  We develop a numerical scheme and code for estimating the energy and momentum
transfer via neutrino pair annihilation ($\nu + {\bar \nu} \to e^{-}+ e^{+}$),
bearing in mind the application to the collapsar models of gamma-ray bursts
(GRBs). To calculate the neutrino flux illuminated from the accretion disk, we
perform a ray-tracing calculation in the framework of special relativity. The
numerical accuracy of the developed code is certificated by several tests, in
which we show comparisons with the corresponding analytical solutions. Using
hydrodynamical data in our collapsar simulation, we estimate the annihilation
rates in a post-processing manner. We show that the neutrino energy deposition
and momentum transfers are strongest near the inner edge of the accretion disk.
The beaming effects of special relativity are found to change the annihilation
rates by several factors in the polar funnel region. After the accretion disk
settles into a stationary state (typically later than
  $\sim 9$ s from the onset of gravitational collapse), we find that the
neutrino-heating timescale in the vicinity of the polar funnel ($\lesssim 80$
km) can become shorter than the hydrodynamical timescale, indicating that the
neutrino-heated outflows can be launched there. We point out that the momentum
transfer can play as important role as the energy deposition for the efficient
acceleration of neutrino-driven outflows.
  Our results suggest that the neutrino pair annihilation has a potential
importance equal to the conventional magnetohydrodynamic mechanism for igniting
the GRB fireballs.
