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A grand unified model for liganded gold clusters

  • Nature Communications 7, Article number: 13574 (2016)
  • doi:10.1038/ncomms13574
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Abstract

A grand unified model (GUM) is developed to achieve fundamental understanding of rich structures of all 71 liganded gold clusters reported to date. Inspired by the quark model by which composite particles (for example, protons and neutrons) are formed by combining three quarks (or flavours), here gold atoms are assigned three ‘flavours’ (namely, bottom, middle and top) to represent three possible valence states. The ‘composite particles’ in GUM are categorized into two groups: variants of triangular elementary block Au3(2e) and tetrahedral elementary block Au4(2e), all satisfying the duet rule (2e) of the valence shell, akin to the octet rule in general chemistry. The elementary blocks, when packed together, form the cores of liganded gold clusters. With the GUM, structures of 71 liganded gold clusters and their growth mechanism can be deciphered altogether. Although GUM is a predictive heuristic and may not be necessarily reflective of the actual electronic structure, several highly stable liganded gold clusters are predicted, thereby offering GUM-guided synthesis of liganded gold clusters by design.

Introduction

Liganded gold clusters have attracted intensive interest over the past 10 years owing to their broad and practical applications in catalysis1, electrochemistry2, quantum electronics3 and biomedicine4. A grand challenge to scientists in this field, however, is the precise determination of atomic structures of liganded gold clusters. To date, atomic structures of tens of liganded gold clusters have been determined via X-ray crystallography5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63. Nevertheless, these revelations appear serendipitous as the structural determination largely hinges on availability of single crystals for the liganded gold clusters. Although density-functional theory computation has been widely applied to predict the structures of many gold clusters64,65,66,67,68,69,70,71,72,73,74,75,76,77,78, ultimate confirmation still requires X-ray crystallography measurements. Theoretical efforts have also been made in the past for more general unification models to comprehend stabilities of liganded gold clusters with apparently very different and seemingly unrelated complex structures. For example, Wade–Mingos counting rules79,80 can provide a simple rationale of various shapes of ‘electron-deficient’ polyhedral clusters in terms of the number of skeletal electron pairs, particularly for borane and carborane clusters. However, few gold clusters can be rationalized by the Wade–Mingos counting rules81, especially for the gold clusters with a high number of interstitial atoms, whose structures have been precisely resolved60,61. The superatom complex (SAC) model proposed by Walter et al.82 suggests that the high stability of several spherical-like ligand-protected gold clusters26,55,60 is due largely to the strong electron shell closures, an important concept that stems from the jellium model83. The total 16 electronic shell-closing ligand-protected gold clusters (all from previous experiments) are summarized in Supplementary Table 1. Cheng et al. developed the super valence bond (SVB) model84 to explain the electronic stability of non-spherical shells of Au38(SR)24 (ref. 53). They suggest that the bi-icosahedral Au23(+9) core of Au38(SR)24 can be viewed as a superatomic molecule. Later, the superatom network (SAN)66 model, coupled with the adaptive natural density partitioning analysis85, have been also invoked by Cheng et al. to explain the high stability of certain low-symmetry ligand-protected gold clusters. A key notion in the SAN model is that the core of Au nanoclusters can be viewed as a network of n-centered two-electron (n=2–6) superatoms. A tetrahedral unit with two valence electrons has also been identified by Jin and co-workers58 through an account of the number of valence electrons and tetrahedral units in a serial of structurally resolved double-helical gold clusters (Au28(SR)20, Au36(SR)24, Au44(SR)28 and Au52(SR)32)44,49,56,58. Recently, the Borromean-ring diagrams for the Au25(SR)18 (refs 42, 43), Au38(SR)24 (ref. 53) and Au102(SR)44 (ref. 60) clusters have been proposed by Pradeep, Whetten and co-workers86 to explain high stabilities of these clusters. All the theoretical models developed thus far are mainly to address stabilities of a subset of gold nanoclusters, rather than the entire set of 71 reported liganded gold clusters. As such, exceptional cases to these independent models abound. Hence, a grand unified model that can go beyond these previously developed models (SAC, SVB, SAN and so on) for understanding stabilities of all ligand-protected gold clusters is called for. Here we present a grand unified model (GUM) that can offer a universal description of the structures of diverse liganded gold clusters.

In this communication, the triangular elementary block Au3(2e) and tetrahedral elementary block Au4(2e) are identified to describe the stabilities of 71 liganded gold nanoclusters (Supplementary Table 2) reported up to date. On the basis of the GUM, deeper insights into structure evolution of the liganded gold nanoclusters can be obtained, namely, the structure evolution of the gold core cannot be viewed simply as addition of Au atoms, but rather as seamless packing of the elementary blocks. In addition, several stable liganded gold clusters are predicted.

Results

GUM development and quark model analogy

The development of the GUM is based on detailed analysis of the structures of all 71 liganded gold clusters (Supplementary Table 2) either determined from previous experiments (54 crystallized structures) or predicted from density-functional theory computation (17 structures) over the past three decades. The scheme of grand conceptual unification of the diverse structures of these 71 liganded gold clusters is motivated from the quark model in particle physics wherein six types of quarks, known as flavours, are conceptualized as a unification scheme for composite particles, such as protons and neutrons, and exotic hadrons, in terms of their valence quarks. For instance, it is known that protons, neutrons alike are not elementary but are viewed as bound states of the elementary valence quarks and antiquarks. All quarks are characterized by a set of quantum numbers, such as fractional electric charge of ±2/3 or ±1/3. In an analogous fashion, here, we assign a gold atom as the ‘elementary particle’ but with one of three ‘flavours’ due to its three possible valence states, that is, 1e, 0.5e and 0e. The three flavours are named as bottom, middle and top, respectively. Through close inspection of the 71 known clusters, we identify two ‘composite particles’, namely, the triangular Au3 and tetrahedral Au4 elementary blocks, in analogy to the protons and tetraquarks, respectively. We find that both elementary blocks satisfy the duet rule, that is, the high tendency of having two electrons in the valence shell. As a result, depending on the flavour of each constituent gold atom, the triangular elementary block can exhibit in total 10 variants of valence states (named as Δ1–Δ10), whereas the tetrahedral elementary block can exhibit in total 15 variants of valence states (named as T1–T15) (Fig. 1). We show that for all 71 liganded gold clusters, once the outer ligands are effectively detached from the inner Au cores (see below), the resulting Au cores are universally packed by the elementary blocks. Hence, the stabilities of the liganded gold clusters are due to the high stability of each individual elementary block.

Figure 1: Valence states of the triangular and tetrahedral elementary blocks.
Figure 1

Ten variants (Δ1–Δ10) of valence states for the triangular elementary block Au3 and 15 variants (T1–T15) of valence states for the tetrahedral elementary block Au4 due to constituent Au atoms having three possible flavours (b for bottom flavour, m for middle flavour and t for top flavour), and the requirement of duet rule (that is, having 2e valence electrons). Colour code of Au atom: magenta (b), dark yellow (m) and yellow (t).

Duet rule

Note that the duet rule elucidated here is akin to the textbook octet rule, a well-known and the first chemical rule of thumb in general chemistry. The octet rule is a valence-electron counting rule for the explanation or prediction of electronic structure and chemical bonding of molecules made of main-group elements. The other two valence-electron counting rules (that is, the second and third rules of thumb), namely, the 18-electron rule and Wade’s rule, are newer chemical rules of thumb for understanding chemical structures of organometallics and polyhedral cluster compounds, respectively. Below, we report that in conjunction with the GUM, the duet rule of the valence shell for the elementary blocks can be treated as the fourth rule of thumb for understanding diverse liganded gold clusters.

Evidences on stabilities of the elementary blocks

In GUM, both elementary blocks triangular Au3 and tetrahedral Au4 entail only two valence electrons [Au3(2e) and Au4(2e)], thereby both having strong electron shell closures. As shown in Fig. 2, the two valence electrons are delocalized in the shell-closing elementary blocks Au3(2e) and Au4(2e), consistent with the SAC model82. Moreover, from an experimental perspective, the Au3 core of crystallized [Au3(IDipp)3]1+ [IDipp=1,3-bis(2,6-diisopropylphenyl)imidazol-2-ylidene] (ref. 5) and the Au4 core of [Au4(PR3)4]2+ (ref. 6) (Supplementary Table 2) are essentially the same as the elementary blocks Au3(2e) and Au4(2e), respectively, supporting the high stabilities of the Au3(2e) and Au4(2e). The high stabilities of Au3(2e) and Au4(2e) structures can be also shown from ab initio computation. In Supplementary Table 3, the formation energies of five isoelectronic species, Au2(2e), Au3(2e), Au4(2e), Au5(2e) and Au6(2e), are listed. Only the triangular Au3(2e) and tetrahedral Au4(2e) exhibit highly negative formation energies, which provides another piece of strong evidence of their high stabilities. In addition, both elementary blocks exhibit larger highest/lowest occupied/unoccupied molecular orbital (HOMO/LUMO) gaps than their isoelectronic counterparts (Supplementary Table 3).

Figure 2: The computed delocalized occupied orbitals (1S2) of Au3(2e) (left) and Au4(2e) (right).
Figure 2

Colour code: Au—yellow.

Electron counting protocols for effective detachment of ligands

All ligand-protected Au clusters are composed of an inner Au core and a number of outer ligands. The first step toward developing the GUM is to find effective protocols to detach the protection ligands from the Au core so that the vast complex factors of the outer ligands can be removed while all ligand-protected Au clusters can be reduced to bare Au cores for structural analysis. Note that different elements or functional groups that are directly bonded with an Au atom at the core surface are in different valence-electron states. As such, if one is to focus on the valence state of the inner Au core, equivalent electron counting is required to effectively detach all protection ligands from the Au core. The following electron-counting protocols for effective detachment of different types of ligands can be undertaken (see Fig. 3 for graphical illustrations):

Figure 3: Electron counting protocols for effective detachment of ligands from inner Au core.
Figure 3

There are four cases for Au atom bonded with (a) gold–thiolate staple motifs, (b) SR, (c) X (X=F, Cl, Br and I) and (d) PR3 functional groups; (e) effective separation of an Au atom shared by two elementary blocks. The blue arrow denotes charge transfer. Colour code: Au—magenta (bottom flavour), dark yellow (middle flavour) and yellow (top flavour); S—dark green; X—light green; P—wine. The R groups are omitted for clarity.

First, each SR group and Au atom embedded in the gold–thiolate staple motifs possess −1e and 1e valence electron, respectively. To effectively detach the smallest staple motif from the Au core, the net number of valence electrons of the staple motif should be converted to zero. To this end, each of the two Au atoms (on the Au core) bonded with SR is considered to transfer 0.5e valence electron to the staple motif. As such, equivalent electron counting for effective detachment of the staple motif is achieved (Fig. 3a). Second, for SR group bonded with two Au atoms on the Au core, again, each of the two Au atoms (on the core) is considered to transfer 0.5e valence electron to the SR group so that equivalent electron counting for effective detachment of the SR group is achieved (Fig. 3b). Third, for halogen ligand X (X=F, Cl, Br and I) bonded with a single Au atom on the Au core, each Au atom is considered to transfer 1e valence electron to the X atom so that equivalent electron counting for effective detachment of the X atom is achieved (Fig. 3c). Fourth, the phosphine group is known as a weak ligand, thereby possessing 0e valence electron. As such, each Au atom bonded with the phosphine group still maintain its original 1e valence electron upon detachment with the phosphine group (Fig. 3d).

In summary, depending on the ligands, for example, PR3, X, SR or gold–thiolate staple motifs, Au atoms on the Au core can exhibit one of the three flavours: bottom (1e), middle (0.5e) and top (0e) after effective detachment of the protection ligands from the Au core. Note also that when two elementary blocks are fused together via sharing a single Au atom, the shared Au atom contributes 0.5e valence electron to each elementary block. As such, the fused elementary blocks can be effectively separated via the protocol shown in Fig. 3e.

Prototypical liganded clusters

As shown in Supplementary Table 2, once the ligands are effectively detached from the Au cores, the Au cores of all 71 ligand-protected gold clusters can be universally decomposed into a number of the triangular Au31–Δ10) and/or tetrahedral Au4 (T1–T15) elementary blocks. Two prototypical structures are analysed here as two examples while other eight representative structures are either briefly illustrated here or elaborated in the Supplementary Figs 1–8. The remaining cases can be analysed in similar fashion.

The first prototype structure we consider is [Au6(dppp)4]2+ (dppp=1,3-Bis(diphenylphosphino)propane)9, which is composed of four dppp ligands (each with 0e valence electron) and an Au62+ core (Fig. 4a). The Au6 core consists of two triangular Au3 blocks. According to the electron-counting protocol, the six Au atoms of the Au6 core are all bonded with phosphine ligands and thus have 1e valence electron or the bottom flavour. Two positive charges are equally distributed in the two triangular Au3 blocks. Thus, both triangular Au3 elementary blocks are in the Δ1 valence state (Fig. 4b).

Figure 4: The structural decomposition of [Au6(dppp)4]2+.
Figure 4

(a) The decomposition of [Au6(dppp)4]2+ cluster into four ligands and an Au core. (b) The decomposition of the inner Au core into two triangular Au3 elementary blocks, each block being in the Δ1 valence state. Colour code: Au—magenta (bottom flavour); C—black; P—wine. The R groups are omitted for clarity.

The second prototype structure we consider is Au40(SR)24 (ref. 56). The Au core of Au40(SR)24 can be viewed as a combination of an Au7 and a Kekulé-like Au18 structure (Fig. 5a). The Au7 core is composed of two tetrahedral Au4 blocks, fused together by sharing an Au atom. The latter contributes 0.5e valence electron to each tetrahedral Au4 block. Other six vertex Au atoms are bonded with the SR group and each contributes 0.5e valence electron to the resident tetrahedral Au4 block. Thus, each of the four vertex Au atoms in the tetrahedral Au4 block has the middle flavour so that each tetrahedral Au4 elementary block is in the T9 valence state (Fig. 5b). Moreover, the Kekulé-like Au18 structure can be viewed as six tetrahedral Au4 blocks fused together in a loop with six sharing Au atoms (Fig. 5c). Again, each of the six tetrahedral Au4 elementary blocks is in the T9 valence state.

Figure 5: The structural decomposition of Au40(SR)24.
Figure 5

(a) The decomposition of Au40(SR)24 cluster into nine ligands and an Au core with two shells. (b) The decomposition of inner shell of the Au core into two fused tetrahedral Au4 elementary blocks, each block, when separated, being in the T9 valence state. (c) The decomposition of the outer shell of the Au core into six fused tetrahedral Au4 elementary blocks in a loop, each block, when separated, being in the T9 valence state. Colour code: Au—magenta (bottom flavour) and dark yellow (middle flavour); S—dark green. The R groups are omitted for clarity.

Finally, eight other representative clusters are considered. The structure decompositions of eight other representative clusters in terms of elementary blocks as well as their corresponding valence states are shown in Supplementary Figs 1–8. Specifically, the structure decomposition of three largest ligand-protected gold clusters, Au102(SR)44 (ref. 60), Au130(SR)50 (ref. 61) and Au144(SR)60 (ref. 78), are given below. According to the ‘divide-and-protect’ formulation87,88, the Au102(SR)44 can be written as Au79[RS-Au-SR]19[RS-Au-SR-Au-SR]2. Each of the two Au atoms in the Au102(SR)44 shared by two [-RS-Au-SR-] staple motifs can be viewed as the Au atom with 0e valence electron (that is, top flavour). As such, the Au79 core of Au102(SR)44 is composed of 29 elementary blocks and their corresponding valence states can be described by [5Δ1, Δ3, 16Δ5, 2T4, 2T5, T7 and 2T9] (Supplementary Fig. 9), giving rise to 58 valence electrons in total (corresponding to the strong electron shell closure, according to SAC model82) for Au102(SR)44. The Au105 core of Au130(SR)50 is composed of 40 elementary blocks and their corresponding valence states include 5Δ3, 11Δ5, 4T4, 3T5 and 17T9 (Supplementary Fig. 10), giving rise to 80 valence electrons in total for Au130(SR)50. The Au114 core of Au144(SR)60 is composed of 42 elementary blocks and their corresponding valence states include 20Δ5, 13T4, 3T5 and 6T9 (Supplementary Fig. 11), giving rise to 84 valence electrons in total in Au144(SR)60.

Discussion

In general, the inner Au cores of all the ligand-protected gold clusters are composed of elementary blocks (Supplementary Table 2 and Fig. 6), each having two valence electrons (Fig. 1), whereas the outer ligands with 0e valence electron provide the geometry constraint and arrangement of valence electrons in each elementary block to satisfy the duet rule. In Supplementary Fig. 12, we show several possible valence states for two fused tetrahedral Au4 blocks (Au7) protected by either SR or PR3 groups in various liganded gold clusters. Our analysis indicates that the fused tetrahedral Au4 elementary blocks in the four predicted stable clusters, Au10(SR)6, [Au9(SR)4(PR3)2]1+, [Au8(SR)2(PR3)4]2+ and [Au7(PR3)6]3+, are in the 2T9, 2T5, 2T4 and 2T2 valence states, respectively, whereas the constructed Au8(SR)2(PR3)4 and Au7(PR3)6 clusters are expected to be less stable due to their violation of the duet rule. Note that a recent experimental investigation of Au25(SR)18 in its three oxidation states, that is, Au25(SR)18−1/0/+1, provides more compiling evidence on the effect of violation of the duet rule (or deviation from the strong electron shell closure) to the symmetry of the Au13 core and stability of the cluster89. The obtained three crystalline structures in the related three oxidation states demonstrate that the structural distortion (in the Au core in particular) increases with the decreased superatomic valence from 1S21P6 to 1S21P4 as the Au25(SR)18−1 cluster has the eight-electron shell-closing configuration82 1S21P6, and thus the highest thermal stability. The other two oxidation states, Au25(SR)180/+1, are less stable due to the incomplete 1P superatomic orbital. Likewise, the structural distortion observed in the two oxidation states, Au25(SR)180/+1, can be understood based on GUM. The Au13 core of Au25(SR)18−1 can be decomposed into four elementary blocks and their corresponding valence states can be described as [2Δ5 and 2T9] (Supplementary Table 2). Clearly, the number of valence electrons in both oxidation states, Au25(SR)180/+1, dissatisfies the duet rule. As a consequence, lower thermal stability and larger structural distortion in the highly symmetric Au13 core are expected, consistent with the experimental evidence89.

Figure 6: Structure evolution of the Au cores (large magenta spheres) with increasing number of elementary blocks.
Figure 6

Colour code: Au—magenta; S—dark green; X—light green; P—wine; C—black; N—blue. The R groups are omitted for clarity. The dotted and solid squares denote the newly and previously predicted structures, respectively. Others are crystallized structures.

The GUM does not only offer a universal structural characterization of all 71 liganded gold clusters, but it also provides deeper insights into structure evolution of the Au clusters. In Fig. 6, a structure evolution map for the Au cores with increasing number of elementary blocks is presented. Indeed, the structure evolution of the Au cores can be understood through various routes of packing the elementary blocks. For example, two elementary blocks with T9 valence state can yield either the Au8 core of Au24(SR)20 (ref. 39) via direct packing or the Au7 core of Au20(SR)16 (ref. 33) via sharing a vertex Au atom. Two elementary blocks with T4 valence states can give rise to the Au6 core of [Au6(PR3)6]2+ (ref. 7) through sharing one common edge. With gradually increasing the number of elementary blocks, larger Au core structures can be formed while adjusting the overall charge of the liganded gold clusters to meet the duet rule. In summary, the structure evolution of the Au core cannot be viewed simply as addition of Au atoms (Supplementary Fig. 13), but rather as seamless packing of the elementary blocks while obeying the duet rule.

In addition to structure unification, the GUM can be utilized for predicting new structures of liganded gold clusters. From the structure evolution map shown in Fig. 6, one can see that there are many vacant spaces, suggesting many missing liganded gold clusters yet to be synthesized. We construct a series of ligand-protected gold clusters (shown in dotted squares in Fig. 6) on the basis of GUM to fill some vacant spaces in Fig. 6. The elementary blocks and their valence states for the Au cores of all constructed clusters are depicted in Supplementary Fig. 14. These clusters exhibit large computed HOMO–LUMO gaps (Supplementary Table 4), suggesting potentially high chemical stabilities.

In particular, a class of hitherto unreported ligand-protected hollow Au clusters are presented here. For example, Au36(SR)12 can be constructed by using the C12 fullerene as a template. C12 fullerene exhibits eight polygons: four quadrilaterals and four pentagons (Fig. 7a). Replacing all C atoms of the C12 fullerene with 12 fused tetrahedral Au4 gives rise to the Au30 core (Fig. 7b) of Au36(SR)12, followed by adding [-RS-Au-SR-] staple motifs on the unfused Au atoms to build the complete Au36(SR)12 cluster (Fig. 7c). The Au30 hollow cage in Au36(SR)12 is composed of 12 fused elementary blocks all at the T9 valence state. The Au36(SR)12 cluster exhibits a large computed HOMO–LUMO gap of 2.20 eV and has no imaginary vibrational frequencies. An ab initio molecular dynamics simulation of the Au36(SR)12 cluster at 355 K for 10 ps suggests high thermal stability of the Au36(SR)12 cluster (Supplementary Fig. 15 and Supplementary Methods). Figure 7f shows another example of ligand-protected hollow Au cluster, namely, Au42(SR)14, constructed by using the C14 fullerene (Fig. 7d) as a template. The Au35 hollow cage (Fig. 7e) in Au42(SR)14 is composed of 14 fused elementary blocks all at the T9 valence state. The computed HOMO–LUMO gap of Au42(SR)14 is 2.00 eV, suggesting high chemical stability. Interestingly, the Au36(SR)12 and Au42(SR)14 can be rewritten as Au30[Au(SR)2]6 and Au35[Au(SR)2]7, respectively, consistent with the ‘divide-and-protect’ formulation87,88.

Figure 7: Two predicted Au36(SR)12 and Au42(SR)14 clusters based on GUM using C12 and C14 fullerenes as templates.
Figure 7

Structures of a C12 fullerene (a), Au30 core (b) and Au36(SR)12 cluster (c), as well as C14 fullerene (d), Au35 core (e) and Au42(SR)14 cluster (f). Colour code: Au—magenta; S—dark green; C—black. The R groups are omitted for clarity.

In conclusion, a grand unified model that can incorporate previously developed independent models (SAC, SVB, SAN and so on.) is developed to address stabilities of all ligand-protected gold clusters. On the basis of the GUM, all 71 liganded gold nanoclusters can be decomposed into several elementary blocks of triangular Au3(2e) and tetrahedral Au4(2e). Although GUM is a predictive heuristic and may not be necessarily reflective of the actual electronic structure, a series of highly stable liganded gold clusters are predicted, which provides a guide to synthesizing new ligand-protected gold clusters. Hence, the GUM can offer not only new insights into the packing and structure evolution of the 71 liganded gold clusters known as of today, but also a systematic route toward rational design and characterization of liganded metal clusters to inspire future experimental synthesis.

Data availability

The authors declare that the data supporting the findings of this study are available within the article and its Supplementary Information files, and all relevant data are available from the authors.

Additional information

How to cite this article: Xu, W. W. et al. A grand unified model for liganded gold clusters. Nat. Commun. 7, 13574 doi: 10.1038/ncomms13574 (2016).

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Acknowledgements

W.W.X. is supported by National Natural Science Foundation of China (11504396). Y.G. is supported by the start-up funding from Shanghai Institute of Applied Physics, Chinese Academy of Sciences (Y290011011), National Natural Science Foundation of China (21273268, 11574340), ‘Hundred People Project’ from Chinese Academy of Sciences and CAS-Shanghai Science Research Center (CAS-SSRC-YJ-2015-01). B.Z. is supported by the development fund for Shanghai talents, Shanghai Natural Science Foundation of China (16ZR443200) and National Natural science foundation of China (11604357). X.C.Z. is supported by a grant from Nebraska Center for Energy Sciences Research and a Qian-ren B (One Thousand Talent Plan B) summer research fund from USTC, and by a State Key R&D Fund of China (2016YFA0200604) to USTC. The computational resources utilized in this research were provided by Shanghai Supercomputer Center, National Supercomputing Center in Tianjin and Shenzhen, special programme for applied research on super computation of the NSFC-Guangdong joint fund (the second phase) and NC3 computer facility in University of Nebraska-Lincoln.

Author information

Affiliations

  1. Division of Interfacial Water and Key Laboratory of Interfacial Physics and Technology, Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai, 201800 China

    • Wen Wu Xu
    • , Beien Zhu
    •  & Yi Gao
  2. Department of Chemistry, University of Nebraska-Lincoln, Lincoln, Nebraska 68588, USA

    • Wen Wu Xu
    •  & Xiao Cheng Zeng
  3. Collaborative Innovation Center of Chemistry for Energy Materials, University of Science and Technology of China, Hefei, Anhui 230026, China

    • Xiao Cheng Zeng
  4. Shanghai Science Research Center, Chinese Academy of Sciences, Shanghai 201204, China

    • Yi Gao

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Contributions

Y.G. created the idea and initiated the model. Y.G. and X.C.Z. designed the whole project. W.W.X., B.Z., X.C.Z. and Y.G. developed the model. X.C.Z. introduced flavour concept. W.W.X. performed all the calculations and plotted the pictures. W.W.X., B.Z., X.C.Z. and Y.G. wrote the manuscript. All authors discussed the results and commented on the manuscript.

Competing interests

The authors declare no competing financial interests.

Corresponding authors

Correspondence to Xiao Cheng Zeng or Yi Gao.

Supplementary information

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    Supplementary Information

    Supplementary Figures 1-15, Supplementary Tables 1-4, Supplementary Methods and Supplementary References.

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