller factor cannot always be obtained.