Exciplexes with Ionic Dopants: Stability, Structure, and ExperimentalRelevance of M+(2P)4Hen (M = Sr, Ba)Massimo Mella*,†,‡ and Fausto Cargnoni*,†,§
‡Dipartimento di Scienza ed Alta Tecnologia, Universita dell’Insubria, via Valleggio 11, 22100 Como, Italy§Istituto di Scienze e Tecnologie Molecolari (ISTM), Consiglio Nazionale delle Ricerche (CNR), via Golgi 19, 20133 Milano, Italy
*S Supporting Information
ABSTRACT: M+(2P)4Hen species, possibly involved in the post 2P ← 2S excitationdynamics of Sr+ and Ba+ in cold 4He gas or droplets, are studied employing both highlevel ab initio calculations to determine the potential energy curves (PEC) and diffusionMonte Carlo (DMC) to obtain information on their ground state structure andenergetics. PEC for the excited M+(2P)He dimers were obtained using MRCI calculationswith extended basis sets. Potential energy surfaces (PES) for M+(2P)Hen were built withthe DIM model including spin−orbit coupling via a perturbative procedure. DMCsimulations indicated several exciplexes (n > 2) to be stable against He dissociationwhatever the ion state, a finding that is at variance with what was previously suggested forBa+(2P1/2) due to the repulsive nature of the interaction potential obtained in [Phys. Rev. A 2004, 69, 042505]. Our results,instead, support the suggestion made in [J. Chem. Phys. 2012, 137, 051102] for the existence of Ba+(2P1/2)Hen exciplexes emittedfollowing the excitation of the barium cation solvated into He droplets. In the 2P1/2 state, the Ba ion also shows a peculiarbehavior as a function of n with respect to the location and binding strength of the attached He atoms compared to Sr+. Althoughthe latter forms the usual equatorial He ring, Ba+ deviates from this geometry for 1 ≤ n ≤ 4, with the radial distribution functionsstrongly depending on the number of solvent atoms. Finally, a putative species is proposed to explain the emission band at 523nm that follows D1 or D2 excitation of Ba+ in superfluid bulk helium.
1. INTRODUCTION
The usage of cold (0.37 K) and superfluid He droplets asnanocryostats for spectroscopically studying interesting speciesin isolation and quenching their internal degrees of freedomscontinues to surprise in terms of unexpected observations,especially when cations are involved. In these cases, thecharge−induced dipole interaction between the ion and thesolvent He atoms should, in principle, ensure cation solvationwhatever the vibrational or electronic state of the latter. Recentexperiments, however, contradicted such common lore,suggesting that both vibrational (for molecular species1−4)and electronic (for simpler monatomic ions5) excitation maylead to ion expulsion from the droplet. Given the fact that ionsseem to be expelled via a nonthermal mechanism, the processought to hinge on the possibility that an excited species findsitself in a less attractive environment than the ground state one,or on its capability of “dumping” the excess energy into arelative translational mode with respect to the droplet center ofmass.In this respect, the recent experiments on the postphotoio-
nization dynamics of barium attached to He droplets and onthe photoexcitation of the resulting Ba+, once solvated by thelatter,5 attract interest due to the extreme simplicity of thedopant. Peculiarly, the analysis of the ejected species clearlyindicated the presence of Ba+Hen clusters, which could beexciplexes according to Zhang and Drabbels’ analysis. If thiswere the case, there would be the need to reconcile the
formation of bound species with the expulsion of the cationfrom the droplet.The observation made in ref 5 appears even more interesting
if one also considers that previous spectroscopic experiments6
have left important details of the post−excitation dynamics ofBa+(2P) in liquid helium unclear. In particular, we refer to thesimultaneous presence of free−atom like 2S1/2 ← 2P1/2 and2D3/2 ←
2P1/2 emission lines and of a third band at 523 nm withan unclear origin. The latter may be due to bound speciesformed during the post−excitation dynamics into thecondensed He environment.Apart from the spectroscopic experiments in bulk helium and
He droplets, the formation (and decomposition) of exciplexeswas observed following the excitation of the D2 line of Ba+ incold (3−20 K) He gas,7,8 perhaps as a byproduct of aninvestigation on fine-structure changing collisional processesinvolving alkali-earth-metal cations and He atoms.9 As a netresult of these experiments, it emerged that Ba+(2P3/2) iscapable of binding a single He atom, the possibility of bindingmore being hampered by the low He density employed duringthe experiments. Indeed, two He atoms would be expected tobind strongly to the cation due to the anisotropic density as
Special Issue: Franco Gianturco Festschrift
Received: March 19, 2014Revised: May 30, 2014Published: May 30, 2014
suggested by Dupont−Roc;10 additional atoms would bind onlyvia weak dispersion forces. The presence of Sr+He excimer wasalso hinted at in ref 7 recording the fluorescence emission afterD2 line excitation in similar conditions. The formation of Heexciplexes with Ba+(2P1/2) was instead deemed to be impossibleon the basis of the repulsive nature of the 2Π1/2 potentialenergy curve (PEC) for excimer; however, no methodologicaldetails were given about the PEC calculation.At the moment, the idea that Ba+(2P1/2) cannot form
exciplexes with He atoms appears inconsistent with the massspectrometry analysis of the Ba+ ejected from the dropletsfollowing both 2P1/2 ←
2S1/2 and2P3/2 ←
2S1/2 excitations, ananalysis that clearly indicates the presence of Ba+Hen. As theresults of experiments in bulk He suggest a rapid fine-structurechange,6 the mass spectrometry data may be interpreted as anindication of the formation of only Ba+(2P1/2)Hen exciplexes.These, however, may represent a secondary product obtainedafter the initial formation of Ba+(2P3/2)Hen species uponexcitation of the D2 line. Such a process seems somewhat morelikely for n = 1 and 2, as the 2P3/2 state of metal atoms areknown to strongly bind no more than two He atoms.In the context just described, we deemed it interesting to
investigate the stability and structural details of M+(2P)Henspecies with M = Sr or Ba. The main goal would be to describethe effects related to the anisotropic interaction between Heatoms and P-state like charged atomic species, with a particularaim toward interpreting the experimental results discussed inthis section. To do so, we need accurate PESs describing theinteraction between the ion and the surrounding He atoms,which we build starting from ab initio calculations on the
relevant dimers and employing the diatomics-in-molecules11
approach to generate many-body surfaces. Diffusion MonteCarlo simulations are subsequently employed to sample theground state wave function Ψ0 of the title systems, and thuscomputing their lowest vibrational eigenvalue E0. With thisdata, we also discuss similarities and differences between thetwo metals and, when useful, compare with previously studiedneutral exciplexes. In particular, we highlight a compellingbehavior shown by Ba+ hinging on the relative strength of theinteraction PECs and of the spin−orbit coupling constant.Finally, we pay some effort toward providing an unified pictureof all the experiments discussed in the literature involving Ba+
interacting with He atoms.
2. POTENTIAL ENERGY CURVES AND AB INITIOMETHODS
In this study, we determined the potential energy curves (PEC)of M+−He dimers (M = Sr, Ba) which asymptotically correlatewith the lowest lying 2P and 2D states of the metal. To obtainthese potentials, we first performed Configurations Interaction(CI) computations on the isolated cations, including excitedconfigurations up to triples. The excited 2P and 2D states areentirely described by single excitations of the outermostelectron (5s in the case of Sr+, 6s in the case of Ba+) to theempty p and d shells. Accordingly, the interaction potentialswith helium have been determined at the MRCI level of theory.The reference wave function consists of the ROHF term plusthe single excitations of the outermost electron of the metalinto the lowest lying empty p and d shells. Electron correlationis accounted for by including CI terms up to double excitations
Figure 1. Interaction potentials between excited Sr+ or Ba+ and a single He atom. All the asymptotes have been set to zero for convenience.Distances in Å, energies in cm−1.
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spanning the entire virtual space, which should be able todescribe properly dispersion effects as well as intramonomercorrelation on both the metal and the He atoms. As for thebasis set, the core electrons of the metal atoms have beendescribed with the Def2 pseudopotentials,12 and the outermost9 electrons are assigned the QZVP gaussians basis sets.13
Helium is described with the aug-cc-pVTZ basis,14 and a3s3p2d set of bond functions15 is located midway between thetwo nuclei. All the interaction energy data have been processedusing the standard counterpoise technique proposed by Boysand Bernardi. Spin−orbit coupling is not described at this stageand is added a posteriori according to the scheme described inref 16. Test computations at the CI singles and doubles level oftheory proved that improving the quality of the basis set on He(up to the aug-cc-pV5Z set) has a very limited effect on theproperties of the excited states PEC. As an example, the welldepth of the highly attractive 2Π states asymptoticallycorrelating with Ba+(2P) increases by roughly 3% as weincrease the basis set quality from aug-cc-pVTZ to the aug-cc-pV5Z, while the equilibrium internuclear distance decreases byabout 0.1 Å.The interaction potentials have been computed at 30
internuclear separations, sampling more finely the regionswhere the PEC undergoes sudden changes. All computationshave been carried out with the GAMESS-US17 code. Theanalytical expressions for the PEC have been determined fittingthe ab initio energies with Morse-like model functions modifiedby using polynomials of exponential functions of the nucleardistance r, and including a short-range damped Cn/r
n term to fitthe PEC long-range regions.The main features of the interaction potentials are reported
in Figure 1, and relevant data are collected in Table 1. As
expected, 2Σ states are almost entirely repulsive due to the factthat the metal atom accumulates its valence electron densityalong the M+−He internuclear axis. Conversely, the heliumatom can interact with the positive core of the metal when thevalence electron of the latter is accommodated into a p orbitalpointing away from the M+−He axis, and a quite deep wellarises. The interaction potentials for Ba+−He are shifted atlarger distances as compared to the corresponding states ofSr+−He, which is clearly an effect of the larger size of thebarium cation compared to the strontium one. The excitedBa+−He PES asymptotically correlating with Ba+(2P) compare
well with recent literature data for this same system18 (see thediscussion and Figure SI1 in the Supporting Information);instead, there are no available potentials for excited Sr+−Helevels and for Ba+−He states correlating with Ba+(2D), at leastto the best of our knowledge.
3. DIFFUSION MONTE CARLO ANDDIATOMICS-IN-MOLECULES POTENTIALS
As deeply fluxional in nature, M+(2P)Hen species require theuse of specialized methods not relying on geometrical referencepoints such as minimum energy structures employed during theharmonic analysis of vibrational modes. Also, the small size ofour systems makes semi-phenomenological methods such asdensity functional theory19 less useful, as they tend to rely on aliquid-like description of the He part. Thus, we opted foremploying DMC,20 which efficiently gives energies and groundstate Ψ0 distributions for highly quantum systems. Theinterested reader would find methodological information inthe extensive literature on the approach (e.g., see ref 21), thuswe avoid lengthy discussions and provide only a few details onthe particular choices made in this work. First, the deep wellpresent in the M+*−He interaction potentials other than in the2Σ+ states allows us to avoid guiding the DMC sampling bymeans of a trial wave function ΨT. Second, we employed thethird-order “on the fly” algorithm developed in ref 22 that usesan intermediate half-step potential evaluation to extrapolate thebranching weights to third order. A time step δt = 100 hartree−1
was found sufficiently small to guarantee a small systematic biasfor all clusters simulated when used in conjunction with a totalpopulation weight around 2000 and uniformly distributed overthe walkers. Third, we collected geometrical parameters whilesampling Ψ0 to obtain structural information; this approach isused despite methods giving Ψ0
2 sampling are available23,24 asthe latter are more expensive or complicate to implement.Besides, the simple approach taken in this work to extractstructural details has already been shown to provide goodquality information for reasonably bound species.25,26
As done previously by several research groups,16,26−33 weemployed the DIM approach11 to build the many-body PES forM+(2P)Hen. This allows one to introduce completely the two-body terms and a part of the three-body contributions to thecomplete description,34 the latter components being mainlyrelated to orbital rotation. Specifically, we used the sameapproach detailed in ref 26, including a treatment of the spin−orbit coupling based on distance-independent couplingsassumed proportional to the atomic splitting values (Δ =801.46 cm−1 for Sr+ and Δ = 1690.84 cm−1 for Ba+). Albeitmore accurate approaches based on the diabatization of thediatomic PEC are available35 to generate a many-body PES, theDIM approach has been shown to provide a sufficiently highlevel of accuracy for the purpose of quantitatively extracting andcomparing energy trends and structural features26,28,33 betweenthe family of species. Besides, applying the diabatizationapproach proposed in ref 35 is made quite complicated bythe CI wave function model used to compute the PEC, whichinclude millions of configurations.
4. RESULTS
In the following sections, we describe the structural details andenergetics of the stable exciplexes formed between He atomsand M+ in the 2P1/2 or
2P3/2 states. Initially, we would discussthe behavior of the spin−orbit coupled PEC for the M+He
Table 1. Relevant Properties of the M+−He InteractionPotentials (M = Sr, Ba)a
aThe asymptotes have been set to zero for convenience. bInternucleardistance at the minimum interaction energy, Å. cMinimum interactionenergy, cm−1. dInternuclear distance where the interaction potentialbecomes repulsive, Å.
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dimers; this is done to allow one to form an educated guess onthe chemical physics of larger systems. Successively, we providethe results of the DMC simulations for M+Hen clusters, basingour discussion on a chosen asymptotic electronic state of themetal ions rather than grouping the results by dopants. Thisapproach has the advantage of stressing similarities anddeviations in behavior that depends on differences in theelectronic structure due to the atomic number. Forconvenience, however, any relevance of our findings to availableexperimental studies will be discussed in the conclusions.Notice, also, that we would focus on studying only cluster sizesthat may be of some help in interpreting dynamical orspectroscopic experiments. In other words, we would mainlyrefer to small systems for which no more than a few He atomsare added to the ones forming the first solvation shell of M+.4.1. Spin−Orbit Coupled 2P States. Figures 2 and 3 show
the behavior of the M+(2P)He PEC after the introduction of
the spin−orbit coupling between P states. As one can notice,the shape and quantitative features strongly depends on thecation, and in particularly on the relative magnitude of Δ, the
spin−orbit splitting, and of the spin-averaged interactioncurves.In particular, one notices the near-disappearance of the
internal potential well in the 2Π1/2 state of Ba+(2P1/2)He, whose
depth decreases to 35 cm−1 from the original 487 cm−1 in the2Π states. Notice, also, that the well of the 2Π3/2 for Ba
+(2P3/2)He remains substantially unmodified compared to the ab initioresults. A similar situation is found for Sr+, albeit the lower Δvalue reduces the PEC couplings, so that the 2Π1/2 state ofSr+(2P1/2)He maintains a deep well and shows a low entrancebarrier. In this respect, the spin−orbit coupled Sr+(2P) curvesare akin to the one shown before for heavy alkali metals such asRb30,31 and for the coinage metal Cu.26 Thus, one is tempted topredict the possible formation of stable exciplexes containingseveral He atoms when Sr+ is in the 2P1/2 state. Conversely, thenarrow and shallow well in the Ba+(2P1/2)He PEC suggests asunlikely the existence of bound species. A similar situation isfound when the previously published PEC18 is used (Figure SI2in the Supporting Information). This conclusion, however,must be carefully checked as there is a second attractive well inthe 5 Å ≤ R ≤ 12 Å range (De ∼ 6 cm−1).Turning to the higher states, one would expect to find, at
least, M+(2P3/2)He2 species to be stable for both ions10 (theattractive wells are quite deep) and no stable species when theelectronic state correlates with the 2Σ1/2 one despite thepresence of a very shallow well (De ∼ 6 cm−1 in both cases) atlarge distances. According to the latter conclusion, we avoid tostudy aggregates with the cations in their highest excited state.
4.2. Exciplexes Formed with Metal Ions in the Spin−Orbit Coupled 2P1/2 State. Figure 4 shows the evaporation
energy for a single He atom from already formed M+(2P1/2)-Hen; the zero of the energy is defined as the free metal ion inthe appropriate spin−orbit coupled state plus the free He atomsresting at infinite distance. Although previous experience byus26 and others28−32 would suggest as likely the formation ofexciplexes with a ring-like He distribution around Sr+, anyprediction for Ba+ is far from being straightforward due to thecoupling between the 2Π and 2Σ states induced by the largespin−orbit splitting. A similar effect has already beenhighlighted for Ag, whose 2P1/2 state is not able to bind anyHe atom due to a shallow and narrow interaction well. Albeit asubstantial depth reduction and width narrowing of the well
Figure 2. Spin−orbit coupled PEC for the Sr+(2P)He (distances in Å,energies in cm−1). The zero of the energy scale is chosen as the spin−orbit averaged energy of the ion plus the He atom. Dissociationenergies De are computed with respect to the asymptotic threshold ofeach PEC.
Figure 3. Spin−orbit coupled PEC for the Ba+(2P)He (distances in Å,energies in cm−1). The zero of the energy scale is chosen as the spin−orbit averaged energy of the ion plus the He atom. Dissociationenergies De are computed with respect to the asymptotic threshold ofeach PEC.
Figure 4. He atom evaporation energy for M+(2P1/2)Hen (in cm−1).
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appears also for Ba+ upon introducing the spin−orbit coupling,the dimer PEC still presents a well that is more than three timesdeeper than the Ag case and a non-negligible external well (seeFigure 3 and the discussion in section 4.1).4.2.1. Sr+(2P1/2)Hen. The data in Figure 4 confirm our
expectation (vide supra section 4.2), suggesting that Sr+(2P1/2)-Hen clusters have all the He atoms interacting in a very similarway with the metal ion. The fluctuations in the evaporationenergy, not particularly wide given the energy scale imposed bythe well depth in the 2Π1/2 dimer state, seem to follow an“odd−even” alternation, with the odd n clusters presenting alower evaporation energy. This rule is violated by the n = 6species; in this case, however, the sixth atom is the last oneaccepted into the inner shell, and one would thus expectenergetic effects due to the compression of the already boundHe atoms. We tested this conclusion adding a seventh andeighth atoms, which were found bound by only a few cm−1.To corroborate our analysis based on the energetic data,
Figure 5 shows configurations randomly extracted during the
DMC simulations for each Sr+(2P1/2)Hen (n = 1−6); fromthese, it becomes evident the nearly planar disposition of theHe atoms around the cation. Also evident, there is thediametrically opposite location of the two He atoms inSr+(2P1/2)He2 with respect to the cationic center, and thatthe Sr+(2P1/2)He3 cluster has a T-like shape with neighbor Heatoms forming a right He−Sr−He angle. These structures differfrom the ones of larger clusters, which show a relative He−Hedistribution that is compatible with the idea that the He atomsoccupy the volume of a torus around the cation and thepresence of a weakly attractive He−He interaction. Indeed, thedistribution of the cosine for the He−Sr−He angle has a verystrong maximum at −1 for both n = 2 and 3, and a secondaryshallow peak around 0 for n = 3 (not shown).
To better appreciate the peculiarities of the Sr+(2P1/2)Henclusters, we stress that isotropic M−He potentials would beexpected to produce either a uniform (apart from the atomicexcluded volume) or slightly peaked He−Sr−He angulardistribution; given the equilibrium Sr−He distance the peakshould be located around a cosine value of 0.38 due to the He−He interaction. Hence, the strong preference shown bySr+(2P1/2)He2 and Sr+(2P1/2)He3 in locating two He atomsopposite to each other seems to indicate that electronic effectsare playing a key role, the details of which are better discussedconsidering Ba+(2P1/2)He2 (vide inf ra section 4.2.2).
4.2.2. Ba+(2P1/2)Hen. Turning to the clusters formed by Ba+,we begin by noticing that the evaporation energy does notbehave smoothly and presents two downward “kinks” at n = 1and 3. These are in stark contrast with the common expectationof a reasonably smooth change in total energy even during shellclosure or opening upon adding He atoms. The results shownfor n > 3, instead, are in line with what is discussed above forthe strontium cation in the 2P1/2 state, the main differencebeing the fact that the filling of the first shell seems to happenat n = 7. This is indicated by the sudden drop by 150 cm−1 inevaporation energy.Retrospectively, the sudden drop in evaporation energy seen
for Ba+(2P1/2)He3 may recall what already shown for Rb in the2P3/2 spin−orbit state;31 in this case, the third and allsubsequent He atoms are more weakly bound than the firstand the second, and this effect is due to the limited volumeavailable around the diametrically opposite regions where thedeep Rb−He interaction wells are present. The remaining Heatoms are thus effectively screened by the tightly bound Heatoms and allowed to interact only weakly with the metal.26,31
The situation just described is, however, not compatibleeither with the very low binding energy for Ba+(2P1/2)Heshown in Figure 4, which should have an evaporation energysimilar to the one of the second He atom,31 or with the fact thatthe evaporation energy of the second He atom is much largerthan the depth of the inner well in Figure 3. The analysis of thedistance distribution for Ba+(2P1/2)He and Ba+(2P1/2)He2 alsohighlights an interesting correlation between the evaporationenergy and the Ba−He distance distribution for the twosystems. Indeed, Ba+(2P1/2)He shows a very diffuse distributionconcentrated in the region of the external well (R = 7 Å, Figure3), whereas Ba+He2 has both He atoms distributed closer to thecation and inside the inner well (see Figure 6).Turning back to the energetics of Ba+(2P1/2)Hen, one notices
that a substantial change in the electronic structure of Ba+
ought to take place upon adding the second He atom for theenergy of the trimer to decrease to the level indicated by thedata; we attribute such a change to the fact that contributionsto the DIM matrix elements involving the 2Π states may besufficiently negative to reduce the mixing of the states due tothe spin−orbit coupling. Supporting this point of view, there isalso the nearly linear geometry (with Ba between the two Heatoms) found visualizing configurations sampled during DMCsimulations (Figure 7) or their He−Ba−He angular distribu-tions. The latter gives a clear indication for a strong preferenceof the He atoms to sit opposite each other while “sandwiching”the Ba cation. Again, this is at variance with commonexpectation for a more triangle-like geometry when isotropicpotentials are involved, suggesting that the many-body PESpresents angular terms connected to the forceful mixing ofelectronic states due to the spin−orbit coupling.
Figure 5. Sampled configurations during DMC simulations ofSr+(2P1/2)Hen. The number of He atoms is indicated as an insetclose to the Sr cation.
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To investigate more deeply the case, we have computed thepotential energy for the Ba+(2P1/2)He2 cluster as a function ofthe distance between the cation and one of the two He atoms
while maintaining the second constrained at a chosen bondlength (R1 = 3, 4, 5, 7 Å). Linear He−Ba−He and right angle(with Ba at the center) geometries have been explored, and theresults are shown in Figure 8.The results shown indicate that, depending on the geometry
and R1, the He atom may experience a potential that is quiteclose to the dimer one (both geometries when R1 = 7 Å), asubstantially repulsive interaction (bent geometry with R1 < 5Å), or a potential becoming increasingly more attractive andlowering its barrier as R1 (<5 Å) decreases for the lineargeometry. It thus seems clear that the more compact and lessenergetic linear structure sampled for Ba+(2P1/2)He2 during theDMC simulations ought to be due to, first, a change in the Ba+
electronic structure possibly induced by the penetration of evena single He atom into the barrier region (see Figure SI4 and SI5in the Supporting Information for a comparison between DIMwave functions for dimers and trimers) and, second, by astrongly angular anisotropy of the potential surface evidencedin the bottom panel of Figure 8.To foster a better understanding, let us write the DIM matrix
elements30 for the trimer with the linear geometry. In thissituation, one has:
= +Π ΠV r V rU ( ) ( )1,1 1 2 (1)
Figure 6. He−Ba+ distance distribution (arbitrary normalization)sampled during the DMC simulations of Ba+(2P1/2)He and Ba
+(2P1/2)-He2. Also shown is the 2Π1/2 potential. The zero of the energy scale isset as the center of mass of the spin coupled states.
Figure 7. Sampled configurations during DMC simulations of Ba+(2P1/2)Hen. Also shown is the He−Ba−He cosine distribution for Ba+(2P1/2)He2.
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= +Σ ΣV r V rU ( ) ( )2,2 1 2 (2)
=U 01,2 (3)
=U 01,3 (4)
From eq 1, it emerges that the quantity U1,1, whichcontributes to the first and second diagonal elements of theDIM matrix, may become strongly negative if one of He atomssits in the attractive well. This is true even in the presence ofthe spin−orbit coupling, whose contribution increases one ofthe two diagonal elements and lowers the other. The net effectof this decrease is of effectively decoupling one of the Ba+ states(i.e., a Π one) from the others, as states with substantiallydifferent energies do not strongly mix, and to make the surfaceoverall attractive. Geometries deviating from the linear onewould be expected to have a higher energy due to acontribution by the 2Σ state proportional to sin2(θ)VΣ(r).
30
As for the unexpected drop in evaporation energy found forBa+(2P1/2)He3, the visualization of configurations generatedduring DMC simulations (Figure 7) indicates that the third Heatom resides further away from the cation than the other two,in a region that is compatible with the bottom of the externalwell shown in Figure 3 while remaining close to one of theinner shell He atoms. Notice, also, that the magnitude of theevaporation energy for the third atom (10.5 cm−1) is
quantitatively compatible with the idea that the third Heatom interacts with the cation via the external well in the 2Π1/2state (∼9 cm−1) and with a single He (well depth of roughly ∼7cm−1), as well as with the fact that all the attempts of runningDMC simulations starting from configuration sets with all threeHe atoms initially located at the bottom of the inner wellinvariably led to the escape of one He atom to the externalminimum.Moving onto larger clusters, it appears that the fourth He
atom, again, induces a strong modification into the electronicstructure of the barium cation, leading to the sampling ofconfiguration characterized by an annular disposition of the Heatoms, all having average distances similar to those found forBa+(2P1/2)He2. These characteristics remain unchanged uponincreasing n up to 7, when a substantial drop in evaporationenergy becomes evident due to shell closure. In fact, anadditional atom was found to be only weakly bound, andoccasionally dissociating during long simulations.
4.3. Exciplexes Formed with Metal Ions in the Spin−Orbit Coupled 2P3/2 State. Figure 9 shows the evaporation
energy of a single He atom from already formed M+(2P3/2)Henclusters; as before, the zero of the energy is defined as the freemetal ion in the appropriate spin−orbit coupled state plus thefree He atoms resting at infinite distance. The 2Π3/2 states ofboth ions with one He atom are characterized by a deep well; inreality, there are usually two wells, diametrically opposite, asdiscussed above. This, usually, gives rise to the strong bindingof two He atoms in a linear geometry, and the evaporationenergy data seem to support this idea. Notice that Sr+ clustershave higher evaporation energies for both He atoms than Ba+
ones, in good agreement with expectation based on the ion sizeand well depths. Also, Figures 10 and 11 show that theexpectation with respect to the structure of the trimers arefulfilled by the results of our simulations.Comparing the He evaporation energy for M+(2P3/2)He and
M+(2P3/2)He2 with the same quantity obtained for clusterscontaining similar size dopants, one notices the effect of the netpositive charge. In fact, dimers and trimers have, respectively, aHe evaporation energy of 24 and 63 cm−1 for Rb(2P3/2),
31 212and 153 cm−1 for Ag(2P3/2), and 99 and 52 cm−1 forAu(2P3/2).
26 Indeed, even for a supposedly smaller dopant asCu, one finds lower evaporation energies (360 and 233 cm−1),whereas both Na and Li in the same electronic state bind morestrongly the He atoms (for both cluster sizes, the He
Figure 8. Potential energy for Ba+(2P1/2)He2 as a function of thedistance between the cation and a single He atom for linear He−Ba−He (top) and right angle (bottom) geometries. The second He atom isconstrained at the distances R1 shown in the legend. Also shown is thedimer potential.
Figure 9. He atom evaporation energy for M+(2P3/2)Hen (in cm−1).
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evaporation energy is roughly 335 and 845 cm−1, respectively,for Na and Li28). In the latter cases, the stronger binding is dueto the small spin−orbit coupling constant, which limits themixing of attractive and repulsive curves in the DIM matrices.Adding more He atoms to M+(2P3/2)He2 produces stable
species that are, however, characterized by a weaker binding, asindicated by a He evaporation energy of roughly 10 cm−1 forboth Ba+ and Sr+. Notice, however, that even for these weaklybound atoms, the binding energy is more than 20 times the oneestimated for similar species involving Rb as central dopant,31 aclear indication of the stronger interaction due to the netpositive charge of the metal. A representation of theM+(2P3/2)Hen structure is provided in Figures 10 and 11,from which one notices the symmetric distribution of He atomsnaturally emerging from DMC simulations. The latter werestarted from walker ensembles with randomly selected initialHe positions. The symmetric distributions are at variance withwhat found previously for Rb(2P3/2)Hen,
31 where a droplet
developed on one side of the cluster for n ≥ 5. In these cases,however, the small binding energy afforded by the externalatoms (less than 0.1 cm−1 for the first two added ones) to thecompact Rb(2P3/2)He2 core made advantageous the formationof an asymmetric distribution around the core from theenergetic point of view.36 This tendency for Rb(2P3/2)Henought to be expected to never subside as the evaporationenergy of a He atom is reported to become of the order ofroughly 4.5 cm−1 upon going toward the limit of large clusters.Thus, the transfer of one He atom from the side bearing theclusters to the bare one would always be energeticallydemanding, requiring an external source of energy to takeplace. In the case of the two cations, instead, the binding energyfor the external solvent atoms (∼10 cm−1) is already higherthan the He chemical potential in small pure clusters, so muchso that the systems prefer to symmetrically distribute them tomaximize the interaction with the partially shielded positivecore. Worth highlighting, it is also the fact that the distribution
Figure 10. Sampled configurations during DMC simulations of Sr+(2P3/2)Hen. n is given close to the cation. Also shown is the Sr−He distribution forSr+(2P3/2)He3.
Figure 11. Sampled configurations during DMC simulations of Ba+(2P3/2)Hen. n is given close to the cation. Also shown is the Sr−He distributionfor Ba+(2P3/2)He3.
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for the cosine of the He−M+−He angle indicates that there isfinite, albeit low, probability of finding the external He atoms inM+(2P3/2)Hen forming a right angle with the ones constitutingthe inner core. This finding indicates that a “side-to-side”migration ought to be reasonably easy and not hampered byhigh energy barriers.
5. DISCUSSION AND CONCLUSIONSIn this work, we have investigated the structural and energeticdetails of exciplexes composed of heavy alkali-earth cations (Sr+
and Ba+) in their 2P electronic state and He atoms. To this end,we simulated their vibrational ground state using DMC.Interaction potentials needed for such study were generatedcomputing counterpoise-corrected ab initio MRCI curves withDef2 pseudopotentials and large basis sets; we exploited thoseresults to build many-body PES by means of the DIM method.Although strontium cation exciplexes behave, overall,
according to our expectations (i.e., tightly bound ringgeometries for the 2P1/2 state and the formation of a compactcore with two opposite He atoms for the 2P3/2 state), thebarium cation demonstrated a variety of changes in behaviorthat, retrospectively, can be traced to the interplay between themagnitude of the 2Π well depth and of the spin−orbit couplingconstant in defining the value of each DIM matrix element.Thus, although we found the dimer to be loosely bound andwith a wide Ba−He distance distribution peaked around 6.2 Åwhen Ba+ is in its 2P1/2 state, the trimer collapsed in a tightstructure (the distribution is peaked around 3.1 Å) more similarto the compact core found for the 2P3/2 species. Moreover, athird He atoms remains outside this core, whereas the additionof a fourth one induces, again, the whole system to collapse intoa ring planar structure with short Ba−He distances.Disregarding temporarily the peculiar behavior just summar-
ized, the stability of exciplexes involving Ba+(2P1/2) found inour investigation is at variance with the prediction made in ref7, where a lack of binding was suggested. Such a difference isimputable, mainly, to the level of the calculations employed inref 7, as can be inferred from a quoted well depth of ∼330 cm−1
for the 2Π3/2 state of Ba+He. As the 2Π3/2 state is usually theleast influenced by the spin−orbit coupling (e.g., compare thewell depths quoted in Figures 2 and 3 and Table 1), our PECsseem to bind more strongly than the PECs previously obtained.This fact has the net effect of lowering both the bottom of theinner well (present in both cases) and the tail of the 2Π1/2 statedue to the charge-induced dipole interaction, thus makingpossible the formation of a stable dimer. Also notice thatindications for a possible underestimation of the interactionstrength were already present as the comparison betweentheoretical and experimental fluorescence spectra in ref 7indicated that computed emission bands were all shifted tohigher energies with respect to the experiments by, at least, 140cm−1. In this respect, our DMC results for the vibrationalground state of the 2Π3/2 state of Ba+He, coupled with therecently computed PEC for 2Σ ground state of the samesystem,18 suggest an emission around 21 027 cm−1, in slightbetter agreement with the experimental 20 910 cm−1 than theprevious calculation.7
Aiming to exploit fluorescence spectra (see refs 7 and 8) todetect exciplexes, the deep well in the 2Π3/2 state would suggestthe D2 line (i.e., the 2P3/2 ←
2S1/2 transition) to represent theexcitation most likely to form such species. Besides, it shouldnot come as a surprise that not much effort was paid todetecting the possible formation of Ba+(2P1/2) exciplexes, either
as product of a direct D1 excitation or as a product of aninternal conversion between the two states, given the repulsive2Π1/2 PEC suggested by theory. Nevertheless, our resultsindicate that it would be indeed possible to form exciplexesrelated to the lowest 2P electronic state, albeit the likelihood offorming them may be rather small due to the small bindingenergy of the dimer. Whether this species can be formed bycollisionally depopulating the 2Π3/2 state into the 2Π1/2 one, assuggested happening in cold He gas, would depend on theefficient dissipation of a large amount of energy correspondingto the 2P1/2 ←
2P3/2 transition. Such possibility would, instead,be more likely in the condensed phase, such as in the case ofthe experiments discussed in ref 6. There, results wereinterpreted as indicating a fast depopulation of the 2P3/2 stateinto the 2P1/2 one, with two of the three fluorescence lines (491and 648 nm) being interpreted as related to the 2S1/2 ←
2P1/2and 2D3/2 ←
2P1/2 transitions. Notice that the suggestion for afast state conversion emerged from the fact that all transitionsappear independently of the line (D1 or D2) excited. Theremaining band at 523 nm (19 121 cm−1) was insteadinterpreted as the de-excitation of p-like Ba+ “in a specialconfiguration of the bubble”,6 without any further clarification.With our results, instead, it seems at least possible to make amore precise suggestion and to indicate the emission involvingthe relaxation of Ba+(2P1/2)He2 to the ground state of thebarium cation as the source of such line. In fact, subtracting thetrimer binding energy (253 cm−1) in Figure 3 and the estimateddestabilization of 1038 cm−1 for the ground state Ba+ due to thetight location of the He density maximum (d(Ba−He) = 3.05Å) in Ba+(2P1/2)He2 from the free-atom frequency gives apossible transition energy of 18 970 cm−1 (527 nm). This valueis compatible, considering the experimental bandwidth and theremaining inaccuracies of the theoretical methods, with theexperimental results, and the vibrational ground state ofBa+(2P1/2)He2 is thus proposed as possible source for suchfluorescence band.As indicated in the Introduction, there is also experimental
evidence for the formation of larger Ba+(2P1/2)Hen species thanthe trimer, such evidence emerging from the droplet adsorptionexperiment carrier out by Zhang and Drabbels.5 The intensitypattern, however, does not agree with what could be expectedfrom our energy calculation, perhaps suggesting that the relativeintensities corresponding to different Ba+(2P1/2)Hen may bemore related to dynamical effects than to energetic ones. Inparticular, the fairly large intensity evidenced for the n = 1species in the experiments is at variance with its low bindingand evaporation energies. The fact that the experimentalpattern in cluster masses may be due to dynamical effects canbe appreciated by considering the expulsion of the Ba+ ionfollowing excitation of the D1 and D2 lines, for which recordedspectra5,6 suggest that the environment the 2P states foundthemselves in is indeed quite repulsive. In particular, the blueshift indicates an additional energy requirement of roughly 490cm−1 in excess of the free excitations, an amount of energy thatcan easily lead to an initial acceleration of the ion in a directionthat depends on the instantaneous fluctuation of the cationicbubble. Slowed down on its way out, the cation maycontinuously pick up and lose He atoms until it reaches thesurface. However, the larger the number of attached He atoms,the larger the collision-induced dissociation cross section,8 thusmaking it less likely for the ion to capture a large number ofsolvent atoms. Whether the dynamical picture just proposed iscorrect can only be decided, obviously, by employing a
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dynamical simulations method. Our group plans future work inthis direction.Concluding, we highlight that our investigation has led to
suggesting the existence of species that have been previouslydeemed unstable, particularly the ones formed by the ions intheir 2P1/2 state; this finding supports the assignment made byZhang and Drabbels5 as far as Ba is concerned. We thus hopethat our results spur additional experimental efforts, perhapsalso in the direction of strontium, which has been partiallyneglected in the past. For strontium, it might be possible tohave two different exciplex formation channels, one involvingdirectly the excitation of the D1 line, and the second requiringthe depopulation of the exciplexes formed following theexcitation of the D2 line.
■ ASSOCIATED CONTENT*S Supporting InformationInteraction energy curves of the Ba+−He excited states, SOcoupled curves for Ba+He, DMC results for the Ba+(2P1/2)Heand Ba+(2P3/2)He systems, and square modulus of the mixingcoefficients. This material is available free of charge via theInternet at http://pubs.acs.org.
■ AUTHOR INFORMATIONCorresponding Authors*M. Mella: e-mail, [email protected]; phone, +390312386625; fax, +39 0312386630.*F. Cargnoni: e-mail, [email protected]; phone, +390312386625; fax, +39 0312386630.Author Contributions†F.C. executed the ab initio calculations at the CI and MRCIlevels, analyzed the ab initio data obtained, and contributed tothe writing of the manuscript. M.M. fitted the potential energycurves, carried out the diffusion Monte Carlo simulationsanalyzing the results, and contributed to the writing of themanuscript.NotesThe authors declare no competing financial interest.
■ ACKNOWLEDGMENTSWe thank Manuel Barranco and Marcel Drabbels for usefuldiscussions and providing information on their work beforepublication. M.M. also thanks the Fondo Ateneo per la Ricerca(FAR) of the Universita degli Studi dell’Insubria for funding.
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