  Positron annihilation in solid state matter can be utilized to detect and
identify open-volume defects. The momentum distribution of the annihilation
radiation is an important observable in positron-based measurements, and can
reveal information on the chemical surroundings of the defect sites. In this
work we present a variational quantum Monte Carlo method for simulation of the
momentum densities of annihilating electron-positron pairs in semiconductors
and insulators. We study finite-size effects, effects of lattice vibrations,
and different levels of trial wave functions. Small simulation cells and simple
wave function forms are found to be sufficient for accurate calculations in
simulation of pristine lattices, enabling cheap accumulation of results. We
compare calculated predictions of the Doppler broadening of the 511-keV
2{\gamma} annihilation line of the in aluminium nitride and silicon against
experimental data measured from reference samples. Our results achieve better
agreement with experiments in the these materials than conventional
state-of-the-art methods, and proves that direct modeling of the
electron-positron correlations is important for a supporting theory of positron
annihilation sprectroscopies
