  Nuclear Berry curvature effects emerge from electronic spin degeneracy and
canlead to non-trivial spin-dependent (nonadiabatic) nuclear dynamics. However,
such effects are completely neglected in all current mixed quantum-classical
methods such as fewest switches surface-hopping. In this work, we present a
phase-space surface-hopping (PSSH) approach to simulate singlet-triplet
intersystem crossing dynamics. We show that with a simple pseudo-diabatic
ansatz, a PSSH algorithm can capture the relevant Berry curvature effects and
make predictions in agreement with exact quantum dynamics for a simple
singlet-triplet model Hamiltonian. Thus, this approach represents an important
step towards simulating photochemical and spin processes concomitantly, as
relevant to intersystem crossing and spin-lattice relaxation dynamics.
