  The diffusion of elements is a key process in understanding the unusual
surface composition of white dwarfs stars and their spectral evolution. The
diffusion coefficients of Paquette et al. (1986) have been widely used to model
diffusion in white dwarfs. We perform new calculations of the coefficients of
inter-diffusion and ionic thermal diffusion with 1) a more advanced model that
uses a recent modification of the calculation of the collision integrals that
is more suitable for the partially ionized, partially degenerate and moderately
coupled plasma, and 2) classical molecular dynamics. The coefficients are
evaluated for silicon and calcium in white dwarf envelopes of hydrogen and
helium. A comparison of our results with Paquette et al. shows that the latter
systematically underestimates the coefficient of inter-diffusion yet provides
reliable estimates for the relatively weakly coupled plasmas found in nearly
all types of stars as well as in white dwarfs with hydrogen envelopes. In white
dwarfs with cool helium envelopes (Teff < 15000K), the difference grows to more
than a factor of two. We also explored the effect of the ionization model used
to determine the charges of the ions and found that it can be a substantial
source of discrepancy between different calculations. Finally, we consider the
relative diffusion time scales of Si and Ca in the context of the pollution of
white dwarf photospheres by accreted planetesimals and find factor of > 3
differences between calculations based on Paquette et al. and our model.
