The prospects for producing ultracold NH 3 molecules by sympathetic cooling: a survey of interaction potentials
aa r X i v : . [ phy s i c s . c h e m - ph ] M a y The prospects for producing ultracold NH molecules by sympathetic cooling:a survey of interaction potentials Piotr S. ˙Zuchowski ∗ and Jeremy M. Hutson † Department of Chemistry, Durham University, South Road, DH1 3LE, United Kingdom (Dated: November 3, 2018)We investigate the possibility of producing ultracold NH molecules by sympathetic cooling in abath of ultracold atoms. We consider the interactions of NH with alkali-metal and alkaline-earthatoms, and with Xe, using ab initio coupled-cluster calculations. For Rb-NH and Xe-NH wedevelop full potential energy surfaces, while for the other systems we characterize the stationarypoints (global and local minima and saddle points). We also calculate isotropic and anisotropic Vander Waals C coefficients for all the systems. The potential energy surfaces for interaction of NH with alkali-metal and alkaline-earth atoms all show deep potential wells and strong anisotropies.The well depths vary from 887 cm − for Mg-NH to 5104 cm − for Li-NH . This suggests that allthese systems will exhibit strong inelasticity whenever inelastic collisions are energetically allowedand that sympathetic cooling will work only when both the atoms and the molecules are already intheir lowest internal states. Xe-NH is more weakly bound and less anisotropic. I. INTRODUCTION
There is great interest at present in producing samplesof cold molecules (below 1 K) and ultracold molecules(below 1 mK). Such molecules have many potential ap-plications. High-precision measurements on ultracoldmolecules might be used to measure quantities of fun-damental physics interest, such as the electric dipole mo-ment of the electron [1] and the time-dependence of fun-damental constants such as the electron/proton mass ra-tio [2]. Ultracold molecules are a stepping stone to ul-tracold quantum gases [3] and might have applications inquantum information and quantum computing [4].There are two basic approaches to producing ultra-cold molecules. In direct methods such as Stark decel-eration [5, 6] and helium buffer-gas cooling [7], preexist-ing molecules are cooled from higher temperatures andtrapped in electrostatic or magnetic traps. In indirect methods [8], laser-cooled atoms that are already ultracoldare paired up to form molecules by either photoassocia-tion [9] or tuning through magnetic Feshbach resonances[10].Indirect methods have already been used extensivelyto produce ultracold molecules at temperatures below 1 µ K. However, they are limited to molecules formed fromatoms that can themselves be cooled to such tempera-tures. Direct methods are far more general than indirectmethods, and can in principle be applied to a very widerange of molecules. However, at present direct meth-ods are limited to temperatures in the range 10-100 mK,which is outside the ultracold regime. There is muchcurrent research directed at finding second-stage coolingmethods to bridge the gap and eventually allow directlycooled molecules to reach the region below 1 µ K where ∗ Electronic address: E-mail: [email protected] † Electronic address: E-mail: [email protected] quantum gases can form.One of the most promising second-stage cooling meth-ods that has been proposed is sympathetic cooling . Thehope is that, if a sample of cold molecules in brought intocontact with a gas of ultracold atoms, thermalization willoccur and the molecules will be cooled towards the tem-perature of the atoms. Sympathetic cooling has alreadybeen used successfully to cool atomic species such as Li[11] and K [12] but has not yet been applied to neutralmolecules.Sympathetic cooling relies on thermalization occurringbefore molecules are lost from the trap. Thermalizationrequires elastic collisions between atoms and molecules toredistribute translational energy. However, electrostaticand magnetic traps rely on Stark and Zeeman splittingsand trapped atoms and molecules are not usually in theirabsolute ground state in the applied field. Any inelastic collision that converts internal energy into translationalenergy is likely to kick both colliding species out of thetrap. The ratio of elastic to inelastic cross sections isthus crucial, and a commonly stated rule of thumb isthat sympathetic cooling will not work unless elastic crosssections are a factor of 10 to 100 greater than inelasticcross sections for the states concerned.Inelastic cross sections for atom-atom collisions aresometimes strongly suppressed by angular momentumconstraints. In particular, for s-wave collisions (end-over-end angular momentum L = 0), pairs of atoms in spin-stretched states (with the maximum possible values of thetotal angular momentum F and its projection | M F | ) canundergo inelastic collisions only by changing L . Crosssections for such processes are very small because, foratoms in S states, the only interaction that can change L is the weak dipolar coupling between the electron spins.However, for molecular collisions the situation is differ-ent: the anisotropy of the intermolecular potential canchange L , and this is usually much stronger than spin-spin coupling.It is thus crucial to investigate the anisotropy of theinteraction potential for systems that are candidatesfor sympathetic cooling experiments. In experimen-tal terms, the easiest systems to work with are thosein which molecules that can be cooled by Stark de-celeration (such as NH , OH and NH) interact withatoms that can be laser-cooled (such as alkali-metal andalkaline-earth atoms). There has been extensive workon low-energy collisions of molecules with helium atoms[13, 14, 15, 16, 17, 18, 19], but relatively little on colli-sions with alkali-metal and alkaline-earth atoms. Sold´anand Hutson [20] investigated the potential energy sur-faces for Rb + NH and identified deeply bound ion-pairstates as well as weakly bound covalent states. Theysuggested that the ion-pair states might hinder sympa-thetic cooling. Lara et al. [21, 22] subsequently calcu-lated full potential energy surfaces for Rb + OH, for bothion-pair states and covalent states, and used them to in-vestigate low-energy elastic and inelastic cross sections,including spin-orbit coupling and nuclear spin splittings.They found that even for the covalent states the potentialenergy surfaces had anisotropies of the order of 500 cm − and that this was sufficient to make the inelastic cross sec-tions larger than inelastic cross sections at temperaturesbelow 10 mK. Tacconi et al. [23] have recently carried outanalogous calculations on Rb + NH, though without con-sidering nuclear spin. There has also been a considerableamount of work on collisions between alkali metal atomsand the corresponding dimers [24, 25, 26, 27, 28, 29].One way around the problem of inelastic collisions isto work with atoms and molecules that are in their abso-lute ground state in the trapping field. However, this isquite limiting: only optical dipole traps and alternatingcurrent traps [30] can trap such molecules. It is thereforehighly desirable to seek systems in which the potentialenergy surface is only weakly anisotropic. The purposeof the present paper is to survey the possibilities for col-lision partners to use in sympathetic cooling of NH (orND ), which is one of the easiest molecules for Stark de-celeration.Even if sympathetic cooling proves to be impracticalfor a particular system, the combination of laser cool-ing for atoms and Stark deceleration for molecules offersopportunities for studying molecular collisions in a newlow-energy regime. For example, experiments are underway at the University of Colorado [31] to study collisionsbetween decelerated NH molecules and laser-cooled Rbatoms.There alkali-metal atom + NH systems have not beenextensively studied theoretically, though there has beenexperimental interest in the spectroscopy of Li-NH com-plex as a prototype metal atom-Lewis base complex [32].Lim et al. [33] recently calculated electrical propertiesand infrared spectra for complexes of NH with alkali-metal atoms from K to Fr and gave the equilibrium struc-tures of their global minima. However, to our knowledge,no complete potential energy surfaces have been pub-lished for any of these systems. The alkaline-earth +NH have been studied even less, and except for an early study of the Be-NH system [34] there are no previousresults available. II.
AB INITIO
METHODS
The interaction energy of two monomers A and B isdefined as E AB int = E AB tot − E A tot − E B tot (1)where E AB tot is the total energy of the dimer and E A tot and E B tot are the total energies of the isolated monomers.Since the interaction energy is dominated at long rangeby intermolecular correlation (dispersion), ab initio cal-culations of the interaction energy must include elec-tronic correlation effects at the highest possible level [35]and must be carried out with large basis sets augmentedby diffuse functions. At present, the coupled-cluster (CC)method with single, double and noniterative triple excita-tions (CCSD(T)) provides the best compromise betweenhigh accuracy and computational cost. In the present pa-per, we carry out coupled-cluster calculations using the Molpro package [36]. All interaction energies are cor-rected for basis-set superposition error (BSSE) with thecounterpoise method of Boys and Bernardi [37].Standard coupled-cluster methods are reliable onlywhen the wavefunction is dominated by a single elec-tronic configuration This is often an issue for molecu-lar systems with low-lying excited states. In order tocheck the reliability of CC calculations, it is necessary tomonitor the norm of T operator [38] (measured by theT1 diagnostic). In the case of metal-NH systems thisis relatively large, especially when the atom approachesthe lone pair of the NH molecule, but the convergence ofthe CC equations is fast and the converged CCSD resultsare very close to benchmark multireference configurationinteraction (MRCI-SD) calculations with size-extensivitycorrections. Thus we consider the CC results reliable.To understand the origin of the intermolecular forceswe also consider the interaction energies obtained at theHartree-Fock level, which neglects electron correlationand thus provides information about the role of disper-sion and other correlation effects. For some systems wealso analyze the components of the intermolecular in-teractions using symmetry-adapted perturbation theory[39] (SAPT). The first-order SAPT corrections (electro-static and exchange terms) are computed at the Hartree-Fock level, while the dispersion energy is evaluated in thecoupled Hartree-Fock approximation [40]. These calcula-tions are carried out using the SAPT2006 [41] program.We are interested principally in the collisions of coldammonia molecules with atoms at energies that are muchtoo low for vibrational excitation to occur. Such colli-sions are governed by an effective potential that is vi-brationally averaged over the ground-state vibrationalwavefunction of NH . For the present purpose it is ade-quate to represent this by a potential calculated with theNH molecule frozen at a geometry that represents theground state. In the present paper we use a geometry de-rived from the high-resolution infrared spectra [42]: themolecule is taken to have C v symmetry with N-H bondlengths of 1.913 a and an H-N-H angle of 106 . ◦ . Inter-molecular geometries are specified in Jacobi coordinates: R is the distance from the center of mass of NH to theatom, while θ is the angle between the intermolecularvector and the C axis of the NH molecule (with θ = 0 ◦ corresponding to the atom approaching towards the lonepair of NH ). Finally, χ is the dihedral angle between theplane containing the C axis and an NH bond and thatcontaining the C axis and the intermolecular vector.Table I gives the lowest excitation energies, dipole po-larizabilities and ionization energies of the atoms studiedin this paper. The neutral alkali-metal and alkaline-earthatoms (denoted below as A and Ae, respectively) haveparticularly low excitation energies, resulting from smallseparations between energy levels corresponding to ns and np or ( n − d configurations. Since the gap betweenthe ground and excited states is small, the atoms havevery large polarizabilities. Hence, we expect particularlystrong induction and dispersion interactions. The alkali-metal and alkaline-earth atoms also have low ionizationenergies E i . Since the atomic orbital wavefunctions van-ish at long range as exp( − E / r ), the wavefunctions anddensities are very diffuse, and this causes large overlapbetween monomers even at relatively large separations.Finally, because of the low ionization energies, alkali-metal and alkaline-earth atoms have a strong tendencyto form charge-transfer complexes.The basis sets used in the ab initio calculations areas follows. For Be, Li, Mg, Na, Ca atoms we use all-electron cc-pVTZ basis sets augmented by even-tempereddiffuse exponents, while for potassium we use the CVTZbasis set of Feller [52]. For Rb, Sr and Xe we han-dle only the outermost electrons explicitly, with the coreelectrons represented by effective core potentials (ECPs).For Rb we use the small-core effective core potentialECP28MWB with a basis set based on that of Ref. 53,which was optimized to recover the static dipole polar-izability. We modified this slightly to account better forintramonomer electronic correlation effects by removing0.07 f and adding 0.001049 s , 0.0024 p , 4.5,0.016667 d ,1.9,0.655 f and 0.95,0.3167 g functions. The basis setfor Sr is taken from Ref. 54. For Xe we use the basis setgiven by Lozeille et al. [50], which was found to be ex-cellent for polarizabilities and hyperpolarizabilities. Foreach system we added a set of midbond functions withexponents sp : 0.9,0.3,0.1, df III. RESULTS AND DISCUSSION
The potential energy surface for an atom-NH systemis a function of the intermolecular distance R and twoangles θ and χ . However, functions of 3 variables are difficult to represent graphically. It is convenient to rep-resent the χ -dependence in the form V ( R, θ, χ ) = ∞ X k =0 V k ( R, θ ) cos 3 kχ. (2)To reduce the computational effort we calculate the in-teraction potential only for χ = 0 ◦ and χ = 60 ◦ andapproximate the leading terms V ( R, θ ) and V ( R, θ ) bysum and difference potentials, V ( R, θ ) = [ V ( R, θ,
0) + V ( R, θ, ◦ )] V ( R, θ ) = [ V ( R, θ, − V ( R, θ, ◦ )] . (3) V can be viewed as the interaction potential averagedover χ , while V describes the leading anisotropy of thepotential with respect to rotation about the C axis ofNH . A. Alkali-metal atom + NH interactions The potential energy surface for Rb-NH is shown inFigure 1. CCSD(T) calculations were carried out at χ = 0 ◦ and 60 ◦ , at values of θ corresponding to a 20-pointGauss-Lobatto quadrature. The grid included R valuesfrom 3.5 to 12 a with a step of 0.5 a , and from 12 to15 a with the step of 1 a . There is a deep minimum(1862 cm − ) at R = 5 . a and θ = 0, corresponding toapproach of Rb towards the NH lone pair. The poten-tial is much shallower at other geometries, with a saddlepoint near θ = 110 ◦ and a shallow secondary minimumat θ = 180 ◦ . The anisotropy with respect to rotation ofNH about the C axis ( χ ) is relatively weak, at leastin the low-energy classically allowed region defined by V ( R, θ ) < potentials isquite similar. In each case there is a deep minimumaround θ = 0 ◦ and a shallow secondary minimum for θ = 180 ◦ . Table II gives the well depths and equilibriumdistances. For the alkali metals the well depth of theglobal minimum decreases down the periodic table, from5104 cm − for Li to 1862 cm − for Rb, and the equilib-rium distance increases from 3.91 a for Li to 5.90 a forRb. The changes in the properties of the shallow sec-ondary minima are much smaller, with well depths closeto 100 cm − for all the alkali metals. Our results forthe species containing K and Rb are in good agreementwith the CCSD(T) calculations of Lim et al. [33]; theyobtained slightly different values of the binding energiesof K-NH and Rb-NH (2210 cm − and 1950 cm − , re-spectively), but their results are not corrected for BSSE.It should also be noted that their binding energies are forrelaxed NH geometries.The deep wells and large anisotropies of the A-NH potentials will produce strong coupling between the dif-ferent NH rotational states during collisions. All thesesystems are therefore likely to have large inelastic cross TABLE I: Properties of alkali-metal, alkaline-earth and Xe atoms important to interaction potentials. Note that for alkali-metalatoms the lowest excitation energy corresponds to S → P / excitation and for alkaline-earth and Xe atoms to S → P .The excitation and ionization energies are take from NIST Handbook of Basic Atomic Spectroscopic Data [43].Atom dipole polarizability C coefficient lowest excitation energy ionization energy( a ) Ref. ( E h a ) Ref. (cm − ) (cm − )Li 164 44 1395 45 14904 43487Na 162 46 1561 45 16956 41449K 293 46 3921 47 12985 35010Rb 319 46 4707 48 12578 33691Be 37.7 45 213 45 21978 75193Mg 71 45 629 45 21850 61671Ca 159 45 2221 49 15157 49305Sr 200 45 3250 45 14317 45932Xe 27.3 50 286 51 67068 97834 − − − − − − − − − −300 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − θ R a ( ) − − − − − − − − − − θ (deg) R ( a )
0 30 60 90 120 150 1805 7 9 111315
FIG. 1: The interaction potential of Rb-NH from CCSD(T)calculations: V ( R, θ ) component (upper panel) and V ( R, θ )component (lower panel). Contours are labelled in cm − . Toaid visualization, V is plotted only in the energetically acces-sible region defined by V < sections. It is thus unlikely that sympathetic coolingof NH with alkali-metal atoms will be successful unlessboth the atoms and the molecules are already are in theirlowest internal states. TABLE II: Equilibrium distances and well depths for alkali-metal atom + NH systems from CCSD(T) calculations. θ = 0 ◦ θ = 180 ◦ R e ( a ) D e (cm − ) R e ( a ) D e (cm − )Li 3.91 5104 7.86 104.8Na 4.73 2359 8.33 98.2K 5.52 2161 8.90 99.6Rb 5.90 1862 8.89 110.2 B. Alkaline-earth atom + NH interactions We originally hoped that the potentials for systemscontaining alkaline-earth atoms would be more weaklybound and less anisotropic than for those containingalkali-metal atoms . However, this proved not to be thecase, at least for the heavier alkaline-earth atoms that aremost suitable for laser cooling. The results for the Ae-NH systems are summarized in Table III. The shapesof the potential energy surfaces are generally similar tothose for A-NH systems. For Ca and Sr, the depthsof the global minima are 3229 and 3141 cm − respec-tively; these are both deeper than for the correspondingalkali-metal atom. For Mg, however, the well depth isconsiderably shallower at only 887.5 cm − . The min-ima corresponding to approach at the hydrogen end ofNH are slightly deeper than for the alkali metals, rang-ing from 115.7 for Mg to 131.6 cm − for Sr. On theother hand, the interaction potential for Be-NH resem-bles those for Ca-NH and Sr-NH more than that forMg-NH : the global minimum is 1973 cm − deep, whilethe dispersion-bound minimum is 100.5 cm − deep. Theequilibrium distance for Be-NH at θ = 0 ◦ (3.57 a ) isalso much shorter than for the other Ae-NH systems,and is comparable to that for Li-NH . TABLE III: Equilibrium distances and well depths foralkaline-earth atom + NH systems from CCSD(T) calcula-tions. θ = 0 ◦ θ = 180 ◦ R e ( a ) D e (cm − ) R e ( a ) D e (cm − )Be 3.57 1973 7.61 100.5Mg 4.83 887.5 8.20 115.7Ca 4.92 3229 8.85 129.1Sr 5.22 3141 9.06 131.6TABLE IV: The interaction energies (in cm − ) for Li-NH ,Ca-NH and Mg-NH at different levels of electronic correla-tion, for geometries corresponding to the global and secondaryminima. GMHF CCSD CCSD(T)Li -4405 -5022 -5104Ca -2152 -2937 -3229Mg 260 -590 -888LMHF CCSD CCSD(T)Li 248 -54 -105Ca 244 -47 -129Mg 155 -54 -116 C. Origin of bonding in metal-atom + NH systems It is important to understand the large difference be-tween the metal–lone pair bond energies between Mg andthe other Group 1 and 2 atoms considered here. Table IVgives the interaction energies in the global and secondaryminima at the Hartree-Fock, CCSD and CCSD(T) levelsfor Li-NH , Mg-NH and Ca-NH . For all these systemsthe Hartree-Fock interaction energies are positive for theshallow secondary minima, indicating that the shallowwells are dominated by dispersion forces. At the globalminima, however, Mg-NH is repulsive at the Hartree-Fock level while the other two systems are strongly at-tractive. There is thus strong chemical bonding in Li-NH and Ca-NH that is absent in Mg-NH .The qualitative differences between Mg and the otheratoms can be understood if we consider how the energyof the highest occupied molecular orbital (HOMO) differsfor the different atom-NH systems. Fig. 2 shows the twohighest occupied molecular orbitals of each system. Aswe separate the monomers to infinity, these two orbitalsbecame HOMOs of the atom and the NH molecule. Forany alkali-metal atom, the strong A-NH bond can beexplained in terms of LCAO-MO theory as a chemicalbond of order one half, since we have a doubly occupiedbonding orbital and a singly occupied antibonding or- FIG. 2: The pattern of molecular orbitals for a) Li-NH , b)Ca-NH , c) Mg-NH near their global minima. The HOMOsof NH and of the metal atoms form bonding and antibondingorbitals. Note the small change in the HOMO energy for theLi-NH and Ca-NH systems and the much larger change forMg-NH . bital [see Fig. 2 a)]. However, this explanation does notapply to the alkaline-earth atoms, where the antibondingorbital is doubly occupied. The net bonding in Ca-NH arises because the bonding orbital is shifted down in en-ergy considerably more than the antibonding orbital isshifted up. Conversely, in Mg-NH , the contributionsfrom the bonding and antibonding orbitals are closelybalanced. The difference can probably be attributed tothe participation of np orbitals; as shown in Table I, the S → P splitting is considerably smaller in Ca than inMg. Thus Mg-NH is bound mainly by dispersion forceswhereas Ca-NH has substantial chemical bonding.Different considerations apply to the Be atom, which isa notoriously difficult case for electronic structure theory[55]. Although the potential energy surfaces are qual-itatively similar for the Be-NH , Ca-NH and Sr-NH systems at the CCSD(T) level, the origin of the strongbonding is probably different in Be-NH . In this casethe Hartree-Fock and CCSD potential energy curves for θ = 0 ◦ show a double-minimum structure, with a shal-low long-range minimum separated from the global min-imum by a barrier. This suggests a sudden change inchemical character as the Be atom approaches N. At theHartree-Fock level the maximum has an energy of 730cm − at R = 4 . a . The long-range minimum at theHartree-Fock level is 18.4 cm − deep at R = 9 . a ,while at the CCSD level it is 138 cm − deep at R = 6 . a . Despite this peculiar behavior, the CC calculationsshowed no convergence problems or unusually large T1diagnostics. However, our results for Be-NH disagreewith those of Cha lasi´nski and coworkers [34], who carriedout fourth-order Moller-Plesset (MP4) calculations andfound a global minimum that corresponds to the outerminimum on the CCSD potential energy curve. They didnot find the inner minimum, which turned out to be theglobal minimum in our calculations.As mentioned before, the feature of the potential en-ergy surfaces that is important for elastic/inelastic col-lision ratios is the anisotropy. In order to understandthe origin of the anisotropies better, we carried out ad-
0 30 60 90 120 150 180−3000−2000−1000 θ (deg) E ( ) e l s t ( c m − ) FIG. 3: Electrostatic energy of Na (squares) and Mg (circles)atoms interacting with NH as a function of θ for R = 6 a .The energy is averaged over χ .
0 30 60 90 120 150 180400050006000 θ (deg) E ( ) e x c h ( c m − ) FIG. 4: First-order exchange energy of Na (squares) and Mg(circles) atoms interacting with NH as a function of θ for R = 6 a . The energy is averaged over χ . ditional calculations based on symmetry-adapted pertur-bation theory (SAPT). Figures 3, 4, and 5 show the elec-trostatic, first-order exchange and dispersion componentsof the interaction energy V for Na-NH and Mg-NH ,averaged over χ as in Eq. (3). The calculations wereperformed at a fixed R value of 6 a , which is in anattractive region for θ = 0 ◦ and a repulsive region for θ = 180 ◦ . Figs. 3, 4 and 5 show clearly that it is thefirst-order interaction energy that is responsible for mostof the anisotropy in the valence overlap region. This iscaused by a very large difference between the electrostaticattraction near the lone-pair site and near hydrogen sites(see Fig. 3). This difference is significantly larger thanthat in the exchange energy. The anisotropy of the dis-persion interaction (plotted in Fig. 5), is even weaker.The anisotropy V of all three components of the inter-action energy with respect to χ is shown for Na-NH andMg-NH in Fig. 6. The exchange energy is very stronglyanisotropic, especially for Mg-NH . The large differencein exchange energy between Mg-NH and Na-NH can beexplained by the closed-shell character of the Mg atomand the much stronger Pauli repulsion between hydro-gens of NH and Mg. The electrostatic and dispersioncontributions to V are much more similar for the twosystems.
0 30 60 90 120 150 180−2500−2000−1500−1000 θ (deg) E ( ) d i s p ( c m − ) FIG. 5: Dispersion energy of Na (squares) and Mg (circles)atoms interacting with NH as a function of θ for R = 6 a .The energy is averaged over χ . θ (deg) ∆ E ( c m − ) electrostaticdispersionexchange
0 30 60 90 120 150 180−20005001000 θ (deg) ∆ E ( c m − ) FIG. 6: Anisotropy of the electrostatic, exchange and disper-sion contributions to the interaction energy, with respect torotation about the C axis of NH , as a function of θ , for R = 6 a , for Na-NH (upper panel) and Mg-NH (lowerpanel). D. Xe + NH interaction All the metal-NH systems investigated above have dis-appointingly large anisotropies. It is likely that all ofthem will exhibit large inelastic cross sections for anyinitial state where inelasticity is possible. We there-fore decided to consider other possible collision partnersfor sympathetic cooling of NH . Barker [56] has sug-gested an experiment in which Xe is first laser-cooledin its metastable P state and then transferred to itsground S state by laser excitation followed by sponta-neous emission. Since ground-state Xe has a fairly largedipole polarizability, it can be held in an optical dipoletrap and might be used for sympathetic cooling. In thissubsection we investigate the Xe-NH interaction in orderto evaluate its potential in this respect.Interactions between noble gases and ammonia havebeen studied extensively. The interaction between Heand NH is important in understanding the spectroscopyof NH molecules in helium nanodroplets [57]. The mostrecent ab initio calculations of Hodges and Wheatley[58, 59] gave a global minimum about 33 cm − deep at R = 6 a , θ = 90 ◦ and χ = 60 ◦ . The interaction of Arwith NH has been studied even more extensively, bothexperimentally [60] and by ab initio methods [61, 62]. In-version of vibration-rotation-tunnelling spectra [60] gavea minimum 147 cm − deep at R = 6 . a , θ = 97 ◦ and χ = 60 ◦ , while the ab initio MP4 (fourth-orderMøller-Plesset) calculations of Tao and Klemperer [62]gave a global minimum 130 cm − deep at R = 6 . a , θ = 90 ◦ and χ = 60 ◦ . The Ne-NH system was inves-tigated through MP4 calculations by van Wijngaardenand J¨ager [63], who obtained a global minimum 63 cm − deep at R = 6 . a , θ = 90 ◦ and χ = 60 ◦ . For Kr-NH , Cha lasi´nski et al. [64] obtained a global minimum108 cm − deep at R = 7 . a , θ = 100 ◦ and χ = 60 ◦ .However their results were based on calculations at theMP2 level and may not reproduce the dispersion energyaccurately.Fig. 7 shows the interaction potential for Xe-NH fromour CCSD(T) calculations. The potential energy sur-face differs qualitatively from those for metal-NH po-tentials studied in the previous subsection, and behavesanalogously to those for other Rg-NH systems. The V surface for Xe-NH has only one minimum, 173.5 cm − deep, at R = 7 . a and θ = 66 ◦ . The global mini-mum for the non-expanded surface is 196.8 cm − deep,at R = 7 . a , θ = 81 ◦ and χ = 60 ◦ . There are saddlepoints at both C v geometries. For θ = 0 the saddle pointis 166.2 cm − deep at R = 7 . a , while for θ = 180 ◦ the saddle point is 134.1 cm − deep at R = 7 . a . Themajor binding arises from the dispersion energy, and atthe Hartree-Fock level we observe only a small attraction(a few cm − ) at large distances, due to weak inductionforces which behave asymptotically as − C R − . Nearthe Van der Waals minimum predicted by CCSD(T), theSCF energy is repulsive.The V surface for Xe-NH system thus has ananisotropy of only about 60 cm − between the poten-tial minimum and the higher of the two saddle points.This is considerably smaller than for Rb-OH or any ofthe metal-NH systems studied here, but still substan-tial compared to the rotational constant of NH , b = 6 . − for rotation about an axis perpendicular to C . E. Long-range forces
Long-range forces are very important in cold and ultra-cold collisions. We therefore carried out separate calcu-lations of the Van der Waals coefficients for the interac-tions. The isotropic C , and anisotropic C , dispersion − − − − − − − − − − − − − − − − − − − − − − − − − − − − − θ R ( a )
0 30 60 90 120 150 18067891011 (deg) − − − − − − − R ( a ) θ (deg) FIG. 7: The interaction potential of Xe-NH from CCSD(T)calculations: V ( R, θ ) component (upper panel) and V ( R, θ )component (lower panel). Contours are labelled in cm − . Toaid visualization, V is plotted only in the energetically acces-sible region defined by V < coefficients for the interaction of atom A and symmetrictop molecule B may be written in terms of the dynamicpolarizabilities of the monomers, evaluated at imaginaryfrequencies, C disp6 , = 3 π Z + ∞ α A ( iu )¯ α B ( iu )d u ; C disp6 , = 1 π Z + ∞ α A ( iu )∆ α B ( iu )d u, (4)where ¯ α = (2 α xx + α zz ) is the isotropic polarizabilityand ∆ α = α zz − α xx is the polarizability anisotropy. Theinduction contributions to the Van der Waals coefficientsare C ind6 , = C ind6 , = α A µ , (5)where the dipole moment µ is 0.579 ea for NH [65].The integrals in Eqs. 4 were evaluated using themethod given by Amos et al. [66]. The dynamic po-larizabilities of NH were obtained using coupled Kohn-Sham theory with the asymptotically corrected PBE0functional [67] and d-aug-cc-pVTZ basis sets. To getthe dynamic polarizabilities for the alkali-metal atoms,we adjusted the fraction of exchange, exact exchange andcorrelation fraction in the PBE0 functional in such a way TABLE V: Van der Waals dispersion and induction coeffi-cients for A-NH and Ae-NH systems. All values are inatomic units, E h a . C disp6 , C disp6 , C ind6 , = C ind6 , Li 224 7.2 55.0Na 258 7.4 54.3K 378 11.6 98.2Rb 416 12.5 106.9Be 121 2.3 12.7Mg 200 4.4 24.0Ca 342 8.4 53.4Sr 413 10.2 67.4Xe 161 0.94 9.1 as to recover the atom-atom C coefficients (see Table I).Our coupled Kohn-Sham program does not allow us touse core potentials to calculate dynamic polarizabilities.For Rb we therefore performed all-electron calculationswith the pVTZ basis set of Sadlej [68] combined with theDouglas-Kroll approximation [69]. The dynamic polariz-abilities obtained in this way were tested by comparing C coefficients for A-Ar and A-Xe systems with those ob-tained by Mitroy and Zhang [70]. The maximum errorwas found to be +6.3% (for Na-Xe) while the averageerror is less than +3%. For alkaline-earth and Xe atomsthe frequency-dependent dipole polarizabilities were ob-tained from time-independent coupled-cluster linear re-sponse functions [71, 72].The resulting C coefficients are shown in Table V. Forthe alkali-metal and alkaline-earth atoms, the isotropicdispersion coefficients C disp6 , are fairly large because ofthe large atomic polarizabilities. The anisotropies in thedispersion coefficients are much smaller, because of thesmall polarizability anisotropy of NH (2.1 a ) comparedto its isotropic polarizability (14.6 a ). The inductionVan der Waals coefficients are large, and account for 10-25% of the total C , and 70-90% of the total C , . Itmay be noted that C disp6 , for Rb-NH is somewhat largerthan C disp6 , for Rb-OH [22]. As one might expect, the Xe-NH long-range interaction has slightly different charac- ter from the A- and Ae-NH systems. The C disp6 , coeffi-cient is still large, but the total anisotropy (in particularthe dispersion anisotropy) is much smaller. IV. CONCLUSIONS
We have investigated the intermolecular potential en-ergy surfaces for interaction of NH with several differentatoms that might be used for sympathetic cooling. Forinteraction with all the alkali-metal and alkaline-earthatoms, we found deep minima and strong anisotropies.The shallowest potential is for Mg-NH , but even therethe anisotropy in the well depth is close to 800 cm − .This is likely to cause strong inelastic collisions for allinitial states for which they are energetically allowed. Ac-cordingly, we consider that none of the alkali metals andalkaline earths are good prospects for sympathetic cool-ing of NH unless both the atoms and the molecules arein their lowest states in the trapping field. This suggeststhat sympathetic cooling would need to be carried out ineither optical or alternating current traps.A somewhat more promising system for sympatheticcooling is Xe-NH , for which the global minimum is cal-culated to be 196.8 cm − deep at an off-axis geometry.The Xe-NH system is relatively weakly anisotropic, withthe saddle points for C v geometries only 30.6 and 62.7cm − higher than the global minimum. In future work wewill use the interaction potential to calculate low-energyelastic and inelastic cross sections, in order to predictwhether sympathetic cooling of NH by Xe is likely to befeasible.Even if sympathetic cooling proves to be impossible forthese systems, there is much to be learnt from collisionsbetween velocity-controlled beams of molecules and laser-cooled atoms. 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