Spin-Orbit Coupling Effects on Ligand-Free Icosahedral Matryoshka

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Spin-Orbit Coupling Effects on LigandFree Icosahedral Matryoshka Superatoms Feiyun Long, Haitao Liu, Dafang Li, and Jun Yan J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b12186 • Publication Date (Web): 22 Feb 2017 Downloaded from http://pubs.acs.org on February 24, 2017

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The Journal of Physical Chemistry

Spin-Orbit Coupling Effects on Ligand-Free Icosahedral Matryoshka Superatoms Feiyun Long, Haitao Liu,∗ Dafang Li, and Jun Yan Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing 100088, China E-mail: [email protected]

∗

To whom correspondence should be addressed

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Abstract With the help of density functional theory, a series of matryoshka superatoms X@Y12 @X20 (X=Ge, Y=Zn; X=Sn, Y=Mg, Mn, Zn or Cd; X=Pb, Y= Mg, Mn, Cd or Hg) with icosahedral symmetry has been extensively studied, to focus on the influence of the spin-orbit coupling on geometries, stabilities, electronic structures and magnetic moments for these clusters. Generally speaking, the effect of spin-orbit coupling is highly correlated with composition elements of these clusters. Ge@Zn12 @Ge20 is little affected by the spin-orbit coupling, while clusters containing Sn atom will generally undergo a moderate influence on their atomization energy, HOMO-LUMO gap and projected density of states. For clusters with Pb atoms, the effect of spin-orbit coupling could be observed distinctly in most cases. Our results demonstrate that the spin-orbit coupling can play a substantial role in superatoms containing heavy elements.


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Introduction In the cluster science, the “superatom” is generally a kind of of well-designed clusters which could exhibit some atom-like characteristics. 1–8 A great deal of attractive properties could be found for specific superatoms, 9–14 and then their enormous potential in massive application fields could be drawn out, such as, the field-effect transistor device, 15 the high-Tc superconductivity, 16,17 the chemical catalysis, 18,19 etc. In addition, the superatoms play a conspicuously important role in the cluster-assembled materials, 4,20–22 which has attracted great attentions in the materials science. 23–26 Therefore, it is an essential task to search for new superatoms with peculiar structures and novel properties. The bimetallic cluster (also called “nanoalloy”) was deemed a feasible pattern to design favorable superatoms because its properties could be tuned easily. 27–30 In particular, our recent research interest was greatly attracted by a kind of bimetallic cluster which owns a novel matryoshka structure and the beautiful icosahedral symmetry. As an important milestone, the successful synthesis of [As@Ni12 @As20 ]3− demonstrated the actual existence of such a kind of matryoshka superatom. 31 Furthermore, the subsequent synthesis of [Sn@Cu12 @Sn20 ]12− enriched the family of matryoshka superatoms. 32 Many theoretical calculations were also carried out, and abundant results were obtained, including the accurate geometrical parameters, vibrational properties, electronic structures, etc. 31–41 Beyond the synthesized superatoms, artificial species were also predicted, especially the neutral clusters which could be isolated in vacuum and be more useful for industrial applications. 37,42–47 For example, Ge@Zn12 @Ge20 was calculated more than a decade ago. 42 Recently, a series of neutral clusters A@B12 @A20 (A=Sn, Pb; B=Mg, Zn, Cd, Mn) was extensively discussed. 45,46 Since many of these clusters contained heavy elements, the relativistic effect is probably not negligible in certain cases. Actually, the relativistic effect in superatoms, including both the scalarrelativistic effect and the spin-orbit coupling (SOC), has been an attractive topic in the cluster science. In particular, Castro et al. concluded that omitting the relativistic effect would lead to an improper description of the electron delocalization in superatoms with heavy elements. 48 In 3 ACS Paragon Plus Environment


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practice, the scalar relativistic effect can be taken into account in an easy manner, but the SOC is more difficult to be calculated and usually ignored in the past. For matryoshka bimetallic clusters mentioned above, only a few calculations including SOC effects have been performed. 41 Thus, it is essential to perform a further assessment of the SOC effect on them. In fact, the importance of the SOC effect on superatoms containing heavy elements has been extensively demonstrated in many cases. For example, the SOC could greatly alter magnetic properties of metal atom encapsulated Pb12 clusters, 49 and the optical absorption of the thiolated gold cluster was greatly changed owing to the splitting of superatomic orbitals by the SOC. 50 In this article, our goal is to present a quantitative description of the SOC effect on the geometrical structure and various properties of a series of neutral matryoshka superatoms in the gas phase, namely, X@Y12 @X20 (X=Ge, Y=Zn; X=Sn, Y=Mg, Mn, Zn or Cd; X=Pb, Y= Mg, Mn, Cd or Hg). More specifically, the accurate geometrical parameters will be investigated at first to clarify if the SOC could be neglected in the geometry optimization. In the next, the stability of these clusters is also an essential issue because there is a lack of synthesis experiments for them so far. Finally, the SOC effect on the fundamental electronic structure is systematically investigated.

Computational methods All calculations in this article were performed with the density functional theory (DFT), by using the Vienna ab initio simulation package (VASP). 51,52 The Perdew-Burke-Ernzerhof (PBE) 53,54 functional based on the generalized gradient approximation (GGA) was chosen since it can usually offer extraordinarily good results at relatively low computational expense. The projector augmented wave (PAW) method 55,56 was adopted to represent the interaction between core and electrons. The spin-orbit coupling term based on the second-order approximation 57 is added to the scalar relativistic DFT Hamiltonian, and has been implemented in the PAW method. 58,59 The kinetic energy cutoff for the plane wave basis set was 400 eV. Tight criteria were chosen to obtain the good convergence in both the electronic self-consistent-field calculation and the geometry opti-


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mization. Specifically, the total energy convergence up to 10−6 should be achieved during the self-consistent-field calculations, and the residual force should be less than 10−3 eV/Å after the geometry optimization was finished. Besides, the periodic boundary condition was used, and the cluster was placed in a cubic supercell to guarantee the vacuum along each axis to be at least 15 Å. Thus, the interaction between the cluster and neighboring images could be negligible. The Γ point was adopted for the Brillouin zone sampling.

Results and discussion Geometrical structures

d

1

d12

d2

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Figure 1: (Color online). Schematization of the icosahedral matryoshka superatom X@Y12 @X20 . X and Y atoms are represented by the small (red) and large (green) balls respectively.

As shown in Figure 1, the matryoshka structure of the X@Y12 @X20 cluster can be viewed as a nested structure with the outermost dodecahedron shell X20 , the intermediate icosahedron shell Y12 and the central X atom. At first, our calculated results without the SOC effect are compared with previously achieved results by other research groups, as a verification of our computational method (see Table S1, Supporting Information). Generally, our calculated atomic distances are in good 5 ACS Paragon Plus Environment


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agreement with those calculated with a different GGA functional and the double numerical basis as in Ref. 45, though our results are slightly shorter. For Ge@Zn12 @Ge20 , our atomic distances are 4-6% shorter than those calculated by Chang et al. 42 Considering the higher quality of our computational settings (e.g., the PAW method and the large plane wave basis set), it is reasonable that our results are supposed to be more reliable. In the next, one of the most important things is to examine if the SOC could affect the geometrical structure significantly. According to geometrical parameters present in Table 1, these clusters can be classified into two groups: (1) For clusters without lead atoms, as well as Pb@Mg12 @Pb20 , the SOC effect can only bring little change to the geometrical parameters. The largest relative deviation is only about 0.1%, viz., the SOC effect could be neglected during the geometry optimization of these clusters. (2) For the rest, evident changes of geometrical parameters have been found. Among them, Pb@Hg12 @Pb20 present the largest relative deviation of -0.9%, corresponding to about 0.035 Å of the change of atomic distance. Therefore, the SOC effect could not be omitted in an accurate calculation for the structure optimization. Clearly, the influence of SOC effects on geometrical parameters is not monotonous for these clusters. More specifically, the atomic distance can become either shorter or longer with the inclusion of SOC effects. Another interesting characteristic is d2 is more significantly influenced than d1 , i.e., the impact of SOC effects on the intermediate shell is more remarkable than that on the outermost Pb20 shell.

Stabilities Because there is no actual synthesis experiments of these matryoshka superatoms in gas phase so far, it is meaningful to perform an extensive investigation on their stability against thermal or electronic perturbations. Two typical physical quantities are focused, namely, the atomization energy and the energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). Firstly, the atomization energies Eat , which is defined as the energy required to divide the cluster into separate atoms, could be used to evaluate the bonding strength of a cluster. Herein Eat = 21 × EX + 12 × EY − EX@Y12 @X20 , and EX , EY and EX@Y12 @X20 6 ACS Paragon Plus Environment

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Table 1: Calculated geometrical parameters (Å) a of X@Y12 @X20 clusters. Name Ge@Zn12 @Ge20 Sn@Mg12 @Sn20 Sn@Mn12 @Sn20 Sn@Zn12 @Sn20 Sn@Cd12 @Sn20 Pb@Mg12 @Pb20 Pb@Mn12 @Pb20 Pb@Zn12 @Pb20 Pb@Cd12 @Pb20 Pb@Hg12 @Pb20

d1 (w/o) 3.020 3.418 3.159 3.259 3.446 3.502 3.261 3.351 3.527 3.545

(w/ ) ∆d1 (%) 3.020 0.00 3.418 0.00 3.159 0.00 3.259 0.01 3.446 0.00 3.502 0.00 3.267 0.17 3.355 0.13 3.533 0.19 3.543 -0.07

d2 (w/o) 2.833 3.136 2.694 2.955 3.273 3.192 2.738 3.023 3.320 3.906

(w/ ) ∆d2 (%) 2.832 -0.01 3.137 0.02 2.695 0.05 2.954 -0.04 3.270 -0.08 3.195 0.11 2.753 0.56 3.010 -0.43 3.297 -0.69 3.871 -0.90

d12 (w/o) 2.654 3.022 2.852 2.891 3.020 3.101 2.958 2.976 3.097 3.025

(w/ ) 2.654 3.021 2.851 2.891 3.021 3.100 2.960 2.984 3.110 3.025

∆d12 (%) 0.00 -0.01 -0.02 0.02 0.02 -0.03 0.06 0.28 0.42 0.02

a

d1 is the nearest atomic distance in the outermost shell, while d2 is the nearest atomic distance in the intermediate shell. d12 is the nearest atomic distance between the outermost shell and the intermediate shell. The denotation “w/o” (“w/”) corresponds to results without (with) the inclusion of the SOC effect. ∆d1 , ∆d2 and ∆d12 are relative deviations of d1 , d2 , and d12 . are the total energies of X atom, Y atom and X@Y12 @X20 cluster, respectively. Secondly, the HOMO-LUMO gap is essential for understanding the electronic stability, since it corresponds to the energy required to stiffly “lift” an electron from HOMO to LUMO. Although it is well-known that the HOMO-LUMO gap is usually underestimated in GGA calculations, our results are still valuable because the main goal in this article is to examine if the SOC could significantly alter the electronic stability. These results are listed in Table 2. Then the atomization energy will be discussed in the following. For Ge@Zn12 @Ge20 , our result is about 0.4 eV higher than the calculated value in Ref. 42 (see Table S2, Supporting Information), in consistent with the shorter bond length predicted in our calculations. However, in comparison with the “binding energy” calculated by Huang et al., 45 our results seem to be about 0.2-1.2 eV/atom lower (see Table S2, Supporting Information). We infer that such a notable discrepancy might be attributed to the different choice of the reference energy for an isolated atom. In the test calculation, if we chose the singlet state instead of the triplet state for Sn and Pb atoms, the atomization energy of Sn@Mg12 @Sn20 , Sn@Cd12 @Sn20 , Pb@Mg12 @Pb20 and Pb@Cd12 @Pb20 would agree with Huang’s results.



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Table 2: The atomization energies Eat (eV/atom) and the HOMO-LUMO gap Egap (eV) of X@Y12 @X20 clusters. Name Ge@Zn12 @Ge20 Sn@Mg12 @Sn20 Sn@Mn12 @Sn20 Sn@Zn12 @Sn20 Sn@Cd12 @Sn20 Pb@Mg12 @Pb20 Pb@Mn12 @Pb20 Pb@Zn12 @Pb20 Pb@Cd12 @Pb20 Pb@Hg12 @Pb20

without SOC Eat 2.42 2.39 3.05 2.23 2.04 2.29 2.81 2.09 1.95 1.73

Egap 1.60 1.35 0.41 1.44 1.30 1.46 0.68 1.56 1.32 1.22

with SOC Eat Egap 2.40 1.51 2.28 1.15 2.94 0.37 2.12 1.20 1.93 1.07 1.69 0.77 2.19 0.40 1.47 0.86 1.34 0.52 1.13 0.17

The systematic trends can be found for the atomization energy of various clusters. At first, the existence of Mn atoms can enhance the atomization energy remarkably. In detail, Sn@Mn12 @Sn20 has the largest atomization energy, as well as Pb@Mn12 @Pb20 has the largest atomization energy among all species containing Pb atoms. In the next, for clusters without Mn atoms, the atomization energy is decreased with the increasing atomic number of the composition elements. Actually, the relationship between the atomization energy and the composition elements could also be found for Sn@Mn12 @Sn20 and Pb@Mn12 @Pb20 . The influence of the SOC on the atomization energy is clear. For Ge@Zn12 @Ge20 , its atomization energy is nearly unchanged after the SOC was included. For any cluster including Sn atoms, the atomization energy is deduced by about 0.1 eV/atom. Furthermore, this reduction of the atomization energy is increased to about 0.6 eV/atom for clusters containing Pb atoms. It implies that the reduction of the atomization energy is mainly correlated with the SOC effect on the atomic energy of Sn and Pb atoms. The HOMO-LUMO gap could reflect the electronic stability of a certain cluster in principle. When the SOC is not included, the HOMO-LUMO gap of these matryoshka clusters without Mn atoms is between 1.2 and 1.6 eV. In view of that GGA calculations usually underestimate the HOMO-LUMO gap, these clusters should have remarkable electronic stability. For


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Sn@Mn12 @Sn20 and Pb@Mn12 @Pb20 , smaller HOMO-LUMO gap could be found. Once the SOC effect is taken into account in the calculation of these matryoshka clusters, their HOMO-LUMO gap will be decreased. Especially for clusters containing Pb atoms, the reduction of the HOMO-LUMO gap is significant. The most remarkable reduction could be observed for Pb@Hg12 @Pb20 . More attention should be paid to Pb@Hg12 @Pb20 , which has the smallest atomization energy. In addition, the SOC effect will greatly reduce the HOMO-LUMO gap to about 0.2 eV. Therefore, an extra examination for its stability should be interesting since the atomization energy is only a general quantity to reveal the average bonding strength, and a subtle analysis upon the removing of an atom is necessary for this case. For the icosahedral structure, two values of Eout and Eint could be defined as the energy to remove a Pb atom from the outermost shell and a Hg atom from the intermediate shell respectively. Including the SOC effect will greatly reduce Eout from 3.64 eV to 2.39 eV. Especially, Eint is only 0.44 eV in the calculation without the SOC effect, and it is even decreased to a negative value (-0.06 eV) after the inclusion of the SOC into the calculation. It implies that it is exothermic to release of a Hg atom from the cluster. Consequently, Pb@Hg12 @Pb20 seems to be unstable and it is hard to be synthesized in experiment.

Electronic structures Two aspects of the electronic structure are investigated for these matryoshka clusters, i.e., the charge transfer and the project density of states (PDOS). As shown in Table 3, the localized atomic charge is obtained via the Bader analysis, 60–62 and accordingly these clusters could be classified into two kinds in this way. For Sn@Mg12 @Sn20 , Pb@Mg12 @Pb20 , and Pb@Hg12 @Pb20 , a remarkable charge transfer between different shells can be detected, implying the existence of the ionic character. For other clusters, the charge transfer is small, and the covalent bonding is supposed to be essential in these clusters. Moreover, the SOC effect on the charge transfer should also be examined. Clearly, the SOC effect has a negligible influence on these clusters without Pb atoms, and a similar phenomenon 9 ACS Paragon Plus Environment


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Table 3: The gross atomic charges (Q) (e) of the central atom (X), the intermediate shell (Y12 ) and the outermost shell (X20 ) of X@Y12 @X20 clusters given by the Bader analysis. Name Ge@Zn12 @Ge20 Sn@Mg12 @Sn20 Sn@Mn12 @Sn20 Sn@Zn12 @Sn20 Sn@Cd12 @Sn20 Pb@Mg12 @Pb20 Pb@Mn12 @Pb20 Pb@Zn12 @Pb20 Pb@Cd12 @Pb20 Pb@Hg12 @Pb20

without SOC

with SOC

Q(X) Q(Y12 ) Q(X20 ) Q(X) Q(Y12 ) Q(X20 ) -0.74 0.87 -0.13 -0.74 0.86 -0.12 -3.36 16.53 -13.18 -3.35 16.53 -13.18 -0.17 0.80 -0.63 -0.17 0.84 -0.67 -0.53 -0.83 1.36 -0.53 -0.78 1.31 -0.42 -1.19 1.61 -0.42 -1.15 1.57 -3.21 15.71 -12.49 -3.20 15.67 -12.47 -0.35 1.12 -0.78 -0.38 1.58 -1.20 -0.57 -0.41 0.97 -0.60 0.04 0.56 -0.42 -0.93 1.35 -0.46 -0.40 0.86 0.38 -5.53 5.16 0.37 -5.18 4.81

could also be found for Pb@Mg12 @Pb20 cluster. After the inclusion of the SOC effect for other four clusters, only the localized atomic charge of the central atom is still unaffected, while the gross atomic charge of the intermediate (outermost) shell is increased (decreased) by about 0.40.5 e, viz., the electron distribution is more diffuse. In the next, to understand the subtle influence of the SOC on the electronic structure of such a kind of matryoshka superatoms, the PDOS deserves a comprehensive discussion. The nonmagnetic clusters will be discussed at first. As shown in Figure 2a, the SOC only brings about a very small affect on the PDOS of Ge@Zn12 @Ge20 . More specifically, the unoccupied states at around -3.3 eV is a single peak in the PDOS without the inclusion of the SOC effect, and it is split into two consecutive peaks after considering the SOC effect. The relative ratio of peaks is also changed. For these clusters contained Sn atoms, as shown in Figure 2b-d, the influence of SOC effect is more evident. The SOC splits both occupied and unoccupied states, to generate more peaks in the PDOS. Furthermore, some peaks tend to overlap with each other to derive a broader “envelope”. It is notable that the splitting of the LUMO is closely related to the significant narrowing of the HOMO-LUMO gap mentioned above. For clusters with Pb atoms, the splitting of energy states is greatly enhanced, as shown in Figure 3 (panels a to d). Consequently, the narrowing of the HOMO-LUMO gap is also remarkable. In addition, the great SOC effect in clusters with


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(a)

(b)

(c)

(d)

Figure 2: (Color online). The PDOS of (a) Ge@Zn12 @Ge20 ; (b) Sn@Mg12 @Sn20 ; (c) Sn@Zn12 @Sn20 ; (d) Sn@Cd12 @Sn20 . In each plot, the upper (lower) panel is corresponding to the result without (with) the inclusion of the SOC effect. Note that a vertical dash line is drawn between the occupied and unoccupied states. 11 ACS Paragon Plus Environment


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(a)

(b)

(c)

(d)

Figure 3: (Color online). The PDOS of (a) Pb@Mg12 @Pb20 ; (b) Pb@Zn12 @Pb20 ; (c) Pb@Cd12 @Pb20 ; (d) Pb@Hg12 @Pb20 . In each plot, the upper (lower) panel is corresponding to the result without (with) the inclusion of the SOC effect. 12 ACS Paragon Plus Environment

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Pb atoms could even change the order of energy states from the calculation without SOC, and a drastic influence on the PDOS is straightforward. Furthermore, the PDOS of the magnetic clusters is present in Figure 4. It is not unusual that the SOC effect influences the PDOS of Sn@Mn12 @Sn20 only in a minor way (Figure 4a), while the PDOS of Pb@Mn12 @Pb20 is greatly changed by the SOC(Figure 4b). In comparison with the PDOS of non-magnetic clusters, an interesting characteristic could be found in the PDOS of these magnetic clusters. To be specific, the intermediate shell (i.e., Mn12 ) of these magnetic clusters contributed mostly in the density of states, and a detailed analysis shows that the Mn-3d states play a major role in the energy states near the HOMO-LUMO gap. Then it is understandable that Sn@Mn12 @Sn20 and Pb@Mn12 @Pb20 have smaller HOMO-LUMO gap than these non-magnetic clusters, as mentioned above. Besides, for all these clusters including Sn or Pb atoms, it is reasonable to suppose an evident SOC influence on their optical properties (e.g., the fluorescence and the absorption), since their energy levels are greatly affected by the SOC. Therefore, an accurate excitation calculation including the SOC effect would be crucial in the future research.

Magnetic moments For these two magnetic clusters (i.e., Sn@Mn12 @Sn20 and Pb@Mn12 @Pb20 ), an additional survey of their magnetic properties is appealing. To omit the SOC effect, the total spin magnetic moment of these clusters is 28 µB , in agreement with Huang’s results. 45 The SOC will give rise to the non-integer magnetic moment. For Sn@Mn12 @Sn20 , the total spin magnetic moment is increased to about 28.1 µB , in corresponding to its modest SOC effect. The total spin magnetic moment of Pb@Mn12 @Pb20 is considerably increased to about 29.5 µB , and the enhancement is mainly contributed by Mn atoms according to an elaborate analysis of local magnetic moments. Furthermore, not only the value of the magnetic moment but also the arrangement of local magnetic moments has been altered by the SOC. As shown in Figure 5, a visible noncollinear magnetic structure can be observed for Pb@Mn12 @Pb20 when the SOC effect is taken into account then. To give a quan13 ACS Paragon Plus Environment


(a)

(b)

X@Y

@X

X@Y

@X

12

20

20

DOS(arb. unit)

DOS(arb. unit)

12

DOS(arb. unit)

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Energy (eV)

Energy (eV)

Figure 4: (Color online). The PDOS of (a) Sn@Mn12 @Sn20 ; (b) Pb@Mn12 @Pb20 . In each plot, the upper (lower) panel is corresponding to the result without (with) the inclusion of the SOC effect.

Figure 5: (Color online). Local magnetic moments of Pb@Mn12 @Pb20 cluster. The small (gray) and large (magenta) balls represent the Pb and Mn atoms respectively. Note that the vector at Pb site is elongated ten times of its length to represent the small magnetic moment on Pb atom clearly. 14 ACS Paragon Plus Environment

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titative description of the noncollinear magnetism, the “averaged degree of noncollinearity” 63 is calculated and a small value of 2.6◦ is obtained then.

Summary and Conclusions In this article, a series of ligand-free icosahedral matryoshka superatoms has been extensively studied based on the density functional theory. In particular, the influence of the SOC on their geometries, stabilities, electronic structures, and magnetic properties has been summarized as follows: • For all clusters without Pb atoms, as well as Pb@Mg12 @Pb20 , their geometrical structures stay nearly unchanged by the SOC effect, while the other four clusters with Pb atoms will undergo an evident influence on their geometrical parameters. • The atomization energy of Ge@Zn12 @Ge20 is nearly the same after the SOC is taken into account, while the SOC effect reduces the atomization energy of other clusters. More specifically, for clusters containing Sn atoms, the SOC effect tends to reduce the atomization energy by about 0.1 eV/atom, and this reduction of the atomization energy is even as large as 0.6 eV/atom for clusters containing Pb atoms. • The electronic stability reflected by the HOMO-LUMO gap has been also decreased by the SOC effect. Primarily, the narrowing of the HOMO-LUMO gap is more significant for clusters including Pb atoms, as a straightforward consequence of the remarkable splitting of energy levels. • Pb@Hg12 @Pb20 seems to be unstable under the external perturbation, so it should be difficult to synthesize it in experiment. • For all clusters containing Pb atoms, except the ionic species Pb@Mg12 @Pb20 , the SOC could make the electron distribution more diffuse.



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• In the PDOS, the SOC will lead to energy level splitting for all clusters, although it is not very notable for Ge@Zn12 @Ge20 . However, for clusters with heavier element like Pb, the remarkable splitting of energy levels changes the PDOS significantly. • The SOC effect increases the magnetic moment of Pb@Mn12 @Pb20 by about 1.5 µB , and also brings slight noncollinearity into the arrangement of local magnetic moments. For Sn@Mn12 @Sn20 , the SOC effect on its magnetic moment is little. In conclusion, the influence of the SOC on these matryoshka superatoms is highly correlated with their composition elements. The SOC effect on Ge@Zn12 @Ge20 is small enough to be neglected mostly. For clusters containing the Sn atoms, their atomization energy, HOMO-LUMO gap and PDOS are affected moderately, but their geometrical parameters and localized atomic charges are nearly unchanged by the SOC effect. For clusters containing the Pb atoms, a significant SOC effect could be observed in most cases, except that the geometry and localized atomic charge of Pb@Mg12 @Pb20 are nearly unchanged by the SOC.

Acknowledgement This work was supported by the National Natural Science Foundation of China(under Grants Nos. 11474034 and 11675024), the Open Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum physics(under Grant No. KF201514), and the Foundation for Development of Science and Technology of China Academy of Engineering Physics(under Grants Nos. 2015B0102020 and 2015B0102022).

Supporting Information Available The comparison between our calculated geometrical parameters, atomization energies and HOMOLUMO gaps and previously achieved results by other research groups. All these calculation results are without the inclusion of the SOC effect.

This material is available free of charge via the

Internet at http://pubs.acs.org/. 16 ACS Paragon Plus Environment

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TOC Graphic 150

Pb@Zn

@Pb

Sn@Zn

@Sn

12

20

100 50 Density of states (arb. unit)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0 150

12

20

100 50 0 150

Ge@Zn

@Ge

12

20

100 50 0 -8

-6

-4

-2

Energy (eV)

w/o spin-orbit coupling w/

spin-orbit coupling


Page 24 of 24

Spin-Orbit Coupling Effects on Ligand-Free Icosahedral Matryoshka

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