How are valence electrons distributed in hyperlithiated carbon clusters? An answer from the electron localization function topology

How are valence electrons distributed in hyperlithiated carbon clusters? 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An answer from the electron localization function topology Bruno Madebène, Julien Pilmé, Peter Reinhardt, Mohammad Esmaïl Alikhani This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8046509/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 29 Dec, 2025 Read the published version in Structural Chemistry → Version 1 posted 9 You are reading this latest preprint version Abstract We reinvestigated some archetypical hyperlithiated mono-carbon clusters within the electron localization function (ELF) topology. We show that despite the high coordination number, the central atom (C) always obeys the octet rule. In the LiC cluster, two delocalized electrons can be localized by adding one or two lithium atoms. For clusters larger than LiC, the coordination number increases from six to seven, with a first coordination sphere formed by lithium cations, while those in the second sphere are lithium anions. Indeed, ELF topology reveals no covalency in any of the clusters studied. ELF topology localized and delocalized electrons lithium ion octet rule hexa- and heptacoordinated carbon Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 1. Introduction It is well accepted by chemists that chemistry without chemical bonds is unthinkable This fundamental concept, introduced by G. N. Lewis in 1916, [ 1 ] is actually based on two rules: the rule of two (pair of electrons) and the rule of eight (the octet rule). [ 2 – 7 ] Although these rules are of great importance for much of our present understanding of the chemical bond and molecular structure, they do not correspond unambiguously to directly measurable physical quantities. The chemical bond is often described by localized molecular orbitals that can provide useful insights into chemical phenomena such as two-electron shared bonds, π delocalization, and electron lone pairs. However, the localized orbital set of a chemical system is not unique and can lead to different chemical interpretations. [ 6 , 8 , 9 ] The 2c-2e (2-center / 2-electron) pattern cannot explain all the varieties of chemical bonding known today. One famous example is the bonding mechanism in the diborane molecule which is better described by two 3c-2e (3-center / 2-electron) bonds. [ 10 – 13 ] The detection and synthesis of new chemical compounds that do not respect the octet rule have prompted chemists to propose a new class of molecules called "hypervalent". [ 4 , 14 – 16 ] While the concept of multicenter bonding is rather well accepted by chemists, the concept of a hypervalent molecule is still debated. [ 17 – 21 ] Quantum theory teaches us that accounting for electron delocalization is essential to our understanding of chemistry. [ 7 , 22 – 25 ] Since in many molecules, bonding electron pairs are not as well localized as Lewis diagrams suggest, the use of resonance structures, i.e., plausible alternative Lewis structures, is still essential to well describe these molecules. [ 26 , 27 ] Lithium compounds are known as such cases, and given the importance of lithium in several industries, including rechargeable batteries and pharmaceuticals, [ 28 – 31 ] the different bonding mechanisms of this monovalent atom have been the subject of several publications. Lithium as a monovalent atom not only forms ionic compounds, but is also capable of forming other types of bonds, such as hydrogen-like bonds, [ 31 – 36 ] or boron-like multicentric bonds, for instance. [ 37 – 42 ] The two hyperlithiated compounds with 10 valence electrons, namely Li 6 C and Li 4 O, illustrate to some extent the difficulty of formally applying the octet rule. The two molecules observed experimentally [ 43 , 44 ] are thermodynamically stable with respect to any possible dissociation reaction, notably that of loss of Li 2 . [ 45 – 50 ] In other words, the expected compounds formally respecting the octet rule (Li 4 C and Li 2 O) are not thermodynamically the most stable species. The two hyperlithiated molecules can indeed be considered as archetypes of molecular electrides. [ 51 – 56 ] Both octahedral Li 6 C (O h ) and tetrahedral Li 4 O (T d ) were first computationally predicted by P. R. v. Schleyer and coworkers in the early 1980s. [ 47 , 48 , 57 ] In their short but profound theoretical analysis, they found that the net charge of carbon does not increase significantly with the addition of lithium atoms: Li 4 C (-0.81 e), Li 5 C (-0.81 e), and Li 6 C (-0.93 e). They therefore concluded that the central atom does indeed carry eight out of the ten valence electrons, and that the remaining two "extra” electrons contribute to the Li-Li bond (precursor to a metal cage) around the central atom. They also highlighted two remarkable properties of hyperlithiation. First, the nature of the central atom is secondary; this phenomenon can occur with all first-row, second-row, as well as heavier elements. Second, lithium is not unique; other alkali metals can also give rise to hypecoordinated compounds. [ 48 ] Since then, numerous studies of lithium-rich binary clusters have been carried out. Examples include Li 6 N, [ 58 ] Li 4 P, [ 59 ] EM 6 (E = C, Si, Ge, Sn, Pb; M = Li, Na, K, Rb, Cs), [ 60 , 61 ] Li n M (M = Mg, Be), [ 62 , 63 ] Li n Si, [ 64 , 65 ] Li n F, [ 66 , 67 ], Li 4 S, [ 45 , 46 ] Li n B, [ 68 , 69 ] Li n Al. [ 70 ] Interestingly, investigation of the evolution of ionization potentials as a function of Li number in Li n O clusters clearly highlighted the existence of distinct steps (“magic numbers”) in cluster growth at n = 10, 22 and 42, corroborating [ 50 ]: “the shell model for metal clusters, provided that the oxygen atom localizes two of the lithium valence electrons while leaving the other valence electrons delocalized in a metal cluster”. This pattern also holds for the Li n C cluster. [ 49 ] In addition to the Mulliken-population-based wave function analysis, the bonding mechanism within Li 6 C has also been investigated by three distinct methods: 1) from the localized natural bond orbital (NBO) analysis, the central ion core, bearing 8 e, appears to donate only about 0.1 e back to each lithium of the surrounding cage, [ 71 ] 2) valence bond study of Li 6 C underlined the presence of two three-center/three-electron bonding units leading to one-electron delocalization and fractional Li-Li bonding, [ 72 ] and 3) comparing SF 6 and Li 6 C, the analysis of the domain-averaged Fermi hole suggested a bonding picture for Li 6 C (hyper coordinated C) dramatically different from the SF 6 (hypervalent S) case. [ 73 , 74 ] Furthermore, to the best of our knowledge, there exists only one publication on the ELF (electron localization function) topological analysis of the bonding pattern within hyperlithiated boron (Li 7 B and Li 8 B + ). [ 69 ] This paper suggests the presence of a perfect electron delocalization in which Li-cage electrons are mainly distributed on eight Li 3 trisynaptic basins. In the present work, we propose to study Li n C cluster with n = 6, 8, 9, and 10 within the ELF topological framework in order to describe the chemical bonding mechanisms. The main objectives of this work are threefold: 1) Does the central atom respect the octet rule? 2) Does ELF topology allow us to distinguish between lone pairs localized on an atom and those delocalized on the periphery of the chemical compound? 3) Do Li-Li bonds exist? If so, do they span over two or three atoms? 2. Computational Details All first-principles calculations were performed using the Gaussian 09 quantum chemical packages. [ 75 ] Optimization of the stationary points on the potential energy surface was performed using density functional theory with the popular B3LYP exchange–correlation functional [ 76 – 78 ] including the D3 version of Grimme’s dispersion with Becke–Johnson damping (GD3BJ). [ 79 ] The Pople triple-z quality basis set extended with polarization and diffuse functions, 6-311 + + G(2d,2p), has been used for all atoms. [ 80 , 81 ] The chemical bonding pattern was described within the ELF (electron localization function) topological framework. [ 82 ] The partition of the molecular space in terms of non-overlapping space-filling domains has been performed using the TopChem2 package. [ 83 , 84 ] The molecular electrostatic potential (MESP) has been investigated using the Multiwfn software. [ 85 ] Visualization of ELF isosurfaces, together with 3D MESP maps projected onto electron-density isosurfaces and corresponding analyses were performed with UCSF Chimera [ 86 ] and VMD. [ 87 ] 3. Results and discussion Optimized geometries have been well established in the literature, [ 49 , 88 ] for pair Li n C clusters (n = 6, 8, and 10). For the Li 9 C cluster, we found that the global minimum is a C 2v structure in which the coordination number of carbon is seven. The global minimal structures of the studied clusters are reported in Fig. 1 . To understand the electronic structure of Li n C clusters, it is helpful to consider first the molecular electrostatic potential (MEP) topography of the Li 6 C cluster. We remind readers that MEP analysis can be performed in two distinct ways: the MEP surface evaluated onto the van der Waals surface defined by the 0.001 a.u. electron density isosurface, [ 89 , 90 ] and the MEP analysis based on the properties of the minima defined by the (3, + 3) critical points (CP). [ 91 – 93 ] The first approach, by circumventing the non-existence of the MEP maxima, allows us to identify the most positive (nucleophilic attack) and the most negative (electrophilic attack) areas on the MEP surface. The second approach by determining the global minima of MEP, (3, + 3) critical points, provides an unambiguous representation of lone pairs and the delocalized electrons in electrides. The MEP evaluated on the van der Waals surface of the Li 6 C molecule is illustrated in Fig. 2 (left panel). The most positive region being blue (V s, max = + 44.7 kcal/mol) whereas the most negative being red (V s, min = -2.2 kcal/mol). The latter is, in fact, the most favorable site to an electrophilic attack. The MEP minima plot depicted in Fig. 2 (right panel) clearly shows two types of (3,+3) CPs: those located close to C (red), labeled as inner minima, and those located outside of each facet formed by three lithium atoms (green), labeled as outer minima. The two classes of minima are distinguished by two properties. First, the total electrostatic potential: at the inner minimum it is larger than at the outer minimum (CP1 = -11.3 kcal/mol; CP2 = -2.8 kcal/mol, respectively). Second, the eigenvalues of the hessian of the electrostatic potential: one of the three eigenvalues ​​at CP1 is significantly larger than the other two (0.115 a.u., vs. 0.088 and 0.088 a.u.), whereas the three eigenvalues of CP2 are of the comparable magnitude (0.0014, 0.0023, 0.0023 a.u.). In accordance with the interpretation proposed by Gadre and coworkers for the case of electrides, [ 92 ] we can consider the lone pairs located around carbon (CP1) are of a highly directional nature, while the additional electrons located on the outer minima (CP2) are almost isotropically delocalized electrons. Before switching to the ELF analysis, let us emphasize the difference between the two vocabularies used by ELF-analysis topologists: valence basin and f -localization domain. By definition, the ELF is normalized to the interval between 0 and 1: η(r) ∈ [0–1]. An ELF basin is an irreducible domain characterized by the presence of one and only one attractor – (3, -3) critical point – where ELF η(r) attains a maximum value, labeled as f max . All the points in space with f max ≥ η(r) ≥ f define the “ f -localization domain” for a given positive number f < f max . A localization domain surrounds at least one attractor; in this case it is called “irreducible”. If it contains more than one attractor it is “reducible”. In Fig. 3 are depicted two f -localization domains with η(r) ≥ 0.670 (left panel) and 0.6782 (right panel). The ELF partitioning of the molecular space clearly shows the existence of two f -localization domains: the monosynaptic domain centered on the central atom (colored in red) and the f -localization domain placed essentially outside the molecular sphere, "outer valence domain", (in green). There is no disynaptic domain for η(r) ≥ 0.670 corresponding to any covalent bond C-Li. At this point, the ELF description agrees fully with the MEP picture. The calculated populations for the C monosynaptic domain gives seven electrons, the three remaining valence electrons are accommodated in the outer valence domain. A detailed analysis of each of the eight outer valence domains shows that this domain contains three attractors when f goes from 0.67 to 0.6782, very close to the ELF value for each of these three attractors (0.6793). This information shows that the eight green domains (Fig. 3 left panel) do not correspond to any trisynaptic basins or bonds. [ 94 ] To evaluate the location of these attractors (represented by the symbol x in Fig. 3 , right panel), we compare the r(C-x) = 2.919 Å distance ​​with the distances r(C-Li) = 1.983 Å and r vdW = 3.416 Å (the radius of van der Waals at ρ = 0.001 a.u.). We note that these attractors are located outside the molecular sphere, but inside the van der Waals sphere. As reported in Table 1 , the three valence electrons (or more precisely 2.8 electrons, because the seven cores contain 14.2 electrons instead of 14 e) are equally distributed over 24 monosynaptic basins centered on the six lithium atoms (Fig. 3 , right panel). In summary, the ELF analysis reveals that the octet rule is indeed verified for the central atom, and that there is no covalent bond between C and Li, nor between Li and Li. The Li 6 C cluster should be considered in fact as (Li 6 + C 3- ) @ 3e - . The last but not least point to be emphasized in the ELF analysis of the Li 6 C cluster is that the three attractors – (3, -3) CPs – of each facet formed during the reduction of f -localization ( f goes from 0.67 to 0.6782) are interconnected via a critical point of index 2 – (3, + 1) CP – located on the separatrix of three monosynaptic basins (see Fig. 3 , right panel). We believe that such a CP corresponds to the most favorable site for the electrophilic attack. [ 95 ] Table 1 Core and valence populations (in e) for four hyperlithiated clusters. Core pop. C valence pop. 1st shell Li valence pop. 2d shell Li valence pop. C(C) C(Li) Total V(C) per Li Total per Li Total Li 6 C 2.03 2.03 14.2 7.0 4 x 0.117 2.8 - - Li 8 C 2.03 2.03 18.3 7.7 - - 2.0 4.0 Li 9 C 2.03 2.03 20.3 7.7 1.0 1.0 2.0 4.0 Li 10 C 2.03 2.03 22.3 7.7 - - 2.0 6.0 Proving this, we suggest to study the Li 8 C cluster (Fig. 1 ). As discussed in the literature, [ 49 ] adding two lithium atoms to the octahedral Li 6 C (O h ) cluster gives a D 3d structure with a 1A 1g ground state. Here again, the core population (18.3 e) is slightly larger than its conventional expectation (18 e). Two extra lithium atoms capture two of the three delocalized electrons of Li 6 C, and push the third one into the C valence basin. In other words, the central atom now carries 7.7 electrons, and the two added lithiums are anions with a charge or oxidation state of -1, i.e., alkalides (see Table 1 ). The Li 8 C cluster is therefore a cationic core, (Li 6 C) 2+ , sandwiched between two lithium anions: (Li 6 C) 2+ @ (Li - ) 2 . (see Fig. 4 ) A significant change occurs when adding a ninth Li: the carbon atom is now hepta-coordinated, and two lithium atoms belong to the second coordination shell. Within the Li 9 C C 2v structure, the ELF core population amounts to 20.3 e, the carbon atom carries 7.7 e, each of the two Li of the second shell is an anion, and the remaining delocalized electron is accommodated in the outer valence domain located on the two Li away from C (Fig. 5 , left panel). The addition of a tenth Li is done naturally on the outer valence domain of Li 9 C in order to capture the single delocalized electron. The cationic subunit (Li 7 + C 4- ) is therefore surrounded by three Li anions (Fig. 5 , right panel). Based on the data obtained from ELF analysis, we can consider Li 9 C and Li 10 C clusters as (Li 7 + C 4- ) @ (Li - ) 2 .e - , and (Li 7 + C 4- ) @ (Li - ) 3 , respectively. 4. Conclusions In this paper, we examined the nature of bonding in the Li n C clusters (n = 6, 8, 9, and 10). The topological ELF analysis allowed us to clarify the following points: The calculated ELF core population is always slightly larger than what is conventionally expected. The central atom can be hexa-coordinated (Li 6 C and Li 8 C) or hepta-coordinated (Li 9 C and Li 10 C). In all cases, it respects the octet rule, because the inner valence domain population corresponds to less than eight electrons, V(C) ≤ 8 e. There are no di- nor trisynaptic basins within the studied clusters: the neutral clusters are composed of ionic subsystems, in perfect electrostatic equilibrium. The valence electrons accommodated in the inner valence domain are well localized, whereas those in the outer valence domain are delocalized. The latter electrons are nearly free electrons (IP < 4 eV) [ 49 ] and are accommodated in islands of several monosynaptic basins located around the Li atoms. Lithium atoms in the second coordination sphere are reduced because they capture delocalized electrons from the first shell. We can give the following relationships by formalizing the data from the ELF analysis: Li 6 C : (Li 6 + C 3- ) @ 3e - Li 8 C : (Li 6 + C 4- ) @ (Li - ) 2 Li 9 C : (Li 7 + C 4- ) @ (Li - ) 2 .e - Li 10 C : (Li 7 + C 4- ) @ (Li - ) 3 Declarations Ethical Approval: not applicable Funding: not applicable Availability data and materials: The data that support the findings of this study are available from the corresponding author upon reasonable request. Author Contribution B.M., J.P. participated in doing calculations and in the discussionP.R. participated in the discussion and writing and correction of the manuscript.M. E.A. did calculations and wrote the main manuscript text Acknowledgement All calculations have been performed on the calculator cluster of the MONARIS laboratory. References Lewis GN, Pitzer KS (1966) Valence and the structure of atoms and molecules. Dover Publications, New York Shaik S (2007) The Lewis legacy: The chemical bond—A territory and heartland of chemistry. 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07:41:19","extension":"html","order_by":16,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":187018,"visible":true,"origin":"","legend":"","description":"","filename":"earlyproof.html","url":"https://assets-eu.researchsquare.com/files/rs-8046509/v1/0774e58f77a4f0d9b12dac95.html"},{"id":96253077,"identity":"55f36708-1d0b-4bee-b44d-f21f00c65a7a","added_by":"auto","created_at":"2025-11-19 07:41:56","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":58709,"visible":true,"origin":"","legend":"\u003cp\u003ethe ground state structures of Li\u003csub\u003en\u003c/sub\u003eC with n = 6, 8-10. Black and purple spheres represent C and Li atoms, respectively.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-8046509/v1/c657b64b793cac71e7ed437b.png"},{"id":96250632,"identity":"dc8938a3-208e-4e56-a167-3c0ec52a2759","added_by":"auto","created_at":"2025-11-19 07:38:48","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":179109,"visible":true,"origin":"","legend":"\u003cp\u003eMolecular electrostatic potential (left and medium panels) and electron localization function for the Li\u003csub\u003e6\u003c/sub\u003eC cluster.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-8046509/v1/48b75629fe626398e7ae6a75.png"},{"id":96251706,"identity":"38608ad4-61f7-4906-96d7-f16563e500ab","added_by":"auto","created_at":"2025-11-19 07:39:56","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":90011,"visible":true,"origin":"","legend":"\u003cp\u003eELF isosurfaces (h(r) = f) illustrating the f-localization domains for the octahedral Li\u003csub\u003e6\u003c/sub\u003eC (Oh).\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-8046509/v1/7dd42f6a501e761e2e969182.png"},{"id":96252705,"identity":"17bd0da3-0eb2-4d58-af2a-ccef3e5b9647","added_by":"auto","created_at":"2025-11-19 07:41:22","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":34620,"visible":true,"origin":"","legend":"\u003cp\u003eELF f-localization domains of Li\u003csub\u003e8\u003c/sub\u003eC for h(r) = 0.7\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-8046509/v1/da8cc49cf90fc80f84e23b78.png"},{"id":96210588,"identity":"13fdca71-1429-49e5-bd4b-2df40c59845f","added_by":"auto","created_at":"2025-11-18 18:26:22","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":97492,"visible":true,"origin":"","legend":"\u003cp\u003eELF isosurfaces (h(r) = 0.72) depicting the f-localization domains for Li\u003csub\u003e9\u003c/sub\u003eC and Li\u003csub\u003e10\u003c/sub\u003eC clusters.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-8046509/v1/238a61716dc253595caed2d2.png"},{"id":99545172,"identity":"c81a28f1-3712-49a0-b5cc-b136d0db4da9","added_by":"auto","created_at":"2026-01-05 16:00:31","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1113225,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8046509/v1/6a8c8a31-4966-4a88-830d-d67ad2759bf8.pdf"},{"id":96210579,"identity":"08100911-04a6-4925-8382-9dd166404aef","added_by":"auto","created_at":"2025-11-18 18:26:21","extension":"png","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":43083,"visible":true,"origin":"","legend":"","description":"","filename":"Graphicalabstract.png","url":"https://assets-eu.researchsquare.com/files/rs-8046509/v1/c597bfb5c62e9c2aa31f44a2.png"}],"financialInterests":"No competing interests reported.","formattedTitle":"How are valence electrons distributed in hyperlithiated carbon clusters? An answer from the electron localization function topology","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eIt is well accepted by chemists that chemistry without chemical bonds is unthinkable This fundamental concept, introduced by G. N. Lewis in 1916, [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e] is actually based on two rules: the rule of two (pair of electrons) and the rule of eight (the octet rule). [\u003cspan additionalcitationids=\"CR3 CR4 CR5 CR6\" citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] Although these rules are of great importance for much of our present understanding of the chemical bond and molecular structure, they do not correspond unambiguously to directly measurable physical quantities. The chemical bond is often described by localized molecular orbitals that can provide useful insights into chemical phenomena such as two-electron shared bonds, π delocalization, and electron lone pairs. However, the localized orbital set of a chemical system is not unique and can lead to different chemical interpretations. [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e] The 2c-2e (2-center / 2-electron) pattern cannot explain all the varieties of chemical bonding known today. One famous example is the bonding mechanism in the diborane molecule which is better described by two 3c-2e (3-center / 2-electron) bonds. [\u003cspan additionalcitationids=\"CR11 CR12\" citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e] The detection and synthesis of new chemical compounds that do not respect the octet rule have prompted chemists to propose a new class of molecules called \"hypervalent\". [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e, \u003cspan additionalcitationids=\"CR15\" citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e] While the concept of multicenter bonding is rather well accepted by chemists, the concept of a hypervalent molecule is still debated. [\u003cspan additionalcitationids=\"CR18 CR19 CR20\" citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e] Quantum theory teaches us that accounting for electron delocalization is essential to our understanding of chemistry. [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e, \u003cspan additionalcitationids=\"CR23 CR24\" citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e] Since in many molecules, bonding electron pairs are not as well localized as Lewis diagrams suggest, the use of resonance structures, i.e., plausible alternative Lewis structures, is still essential to well describe these molecules. [\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e, \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e]\u003c/p\u003e\u003cp\u003eLithium compounds are known as such cases, and given the importance of lithium in several industries, including rechargeable batteries and pharmaceuticals, [\u003cspan additionalcitationids=\"CR29 CR30\" citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e] the different bonding mechanisms of this monovalent atom have been the subject of several publications. Lithium as a monovalent atom not only forms ionic compounds, but is also capable of forming other types of bonds, such as hydrogen-like bonds, [\u003cspan additionalcitationids=\"CR32 CR33 CR34 CR35\" citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e] or boron-like multicentric bonds, for instance. [\u003cspan additionalcitationids=\"CR38 CR39 CR40 CR41\" citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e]\u003c/p\u003e\u003cp\u003eThe two hyperlithiated compounds with 10 valence electrons, namely Li\u003csub\u003e6\u003c/sub\u003eC and Li\u003csub\u003e4\u003c/sub\u003eO, illustrate to some extent the difficulty of formally applying the octet rule. The two molecules observed experimentally [\u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e43\u003c/span\u003e, \u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e44\u003c/span\u003e] are thermodynamically stable with respect to any possible dissociation reaction, notably that of loss of Li\u003csub\u003e2\u003c/sub\u003e. [\u003cspan additionalcitationids=\"CR46 CR47 CR48 CR49\" citationid=\"CR45\" class=\"CitationRef\"\u003e45\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e] In other words, the expected compounds formally respecting the octet rule (Li\u003csub\u003e4\u003c/sub\u003eC and Li\u003csub\u003e2\u003c/sub\u003eO) are not thermodynamically the most stable species. The two hyperlithiated molecules can indeed be considered as archetypes of molecular electrides. [\u003cspan additionalcitationids=\"CR52 CR53 CR54 CR55\" citationid=\"CR51\" class=\"CitationRef\"\u003e51\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR56\" class=\"CitationRef\"\u003e56\u003c/span\u003e]\u003c/p\u003e\u003cp\u003eBoth octahedral Li\u003csub\u003e6\u003c/sub\u003eC (O\u003csub\u003eh\u003c/sub\u003e) and tetrahedral Li\u003csub\u003e4\u003c/sub\u003eO (T\u003csub\u003ed\u003c/sub\u003e) were first computationally predicted by P. R. v. Schleyer and coworkers in the early 1980s. [\u003cspan citationid=\"CR47\" class=\"CitationRef\"\u003e47\u003c/span\u003e, \u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e48\u003c/span\u003e, \u003cspan citationid=\"CR57\" class=\"CitationRef\"\u003e57\u003c/span\u003e] In their short but profound theoretical analysis, they found that the net charge of carbon does not increase significantly with the addition of lithium atoms: Li\u003csub\u003e4\u003c/sub\u003eC (-0.81 e), Li\u003csub\u003e5\u003c/sub\u003eC (-0.81 e), and Li\u003csub\u003e6\u003c/sub\u003eC (-0.93 e). They therefore concluded that the central atom does indeed carry eight out of the ten valence electrons, and that the remaining two \"extra\u0026rdquo; electrons contribute to the Li-Li bond (precursor to a metal cage) around the central atom. They also highlighted two remarkable properties of hyperlithiation. First, the nature of the central atom is secondary; this phenomenon can occur with all first-row, second-row, as well as heavier elements. Second, lithium is not unique; other alkali metals can also give rise to hypecoordinated compounds. [\u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e48\u003c/span\u003e] Since then, numerous studies of lithium-rich binary clusters have been carried out. Examples include Li\u003csub\u003e6\u003c/sub\u003eN, [\u003cspan citationid=\"CR58\" class=\"CitationRef\"\u003e58\u003c/span\u003e] Li\u003csub\u003e4\u003c/sub\u003eP, [\u003cspan citationid=\"CR59\" class=\"CitationRef\"\u003e59\u003c/span\u003e] EM\u003csub\u003e6\u003c/sub\u003e (E\u0026thinsp;=\u0026thinsp;C, Si, Ge, Sn, Pb; M\u0026thinsp;=\u0026thinsp;Li, Na, K, Rb, Cs), [\u003cspan citationid=\"CR60\" class=\"CitationRef\"\u003e60\u003c/span\u003e, \u003cspan citationid=\"CR61\" class=\"CitationRef\"\u003e61\u003c/span\u003e] Li\u003csub\u003en\u003c/sub\u003eM (M\u0026thinsp;=\u0026thinsp;Mg, Be), [\u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e62\u003c/span\u003e, \u003cspan citationid=\"CR63\" class=\"CitationRef\"\u003e63\u003c/span\u003e] Li\u003csub\u003en\u003c/sub\u003eSi, [\u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e64\u003c/span\u003e, \u003cspan citationid=\"CR65\" class=\"CitationRef\"\u003e65\u003c/span\u003e] Li\u003csub\u003en\u003c/sub\u003eF, [\u003cspan citationid=\"CR66\" class=\"CitationRef\"\u003e66\u003c/span\u003e, \u003cspan citationid=\"CR67\" class=\"CitationRef\"\u003e67\u003c/span\u003e], Li\u003csub\u003e4\u003c/sub\u003eS, [\u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e45\u003c/span\u003e, \u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e46\u003c/span\u003e] Li\u003csub\u003en\u003c/sub\u003eB, [\u003cspan citationid=\"CR68\" class=\"CitationRef\"\u003e68\u003c/span\u003e, \u003cspan citationid=\"CR69\" class=\"CitationRef\"\u003e69\u003c/span\u003e] Li\u003csub\u003en\u003c/sub\u003eAl. [\u003cspan citationid=\"CR70\" class=\"CitationRef\"\u003e70\u003c/span\u003e]\u003c/p\u003e\u003cp\u003eInterestingly, investigation of the evolution of ionization potentials as a function of Li number in Li\u003csub\u003en\u003c/sub\u003eO clusters clearly highlighted the existence of distinct steps (\u0026ldquo;magic numbers\u0026rdquo;) in cluster growth at n\u0026thinsp;=\u0026thinsp;10, 22 and 42, corroborating [\u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e]: \u0026ldquo;the shell model for metal clusters, provided that the oxygen atom localizes two of the lithium valence electrons while leaving the other valence electrons delocalized in a metal cluster\u0026rdquo;. This pattern also holds for the Li\u003csub\u003en\u003c/sub\u003eC cluster. [\u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e49\u003c/span\u003e]\u003c/p\u003e\u003cp\u003eIn addition to the Mulliken-population-based wave function analysis, the bonding mechanism within Li\u003csub\u003e6\u003c/sub\u003eC has also been investigated by three distinct methods: 1) from the localized natural bond orbital (NBO) analysis, the central ion core, bearing 8 e, appears to donate only about 0.1 e back to each lithium of the surrounding cage, [\u003cspan citationid=\"CR71\" class=\"CitationRef\"\u003e71\u003c/span\u003e] 2) valence bond study of Li\u003csub\u003e6\u003c/sub\u003eC underlined the presence of two three-center/three-electron bonding units leading to one-electron delocalization and fractional Li-Li bonding, [\u003cspan citationid=\"CR72\" class=\"CitationRef\"\u003e72\u003c/span\u003e] and 3) comparing SF\u003csub\u003e6\u003c/sub\u003e and Li\u003csub\u003e6\u003c/sub\u003eC, the analysis of the domain-averaged Fermi hole suggested a bonding picture for Li\u003csub\u003e6\u003c/sub\u003eC (hyper coordinated C) dramatically different from the SF\u003csub\u003e6\u003c/sub\u003e (hypervalent S) case. [\u003cspan citationid=\"CR73\" class=\"CitationRef\"\u003e73\u003c/span\u003e, \u003cspan citationid=\"CR74\" class=\"CitationRef\"\u003e74\u003c/span\u003e]\u003c/p\u003e\u003cp\u003eFurthermore, to the best of our knowledge, there exists only one publication on the ELF (electron localization function) topological analysis of the bonding pattern within hyperlithiated boron (Li\u003csub\u003e7\u003c/sub\u003eB and Li\u003csub\u003e8\u003c/sub\u003eB\u003csup\u003e+\u003c/sup\u003e). [\u003cspan citationid=\"CR69\" class=\"CitationRef\"\u003e69\u003c/span\u003e] This paper suggests the presence of a perfect electron delocalization in which Li-cage electrons are mainly distributed on eight Li\u003csub\u003e3\u003c/sub\u003e trisynaptic basins. In the present work, we propose to study Li\u003csub\u003en\u003c/sub\u003eC cluster with n\u0026thinsp;=\u0026thinsp;6, 8, 9, and 10 within the ELF topological framework in order to describe the chemical bonding mechanisms. The main objectives of this work are threefold: 1) Does the central atom respect the octet rule? 2) Does ELF topology allow us to distinguish between lone pairs localized on an atom and those delocalized on the periphery of the chemical compound? 3) Do Li-Li bonds exist? If so, do they span over two or three atoms?\u003c/p\u003e"},{"header":"2. Computational Details","content":"\u003cp\u003eAll first-principles calculations were performed using the Gaussian 09 quantum chemical packages. [\u003cspan citationid=\"CR75\" class=\"CitationRef\"\u003e75\u003c/span\u003e] Optimization of the stationary points on the potential energy surface was performed using density functional theory with the popular B3LYP exchange\u0026ndash;correlation functional [\u003cspan additionalcitationids=\"CR77\" citationid=\"CR76\" class=\"CitationRef\"\u003e76\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR78\" class=\"CitationRef\"\u003e78\u003c/span\u003e] including the D3 version of Grimme\u0026rsquo;s dispersion with Becke\u0026ndash;Johnson damping (GD3BJ). [\u003cspan citationid=\"CR79\" class=\"CitationRef\"\u003e79\u003c/span\u003e] The Pople triple-z quality basis set extended with polarization and diffuse functions, 6-311\u0026thinsp;+\u0026thinsp;+\u0026thinsp;G(2d,2p), has been used for all atoms. [\u003cspan citationid=\"CR80\" class=\"CitationRef\"\u003e80\u003c/span\u003e, \u003cspan citationid=\"CR81\" class=\"CitationRef\"\u003e81\u003c/span\u003e]\u003c/p\u003e\u003cp\u003eThe chemical bonding pattern was described within the ELF (electron localization function) topological framework. [\u003cspan citationid=\"CR82\" class=\"CitationRef\"\u003e82\u003c/span\u003e] The partition of the molecular space in terms of non-overlapping space-filling domains has been performed using the TopChem2 package. [\u003cspan citationid=\"CR83\" class=\"CitationRef\"\u003e83\u003c/span\u003e, \u003cspan citationid=\"CR84\" class=\"CitationRef\"\u003e84\u003c/span\u003e] The molecular electrostatic potential (MESP) has been investigated using the Multiwfn software. [\u003cspan citationid=\"CR85\" class=\"CitationRef\"\u003e85\u003c/span\u003e] Visualization of ELF isosurfaces, together with 3D MESP maps projected onto electron-density isosurfaces and corresponding analyses were performed with UCSF Chimera [\u003cspan citationid=\"CR86\" class=\"CitationRef\"\u003e86\u003c/span\u003e] and VMD. [\u003cspan citationid=\"CR87\" class=\"CitationRef\"\u003e87\u003c/span\u003e]\u003c/p\u003e"},{"header":"3. Results and discussion","content":"\u003cp\u003eOptimized geometries have been well established in the literature, [\u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e49\u003c/span\u003e, \u003cspan citationid=\"CR88\" class=\"CitationRef\"\u003e88\u003c/span\u003e] for pair Li\u003csub\u003en\u003c/sub\u003eC clusters (n\u0026thinsp;=\u0026thinsp;6, 8, and 10). For the Li\u003csub\u003e9\u003c/sub\u003eC cluster, we found that the global minimum is a \u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003e2v\u003c/em\u003e\u003c/sub\u003e structure in which the coordination number of carbon is seven. The global minimal structures of the studied clusters are reported in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eTo understand the electronic structure of Li\u003csub\u003en\u003c/sub\u003eC clusters, it is helpful to consider first the molecular electrostatic potential (MEP) topography of the Li\u003csub\u003e6\u003c/sub\u003eC cluster. We remind readers that MEP analysis can be performed in two distinct ways: the MEP surface evaluated onto the van der Waals surface defined by the 0.001 a.u. electron density isosurface, [\u003cspan citationid=\"CR89\" class=\"CitationRef\"\u003e89\u003c/span\u003e, \u003cspan citationid=\"CR90\" class=\"CitationRef\"\u003e90\u003c/span\u003e] and the MEP analysis based on the properties of the minima defined by the (3, +\u0026thinsp;3) critical points (CP). [\u003cspan additionalcitationids=\"CR92\" citationid=\"CR91\" class=\"CitationRef\"\u003e91\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR93\" class=\"CitationRef\"\u003e93\u003c/span\u003e] The first approach, by circumventing the non-existence of the MEP maxima, allows us to identify the most positive (nucleophilic attack) and the most negative (electrophilic attack) areas on the MEP surface. The second approach by determining the global minima of MEP, (3, +\u0026thinsp;3) critical points, provides an unambiguous representation of lone pairs and the delocalized electrons in electrides.\u003c/p\u003e\u003cp\u003eThe MEP evaluated on the van der Waals surface of the Li\u003csub\u003e6\u003c/sub\u003eC molecule is illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e (left panel). The most positive region being blue (V\u003csub\u003es, max\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;+\u0026thinsp;44.7 kcal/mol) whereas the most negative being red (V\u003csub\u003es, min\u003c/sub\u003e = -2.2 kcal/mol). The latter is, in fact, the most favorable site to an electrophilic attack.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe MEP minima plot depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e (right panel) clearly shows two types of (3,+3) CPs: those located close to C (red), labeled as inner minima, and those located outside of each facet formed by three lithium atoms (green), labeled as outer minima. The two classes of minima are distinguished by two properties. First, the total electrostatic potential: at the inner minimum it is larger than at the outer minimum (CP1 = -11.3 kcal/mol; CP2 = -2.8 kcal/mol, respectively). Second, the eigenvalues of the hessian of the electrostatic potential: one of the three eigenvalues ​​at CP1 is significantly larger than the other two (0.115 a.u., vs. 0.088 and 0.088 a.u.), whereas the three eigenvalues of CP2 are of the comparable magnitude (0.0014, 0.0023, 0.0023 a.u.). In accordance with the interpretation proposed by Gadre and coworkers for the case of electrides, [\u003cspan citationid=\"CR92\" class=\"CitationRef\"\u003e92\u003c/span\u003e] we can consider the lone pairs located around carbon (CP1) are of a highly directional nature, while the additional electrons located on the outer minima (CP2) are almost isotropically delocalized electrons.\u003c/p\u003e\u003cp\u003eBefore switching to the ELF analysis, let us emphasize the difference between the two vocabularies used by ELF-analysis topologists: valence basin and \u003cem\u003ef\u003c/em\u003e-localization domain. By definition, the ELF is normalized to the interval between 0 and 1: \u003cem\u003eη(r)\u003c/em\u003e \u0026isin; [0\u0026ndash;1]. An ELF basin is an irreducible domain characterized by the presence of one and only one attractor \u0026ndash; (3, -3) critical point \u0026ndash; where ELF \u003cem\u003eη(r)\u003c/em\u003e attains a maximum value, labeled as \u003cem\u003ef\u003c/em\u003e\u003csub\u003e\u003cem\u003emax\u003c/em\u003e\u003c/sub\u003e. All the points in space with \u003cem\u003ef\u003c/em\u003e\u003csub\u003e\u003cem\u003emax\u003c/em\u003e\u003c/sub\u003e \u0026ge; \u003cem\u003eη(r)\u003c/em\u003e \u0026ge; \u003cem\u003ef\u003c/em\u003e define the \u0026ldquo;\u003cem\u003ef\u003c/em\u003e-localization domain\u0026rdquo; for a given positive number \u003cem\u003ef\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003ef\u003c/em\u003e\u003csub\u003e\u003cem\u003emax\u003c/em\u003e\u003c/sub\u003e. A localization domain surrounds at least one attractor; in this case it is called \u0026ldquo;irreducible\u0026rdquo;. If it contains more than one attractor it is \u0026ldquo;reducible\u0026rdquo;.\u003c/p\u003e\u003cp\u003eIn Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e are depicted two \u003cem\u003ef\u003c/em\u003e-localization domains with η(r) \u0026ge; 0.670 (left panel) and 0.6782 (right panel).\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe ELF partitioning of the molecular space clearly shows the existence of two \u003cem\u003ef\u003c/em\u003e-localization domains: the monosynaptic domain centered on the central atom (colored in red) and the \u003cem\u003ef\u003c/em\u003e-localization domain placed essentially outside the molecular sphere, \"outer valence domain\", (in green). There is no disynaptic domain for η(r) \u0026ge; 0.670 corresponding to any covalent bond C-Li. At this point, the ELF description agrees fully with the MEP picture. The calculated populations for the C monosynaptic domain gives seven electrons, the three remaining valence electrons are accommodated in the outer valence domain. A detailed analysis of each of the eight outer valence domains shows that this domain contains three attractors when \u003cem\u003ef\u003c/em\u003e goes from 0.67 to 0.6782, very close to the ELF value for each of these three attractors (0.6793). This information shows that the eight green domains (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e left panel) do not correspond to any trisynaptic basins or bonds. [\u003cspan citationid=\"CR94\" class=\"CitationRef\"\u003e94\u003c/span\u003e] To evaluate the location of these attractors (represented by the symbol x in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, right panel), we compare the r(C-x)\u0026thinsp;=\u0026thinsp;2.919 \u0026Aring; distance ​​with the distances r(C-Li)\u0026thinsp;=\u0026thinsp;1.983 \u0026Aring; and r\u003csub\u003evdW\u003c/sub\u003e = 3.416 \u0026Aring; (the radius of van der Waals at ρ\u0026thinsp;=\u0026thinsp;0.001 a.u.). We note that these attractors are located outside the molecular sphere, but inside the van der Waals sphere. As reported in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, the three valence electrons (or more precisely 2.8 electrons, because the seven cores contain 14.2 electrons instead of 14 e) are equally distributed over 24 monosynaptic basins centered on the six lithium atoms (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, right panel). In summary, the ELF analysis reveals that the octet rule is indeed verified for the central atom, and that there is no covalent bond between C and Li, nor between Li and Li. The Li\u003csub\u003e6\u003c/sub\u003eC cluster should be considered in fact as (Li\u003csub\u003e6\u003c/sub\u003e\u003csup\u003e+\u003c/sup\u003eC\u003csup\u003e3-\u003c/sup\u003e) @ 3e\u003csup\u003e-\u003c/sup\u003e.\u003c/p\u003e\u003cp\u003eThe last but not least point to be emphasized in the ELF analysis of the Li\u003csub\u003e6\u003c/sub\u003eC cluster is that the three attractors \u0026ndash; (3, -3) CPs \u0026ndash; of each facet formed during the reduction of \u003cem\u003ef\u003c/em\u003e-localization (\u003cem\u003ef\u003c/em\u003e goes from 0.67 to 0.6782) are interconnected via a critical point of index 2 \u0026ndash; (3, +\u0026thinsp;1) CP \u0026ndash; located on the separatrix of three monosynaptic basins (see Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, right panel). We believe that such a CP corresponds to the most favorable site for the electrophilic attack. [\u003cspan citationid=\"CR95\" class=\"CitationRef\"\u003e95\u003c/span\u003e]\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eCore and valence populations (in e) for four hyperlithiated clusters.\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"9\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e\u003cth align=\"left\" colspan=\"3\" nameend=\"c4\" namest=\"c2\"\u003e\u003cp\u003eCore pop.\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e\u003cp\u003eC valence pop.\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colspan=\"2\" nameend=\"c7\" namest=\"c6\"\u003e\u003cp\u003e1st shell Li valence pop.\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colspan=\"2\" nameend=\"c9\" namest=\"c8\"\u003e\u003cp\u003e2d shell Li valence pop.\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eC(C)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003eC(Li)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003eTotal\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003eV(C)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003eper Li\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003eTotal\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003eper Li\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003eTotal\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLi\u003csub\u003e6\u003c/sub\u003eC\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e2.03\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e2.03\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e14.2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e7.0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e4 x 0.117\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e2.8\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLi\u003csub\u003e8\u003c/sub\u003eC\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e2.03\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e2.03\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e18.3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e7.7\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e2.0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e4.0\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLi\u003csub\u003e9\u003c/sub\u003eC\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e2.03\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e2.03\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e20.3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e7.7\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e1.0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e1.0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e2.0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e4.0\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLi\u003csub\u003e10\u003c/sub\u003eC\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e2.03\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e2.03\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e22.3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e7.7\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e-\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c8\"\u003e\u003cp\u003e2.0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c9\"\u003e\u003cp\u003e6.0\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003eProving this, we suggest to study the Li\u003csub\u003e8\u003c/sub\u003eC cluster (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). As discussed in the literature, [\u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e49\u003c/span\u003e] adding two lithium atoms to the octahedral Li\u003csub\u003e6\u003c/sub\u003eC (O\u003csub\u003eh\u003c/sub\u003e) cluster gives a D\u003csub\u003e3d\u003c/sub\u003e structure with a 1A\u003csub\u003e1g\u003c/sub\u003e ground state. Here again, the core population (18.3 e) is slightly larger than its conventional expectation (18 e). Two extra lithium atoms capture two of the three delocalized electrons of Li\u003csub\u003e6\u003c/sub\u003eC, and push the third one into the C valence basin. In other words, the central atom now carries 7.7 electrons, and the two added lithiums are anions with a charge or oxidation state of -1, i.e., alkalides (see Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). The Li\u003csub\u003e8\u003c/sub\u003eC cluster is therefore a cationic core, (Li\u003csub\u003e6\u003c/sub\u003eC)\u003csup\u003e2+\u003c/sup\u003e, sandwiched between two lithium anions: (Li\u003csub\u003e6\u003c/sub\u003eC)\u003csup\u003e2+\u003c/sup\u003e @ (Li\u003csup\u003e-\u003c/sup\u003e)\u003csub\u003e2\u003c/sub\u003e. (see Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e )\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eA significant change occurs when adding a ninth Li: the carbon atom is now hepta-coordinated, and two lithium atoms belong to the second coordination shell.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eWithin the Li\u003csub\u003e9\u003c/sub\u003eC C\u003csub\u003e2v\u003c/sub\u003e structure, the ELF core population amounts to 20.3 e, the carbon atom carries 7.7 e, each of the two Li of the second shell is an anion, and the remaining delocalized electron is accommodated in the outer valence domain located on the two Li away from C (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, left panel). The addition of a tenth Li is done naturally on the outer valence domain of Li\u003csub\u003e9\u003c/sub\u003eC in order to capture the single delocalized electron. The cationic subunit (Li\u003csub\u003e7\u003c/sub\u003e\u003csup\u003e+\u003c/sup\u003eC\u003csup\u003e4-\u003c/sup\u003e) is therefore surrounded by three Li anions (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, right panel). Based on the data obtained from ELF analysis, we can consider Li\u003csub\u003e9\u003c/sub\u003eC and Li\u003csub\u003e10\u003c/sub\u003eC clusters as (Li\u003csub\u003e7\u003c/sub\u003e\u003csup\u003e+\u003c/sup\u003eC\u003csup\u003e4-\u003c/sup\u003e) @ (Li\u003csup\u003e-\u003c/sup\u003e)\u003csub\u003e2\u003c/sub\u003e.e\u003csup\u003e-\u003c/sup\u003e, and (Li\u003csub\u003e7\u003c/sub\u003e\u003csup\u003e+\u003c/sup\u003eC\u003csup\u003e4-\u003c/sup\u003e) @ (Li\u003csup\u003e-\u003c/sup\u003e)\u003csub\u003e3\u003c/sub\u003e, respectively.\u003c/p\u003e"},{"header":"4. Conclusions","content":"\u003cp\u003eIn this paper, we examined the nature of bonding in the Li\u003csub\u003en\u003c/sub\u003eC clusters (n\u0026thinsp;=\u0026thinsp;6, 8, 9, and 10). The topological ELF analysis allowed us to clarify the following points:\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003eThe calculated ELF core population is always slightly larger than what is conventionally expected.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eThe central atom can be hexa-coordinated (Li\u003csub\u003e6\u003c/sub\u003eC and Li\u003csub\u003e8\u003c/sub\u003eC) or hepta-coordinated (Li\u003csub\u003e9\u003c/sub\u003eC and Li\u003csub\u003e10\u003c/sub\u003eC). In all cases, it respects the octet rule, because the inner valence domain population corresponds to less than eight electrons, V(C) \u0026le; 8 e.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eThere are no di- nor trisynaptic basins within the studied clusters: the neutral clusters are composed of ionic subsystems, in perfect electrostatic equilibrium.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eThe valence electrons accommodated in the inner valence domain are well localized, whereas those in the outer valence domain are delocalized. The latter electrons are nearly free electrons (IP\u0026thinsp;\u0026lt;\u0026thinsp;4 eV) [\u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e49\u003c/span\u003e] and are accommodated in islands of several monosynaptic basins located around the Li atoms.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eLithium atoms in the second coordination sphere are reduced because they capture delocalized electrons from the first shell.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eWe can give the following relationships by formalizing the data from the ELF analysis:\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003eLi\u003csub\u003e6\u003c/sub\u003eC : (Li\u003csub\u003e6\u003c/sub\u003e\u003csup\u003e+\u003c/sup\u003eC\u003csup\u003e3-\u003c/sup\u003e) @ 3e\u003csup\u003e-\u003c/sup\u003e\u003c/p\u003e\u003cp\u003eLi\u003csub\u003e8\u003c/sub\u003eC : (Li\u003csub\u003e6\u003c/sub\u003e\u003csup\u003e+\u003c/sup\u003eC\u003csup\u003e4-\u003c/sup\u003e) @ (Li\u003csup\u003e-\u003c/sup\u003e)\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e\u003cp\u003eLi\u003csub\u003e9\u003c/sub\u003eC : (Li\u003csub\u003e7\u003c/sub\u003e\u003csup\u003e+\u003c/sup\u003eC\u003csup\u003e4-\u003c/sup\u003e) @ (Li\u003csup\u003e-\u003c/sup\u003e)\u003csub\u003e2\u003c/sub\u003e.e\u003csup\u003e-\u003c/sup\u003e\u003c/p\u003e\u003cp\u003eLi\u003csub\u003e10\u003c/sub\u003eC : (Li\u003csub\u003e7\u003c/sub\u003e\u003csup\u003e+\u003c/sup\u003eC\u003csup\u003e4-\u003c/sup\u003e) @ (Li\u003csup\u003e-\u003c/sup\u003e)\u003csub\u003e3\u003c/sub\u003e\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eEthical Approval: \u003c/strong\u003enot applicable\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding:\u003c/strong\u003e not applicable\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAvailability data and materials: \u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe data that support the findings of this study are available from the corresponding author upon reasonable request. \u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eB.M., J.P. participated in doing calculations and in the discussionP.R. participated in the discussion and writing and correction of the manuscript.M. E.A. did calculations and wrote the main manuscript text\u003c/p\u003e\u003ch2\u003eAcknowledgement\u003c/h2\u003e\u003cp\u003eAll calculations have been performed on the calculator cluster of the MONARIS laboratory.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eLewis GN, Pitzer KS (1966) Valence and the structure of atoms and molecules. Dover Publications, New York\u003c/li\u003e\n\u003cli\u003eShaik S (2007) The Lewis legacy: The chemical bond\u0026mdash;A territory and heartland of chemistry. J Comput Chem 28:51\u0026ndash;61. https://doi.org/10.1002/jcc.20517\u003c/li\u003e\n\u003cli\u003eServos JW (1984) G. N. Lewis: The disciplinary setting. J Chem Educ 61:5. https://doi.org/10.1021/ed061p5\u003c/li\u003e\n\u003cli\u003eJensen WB (1984) Abegg, Lewis, Langmuir, and the octet rule. J Chem Educ 61:191. https://doi.org/10.1021/ed061p191\u003c/li\u003e\n\u003cli\u003eGillespie RJ, Robinson EA (2007) Gilbert N. 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Phys Chem Chem Phys 16:2430\u0026ndash;2442. https://doi.org/10.1039/C3CP54208D\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"structural-chemistry","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"stuc","sideBox":"Learn more about [Structural Chemistry](https://www.springer.com/journal/11224)","snPcode":"11224","submissionUrl":"https://submission.nature.com/new-submission/11224/3","title":"Structural Chemistry","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"ELF topology, localized and delocalized electrons, lithium ion, octet rule, hexa- and heptacoordinated carbon","lastPublishedDoi":"10.21203/rs.3.rs-8046509/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8046509/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"We reinvestigated some archetypical hyperlithiated mono-carbon clusters within the electron localization function (ELF) topology. We show that despite the high coordination number, the central atom (C) always obeys the octet rule. In the LiC cluster, two delocalized electrons can be localized by adding one or two lithium atoms. For clusters larger than LiC, the coordination number increases from six to seven, with a first coordination sphere formed by lithium cations, while those in the second sphere are lithium anions. Indeed, ELF topology reveals no covalency in any of the clusters studied.","manuscriptTitle":"How are valence electrons distributed in hyperlithiated carbon clusters? 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