Magic Numbers for the Photoelectron Anisotropy in Li-Doped Dimethyl Ether Clusters
1 Magic Numbers for the Photoelectron Anisotropy in Li-Doped Dimethyl Ether Clusters
Jonathan V. Barnes, Bruce L. Yoder, and Ruth Signorell*
ETH Zürich, Laboratory of Physical Chemistry, Vladimir-Prelog-Weg 2, CH-8093, Zürich, Switzerland * To whom correspondence should be addressed. E-mail: [email protected] ABSTRACT
Photoelectron velocity map imaging of Li(CH OCH ) n clusters (1 ≤ n ≤ 175) is used to search for magic numbers related to the photoelectron anisotropy. Comparison with density functional calculations reveals magic numbers at n=4, 5, and 6, resulting from the symmetric charge distribution with high s-character of the highest occupied molecular orbital. Since each of these three cluster sizes correspond to the completion of a first coordination shell, they can be considered as “isomeric motifs of the first coordination shell”. Differences in the photoelectron anisotropy, the vertical ionization energies and the enthalpies of vaporization between Li(CH OCH ) n and Na(CH OCH ) n can be rationalized in terms of differences in their solvation shells, atomic ionization energies, polarizabilities, metal-oxygen bonds, ligand-ligand interactions, and by cooperative effects.
1. Introduction
Magic numbers play a central role in cluster science (see references on molecular clusters ). Usually, these magic numbers are related to the high stability of clusters of certain sizes. By contrast, reports on magic numbers related to photoelectron anisotropy are comparatively sparse.
10, 11, 13, 14
This is because measurements of photoelectron angular distributions (PADs) of clusters are not so common and the modelling of cluster PADs is demanding.
5, 10, 11, 13-29
Typically, a prerequisite for the observation of magic numbers in the photoelectron anisotropy is a high cluster symmetry that results in orbitals with high fractional s-character.
10, 13, 14, 24
In our recent studies,
10, 24 we reported the first observation of magic numbers in the photoelectron anisotropy of solvated electrons in Na-doped clusters of dimethyl ether, ammonia, methanol and water. The studies have revealed that in clusters of high symmetry the solvated electron can delocalize over extended regions, forming symmetric charge distributions of high s-character. However, they have also shown that the direct experimental observation of magic clusters can be hindered by several factors. An important factor is the lack of size selection for the neutral clusters under investigation. This results in PADs that are averages of several cluster sizes, making the detection of magic numbers more difficult. Furthermore, many structural isomers with similar energies can occur in these weakly-bound systems, again making the observation of magic numbers less likely compared with systems that exhibit fewer structural isomers. Our calculations showed that in particular the strong hydrogen bonds in the Na-doped methanol and water clusters result in a large number of isomers. In addition, these systems tend to prefer non-symmetric structures with the Na and the electron pushed to one side of the cluster to minimize the perturbation of the hydrogen bond network. Given these facts, it is thus not so surprising that the clearest experimental result for a magic number cluster was found for Na-doped dimethyl ether clusters, namely for the hexamer Na(CH OCH ) which has near T h symmetry with an octahedral coordination of Na by the CH OCH molecules. The lack of strong hydrogen bonding in these clusters strongly reduced the number of isomers, and in addition the hexamer was also found to be a particularly stable structure, i.e. it is also a magic number cluster with respect to stability. High level quantum chemical calculations for Na(CH OCH ) n and Na(NH ) n clusters by Gunina and Krylov are in agreement with our previous experimental results
10, 24 and provide a detailed understanding of the underlying phenomena regarding the character of the electronic structure and the influence of structural fluctuations on the electronic properties. The present study focuses on magic numbers in the photoelectron anisotropy of Li-doped dimethyl ether clusters (Li(CH OCH ) n ). Many aspects of Li-doped molecular clusters have been investigated in detail (see refs.
9, 30-36 and references therein) but to the best of our knowledge no angle-resolved photoelectron spectra have been reported so far. Li is smaller and less polarizable than Na, which, for example, lets one expect that the almost perfect T h symmetry with octahedral coordination of the Na core in Na(CH OCH ) might be distorted in the Li-doped hexamer so that the magic number cluster might shift to another cluster size than the hexamer. The goal of the present work is to unravel how the substitution of the alkali metal in dimethyl ether clusters influences the energetics, structure, and magic numbers by a combination of experimental data and density functional theory (DFT) calculations.
2. Experiment
The experimental setup, the measurement procedures, and the data analysis are essentially identical to those used in our previous investigations of Na(CH OCH ) n clusters.
10, 24
For convenience, we repeat here the main aspects as provided in the experimental part of West et al. The experimental setup has been previously described in detail.
All measurements were performed in a velocity map imaging (VMI)
40, 41 photoelectron spectrometer, which can also function as a time-of-flight mass spectrometer. (CH OCH ) n solvent clusters were generated by pulsed supersonic expansion of a He/CH OCH gas mixture into vacuum. The solvent cluster size was varied from one molecule up to a maximum of approximately 175 molecules per cluster by varying the expansion conditions (backing pressure, gas composition, pressure, nozzle temperature) and oven temperature. The solvent clusters were doped with a single Li atom in a Li oven, which was heated to a temperature of 650 K. The resulting Li(CH OCH ) n clusters were ionized with a 266 nm pulse from an Nd:YAG laser (photon energy of 4.66 eV), which exclusively ionized the unpaired (solvated) electron. The cluster size distributions were determined by mass spectrometry, which through the cluster mass provides information on the number of solvent molecules n per cluster.
21, 24, 37
For small clusters (n≤4) we use the actual number of molecules n to assign a cluster size, while the cluster size distributions for large clusters are characterized by the average cluster size
3. DFT calculations
The experimental results are compared with various quantities ( β -parameters, vertical ionization energies IE vert , enthalpies of vaporization H vap , and dipole moments) obtained from calculations with the Gaussian program package using the dispersion corrected ωB97XD density functional with a 6-31+G* basis set. The calculations are analogous to those for Na(CH OCH ) n clusters,
10, 24 the most important aspects of which we repeat here for convenience. H vap is calculated for the neutral clusters as the total dissociation energy divided by the number of solvent monomer units. It is used here to compare cluster stabilities for different cluster sizes. The calculated total dipole moment of the different neutral clusters is used as a simple but very sensitive measure of the displacement of the charge distribution. IE vert are compared with the experimental IE max . IE vert are obtained by subtracting the energy of the neutral cluster from the energy of the ionic cluster with the same geometry. The calculated β -parameters are determined as previously explained in detail and in the supporting information of West et al. Briefly, the highest occupied molecular orbital (HOMO) is expanded in terms of atomic natural orbitals (ANOs). In order to account for the polarization of the HOMO upon solvation we use an expanded valence shell including 2p functions on Li for the ANO analysis (NBO program version 3.1). The normalized angular momentum ( ℓ ) character c of the HOMO is calculated as the sum over ANO contributions of the same ℓ . The β -parameters are then obtained from, c Eq. (2) with β ℓ determined from the Cooper–Zare formula,
22 22
RR RRRR . Eq. (3) R is the relative radial dipole matrix element of the ( ℓ +1) partial wave. We neglect the phase shift between outgoing partial waves. Furthermore, we provide here the results for R = 0.5 and for radial matrix elements that vanish at all centers except at the Li atom. We have previously shown for Na-doped clusters that the size-dependence of β (not the actual values) is almost independent of the choice of the parameters (i.e. other limiting cases for R and for radial matrix elements for all atomic centers). In Li-doped and Na-doped clusters, the unpaired electron largely retains the character of the Li and Na valence electron, respectively. The above-mentioned robustness with respect to the model parameters derives from precisely this special property of the unpaired electron in the clusters, and enables us to derive meaningful results from the simple approach in Eq. (2) and (3). Note that trends in the size-dependence of calculated and experimental β -parameters can be compared, even though their actual values cannot.
4. Results for Li(CH OCH ) n clusters As an example Figure 1 shows a photoelectron image for small Li(CH OCH ) n clusters with n≤4 together with the corresponding energy-dependent β -trace (full black line) and the photoelectron spectrum (dotted red line) as a function of the eBE. For these small clusters, the different rings in the image, the different bands in the energy-dependent β -trace, and the resolved bands in the photoelectron spectrum can be assigned to specific cluster sizes. The image and the β -trace show that the PAD remains clearly anisotropic (large values of the β -parameters at the band maxima), while the photoelectron spectrum reveals a very strong decrease in IE max by around 2eV with increasing cluster size from n=1 to n=4. For Li-doped clusters, truly size-resolved data could only be obtained up to n=4. As shown in Figure 2a, the photoelectron bands of larger clusters lie too close in energy for the current experiment to resolve specific cluster sizes. We thus assign an average cluster size
Inset: reconstructed photoelectron images of Li(CH OCH ) n clusters with n=1-4 solvent molecules. Full black line: β -trace as a function of the electron binding energy (eBE). Dotted red line: Photoelectron spectrum as a function of eBE. Figure 2: a) Photoelectron spectra of Li(CH OCH ) n clusters as a function of eBE. Dashed black line: n=1-4; Dotted green line:
Properties of
Li(CH OCH ) n clusters as a function of the number of solvent molecules n: a) calculated β -parameters, b) calculated enthalpies of vaporization H vap , c) calculated vertical ionization energies IE vert , d) calculated dipole moments, e) experimental β -parameters, f) representative mass spectrum for
Comparison of Li(CH OCH ) n (open circles) and Na(CH OCH ) n (open triangles) cluster data a) Calculated enthalpies of vaporization H vap b) Experimental ionization energies IE max c) Experimental β -parameters. The next two larger cluster sizes, Li(CH OCH ) and Li(CH OCH ) , have similarly high β -parameters as the tetramer (Figure 3a). This is rather surprising and different from the Na-case in West et al. However, it supports our hypothesis that several clusters with sizes larger than n=4 and high β -parameters are the reason for the experimentally observed trend of high β -values even up to
Isosurfaces of the HOMO of the most stable isomers of a) the Li(CH OCH ) cluster, b) the Li(CH OCH ) cluster, and c) the Li(CH OCH ) cluster. The calculated s-character of these HOMO are 100%, 95%, and 100%, respectively. The PADs of even larger clusters are still anisotropic but with β values clearly below those of n=4,5 and 6 (Figure 3a). These clusters are less symmetric than the smaller ones with correspondingly lower β values and larger dipoles. The less symmetric structures – typically with the Li and its electron pushed to one side of the cluster (see Figure S1 in the Supporting Information for n=20) allow the perturbation of the solvent molecules to be minimized, while keeping the unpaired electron close to the Li core and maximizing the strong favorable Li-O interactions in the 1 st solvation shell. Table S2 in the Supplementary Information lists the properties of some higher lying isomers. Among them are also highly symmetric isomers with high β values, such as isomer (b) for n=10. For Li-clusters, H vap decreases almost continuously with increasing cluster size (Figure 3b). As expected, for very large clusters it converges to the calculated bulk value of pure (without Li) dimethyl ether of about 0.23eV (experimental bulk value around 0.29eV) . For small clusters with one ligand shell (up to n~6), H vap is comparatively high because the strong Li-O bond dominates – partially counterbalanced by the steric interaction in the increasingly crowded ligand shell. With more ligands (n ≳ vap is increasingly ‘diluted’ by the much weaker ligand-ligand interaction in the outer shells and gradually converges to the bulk value. For IE vert , pronounced changes are only observed until the completion of the first solvation shell at n=4. The extension of the 1 st shell in n=5 and 6 retains a balance between the increase in the number of strong Li-O interactions and their weakening as a consequence of ligand crowding (bond lengthening). The further slow decrease of IE vert beyond n=6 can be attributed to increasing polarization effects (as the cluster’s polarizability increases with its size).
5. Comparison of Li(CH OCH ) n and Na(CH OCH ) n clusters Figure 4 provides a comparison of Li(CH OCH ) n and Na(CH OCH ) n cluster data. The behavior of H vap for the Na-clusters differs pronouncedly from that of the Li-clusters (Figure 4a). Small Na-clusters have a lower H vap that increases with cluster size, while small Li clusters have a higher H vap that decreases with cluster size. The maxima for H vap are reached at the hexamer of Na and at the monomer for Li. The generally lower H vap for small Na-clusters can likely be attributed to the weaker Na-O bond compared with the Li-O bond. As mentioned in section 4, the decrease of H vap is consistent with a weakening (i.e. lengthening) of the Li-O bond because of the increased crowding of ligands in the 1 st solvation shell. Given the much larger atomic radius of sodium ligand crowding plays a less important (if any) role in small Na(CH OCH ) n clusters. This would lead to the expectation of a roughly constant H vap until the 1 st solvation shell is complete at n=6. The increase of H vap observed instead points toward significant cooperative effects, possibly resulting in part from (weak) hydrogen bonding interactions between the ligands. For larger clusters (n>6), H vap decreases again, but more slowly than for Li-clusters. This is in part a trivial consequence of the smaller difference between the Na-O bond strength and the ligand-ligand interaction (the “dilution” per ligand added is less in Na- than in Li-clusters). Another contributing factor is the larger polarizability of the 3s unpaired electron of Na as compared with the 2s electron of Li. The former more easily deforms to adapt to its position on the cluster surface. In contrast to the trends in H vap , the trends in IE max are qualitatively identical for Na- and Li-clusters (Figure 4b). Strong decreases are only observed before the closure of the first solvation shells (at n=4 for Li and n=6 for Na), while for larger clusters the values of IE max drop only very slowly as a result of the increasing overall polarizability of the cluster. For larger Li clusters, the absolute values of IE max observed experimentally lie systematically above those of the Na clusters by about 0.3 eV. This difference approximately equals the difference between the ionization energies for atomic Na and Li (5.14 eV and 5.39 eV, respectively). The unpaired (surface-solvated) electron in the cluster apparently still feels the core it belonged to. This is consistent with the results of our DFT calculations. As mentioned above, the most stable larger clusters tend to have the metal core and the electron located at one side of the cluster close to the surface. The electron is thus still close to the respective metal core, which might explain the conservation of the shift between cluster and atomic metal. Finally, Figure 4c compares the β -parameters for the two cases. The occurrence of magic clusters related to the anisotropy at n=6 for Na clusters and at n=4,5, and 6 for Li clusters was already discussed in West et al. and in section 4. Here, we additionally point out the general downshift of β of Na-clusters compared with Li-clusters observed experimentally for essentially all cluster sizes. This phenomenon is reproduced at least qualitatively by the DFT calculations. A lowering of β results from the polarization of the HOMO upon solvation, which gives rise to higher angular momentum components l (essentially l = 1) of the HOMO. A more polarizable atomic orbital is more easily distorted (polarized), i.e. more easily acquires higher l components upon solvation of the atom. The lower β values of the Na-clusters can thus be explained by the higher polarizability of the 3s electron compared with the 2s electron of Li.
6. Summary
This paper compares properties of neutral Li(CH OCH ) n and Na(CH OCH ) n clusters with a focus on magic numbers related to the photoelectron anisotropies of the highest occupied molecular orbital; i. e. the solvated electron which can delocalized over extended cluster regions. In Li-doped clusters, magic numbers are observed at n=4,5, and 6 as a result of the completion of the first solvation shell. Such “isomeric motifs of the first coordination shell” were not observed for Na-doped clusters, which showed a distinct magic cluster at n=6. The difference between the two alkali metals seems to arise from a balance between the electronic stabilization by the metal-oxygen bonds and the steric destabilization due to crowding of ligands. The general lowering of the β parameters of around 0.25 for Na-clusters compared with Li-clusters for larger clusters with up to
9, 50-61
ACKNOWLEDGMENT
Financial support was provided by the Swiss National Science Foundation under project no. 200020_172472 and by the ETH Zürich.
This project has received funding from the European Union’s Horizon 2020 research and innovation program from the European Research Council under the Grant Agreement No 786636.
We are very grateful to David Stapfer and Markus Steger from our workshop for their help in the setup of the Li oven, and to Dr. David Luckhaus for his help with the calculations. REFERENCES (1) Vafayi, K.; Esfarjani, K. Abundance of Nanoclusters in a Molecular Beam: The Magic Numbers for Lennard-Jones Potential.
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Jonathan V. Barnes, Bruce L. Yoder, and Ruth Signorell*
ETH Zürich, Laboratory of Physical Chemistry, Vladimir-Prelog-Weg 2, CH-8093, Zürich, Switzerland * To whom correspondence should be addressed. E-mail: [email protected] Table S1 : Experimental Results of the reconstructed images using MEVIR. n,
0 5.39 - 1 4.30 1.39 2 3.54 1.00 3 2.95 0.83 4 2.31 1.25 12 18 2.11 1.10 14 21 2.06 1.08 20 52 1.89 1.12 25 63 1.85 0.88 27 70 1.84 0.97 33 83 1.74 1.07 52 142 1.72 0.94 63 175 1.70 1.01 3
Table S2 : Results of the DFT calculations (wB97XD/6-31+G*) using Gaussian 09. Enthalpies of vaporization H vap , dipole moments, vertical ionization energies IE vert , and β -parameters. The table has been divided into two sections in order to display it more effectively. n=2: bent. n=3: flat trigonal pyramid. n=4: tetrahedron. n=5: trigonal bipyramid. (a) n=4 (tetrahedron) with one ligand added in the 2 nd shell. n=6: octahedron. (a): n=5 (trigonal bipyramid) with one ligand added in the 2 nd shell. (b) n=4 (tetrahedron) with two ligands added in the 2 nd shell. n=7: n=6 (octahedron) with one ligands added in the 2 nd shell. (a) n=5 (trigonal bipyramid) with two ligands added in the 2 nd shell. n=8: n=5 (trigonal bipyramid) with three ligands added in the 2 nd shell. (a) n=6 (octahedron) with two ligands added in the 2 nd shell. n=9: n=5 (trigonal bipyramid) with four ligands added in the 2 nd shell. (a) n=6 (octahedron) with three ligands added in the 2 nd shell. n isomer H vap (eV) dipole (D) IE vert (eV) β
0 - - 5.37 2.00 1 0.428 6.3 4.19 1.76 2 0.429 7.8 3.38 1.42 3 0.421 8.8 2.77 1.08 4 0.405 0.0 1.95 2.00 5 0.413 8.5 1.81 1.86 a 0.384 13.1 1.82 1.49 6 0.387 0.1 1.70 2.00 a 0.380 13.8 1.76 1.44 b 0.365 15.5 1.72 1.34 7 0.371 14.3 1.68 1.45 a 0.363 16.0 1.70 1.21 8 0.356 17.4 1.64 1.14 a 0.353 15.9 1.62 1.29 9 0.346 17.5 1.60 1.12 a 0.344 16.3 1.60 1.19 4
Table S2 (continued) : Results of the DFT calculations (wB97XD/6-31+G*) using Gaussian 09. Enthalpies of vaporization H vap , dipole moments, vertical ionization energies IE vert , and β -parameters. The table has been divided into two sections in order to display it more effectively. n=10: n=6 (octahedron) with four ligands added in the 2 nd shell on one side (Li near the surface of the cluster). (a) similar to 10, but Li further away from the surface of the cluster. (b) n=6 (octahedron) with four ligands distributed symmetrically in the 2 nd shell (Li at the center of the cluster). n=12: n=6 (octahedron) with six ligands added in the 2 nd shell on one side (Li near the surface of the cluster). (a) similar to n=12, but one ligand of the 2 nd shell moved closer to the Li-side. (b) n=10 with one ligand added on the far side and one on the near side of Li. (c) similar to n=12b, but one ligand in the 2 nd shell rearranged so that Li is more centered. (d, e) n=10b with two ligands added in the 2 nd shell, distorting the symmetry. n=14: n=6 (octahedron) with eight ligands added in the 2 nd and 3 rd shell on one side (Li near the surface of the cluster). (a) similar to n=12d, e with two more ligands added in the 2 nd shell, Li slightly off-center. (b) n=14 with one ligand moved from the 2 nd to the 3 rd shell. n isomer H vap (eV) dipole (D) IE vert (eV) β
10 0.342 19.0 1.56 1.21 a 0.335 18.1 1.52 0.98 b 0.323 2.5 1.40 2.00 12 0.331 21.1 1.47 1.16 a 0.329 18.6 1.51 0.79 b 0.322 19.2 1.49 0.91 c 0.321 18.4 1.46 0.71 d 0.318 18.0 1.47 0.68 e 0.318 17.4 1.48 0.66 13 0.326 20.8 1.48 1.07 14 0.322 21.9 1.46 1.00 a 0.318 15.0 1.50 0.77 b 0.312 22.2 1.43 0.84 15 0.319 22.7 1.39 0.83 a 0.318 21.1 1.45 0.90 b 0.232 7.0 3.97 1.75 18 0.312 20.3 1.33 0.43 a 0.308 18.5 1.38 0.38 20 0.318 21.1 1.35 0.52 a 0.312 19.2 1.33 0.32 5 n=15: n=6 (octahedron) with nine ligands added in the 2 nd and 3 rd shell on one side (Li near the surface of the cluster). (a) similar to n=15, but ligands in the outer shell rearranged to yield an overall flatter cluster. (b) crystalline slab (monolayer) with Li at one edge. n=18: n=6 (octahedron) with twelve ligands added in the 2 nd and 3 rd shell to yield a compact slightly flattened cluster shape with an off center Li closer to one surface. (a) similar to n=18, but more spherical in shape with Li closer to the center. n=20: n=18 with two ligands added on the far side of Li. (a) n=18a with two ligands added, Li closer to the center. Figure S1 : Isosurface of the HOMO of the most stable isomer of the Li(CH OCH )20