Calculation of $d$-wave Feshbach resonance in $^{87}$Rb-$^{85}$Rb cold collisions by a multichannel computational method

We calculate recently observed $d$-wave Feshbach resonance [Phys. Rev. Lett. {\bf 119}, 203402 (2017)] and associated triplet structure in $^{87}$Rb-$^{85}$Rb cold collisions. We resort to a computational technique combining some features of close-coupling (CC) and $R$-matrix theory. In the long-range part, we numerically calculate a pair of base or reference functions for each channel ignoring all interchannel couplings, ensuring the linear independence of these base functions. The short-range part of the wave function is calculated by CC method. The multichannel wave functions are obtained by matching the short- and long-range parts at a suitably chosen matching point. We first verify the method by calculating cold collisional properties of $^{85}$Rb and $^6$Li in the presence of external magnetic fields tuned across specific $s$-wave Feshbach resonances and thereby reproducing known results. Our numerical results on the $d$-wave resonance agree reasonably well with the experimental ones. Furthermore, our method enables one to readily construct the real space Green functions in order to calculate perturbatively the effects of any residual inter-channel couplings in the long range.