Solubility of Metal Atoms in Helium Droplets: Exploring the Effect of the

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Solubility of Metal Atoms in Helium Droplets: Exploring the Effect of the Well Depth Using the Coinage Metals Cu and Ag Fausto Cargnoni*,† and Massimo Mella*,‡,§ †

CNR
ISTM, via Golgi 19, 20133 Milano (I), Italy School of Chemistry, Cardiff University, Main Building, CF10 3AT Cardiff, United Kingdom § Dipartimento di Scienze Chimiche ed Ambientali, Universit
a degli Studi dell’Insubria, via Lucini 3, 22100 Como (I), Italy ‡

ABSTRACT: We report a theoretical investigation of the solution properties of Cu and Ag atoms dissolved in He clusters. Employing our recent ab initio ground state pair potential for Me-He (Me = Ag, Cu), we simulated the species Me@He n (n = 2
100) by means of diffusion Monte Carlo (DMC) obtaining exact information on their energetics and the structural properties. In particular, we investigated the sensitivity of structural details on the well depth of the two interaction potentials. Whereas Ag structures the first He solvation layer similarly, to some extent, to a positive ion such as Naþ, Cu appears to require the onset of a second solvation shell for a similar dense structure to be formed despite an interaction well of 28.4 μhartree. An additional signature of the different solution behavior between Ag and Cu appears also in the dependence of the energy required to evaporate a single He atom on the size of the MeHen clusters. The absorption spectrum for the 2P r 2S excitation of the metals was also simulated employing the semi-classical Lax approximation to further characterize Me@Hen (n = 2
100) using novel accurate interaction potentials between He and the lowest 2 P state of Ag and Cu in conjunction with the Diatomic-in-Molecules approach. The results indicated that Ag exciplexes should not form via a direct vertical excitation into an attractive region of the excited manifolds and that there is an interesting dependence of the shape of the Cu excitation bands on the local structure of the first solvation shell.

’ INTRODUCTION The usage and investigation of superfluid helium droplets by theoreticians and experimentalists alike have seen a substantial increase in recent years thanks to peculiar properties1 capable of paving the way for interesting applications. For instance, the soft nature of He droplets has allowed their use as transparent and weakly interacting matrices in which dopants can be trapped and investigated spectroscopically. In this context, they allowed one to study the evolution of infrared (IR) bands of ammonia aggregates toward the solid state,2 the recording of the electronic spin resonance spectrum of very cold radical systems such as alkali atoms,3 and the possibility of modifying the fragmentation pattern of solubilized molecules exploiting a charge-migration mechanism through the quantum solvent.4 Noteworthy, it is also the fact that the spectroscopic investigation of van der Waals aggregates embedded in Hen has provided strong evidence for the presence of intermolecular structures very different from the minimum energy ones, a finding that has been attributed to the kinetic control imposed on the cluster formation by the quantum matrix. In alternative to their use as matrices, He droplets can also serve as nano-cryostats. In this case, a strictly regulated number of molecular species is attached to Hen and their interaction or reactivity studied under controlled conditions. To mention a few examples of such applications, we recall a study on H2 clusters that has highlighted their strong fluxionality even at the temperature of r 2011 American Chemical Society

He droplets (∼0.4 K),5 the investigation of the change in acid behavior of HCl as a function of the number of added water molecules,6 and the reaction between alkali
metal clusters and water7 or between Mg clusters and oxygen molecules.8 In the latter example, however, the quantum cryostat seems to play an important role in the studied process, since the observed chemiluminescence decreases upon increasing the droplet size. Whether the reduction of the chemiluminescence may be due to the scarce solubility of the oxygen molecule in superfluid He or to a size-dependent nonradiative energy dissipation mechanism is still unclear at the moment. Despite the weakly interacting nature of He compared with heavier rare gases, its presence as a solvent may have unexpected and often quite interesting side effects. For instance, doping He droplets with several Mg atoms picking them up one by one seems to generate a “magnesium foam”, with Mg atoms not forming stable Mgn clusters despite the Mg
Mg interaction being much stronger than the Mg
He one.9 Similarly, the excitation spectrum of the lowest triplet state of Ag2, 3Σu, has recently been recorded regardless of the fact that the solubility of Ag in Hen is expected to favor the Special Issue: J. Peter Toennies Festschrift Received: December 31, 2010 Revised: March 4, 2011 Published: March 22, 2011 7141

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The Journal of Physical Chemistry A formation of the more stable singlet.10 Moreover, the large preference for C
H bond dissociation in ionized species such as simple alcohols seems to be well explained by the caging effects produced by the electrostriction of the first He solvation layer.4,11 Whether or not He can be considered to the full extent a weakly interacting quantum solvent, it appears from the latter few examples that its presence may modify in unpredictable ways the behavior of dopants due to solvation. Despite the important bearing that such an idea may have on future experiments, the characteristics exhibited by He as quantum solvent have only been marginally explored. At the moment, it seems clear that molecules, especially polar ones, tend usually to interact strongly enough with the He matrix to be solubilized and to show a larger moment of inertia (or lower rotational constants) when probed spectroscopically, with heavier rotors usually presenting a stronger increase in their inertia moments than lighter ones due to an almost adiabatical following.12 Also, strongly interacting species such as alkali13 and lead ions14 are expected to solubilize while “freezing”, at least, the first solvation shell due to the chargeinduced dipole interaction. Due to the lack of strong intermolecular forces, metals atoms have been used in the past to explore some of the solvent properties of He. In this respect, it is well established that alkali metals sit on the droplet surface mainly due to their large van der Waals radius, that heavy alkali-earths show a similar behavior, while metal atoms such as Ag, In, Eu,15 and Au16 are easily solubilized. Often, these finding are interpreted using the model by Ancillotto et al.,17 even though it is now clear that we need somewhat more sophisticated approaches for borderline cases such as Mg.18,19 For the latter, an extremely delocalized distribution of the dopant inside Hen and a strong sensitivity to the details of the Mg
He potential have been highlighted by recent theoretical studies.18,19 In any case, details on the solvation of metals inside He droplets can only be indirectly extracted from experiments, often in the form of broadenings and shifts of the free atom absorption lines. Computations, instead, allow one to probe directly and quantitatively the diverse solvation features, provided accurate interaction potentials are used. They permit also the exploration of model systems that, albeit artificial, may shed light on the intricate dependency between interactions, energetic effects and solubility. As such, theory represents a much needed complement to experimental investigations. In this respect, our recent work on Me
He systems (Me = Cu, Ag, and Au)20 has highlighted a peculiar similarity between the Cu
He and Ag
He interatomic potentials that we aim to exploit to improve our grasp on the matter. In particular, the two systems present almost identical equilibrium and crossing distances, the latter being the classical inner turning point for the dimer with an interaction energy of zero. This similarity in shape would permit an easier comparison between the two metals, hopefully allowing one to interpret different behaviors mainly on the basis of their well depth. In particular, we aim to explore possible differences in solvent structure, energetics, and excitation spectra, relating them with dissimilarities in the interaction potentials. Besides, previous work from us18,20 has also stressed the importance of high level electronic structure theory (i.e., CCSD(T) and beyond) for an accurate description of metal
helium dimers, especially in the case of open shell systems such as the coinage metals, for which lower level approaches such as MP221 may still contain a strong spin contamination. In this respect, we also aim to compare the results obtained previously22 for AgHen with the one achieved using our new interaction potential.20 Whether such sensitivity to the theoretical

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level is a peculiarity only of S-type state or is shared with atoms having P- or D-like states (e.g., Al and Sc) is an issue worthy of future explorations. The plan of this work is as follows. Excited States Interaction Potentials provides a description of the methods used to compute the interaction potentials (both 2Π and 2Σ) between the excited 2P metal atoms and He. It also contains a discussion on the results obtained. In Simulation Methods, we provide a brief description of the quantum Monte Carlo and Diatomic-inMolecules (DIM) approaches employed to investigate the vibrational ground state and excitation spectra of MeHen (Me = Cu, Ag). Results and Discussion instead bestows a detailed description of the energetic, structural and spectroscopic results obtained for the doped MeHen (n = 2
100) clusters, emphasizing the differences in behavior between the two metals. Finally, the Conclusions section provides our conclusions and a discusses future directions to investigate on this subject.

’ EXCITED STATES INTERACTION POTENTIALS We computed the potential energy surfaces (PES) of the lower lying 2P excited states of the Ag
He and the Cu
He dimers by means of ab initio Configuration Interaction (CI) calculations. First, we adopted a medium-sized Gaussians basis set and performed extensive tests on both intermolecular complexes to establish the appropriate level of theory for these systems. Second, we computed the interaction potentials with a much larger basis set and, third, we combined the results obtained in the first two steps to define the PES used in the dynamical simulations. In all calculations, the inner electrons of Ag and Cu have been described with the relativistic small-core pseudo-potentials (PP) proposed by Figgen et al.23 We included in the effective potential 28 core electrons in the case of Ag and 10 electrons in the case of Cu. The Me
He (Me = Ag, Cu) dimers are then treated as 21-electron systems: two 1s2 electrons on helium and the 19 outermost electrons on the metal atoms, 3s23p63d104s1 for Cu and 4s24p64d105s1 for Ag. At this stage we neglected spin
orbit coupling effects, and we included them in the potentials a posteriori, according to the scheme adopted in a previous investigation on this same subject.22 All the computations have been conducted using the GAMESS-US suite of programs.24 For each Me
He complex we considered three excitations, consisting in the promotion of an electron from the outermost s shell of the metal atom to its empty p shell. We thus generated the lower lying 2P states, which correlate asymptotically with the 2Σþ and 2Π states of Me
He dimers. Roughly speaking, in the 2Σþ state the outermost electron of the metal atom occupies a p orbital pointing toward helium, and hence, the electronic repulsion becomes dominant even at large internuclear distances. Accordingly, the interaction potential has no relevant well depth, and is more steeply repulsive than the ground state. Conversely, in the degenerate 2Π states the outermost electron of Me is promoted into p orbitals orthogonal to the Me
He internuclear axis, and the helium atom interacts with the positive core of the metal atom. These PESs exhibit a well depth much larger than in the ground state, and their repulsive walls are shifted at shorter internuclear distances. A thorough discussion along these lines can be found in ref 21. As a practical mean for the analysis of energy data in the discussion below, we set to zero the asymptotic values of all interaction potentials, and hence expressions such as “attractive” 7142

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The Journal of Physical Chemistry A and “repulsive” region of the PES should be interpreted according to this choice. Clearly, to perform the dynamical simulations each excited state PES will be translated to the proper experimental asymptote. In test computations we assigned to Ag and to Cu the ccpVDZ25 basis set, with the exclusion of the single set of f functions; for the He atom we adopted the aug-cc-pVDZ26 basis set, and we placed a 1s1p set of bond functions at midway between the Me-He nuclei. The overall basis set for Me-He complexes (Me = Ag, Cu) contains just 37 Cartesian Gaussian functions, with no diffuse functions on the metal atom, and from now on will be referred to as small basis set. We do not expect the small basis to predict interaction potentials for the excited states of Cu-He and Ag-He to a high degree of accuracy. However, present and previous20 computations proved that this basis set is flexible enough to recover correctly the main features of the PES under investigation. At the same time, the limited number of functions allows to test a wide spectrum of computational schemes, as discussed in the following. All the electronic ab initio computations have been conducted within a CI approach, assuming the Restricted Open-Shell Hartree-Fock (ROHF) solution as reference wave function. We accounted for electron correlation at three levels of accuracy: ROHF plus single excitations (CIS); ROHF plus singles and doubles (CISD); ROHF plus single, double, and triple excited configurations (CISDT). We varied the number of correlated electrons from just 3 (two 1s2 electrons on He, the outermost one of Cu or Ag) to 13 (two 1s2 on He, the outermost one of Me plus the entire d10 shell) up to 21 (1s2 on He, ns2np6nd10(n þ 1)s1, with n = 3 for Cu and n = 4 for Ag). From now the theoretical schemes will be labeled as CI A(ne), where A refers to the excitation level and n represents the number of correlated electrons. In all computations, and along the entire range of internuclear distances considered, the excited states of Cu-He and Ag-He retain strong atomic character, with the orbital components related to the promoted electron entirely localized onto the metal atom. We collected in Figure 1 the PES of 2Σþ and 2Π states of AgHe, along with the differences predicted by different computational schemes. All the interaction energy data have been corrected using the standard counterpoise technique proposed by Boys and Bernardi.27 The main results of our data analysis can be schematized as follows: • The lowest theoretical level approach, CIS(13e), is already capable of describing the gross features of the potentials: the 2 þ Σ state PES is entirely repulsive, and becomes relevant from the internuclear distance of 7 Å inward; the 2Π potentials are attractive up to about 2.5 Å, and at shorter Ag
He distances present a steeply repulsive wall. • The main contribution of electron correlation to the interaction energy is well described by single and double excitations: CISD(13e) is quite different from CIS(13e) and very similar to CISDT(13e). • The contribution of triple excitations is repulsive at large distances, and becomes attractive as the two atoms approaches. In fact, at large Ag
He separation, the CISD(13e) potentials are more attractive than the CISDT(13e) ones, and this behavior is reversed from about 3 and 2.5 Å inward for 2Σþ and 2Π states, respectively. • The correlation contributions coming from the inner s and p shells of the metal atom have small effects on the interaction potentials. The differences between CISD(21e) and

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CISD(13e) interaction energies are not significant along the entire range of Me
He distances considered. • The CISDT(3e) potentials are more repulsive (or less attractive) than CISDT(13e) ones, indicating that the overall contribution of the d10 shell to the interaction energy is always attractive at this level of theory. In the 2Π states, the CISDT(13e) potential is closer to the CISDT(3e) than to the CISD(13e) one at large internuclear separations up to 3.25 Å, while at shorter distances the behavior is reversed. In the 2Σþ state the CISDT(13e) interaction potential is always closer to the CISD(13e) than the CISDT(3e) one. In summary, CISDT(13e) computations are fully adequate for describing of the excited states of Ag-He under investigation, and this same conclusion was drawn from extensive tests on Cu-He. However, to adopt basis sets much larger than the small one at this level of theory would require a huge amount of computational resources, and hence, a downgrade of the theoretical level is mandatory. In the Ag-He dimer, the best choice for the 2Σþ state is to keep fixed the number of correlated electrons and to reduce the excitation level, thus, adopting the CISD(13e) scheme. Concerning the 2Π potentials, at large internuclear separations, the CISDT(3e) results are slightly better than CISD(13e) ones. From 3.25 Å inward, the contribution coming from the 4d10 shell becomes relevant, and hence, the CISD(13e) approach becomes more accurate than the CISDT(3e) one. Test calculations conducted on the 2Σþ state of the Cu-He dimer support the same conclusions drawn for Ag-He. Conversely, the interaction potential of 2Π states for Cu-He computed at the CISDT(13e) level of theory is closer to CISDT(3e) results than to CISD(13e) ones along the entire range of internuclear distances considered. Probably due to the weaker relativistic effects that usually expand the d shells, the 3d10 electrons of copper come into play at much shorter internuclear separations than happens for the 4d10 electrons in Ag-He. Consistently, the CISDT(3e) curves are nearly indistinguishable from CISDT(13e) ones up to the internuclear distance of 2.5 Å, a value well below the range relevant to our dynamical simulations. At shorter Cu-He separations, the difference between CISDT(3e) and CISDT(13e) increases very steeply. As a second step in the determination of the excited state potentials for Cu-He and Ag-He, we selected a basis set with much higher flexibility than the small one. We assigned to the metal atoms the aug-cc-pVTZ basis,25 to He the d-aug-ccpVTZ26 and placed a 3s3p2d28 set of bond functions at midway between the Me-He nuclei (Me = Cu, Ag). The resulting set (from now on referred to as large) consists of 173 Cartesian Gaussian functions and proved capable of recovering about 99% of the electron correlation contribution to the interaction energy of these same complexes in their electronic ground states.20 We computed the PES of excited states for Ag-He at the CISD(13e) level of theory. We considered 33 internuclear distances ranging from 2 to 10 Å, plus the asymptote at 100 Å, sampling finely the regions where the interaction potential undergoes sudden changes. In the Cu-He complex, we computed the PES at the CISDT(3e) level of theory, which describes accurately the 2Π states at internuclear separations larger than 2.5 Å, and gives an adequate curve also for the 2Σþ state. We considered 32 internuclear distances in the range 1.6 to 10 Å, plus the asymptote at 100 Å. Quite interestingly, the well depth of 2Π states in Ag-He predicted at CISD(13e) level with the small basis is about 75% of 7143

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Figure 1. Energetics of the excited states of the Ag
He complex obtained adopting the small basis set. Left columns: potential energy surfaces computed at various levels of theory; right columns: interaction energy differences between a given computational approach and the most accurate one, CISDT(13e). Energy in μhartree, distances in Å.

the value obtained with this same theoretical approach and the large basis set, while the internuclear distance of the minimum interaction energy is overestimated by just 0.10 Å. This is even more favorable considering CISDT(3e) computations on 2Π states of Cu-He, where the small basis recover more than 80% of the well depth as obtained with the large set. We assumed that this same level of transferability of interaction energy between small and large basis sets applies also to CISDT(13e) computations. Accordingly, in Ag-He we summed up the CISD(13e) curve obtained with the large basis set, and the CISDT(13e) to CISD(13e) energy difference computed in a limited set of 21 internuclear distances with the small basis set. For Cu-He, we determined the interaction potentials summing up the CISDT(3e) interaction energies obtained with the large basis, and the CISDT(13e) to CISDT(3e) energy difference determined with the small basis at 27 Cu
He distances.

The energy corrections described so far, the interaction energy curves computed with the large basis set, and the excited states PES resulting from the combination of the two terms are reported in Figure 2, along with previous results computed by Jakubek and Takami.21 To the best of our knowledge, no interaction potential for the excited states of the Cu-He dimer is available in the literature. In the Ag-He dimer, the CISD(13e) potentials computed with the large basis set are only slightly affected by the energy correction terms due to the CISDT(13e) to CISD(13e) difference, and we feel confident that the PES we propose for 2Σþ and 2Π states have a very high numerical accuracy along the entire range of internuclear distances considered. We registered two main differences with respect to the PES computed by Jakubek and Takami.21 First, the repulsion wall of the 2Σþ potential proposed by Jakubek and Takami increases more steeply. Second, the well depth of 2Π states is deeper and shifted at larger distances. More precisely, the minimum interaction 7144

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In the Cu-He dimer the CISDT(13e) to CISDT(3e) correction computed with small basis set plays a relevant role in the definition of the 2Σþ and 2Π potentials. As can be seen in Figure 2, the repulsive wall of the 2Σþ potential becomes steeper than found at CISDT(3e) level with the large basis, and the correction becomes relevant from about 5.5 Å inward. The PES of 2Π states is affected by the energy correction at much shorter internuclear distances (from 2.5 Å inward), and the attractive well becomes much deeper, from about 1400 to 2100 μhartree. At large internuclear separations the 2Σþ potential increases more steeply in Cu-He than found in Ag-He. However, from 4.75 Å inward, this Cu-He potential becomes softer, and at about 2 Å the PES of Ag-He becomes more repulsive. The PES of 2Π states behave similarly: the values computed for Ag-He are slightly more attractive than the ones for Cu-He up to 2.5 Å, where the two PES crosses and Cu-He reaches a minimum interaction energy of 2100 μhartree at the internuclear distance of 2.03 Å, where Ag-He is already repulsive. The repulsive wall of the 2Π states for Cu-He is shifted at shorter distances by about 0.2 Å as compared to the case of Ag-He. We remark here that the differences between the interaction potentials of Cu-He and Ag-He should not be attributed to the different computational strategies, namely, CISDT(3e) versus CISD(13e). In fact, we found essentially these same differences in the direct comparison of CISDT(13e) results computed with the small basis. In the dynamical simulations of Cu-Hen and Ag-Hen the internuclear Me-He distance never becomes smaller than 3 Å. Accordingly, there are just three features that are due to play some role: (i) the 2Σþ potential for Ag-He computed by Jakubek and Takami is more repulsive than our; (ii) the well depth of the 2 Π potential computed by Jakubek and Takami is larger than our, and shifted at larger distances; (iii) the 2Σþ potential for Cu-He is more repulsive than the corresponding one in Ag-He.

Figure 2. Energetics of the excited states of the Me-He complexes (Me = Cu, Ag). CISD(13e) and CISDT(3e) computations have been carried out with the large basis set. Energy correction terms are determined with the small basis set. The Potential Energy Surfaces are determines by combining these two curves (see text for details). We plot also theoretical data available in the literature for Ag-He21 (energy in μhartree, distances in Å).

energy computed by Jakubek and Takami is about 1600 μhartree at the Ag-He distance of 2.73 Å, to be compared with our result of 1240 μhartree at 2.52 Å. Quite interestingly, we register no significant differences in the location and slope of the repulsive wall of the 2 Π states. Jakubek and Takami adopted a basis set of high quality, and hence, these differences should be attributed to their quantum chemical approach. These authors performed just a CIS diagonalization, followed by perturbation theory MP2 computations, a method that should always be double checked using higher quality schemes, especially in studies on weakly interacting complexes.18,20

’ SIMULATION METHODS As it is well-known, pure and doped He clusters are characterized by a highly quantum nature. This feature induces a small total binding energy and a wide anharmonic motion of both the doping impurity and the He atoms. As a consequence of the intrinsic anharmonicity and of the experimental size of these clusters (usually of the order of several thousands atoms), the possibility of using either the harmonic approximation or more accurate basis set/ grid-base approaches are usually hindered. To describe at atomistic level the solvation properties of doped clusters and to compute their excitation spectra, the quantum Monte Carlo (QMC) methods are the best suited techniques. Because these methods are well described in the literature,29 we restrain ourselves from presenting long discussions, except for the technical details relevant to the present work. Here, we employed variational Monte Carlo (VMC) to optimize a trial wave function ψT(R) and diffusion Monte Carlo (DMC) to correct the remaining deficiencies of ψT(R), projecting out all the excited state components and sampling f(R) = Φ0(R)ψT(R), where Φ0(R) is the exact ground-state wave function for the nuclear motion of the system. In both cases a description of the vibrational ground state is sought, the low temperature of the clusters (0.37 K) and the large energy gap between vibrational excited states in Hen suggest that thermal excitations should not play a relevant role (for a discussion on this topic see ref 22). 7145

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In atomic units, the Hamiltonian operator for MeHen (Me = Ag, Cu) reads as ! 1 n r2i r2Me H ¼
þ þ V ðRÞ ð1Þ 2 i ¼ 1 m4 He mMe

∑

As commonly done, we assume a pair potential of the form V(R) = ∑ni 20 may then be interpreted either as the formation of a new and more external solvation shell, or as due to a dipolar deformation of the droplet upon which He atoms aggregate more likely on one side of the cluster, pushing Cu away from the He geometrical center RGC. Notice that both interpretations are consistent with the absolute value shown by Δ0(n) for Cu at large n values. However, a direct comparison with the same quantity computed for Mg (also shown in Figure 3) when its interaction with He is described by a MP4 based potential18 suggests the first interpretation as the most likely one. In fact, it would be difficult to explain otherwise the similarity in the values of Δ0(n) for Mg and Cu at high n values given the fact that the alkali-earth metal is “soluble” in He droplets. This conclusion is also supported by the fluctuating behavior of the average kinetic energy slope seen in the range 20 e n e 50
60, which would not fit well with the smoother evolution expected for a cluster with a non solvating impurity.13 Structure. As mentioned in the Simulation Methods, we computed several probability density functions (PDFs) to get information on the structure of the MeHen clusters. The Me
He distance, the He
Me
He angular and the Me- or the He-RGC PDFs are of particular relevance to study the solubility of the Ag and the Cu atoms. Whereas the radial PDFs are informative on the global structure and energetics of the MeHen species, the He
Me
He angle distribution is appropriate to characterize the degree of anisotropy of the He droplet with respect to the dopant. To facilitate the interpretation of these PDFs, we use the following normalization condition for the distance dependent ones (i.e., p(RMeHe) and g(RA-GC) with A indicating either Me or He): Z ¥ 2 4π dRMeHe RMeHe pðRMeHe Þ ¼ n ð5Þ 0

and

Z

¥

4π 0

2 dRA-GC RA-GC gðRA-GC Þ ¼ 1

ð6Þ

With a similar intent, we collected the angular distribution h as a function of the cosine of the He
Me
He angle, θ; within this choice there is no need to introduce the angular volume element sin θ and one can use the uniform distribution over cos θ as a simple and convenient reference to compare. In this case, we normalize h(cos θ) such that Z 1 ð7Þ dðcos θÞhðcos θÞ ¼ 1
1

We begin discussing the structural features of MeHen describing the evolution of p(RMeHe) with respect to the total number of He atoms, n. Figure 4 shows the SOE results for the two species in the range 25 e n e 100. We omitted to plot p for n < 25 as a way to limit the clutter in the graphs. Despite the strong resemblance of p as a function of n for the two species, still there are few interesting differences. First, the p for AgHen presents a local He density at RAgHe ∼ 10 bohr that is already higher than the bulk He density for n = 25. This clearly indicates that the AgHe interaction is strong enough to force the “piling up” of the He atoms around the dopant even for relatively small clusters, albeit the effect is not as strong as found in the cases of cationic13,14 or anionic41 impurities. Conversely, n must be at least equal to 40 before a similar “piling up” of He atoms around

Figure 4. Probability distribution function p for CuHen (top) and AgHen (bottom) as a function of the Me
He distance (in bohr) and of n. The horizontal lines represent the bulk He density at similar thermodynamics conditions. The normalization of p is as indicated in eq 5.

Cu is seen, a cluster size for which is already apparent the formation of a marked shoulder indicating the possible onset of a second solvation shell. In other words, while Ag is sufficiently attractive to “condense” a layer of He atoms denser than in the bulk and thus inducing a non monotonic behavior for Δ0(n) (vide supra) even when n e 25, a similar effect becomes apparent for Cu only when the presence of a more external shell generates a “pressure” on the internal one due to the lack of compensating interactions on the cluster surface. The consequences of the weaker Cu
He interaction as compared to the Ag
He interaction manifests also in the lower height of the first peak at RAgHe ∼ 10 bohr and in a slightly retarded onset with respect to n of the shoulder around 12 bohr, indicating the more external He shell. Turning to the distribution of cos θ, Figure 5 shows the behavior of h(cos θ) for a few cluster sizes. As a first remark, we notice that already at n = 25 both metals appear to be surrounded by a nearly uniform distribution of He atoms. This is evidenced by the constant value presented by h(cos θ) for cos θ e 0.5. The two other features apparent in the graphs are the weak peak centered around cos θ ∼ 0.75 and the rapid drop in h(cos θ) at higher value of cos θ. The latter is another indication of the He-He excluded volume (i.e., of the nonoverlapping of two vicinal He atoms), whereas the former appears due to the attractive well between two solvent atoms. Also noteworthy is the fact that the higher peak lowers and moves toward higher cos θ upon increasing n, while the dimple in probability raises due to the formation of the external layer.22 7148

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Figure 5. Probability distribution function h for CuHen (top) and AgHen (bottom) as a function of the cosine of the He
Me
He angle for several cluster sizes n. The normalization of h is as indicated in eq 7.

Figure 6. Probability distribution function g for CuHe100 and AgHe100 as a function of the distance between Me or He and the geometrical center of the He moiety. The He
GC distributions refer to the left y-axis, while the PDF of the two metals refers to the right y-axis and are plotted using a logarithmic scale. The normalization of g is as indicated in eq 6.

In substance, the behavior of h versus n is strongly suggestive of a complete solubility of both metals in He droplet despite the different behavior in Δ0(n). This is definitively ascertained by the PDF g, as shown in Figure 6, for MeHe100. Clearly, the close

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Figure 7. Vertical excitation spectra for CuHen (top) and AgHen (bottom) computed using the semi-classical Lax approximation. The spectra have been normalized to unit area. The vertical lines with symbol on top represent the location of the D1 and D2 free atom lines.

similarity between g and p for the He atoms would already provide a definitive proof for the solubility of the two metals inside He droplets, suggesting that helium is “kept away” from its geometrical center by the impurities. Even more suggestive is the behavior of log g for the two metals, which might be accurately fitted by means of a parabola centered at R = 0. In other words, the metal atoms inside the helium droplet seem to experience a strongly localizing environment, capable of substantially restraining their geometrical fluctuations with respect to the droplet center42,43 (notice that g drops by ∼2 orders of magnitude for a displacement of only 2 bohr from the center of the He moiety). Excitation Spectra of Me in Hen. The results obtained computing the vertical excitation spectra for selected AgHen and CuHen clusters are shown in Figure 7. In both cases, we have also indicated the absorption wavelengths that excite a free metal atom into the 2P1/2 (D1) and 2P3/2 (D2) states as vertical lines. In general terms, one can notice that upon increasing the number of He atoms in a cluster the position of the maxima of the absorption bands decreased in wavelength while their full widths at half-maximum (fwhm) increase. This finding is a clear indication that, upon increasing the cluster size, the environment experienced by Cu and Ag atoms after excitation becomes more repulsive. The same effect was already highlighted in previous calculations on Ag,22 Mg,18,19 and Ca,44 albeit the latter metal is thought to reside on a deep dimple at the surface. Similarly,18,22 7149

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The Journal of Physical Chemistry A the substantial blueshift of the Cu and Ag absorption in He clusters with respect to the free atom lines suggest that the metals should be “spit out” following the excitation. Besides, we interpret the monotonic shift shown by both Cu and Ag upon increasing n for n g 25 as suggesting that also the atoms of the second shell play a role in defining the energy of the excited metals. As suggested before,18,22 this may be due to both an overlap between the second shell distribution and the tail of the 2Σþ state, with the effect of increasing the value of the corresponding matrix element in the DIM approach, and to an increase in the He density at short Me-He distances (see Figure 4). Differentiating instead the two set of results, one notices that while the silver spectra maintain a clear evidence for the spin
orbit splitting of the two excited states at all n investigated, the D1 and D2 absorption bands for Cu merge into a single broader band when n = 100. In this respect, the difference in behavior between the two metals does not seem to derive from an increasing blueshift of the copper 2P1/2 state with respect to the 2P3/2 one; in fact, we found the distance in frequency space of the two peaks to increase upon increasing n. Thus, the overlap between the two bands is connected to a monotonic increase of their widths, induced by the formation of the more compact solvation shell in larger clusters, and facilitated by the small spin
orbit splitting between the 2P1/2 and 2P3/2 state in copper (248.38 cm
1 compared to 920.66 cm
1 in Ag). Confronting the curves for the excited state potentials shown in Figure 2, one may also notice that the excited 2Σþ state of CuHe is more repulsive than the Ag-He one and that the tail of the two Π states are instead rather similar. As such, these observations suggest that the merging of the copper D1 and D2 lines due to the width increase should be mainly related to the more repulsive nature of the Cu-He 2Σþ state potential and to its effects on the DIM matrix elements used to estimate the energy of the excited manifolds. In discussing the expected accuracy of the computed spectra, it is important to reiterate that their calculation was carried out explicitly assuming a vertical electronic transition, that is, that the He atoms in the clusters do not readjust their distribution during the photon absorption process. This assumption is clearly reasonable on the basis, at least, of the difference in energy scale between the electronic transition (hundreds of thousand of cm
1) and a possible vibrational readjustment (requiring at most a few cm
1, see Δ0 in Table 1). It also seems sensible for Ag, whose spectrum in the largest cluster (n = 100) closely resemble the one recorded in He bulk45 and droplets15 despite the fact that the Ag spin
orbit coupling (920.66 cm
1) is much closer in magnitude to any possible vibrational excitation. In fact, we found a blueshift of only ∼1.5 (∼0.5) nm for the computed D1 (D2) line with respect to the experiments,15 a difference that can be easily explained by the fact that the Lax approximation tends to slightly overestimate the energy gaps due to the lack of zero point energy corrections.37 These small discrepancies notwithstanding, the overall shape and location of the Ag excitation spectrum computed for AgHe100 remains in good agreement with experiments strongly supporting the validity of the Lax approximation in this context. A more detailed discussion on the limited effects expected on the spectra upon improving the DIM approximation can be found in our previous work.22 With respect to the results previously presented by one of us,22 one can notice that the new Ag-He potentials (both the ground20 and excited state ones) have a measurable impact on the computed excitation spectra. Most notably, all absorption maxima are blueshifted with respect to the free Ag lines when the new potentials are used; a substantial red-shift was instead previously found for n e 15,

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Figure 8. Vertical P r S excitation spectra for AgHen (black) and CuHen (red) computed using the semi-classical Lax approximation but using the distribution sampled for the other metal. The spectra have been normalized to unit area.

suggesting the possible formation of Ag exciplexes containing more than two He atoms, as in the case of Rb.46 Conversely, the results in this work suggest that, whether formed, Ag exciplexes can only be produced by some post-excitation mechanism. At a more detailed level, we also notice that the D1 (D2) line in AgHen (n g 20) is usually slightly (∼1 nm) blue-shifted (red-shifted) when computed with the excited potentials introduced in this work compared to the one by Jakubek and Takami. We attribute the dissimilarities in the spectra to the observed differences in the 2Π Ag-He excited potentials (vide supra) and in particular to the shallower and less wide well presented by the curve in Figure 2 than by the old CIS/MP2 one.21 In the case of Cu one may need to keep into consideration the fact that its spin
orbit splitting is roughly four times smaller than the Ag one and that copper is known to present an unusual triplet in the absorption spectrum when embedded in Ar, Kr, and Xe matrices47,48 due to a coupling with highly symmetric matrix vibrational modes. The theoretical analysis carried out on this effect47,48 suggests however that the vibrational modes must have frequencies of the same order of magnitude of the spin
orbit splitting to satisfactorily account for this phenomenon. Given the fact that both the results in Table 1 and the numerical estimate for the first few vibrational excitations in Hen made by Chin and Krotscheck49 indicates vibrational gaps of a few cm
1 at most for pure and doped He clusters, a similar coupling should not be expected to play any role in superfluid He matrices. As an additional attempt to improve our understanding of the solvation in He droplets, we have investigated the effects on the Cu and Ag excitation spectra of the details of their ground state solvation structure. To somewhat differentiate the effects of the MeHe distribution and excited potential energy curves on the shape of the excited manifolds, we have in practice carried out a computational experiment by sampling the atomic distribution for one of the two metals while computing the excitation spectrum using the interaction potentials and spin
orbit splitting for the other one (e.g., sampling the distribution for AgHen and computing the Vexc
Vgs gap with the Cu-He potentials). The results of this in silico experiment are shown in Figure 8 for n = 12, 40, and 100. From the latter figure, one notices that for Ag there are only minor differences in the shape of the spectra for all values of n. In fact, we have found that the peak maxima are generally red-shifted by ∼0.5 nm with respect to the same quantities obtained sampling the AgHen ground state. In other words, these 7150

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results indicate that the perturbation on the Ag excitation spectra is weak and predictable simply on the basis of the different overlap between the first solvation shell and the 2Σþ potential by the change in the ground state solvation shell despite the substantial quantitative differences seen in Figure 4 especially for n = 12. As for Cu, it is found that the peak maxima are all blue-shifted by at least 1 nm while using the AgHen distributions, an effect that can be attributed again to the more repulsive 2Σþ potential and to the denser first solvation shell. Perhaps more interesting, the shape of the excitation spectrum for the case n = 100 appears modified with respect to the one obtained using the CuHen distributions, with the peak associated to the D2 line still being clearly visible and blue-shifted by roughly 2 nm. An additional shoulder at low wavelengths also appears on the D2 band. In other words, it seems that the shape of the excitation spectrum of the fully solvated Cu is critically dependent on the local density in the first solvation shell, at least within the range of validity of our semiclassical approach. In our view, this finding suggests that interesting effects may be expected investigating spectroscopically the Cu atom in He matrices under varying pressure.

albeit slightly blue-shifted by the very nature of the semiclassical approximation. The new spectra also indicate that the formation of the exciplexes previously suggested would require an indirect formation mechanism instead of a direct one. A comparative analysis of the sensitivity of both Ag and Cu spectra also highlighted a “pressure” effect on the shape of the Cu absorption bands. This is probably due to an increase in the overlap between the first solvation shell and the highly repulsive 2Σþ Cu*-He potential. We conclude with a brief mention on the new directions envisaged by the present investigation. We suggest the possibility of quantitatively investigating the binding of He atoms to the 2P1/ 2,3/2 states of the metal atoms, as a way to extract further interesting differences between Cu and Ag due the different shapes of the interaction surfaces. Similarly, it may of interest to study the indirect formation of exciplexes starting from either doped He bulk or our clusters. Furthermore, it would be interesting to investigate the solvation properties of atoms with P and D ground states such as Al and Sc, for which directional effects should also be considered.

’ CONCLUSIONS In this work, we have carried out an investigation on the solvation of the two coinage metals Ag and Cu in Hen clusters (n = 2
100) focusing on their ground state structure and energetics, as well as on the absorption of the two metal atoms in the UV range. To tackle the latter task, we computed the potential energy curves for the lower lying 2Σþ and 2Π excited Me*-He states that asymptotically correlate with the 2P state of the metal atoms, using high level electronic structure approaches. The main drive for this investigation comes from the observation that the ground state interaction potentials of these metals with He exhibit very similar equilibrium distances and inner turning points. This provides an ideal starting point for a theoretical investigation aiming at comparing how the solution properties of a doping element may depend on subtle features of its interaction potential with He. For silver and copper, these manifest as a different behavior of the He evaporation energy Δ0(n), with the more strongly binding Ag showing a nonmonotonic trend that can be interpreted as the completion of a compact first solvation shell at n ∼ 25. Differently, Cu shows a monotonically decreasing trend in Δ0(n), suggesting that the first solvation shell, if present, is fairly less compact than for Ag. To further investigate these differences, we estimated the expectation value of the cluster kinetic energy and its slope. The very different slope of the kinetic energy in the range 12 e n e 18 characterizes the dissimilar behavior shown by Cu and Ag. The inspection of several quantum distributions related to the geometrical structure of MeHen fully supports the analysis of the different energetics provided by the two metals, with copper clearly appearing as solvated by He, but being a weaker “structure maker” than silver. In fact, it also becomes apparent that the density of the first solvation shell in CuHen depends on the “pressure” exerted by the forming second shell, an effect that is particularly evident between n = 30 and n = 40. Besides, the shallower ground state of Cu-He as compared to Ag-He emerges also from the wider amplitude of its vibrations with respect to the geometrical center of the Hen moiety shown in Figure 6. Finally, the vertical excitation spectra computed using the Lax semiclassical approximation for Ag in the largest helium cluster turns out to be in good agreement with the experimental ones,

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]; mellam@cardiff.ac.uk.

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