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We show that the integral cross sections for state-to-state quantumscattering of cold molecules in a magnetic field can be efficiently computedusing the total angular momentum representation despite the presence ofunphysical Zeeman states in the eigenspectrum of the asymptotic Hamiltonian. Wedemonstrate that the unphysical states arise due to the incompleteness of thespace-fixed total angular momentum basis caused by using a fixed cutoff value$J_\text{max}$ for the total angular momentum of the collision complex $J$. Asa result, certain orbital angular momentum ($l$) basis states lack the fullrange of $J$ values required by the angular momentum addition rules, resultingin the appearance of unphysical states. We find that by augmenting the basiswith a full range of $J$-states for every $l$, it is possible to completelyeliminate the unphysical states from quantum scattering calculations onmolecular collisions in external fields. To illustrate the procedure, we usethe augmented basis sets to calculate the state-to-state cross sections forrotational and spin relaxation in cold collisions of$^{40}$CaH$(X^2\Sigma^+,v=0,N=1, M_N=1, M_S=1/2)$ molecules with $^4$He atomsin a magnetic field. We find excellent agreement with benchmark calculations,validating our proposed procedure. We find that $N$-conserving spin relaxationto the lowest-energy Zeeman state of the $N=1$ manifold,$|11\frac{1}{2}\rangle\to |1-1-\frac{1}{2}\rangle$ is nearly completelysuppressed due to the lack of spin-rotation coupling between the fullyspin-stretched Zeeman states. Our results demonstrate the possibility ofrigorous, computationally efficient, and unphysical state-free quantumscattering calculations on cold molecular collisions in an external magneticfield.

Total angular momentum representation for state-to-state quantum scattering of cold molecules in a magnetic field