Quantum spin state selectivity and magnetic tuning of ultracold chemical reactions of triplet alkali-metal dimers with alkali-metal atoms
Rebekah Hermsmeier, Jacek Klos, Svetlana Kotochigova, Timur V. Tscherbul
QQuantum spin state selectivity and magnetic tuning of ultracold chemical reactionsof triplet alkali-metal dimers with alkali-metal atoms
Rebekah Hermsmeier , Jacek Kłos , , Svetlana Kotochigova and Timur V. Tscherbul Department of Physics, University of Nevada, Reno, NV, 89557, USA Department of Physics, Joint Quantum Institute,University of Maryland College Park, College Park, Maryland, 20742, USA Department of Physics, Temple University, Philadelphia, PA 19122, USA (Dated: February 12, 2021)We use coupled-channel statistical calculations to explore the effects of external magnetic fieldsand hyperfine interactions on the ultracold chemical reactions of triplet alkali-metal dimers withalkali-metal atoms. Using the Na + NaLi( a Σ + ) → Na ( Σ + g ) + Li reaction as a representativeexample, we find that the reaction rates are sensitive to the initial hyperfine states of the reactants,and that the chemical reaction of fully spin-polarized NaLi( a Σ + , v = 0 , N = 0) molecules with Naatoms is suppressed by a factor of ’ Introduction.
Recent experimental advances in molec-ular cooling and trapping have opened up new avenuesof research into controlling chemical reactivity with ex-ternal electromagnetic fields [1–3], the idea that fasci-nated scientists for decades, and led to the developmentof new research frontiers at the interface of chemistryand physics, such as mode-selective chemistry [4, 5],quantum coherent control [6], and attochemistry [7]. Inparticular, the production and trapping of ground-statemolecular radicals NaLi(a Σ + ), SrF( Σ + ), CaF( Σ + ),YO( Σ + ), YbF( Σ + ) [8–14] and studies of their colli-sional properties at µ K temperatures [15, 16] suggestedthe possibility of using the reactants’s electron spin de-grees of freedom to tune ultracold reaction dynamics bymagnetic fields.The prospect of using magnetic fields as a tool tocontrol chemical reactivity is central to ultracold chem-istry [1, 2] and a very important one in chemical kinetics[17] and biological magnetoreception [18], where radicalpair reactions in cryptochrome proteins are thought toplay a key role in magnetic-field-guided orientation ofbirds and insects [19, 20]. However, despite the long-standing significance of this question and the recent ex-perimental observation of inelastic collisions in an ultra-cold Na-NaLi(a Σ + ) mixture [16], no theoretical stud-ies have been reported on ultracold reaction dynamicsinvolving ground-state alkali-metal dimers and atomsin the presence of external magnetic fields and hyper-fine interactions. This is because such reactions occurthrough the formation of a deeply bound reaction com-plex [21–23], whose numerous strongly coupled boundand resonance states defy rigorous quantum scatteringcalculations [22–24].Here, we explore the dynamics of the ultracold chemi-cal reaction Na + NaLi( a Σ + ) → Na ( Σ + g ) + Li in thepresence of magnetic fields and hyperfine interactionsusing the extended coupled-channel statistical (CCS)model [25] parametrized by ab initio calculations. The model assumes the existence of a long-lived reactioncomplex at short range, whose properties can be mod-eled statistically ( i.e. using classical probabilities) [26–28]. Statistical (or universal) models [26–40] have beensuccessfully applied to calculate the rate of ultracoldchemical reactions of alkali-metal dimers [30, 32–34, 37]and the density of states of the (KRb) reaction com-plex [41]. However, the previous calculations have beenlimited to the case of zero magnetic field and did notaccount for electron spins, hyperfine interactions, andnon-adiabatic effects, all of which we will consider inthe present work.A promising scenario for controlling ultracold chem-istry explored here is based on spin polarizing the re-actants to create a nearly pure high-spin state of thereaction complex [1, 42, 43]. Since high-spin electronicstates are often nonreactive, the chemical reaction ofspin-polarized reactants can only occur via nonadiabatictransitions to low-spin (reactive) electronic states whichare typically mediated by weak relativistic spin-orbitand spin-spin interactions (in light molecular systems)[44–48] leading one to expect suppressed chemical re-activity of spin-polarized reactants [1, 42, 43]. Indeed,we find that the fully spin-polarized spin states of NaLiand Na are ∼ a Σ + ).Our results show that chemical reactions of triplet-statealkali-metal dimers with alkali-metal atoms can be effi-ciently controlled by tuning the hyperfine states of thereactants and by applying an external magnetic field. Theory: Ab initio calculations and extended CCSmodel.
To describe ultracold reactive collisions betweenNa atoms and NaLi molecules in the metastable a Σ + electronic state, we performed ab initio calculations ofthe electronic potential energy surfaces (PESs) of the a r X i v : . [ phy s i c s . c h e m - ph ] F e b E n e r g y Nuclear coordinate(s) Na +LiNaLi+Na ground 1 A’ state excited 1 A’ state excited 2 A’ state NaLi ( Σ ) +3 NaLi ( Σ ) +1 Na ( Σ ) +32 u Na ( Σ ) +12 g (a)(b) FIG. 1. (a) Schematic of the Na + NaLi(a Σ + ) reactive scat-tering through a CI between the A PESs leading to eitherground state NaLi(X Σ + ) or Na (X Σ + g ) molecules. The CIis indicated by the red/blue cone. (b) Ab initio adiabaticPESs for Na-NaLi as functions of the Na-to-NaLi separation R and of the bending angle θ with r = 9 . a , close to theequilibrium distance of the NaLi(a Σ + ) potential. The blue(1 A ) and red (2 A ) PESs have a CI, where two PESs ofthe same electronic symmetry touch. The green surface isthe spin-polarized, nonreactive PES of the 1 A symmetry. long-lived intermediate Na Li reaction complex. Thecomplex is characterized by two A and one A trimerelectronic states. The potential landscape of these bar-rierless PESs is shown in Fig. 1. The PESs are expressedin the Jacobi coordinates R —the atom-molecule sepa-ration vector and r —the vector joining the nuclei of thediatomic molecule. For our purposes it is sufficient todetermine the PESs, which are only functions of R and θ (the angle between R and r ) in the two-dimensionalplane with the internuclear distance of NaLi fixed at itsequilibrium value ( r = r e ) [25]. Our ab initio calcu-lations of the two-state A PESs reveal a conical in-tersection (CI) between the two doublet states whichis located at R ’ . a and θ = 70 o . The relevantmulti-dimensional PESs have been determined using theinternally-contracted multi-reference configuration in-teraction (MRCI) method [49] with single and doubleexcitations and Davidson correction [50] as further de-scribed in the Supplemental Material [51]. From the energetics of the relevant molecular statesin the entrance and exit reaction channels we deter-mine that the production of the Na (a Σ + u ) moleculein the Na( S) + NaLi(a Σ + ) reaction is endothermicby about 50 cm − without the zero-point vibrationalenergy (ZPE) correction. The inclusion of the ZPE de-termined by quantizing the Na (a Σ + u ) and NaLi(a Σ + )potentials in the discrete variable representation, low-ers the reaction endothermicity to 41.7 cm − . Thissuggests that the vibrational excitation of the reactantNaLi(a Σ + ) molecule to the v ≥ products.However, the CI allows for an efficient transfer of the re-actant NaLi(a Σ + ) molecules into either NaLi (X Σ + )or Na (X Σ + g ) states of the ground electronic config-uration. A schematic depiction of reactive scatteringbetween Na atoms and NaLi(a Σ + ) molecules througha CI is demonstrated in Fig. 1(a). The reactants startout on the asymptotically degenerate 2 A and 1 A ex-cited PESs. The reaction flux on the 2 A PESs canreach the CI and make a transition to the ground 1 A PES leading to ground-state reaction products. Onlythe 2 A and 1 A PESs are included in our CCS cal-culations, which is justified by the fact that the CI islocated deeply inside the reaction complex region notexplicitly included in the calculations [51].The extended CCS model [25] describes the quan-tum dynamics of barrierless chemical reactions in thepresence of external magnetic fields and hyperfine in-teractions. Assuming the existence of a long-lived re-action complex, whose formation from the reactantsor decay to products can be treated as independentevents [26, 27], the state-to-state reaction probabil-ity between the reactant and product states r and p , P r → p ( E ) = p p ( E ) p r ( E ) P c p c ( E ) , where p r ( E ) and p p ( E ) are theenergy-dependent capture probabilities of the reactantsand products into the reaction complex obtained bysolving the time-independent Schrödinger equation inthe entrance reaction channel subject to a short-rangecapture boundary condition for the reactive 2 A PESand a regular boundary condition for the nonreactive A PES [25, 51].
Ultracold reaction dynamics in a magnetic field.
Webegin by describing the hyperfine energy level struc-ture of the reactant Na( S) atom and the rovibrationalground state of the NaLi(a Σ + ) molecule in a magneticfield. Figures 2(b) and (c) show the Zeeman levels ofNa and NaLi( a Σ + , v = 0 , N = 0) obtained by diag-onalization of the atomic and molecular Hamiltonians[51]. There are a total of 36 molecular energy levels inthe N = 0 manifold of NaLi(a Σ + ), which can be clas-sified in the weak-field limit by the values of the totalangular momentum of the molecule F and its projectionon the field axis M F [52, 53]. The calculated zero-fieldhyperfine splittings are in good agreement with recentexperimental measurements [8, 51]. In the strong fieldlimit we observe a clear hierarchy of energy level split-tings. The largest splitting is caused by the Zeeman Magnetic Field (G) E n e r gy ( c m - ) N = 1 N = 0 36 (b) Magnetic Field (G) -0.04-0.0200.020.04 (c)
10 100 1000 10000
Magnetic field (G) -12 -11 -10 -9 R eac ti on r a t e c o e ff i c i e n t ( c m / s ) Na(2) + NaLi(3)Na(7) + NaLi(36)Na(8) + NaLi(36) (a)
FIG. 2. (a) Magnetic field dependence of the Na + NaLireaction rate coefficient calculated for the fully spin-polarized (8,36) [diamonds] and non-fully spin polarized(7,36) [squares] initial states at T = 2 µ K. The numbers inparentheses represent energy-ordered hyperfine states of Naand NaLi marked in panels (c) and (b). Also shown are theresults for the Na(2) + NaLi(3) spin-polarized initial statein the absence of the hyperfine structure (dots). Panels (b)and (c) show the Zeeman energy levels of NaLi(a Σ + , v =0 , N = 0) and of the ground-state Na atom. interaction, which splits all N = 0 levels into three Zee-man manifolds according to the values of M S A . Thenext largest splitting is caused by the hyperfine struc-ture of the Na atom within the NaLi(a Σ + ) molecule(the ratio of atomic hyperfine constants a Na /a Li = 5 . M I Na = − / , − / , / , /
2, the projection of the nu-clear spin of Na on the field axis. The smallest-scale splittings in each of the fixed- M I Na manifolds arecaused by the hyperfine interaction of fermionic Li with M I Li = − , , | i of the N = 0 manifold with Na atoms in the hyperfine states | i and | i [see Figs. 2(b) and (c)]. Note that state | i is a triply spin-polarized state of NaLi, where all of thespins in the molecule are aligned along the magneticfield. Similarly, state | i of Na is doubly spin-polarized( | F = 2 , m F = 2 i ), in contrast to state | i , in which the nuclear spin is not fully parallel to the magnetic field.In the absence of the hyperfine structure, the Zeemanstates of NaLi and Na shown in Fig. 2 reduce to 3 molec-ular states | S A M S A i ( M S A = 0 , ± | S B M S B i ( M S B = ± / | i and | i .In Fig. 2(a) we plot the magnetic field dependence ofthe reaction rates for the (8,36) and (7,36) initial statesof Na + NaLi(a Σ) at T = 2 µ K. The rates are nearlytemperature independent, as expected for a two-bodyinelastic process near an s -wave threshold [54]. Theorder of magnitude of the unpolarized reaction rate (5-9 × − cm /s) is typical of an s -wave capture process,such as a barrierless chemical reaction [30, 32, 38].More significantly, we observe that the chemical reac-tivity of fully spin-polarized reactants Na(8) + NaLi(36)is suppressed by a factor of ’ B ’
500 G.The latter trend is similar to that observed in Ref. [25]and can be explained by referring to Eq. (1): the weight c ( B ) of the “reactive” electron spin state | , − i in thehyperfine state | i of Na | i = c ( B ) |
12 12 i|
32 12 i + c ( B ) | , − i|
32 32 i (1)decreases with increasing magnetic field, as the statetends to the unreactive spin-polarized state |
12 12 i|
32 12 i inthe large-field limit (where |
12 12 i|
32 12 i denotes the Zee-man state with S B = M S B = 1 / I B = 3 /
2, and M I B = 1 / | i of Na becomesless and less reactive towards NaLi with increasing fieldbecause the reactive weight c ( B ) ’ B − [25].A similarly dramatic sensitivity of ultracold reactionrates to the initial states of the reactants was mea-sured for ultracold reactions of KRb molecules [55]. Thephysical reason behind the quantum state selectivity inthat experiment is the quantum statistics of identicalfermions, which does not apply to nonidentical reac-tants. Here, the suppression of chemical reactivity ofspin-polarized molecules is due to a different, more gen-eral mechanism [42, 43] based on approximate conser-vation of the total spin of the reaction complex.Specifically, if the electron spins of the reactants arecompletely polarized, the reaction complex is initializedin the nonreactive state of total spin S = 3 / A PES (see Fig. 1). Thus, in the absence ofspin-nonconserving interactions, such as the intramolec-ular spin-spin or intermolecular magnetic dipole inter-actions, the value of S must be the same for the reac-tants and products. The energetically allowed productsof the Na + NaLi reaction—molecular Na ( Σ + g ) andatomic Li( S / )—correspond to S = 1 /
2. As a result,the spin-polarized chemical reaction Na + NaLi( a Σ + ) → Na ( Σ + g ) + Li requires a spin-changing transition S = 3 / → / → Na + Li, while having littleeffect on the reactivity of the initial state (7,36). Dueto the small magnitude of these interactions [44, 47, 48],intersystem crossing transitions are slow, leading one toexpect the chemical reaction of spin-polarized reactantsto be suppressed compared to that of unpolarized reac-tants, as is indeed observed in Fig. 2(a).The increase of the spin-polarized reaction rate above B = 1000 G shown in Fig. 2(a) is an interesting trendnot observed in our previous CCS calculations on theLi( S) + CaH( Σ + ) reaction [25]. We attribute thistrend to the crossings between the ground ( N = 0) andfirst excited ( N = 1) rotational states of NaLi( a Σ + ) at B ’ B ’ .
45 T avoided crossings oc-cur between the N = 0 and N = 2 Zeeman levels ofNaLi induced by the intramolecular spin-spin interac-tion [51, 57]. Near such crossings, the character of theinitial molecular state | i changes abruptly from spin-polarized to unpolarized, triggering the chemical reac-tion. Accordingly, as shown in Fig. 2(a), the reactionrates for the different initial hyperfine states becomenearly identical, signaling a loss of quantum spin stateselectivity.We note that the spin-polarized reaction rates calcu-lated with and without the hyperfine structure of Naand NaLi taken into account [see Fig. 2(a)] are similarin magnitude and magnetic field dependence. Indeed,the fully spin-stretched hyperfine states | i of NaLiand | i of Na can be written as direct products of theelectron and nuclear spin states. For these states, thenuclear spin degree of freedom only causes a slight shiftin threshold energies, but otherwise plays the role of aspectator. In contrast, the hyperfine mixing between theelectron and nuclear spin functions [see Eq. (1)] playsa major role in ultracold reaction dynamics of unpolar-ized hyperfine states, as can be seen by comparing thereaction rates of the (8,36) and (7,36) initial states inFig. 2(a). Both of these states correspond to the same(2,3) Zeeman state in the absence of hyperfine structure.To gain further insight into the mechanism of thespin-polarized chemical reaction Na + NaLi( a Σ + ) weplot in Fig. 3 the adiabatic eigenvalues (cid:15) i ( R ) of theatom-molecule Hamiltonian [29, 31]. The adiabatic po-tentials allow us to visualize how the atom-molecule in-teractions couple the different spin states of the reactioncomplex, ultimately leading to the reaction [58–60]. Atlarge R , where the atom-molecule interaction vanishes,the adiabatic curves tend to the atom-molecule thresh-olds, approaching them from either below (for ‘ = 0) orfrom above (for ‘ ≥
1) as shown in Figs. 3(b)-(c).Consider the S = 3 / ‘ = 0 diabatic potential ob-tained by following the corresponding adiabatic curves
12 14 16 18 R ( a ) -400-2000200400 E n e r gy ( c m - )
200 400 600
200 400 600 (a) (b)(c)
FIG. 3. (a) Adiabatic potentials of the Na-NaLi reactioncomplex in the absence of the hyperfine structure at B =0 .
01 T and M = 9 /
2. The nonreactive ( S = 3 / , ‘ = 0)and reactive ( S = 1 / , ‘ = 2) diabatic states are shown bythe light blue (light grey) and red (grey) lines. The large- R limits of these states are shown in panels (b) and (c). Theinset in panel (a) is an expanded view of the largest avoidedcrossing. The basis set is restricted to N max = ‘ max = 2. through a series of avoided crossings shown in Fig. 3.The potential is repulsive at short range with a welldepth of ’
200 cm − , and correlates with the fullyspin-polarized initial state of Na(2)-NaLi(3). It experi-ences several crossings with the S = 1 / S -nonconserving interactions, predominantly by the in-tramolecular spin-spin interaction of NaLi( a Σ). Thechemical reaction is then naturally associated with non-adiabatic transitions from the initial repulsive S = 3 / S = 1 / N = 0 and N = 1rotational levels cross. As a result, the avoided crossingsbecome wider and shift toward higher R leading to anincrease of the S = 3 / → S = 1 / Σ + ) molecules in their ground rovibrationalstates in the presence of external magnetic fields andhyperfine interactions. This reaction is representativeof a wide class of ultracold chemical reactions of alkali-dimer molecules with alkali-metal atoms currently stud-ied by several experimental groups [3, 16]. To thisend, we extended the recently developed CCS method-ology [25] to include the fine and hyperfine structureof NaLi(a Σ + ) and carried out ab initio calculations ofthe interaction potentials of the Na Li reaction com-plex. Our calculations reveal a substantial degree ofquantum state selectivity in the dependence of the re-action rate on the initial states of the reactants (fullyspin-polarized vs. unpolarized). In the adiabatic pic-ture, the Na + NaLi(a Σ + ) reaction occurs due to non-adiabatic intersystem crossing transitions between thehigh-spin and low-spin states of the reaction complexmediated by the inramolecular spin-spin interaction ofNaLi(a Σ + ). These transitions are slow due to the small magnitude of the spin-spin interaction, and hence thespin-polarized reaction rate is suppressed by a factor of10-100 compared to the universal limit. 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