Dimensionality control of coupled scattering equations using partitioning techniques: The case of two molecules

A method of variable reduction of the dimensionality of the coupled equations for inelastic scattering is presented. The method is based upon a projection operator $P$ with restricted ranges of values for the orbital angular momentum and coupled rotational angular momenta of the two molecules. For rotational states restricted by $0\le j_1\le j_1^{*}$ and $0\le j_2\le j_2^{*}$ and total angular momentum large, the coupled equations have dimensionality $2(j_1^{*}+1) (j_2^{*}+1)-1\mathrm{}\le N\le {(j_1^{*}+1)}^2{(j_2^{*}+1)}^2$, where $N$ is controlled by the choice of $P$. This is in contrast to conventional partitioning techniques which utilize further restrictions on the important molecular rotational states. In both cases the well-known parity conservation causes blocking of the equations into two sets, each of which is essentially half the size of the original set. The dynamics in the $P$ subspace and its complementary $Q$ subspace are decoupled by various approximations on the equation of motion of $Q{\psi }_{\mathrm{scat}}$. Information about scattering into the $Q$ subspace is retained within these approximations and is reintroduced at the end of the computation with little additional labor. An expression for the error in the calculation of the resultant approximate reactance operators is derived and its implications discussed. The general formal equations are then applied to the case of scattering of two rigid rotors, although the inclusion of vibrational modes would in no way affect the procedures outlined. Various possible choices for $P$ are presented, and additional constraints on $P$ for the case of two indistinguishable molecules are discussed. A method of solution for the $P{\psi }_{\mathrm{scat}}$ equation is suggested, and it is shown to lead to the possibility of substantial savings in computational labor.