Chirality-induced spin polarization: some fundamental ingredients.

This paper examines some fundamental ingredients that produce enantiomer-specific spin-polarization in chiral systems, leading to chirality-induced spin selectivity (CISS). In particular, it examines parity and time-reversal symmetries of the key quantities, such as linear and angular momenta and spin. Enantiomer-specific spin-polarization emerges because of the symmetry relationship between the two mirror images that results in different transformation properties for axial and polar vectors. It is best described by pseudoscalars. The analysis of the structure of the spin-orbit coupling (SOC) operator shows that the chirality of the system enters the picture via the shapes of molecular orbitals, which are determined by the (electrostatic) molecular field. The spin-orbit operator detects the handedness of the orbitals and reports on it by means of the phase of the individual matrix elements. Although the phase of an individual interstate matrix element depends on the chosen state convention, rephasing-invariant combinations of the SOCs and the resulting spin and magnetic observables retain the enantiomer-specific information. This explains how a short-range operator can sense the effect of a distant molecular field. The symmetry of the SOC also explains the different electronic population dynamics observed in chiral and non-chiral molecules. Drawing from the field of molecular magnetism, we connect spin-polarization with magnetic anisotropy and show that the latter is a necessary but not a sufficient condition for the former. The anisotropy requires a non-zero expectation value of the angular momentum operator in the spin-orbit-perturbed states and lifting electronic degeneracy within the multiplet. Formal theoretical analysis is supplemented by numerical examples.