  Interatomic hopping mediated by spin-orbit coupling (SOC) entangles spin,
orbital and sublattice degrees of freedom of electrons, leading to the
emergence of intriguing phenomena such as novel topological insulators and
exotic spin-dependent transport including chirality-induced spin selectivity
(CISS). Despite these effects, a comprehensive microscopic formalism to
describe the spin-dependent hopping remains insufficiently established. In this
study, we systematically investigate SOC hopping by analytically deriving the
hopping integrals within a two-center approximation based on atomic orbitals.
Introducing independent parameters, or extended Slater-Koster symbols, that
characterize SOC hopping, we explicitly determine the form of the hopping for
$s$, $p$ and $d$ orbitals in the arbitrary hopping directions. Our formalism is
then implemented in tight-binding models on several lattices. Furthermore, we
examine the effect of SOC on band dispersion by employing a multipole
decomposition for the SOC Hamiltonian, providing a fundamental understanding of
SOC-induced phenomena. In particular, we derive an explicit expression for the
SOC Hamiltonian that causes unique spin splitting in chiral systems by
considering a triangular helical chain. Most importantly, the obtained SOC
Hamiltonian does not contain a term that has the symmetry of electric toroidal
monopole $G_0$ but rather an electric toroidal quadrupole $G_u$, which is the
origin of chirality in this case.
