Ultracold chemistry with alkali-metal–rare-earth molecules

A first principles study of the dynamics of ${}^6\mathrm{Li}\left({}^{2}S\right)+{}^6\mathrm{Li}{}^{174}\mathrm{Yb}\left({}^2{\Sigma }^{+}\right){\to }^6{\mathrm{Li}}_2{(}^1{\Sigma }^{+})\hspace{0.17em}+\hspace{0.17em}{}^{174}\mathrm{Yb}({}^{1}S)$ reaction is presented at cold and ultracold temperatures. The computations involve determination and analytic fitting of a three-dimensional potential energy surface for the ${\mathrm{Li}}_2\mathrm{Yb}$ system and quantum dynamics calculations of varying complexities, ranging from exact quantum dynamics within the close-coupling scheme, to statistical quantum treatment, and universal models. It is demonstrated that the two simplified methods yield zero-temperature limiting reaction rate coefficients in reasonable agreement with the full close-coupling calculations. The effect of the three-body term in the interaction potential is explored by comparing quantum dynamics results from a pairwise potential that neglects the three-body term to that derived from the full interaction potential. Inclusion of the three-body term in the close-coupling calculations was found to reduce the limiting rate coefficients by a factor of two. The reaction exoergicity populates vibrational levels as high as $v=19$ of the ${}^6\mathrm{Li}{}_2$ molecule in the limit of zero collision energy. Product vibrational distributions from the close-coupling calculations reveal sensitivity to inclusion of three-body forces in the interaction potential. Overall, the results indicate that a simplified model based on the long-range potential is able to yield reliable values of the total reaction rate coefficient in the ultracold limit but a more rigorous approach based on statistical quantum or quantum close-coupling methods is desirable when product rovibrational distribution is required.

Figure 1

Energetics of the LiYb + Li $\to$ ${\mathrm{Li}}_2$ + Yb reaction. The $j=0$ vibrational levels of the $X$ ${}^2{\Sigma }^{+}$ potential of the reactant ${}^6\mathrm{Li}{}^{174}\mathrm{Yb}$ molecule are shown on the left as horizontal red lines. The $j^{\prime }=0$ vibrational levels of the $X$ ${}^1{\Sigma }_g^{+}$ potential of the product ${}^6\mathrm{Li}{}_2$ molecule are shown on the right as horizontal blue lines. The zero of energy is located at the $v=0,j=0$ level of the ${}^6\mathrm{Li}{}^{174}\mathrm{Yb}$ molecule. (Energies are divided by Planck's constant $h$ and the speed of light $c$.) The inset shows a blowup of the energy levels near the $v=0$ level of ${}^6\mathrm{Li}{}^{174}\mathrm{Yb}$. For clarity the rotational progressions are not shown.

Figure 2

A three-dimensional view of the PES in atomic units for the reaction $\mathrm{Li}\left(1\right)\mathrm{Yb}+\mathrm{Li}\left(2\right)\to \left[\mathrm{LiYbLi}\right]\to {\mathrm{Li}}_2+\mathrm{Yb}$ as a function of bond lengths $R_{\mathrm{LiYb}}$ and $R_{\mathrm{Li}\left(1\right)\mathrm{Li}\left(2\right)}$. The angle between $R_{\mathrm{Li}\left(1\right)\mathrm{Li}\left(2\right)}$ and $R_{\mathrm{Li}\left(2\right)\mathrm{Yb}}$ is fixed at ${180}^{\circ }$. The zero of energy corresponds to the three separated atoms. Topographical contours of equal energies are shown on the base of the figure. From inside out their energies are $-0.04E_h,-0.03E_h,-0.02E_h$, and $-0.0075E_h$, respectively.

Figure 3

(Top) The EQM reaction rate coefficient for the collision of the $v=0,j=0$ rovibrational level of ${}^6\mathrm{Li}{}^{174}\mathrm{Yb}$ with a ${}^6\mathrm{Li}$ atom as a function of relative collision energy $E$ for total angular momentum $J=0$. (Bottom) The thermally averaged reaction rate coefficient summed over total angular momenta $J$ using the $J$-shifting approach as a function of temperature $T$. In both panels black and blue lines correspond to rate coefficients to form even and odd rotational levels of the ${}^6\mathrm{Li}{}_2$ product molecule, respectively. Solid and dashed lines are rate coefficients from calculations using the full trimer and the additive pairwise potential, respectively.

Figure 4

The $J=0$ state-to-state EQM reaction rate coefficient as a function of initial relative collision energy $E$ and vibrational state of the ${}^6\mathrm{Li}{}_2$ product molecule. Left panel corresponds to the results based on the calculation with full trimer potential, whereas the right panel shows rate for the additive pairwise potential.

Figure 5

The EQM reaction rate coefficient as a function of the even rotational quantum number of the $v^{\prime }=15$ vibrational level of the ${}^6\mathrm{Li}{}_2$ molecule. The total angular momentum is $J=0$. Upper and lower panels show the rate coefficient for an initial collision energy of $E/k={10}^{-4}$ K and 1 K, respectively. The black and red bars in both panels correspond to results of a calculation with the full trimer potential and additive pairwise potential, respectively.

Figure 6

The EQM reaction rate coefficient as a function of the odd rotational quantum number of the $v^{\prime }=15$ vibrational level of the ${}^6\mathrm{Li}{}_2$ molecule. The total angular momentum is $J=0$. Upper and lower panels show the rate coefficient for an initial collision energy of $E/k={10}^{-4}$ K and 1 K, respectively. The black and red bars in both panels correspond to results of a calculation with the full trimer potential and additive pairwise potential, respectively.

Figure 7

The state-to-state SQM reaction rate coefficient for vibrational states $v^{\prime }$ of ${}^6\mathrm{Li}{}_2$ as a function of relative collision energy. The left panel corresponds to the results restricted to total angular momentum $J=0$ and the right panel shows the rate coefficients summed over all $J$. The results are based on the full trimer potential.

Figure 8

The state-to-state SQM reaction rate coefficient for the rotational distribution in the $v^{\prime }=0$ (upper panel), $v^{\prime }=5$ (middle panel), and $v^{\prime }=15$ (lower panel) vibrational levels of the ${}^6\mathrm{Li}{}_2$ molecule as a function of the rotational quantum number. The calculation is for $J=0$ and $E/k={10}^{-4}$ K and is based on the full trimer potential.

Figure 9

The reaction rate coefficient for the $v=0,j=0$ rovibrational level of ${}^6\mathrm{Li}{}^{174}\mathrm{Yb}$ colliding with ${}^6\mathrm{Li}$ as a function of collision energy, restricted to total angular momentum $J=0$ (left panel) and the thermally averaged rate coefficient summed over total angular momenta $J$ as a function of temperature (right panel). The blue, red, and green curves correspond to the exact (EQM), statistical (SQM), and universal quantum-defect model (QDT), respectively. Solid and dashed lines for the EQM and SQM calculations correspond to calculations based on the full trimer potential and the pairwise additive potentials, respectively. We used $C_6=3086E_h{a}_0^6$ in the UM.