Antisymmetric Berry frictional force at equilibrium in the presence of spin-orbit coupling

We analytically calculate the electronic friction tensor for a molecule near a metal surface in the case where the electronic Hamiltonian is complex valued, e.g., the case where there is spin-orbit coupling and/or an external magnetic field. In such a case, even at equilibrium, we show that the friction tensor is not symmetric. Instead, the tensor is the real-valued sum of one positive definite tensor (corresponding to dissipation) plus one antisymmetric tensor (corresponding to a Berry pseudomagnetic force). Moreover, we find that this Berry force can be much larger than the dissipational force, suggesting the possibility of strongly spin-polarized chemicurrents or strongly spin-dependent rate constants for systems with spin-orbit coupling.