On algebraic properties of the sub-block of zero field hyperfine Hamiltonian with penultimate total spin projection for arbitrary hyperfine structure, and field dependence of radical pair recombination probability in the vicinity of zero field.

Basic algebraic arguments demonstrate that the probability of radical pair recombination in low field for an arbitrary pair with Hamiltonian confined to Zeeman and isotropic hyperfine interactions contains two additive contributions linear with applied field with equal, but opposite in sign, proportionality factors. Their weights are determined by the probability of having all nuclear spins along the field in the initial electron-singlet state of the pair, and in the case of equilibrium with respect to nuclear spins, the two contributions completely compensate the field dependences of each other, producing an additive term in the singlet yield with zero derivative. However, if the nuclear set is polarized, a linear skew proportional to polarization appears, introducing anisotropy in the intrinsically spherically symmetric system. The key element in this derivation is guaranteed nondegeneracy of the eigenvalues of the "penultimate" (M - 1) block of the Hamiltonian for a radical with any number of distinct spin-1/2 nuclei in zero field, which leads to guaranteed applicability of the first-order nondegenerate perturbation theory with nonvanishing linear in field admixture of states, persisting all the way into the final expression for singlet yield for a pair with an arbitrary isotropic hyperfine structure. We argue that this behavior of the field dependence of recombination yield is representative of a radical pair of an arbitrarily complex hyperfine structure; this may be a possible mechanism for anisotropic response ("chemical compass") for an isotropic radical pair based system with isotropic-only internal interactions and anisotropy introduced via the initial state of nuclei, while such an approach may complement the usually needed numerical simulations.