Multichannel quantum defect theory for rovibrational transitions in ultracold molecule-molecule collisions

Multichannel quantum defect theory (MQDT) has been widely applied to resonant and nonresonant scattering in a variety of atomic collision processes. In recent years, the method has been applied to cold collisions with considerable success, and it has proven to be a computationally viable alternative to full close-coupling (CC) calculations when spin, hyperfine, and external field effects are included. In this paper, we describe a hybrid approach for molecule-molecule scattering that includes the simplicity of MQDT while treating the short-range interaction explicitly using CC calculations. This hybrid approach, demonstrated for ${\mathrm{H}}_2\text{−}{\mathrm{H}}_2$ collisions in full dimensionality, is shown to adequately reproduce cross sections for quasiresonant rotational and vibrational transitions in the ultracold $\left(1\mu \mathrm{K}\right)$ and $\sim 1-10\mathrm{K}$ regime spanning seven orders of magnitude. It is further shown that an energy-independent short-range $K$ matrix evaluated in the ultracold regime $\left(1 \mu \mathrm{K}\right)$ can adequately characterize cross sections in the mK-K regime when no shape resonances are present. The hybrid CC-MQDT formalism provides an alternative approach to full CC calculations at considerably less computational expense for cold and ultracold molecular scattering.

Figure 1

Effective potential energy curves for para-${\mathrm{H}}_2$ scattering as defined in Eq. (29). The labels for nearly degenerate curves are separated by a colon, where the highest lying threshold is on the right. The panel on the right shows the CMSs included in the basis set.

Figure 2

Elastic (left) and inelastic (right) cross sections for (1,0,0,2)–(1,2,0,0) quasiresonant scattering in para-${\mathrm{H}}_2$. The thick solid black curves correspond to the full CC calculation, while the other curves correspond to the MQDT result with different matching radii, $R_m$.

Figure 3

This figure shows the elastic cross section for ${\mathrm{H}}_2(v=1,j=1)+{\mathrm{H}}_2(v=0,j=3)$ collisions (left) and the inelastic cross section for ${\mathrm{H}}_2(v=1,j=1)+{\mathrm{H}}_2(v=0,j=3)\to {\mathrm{H}}_2(v=1,j=3)+{\mathrm{H}}_2(v=0,j=1)$ quasiresonant process (right).

Figure 4

Diagonal elements of the short-range K matrix corresponding to the quasiresonant transition [(1,3,0,1) $\to$ (1,1,0,3)] as a function of the collision energy. The isotropic part of the diabatic potential matrix elements is used for the reference potential and the matching radius $R_m=9.5a_0$.

Figure 5

Elastic (left) and inelastic (right) cross sections for the scattering of ortho- and para-${\mathrm{H}}_2$. The solid black curves represent the full CC calculation, while the other curves denote different MQDT results as described in the text.