Cold collisions of heavy ^2Σ molecules with alkali-metal atoms in a magnetic field: Ab initio analysis and prospects for sympathetic cooling of SrOH(^2Σ) by Li(^2S)
Masato Morita, Jacek K?os, Alexei A. Buchachenko, Timur V. Tscherbul
aa r X i v : . [ phy s i c s . a t o m - ph ] F e b Cold collisions of heavy Σ molecules with alkali-metal atoms in amagnetic field: Ab initio analysis and prospects for sympatheticcooling of SrOH ( Σ) by Li( S) Masato Morita, Jacek K los, Alexei A. Buchachenko,
3, 4 and Timur V. Tscherbul Department of Physics, University of Nevada, Reno, NV, 89557, USA Department of Chemistry and Biochemistry,University of Maryland, College Park, MD, 89557, USA Skolkovo Institute of Science and Technology,Skolkovo Innovation Center, Building 3, Moscow 143026, Russia Department of Chemistry, M. V. LomonosovMoscow State University, Moscow 119991, Russia (Dated: May 16, 2018) bstract We use accurate ab initio and quantum scattering calculations to explore the prospects forsympathetic cooling of the heavy molecular radical SrOH( Σ) by ultracold Li atoms in a magnetictrap. A two-dimensional potential energy surface (PES) for the triplet electronic state of Li-SrOH is calculated ab initio using the partially spin-restricted coupled cluster method with single,double and perturbative triple excitations and a large correlation-consistent basis set. The highlyanisotropic PES has a deep global minimum in the skewed Li-HOSr geometry with D e = 4932 cm − and saddle points in collinear configurations. Our quantum scattering calculations predict low spinrelaxation rates in fully spin-polarized Li + SrOH collisions with the ratios of elastic to inelasticcollision rates well in excess of 100 over a wide range of magnetic fields (1-1000 G) and collisionenergies (10 − − . × − cm /s to a limiting valueof 3 . × − cm /s with decreasing temperature from 0.1 K to 1 µ K. . INTRODUCTION Ultracold molecular gases offer a wide range of research opportunities, extending fromquantum simulation of many-body systems with long-range dipolar interactions [1–3] to ex-ternal field control of chemical reaction dynamics [2, 4, 5], precision measurement of molec-ular energy levels to uncover new physics beyond the Standard Model [6–9], and quantuminformation processing with molecular arrays in optical lattices [10]. At present, coher-ent association of ultracold alkali-metal atoms remains the only experimental technique toproduce ultracold molecular gases of KRb and NaK [11–13]. Recent advances in laser cool-ing and magneto-optical trapping [14–18], single-photon cooling [19], Sisyphus laser cooling[20, 21], and optoelectrical cooling [21, 22] made it possible to control and confine molecularspecies such as SrF, CaF, SrOH, YO, CH F, and H CO in electrostatic and magnetic trapsat temperatures as low as a fraction of a milliKelvin [14–19, 21, 22]. Due to the intrinsiclimitations of optical cooling, it is necessary to employ second-stage cooling techniques tofurther reduce the temperature of a trapped molecular gas to < ab initio and quantum scattering calculations. In particular, Lara et al. showed that inelastic relaxation in cold Rb + OH collisions occurs at a high rate, therebyprecluding sympathetic cooling of magnetically trapped OH by Rb atoms [23]. The spin3elaxation cross sections for collisions of polar radicals NH and OH with spin-polarized Nand H atoms were found to be small owing to the weak anisotropy of the high-spin NH-N,NH-H, OH-H interactions, making atomic nitrogen and hydrogen promising coolant atoms[24–27]. The same conclusion was reached for ground-state Mg atoms colliding with NH( Σ)molecules [28]. We showed that despite the strong angular anisotropy of the interactionsbetween Σ molecular radicals and alkali-metal atoms, the inelastic cross sections for inter-species collisions are strongly suppressed due to the weakness of the spin-rotation interactionin Σ molecules [29]. Small polyatomic molecular radicals such as methylene (CH ), methyl(CH ), and amidogen (NH ) were found to have small spin relaxation cross sections with S -state atoms, and hence suggested as promising candidates for sympathetic cooling exper-iments in a magnetic trap [30, 31].The vast majority of atom-molecule combinations proposed for sympathetic cooling ex-periments included light hydrogen-containing molecules such as NH, OH, and CaH. Thesemolecules have large rotational level spacings and low densities of rovibrational states, facil-itating accurate quantum scattering calculations [29, 32]. In contrast, the heavy molecularradicals produced and studied in recent experiments (CaF, SrF, YO, and SrOH) have smallrotational constants and dense spectra of rovibrational states. While the possibility of co-trapping and sympathetic cooling of Σ molecular radicals with ultracold alkali-metal atomshas been suggested [19, 20, 29], numerically exact quantum scattering calculations of theircollisional properties are challenging [29] due to the strongly anisotropic atom-molecule in-teractions, which couple a large number of rovibrational states and field-induced mixingbetween different total angular momenta (see Ref. [33] for a detailed discussion). As aresult, it remains unclear whether heavy molecular radicals trapped in recent experiments[14–19] have small enough inelastic collision rates with ultracold alkali-metal atoms to allowfor efficient sympathetic cooling in a magnetic trap.Cooling and trapping polyatomic molecular radicals is expected to provide new insightsinto many-mode vibrational dynamics, photochemistry, and chemical reactivity at ultralowtemperatures [20, 30, 34–36]. Recently, Kozyryev et al. used buffer-gas cooling to preparea cold sample of the strontium monohydroxide radical [SrOH ( X Σ)] in the ground andfirst excited vibrational states and to observe vibrational energy transfer between the statesinduced by collisions with He atoms at 2 K [34]. The highly diagonal array of Franck-Condon factors between the ground ˜ X Σ + and the first excited ˜ A Π / electronic states of4rOH enables efficient photon cycling, making SrOH an attractive candidate for molecularlaser cooling and trapping. In a series of recent experiments, Kozyryev et al. observed theradiation pressure force and demonstrated Sisyphus laser cooling of SrOH to below 1 mK inone dimension [20, 35].Here, we use accurate ab initio and quantum scattering calculations to explore the pos-sibility of sympathetic cooling of SrOH( Σ) with ultracold Li( S) atoms in a magnetic trap.To this end, we develop an ab initio potential energy surface (PES) for the triplet electronicstate of Li-SrOH (Sec. IIA) and employ it in multichannel quantum scattering calculationsusing a computationally efficient total angular momentum representation for molecular col-lisions in magnetic fields [37] (Sec. IIB). In Sec. IIIA we show that inelastic spin relaxationof spin-polarized SrOH molecules in collisions with spin-polarized Li atoms occurs 100-1000times slower than elastic collisions over a wide range of collision energies and magneticfields, suggesting good prospects of sympathetic cooling of SrOH molecules with ultracoldLi atoms in a magnetic trap. We find broad resonance features in the magnetic field de-pendence of atom-molecule scattering cross sections and show that spin relaxation in coldLi + SrOH collisions occurs predominantly due to the magnetic dipole-dipole interaction(direct mechanism) rather than via the intramolecular spin-rotation interaction combinedwith the anisotropy of the interaction potential (indirect mechanism). In Sec. IIIB weuse adiabatic capture theory to estimate the upper limit to the rate of the Li + SrOH → LiOH + Sr chemical reaction. The paper concludes in Sec. IV with a summary of mainresults and a brief outline of future research directions. Atomic units are used throughoutthe rest of the paper unless otherwise stated.
II. THEORY AND COMPUTATIONAL METHODOLOGYA. Ab initio calculations
The SrOH radical is a linear molecule in its ground electronic state of Σ symmetry [38].The interaction with a ground-state Li( S) atom gives rise to two adiabatic PESs of singletand triplet spin multiplicities. The triplet-singlet couplings have been shown to be negli-gible in a closely related Li-CaH system [39] and since our interest here is in collisions offully spin-polarized Li and SrOH which occur on the triplet PES, we make the common as-5umption of neglecting the difference between the singlet and triplet PESs [24–27, 29]. Thishas the added advantage that single-reference electronic structure methods can be used todescribe the triplet state of the Li-SrOH collision complex. To compute the triplet PES,we thus employ the partially spin-restricted coupled cluster method [40] with single, dou-ble and perturbative triple excitations (RCCSD(T)) with the reference wavefunction takenfrom the restricted Hartree-Fock (RHF) approach. The RHF wavefunction was calculatedusing pseudocanonical orbitals from multi-reference self-consistent field [41, 42] (MCSCF)calculations with valence active space as a starting point.The geometry of the complex is described by the Jacobi coordinates R – the distancebetween Li and the center of mass of SrOH, r – the SrOH bond length, and θ – the anglebetween the Jacobi vectors R and r . The origin of the coordinate system is taken at thecenter of mass of SrOH. The geometry of SrOH is kept linear and fixed throughout thecalculations. The position of the center of mass was calculated using the exact mass of themost abundant isotope Sr. The Jacobi angle θ describes the angular dependence of thePES and the θ = 0 ◦ geometry describes the Li–H-O-Sr collinear arrangement. The linear Sr–O–H geometry is described by the bond lengths r (SrO) = 2 . r (OH) = 0 . − , the Sr-O stretching mode at 534 cm − andthe OH stretching vibration at 3919 cm − .For the Sr atom, we use a pseudopotential-based augmented correlation-consistentquintuple-zeta basis (aug-cc-PV5Z-PP) of Peterson and coworkers [43] with Stuttgart/Cologneeffective core potential (ECP) (ECP28MDF) [45]. The remaining Li, O, and H atoms aredescribed by core-valence Dunning’s aug-cc-pCVTZ basis functions [46]. The basis set usedin the calculations of the Li-SrOH complex is augmented with a set of 3 s p d f g bond-functions placed on an ellipsoid as shown in the inset of Fig. 1 [44]. The bond functionswere represented using the following exponents: sp = (0 . , . , . df = (0 . , .
23) and g = (0 . R .We calculate the triplet PES on a two-dimensional grid in R and θ within a supermolecularapproach and correct for the basis set superposition error using the counterpoise correction6rocedure of Boys and Bernardi [47]: V ( R, θ ) = E Li − SrOH ( R, θ ) − E Li − ghost ( R, θ ) − E SrOH − ghost ( R, θ ) . (1)The ghost in the above equation denotes the presence of dimer-centered basis functionsduring the calculations of monomer energies. The R Jacobi coordinate is represented by theradial grid of 105 points spanning distances from R = 2 . a to R = 40 a with a variablestep from 0.05 a in the medium-range to 0.5-2.0 a in the long-range. The angular variable θ was represented on a grid of 26 points, with a step of 5 degrees from 0 to 70 degrees andwith the step of 10 degrees in the remaining interval to 180 degrees. This gives around2700 points representing the triplet Li-SrOH PES for a fixed SrOH geometry. All electronicstructure calculations have been performed with the MOLPRO suite of programs [48, 49].The calculated PES data points are expanded in Legendre polynomials V ( R, θ ) = X λ =0 V λ ( R ) P λ (cos θ ) . (2)Figure 1(a) shows a contour plot of the triplet Li-SrOH PES. The potential is extremelyanisotropic, varying from strongly attractive (thousands of cm − ) in the region of the globalminimum to weakly attractive (-100-200 cm − ) near the collinear saddle points at θ = 0 ◦ or θ = 180 ◦ and R ≈ a . The high anisotropy is also manifested in the large magni-tude of the first few anisotropic Legendre moments V λ ( R ) shown in Fig. 1(b) at mediumand short R . Higher-order Legendre terms become progressively less important at largeratom-molecule separations. The long-range fit is performed using the analytical formula V LR ( R, θ ) = − P n,l C nl R n P l (cos θ ) including the dispersion coefficients from C to C . Theisotropic van der Waals dispersion coefficient of the triplet PES is estimated from the long-range fit to be C = 1 . × cm − a . The long-range fit is smoothly joined with theexpansion fit [Eq. (2)] by the hyperbolic tangent switching function. The radial V λ coef-ficients are fit using the Reproducing Kernel Hilbert Space (RKHS) interpolation methodwith a one-dimensional radial kernel with n = 2 and m = 5 [50, 51]. A Fortran routine forthe Li-SrOH PES is available in the Supplemental Material [52].The global minimum of the triplet Li-SrOH PES is located at R e = 5 . a and θ e =43 . ◦ with a well depth of 4931.94 cm − . As shown in Fig. 1 the global minimum of theLi-SrOH complex corresponds to a skewed Li–HOSr geometry with the Li–H distance of3.818 a and the Sr-O-H–Li angle of ≈ . ◦ .7 . Quantum scattering calculations In order to solve the quantum scattering problem for Li-SrOH, we numerically integratethe close-coupling (CC) equations in the body-fixed (BF) coordinate frame [29, 37]. Mo-tivated by the need to reduce the computational cost of quantum scattering calculations,we assume that SrOH remains frozen at its ground-state equilibrium configuration, therebyinvoking the rigid-rotor approximation [29]. The energy gap between the ground and thelowest excited vibrational states of SrOH (386 cm − ) is small compared to the Li-SrOH po-tential strength (4931.9 cm − ), which may lead to a temporary excitation of the vibrationalmodes, giving rise to a resonance structure in the energy and field dependence of scatteringcross sections. At collision energies far detuned from the resonances, the coupling betweenthe different vibrational modes of SrOH induced by the interaction with the incident Li atomis small and the rigid-rotor approximation is expected to hold. We therefore expect thatour calculations provide a sufficiently accurate description of cold Li + SrOH backgroundscattering.The effective Hamiltonian for low-energy collisions between an atom A ( S) and a diatomicmolecule B ( Σ) in the presence of an external magnetic field may be written [29, 37]ˆ H = − µ R − d dR R + ( ˆ J − ˆ N − ˆ S A − ˆ S B ) µR + ˆ H A + ˆ H B + ˆ H int (3)where µ is the reduced mass of collision complex defined by µ = m A m B / ( m A + m B ), ˆ H A and ˆ H B describe separated A and B in an external magnetic field, and ˆ H int describes theinteraction between the collision partners. As mentioned in Sec. IIA the collision complex isdescribed by the Jacobi vectors R and r in the BF frame. The embedding of the BF z axisis chosen to coincide with the vector R , and the BF y axis is chosen to be perpendicular tothe plane defined by the collision complex.In Eq. (3), ˆ J is the operator for the total angular momentum of the collision complex, ˆ N isthe rotational angular momentum operator for molecule B, and ˆ S A and ˆ S B are the operatorsfor the spin angular momenta of atom A and molecule B, respectively. The orbital angularmomentum operator of the collision complex in the BF frame is given by ˆ l = ˆ J − ˆ N − ˆ S A − ˆ S B .The separated atom Hamiltonian in the presence of an external magnetic field is given asˆ H A = g e µ B ˆ S A ,Z B , where g e is the electron g-factor, µ B is the Bohr magneton, ˆ S A ,Z gives theprojection of ˆ S A onto the magnetic field axis and B is the magnitude of the external magnetic8eld. For a linear molecule such as SrOH( X Σ), ˆ H B = B e ˆ N + γ SR ˆ N · ˆ S B + g e µ B ˆ S B ,Z B ,where B e is the rotational constant, and γ SR is the spin-rotation constant. The last term inEq. (3) describes the atom-molecule interaction, including both the electrostatic interactionpotential ˆ V and the magnetic dipole-dipole interaction ˆ V dd between the magnetic momentsof the atom and the molecule. The interaction potential ˆ V may be writtenˆ V ( R, θ ) = S A + S B X S = | S A − S B | S X Σ= − S | S Σ i ˆ V S ( R, θ ) h S Σ | , (4)where total electronic spin S is defined as ˆ S = ˆ S A + ˆ S B . Our interest here is in colli-sions between rotationally ground-state SrOH molecules ( N = 0) with Li atoms initiallyin their maximally stretched Zeeman states M S A = M S B = 1 /
2, where M S A and M S B arethe projections of ˆ S A and ˆ S B onto the space-fixed Z -axis. Following our previous work onLi-CaH [29, 39] we assume that the non-adiabatic coupling between the triplet ( S = 1) andthe singlet ( S = 0) Li-SrOH PESs can be neglected, and that the PESs are identical, i.e. ˆ V S =0 ( R, θ ) = ˆ V S =1 ( R, θ ). The dipolar interaction between the magnetic moments of theatom and molecule may be written [32]ˆ V dd = − g e µ r π α R X q ( − ) q Y ∗ , − q ( ˆ R )[ ˆ S A ⊗ ˆ S B ] (2) q , (5)where µ is the magnetic permeability of free space, α is the fine-structure constant and[ ˆ S A ⊗ ˆ S B ] (2) q is the spherical tensor product of ˆ S A and ˆ S B .Following previous studies [29, 32, 37], the total wave function of the Li-SrOH collisioncomplex is expanded in a set of basis functions | J M Ω i| N K N i| S A Σ A i| S B Σ B i . (6)Here, Ω, K N , Σ A and Σ B are the projections of J , N , S A and S B onto the BF quantizationaxis z , and Ω = K N + Σ A + Σ B is satisfied. Unlike Ω, the projection of J onto the space-fixedquantization axis M is rigorously conserved for collisions in a DC magnetic field [1, 53], sothe CC equations can be constructed and solved independently for each value of M . InEq. (6) | J M Ω i = p (2 J + 1) / π D J ∗ M Ω ( ¯ α, ¯ β, ¯ γ ) is an eigenfunction of the symmetric top,and the Wigner D -functions D J ∗ M Ω ( ¯ α, ¯ β, ¯ γ ) depend on the Euler angles ¯ α , ¯ β and ¯ γ , whichspecify the position of the BF axes x , y and z in the SF frame. The rotational degreesof freedom of SrOH in the BF frame are described by the functions | N K N i , which can be9 ABLE I. Spectroscopic constants of SrOH (in cm − ) and masses of the collision partners(in a.m.u.) used in scattering calculations.Parameter Value B e γ SR . × − m SrOH m Li expressed using the spherical harmonics as √ πY NK N ( θ, J max and N max which showthe maximum quantum numbers of the total angular momentum J of the collision complexLi-SrOH and the rotational angular momentum N of SrOH in the basis set. Unless statedotherwise, all scattering calculations are carried out with J max = 3. The convergence prop-erties with respect to J max are examined in the Appendix. Due to the strong anisotropy ofthe Li-SrOH interaction, a large number of rotational channels must be included in the basisset to obtain converged results. Furthermore, the rotational constant of SrOH is ∼
20 timessmaller than that of CaH, which results in larger values of N max for Li-SrOH compared toLi-CaH ( N max = 55 [29]). Indeed, we found that using N max = 115 is necessary to obtainthe cross sections converged to within 2% (see the Appendix).The numerical procedures used in this work are essentially the same as those employedin our previous study of Li-CaH collisions [29] as explained in detail elsewhere [32, 37]. Inbrief, the CC equations are solved numerically using the log-derivative propagator method[54, 55] on an equidistant radial grid from R min = 4 . a to R mid with R mid = 9 . a for B >
10 G and R mid = 22 . a for B ≤
10 G using a step size of 0 . a . Airy propagationis employed for R mid ≤ R ≤ R max with R max = 280 . a for B >
10 G and R max = 1322 . a for B ≤
10 G. Propagating the log-derivative matrix out to very large values of R max isnecessary to maintain the numerical accuracy of quantum scattering calculations on systemswith long-range anisotropic interactions at low magnetic fields [56].10fter propagating the log-derivative matrix out to a sufficiently large R = R max wherethe interaction potential becomes negligible, the matrix is transformed from the total an-gular momentum representation to a basis set in which ˆ H A , ˆ H B and ˆ l are diagonal. Theresultant log-derivative matrix is matched to the scattering boundary conditions to obtainthe S -matrix, and the elastic and inelastic cross sections are extracted from the S -matrix asdescribed in Ref. [37]. C. Quantum capture calculations
The Li( S) + SrOH( Σ) → LiOH + Sr chemical reaction can proceed through the singletand triplet pathways. The singlet pathway gives the ground-state products and is exothermicby ∼ A PES (the entrance channel of whichis calculated in Sec. IIA) and correlates to the Sr atom excited to the P state, which lies1.8 eV above the ground state. Thus, the chemical reaction of spin-polarized reactants isenergetically forbidden at low collision energies. However, spin-nonconserving interactions(such as spin-orbit and hyperfine) may induce non-adiabatic transitions at short range [39],which are not accounted for within our reduced-dimensional single-state model. An upperbound to the rate of these transitions is given by the capture rate, i.e. , the rate of reactantpenetration to the short-range region as defined classically by the Langevin model. Toestimate this rate, we applied here a quantum version of the statistical adiabatic channelmodel [58] implemented as described in Ref. [59].In brief, we use a simplified atom-molecule Hamiltonian (3) without the spin-rotationcoupling and Zeeman interactions. The adiabatic channel potentials are obtained by diago-nalizing the Hamiltonian, at fixed atom-molecule separations R , in the symmetry-adaptedrigid rotor function basis set. Since we are only interested in the channels correlating to theground rotational state of the SrOH reactant, the single lowest-energy root was retained foreach total angular momentum quantum number J ( l ≡ J in this case). We found that 50basis functions with N max = 49 give results converged to within 2% for the desirable N = 0adiabatic channel near the bottom of the potential well at R = 5 . a .To calculate the quantum capture probabilities for J ≤
20, we use the modified Truhlar-Kupperman finite difference method [60] as described in Refs. [58, 59] on a grid of collisionenergies extending from 10 − to 1000 cm − . Inner capture boundary conditions are applied11t R = R within the short-range region. We used 6 values of R ∈ [5 . , . a to obtainthe average capture probability at each collision energy. The classical capture probabilitiesare determined from the height of the centrifugal barrier for each J ≤
40 [59].
III. RESULTSA. Elastic and inelastic cross sections
Figure 2 (a) shows the elastic ( σ el ) and inelastic ( σ inel ) cross sections for fully spin-polarized Li + SrOH collisions as functions of collision energy for the external magneticfields of 1 G, 10 G, 100 G and 1000 G. Due to the very weak magnetic field dependence ofthe elastic cross section, only the B = 1000 G result is shown in the figure. The inelastic crosssection increases significantly as the magnetic field increases from 1 G to 100 G, especially inthe ultracold s -wave regime (the field dependence will be explored later in this section). Theinelastic cross sections as a function of collision energy remain smooth and small over theentire energy range considered. As mentioned in the Introduction, for sympathetic coolingto be effective, the ratio of elastic to inelastic cross sections γ = σ el /σ inel must exceed 100.Figure 2(b) shows that the calculated values of γ do exceed 100 throughout the whole energyrange except in the vicinity of E C = 5 . × − cm − .Figure 3 shows the temperature dependence of the rate constants for elastic scatteringand spin relaxation. The rate constant is an energy averaged property obtained by theconvolution of the cross sections with the Maxwell-Boltzmann distribution function. Assuch, the behavior of the rate constant as function of temperature tends to be monotonous.Importantly, collision-induced spin relaxation in Li + SrOH collisions occurs more thantwo orders of magnitude slower than elastic scattering, suggesting favorable prospects forsympathetic cooling of SrOH molecules with Li atoms in a magnetic trap.As shown in Fig. 2(a), the inelastic cross section decreases dramatically as the magneticfield is reduced from 100 G to 1 G in the ultracold s -wave regime. The suppression ofspin relaxation is a consequence of conservation of parity and the total angular momentumprojection M [61], which dictate that inelastic spin relaxation of the incoming s -wave channelmust be accompanied by a change of the orbital angular momentum from l = 0 to l = 2. Ifthe energy difference between the initial and final channels is small enough due to the small12eeman splitting in a weak magnetic field, the height of the d -wave centrifugal barrier in thefinal channel can be larger than the initial kinetic energy in the incoming channel. Undersuch conditions, spin relaxation occurs by tunnelling under the d -wave centrifugal barrier,and is strongly suppressed. We note that, as discussed in Sec. IIB below, this mechanism onlyapplies to indirect spin relaxation induced by the intramolecular spin-rotation interaction.In Fig. 4, we plot the magnetic field dependence of the inelastic cross sections calculatedfor the collision energy of 10 − cm − . We observe two broad asymmetric resonance profiles,around which the inelastic cross sections are reduced dramatically. This suggests the pos-sibility of suppressing spin relaxation in Li + SrOH collisions by tuning the DC magneticfield as noted previously for He-O [62]. We note that, despite the high density of rovibra-tional states of the Li-SrOH collision complex, only a few resonances are observed in theinelastic cross section below 2000 G. This suggests that most of the states of the complexare decoupled from the incident spin-polarized collision channel. A similar magnetic fielddependence is observed in ultracold collisions of spin-polarized alkali-metal atoms [63, 64]and O ( Σ) molecules [65, 66], which display a lower resonance density in non-spin-polarizedinitial channels.
B. Direct vs. indirect spin relaxation mechanisms
In general, inelastic spin relaxation in cold collisions of Σ molecules in their ground rota-tional states with S atoms is mediated by two mechanisms, direct and indirect. The directmechanism is due to the long-range intermolecular magnetic dipole-dipole interaction ˆ V dd given by Eq. (5). The indirect mechanism is due to a combined effect of the intramolecularspin-rotation interaction and the coupling between the rotational states of the molecule in-duced by the anisotropy of the interaction potential [67]. As shown in Fig. 1, the anisotropyof the interaction potential is strong in the range of small atom-molecule distances R ; thusthe indirect mechanism operates at short range.In order to compare these mechanisms, we plot in Figs. 4 and 5(a) the inelastic crosssections calculated with and without the magnetic dipole-dipole interaction. Omitting themagnetic dipole-dipole interaction leads to a dramatic reduction of the inelastic cross sectionover the entire magnetic field range (including near scattering resonances), which stronglysuggests that spin relaxation in spin-polarized Li + SrOH collisions is driven by the direct13echanism. As shown in Fig. 5(a), the indirect mechanism becomes more efficient withincreasing collision energy; however, the direct mechanism dominates even at the highestcollision energy considered.It is instructive to compare the efficiency of the indirect spin relaxation mechanism incollisions of light (CaH) and heavy (SrOH) molecular radicals with Li atoms. The inelasticcross sections calculated in the absence of the magnetic dipole-dipole interaction are similarin magnitude (5 . × − ˚A for Li + SrOH and 10 − ˚A for Li + CaH [29] at E C = 10 − cm − and B = 0 . ∼
120 excited rotationalstates contribute to the indirect spin relaxation mechanism for Li + SrOH, as opposed toonly ∼
50 rotational states for Li + CaH. As a result, the number of third (and higher)-ordercontributions to the Li + SrOH inelastic scattering amplitude is expected to be significantlylarger than for Li + CaH, leading one to expect the indirect spin relaxation mechanism tobe more efficient for Li + SrOH. However, the spin-rotation constant of SrOH is 10 timessmaller than that of CaH, so each contribution to the Li + SrOH scattering amplitudeis suppressed by a factor of 10. This suppression compensates for the larger number ofcontributing terms for Li + SrOH, providing a qualitative explanation for the comparableefficiency of indirect spin relaxation mechanisms in collisions of light and heavy molecularradicals.Figures 5(b) and (c) compare the incoming partial wave contributions to the inelastic crosssections calculated with and without the magnetic dipole-dipole interaction. For the indirectspin relaxation mechanism, the incoming p -wave contribution tends to exceed the incoming s -wave contribution in both the s -wave and multiple-partial-wave regimes as discussed inRefs. [53, 61]. In contrast, the cross sections calculated with the magnetic dipole-dipoleinteraction included display a more conventional partial wave structure, with the incoming s -wave contributions being dominant below E C = 10 − cm − and all incoming partial wavecomponents becoming comparable at higher collision energies. This explains the diminishingrole of the indirect spin relaxation mechanism with decreasing collision energy evident inFig. 5(a). 14 . Quantum capture rates Figure 6 shows the Li + SrOH capture rate constant as a function of temperature, withquantum and classical results shown by lines and symbols, respectively. At T →
0, thequantum rate obeys the Wigner threshold law for s -wave scattering. The crossover tomultiple scattering regime, which occurs at ca. µ K, manifests itself as a shallow minimumin the temperature dependence of the total capture rate. The total classical capture rateexhibits the expected divergence as T → s -wave scattering channel. On the other hand, neglecting tunnelling leads to afaster decline of the contributions from higher partial waves as the temperature decreases. Acombination of these two effects makes the classical capture approximation quite reasonabledown to the temperatures as low as 300 µ K. Overall, the magnitude of the Li + SrOHreaction rate and its temperature dependence are very similar to those calculated previouslyfor the Li + CaH → LiH + Ca chemical reaction [59]. However, the larger reduced massof the Li-SrOH collision complex and its stronger long-range dispersion forces make thecrossover effect more pronounced and the classical approach more reliable for the
J > → LiOH + Sr chemicalreaction should be the dominant loss channel for the reactants colliding in non fully spin-polarized initial states. Assuming that the long-range behavior of the singlet and tripletPESs is identical, we obtain an upper bound to the reaction rate as 3 × − cm /s, onethird of the value shown in Fig. 6, and 3-4 orders of magnitude larger than the spin relaxationrate for fully spin-polarized Li + SrOH collisions shown in Fig. 3. Thus, spin polarizationof the reactants can be considered as a way to suppress inelastic and reactive losses in coldLi + SrOH collisions. IV. SUMMARY AND OUTLOOK
We have studied the collisional properties of ultracold spin-polarized mixtures of SrOHmolecules with Li atoms using reduced-dimensional quantum scattering calculations and15 newly developed, highly anisotropic triplet PES of the Li-SrOH collision complex. Wepresent the elastic and inelastic collision cross sections over a wide range of collision energies(10 − -1 K) and magnetic fields (1-1000 G) along with the quantum and classical capturerates, which give an upper limit to the total Li+SrOH reaction rate. We find that inelasticspin relaxation in fully spin-polarized Li + SrOH collisions is strongly suppressed (withthe ratio of elastic to inelastic collision rates γ > -10 ), suggesting good prospects forsympathetic cooling of spin-polarized SrOH molecules with Li atoms in a magnetic trap. Inthe context of rapid experimental progress in buffer-gas cooling and Sisyphus laser cooling ofpolyatomic radicals [20, 34, 35], our results open up the possibility of sympathetic cooling ofpolyatomic molecules with magnetically co-trapped ultracold alkali-metal atoms, potentiallyleading to new advances in low-temperature chemical dynamics and spectroscopy of largemolecules in the gas phase [30, 36].In future work, we intend to explore the sensitivity of scattering observables to smalluncertainties of the Li-SrOH interaction PES (preliminary calculations indicate that themain conclusions of this work are robust against the uncertainties). It would also be in-structive to study the effect of the SrOH vibrational modes and singlet-triplet interactionsneglected here [39] on cold collisions of SrOH molecules with alkali-metal atoms in arbitraryinitial quantum states. Such interactions could be particularly important for heavier coolantatoms, such as K and Rb, whose use in sympathetic cooling experiments may be preferablefor experimental reasons. ACKNOWLEDGEMENTS
We are grateful to John Doyle and Ivan Kozyryev for stimulating discussions. This workwas supported by the NSF (grant No. PHY-1607610).
Appendix A: Basis set convergence of scattering observables
Here, we explore the convergence of Li + SrOH scattering cross sections with respect tothe truncation parameters N max and J max . First, we check the convergence with respect tothe maximum rotational state included in the basis set N max . As pointed out in Sec. IIB,the small rotational constant of SrOH along with the large well depth and strong anisotropy16f the Li-SrOH interaction lead to a large value of N max required for convergence. Figure 7shows the cross sections as a function of N max at the collision energy of 1 . × − cm − andthe magnetic field of 100 G with J max = 1. We observe rapid oscillations in the calculatedcross sections until N max ∼
95. Even after the oscillations cease at N max > N max = 105. We note thatthere seems to be no correlation between the behavior of the elastic and inelastic crosssections as a function of N max in the region of strong oscillations (60 < N max < e.g. − cm − ) resemble thoseshown in Fig. 7, with the oscillations becoming less pronounced. We find that using N max =115 gives both the elastic and inelastic cross sections converged to within 2%.To test the convergence of scattering observables with respect to the maximum value ofthe total angular momentum in the basis set J max , we plot in Fig. 8 the elastic and inelasticcross sections as a function of collision energy calculated for J max = 3 and 4. Note that sincethe couplings between the adjacent J -blocks become stronger with increasing the B -field[37], using B = 1000 G provides a more stringent convergence test than using B = 100 G.As the computational cost of the J max = 4 calculations is very large, we limit the calculationsto 7 representative collision energies spanning the range 10 − − − K. Figure 8 shows thatadequate convergence of the inelastic cross sections is achieved with J max = 3 at all collisionenergies. The observed convergence for J max = 3 implies the smallness of the incoming f -wave contributions to the inelastic cross sections (described by adding the J = 4 block inthe basis set). It also implies that the couplings between the incoming f -wave and p -wavescattering states in the entrance and exit collision channels are not critically important.17 / degrees R / a − − − − − − − − − − − − − − − − − − −6 R / a -1500-1000-500050010001500200025003000 V λ ( R ) / c m - V V V V V V (a)(b) Sr O Li Li c.o.m.
FIG. 1. (a) A contour plot of the ab initio potential energy surface for Li-SrOH in its tripletelectronic state (in units of cm − ). The θ = 0 ◦ geometry corresponds to the collinear Li–H-O-Srarrangement. (b) The radial dependence of the first few Legendre expansion coefficients V λ ( R ).The insert shows the ellipsoid along which the bond functions are placed. The center of theellipsoid is located at the center of mass of SrOH, and its horizontal and vertical axes are givenby r b = R Li-H / r H-X and r a = R/
2, where r H-X is the distance from H to the center of mass ofSrOH. − − − − − Collision energy (cm − )10 − C r o sss ec t i o n ( ˚A ) (a) σ el ( B = 1000 G) σ inel ( B = 1000 G) σ inel ( B = 100 G) σ inel ( B = 10 G) σ inel ( B = 1 G) − − − − − Collision energy (cm − )10 E l a s t i c - t o - i n e l a s t i c r a t i o : γ (b) γ ( B = 1000 G) γ ( B = 100 G) γ ( B = 10 G) γ ( B = 1 G) FIG. 2. (a) Cross sections for elastic scattering and inelastic spin relaxation in spin-polarizedLi + SrOH collisions plotted as functions of collision energy for the external magnetic field of 1 G(diamonds), 10 G (triangles), 100 G (squares), 1000 G (crosses). The elastic cross section displaysa very weak magnetic field dependence. (b) The ratios of elastic and inelastic cross sections asfunctions of collision energy for the same values of the magnetic field as in (a). − − − Temperature (K)10 − − − − − − − R a t ec o n s t a n t : K ( c m / s ) K el ( B = 1000 G) K inel ( B = 1000 G) K inel ( B = 100 G) K inel ( B = 10 G) K inel ( B = 1 G) FIG. 3. Temperature dependence of the rate constants for elastic scattering (circles) and inelasticspin relaxation in spin–polarized Li + SrOH collisions calculated for the magnetic field values of 1G (diamonds), 10 G (triangles), 100 G (squares), 1000 G (crosses).
500 1000 1500 2000Magnetic field (G)10 − − − − − − − C r o sss ec t i o n ( ˚A ) σ inel σ inel (No dipolar) FIG. 4. Magnetic field dependence of the inelastic cross sections for Li + SrOH calculated with (fullline with circles) and without (dashed line with squares) the magnetic dipole-dipole interaction.The collision energy of 1 . × − cm − . − − − − − Collision energy (cm − )10 − − − − C r o sss ec t i o n ( ˚A ) (a) σ inel σ inel (No dipolar)10 − − − − − Collision energy (cm − )10 − − − C r o sss ec t i o n ( ˚A ) (b) σ inel σ inel , s σ inel , p σ inel , d − − − − − Collision energy (cm − )10 − − − − − − − C r o sss ec t i o n ( ˚A ) (c) σ inel σ inel , s σ inel , p σ inel , d FIG. 5. (a) Collision energy dependence of the inelastic cross section calculated with (crosses)and without (diamonds) the magnetic dipole-dipole interaction for the magnetic field of 100 G. (b)Incoming partial wave decomposition of the inelastic cross section. (c) Same as in panel (b) butcalculated without the magnetic dipole-dipole interaction. -6 -5 -4 -3 -2 -1 -11 -10 -9 -8 C a p t u r e r a t e c on s t a n t K ( c m / s ) Temperature (K)
FIG. 6. Adiabatic capture rate constant for the Li + SrOH chemical reaction calculated as afunction of collision energy in the absence of an external magnetic field. Quantum calculations:total rate constant (black solid line), s -wave rate constant (red dotted line) and the higher partialwave contribution (blue dashed line). Classical calculations: total rate constant (dots) and the J > IG. 7. Convergence of the elastic and inelastic cross sections for spin-polarized Li + SrOHcollisions with respect to the number of rotational states included in the basis set at the collisionenergy of 1 . × − cm − . The magnetic field is 100 G and J max = 1.FIG. 8. Convergence of the elastic and inelastic cross sections for spin-polarized Li + SrOHcollisions with respect to the number of total angular momenta included in the basis set: J max = 3(circles and squares) and J max = 4 (pluses and crosses). The magnetic field is 1000 G.
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