  We study the effect of a static electric field on lithium donor spins in
silicon. The anisotropy of the effective mass leads to the anisotropy of the
quadratic Stark susceptibility, which we determined using the Dalgarno-Lewis
exact summation method. The theory is asymptotically exact in the field domain
below Li-donor ionization threshold, relevant to the Stark-tuning electron spin
resonance experiments. To obtain the generalized Stark susceptibilities at
arbitrary fields, we propose a new variational wave function which reproduces
the exact results in the low-field limit. With the calculated susceptibilities
at hand, we are able to predict and analyze several important physical effects.
First, we observe that the energy level shifts due to the quadratic Stark
effect for Li donors in Si are equivalent to, and can be mapped onto, those
produced by an external stress.
  Second, we demonstrate that the Stark effect anisotropy, combined with the
unique valley-orbit splitting of a Li donor in Si, spin-orbit interaction and
specially tuned external stress, may lead to a very strong modulation of the
donor spin $g$-factor by the electric field. Third, we investigate the
influence of random strains on the $g$-factor shifts and quantify the random
strain limits and requirements to Si material purity necessary to observe the
$g$-factor Stark shifts experimentally. Finally, we discuss possible
implications of our results for quantum information processing with Li spin
qubits in Si.
