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Letter pubs.acs.org/JPCL
Ab Initio Confirmation of a Harpoon-Type Electron Transfer in a Helium Droplet María Pilar de Lara-Castells,*,† Andreas W. Hauser,*,‡ and Alexander O. Mitrushchenkov¶ †
Instituto de Física Fundamental, CSIC, Serrano 123, 28006 Madrid, Spain Institute of Experimental Physics, Graz University of Technology, Petersgasse 16, 8010 Graz, Austria ¶ Laboratoire Modélisation et Simulation Multi Echelle, Université Paris-Est, MSME UMR 8208 CNRS 5 bd Descartes, 77454 Marne-la-Vallée, France ‡
S Supporting Information *
ABSTRACT: An ab initio study of a long-range electron transfer or “harpoon”-type process from Cs and Cs2 to C60 in a superfluid helium droplet is presented. The heliophobic Cs or Cs2 species are initially located at the droplet surface, while the heliophilic C60 molecule is fully immersed in the droplet. First, probabilities for the electron transfer in the gas phase are calculated for reactants with velocities below the critical Landau velocity of 57 m/s to account for the superfluid helium environment. Next, reaction pathways are derived that also include the repulsive contribution from the extrusion of helium upon the approach of the two reactants. Our results are in perfect agreement with recent experimental measurements of electron ionization mass spectroscopy [Renzler, M.; et al., J. Chem. Phys. 2016, 145, 181101], showing a high possibility for the formation of a Cs2−C60 complex inside of the droplet through a direct harpoon-type electron transfer involving the rotation of the molecule but a negligibly low reactivity for atomic Cs.
T
distinguish between a T-shaped geometry, with the Cs−Cs axis oriented perpendicularly to the C3 axis, and a collinear geometry with coinciding inter- and intramolecular axes. As in a previous work,11 we use the Def2-QZVP basis set for Cs atoms12 in combination with the ECP46MDF effective core potential13 and the polarized correlation-consistent double-ζ basis set of Dunning and collaborators14 for carbon atoms. When estimating the basis set superposition error (BSSE), the Boys−Bernardi counterpoise scheme10 was applied as detailed in section S7 of the Supporting Information. All electronic structure calculations were performed with the Molpro program package.15 For the Cs2 dimer, the spin-parallel configuration, that is, a triplet state, was chosen for the analysis, which is expected to be the dominating spin state on He droplets.16,17 For this spin state, the T-shaped arrangement can be considered as the typical adsorption geometry on the surface of helium droplets.18,19 In Figure 1, we analyze one-dimensional scans of the potential energy surfaces for Cs−C60 and for Cs2−C60 (T-shaped and collinear geometry) in a diabatic picture. The corresponding energies are plotted as a function of the distance between the centers of mass of both fragments, which can be interpreted as a first approximation to the actual reaction pathway. The ionic state, which is the ground state at equilibrium distance, is characterized by the strong Coulomb interaction between a negatively charged fullerene and a positively charged Csx (x =
his Letter is motivated by recent experiments based on electron ionization mass spectroscopy, which indicated that the formation of Cs+ takes place on the droplet surface via Penning ionization while Cs2 is ionized inside of the He droplet.1 Renzler et al. interpreted the experimental data as indirect confirmation of a harpoon-type reaction inside of the droplet: When the Cs2 and C60 reactants are close enough, the hopping of an electron from Cs2 to C60 can form a cationic Cs+2 and an anionic C−60 species. The charged fragments experience Coulomb attraction, leading to the formation of a strongly bound Cs2−C60 molecule. Alternatively, the dispersion forces between the neutral Cs2 and C60 species leads to the formation of a weakly bound van der Waals (vdW) Cs2−C60 complex. Harpoon reactions2 have been studied in the gas phase3−6 and in rare-gas matrixes, induced by cooperative absorption,7 but not in superfluid helium, where two antagonistic effects play an important role. On one hand, the reactant species are expected to move through the liquid below the so-called critical Landau velocity8 due to frictional energy loss at higher speed. Its low value of only 57 m/s favors the electron hopping process because more time is spent at the crossing region between the two relevant electronic states that are asymptotically correlated to either neutral and ionic fragments. On the other hand, the repulsive energy contribution from the extrusion of helium is expected to add an energetic barrier near the crossing region that quenches the overall reactivity for either channel. Both effects need to be taken into consideration to provide a definite answer. In our simulation, the distance between the fullerene and the Cs or Cs2 species is varied along a C3 axis of the fullerene. We © 2017 American Chemical Society
Received: July 24, 2017 Accepted: August 25, 2017 Published: August 25, 2017 4284
DOI: 10.1021/acs.jpclett.7b01910 J. Phys. Chem. Lett. 2017, 8, 4284−4288
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The Journal of Physical Chemistry Letters
nanodroplets consisting of 2000 He atoms (He2000). Using a slightly modified version of the Orsay−Trento density functional,22,23 the free energy of a He2000 droplet is minimized with respect to a given arrangement of dopants within the droplet and on its surface. The free energy F[ρ], a functional of the helium density ρ, is given by F[ρ] = E[ρ] + Uext[ρ] − μN[ρ] − F ·R[ρ]
(1)
with the Orsay-Trento density functional E[ρ] and an external potential Uext[ρ] that introduces the interaction between the helium environment and the dopants. The two last terms of the equation allow for the conservation of N, the total particle number, and R, the He droplet mass center, with μ as the chemical potential and F as the retaining force.22 The threedimensional potential Uext[ρ] is constructed from ab initio scans of the corresponding potential energy surfaces of C60, Cs, and Cs2 with a single He atom via a summation over pairwise interactions. Details of this procedure are given in ref 11, which also contains information on the more sophisticated construction of a suitable He−C60 potential. The latter has been derived from a recently developed, highly accurate potential model that describes the interaction of He with nonmetallic structures. 24,25 In order to obtain the desired energy corrections,11,26 we evaluate the total energy of a multiply doped He2000 droplet as a function of the distance between the heliophobic Cs or Cs2 molecule and the heliophilic fullerene in one-dimensional scans ranging from 5 to 40 Å. With zero energy set to the asymptotic value at infinite distance between the Csx and the fullerene inside of the He droplet, this energy correction, which originates exclusively from perturbations of the He density, is added to the gas-phase scans of the Csx−C60 interaction in Figure 1. The barrier for a single Cs atom is smallest, with a height of about 400 cm−1, whereas the dimer barriers are approximately three times larger. However, when compared to the actual interaction energies, the additional energy cost due to the helium perturbation seems almost negligible as it barely affects the curvatures at the energy scale used in Figure 1. At the more accurate MP2 level of theory, the crossing between the ionic and the neutral potential occurs at 12.1 Å for a single Cs atom but at the much larger distance of 22− 23 Å for a Cs2 dimer arranged in either orientation. These values agree well with the estimations of 12 and 21 Å using experimental data for the ionization energy of the Csx fragment27,28 and the electron affinity of the C60 molecule29 (see section S4 of the Supporting Information). As can be seen from Table 1, the strongest interaction (about 20000 cm−1) is obtained for ionic Cs2−C60 in a T-shaped arrangement. The neutral curves show much lower potential minima of similar depth (1700−4400 cm−1) in all cases, with estimated BSSE corrections making the vdW interaction even weaker (see Table 1 and section S7 of the Supporting Information). Additional calculations at CCSD(T) level on the Cs2−benzene model system (see section S6 of the Supporting Information) point out that the MP2 method overestimates the interaction energies by about 19% and that the SCS-MP2 treatment9 corrects the MP2 overbinding. Application of the SCS-MP2 approach to the Cs2−C60 system (see Table 1) also indicates that the well-depth calculated with the MP2 method is about 19% too deep. However, calculations on Cs2−benzene show that the MP2 overestimation is almost compensated by the incompleteness of the basis set used for Csx−C60. Hence, our choice is considered to be a good compromise between accuracy and feasibility.
Figure 1. Cs−C60 (upper panel) and Cs2−C60 (middle and lower panels) interaction energies in neutral and ionic states using a diabatic representation, before (full lines) and after (dashed lines) He-induced corrections.
1,2) species and is asymptotically correlated to a separation in charged fragments C+60 and Cs−x . In contrast, the excited state corresponds to the interaction between neutral fragments and leads to a separation into C60 and Csx. Using a similar strategy to that applied in ref 20, these two potential curves are first obtained at the Hartree−Fock (HF) level via a careful, pointwise energy evaluation using neighboring geometries as the starting guess. The HF wave functions perfectly conserve their neutral or ionic character but are neither orthonormal nor noninteracting. Therefore, we apply a diabatization procedure to obtain rigorously diabatic HF wave functions and evaluate the corresponding hopping matrix elements H12 between these wave functions. The diabatization procedure is based on rotation of the two wave functions at a given geometry in such a way that their overlap with the diabatic states at the next larger distance is maximized (see refs 20 and 21 and section S1 of the Supporting Information). Finally, the electron correlation is recovered by applying second-order Møller−Plesset perturbation theory (MP2). The core 1s orbitals are kept frozen while all other orbitals are correlated in the MP2 calculations. The second ingredient in our ab initio study is the repulsive contribution to the interaction energy due to spatial hindrance caused by the superfluid helium environment. Here we apply helium density functional theory (He-DFT) to doped helium 4285
DOI: 10.1021/acs.jpclett.7b01910 J. Phys. Chem. Lett. 2017, 8, 4284−4288
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The Journal of Physical Chemistry Letters Table 1. Characteristics of the Csx−C60 Interaction at the Potential Minima As Calculated with the MP2 Method (See Figure 1)a interaction Cs−C60 Cs−C60 Cs2−C60 Cs2−C60 Cs2−C60
(neutral) (ionic) (neutral, T-shaped) (ionic, T-shaped) (neutral, collinear)
Cs2−C60
(ionic, collinear)
rmin, Å
Emin, cm−1
6.67 6.25 6.01 5.72 9.84 [10.1] 9.46 (9.61) [9.55]
1675 15655 4322 19830 2446 [710] 17414 (14270) [14564]
probabilities, with the value of the orientational average being 48% (see section S2 of the Supporting Information). Therefore, within the Landau−Zener model, electron hopping probabilities for both Cs−C60 and Cs2−C60 are significant at the Landau velocity. Work is in progress to provide the details of the actual (multidimensional) nuclear dynamics in this long-range electron-transfer process. To explain the experimental findings, it is also necessary to account for the extrusion of He upon penetration of Cs or Cs2 species into the droplet. To this end, Figure 3 shows the solvation energy pathways in the diabatic representation along with the maximum kinetic energy at the Landau velocity (see also Table 2). After reaching the crossing region and once the hopping to the ionic state has happened, the repulsion energy contribution due to hindered mobility in the He droplet is marginal in comparison to the strong Coulomb attraction between the charged fragments Cs+x and C−60. In contrast, if the system stays in the neutral vdW state after passing the crossing point, the repulsion correction inside of the droplet is not compensated by attractive contributions and the vdW Csx−C60 complex cannot be formed despite its stability in the gas phase (see Table 1). The global picture is very similar when the dispersion contribution is extrapolated from counterpoise-corrected10 CCSD(T) calculations in vdW Csx−benzene complexes, using the Das function of Szalewicz and collaborators,32,33 and then added to the diabatic HF energies in the neutral state (see Figure S4); the energetic barrier at the crossing region is overcome for the Cs2−C60 system in a collinear approach at the Landau velocity but not for the Cs nor the Cs2−C60 complex at a Tshaped configuration. Also, in agreement with the results obtained at MP2 level (see Figure 3), the energetic barriers after the crossing region hinder the stabilization of the Csx−C60 complexes at the vdW potential minima. Additional estimated BSSE corrections leave the crossing region unperturbed but make the barrier to the vdW potential minimum even larger (see section S7 of the Supporting Information). The same conclusion on the destabilization of vdW complexes inside of a He droplet was reported in our previous study.11 This confirms that the experimentally observed signal is related to the formation of the − Cs2+−C60 molecule through a long-range electron-transfer process and not to an indirect electron relaxation mechanism via a vdW intermediate state. Another important finding is that the energy barrier at the crossing region for the Cs2−C60 system can be overcome only if the Cs2 molecule rotates into a collinear arrangement. From the usual pickup process in the experimental synthesis of the system, where the Cs2 is residing on the surface, the T-shaped configuration is most reasonable as a starting point. However, He-induced extra costs upon penetration are comparably small; therefore, a rotation of the dimer will take place as soon as the interaction potential dictates it. In comparison with Cs−C60, the Cs2−C60 collinear reaction features a smaller barrier in combination with an early crossing of vdW and ionic potential energy curves, which explains why the reaction takes place in this case. Contrarily, mainly due to the larger ionization energy of Cs as compared to Cs2 (about 4300 cm−1 larger), the potential crossing for atomic cesium occurs at a much shorter distance (12.1 vs 22−23 Å; see Table 2). Note that the attractive Cs−C60 interaction is not yet strong enough at this distance to compensate the extra energy costs due to spatial hindrance, which keeps the reaction path above the maximum kinetic energy of a Cs atom moving at Landau speed. This result is also fully consistent with experimental findings indicating that reactions
a
Emin and rmin stand for the well-depth and the position of the potential minima. Values obtained with the spin-component-scaled9 SCS-MP2 treatment are indicated in parentheses, while counterpoisecorrected10 interaction energies are quoted in brackets.
For the long-range electron-transfer process to occur, it is important to confirm that the electron hopping probabilities between the neutral and ionic states are sufficiently high at velocities below the Landau limit. Here we apply the Landau− Zener model,30,31 a well-known, one-dimensional semiclassical model that delivers reasonable estimates of probabilities PLZ for nonadiabatic transitions via the approximation PLZ ≈ exp[( −2π H12 2)/(F12v)]
(2)
with v as the relative velocity of the fragments and F12 as the difference between the two slopes F1 and F2 of the diabatic potential energy curves at the intersection. The probability of electron hopping between neutral and ionic states (i.e., the electron-transfer probability) is defined as 1 − PLZ. In the above expression, H12 is the off-diagonal matrix element of the electronic Hamiltonian evaluated explicitly with the diabatic HF wave functions. We note that our model implicitly assumes that the values of the electronic coupling are unperturbed by the He environment. Figure 2 shows that the probability of electron
Figure 2. Probability of electron transfer as a function of the relative velocity between Csx and C60 species. The value of the Landau velocity is also indicated.
transfer from Cs to C60 is about 48% at the Landau velocity and decreases smoothly as the velocity increases. For the Cs2−C60 system in a collinear geometry, the probability lies at 71%, while a probability of only 33% is found for the T-shaped configuration. The orientational average of the electron-transfer probability from Cs2 to C60 is about 52%. An alternative approach based on the multistate complete-active-space second-order perturbation theory (MS-CASPT2) treatment provides consistent hopping 4286
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Figure 3. Solvation-corrected reaction paths in the diabatic representation for the formation of either the vdW Csx−C60 or ionic Cs+x −C−60 complexes at the MP2 level.
Table 2. Characteristics of the Csx−C60 Interaction at the Crossing Region in the Inside of the He2000 Droplet (See Also Figure 3)a Rx, Å
Ex, cm−1
−1 Emax kin , cm
Cs−C60
12.1
104
Cs2−C60 (T-shaped)
22.3
Cs2−C60 (collinear)
23.4
152.6 (190.7) 250.3 (251.4) 63.3 (65.0)
interaction
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AUTHOR INFORMATION
Corresponding Authors
*E-mail:
[email protected] (M.P.d.L.-C.). *E-mail:
[email protected] (A.W.H.).
121
ORCID 121
María Pilar de Lara-Castells: 0000-0001-8697-5770 Andreas W. Hauser: 0000-0001-6918-3106
a
Rx stands for the position of the crossing while Ex and Emax kin denote the total energy at Rx, that is, the sum of the MP2 interaction energy and the repulsive correction (indicated in parentheses) resulting from the extrusion of helium. Emax kin stands for the maximum kinetic energy at the Landau velocity.
Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work has been partly supported by the Spanish Agencia Estatal de Investigación (AEI) and the Fondo Europeo de Desarrollo Regional (FEDER, UE) under Grant No. MAT201675354-P and COST Action CM1405 “Molecules in Motion” (MOLIM). M.P.d.L.-C. is greatly thankful to Alexandre Zanchet, Carlos Cabrillo, and Pablo Villarreal for very helpful discussions and the CTI (CSIC) and CESGA supercomputer facilities (Spain) for the resources provided.
with C60 also occur for lighter alkali metal atoms such as sodium,1 where the repulsive He correction is expected to be much smaller. To summarize, our ab initio study clearly confirms the occurrence of a long-range electron-transfer process or “harpoon”-type reaction between a Cs2 and a C60 species inside of a He droplet. It also demonstrates a delicate balance between the distance-dependent energy costs, caused by perturbation of the helium environment, and the actual position of the crossing of neutral and ionic potential energy curves that determines the fate of the proposed reaction: It can take place for Cs2 if the molecule also rotates upon approaching the fullerene, but it is inhibited for atomic Cs, being in perfect agreement with the experimental observations.
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poise-corrected interaction potentials, and reaction energy pathways (PDF)
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REFERENCES
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.7b01910. Details of the diabatization technique, MS-CASPT2-based determinations of hopping probabilities, calculations using helium density functional theory, experimental-based estimations of crossing points, reaction energy pathways derived from a HF + Das[CCSD(T)] ansatz, test calculations on the Cs2−benzene model system, counter4287
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