Invariance of quantum scattering rate coefficients to anisotropy of atom-molecule interactions

TL;DR

The paper tackles the high computational cost of quantum scattering in atom–molecule systems by testing whether thermally averaged total rate coefficients κ_tot(T) are invariant to PES anisotropy. It combines rigorous close-coupling calculations for Rb–N$_2$ and Rb–H$_2$ with Gaussian process regression to map κ_tot onto a five-parameter PES space, including short- and long-range anisotropy terms. The authors show that κ_tot is invariant to both short-range anisotropy (varying D_e,λ) and long-range anisotropy (varying γ) within percent-level changes, and that Maxwell–Boltzmann averaging coupled with elastic–inelastic compensation explains this robustness. The practical implication is a substantial reduction in computational effort: total rates can be accurately computed from isotropic PES using a single uncoupled differential equation, benefiting metrology and related applications where precise collision rates are essential.

Abstract

Quantum scattering calculations for strongly interacting molecular systems are computationally demanding due to the large number of molecular states coupled by the anisotropy of atom - molecule interactions. We demonstrate that thermal rate coefficients for total (elastic + inelastic) atom - molecule scattering are insensitive to the interaction anisotropy of the underlying potential energy surface. In particular, we show that the rate coefficients for Rb-H$_2$ and Rb-N$_2$ scattering at room temperature can be computed to 1% accuracy with anisotropy set to zero, reducing the complexity of coupled channel quantum scattering calculations to numerical solutions of a single differential equation. Our numerical calculations and statistical analysis based on Gaussian process regression elucidate the origin and limitations of the invariance of the total scattering rate coefficients to changes in atom - molecule interaction anisotropy.

Figures (9)

• Figure 1: Comparison of $\kappa_{\text{tot}}$ predicted by the 5D GP model with the results of CC calculations. The CC calculations for training and testing the GP models are performed with the reduced mass, $C_{6,\lambda=0}$ and $R_{\text{e}, \lambda = 0}$ corresponding to the $\text{Rb--H}_2$ (upper) and $\text{Rb--N}_2$ (lower) collision systems. The rRMSE (%) in Eq. \ref{['rRMSE']} and maximum error (%) of the GP predictions relative to the results of CC calculations are indicated in each panel.
• Figure 2: CC calculations of the relative deviation (17) for collisions of Rb with H$_2$ (left) and N$_2$ (right). Upper panels: Relative deviation (\ref{['eq:rel_err']}) for collisions of molecules in the ground rotational state as functions of the well depth magnitude for the isotropic $V_{\lambda=0}$ (circles) and leading anisotropic $V_{\lambda=2}$ (squares) Legendre expansion coefficients of the full PES. Lower panels: The relative deviation (\ref{['eq:rel_err']}) as functions of $D_{\mathrm{e},\lambda=2}$ for collisions of molecules in the rotational states $j=0$ (circles), $j=2$ (squares), $j=4$ (triangles), $j=6$ (inverted triangles). The reference values of $D_{\mathrm{e},\lambda=2}$ given by the MLR fits (\ref{['eq:mlr']}) are shown by the vertical dotted lines.
• Figure 3: Rate coefficients $\kappa_{\rm{tot}}$ from the GP models as functions of $D_{\mathrm{e},\lambda=2}$ (upper panels) and $R_{\text{e}, \lambda = 2}$ (lower panels), with the corresponding reference values from the MLR fits (\ref{['eq:mlr']}) shown by the vertical dotted lines. The change of $\kappa_{\rm{tot}}$ within 1% is shown by the shaded regions. The GP models are trained by the CC calculations performed with the reduced mass, $C_{6,\lambda=0}$ and $R_{{\rm e},\lambda=0}$ corresponding to the $\text{Rb--H}_2$ (left) and $\text{Rb--N}_2$ (right) collision systems. The lines with the symbols represent $\kappa_{\rm{tot}}$ marginalized over the variables not shown on the $x$-axis. The dashed and dotted lines without symbols show samples of ${\kappa}_{\rm tot} = f(R_{\text{e}, \lambda = 2}, D_{\text{e}, \lambda = 2}, C_{6,\lambda=2}, D_{\text{e},\lambda=0}, B_{\rm r} )$ for random combinations of the variables not shown on the $x$-axis.
• Figure 4: Rate coefficients for total, elastic, and inelastic scattering of Rb with N$_2$($j=0$) as functions of $R_{\mathrm{e},\lambda=2}$ and $D_{\mathrm{e},\lambda=2}$ from CC calculations. The inelastic rate coefficients increase up to 30% of the elastic rate coefficients with the variations in interaction anisotropy. However, the total rate coefficients are largely insensitive to the variations due to the mutual compensation of the inelastic and elastic rate coefficients.
• Figure 5: Upper panels: The relative deviation (\ref{['eq:rel_err']}) as a function of $\gamma= C_{6,\lambda=2}/C_{6,\lambda=0}$ for collisions of Rb with H$_2$ (left) and N$_2$ (right) initially in the rotational state $j=0$ (circles), $j=2$ (squares), $j=4$ (triangles), $j=6$ (inverted triangles). The reference values of $\gamma$ are shown by the vertical dotted lines. Lower panels: Rate coefficients $\kappa_{\rm{tot}}$ from the GP models as functions of $\gamma$. The lines with the symbols represent $\kappa_{\rm{tot}}$ marginalized over $R_{\text{e}, \lambda = 2}, D_{\text{e}, \lambda = 2}, D_{\text{e},\lambda=0}$ and $B_{\rm r}$. The dashed and dotted lines without symbols show samples of ${\kappa}_{\rm tot} = f(R_{\text{e}, \lambda = 2}, D_{\text{e}, \lambda = 2}, C_{6,\lambda=2}, D_{\text{e},\lambda=0}, B_{\rm r} )$ for random combinations of $R_{\text{e}, \lambda = 2}, D_{\text{e}, \lambda = 2}, D_{\text{e},\lambda=0}$ and $B_{\rm r}$. The GP models are trained by the CC calculations performed with the reduced mass, $C_{6,\lambda=0}$ and $R_{{\rm e},\lambda=0}$ corresponding to the $\text{Rb--H}_2$ (left) and $\text{Rb--N}_2$ (right) collision systems.