  The electron-doping-driven collapse of the charge gap and staggered
magnetization of the spin-orbit-assisted Mott insulator Sr$_{3}$Ir$_{2}$O$_{7}$
is explored via first-principles computational methods. In the
antiferromagnetic phase, the gap and magnetization are observed to decrease
slowly with increasing doping, with an abrupt collapse of both the gap and the
magnetization at an electron concentration corresponding to 4.8\% substitution
of Sr with La, in excellent agreement with experiment. Additionally, we
describe the structural effects of electron doping in Sr$_{3}$Ir$_{2}$O$_{7}$
via a competition between the steric effect from smaller La atoms substituted
within the lattice and the dominant doping-driven deformation-potential effect.
Curiously, our first-principles calculations fail to capture the
low-temperature structural distortion reported in the low-gap phase of
Sr$_{3}$Ir$_{2}$O$_{7}$, supporting the notion that this distortion arises as a
secondary manifestation of an unconventional electronic order parameter in this
material.
