  The marked interplay between the crystalline, electronic, and magnetic
structure of atomically thin magnets has been regarded as the key feature for
designing next-generation magneto-optoelectronic devices. In this respect, a
detailed understanding of the microscopic interactions underlying the magnetic
responses of these crystals is of primary importance. Here, we combine model
Hamiltonians with multi-reference configuration interaction wavefunctions to
accurately determine the strength of the spin couplings in the prototypical
single-layer magnet CrI$_3$. Our calculations identify the (ferromagnetic)
Heisenberg exchange interaction $J = -1.44$ meV as the dominant term, being the
inter-site magnetic anisotropies substantially {weaker}. We also find that
single-layer CrI$_3$ features an out-of-plane easy axis ensuing from a
single-ion anisotropy $A = -0.10$ meV, and predict $g$-tensor in-plane
components $g_{xx} = g_{yy} = 1.90$ and out-of-plane component $g_{zz} = 1.92$.
In addition, we assess the performance of a dozen widely used density
functionals against our accurate correlated wavefunctions {calculations} and
available experimental data, thereby establishing reference results for future
first-principles investigations. Overall, our findings offer a firm theoretical
ground to experimental observations.
