  We calculate the bulk photovoltaic response of the ferroelectrics BaTiO$_3$
and PbTiO$_3$ from first principles by applying "shift current" theory to the
electronic structure from density functional theory. The first principles
results for BaTiO$_3$ reproduce eperimental photocurrent direction and
magnitude as a function of light frequency, as well as the dependence of
current on light polarization, demonstrating that shift current is the dominant
mechanism of the bulk photovoltaic effect in BaTiO$_3$. Additionally, we
analyze the relationship between response and material properties in detail.
The photocurrent does not depend simply or strongly on the magnitude of
material polarization, as has been previously assumed; instead, electronic
states with delocalized, covalent bonding that is highly asymmetric along the
current direction are required for strong shift current enhancements. The
complexity of the response dependence on both external and material parameters
suggests applications not only in solar energy conversion, but to
photocatalysis and sensor and switch type devices as well.
