Alkali Atom-Dimer Exchange Reactions: Na + Rb, BY DAVID J. MASCORD, PETER A. GORRY AND ROGER GRICE* Department of Theoretical Chemistry, Cambridge University, Cambridge CB2 1EW Received 5th May, 1976 Velocity analysis measurements of reactive scattering from the reaction Na + Rbt 3 NaRb + Rb are reported. The product translational energy distribution is close to that predicted by a long- lived collision complex model with an average product translational energy ITav - 30% of the total available energy. A limited range of angular forms for the differential reaction cross section is compatible with the experimental data. The total reaction cross section is large Q = 120 &- 20 A', indicating that a major fraction of collisions captured by the long-range van der Waals attraction, leads to reaction.These features of the reactive scattering are discussed in terms of the form of the reaction potential energy surface, with particular regard to the orientation dependence of reaction. The exchange reactions of alkali atoms with alkali dimers M' + Mz 3 M'M + M (1) have previously been observed in a molecular beam angular distribution study1 of reactive scattering for M' = Na, K and M2 = Cs2, Rb2, K2. These measurements indicated that the reactions have large total reaction cross sections Q - 100 A2 and dispose only a small fraction -20% of the total available energy into product trans- lation. However, the measurements did not permit the form of the differential reaction cross section to be determined. In this paper, we give a brief account of a velocity analysis study of the reaction Na + Rbz + NaRb + Rb (2) which has been undertaken to improve the resolution of the differential reaction cross section.A detailed account of these velocity analysis experiments and their analysis has been given elsewhere., Large total reaction cross sections Q - 100 Hi2 are also indicated3 for the symmetrical exchange reactions with M = M' = Rb, Cs, in order to explain the nuclear spin relaxation times of alkali dimers observed in the n.m.r. of optically pumped alkali vapours. The potential energy surfaces for alkali atom-dimer exchange reactions offer an interesting example of interaction between three highly polarisable atoms, involving little change in the nature of the chemical bonding. Semi-empirical potential energy surfaces have been calculated both for the Li3 trimer4 and for all the triatomic com- bination~~ of Li and Na, and indicate the existence of bound triatomic molecules.Having only three valence electrons, the triatomic alkali systems also invite quantum mechanical calculation. Ab initio calculations on the Li, trimer have been performed using both the molecular orbital6 and the valence bond' methods. These calcula- tions confirm the stability of the Li3 molecule but raise a question as to its most stable geometry and electronic state, The pseudopotential method has been employed 8-10 for calculations involving heavier alkali atoms. The Na,, K3 and Cs, trimers have been observed" in molecular beams from supersonic nozzle expansions of the alkali256 ALKALI ATOM-DIMER EXCHANGE REACTIONS : Na+Rb, vapour.The e.s.r. spectra of matrix isolated Na, and K3 indicate l2 that these trimers are chemically bound rather than being simply van der Waals complexes. Finally, extensive Monte Carlo trajectory c a l c ~ l a t i o n s ~ ~ ~ ~ ~ have been carried out on the semi- empirical potential energy surfaces' for the reactions Li + Li2, Li + Na2 and Na + Liz. EXPERIMENTAL The apparatus employed in these experiments, shown in fig. 1, is the same as that used in the angular distribution study,l augmented by a small slotted disc velocity selector. A supersonic nozzle beam source gives a mixed beam of Rb atoms and Rb, dimers with -25 mole per cent15 of Rb, dimers. An inhomogeneous magnetic field deflects the Rb atoms, FIG.1.-Schematic diagram of the apparatus: A, alkali dimer oven; B, alkali atom oven; D, detector and velocity selector; F, beam flag; M, magnet; S, collimating slits; T, tantalum radiation shield; LN2, liquid nitrogen cooled cold shield; H20, water cooled cold shield. while the undefiected Rb, dimer beam passes into the scattering chamber with a residual impurity Q 10% of Rb atorns.l6 The Na atom beam effuses from the multichannel source of the cross beam oven. The head temperature of the oven is maintained -100 K hotter than the body in order to suppress the density of the Na2 dimer impurity to 70.2% of the Na atom density. The detector, consisting of a quadrupole mass spectrometer equipped with a surface ionization ion-source, is mounted on the rotatable lid of the apparatus.The small velocity sele~tor,'~ mounted in front of the detector, has a very compact disc assembly. This permits the detector to be maintained in the same position as that used in the angular distribution study1 and does not significantly restrict the angular range accessible to scattering measurements. The velocity distributions of the incident beams were measured with the velocity selector in the same configuration as that used for the scattering measurements. The Rb, dimer beam has a narrow velocity distribution, -20% full width at half maximum intensity, and a peak velocity, vpk = 655 m s-'. The Na has a Maxwell-Boltzmann velocity distribution with a most probable number density velocity, a, = 753 m s-l. RESULTS Since the Pt/W alloy ribbon of the surface ionisation ion-source ionises both reactively scattered NaRb, Rb and elastically scattered Rb2 with near unit efficiency to yield Rb + ions, the quadrupole mass spectrometer cannot directly distinguish between reactive and elastic scattering.This distinction may be achieved l v 2 by kinematic discrimination, since a heavy Rb2 dimer undergoes only a small deflection in laboratory coordinates due to an elastic collision with a light Na atom. ThusDAVID J. MASCORD, PETER A . GORRY AND ROGER GRICE 257 Rb2 elastic scattering is constrained close to the Rb, beam at laboratory scattering angle 0 = 90"; the Na beam is at 0 = 0". The reactive scattering by eqn (2), separates the heavy Rb atoms. Thus the NaRb, Rb reaction products can be disposed over a much wider range of laboratory scattering angles. Detailed analysis shows that Rb+ intensity at laboratory scattering angles 0 3 100" can be identified as reactive scattering.Velocity distributions of Rb+ intensity measured at 0 = loo", 105", 107.5", 110" are shown in fig. 2 and 3. The velocity distribution measurements and the previous angular distribution measurements' have been analysed, by the stochastic method of Entemann l8 which determines the differential reaction cross section in the factorised form Zcn,(9,u) = T(9)U(u) = T(B)uP(E') (3) lo-' -. U t I tr- L- \ 9 100 300 500 700 900 1100 0 1300 1500 laboratory velocity / m s-' FIG. 2.-Laboratory flux density velocity distributions of Rb+ for Na + Rb2 showing fits calculated for loose complex models. Loose complex, - a = 0.834, b = 0.083, * a = 0.5, b = 0.25; loose isotropic, --- a = 1.0, b = 0.0.(a) 0 = 110," (b) 0 = 107.5", (c) 0 = 105", (d) 0 = 100".258 ALKALI ATOM-DIMER EXCHANGE REACTIONS : Na+Rb, FIG. 3.-Laboratory flux density velocity distributions showing fits calculated for a tight stripping model (-, a = 0.5, b = 0.25) and a loose sideways peaked complex (---, H* = 70"). (a) 0 = 110", (b) 0 = 107.5", (c) 0 = 105", (d) 0 = 100". where 8 and u are the centre-of-mass scattering angle and velocity of the detected product and E' is the product translational energy. The RRKM-AM of scattering via a long-lived collision complex requires that the angular function T(8) be symmetric about 0 = 90" and gives the product translational energy distribution in the form P(E') = (E'/B',)2'3(Etot - E')' E' < B', = (E,,, - E')4 E'> B', (4) where B', = (,U/,U')~/~(C/C')~EDAVID J .MASCORD, PETER A . GORRY AND ROGER GRICE 259 and p,p' denote the reactant and product reduced masses; C,C' the van der Waals coefficients. The table lists the values, appropriate to the most probable Newton diagram, for the reactant translational energy E, the total available energy Etot and the maximum centrifugal barrier to complex dissociation Btm. The total available TABLE EN ENERGIES (kJ mol-l) FOR MOST PROBABLE NEWTON DIAGRAM energy is given by Etot = E + AD, where AD, denotes the reaction exoergicity. As shown in fig. 2, the RRKM-AM model for a loose complex (parameter q = 1) gives a good fit to the velocity distribution data, when the angular functions T(8) shown in fig.4 are employed, which range from sharply forward-backward peaked to iso- tropic. The best fit is found for a mildly forward-backward peaked angular function. t + I 1 1 1 1 1 1 1 1 1 1 1 1 0 30 60 90 120 150 30 60 90 120 150 centre of mass scatterkg cingle 8 FIG. 4.-Angular functions T(0) employed in the stochastic kinematic analysis : (a) isotropic a = 1 .O, b = 0.0; (b) mildly peaked complex n = 0.834, b = 0.083; (c) sharply peaked complex n = 0.5, b -- 0.25; (d) sideways, 0" = 70". For the functional forms see ref. (2). Since the surface ionization detector yields Rb" from both NaRb and Rb reaction products, which have nominally equal mass, conservation of linear momentum in centre-of-mass coordinates demands that the observed Rb + intensity distribution be nearly symmetric about 6 = go", independent of the lifetime of the collision complex.Consequently, the observed reactive scattering data will be insensitive to the symmetry of the differential reaction cross section about 6 = 90". Fig. 3 shows that a sharply peaked stripping differential cross section with NaRb recoiling in the forward hemi-260 ALKALI ATOM-DIMER EXCHANGE REACTIONS : Na+Rb, sphere I9 < 90" and Rb in the backward hemisphere 19 > 90°, also gives a reasonable fit to the data, when the parameter q = 2 appropriate to a tight complex is employed. However, a rebound distribution with Rb recoiling forward and RbNa backward does not give an acceptable fit.2 Fig. 3 also shows that a sideways peaked differential cross section, as illustrated in fig.4, does not give an acceptable fit to the velocity t ( a ' 0 ? I - .- s o-*C a c Y c' 1 0 0 t o I 1 110 108 106 104 102 100 98 laboratory scattering angle @ FIG. 5.-Fits to laboratory angular distribution of Rb+ for Na + Rbz, for three loose complex models (a) loose isotropic, a = 1.0, b = 0.0; (b) loose complex, a = 0.834, b = 0.083; (c) loose complex, a = 0.5, b = 0.25 (see fig. 2) and (d) tight stripping model a = 0.5, b = 0.25 (see fig. 3). distribution measurements, being shifted to lower velocity. The three long-lived collision complex models and the stripping model, which give the best fits to the velocity distribution measurements, are compared in fig. 5 with the angular distribu- tion measurements,l and give good fits in all cases.Consequently, the angular distribution measurements do not distinguish further between these differential reac- tion cross sections. Thus the velocity and angular distribution measurements doDAVID J . MASCORD, PETER A . GORRY AND ROGER GRICE 26 1 not permit the lifetime of the collision complex to be determined, and admit a range of possible angular functions between sharply peaked and isotropic. The best fit to the data is obtained for a long-lived collision complex with an angular function exhibiting mild forward and backward peaks and a loose transition state for complex dissociation. The data determine the admissible form of product velocity distribution U(u) much more closely than the angular function T(8) in the differential reaction cross section.A product translational energy distribution P(E') close to that predicted by the RRKM-AM model of eqn (4), ( 5 ) is required in fitting the data. The energies listed in the table for a loose complex, corresponding to the most probable Newton diagram, indicate that the peak of the P(E') distribution given by B', occurs at f'pk = 0.16 and the average product translational energy Elav at ffav = 0.33, where f ' = E'/Etot denotes the fraction of the total available energy disposed into translation. These values show that the product NaRb molecules have substantial internal energy. If the average internal excitation Ei,, given in the table, were mainly vibration, it would correspond to -25% of the NaRb bond energy. The limit, in which the total available energy is disposed into NaRb vibration, corresponds to -40% of the NaRb bond energy.The total reaction cross section Q can also be determined's2 by normalizing the observed reactive scattering to the small angle Rbz elastic scatter- ing. This procedure l8 involves integration over the differential reaction cross section and is relatively insensitive2 to the assumed angular function T(9) in these experiments. An average value is estimated2 as Q = 120 & 20 A2. DISCUSSION The large magnitude of the total reaction cross section Q - 120 A2 for Na + Rb,, indicates that reaction must occur in collisions at large impact parameters b - 6 A, where the intermolecular potential is due to van der Waals interaction. The capture cross section Qcap for collisions, which are attracted by the van der Waals potential and spiral into small internuclear distance, is given20 by Qcap = zb&b = (3~/2)(2C/E)"~ where borb is the orbiting impact parameter.This yields Qcap - 160 A2. Thus collision trajectories drawn into small internuclear distance by the van der Waals interaction lead to reaction for a majority of initial Rb2 orientations. This gives an estimate of the steric factor2' for reaction, p - 0.75. The form of product translational energy distribution for Na + Rb2, which is close to that predicted by the RRKM-AM modellg for a long-lived collision complex, indicates that the potential energy surface remains attractive at smaller internuclear distance inside the range of van der Waals interaction. However, our inability using present data to determine the relative magnitude of the reactive intensity in the forward direction compared with the backward direction, precludes a determination of the depth (or even the existence) of a well in the potential energy surface for the exoergic Na + Rb, reaction. The potential energy surface for the Li + Na, reaction, for which semi-empirical calculations5 have been performed, should be similar to that for Na + Rb2, since both reactions have the similar exoergicities.Comparison of the semi-empirical calculations5 for Li3 with ab initio valence bond calculations' in the collinear con- figuration shows good qualitative agreement; potential well depth = 37 kJ mol-' (semi-empirical), 28 kJ mol-' (adjusted ab initio). Comparison with more limited molecular orbital calculations6 in the bent configuration indicates a similar level of262 ALKALI ATOM-DIMER EXCHANGE REACTIONS : Na+Rb2 qualitative agreement.Consequently, we may expect the semi-empirical potential energy surfaces for Li + Na2 to exhibit the correct qualitative behaviour. As fig. 9 of the angular distribution study1 illustrates, the collinear LiNaNa potential energy surface exhibits a shallow well (depth - 17.5 kJ mol-' with respect to products) at small internuclear distance, which blends smoothly into the van der Waals attraction without an energy barrier in either the entrance or exit valleys. A second well in the potential energy surface for the NaLiNa collinear configuration, is -8 kJ mol-' more stable than the well for the LiNaNa configuration.Moreover, the dynamics of the Li + Na, reaction should be similar to the Na + Rbz reaction, since the atomic masses are in similar proportion. Monte Carlo calculations have been performed by Whiteheadi4 using the semi-empirical Li + Naz potential energy surface. At the initial translational energy of the present experi- ments, they show a product angular distribution peaking equally in the forward and backward directions and a product translational energy distribution in close accord with the RRKM-AM model for a loose complex. These results are thus compatible with the best fit found in the kinematic analysis of the Na + Rb, experimental data. In addition, the Monte Carlo calculations find a probability of reaction P(b) -0.5 out to an impact parameter b - 8 A.This indicates a high steric factor p > 0.5 in accord with our experimental conclusions. It is of interest to examine the form of the potential energy surface for these alkali-atom dimer exchange reactions, in order to establish the reason for the high steric factors. Fig. 6 shows a contour map of the semi-empirical potential energy R / A FIG. 6.-Potential energy contour map for approach of a Li atom to a Naa molecule held at its equilibrium bond length. surface for approach of Li to Na, as a function of the orientation angle cc and distance of approach R, when the Na, bond length is held rigid at the equilibrium value. This potential energy map is attractive for all directions of approach, particularly for the collinear orientation a = 0". The potential energy surface for the collinear LiNaNa configuration mentioned above, develops from this map with cc = 0" when the Na, bond is allowed to extend, and is shown in fig.9 of our previous angular distribution study.' The map of fig. 6 suggests that approach in bent orientations a 40" would lead to reaction in the nearly collinear LiNaNa configuration. Fig. 7(a) shows the minimum energy profile on the contour map of fig. 6 for the Li atom to moveDAVID J . MASCORD, PETER A . GORRY AND ROGER GRICE 263 -1 4 - 18 -20 -22 -24 - 2 6 -28 -30 f- - -2E* _- 10 20 30 A0 50 SO 70 80 90 ct E+ Li -20 - -4 01 reaction path O\ ? P - FIG. 7.-Potential energy profiles for the Li + Naz surface. Energy zero refers to the Li+ Naz asymptotic energy, 0 Li, Na; (a) Li + Nap (fixed Na-Na bond length), (6) Li + Naz, (c) Li+Naz.-32 -34 -36 -38 ’264 ALKALI ATOM-DIMER EXCHANGE REACTIONS : Na+Rb2 round from the perpendicular (a = 90") to collinear (a = 0') orientation. This profile shows a monotonic decline, which would reorient a Li atom from a bent toward a collinear orientation on approach to a rigid Na, molecule. However, the effect on these conclusions of allowing the Na, bond to respond to the approaching Li atom must also be considered. This is likely to be most important in the more strongly bent configurations. Fig. 7(b) shows the minimum potential energy profile for perpendicular (a = 90") approach of a Li atom to an unconstrained Na, molecule. The attraction is much stronger than in the perpendicular approach to a rigid Na, molecule. The descent of this profile for the lower E- potential energy surface is interrupted by an upward cusp before descending sharply into the potential energy well corresponding to the collinear NaLiNa configuration.This cusp is a section through the lower cone of a conical intersection22 of the lower E - and upper E+ potential energy surfaces.' For perpendicular approach with C,, symmetry, the electronic wavefunction of the lower E- surface belongs to the representation ,Al for R > R, and to ,B2 for R < R,, where R, denotes the internuclear distance at the intersection. The continuation of the electronic wavefunctions onto the cone of the upper E , potential energy surface is shown by dashed curves in fig. 7(b). Clearly, a trajectory for Li approaching Na, which passes through or very close to such a point of intersection,? must propagate diabatically onto the upper E+ potential energy surface and experience strong repulsion. However, for directions of approach other than cc = 90" the system no longer has CZ0 symmetry.The electronic states of both the upper and lower potential energy surfaces belong to the representation ,A'. The surfaces have an avoided intersection corresponding to the flanks of the double cone. Consequently, most collisions approaching with a broadside orientation (say a 3 50') will adiabatically follow the lower E- potential energy surface, " side-stepping " the conical inter~ection,~~ and reach the potential well corresponding to the NaLiNa configuration, i.e., insertion into the Na, bond. The propensity of trajectories to reach this most stable NaLiNa configuration is illustrated by the potential energy profile of fig.7(c). This shows the minimum potential energy as the Li atom is brought round from the collinear LiNaNa con- figuration with increasing a, just as in the profile of fig. 7(a) but allowing the Na, bond to extend. The profile now shows only a hump before descending rapidly into the potential energy well of the NaLiNa configuration. Thus even trajectories which approach initially in the near collinear orientation, may insert the Li into the Na, bond if substantially bent configurations are attained during the motion. Indeed, trajectories which are fairly long-lived may involve multiple transitions between the potential energy wells corresponding to the LiNaNa, NaLiNa and NaNaLi configura- tions.Such mobility seems likely to be most pronounced for the symmetrical alkali atom-dimer exchange reactions, where M = M' and all the potential energy wells are equivalent. In this case, the collision complex would have a very " floppy" structure and this should be apparent in the Monte Carlo calculation^.^^*^^ Thus, it is apparent that the form of the potential energy surfaces for alkali atom- dimer exchange reactions favours reaction for all orientations of approach, except a narrow region near a = 90". This is in accord with the high steric factor estimated from our experiments. The steric factor for reaction may be less than unity partly because of dynamical motion which reflects some trajectories back into the entrance valley of the potential energy surface.This will be of increasing importance at higher initial translational energies when trajectories increasingly sample the repulsive wall 7 The locus of all such points of intersection describes a line in the four dimensional space of the full potential energy surface. The line is generated for ci = 90" by the variation of R, with the Na2 bond length.DAVID J . MASCORD, PETER A. GORRY AND ROGER GRICE 265 of the potential energy surface at small internuclear distances. The Monte Carlo calculation^^^ for Li + Na, exhibit just this effect. At low initial translational energies the reaction probability P(b) > 0.5 is high but decreases with increasing initial translational energy at all but the smallest impact parameters.The topographical structure of the Li + Na2 potential energy surface is shown by the contour map of fig. 8, which is drawn in the coordinate system commonly FIG. 8.-Normal coordinate representation of the Li + Naz potential energy surface. Dashed line shows qualitatively the locus of profile (6) of fig. 7; dotted line profile (c). to display Jahn-Teller distortion of molecules of D,,, symmetry. The coordin- ates Ql,Q2, are the degenerate components of the E’ normal mode of vibration of a D,,, molecule, which treat all the bonds of the Li + Na, system equivalently. This contour map for Li + Na, may be compared with the same representati~n~~ of the H + H2 potential energy surface. This demonstrates that the H + H2 reaction is confined to the collinear configuration at low energies because the conical intersection for H, occurs at energies well above that of the H + H2 asymptotic energy.Conse- quently the lower cone provides a substantial barrier to reaction in the broadside orientation for H + H2 in contrast to the Li + Na, reaction where the tip of the cone lies below the Li + Na, asymptotic energy. of such conical intersections confirms that this is the dominant effect in determining the orientation dependence of atom-diatomic molecule reactions at low energy. Examination of other Support of this work by the Science Research Council is gratefully acknowledged.266 ALKALI ATOM-DIMER EXCHANGE REACTIONS : Na+Rb, J. C. Whitehead and R. Grice, Furuduy Disc. Chem. SOC., 1973, 55, 320, 374. D. J. Mascord, H.W. Cruse and R. Grice, MoZ. 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