  Recent experimental efforts have led to considerable interest in donor-based
localized electron spins in Si as viable qubits for a scalable silicon quantum
computer. With the use of isotopically purified $^{28}$Si and the realization
of extremely long spin coherence time in single-donor electrons, the recent
experimental focus is on two-coupled donors with the eventual goal of a
scaled-up quantum circuit. Motivated by this development, we simulate the
statistical distribution of the exchange coupling $J$ between a pair of donors
under realistic donor placement straggles, and quantify the errors relative to
the intended $J$ value. With $J$ values in a broad range of donor-pair
separation ($5<|\mathbf{R}|<60$ nm), we work out various cases systematically,
for a target donor separation $\mathbf{R}_0$ along the [001], [110] and [111]
Si crystallographic directions, with $|\mathbf{R}_0|=10, 20$ or 30 nm and
standard deviation $\sigma_R=1, 2, 5$ or 10 nm. Our extensive theoretical
results demonstrate the great challenge for a prescribed $J$ gate even with
just a donor pair, a first step for any scalable Si-donor-based quantum
computer.
