Microsolvation of Ca2+ cation in small Xen clusters: Structures and relative stabilities

Journal Pre-proof Microsolvation of Ca

2+

cation in small Xen clusters: Structures and relative stabilities

Safa Mtiri, Maha Laajimi, Houcine Ghalla, Brahim Oujia PII:

S0921-4526(19)30736-7

DOI:

https://doi.org/10.1016/j.physb.2019.411849

Reference:

PHYSB 411849

To appear in:

Physica B: Physics of Condensed Matter

Received Date: 29 July 2019 Revised Date:

27 September 2019

Accepted Date: 1 November 2019

2+ Please cite this article as: S. Mtiri, M. Laajimi, H. Ghalla, B. Oujia, Microsolvation of Ca cation in small Xen clusters: Structures and relative stabilities, Physica B: Physics of Condensed Matter (2019), doi: https://doi.org/10.1016/j.physb.2019.411849. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.

Microsolvation of Ca2+ cation in small Xen clusters: structures and relative stabilities Safa Mtiri1*, Maha Laajimi1, Houcine Ghalla1, and Brahim Oujia1,2 1

University of Monastir, Faculty of Sciences, Laboratory of Quantum and statistical Physics LR18ES18, Monastir, Tunisia 2

∗

University of Jeddah, Faculty of Science, Physics Department, Jeddah, Kingdom of Saudi Arabia

Corresponding author:

S. Mtiri University of Monastir, Faculty of Sciences, Laboratory of Quantum and statistical Physics LR18ES18, Monastir, 5000, Tunisia. Tel : +216 73 500 276 ; Fax : +216 73 500 278. E- mail address: [email protected]

1

Abstract The structures and relative stabilities of the Ca2+Xen (n=1-18) clusters have been carried out using two methods: the pairwise/Monte Carlo Basin-Hopping and density functional theory methods. The lowest energy structures have been determined using classical energy minimization method where the total interactions have been obtained as a sum of pairwise potentials. The DFT calculations have been performed using the dispersion-corrected functional B97D3. For both methods the stable structures are characterized by the Ca2+ being coated by a shell of xenon atoms. For the smallest sizes n<4, we show that the many-body contributions in the DFT calculations stabilize the linear or planar structures while the twobody contributions in the pairwise calculations yield three-dimensional structures. For larger sizes, capped square antiprism (CSA) packing is dominantly found and the first solvation shell is saturated at n=10. The stability analyses have shown magic numbers which are consistent with the CSA growth sequence.

Keywords: Ca2+Xen; Solvation; Stability; pairwise; DFT; dispersion-correction

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1. Introduction The interest in cluster physics has been growing progressively over the recent years [1-8] and has attracted a large number of experimental as well as theoretical investigations. This is firstly due to the basic physical problems raised by aggregates like the transition species between molecules and solids, and secondly due to their technological implications and beneficial applications in science. Within, the inspirational subject, the mass spectrometry seems as one of the most useful techniques to investigate unsupported clusters: mostly abundant cluster sizes called magic numbers giving rise to stable structures and reflecting closed electronic shells and electronic sub-shells. The noble gas clusters may serve as prototypes for numerous structural and optical properties' studies of the solvated systems, thanks to the essentially nonbonding character of the rare gas atom. As a consequence, the geometrical structure of the atomic Van der Waals' clusters could be deduced from hard sphere packing models. A series of experimental observations are available concerning the rare gas clusters because of their filled band electronic character. The appearance of magic numbers shown in mass-spectrometric studies on Xen [9] and Arn [10] may be explained by the adaptation of icosahedral sphere packing. These results were also proved by the electron diffraction study of Arn clusters [11]. In these systems, the atoms are organized in shells around a central atom. For these systems, the closing-shell occurs when the total number of atoms is n= 13, 55, 147, etc, giving rise to stable clusters with icosahedral geometry [12]. In addition, the sub-shells are filled for n= 19, 23, 26, 29, 32, etc. and have poly-icosahedral geometry [13]. Every additional atom occupies the available position offered by the neighboring atoms until the completion of shells and sub-shells. A number of experiments have been previously performed on rare gas clusters doped with metal ions such as Na+, K+, Cs+, Mg+, Sr+, Al+ and In+ [14-18]. The magic numbers are found to be identical to those observed in pure rare gases clusters with icosahedral growth scheme

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when the size of the ion is comparable to that of the rare gas atom. However, different magic numbers are found if the ion is relatively smaller than the rare gas. The non-icosahedral geometry could be explained by a simple sphere packing model [14]. The CSA packing was deduced for Na+Rgn (Rg = Ar, Kr) [14] and Mg+Rgn (Rg = Kr, Xe) [15]. Octahedral structures were observed on the mixed ionic clusters Li+Rgn (Rg= Ne, Ar, Kr) [18]. While icosahedral packing was associated for Al+Rgn (Rg = Ar, Kr) [14], In+Rgn (Rg = Ar, Kr) [14], K+Rgn (Rg = Ar, Kr) [15], Mg+Arn [16] Sr+Arn [17]. The theoretical studies on the structures and stabilities of clusters require, necessarily, the determination of the potential energy surface. Therefore, two alternatives of calculation may be in competition. (i) The first alternative consists of using ab-initio calculations (HartreeFock, post-Hartree-Fock, DFT) [19-24]. Currently, DFT methods have been widely used to investigate covalent or metal clusters, while their difficulty to account properly for the dispersion forces precludes from using them with confidence for Van der Waals clusters. Fortunately, thanks to the long-range dispersion correction carried out by Grimme [25] the DFT calculations can be successfully be applied on Van der Waals clusters. Actually, the execution of DFT calculations with a well-chosen functional and a sufficiently extended atomic orbital generally makes it possible to calculate geometric structures with a high precision. However, it is too demanding in terms of CPU-time for the larger sizes. (ii) The second alternative consists of modeling the potential energy surface by additive analytical potentials constructed from ab-initio or experimental data [26-30]. In the present work, we study the microsolvation of Ca2+ cation in Xen clusters up to n=18 by using both pairwise and DFT approaches. It is worthily noticeable that the pairwise model deals with 2-body interactions Ca2+-Xe and Xe-Xe where the charge remains essentially localized on the impurity. Whereas the DFT model includes the many-body effects, especially the interaction between the dipoles induced by the cation on the xenon atoms. For both

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approaches the most stable structures of Ca2+Xen clusters are determined and the stabilities of these clusters are discussed based on structural properties and energetic indicators such as the bending energy per number of xenon atoms e(n), the first and second order difference in energy (∆E and ∆2E) and the HOMO-LUMO energy gap. The paper is organized as follows: First, we explain the theoretical methods in section 2, then we present and discuss the obtained results in section 3 and we finish with the conclusion in section 4. 2. Theoretical methods 2.1 Pairwise method The total potential energy V(r) of Ca2+Xen clusters is expressed as the sum of Ca2+-Xe and Xe-Xe interaction potentials: n

V(r) = ∑ VCa i =1

n

2+

−Xei

+ ∑∑ VXe −Xe j

(1)

i

i =1 i < j

For the Ca2+-Xe interaction, we used the analytical form potential of Tang and Toennies that contains a short-range repulsion, a dispersion expansion involving Ck/rk terms, and a 1/2r4 polarization contribution from the xenon atom [31]: VCa2+ −Xe(r) = Aexp(− br) − C4αXe / 2r4 − C6 / r6 − C8 / r8 −C10 / r10 − C12 / r12

(2)

Where the parameters A, b, Ck, k=4, 6, 8, 10, 12 adjusted to reproduce the Ca2+-Xe potential calculated by Gardner et al. [32] at the coupled cluster level with single, double and perturbative triple excitations. In Fig. 1, we have plotted the reproduced Ca2+-Xe potential compared with the Gardner’s one. We can see that the two curves are quite consistent. Thus, the parameters reproduced the training set of Gardner with good accuracy and they localized very well the equilibrium distance (Re = 3.040 Å) and the well depth (De = 9550 cm-1). For the Xe-Xe interactions, we have employed the pairwise additive Lenard-Jones (LJ) potential [33]:  σ 12  σ 6  VXe−Xe (rij ) = 4ε   −     rij   rij    

5

(3)

Where ε represents the well depth and σ is the finite distance at which the inter-particle potential is zero, which are equal to 0.000882 Hartree and 7.295 a0, respectively. To find the geometry corresponding to the lowest energy, we have used the Monte Carlo technique; it gives a high likelihood of finding the global minimum by exploring the potential energy surface [34, 35]. Indeed, the system can pass from a potential well to another, crossing the potential barriers and searching the lowest energy localizing all the local minimums until finding the global minima. This simulation is mostly used in the case of small clusters. We have employed the Wales and Doye [34] numerical optimization algorithms named “BasinHopping”. By the use of the conjugate gradient technique, for each size n we performed 10.000 local minimizations, with about 106 cycles. The first step in our calculation is to test the optimization procedure for the diatomic case, Ca2+Xe. We found that the equilibrium distance Re and the dissociation energy De are about 3.038 Å and 9550 cm-1, respectively, which are in excellent agreement with the used potential. This fact confirms the accuracy of our theoretical method. 2.2 DFT calculations The DFT calculations are considered powerful tools for studying the structural and electronic properties of molecular clusters. All calculations are performed using Gaussian 09 program [36] and GaussView as a visualization package [37]. The most stable geometries of the Ca2+Xen (n=1-18) clusters are optimized by using the DFT dispersion-corrected functional B97D3 [38] (B97 with Grimme's D3BJ empirical dispersion correction involving Becke– Johnson damping) [39] with ZPVE correction, and without any symmetry constraints. For the calcium atom, we have employed the polarized basis set 6-311++g(3df,3pd). For the xenon atom, we have used Stuttgart-Dresden (SDD) effective core potential (ECP) and basis set. The stability of each size of clusters is confirmed by the absence of any imaginary frequency within the vibrational calculation. In order to evaluate the accuracy of the adopted DFT

6

scheme, we have carried out calculations on Ca2 and Xe2 dimers. The calculated Ca-Ca and Xe-Xe bond lengths are equal to 4.436 Å and 4.512 Å, respectively, which are in fair agreement with corresponding experimental values (Ca-Ca=4.282; 4.279; 4.277 Å [40-42], and Xe-Xe=4.362 Å [43]). At the same level of theory, we have calculated the Ca2+-Xe bond length. The value is found around 3.033 Å, which is in good agreement with the theoretical data 3.040 Å [32]. These results indicate that our DFT scheme is reliable enough to be applied for determining the properties of small Ca2+Xen cluster sizes. 3. Results and discussions 3.1 Lowest-energy structures The lowest-energy structures of the Ca2+Xen clusters obtained by both pairwise and DFT/ B97D3 methods are presented in Figs. 2 and 3 along with the point groups. The most stable structure of Ca2+Xe2 cluster, obtained by pairwise calculation, is triangular with Ca2+ occupying the summit position with a bond angle (XeCa2+Xe) of 91.02°. While the DFT calculation produces a linear geometry with solvated ion core. The Ca2+Xe3 cluster presents trigonal pyramid geometry with the pairwise calculation while it holds a planar geometry with the DFT calculation. For Ca2+Xe4, we found two trigonal pyramids which share a common side by the pairwise calculation; whereas we found a more regular geometry, by the DFT calculation, where the cation Ca2+ is fully covered by the xenon atoms. For Ca2+Xe5 cluster, a square pyramid structure is suggested by the pairwise calculation; while a trigonal bi-pyramidal structure with solvated ion core is suggested by the DFT calculation. For Ca2+Xe6 cluster, both methods gave a regular octahedron with high symmetry Oh. The pairwise model showed that the xenon atoms are progressively coating the cationic impurity Ca2+ and the growth mechanism at the smallest sizes n≤5 is predominated by the attractive effect of Xe-Xe bonds. At the level of n=6, the cation Ca2+ appears totally solvated by the xenon atoms and the solvation process becomes more isotropic and predominated by

7

the Ca2+-Xe bonds. According to the DFT model, the solvation process is more regular and more isotropic where the xenon atoms are placing themselves away from each other as possible in order to optimize the dipolar interactions in such a way that the Ca2+ cation appears always covered by them. For the upper sizes n≥6, both models yield similar geometries. The Ca2+Xe7 cluster presents capped trigonal prism geometry. One more xenon atom leads to more stable cluster Ca2+Xe8 with square antiprism geometry. The latter structure has other than the triangular faces, two quadratic faces which are able to accept capping atoms and so leads to a more closed structure. At the size n=9, a tri-capped trigonal prism structure was obtained. Concerning the Ca2+Xe10 cluster, we found bi-capped square antiprism (bi-CSA) geometry. Among the geometries discussed previously, the sizes n=6, 8 and 10 are relatively the most stable structures from a symmetric point of view. However, we cannot associate the completion of a solvating shell to the clusters Ca2+Xe6 and Ca2+Xe8 because they do not constitute as yet a possible core around which the larger clusters grow. Otherwise, the Ca2+Xe10 cluster forms the basic core for the next sizes and the extra atoms are occupying the triangular faces of the bi-CSA structure. As shown in Fig. 3, the growth pattern occurs around the Ca2+Xe10. The additional atoms are now being placed further away from the impurity without perturbing the core structure Ca2+Xe10. Furthermore, after the completion of the first solvation shell, the additional xenon atoms have to bind to the other xenon atoms and will be there a competition between first-neighbor Xe-Xe bonds and Ca2+-Xe bonds of the second shell. So, the growth mechanism will be predominated by the Xe-Xe bonds. The series of Ca2+Xen is completed with the Ca2+Xe18, where a sub-shell is filled. Herein, the eight atoms occupy the lateral faces of the bi-CSA geometry. The main source of error in the pairwise model is that the charge remains essentially localized on the cation, hence the repulsive interaction between the dipoles induced by the cation on the

8

xenon atoms especially those which are located near the cationic center is neglected. So, the anisotropic growth mechanism shown in the small structures given by the pairwise model is naturally expected as the consequence of Xe-Xe interactions being more attractive. The completion of shells and sub-shells occurs at a certain sequence, that every additional rare gas atom occupies the position which offers the maximum number of neighbors. The lowest energy geometries illustrate the main growth regime of Ca2+Xen clusters. For both methods the stable structures are characterized by the calcium cation being coated by a shell of xenon atoms, as expected from simple energetic arguments. CSA packing is dominantly found and the additional atoms are occupying sites on Ca2+Xe10 cluster. 3.2 Relative stabilities of Ca2+Xen clusters The stabilities of the Ca2+Xen clusters are discussed on the basis of the bending energy per number of xenon atoms e(n), the first and second difference in energy ∆E and ∆2E, and the HOMO-LUMO energy gap Eg. Fig. 4 illustrates the size dependence of the binding energy per number of xenon atoms computed with both methods. The e(n) curve, calculated by the pairwise model, presents two regimes with different slopes for n<6 and n>6. The energy values increase until n=6, each additional xenon atom gaining energy due to its neighboring Xe-Xe contacts. Above 6 xenon atoms this gain is lost due to the crowding in the solvation shell. We notice that the first regime is associated with the predominance of the attractive interactions between xenon atoms and the growth mechanism is anisotropic. At n=6, the cation is appeared totally solvated with the formation of a regular octahedron and the growth mechanism becomes more isotropic. However, With the DFT calculations the Xe-Xe interactions are less attractive due to the repulsive effect of the interaction between the dipoles induced by the Ca2+ on the xenon atoms. Consequently, the growth mechanism of Ca2+Xen clusters is more regular and the cation is always found covered by the xenon atoms.

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To more evaluate the relative stabilities of Ca2+Xen clusters, it is preferable to compare a cluster having n xenon atoms with the near neighbors (n-1) and (n+1). In Fig. 5, we have plotted the first difference in energy ∆E calculated by pairwise and DFT methods. The ∆E variations calculated by the pairwise method describe the three regimes discussed above. A little size effect in the size range n<6, this regime is associated with the Xe-Xe attractive forces. A regular variation in the size range 6≤n≤10, this regime is associated with the predominance of the Ca2+-Xe bonds and the solvation process becomes more regular and more isotropic. Then, above n=10, the effect of the Ca2+-Xe bonds decreases and the Xe-Xe bonds will be the predominant and therefore the energy differences ∆E show a little size effect in this size range. Whereas, the ∆E variations calculated by the DFT method describe two regimes. A regular variation in the size range n≤10, where the growth mechanism is associated with the predominance of the Ca2+-Xe bonds and the solvation process is isotropic. After the completion of the first solvation shell, the growth mechanism is associated with the predominance of the Xe-Xe bonds and a little size effect is observed in this size range. The stabilities of the Ca2+Xen are worthy discussed in term of the second difference in energy ∆2E. This indicator magnifies the special stabilities of experimental relevance in mass spectrometry abundances. The size dependence of ∆2E is depicted in Fig. 6. Same features of ∆E are noted. Three regimes with pairwise approach and two regimes with DFT approach. Moreover, we notice magic numbers at sizes n=6, 8, 9, and 10 which prove the stable character of these clusters. A peak at n=4 is observed in ∆2E curve given by DFT method related to the tetrahedron regular geometry. As a final comment on those energetic properties, the variations of the first and the second differences in energy confirm both the repulsive nature of the interaction between induced dipoles considered in the DFT calculation, and their decreasing quantitative importance once the first solvation shell is saturated.

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Another energetic indicator of stability, the HOMO-LUMO gap is calculated for the lowest energy structures of Ca2+Xen clusters. The size dependence of the HOMO-LUMO energy gap of Ca2+Xen clusters is plotted in Fig.7. The curve shows two regimes n≤10 and n>10. The clusters with sizes n≤10 are more stable and the Ca2+Xe4 and Ca2+Xe6 clusters exhibit the largest HOMO-LUMO gap which indicates the stable character of the tetrahedron and the octahedron. After the completion of the first solvation shell at n=10, no significant variation was observed. According to the energetic analysis, one may conclude that the size dependence of the different energetic quantities support geometrical results. 3.3 Ca2+Xen clusters via hard sphere packing model Several theoretical and experimental studies demonstrate the icosahedral symmetry for the pure rare gas clusters [9-11]. Whereas, herein the structural and energetic examination of the Ca2+Xen clusters using the pairwise and DFT/ B97D3methods has shown that CSA packing is more favorable. This result could be explained by the hard sphere packing model reported by Lüder et al. [14]. In fact, for close packing of spheres, the most stable clusters expected are those which have closed shells of atoms around a central one. For rare gas clusters doped with one metal ion complexes, we can assume that the metal ion is confined in the center and surrounded by the rare gas atoms. To obtain high symmetry and dense packing, the arrangement of the rare gas atoms around the metal ion is represented by polyhedral geometries. These polyhedra exhibit stable structures only if the size of the central atom fits into the cavity dimensions formed by the rare gas atoms. This condition is executed if the radius RM of metal and that of the rare gas Rx should fulfilled the following condition:

RM RX

π 2π    (2 + cos( k ) − cos( k ))  ≤  2π   (1 − cos( )  k 

with k=n/2 represents the number of atoms per ring. 11

1/ 2

−1

(4)

If the ratio RM/RX is in the range of 0.414-0.645, the CSA is preferred. Otherwise, if the ratio RM/RX is in the range of 0.645-0.902 the icosahedral packing is preferred. In this context, the ionic and the Van der Waals radii of Ca2+ cation and the Xe atom are respectively 1.16 Å and 2.20 Å [44, 45]. We have obtained RM/RX= 0.527 which proves the CSA growth for Ca2+Xen clusters. 4. Conclusion The lowest energy structures of Ca2+Xen clusters have been computed using pairwise and DFT methods. The DFT calculations have been performed by using the B97D3 dispersed functional including the dispersion corrections (D3). The structures obtained by the two methods are different in small sizes. Within the DFT method, the solvation process is more isotropic and the Ca2+ impurity was found to be always solvated by the xenon atoms. While according to the pairwise method, the solvation process has started at n=6. This difference is due to the repulsive effect of the interaction between dipoles induced by the impurity on the xenon atoms which is considered in the DFT calculation. However, this effect decreases once the first solvation shell is completed at n=10 with the formation of a bi-CSA. Consequently, similar geometries are found for the upper sizes n≥10 where the additional atoms are occupying the triangular faces of the bi-CSA. The solvation process is described by two regimes. The first one is predominated by the impurity-xenon bonds which are responsible for the built of the first solvation shell. The second regime is predominated by the xenon-xenon bonds. The relative stabilities of the Ca2+Xen clusters were examined by computing the bending energy per number of xenon atoms, the first and second order difference in energy and the HOMO-LUMO energy gap. Accordingly, magic number were found at n= 6, 8, 9 and 10. Finally, the CSA growth for the Ca2+Xen clusters was proved by using the hard sphere packing model. The present theoretical trends can be used as a support to perform any experimental characterization of the Ca2+Xen clusters.

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[32] A.M. Gardner, C.D. Withers, T.G. Wright, K.I. Kaplan, C.Y.N. Chapman, L.A. Viehland, E.P.F. Lee, W.H. Breckenridge, Theoretical study of the bonding in Mn+RG complexes and the transport of Mn+ through rare gas (M=Ca, Sr, and Ra; n=1 and 2; and RG=He–Rn), J. Chem. Phys. 132 (2010) 054302. [33] J.E. Jones, A.E. Ingham, On the calculation of certain crystal potential constants, and on the cubic crystal of least potential energy, Proc. Royal Soc. A 107 (1925) 636-653. [34] D.J. Wales, J.P. Doye, Global Optimization by Basin-Hopping and the Lowest Energy Structures of Lennard-Jones Clusters Containing up to 110 Atoms, J. Phys. Chem. A 101 (1997) 5111-5116. [35] D.J. Wales, H.A. Scheraga, Global Optimization of Clusters, Crystals, and Biomolecules, Science 285 (1999) 1368. [36] M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, H. Nakatsuji, Gaussian 09, Revision D. 01, Gaussian, Inc., Wallingford, CT, 2010. [37] R.I. Dennington, T. Keith, J. Millam, GaussView, Version 5.0.8, Semichem. Inc. Shawnee Mission, KS, 2008. [38] A. Austin, G.A. Petersson, M.J. Frisch, F.J. Dobek, G. Scalmani, K. Throssell, A density functional with spherical atom dispersion terms. J. Chem. theory Comput. 8 (2012) 49895007. [39] S. Grimme, S. Ehrlich, L. Goerigk, Effect of the damping function in dispersion corrected density functional theory, J. Comp. Chem. 32 (2011) 1456-1465. [40] W.J. Balfour, R.F. Whitlock, The Visible Absorption Spectrum of Diatomic Calcium, Can. J. Phys. 53 (1975) 472-485. [41] O. Allard, C. Samuelis, A. Pashov, H. Knöckel, E. Tiemann, Experimental study of the Ca2 1S+1S asymptote, Eur. Phys. J. D 26 (2003) 155-164.

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17

18000

Gardner potential Our potential

15000 12000

-1

Energy (cm )

9000 6000 3000 0 -3000 -6000 -9000 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

R (Å)

Fig. 1. The potential of Ca2+-Xe in comparison with the Gardner’s one.

1

Pairwise

B97D3

1- C∞v

2- C2v

1- C∞v

2- D∞h

3- C3v

4- C2v

3- D3h

4- Td

5- C4v

6- Oh

5- C2v

6- Oh

7- C2v

8- D4d

7- C2v

8- D4d

9- D3h

10- D4d

9- D3h

10- D4d

Fig. 2. Minimum energy structures of the Ca2+Xen clusters up to n=10 calculated by pairwise and DFT/B97D3 methods.

2

Pairwise

B97D3

11- Cs

12- Cs

13- C1

14- C1

15- Cs

16- C1

17- Cs

18- D4d

Fig. 3. Minimum energy structures (Front and Top views) of the Ca2+Xen clusters (n=11-18) calculated by pairwise and DFT/B97D3 methods. 3

DFT Pairwise

-1

e(n) (cm )

9000

6000

3000 0

2

4

6

8

10

12

14

16

18

Size n

Fig. 4. Size dependence of the binding energy per number of Xe atoms for the lowest-energy structures of Ca2+Xen clusters.

4

10000

Pairwise DFT

-1

∆E (cm )

8000

6000

4000

2000

0

2

4

6

8

10

12

14

16

18

Size n

Fig. 5. Size dependence of the first difference in energy ∆E for the lowest-energy structures of Ca2+Xen clusters.

5

-1

∆2E (cm )

3000

Pairwise DFT

2000

1000

0

2

4

6

8

10

12

14

16

18

Size n Fig. 6. Size dependence of the second difference in energy ∆2E for the lowest-energy structures of Ca2+Xen clusters.

6

45000

-1

Eg (cm )

40000

35000

30000

25000

0

2

4

6

8

10

12

14

16

18

Size n

Fig. 7. Size dependence of the HOMO-LUMO energy gap for the lowest-energy structures of Ca2+Xen clusters.

7

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: