An Antisymmetric Berry Frictional Force At Equilibrium in the Presence of Spin-Orbit Coupling

We calculate the electronic friction tensor for a molecule near a metal surface in the case that the electronic Hamiltonian is complex-valued, e.g. the case that there is spin-orbit coupling and/or an external magnetic field. In such a case, {\em 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 reveal that this Berry force can be much larger than the dissipational force, suggesting the strong need to consider such terms for systems with spin-orbit coupling.