  This paper presents a first-principles study of the Debye-Waller factor and
the Debye temperature for amorphous silicon ($a$-Si) from lattice-dynamical
calculations and direct molecular-dynamics simulations using density-functional
theory (DFT). The effects of temperature and structural disorder on the
intensity of the diffraction maxima and the vibrational mean-square
displacement (MSD) of Si atoms are studied in the harmonic approximation, with
particular emphasis on the bond-length disorder, the presence of coordination
defects, and microvoids in $a$-Si networks. It has been observed that the MSDs
associated with tetrahedrally-bonded Si atoms are considerably lower than their
dangling-bond counterparts -- originating from isolated and vacancy-induced
clustered defects -- and those on the surface of microvoids, leading to an
asymmetric non-gaussian tail in the distribution of atomic displacements. An
examination of the effect of anharmonicity on the MSD at high temperatures
using direct $ab$ $initio$ molecular-dynamics simulations (without the harmonic
approximation) suggests that the vibrational motion in $a$-Si is practically
unaffected by anharmonic effects at temperatures below 400 K, as far as the
present DFT calculations are concerned. The Debye temperature of $a$-Si is
found to be in the range of 488--541 K from specific-heat and MSD calculations
using first-principles lattice-dynamical calculations in the harmonic
approximation, which matches closely with the experimental value of 487--528 K
obtained from specific-heat measurements of $a$-Si at low temperatures.
