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Cold Chemistry with Ionic Partners: Quantum Features of HeH+(1Σ) with H(1S) at Ultralow Energies S. Bovino, M. Tacconi, and F. A. Gianturco* Department of Chemistry, “Sapienza” University of Rome, P. le A. Moro 5, 00185 Rome, Italy ABSTRACT: Quantum reactive calculations are presented for an ionatom reaction involving the HeH+cation and its destruction via a barrierless interaction with H atoms. The range of collision energies considered is that of a cold trap regime (around and below millikelvin) where the ionic partner could be spatially confined. Specific resonant features caused by the interplay of the strong ionic interaction with the very slow partners’ dynamics are found and analyzed. Indications are also given on the consequences of the abstraction mechanism that acts for this reaction at low energies.
I. INTRODUCTION The interest in the production and in the transformations of cold and ultracold molecular samples with temperatures well below 1 K and 1 mK has certainly increased in the past decade. The initial thrust on this area has come from the atomic optical and molecular physics community because of the great variety of applications that such samples are expected to have.15 At such low temperatures, in fact, quantum effects are expected to dominate the scattering processes because the collisions are strongly affected by long-range forces and orbiting resonances, which in turn provide, through cross section behavior, detailed information on the reactive region of the potential energy surface driving the reaction.6,7 Many of the familiar, gas-phase chemical reactions occur as thermally activated processes and therefore their behavior could be largely understood using conventional transition-state theory.8,9 On the other hand, at low and ultralow temperatures the thermal fluctuations that produce collisional events to occur with sufficient energy to activate a complex become increasingly rare. Hence, even the most modest barriers prevent most chemical reactions from taking place,9 unless nonclassical effects arising from quantum mechanical tunneling are dominating the process. The low- and ultralow-T regimes, therefore, have to be characterized by barrierless reaction pathways between reactants, as is often the case for ionic reactions with long-range, attractive polarization forces. A distinguishing feature of these types of systems is the existence of deep wells in the reaction regions, which can then describe short-lived chemical complexes often formed by insertion of an atom or molecule into an existing chemical bond. The astrophysical importance of barrierless reaction processes at temperatures below a few kelvins has further driven the development of accurate studies of both ionic and neutral reactions within astrochemical networks10 and also enticed the experimental setups of cold traps for ionic partners down to a few kelvins using radiofrequency fields11 or using a 22-pole trap.12 r 2011 American Chemical Society
As the temperature is lowered into the millikelvins range, a phase transition may occur to produce “Coulomb Crystals”13 in which ordered structures of ions are observed with very low translational temperatures. In the case of molecules, further internal cooling of the trapped species has been suggested to come from He-buffer cooling14 or using spectroscopic pumping schemes.15 It turns out that the ionic reactions involving one neutral partner species that could be more easily studied by taking advantage of the “Coulomb crystal” environment are those involving the direct reaction of the laser cooled atomic ions, although more recent studies have looked at other reactions that involve sympathetically cooled molecular ions.13,16 One of the earliest experiments had CaO+ ions produced by reactions of laser-cooled Ca+ ions and neutral O2,17 while also a small range of direct reactions has been studied, e.g., that of Mg+ + H2 f MgH+ + H and its deuterated anologues.1820 In the present computational work, therefore, we have decided to address the analysis of a very similar reaction, i.e., that of He+ reacting with H2, which we intend to examine in the low- and ultralow-energy regimes, therefore considering the exothermic direction of that process: HeH+ + H f He + H2+. This reaction also bears importance within the chemical network that is essential to know in molecular abundance studies of the early universe evolution21,22 and which we have also considered in recent work.23 The following section (II) will therefore describe some of the features of the reactive potential energy surface (RPES). The latter RPES has been described many times in the literature and therefore we will only sketch those features that help readers understand the use of it in our present work. Section III will report some details of the computational quantum method, also given by us in earlier work, and section IV shall present the results. Section V will finally give our conclusions. Received: April 4, 2011 Revised: June 14, 2011 Published: June 21, 2011 8197
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Figure 1. Computed energy profile of the RPES of reaction 2. See main text for details.
II. REACTIVE POTENTIAL: AN OUTLINE Because of the diffused interest in the reaction involving HeH+: He + H2 + f HeH+ + H
ð1Þ
the corresponding RPES has been computed several times over the years. The earliest calculations involved the collinear geometry of the triatomic species,24 while the use of the semiempirical diatomics-in-molecules method allowed earlier authors to construct later on a surface for the nonlinear geometries,25 which was further fitted and analyzed in an ensuing paper.26 Several different attempts followed, aimed at generating a more accurate RPES using configuration interaction (CI) methods2730 and the quality of the calculations was progressively extended up to the most recent results that included the Davidson corrections and produced a fitted form with a rootmean square (rms) deviation of 4 meV.31 The most recent revisitation of the RPES of eq 1 was done using a full CI extension32 of the earlier set of configurational points generated by ref 33, finding very little differences with those earlier data in reference 33, while also generating a many-body expansion fit with an rms deviation of 6 meV. The astrophysical interest chiefly resides on the inverse of reaction 1, whereby one needs to analyze the chemical destruction mechanism of the HeH+ molecules formed by radiative association34 and stimulated radiative association, reacting with the H atoms of the early universe environment35 HeH+ + H f He + H2 +
ð2Þ
The above process was found to be markedly exothermic, leading to rapid depletion of the primary molecule, which, however, turns out to be able to survive more efficiently than its other ionic counterpart: the LiH+ species.23 In the present study of reaction 2 we have employed the latest fitting of the RPES given by ref 33, the same which has been used by us in the quantum study of its abundances in early universe environments.23
The data of Figure 1 give us a pictorial view of the energy path along the most reactive direction of the process, i.e., the collinear arrangements of the triatomic species.33 A perusal of the behavior of the exothermic reactive channel shown in the figure allows us to verify the following features of this RPES: the energy gain produced by destruction of the polar cation is rather substantial (0.77 eV), while the intermediate complex is also seen to form with an energy gain from the asymptotic products of about 350 meV, thereby suggesting the formation of several bound states within the intermediate complex.28 The relative consequences of such properties will be further discussed in the following sections.
III. QUANTUM REACTIVE METHOD The method employed in our calculations is based on the use of a negative imaginary potential (NIP), as originally introduced by Baer and co-workers.36 The leading idea is to use the NIP to convert a multiarrangement reactive system into a system where a subreactive (inelastic) problem is apparently solved while a reactive system is really being analyzed. Our approach combines the use of a NIP with a coupled states (CS) approximation dynamics.37 Within the CS approximation the orbital angular momentum operator ^l2 is assumed to be equal to (^J 2 + ^j2 2^J z^jz)/2μR2 so that the couplings between different projections of the rotational angular momentum Ω along the body fixed axis are neglected, thereby reducing the dimensionality of the problem. The method provides a good accuracy with less computational cost and in the j = 0 case is equivalent to the full coupled-channel (CC) formulation, once only inelastic processes are considered: our present handling of the reaction through the NIP algorithm is indeed transforming the reactive dynamics into a ”flux modified” inelastic collision. We have to solve the time-independent Schr€odinger equation in Jacobi coordinates at fixed values of total angular momentum J and Ω: ^ J;Ω ðR;r;ϑÞ ¼ EΨJ;Ω ðR;r;ϑÞ HΨ 8198
ð3Þ
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Figure 2. Computed reactive probabilities with the method of section II down to the ultralow energy range. See main text for details.
where ^2 ^2 ^^ ^2 ^ ¼ 1 ∇R 2 + J + j 2J z jz 1 ∇r2 + j H 2μ 2m 2μR 2 2mr 2 NIP + V^ ðr;R;ϑÞ + V^ ðr;R;ϑÞ
ð4Þ
The V^ (r,R,ϑ) term in the equation describes our absorbing negative imaginary potential. Several model forms of NIP have been proposed over the years.36,3841 One of the most usual forms is given by a monomial of order n: r rmin n rmin e r e rmax V^ NIP ðr;R;ϑÞ ¼ iu0 rmax rmin
are enforced, thereby providing in the end the scattering matrix (S-matrix). Because of the flux-absorbing effect of the NIP, the resulting final S-matrix is nonunitary and its default to unitarity gives us the reactive probability for obtaining any final state of the products: JΩ
Pða f allÞ ¼ 1
NIP
^hR̅ ja ðr;ϑ;R̅ k Þ ¼ εa ja ðr;ϑ;R̅ k Þ k
ð6Þ
where the subscript “a” is a collective index that indicates a given roto-vibrational state of the target molecule and ^2 ^hR̅ ¼ 1 ∇r 2 + j + V^ ðr;ϑ;R̅ k Þ k 2m 2mr 2
ð7Þ
The total wave function ΨJ,Ω(R,r,ϑ) is then expanded over the ja(r,ϑ;R k) functions: ΨJ;Ω a ðr;R;ϑÞ ¼
̅ k Þ YJΩ ðϑ;φÞ ∑a GJ;Ω a ðRÞ ja ðr;ϑ; R
ð8Þ
J,Ω where YΩ J (ϑ,ϕ) are the spherical harmonics and Ga (R) are the unknown translational scattering wave functions. Substituting eq 8 in eq 3, we obtain the usual coupled-channel equations that are solved using an R-matrix propagator, generalized to deal with the complex algebra induced by the presence of the NIP potential form (as discussed in refs 43 and 44). The radial R-matrix propagation is extended from the origin out to the asymptotic region, where the asymptotic matching conditions
ð9Þ
From the reaction probability one can in turn obtain the reactive cross section in a straightfoward manner by applying the formula: σ ða f allÞ ðEÞ ¼
ð5Þ To solve eq 3, we divide the range of integration over the R coordinate into N sectors. At the midpoint (R k) of each sector we construct a local roto-vibrational adiabatic basis set by solving the molecular Schr€odinger equation:
2 ∑b jSJΩ ab j
π ð2ja + 1Þka 2
∑J ∑Ω ð2J + 1ÞPaJΩf all
ð10Þ
IV. REACTION FEATURES AT ULTRALOW ENERGIES As already discussed in ref 23, our computed cross sections for the depletion reaction 2 turn out to agree rather well with existing experiments45 and earlier extrapolations of those data.46 We therefore carried out additional calculations, under the same numerical details reported in ref 23, to generate depletion cross sections down to energies of a few microelectronvolts: the results of Figure 2 report the behavior of the reaction probabilities as a function of the initial vibrational state of the molecular ion HeH+. One clearly sees, from the computed probabilities for the process of eq 2, the likely effects from starting the reaction with the HeH+ molecule in different vibrational states. The corresponding probabilities, in fact, change very little as the internal energy content is increased and the presence of oscillatory structures for each initial channel (which we shall discuss later on) remains largely the same. This result indicates, at least in qualitative terms, that the dominant reaction path is very likely to be an extraction mechanism, whereby the H2+ is formed with the ”external” H+ of the ionic partner by the incoming H atom, with little influence from the He “spectator”, therefore mostly involving a process that depends only marginally from the vibrational state of the initial, ionic molecular partner. This aspect has been discussed in greater detail for the endothermic channel of eq 1, 8199
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Figure 3. Computed depletion cross sections of reaction 2 down to the millikelvin regimes. The reported experiments are from ref 45, and the extrapolated data, from ref 46.
Figure 4. Computed reaction probabilities for process (2) at ultralow energies. The dominant J = 0 opacity contribution is shown in the figure.
i.e., for the H2+ destruction reaction, but less frequently for the exothermic, HeH+ destruction counterpart,28,31 so that the present finding constitutes a useful addition to our overall understanding of the reactive behavior of the present system. To further explore the cross sections down to the microelectronvolts of energy, we report in Figure 3 the (v = 0) reactive cross sections of the depletion process well beyond the range of the existing experimental data45,46 and closer to possible conditions in magneto-optical traps, as discussed in the Introduction. The calculations were carried out following the numerical details of the present method already given in ref 23, so we refer the reader to that work for more information on them. What we clearly see from those data is the appearance of a very strong, and
sharp, increase of the reaction cross sections around 100 mK of the temperature, i.e., a range of possible values for experimental detection.1113 We also see that a broad maximum is observed around a region centered at about 10 meV and we wish to analyze both features more specifically by looking at the corresponding reactive probabilities associated with the dominant angular momentum of the low- and ultralow-energy dynamics: the J = 0 contribution to the reactive opacity.7 From the data reported by Figure 4, where the energy range is extended down to 106 eV, we see even more dramatically the presence of the strong, narrow peak appearing in the reaction cross sections at energies around a few microelectronvolts, i.e., at a possible temperature where trap processes could be analyzed. 8200
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Figure 5. Upper panel: reactive probabilities at vanishing kinetic energy for reaction 2. Lower panel: computed behavior of the associated real part of the scattering length. See main text for details.
It is interesting to note here that in a previous study on the behavior in the limit of zero kinetic energy of a chemical reaction involving neutral partners47 F + H2 f HF + H
ð11Þ
we had also found that modifications of the RPES induced via artificial mass changes were able to show the appearance of a similarly strong peak in the products’ channel at an energy of about 3 105 eV, i.e., very close to our findings in Figure 4. The analysis of the scattering matrix S(k), with k being the associated wavenumber48 for the process above, linked that behavior with the occurrence of a virtual state as the relative kinetic energy approaches zero.47,48 The latter quantity appears when a pole of the S(k) matrix exists in the nonphysical part of the Riemann surface (for Im(k) < 0).49 In the case of multichannel scattering, therefore, the virtual state may decay into one of the open channels and have a marked influence on the scattering observables (e.g., see ref 48). The associated scattering length for the multichannel problem becomes a complex quantity: a = R iβ, and the two parameters are then related to the full elastic S-matrix as k tends to zero50 R ¼ lim
kf0
ImðSel Þ 2k
β ¼ lim
kf0
1 ReðSel Þ 2k
ð12Þ
when R > 0 the associated pole of the S-matrix yields an energy location for which the real part of the latter is the binding energy of the least bound resonance, while for R < 0 the energy is that of the virtual state of least energy: R will therefore show divergent behavior when moving from one situation to the other.48,49 Analogously, the corresponding time delay51 Δt can be related to the above quantities in the limit of vanishing energy50 Δt ∼
2μ R k 1 4kβ
ð13Þ
Hence, we can see that, in the elastic scattering channels of the multichannel process, the time delay will be positive for a virtual state and negative for a resonant state.
Because we just established that virtual states of a reactive system will be seen from the presence of divergent behavior of R, we have carried out just such type of calculation for the analysis of the strong peak seen in Figure 4. In this case we have scanned the range of vanishing energy from 104 to 106 eV, ensuring all along that the scattering process is dominated by s-wave contribution49 The results for the corresponding behavior of the real part of the scattering length are given in the panels of Figure 5. They indicate that indeed such a quantity diverges exactly at the energy where the strong feature of the reactive cross section occurs. Furthermore, we see that the R(k) approaches the low-energy behavior as a positive quantity of increasing size, thus indicating there the existence of a resonant state of the complex. The scattering length then changes abruptly to minus infinity as the earlier resonance turns into a virtual state and the R(k) decreases to large negative values. As the energy keeps decreasing, we see that the scattering length changes sign passing through a zero energy resonance, just when the reaction cross section gives rise to a marked maximum in its size.48,49 In other words, we see that the indicator of the occurrence of a zero-energy resonance followed by a virtual state of the complex is then the changing size of the reaction cross section for the destruction process associated to the HeH+ ionic partner. Because this feature is very prominent and occurs at energies amenable, at least in principle, to experimental observations in traps, one could hope that some future observational confirmation could be found. It is worth noting here that the behavior of the energy dependence of the scattering length outside the range of its new feature is shown by Figure 5 to be constant, indicating therefore that we have clearly reached the range of validity of its definition by eq 12. Another interesting set of structures in the computed reactive cross sections at ultralow energies is given by the sequence of close-by maxima reported by Figure 4. We see there, in fact, the occurrence of a series of marked peaks of varied intensity that will merge into the unstructured maximum of Figure 3 once the correct total cross section is considered. However, we know that the J = 0 probabilities already give us very useful details on the effect of interchannel couplings due to the occurrence of resonant states during the reactive evolution into the products. 8201
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Figure 6. Computed, radial adiabatic potential curves Vi(R), obtained from the RPES employed for the present reaction. The low-energy range spans the region shown by Figure 4. See main text for details.
In particular, we wish to argue that the visible peak structure is the consequence of Feshbach-type, closed-channel interaction that makes accessible, during the scattering process, the transition-state region of the triatomic complex, thereby giving occurrence to local bound states created by the marked well of 350 meV indicated by Figure 1. To qualitatively support this possibility, we report in Figure 6 the sequence of adiabatic potential curves (Vi(R)), after integration over the molecular (HeH+) internal degrees of freedom (r, θ, and ϕ), which are given by the full RPES along the reaction coordinate. The energy posts reported in that figure tell us that (i) the binding energy of the reagent ion is about 2.043 eV, thus well above the energies of interaction, (ii) the product formation (H2+) is more than 0.7 eV below, thereby providing a marked energy gain within the reaction region, and (iii) the transition intermediate is below the products’ asymptote by about 350 meV and therefore provides, at vanishing kinetic energy, the additional energy needed to locally couple the very numerous pseudobound states of the many curves shown in the figure that are instead asymptotically closed. In other words, we see that each of the many adiabatic curves shown in the reaction region supports a number of bound states that can become locally open at some very low value of relative kinetic energy of the reactive partners. For some of this extremely dense structure of adiabatic potentials these corresponding bound states thus become accessible during the reaction and therefore give rise to the maxima on the reactive cross sections shown by Figure 4. As the collision energy increases, however, many more J values become important and the greatest majority of such local bound states are no more accessible while the number of possible open channel resonances increases, thus giving rise to the broad maximum in the reaction cross sections shown in Figure 3. Such features were observed earlier on in several numerical studies of low-energy resonances for the F + H2 system5254 and largely confirm, for a neutral system with weaker interaction forces, the present findings.
V. PRESENT CONCLUSION The work reported in this paper is addressing the question of the expected behavior for strongly reactive partners, e.g., as it occurs in exothermic, barrierless ionic reactions, once the relative
collision energy tends to zero and the reaction occurs at low and ultralow temperatures. The experimental conditions are expected to be those of a radio frequency trap involving lasercooled collisional partners (e.g., see refs 1113), and one therefore hopes that an accurate theoretical and computational analysis could bring out strong features at specific energy values that could be amenable to observation and therefore could be employed as experimental checks on the quality of the RPES employed by the calculations. Our study has, in fact, employed an accurately generated ab initio RPES for the exothermic reaction yielding H2+ formation (reaction 2 in the present work) and has further carried out the quantum scattering study by using an in-house elaboration of the NIP approach described earlier in the literature.38,39,42 The results of the calculations down to below the millikelvin regime have shown rather conclusively the presence of two sets of structuring in the total reactive cross sections at the energies of the above regime: (i) a strong, narrow peak around 100 mK due to the presence of a virtual-state formation (near a zero-energy resonance feature), which therefore makes the reactive process much more likely to occur, and (ii) the presence of a very marked set of fairly narrow maxima in the reactive cross section that occur roughly in the 100 K region of low temperatures. Such maxima are shown to be associated with a series of very numerous and narrow Feschbach resonances that, at the low collision energy regime, are created by the local opening of a very large number of bound states of the triatomic complex caused by the existence of the strong ionic interaction within the short-range of partners’ relative distances. Such features are seen to merge into a broad maximum in the total reactive cross section but are still due to the narrow, closed-channel resonant features that are seen when the simpler J = 0 reactive opacity contribution is examined. In conclusion, the present quantum study of the reactive behavior of a realistic ionic system at low- and ultralow-T ranges has clearly shown how important the quantum effects are for the understanding and the prediction of specific structures in the reactive cross sections once the temperatures for the reactions are brought down to the millikelvin range. 8202
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’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT We thank the CINECA and CASPUR Consortia for the awarding of computational grants to carry out the present project. M.T. also thanks the Caspur Consortium for a training research fellowship and we are all grateful to Dr. T. Stoecklin for introducing us to the intricacies of the NIP method. Finally, the financial support of the University of Rome “Sapienza” Research Committee is gratefully acknowledged. ’ REFERENCES (1) Doyle, J.; Friederich, B.; Krems, R. V.; Masnou-Seeuws, F. Eur. Phys. J. D 2004, 31, 149. (2) Krems, R. V. Int. Rev. Phys. Chem. 2005, 24, 99. (3) Bethlem, H. L.; Meijer, G. Int. Rev. Phys. Chem. 2003, 22, 73. (4) Ferlaino, F.; Knoop, S.; Mark, M.; Berninger, M.; Schoebel, H.; Naegerl, H.-C.; Grimm, R. Phys. Rev. Lett. 2008, 101, 023201. (5) De Mille, D. Phys. Rev. Lett. 2002, 88, 067901. (6) Chandler, D. W. J. Chem. Phys. 2010, 132, 110901. (7) Bodo, E.; Gianturco, F. A.; Dalgarno, A. J. Phys. B 2002, 35, 2391. (8) E.g., see: Truhlar, D. G.; Garrett, B. C; Klippenstein, S. J. Phys. Chem. 1996, 100, 12771. (9) Smith, I. W. M. Chem. Soc. Rev. 2008, 37, 812. (10) Bovino, S.; Tacconi, M.; Gianturco, F. A.; Galli, D.; Palla, F. Astrophys. J. 2011, 731, 107. (11) Gerlich, D. In State-selected and state-to-state ionmolecule reaction dynamics; Ng, C. Y., Baer, M., Eds.; Wiley, New York, 1992. (12) Gerlich, D. Phys. Scr. 1995, T59, 256. (13) Willitsch, S.; Bell, M. T.; Gingell, A. D.; Softley, T. P. Phys. Chem. Chem. Phys. 2008, 10, 7200. (14) Gonzalez-Sanchez, L. G.; Tacconi, M.; Gianturco, F. A. Eur. Phys. J. D 2008, 49, 85. (15) Vogelius, I. S.; Madsen, L. B.; Drewsen, M.; Softley, T. P. J. Phys. B 2006, 39, S1241. (16) Baba, T.; Waki, I. J. Chem. Phys. 2002, 116, 1858. (17) Drewsen, M.; et al. AIP Conf. Proc. 2002, 606, 135. (18) Roth, B.; Offenberg, D.; Zhang, C. B.; Schiller, S. Phys. Rev. A 2006, 73, 042712. (19) Molhave, K.; Drewsen, M. Phys. Rev. A 2000, 62, 011401(R). (20) Staanum, P. F.; Hobybeere, K.; Wester, R.; Drewsen, M. Phys. Rev. Lett. 2008, 100, 243003. (21) Bovino, S.; Stoecklin, T.; Gianturco, F. A. Astrophys. J. 2010, 708, 1560. (22) Bovino, S.; Tacconi, M.; Gianturco, F. A.; Stoecklin, T. Astrophys. J. 2010, 724, 126. (23) Bovino, S.; Tacconi, M.; Gianturco, F. A.; Galli, D. Astron. Astrophys. 2011, 529, A140. (24) Edmiston, C.; Doolittle, J; Murphy, K.; Tang, K. C.; Wilson, W. J. Chem. Phys. 1970, 52, 3419. (25) Kuntz, P. J. Chem. Phys. Lett. 1972, 16, 581. (26) Sathyamurthy, N.; Ragaryan, R.; Raff, L. M. J. Chem. Phys. 1976, 64, 4606. (27) McLaughlin, D. R.; Thompson, D. L. J. Chem. Phys. 1979, 70, 2748. (28) Joseph, T.; Sathyamurthy, N. J. Chem. Phys. 1992, 96, 1265. (29) Aguado, A.; Paniagua, M. J. Chem. Phys. 1992, 96, 1265. (30) Aquilanti, V.; et al. Chem. Phys. Lett. 2000, 318, 619. (31) Xu, W.; Liu, X.; Luan, S.; Zhang, Q.; Zhang, P. Chem. Phys. Lett. 2008, 464, 92. (32) Ramachandran, C. N.; DeFazio, D.; Cavalli, S.; Tarantelli, F.; Aquilanti, V. Chem. Phys. Lett. 2009, 469, 26. (33) Palmieri, P.; et al. Mol. Phys. 2000, 98, 1835.
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