arXiv:1004.3317v1 [physics.atom-ph] 19 Apr 2010 ∗ hnoepeae n aiuae the reaction. manipulates chemical the and possible is prepares what excited one suggest advances when ro-vibrational These any interest. to from of state state molecules fur- ground the reactions prepared transferring probe the imagine To could [5]. one else ﬁeld or ther, electric [4] molecules an reactant altering applying the simply by of state by spin rates, nuclear of the reaction order chemical the over on control temperatures ve- gaseous of by µ more- range set are narrow molecules as extremely The locities, an 3]. by down [2, characterized freedom, state molecule over of spin a degrees nuclear all prepare the in to to state possible quantum single now its a is to in It technology beam limit. molecular ultimate pushed has molecules [1]. which inferred the from be of information can dynamics of states reaction wealth vibrational a and providing rotational products, states, diﬀerent abundance ﬁnal relative For the the of select motion. can of methods state spectroscopic relative con- and their molecules, over reactant the trol in freedom inter- of of degrees molec- nal selection of excellent use provides the techniques states, beam initial ular For all of states. determination ﬁnal complete and complete the initial via as provide possible to as is picture a goal the fully experiments, For sur- even these faces. on or calculations scattering semiclassical, mechanical quantum classical, construction surfaces, theory the by energy requiring the complemented potential task, multidimensional On daunting elaborate a of reaction. is product this chemical into side, a molecules during reactant molecules of transformation the loe,o leeegtclydsloe,a ucinof function a as ﬁeld disallowed, electric energetically else or energetically either allowed, become can channels exit Speciﬁcally, nal otx fteH+LiF the of context rudta h xtemct fterato ol be could reaction the of exothermicity the that argued lcrncades [email protected] address: Electronic .Ti icmtnehsealdorders-of-magnitude enabled has circumstance This K. nti etr esgetta aiuaino the of manipulation that suggest we Letter, this In ultracold of attainment experimental the Recently, understand to is physics chemical of business The ttsmyas epsil ncrancircumstances. certain in possible be also may states IA ITadUiest fClrd,Dprmn fPhysic of Department Colorado, of University and NIST JILA, E ecin2KRb reaction hntosc oeue old na lrcl a,w n th ﬁnd we K gas, species trimer ultracold the an form in to reacting collide from molecules such two When ASnmes 34.20.Gj,34.50.Ez,34.50.Lf numbers: PACS imply reaction. ﬁelds, chemical microwave a or electric in applying channels by reaction zero to set be hsie a dacdpeiul,i the in previously, advanced was idea This . ecnie lrcl,ceial eciesatrn col scattering reactive chemically ultracold, consider We → rdc tt oto fb-laiceia reactions chemical bi-alkali of control state Product i+H ecin hr twas it where reaction, HF + Li → K 2 Rb + dudR Meyer R. Edmund 2 oevr h iyeeg eesdi hsrato a np in can reaction this in released energy tiny the Moreover, . initial Dtd coe 0 2018) 30, October (Dated: ttsof states 2 bo Rb or Rb ﬁ- ∗ n onL Bohn L. John and ecnie olsoso aro R oeue,pre- molecules, KRb states of rotational pair particular a in Speciﬁcally, of ter- pared reactants. collisions not the are consider from that, reaction we energy of fact in states the diﬀerent ﬁnal by ribly the aﬀorded dimers, is alkali possibility completely for This be conceivable can oﬀ. is reaction for turned exothermic it that, previously out temperature, a point ultralow that we at Here dimers [6]. alkali ﬁelds electric via shifted h omto ftiesvia reactions: trimers of of kinds two formation to the subject general in are molecules n h odsapn reaction bond-swapping the and y∆ by ic h od r l oaetadvr iia.I zero with In (2) similar. diﬀerences, very reaction and energy the covalent all small ﬁeld are very bonds the produce since (2) bond-rearrangement the in contrast, By reactions 2) en- kinetic gas. the translational in than ergies larger magnitude nine numbers, of wave orders thousand several – large and negative 1 htfr rmr r nreial ialwda ul- ∆ at released, disallowed energy the energetically as are temperatures, tracold trimers form that (1) ed.Teeoe h eaieeeg ∆ energy high relative comparatively the at Therefore, polarized ﬁelds. be only can products the r oa hra h rdcsK products the whereas polar are ecat n rdcsi,i rnil,afnto fthe of function a ﬁeld principle, electric in applied is, products and reactants several a obn ofr rmrwt oa spin total with trimer a form to combine can from them calculate must initio we energies measured, binding been trimer not the have As forbidden. energetically is thereby states. product disallowed, of be distribution can possible the states high- changing this, ﬁnal than smaller rotational ﬁelds At lying altogether. oﬀ turned be tt) eea acltoso obe ttshv been K have trimers states homonuclear doublet the for of achieved, calculations Several state). dultsae ra xie tt with state excited an or state) (doublet 2 ,hnecnol ec i h bond-swapping the via react only can hence K, KRb( erpr eetocrusacs )Tereactions The 1) circumstances: two here report We ebgnwt h rtpit httie omto (1) formation trimer that point, ﬁrst the with begin We KRb( KRb( E ehd.I eea,tresi-/ laiatoms alkali spin-1/2 three general, In methods. n dimer iin ftedaoi oeue KRb. molecules diatomic the of lisions × n en fcnrligteavailable the controlling of means a ing n n 1 10 KRb( + ) ,Budr ooao83904,USA 80309-0440, Colorado Boulder, s, 1 1 KRb( + ) KRb( + ) 5 tte r nreial forbidden energetically are they at 10 = /m ∆ V/cm, . cm 4 n n n 2 ) E 2 2 ) ) ned tﬁlso h re of order the on ﬁelds at Indeed, . E → − → → bs 1 K oevr h ecat KRb reactants the Moreover, K Rb aihsadterato can reaction the and vanishes 2 n ( 2 n 1 b+R ∆ + Rb + Rb 2 1 ′ ∆ + K + K = Rb + ) 2 n n Rb and 2 rinciple n sexothermic is 0 = E 2 1 ( S dimer n 3 and 2 ′ 3 = E 7 n Li and [7] 2 ∆ + ) E trimer r o,thus not, are ( trimer / E n (quartet 2 E between ) 2 S E These . trimer bs 1 = 3 . (1b) (1a) [8], (2) is , / ab 2 2

TABLE I: Molecular properties of the diatomic molecules K2, -500 2 7. B2 KRb, and Rb2. Equilibrium bond lengths Re are in ˚A and −1 6.5 dissociation energies De are in cm . (1) denotes a calculation L

Þ 6. performed at the MCSCF+MRCI level while (2) denotes a H 5.5 geometry optimization at the RHF-CCSD(T) level accounting RbK for BSSE. R 5. - 4.5 6350 -5750 Molecule Re (1) De (1) Re (2) De (2) Re Expt. De Expt. -5000 2 4. K 4.16 -4293 3.92 -4328 3.92 -4450.78(15) -3500 -2000 KRb 4.33 -4039 4.05 -4062 4.05 -4217.30(15) -4250 -3500 2 7. A1 Rb2 4.50 -3729 4.18 -3741 4.17 -3993.53(6) -5000 6.5 L

Þ 6. H -5750 5.5 -2750

RbK - 2000 6100

R 1250 and for certain molecular Li2X systems, where X is an 5005. alkali-metal atom [9]. Sold´an and co-workers have exam- 4.5 - -2000 1250 - ined the two doublet surfaces of K3 at C2v geometries as 4. 500 well as the conical intersection at the equilateral triangle 4 5 6 7 geometry described by the D3h group [7]. Þ RK2 H L We employ similar computational techniques as in Refs. [7, 9] to compute doublet ground states, within the molpro suite of molecular structure codes [10]. We use the eﬀective core potentials and basis sets of the Stuttgart FIG. 1: (Color Online) Contour plots for the doublet ground state PES’s of K2Rb, versus interatomic spacings in an isosce- group for the K (ECP10MDF) and Rb (ECP28MDF) 2 les triangle geometry. The top panel shows the B2 surface atoms [11]. In addition, to adequately model three-body 2 while the bottom panel shows the A1 surface. Contours are forces, we augment these basis functions with a diﬀuse −1 labeled in increments of 750 cm from -5750 – -500 with an function for each of the s, p, and d orbitals in an even additional contour near the minimum of each surface. Arrows tempered manner, as well as a g-function for the K atom. denote the singlet (red) and triplet (green) equilibrium bond We compare two computational approaches, testing lengths of the diatomic species. them on diatomic alkali molecules. For both approaches, we ﬁrst perform a spin-restricted Hartree-Fock (RHF) calculation on the singlet conﬁguration. The ﬁrst ap- alkali-metal systems. By contrast, approach 2 does just proach uses the RHF wave function as the foundation for as well at predicting dissociation energies, but provides a multi-conﬁguration self consistent ﬁeld (MCSCF) cal- far better bond lengths. In either method, binding en- culation [12, 13] with an active space that includes the ergies are clearly reproduced to within several hundred ﬁrst excited p orbital of each atom. In addition, we in- cm−1. We take this as an empirical measure of the cal- clude all states resulting from the ns0 np1 conﬁgurations culations’ accuracy. of each atom. All remaining orbitals are closed, meaning We next apply the same methods to determine the they are energy optimized with the restriction that they minima of the three-body potential energy surfaces. The remain doubly occupied. After each MCSCF calculation, minimum of the potential energy surface should lie in the we perform an internally contracted multi-reference con- C2v point group i.e., the odd-atom-out is expected to lie ﬁguration interaction (MRCI) [14, 15] calculation with on the perpendicular bisector of the line joining the two the same active space as in the MCSCF. The minimum like atoms. Thus we initially restrict ourselves to this energy is then compared to the dissociation energy eval- geometry. As a case study, we will look in detail at the uated in the separated-atom limit. K2Rb surfaces, but the Rb2K surfaces are similar. In the second approach, we perform a coupled As a ﬁrst approach we used the RHF-MCSCF+MRCI clusters with single, double and non-iterative triples method due to its time eﬃciency To be consistent with 1 (CCSD(T)) [16] on the X Σ state at the minimum Re the method employed for dimers, we keep the same active value obtained from the MCSCF+MRCI calculation. We space, which included the ﬁrst excited p-orbital of each 2 2 then performed a geometry optimization followed by a atom. We included 4 states each of B2 and A1 symme- 2 2 4 basis set superposition error correction (BSSE)[17] to ex- tries, 2 B1, 1 A2, and 1 B2 states in the MCSCF. In 2 2 tract the bond length and dissociation energy of the sys- each MRCI calculation of the B2 and A1 surfaces, we tems at the RHF-CCSD(T)+BSSE level of theory. included 4 states as well as the reference symmetries of To check the adequacy of these computational meth- the other doublets. ods, we compare their results with the known binding en- The results of the RHF-MCSCF+MRCI calculations ergies of alkali-metal dimers [18–22], as shown in Table I. are presented in Fig. 1 as contour plots of the poten- These results show that approach 1 gets close to the dis- tial energy surface (PES) in the two independent bond sociation energy of each diatomic species, but noticeably lengths RRbK and RK2 , for the isosceles triangle geom- 2 overestimates the bond length, especially for the heavier etry. These surfaces show that the B2 surface (upper 3 panel) possesses the global minimum, hence represents the ground state of the trimer. The minimum energy oc- TABLE II: Molecular properties of the alkali-metal triatomic molecules containing K and Rb. Bond lengths are in ˚A, an- curs near the singlet bond length of the KRb dimer (de- gles are in degrees, and dissociation energies, relative to the noted by a horizontal red arrow), but is intermediate be- complete breakup into three free atoms, are in cm−1. tween the singlet and triplet bond lengths of the K2 dimer Molecule Symmetry rhetero rhomo θ De (red and green vertical arrows, respectively). In C2v sym- 2 K3 C2v B2 4.10 5.11 90.0 -6059 metries, B2 corresponds to an odd reﬂection about the 2 K2Rb C2v B2 4.26 4.99 90.0 -5817 bisector of the isosceles triangle. This means that along 2 Rb2K C2v B2 4.26 5.63 90.0 -5533 the K2 bond the electronic wave function should roughly 2 ′ K2Rb Cs A 4.25 4.06 73.23 -5982 resemble the K2 triplet wave function, and so should the 2 ′ Rb2K Cs A 4.24 4.41 75.00 -5611 the bond length. There is then diminished electron den- 2 sity at the center between the two K atoms, and this Rb3 C2v B2 4.40 5.59 90.0 -5273 allows the Rb atom to ﬁll this space, bringing the two K atoms slightly closer together than they would be in the 2 K2 triplet state alone. HaL 2 0 Similarly, the minimum of the A1 surface in the lower -2 panel of Fig. 1 is located near the K2 singlet bond length È0,0\È0,0\ L 1 - (vertical red arrow), but in between the singlet and triplet - 4 È0,0\ bonds of the KRb system (horizontal red and green ar- cm H - È1,0\ 2 6 rows). The A1 surface requires an even reﬂection in the E È2,0\ -8 electronic wave function across the bisector of the isosce- È3,0\ les triangle, therefore preferring a singlet-like bond in -10 RK2 . Therefore, the Rb atom does not quite know which -12 spin to take since it can form a triplet or singlet with 101 102 103 104 105 106 one or the other K atoms, but not both. This frustra- Electric Field HVcmL tion manifests itself with a bond somewhere intermediate 2 between singlet and triplet bonds in this coordinate. HbL 0 Knowing that the the RHF-CCSD(T)+BSSE results - in the dimer case are markedly better when compared to 2 L

1 - experimental values, we choose to characterize the min- - 4 cm ima of the PES’s at RHF-UCCSD(T)+BSSE level of the- H -6 E

ory, including a basis set superposition correction. The D U in UCCSD(T) refers to the spin-unrestricted formal- -8 2 Rb ism due to the open-shell nature of the trimer systems. -10 @ N Starting near the parameters obtained above for the 2B N @K2D 2 -12 surface, we ﬁnd the optimum geometry by the method of steepest descents. Binding energies are determined at the RHF-UCCSD(T) level of theory, accounting for basis FIG. 2: (Color Online) Panel (a) shows the Stark shift for set overlap errors. The results are presented in Table II the ground and ﬁrst few rotationally excited states for a pair for all the alkali-metal trimers containing K and Rb. In of KRb molecules (colored lines). The black line shows the n′ n′ the case of K3, where results have previously been com- energy of the ﬁnal states K2( 1 = 0) + Rb2( 2 = 0). Panel puted, we ﬁnd excellent agreement with the reported val- (b) shows the rotational states energetically allowed at zero electric ﬁeld. ues of the isosceles bond length riso = 4.10 A˚ and apex ◦ angle θapex = 77.13 [7]. Based on these results, we con- clude that the formation of the K Rb trimer would re- 2 to the shorter of the two bonds between heteronuclear quire ∆E = 2(4217) − 5982 = 2452 cm−1 of energy, trimer pairs when in C symmetry. θ is the angle made by A- certainly rendering this reaction impossible at ultralow s q-B, where A and B are diﬀering atoms and q is the temperatures. midpoint between the homonuclear bond. While a minimum of the surface appears to be in the We therefore now consider the bond-swapping reaction B symmetry of the C point group, an examination of 2 2v (2), and in particular the variation of ∆Ebs with electric the trimers K2Rb and Rb2K away from this symmetry ﬁeld. To do so, we look at the energy eigenvalues of the is needed in order to determine whether this is a global reactant and product molecules, as given by the Hamil- minumum. We looked at the systems in the Cs point tonian group and found minimums which correspond to bent 2 4 geometries using similar optimaization procedures. In H = BeN~ − D N~ − d~ · E~ (3) both trimers the bent geomtry minimum is found to be the global minimum making the B2 minimum a saddle where Be is the molecule’s rotational constant, D is cen- point. The parameters are given in Table II. rhetero refers trifugal distortion constant, d~ is the body-ﬁxed molecular 4 dipole moment (which is zero for the products), and E~ oﬀ all reactions completely, it shuts oﬀ ﬁrst higher-n′, is the applied electric ﬁeld. In addition, both kinds of then successively lower-n′ states. Recall that the n′- molecules also experience a shift due to their electronic distribution of the products is one of the key observ- polarizability α. Using the known KRb dipole moment ables of physical chemistry. The ability to allow only [2] and computed molecular polarizabilities [23], we can certain values of n′ into this distribution will likely pro- compute the relative energies of the reactants and prod- vide an even greater wealth of information from such ex- ucts. periments. The detailed eﬀect on chemistry of shutting In Fig.2a) we plot the energy of the ground state (red) oﬀ successively lower rotational exit channels remains to and several rotationally excited states of KRb+KRb as a be explored. It may be hoped, for example, that the new function of electric ﬁeld. Also shown (black) is the ground information gleaned would shed additional light on the state energy of the products, K2(n = 0) + Rb2(n = 0). role in these reactions of conical intersections. −1 In zero ﬁeld, the reactants are their natural 10.4 cm So far we have focused on KRb, since it is the molecule above the products. As the ﬁeld increases, the energy of for which ultracold chemistry has recently been demon- the polar reactant states decreases rapidly, but that of strated experimentally. However, there are other reason- the non-polar product states decreases far more slowly. able candidates for these experiments, notably RbCs [27], ∼ Therefore, at a ﬁeld beyond 625 kV/cm, the reactants whose reaction is endothermic by 28.7 cm−1. Such a re- are actually lower in energy than the products, ∆Ebs action would proceed only by placing the reactant RbCs becomes negative, and the reaction is completely shut molecules into excited states, which could certainly be oﬀ. While a static electric ﬁeld this large is probably im- done. An applied electric ﬁeld could then still dictate practical to implement, it may be possible to achieve the which ﬁnal channels are available, by moving these rota- required energy shifts in a suitably designed microwave tionally excited states relative to the Rb2 + Cs2 products. cavity, such as those proposed for trapping polar species [24]. Theory would then naturally have to account for Note added in preparation: New work shows that the collisions of the ﬁeld-dressed states [25, 26]. timer reactions (1) are energetically forbidden for all al- As a point of reference, Fig. 2b) shows the 235 ener- kali dimer collisions [28]. getically allowed states of the products in zero electric The authors acknowledge useful discussions with G. ′ ﬁeld, indexed by their rotational quantum numbers n1 Qu´em´ener and C. H. Greene, as well as funding from the ′ and n2. It is clear that, before the electric ﬁeld shuts NSF.

[1] R. D. Levine, Moleculr Reaction Dynamics (Cambridge 115, 259 (1985). University Press, 2009). [14] H.-J. Werner and P. J. Knowles, J. Chem. Phys. 89, 5803 [2] K.-K. Ni, S. Ospelkaus, M. Miranda, A. Peer, B. Neyen- (1988). huis, J. Zirbel, S. Kotochigova, P. Julienne, D. Jin, and [15] P. J. Knowles and H.-J. Werner, Chem. Phys. Letters J. Ye, Science 322, 231 (2008). 145, 514 (1988). [3] J. G. Danzl, M. J. Mark, E. Haller, M. Gustavsson, [16] C. Hampel, K. Peterson, and H.-J. Werner, Chem. Phys. R. Hart, J. Aldegunde, J. M. Hutson, and H.-C. N¨agerl, Letters 190, 1 (1992). arXiv:0909.4700 (2009). [17] S. F. Boys and F. Bernardi, Mol. Phys. 19, 553 (1970). [4] S. Ospelkaus, K.-K. Ni, D. Wang, M. H. G. de Miranda, [18] C. Amiot, J. Verg´es, and C. Fellows, J. Chem. Phys. 103, B. Neyenhuis, G. Qu´em´ener, P. S. Julienne, J. L. Bohn, 3350 (1995). D. S. Jin, and J. Ye, Science 327, 853 (2010). [19] C. Amiot and J. Verg´es, J. Chem. Phys. 112, 7068 [5] K.-K. Ni, S. Ospelkaus, D. Wang, G. Qu´em´ener, (2000). B. Neyenhuis, M. H. G. de Miranda, J. L. Bohn, J. Ye, [20] J. Seto, R. le Roy, J. Verg´es, and C. Amiot, J. Chem. and D. S. Jin, arXiv:1001.2809 (2010). Phys. 113, 3067 (2000). [6] T. V. Tscherbul and R. V. Krems, J. Chem. Phys. 129, [21] C. Fellows, R. Gutterres, A. Campos, J. Verg´es, and 034112 (2008). C. Amiot, J. Mol. Spectr. 197, 19 (1999). [7] A. Hauser, C. Callegari, P. Sold´an, and E. Ernst, J. [22] C. Amiot and O. Dulieu, J. Chem. Phys. 117, 5155 Chem. Phys. 129, 044307 (2008). (2002). [8] J. N. Byrd, J. A. Montgomery, H. H. Michels, and [23] J. Deiglmayr, M. Aymar, R. Wester, and M. Wei- R. Cˆot´e, Int. J. Quant. Chem. 109, 3112 (2009). dem¨uller, J. Chem. Phys. 129, 064309 (2008). [9] P. Sold´an, Phys. Rev. A 77, 054501 (2008). [24] D. Demille, D. R. Glenn, and J. Petricka, Eur. Phys. J. [10] H.-J. Werner, P. J. Knowles, R. Lindh, F. R. Manby, D 31, 375 (2004). M. Sch¨utz, et al., Molpro, version 2008.3, a package of [25] A. V. Avdeenkov, New J. Phys. 11, 055016 (2009). ab initio programs (2008), see http://www.molpro.net. [26] S. V. Alyabyshev and R. V. Krems, Phys. Rev. A 80, [11] I. Lim, P. Schwerdtfeger, B. Metz, and H. Stoll, J. Chem. 033419 (2009). Phys. 122, 104103 (2005). [27] J. Sage, S. Sainis, T. Bergeman, and D. DeMille, Phys. [12] H.-J. Werner and P. J. Knowles, J. Chem. Phys. 82, 5053 Rev. Lett. 94, 203001 (2005). (1985). [28] P. S. Zuchowski˙ and J. M. Hutson, arXiv:1003.1418 [13] P. J. Knowles and H.-J. Werner, Chem. Phys. Letters (2010).