Complexes of Li, Na, and Mg with 2D allotropies of second and third period: a theoretical study

Complexes of Li, Na, and Mg with graphene, silicene, phosphorene nanoflakes (NFs), and their 2D allotropies have been studied at dispersion corrected TPSS/def-TZVP level of theory. The energy partition analysis of the complexes revealed that for most of the complexes exchange and correlation energies represent dominant contributions to the binding with strong charge transfer from the metal atom to a NF. The exceptions are Mg complexes of graphene and phosphorene NFs where binding is due to dispersion and correlation terms. This difference is also reflected in large Mg-NF distances suggesting weak intermolecular interactions in these complexes. The calculated activation energies for metal hopping are easily achievable at room temperatures for carbon and silicon allotropies. However, they are significantly higher for phosphorus allotropies reaching almost 18 kcal/mol. Generally, activation energies for hopping increase with binding energies for graphene, silicene, and phosphorene NFs. This trend does not observe however for graphene, silicene, and phosphorene 2D allotropies.


Introduction
Although 2D materials have become a mainstream in material chemistry, only after graphene discovery in 2004 [1], the first 2D material, molybdenum disulfide, was obtained and studied, back in 1986 [2]. 2D materials fill the gap between 1 and 3D materials possessing valuable properties for applications especially in micro-and optoelectronics [3]. Graphene discovery pushed forward an extensive search for 2D allotropies of other elements. Following the graphene discovery, 2D allotropies of boron, phosphorus, silicon, and germanium have been obtained too [4][5][6][7][8][9]. Although all 2D allotropies share the same topology, possessing honeycomb structure, their physical structure, and chemical properties are very different.
Graphene and borophene are plane due to sp 2 hybridization common for 2nd row elements, while Si, P, and Ge honeycomb allotropies have buckled geometry. The electronic properties of the allotropies change too from metallic for borophene to semiconductor for phosphorene. Although honeycomb allotropies are the most stable 2D allotropies for all these elements, they are not the only possible ones. Thus, carbon allotropies consisting of fused azulene rings known as "haeckelites" [10] (Fig. 1) are dynamically stable and occur naturally as Stone-Wales defects in graphene [11]. Moreover, it has been recently demonstrated theoretically that silicon, phosphorus, and germanium, but not boron, are also capable of forming stable 2D allotropies of "haeckelite" topology [12]. Moreover, inorganic haeckelites were found to be relatively more stable than honeycomb structures compared to the corresponding carbon counterparts. It has been found that haeckelites have a much higher affinity to Li atoms than graphene which make them promising anode materials for Li-ion batteries [13].
Although Li-ion [14] batteries have the second-highest possible theoretical energy densities after proton batteries, lithium is expensive and chemically very active. A possible alternative to Li-ion batteries is Na-and Mg-ion batteries [15,16]. Sodium is much cheaper than Li, although it is more chemically active and offers lower theoretical energy density. Magnesium, on the other hand, is not only much more accessible than Li; it is also more environmentally stable and offers higher energy density than Na. Haeckelites, including new inorganic systems, are potentially interesting anode materials for Li-ion, Na-ion, and Mg-ion batteries; therefore, the understanding of the interaction nature between 2D systems and metal atoms is of great importance. The aim of this manuscript is to study the interaction nature of Li, Na, and Mg atoms with organic and inorganic haeckelites and to compare the results with these for honeycomb allotropies. Moreover, the mobility of metal atoms along 2D surfaces has also been explored.

Computational details
2D allotropies were modeled as passivated by hydrogen atoms nanoflakes (NFs) containing 96 heavy atoms (Fig. 2). All calculations were carried out using gradient corrected meta TPSS functional in combination with triple zeta quality def-TZVP basis set [17,18]. Grimm D3bj dispersion correction was applied in all cases [19]. Dispersion correction is important for complex modelling where dispersion may play an important role since modern functionals do not account well for dispersion interaction. TPSS functional offers a reasonable tradeoff between speed and accuracy [17]. Frequency calculations were run for all structures to ensure that minimum or transition state was found. All positive vibrational frequencies characterized a minimum, and for transition states, one imaginary mode was found connecting initial and final states. All calculations were carried out using TURBOMOLE 7.5 code [20]. To obtain deeper insight into the interaction nature between a NF and metal atom, energy decomposition analysis (EDA) was used [21]. The total binding energy (E bind ) was partitioned into physically meaningful contributions.
where E el is electrostatic energy, E ex is exchange energy, E orb is the orbital relaxation energy, E corr is the correlation energy, and E disp is the dispersion energy. In our analysis, E bind = E e1 + E ex + E orb + E corr + E disp we neglected the E orb term since in all cases it was smaller than 0.001 kcal/mol. First geometries of the complexes were optimized, and then, EDA analysis had been performed.
Metallic complexes with honeycomb NFs are denoted as X-Me, where X = G for graphene, X = Si for silicene, X = P for phosphorene, and Me-for metal atom type. In case of haeckelite allotropies, the metal complexes are marked as HX5-Me or HX7-Me where X has the same meaning as in honeycomb NFs, Me-for the metal atom type, and 5 or 7 is the complexation site: pentagon or heptagon.

Results and discussion
Geometry Figure 2 depicts the topology of studied metal complexes. In almost all cases, the metal atom is in the center and slightly above the polygon. The exception is HSi5 complexes where in a stable structure the metal atom interacts with two pentagons at once. Corresponding optimized geometries are shown in Fig. 3. Optimized Cartesian coordinates of the complexes are in supporting information. Figure 4 shows shortest distances in complexes between the metal and the NF atoms. For Li and Na, there is a clear correlation of shortest distances with both valence radii of metal atoms and these of NF atoms. Thus, the largest distances are detected for Si-Na, HSi5-Na, and HSi7-Na complexes in accordance with valence radii of Na and Si which are the biggest among the metal atoms and the NF atoms, respectively. However, complexes with magnesium do not follow this trend, especially in the case of the G-Mg complex with the longest distance out of all complexes. We will discuss this exception later. Studied NFs have 5-, 6-, and 7-member rings as complexation sites and the complexation preference depends on the metal atom size and the complex shape. For nearly plane H and Si systems, metal-NF distance in complexes increases with the atom size. The situation becomes more complex for HSi, P, and HP NFs which are not plane; the correlation between the atom size and the atom distance is lost for these complexes. The analysis of the interaction nature is helpful for these cases. Figures 5,6,7,8,and 9 show the total binding energies and their components for studied complexes: electrostatic, exchange, correlation, and dispersion, respectively. As seen from Fig. 5, for all complexes, Mg shows the lowest binding energies except for the HP7-Mg system, while Li binding energies are the highest. Graphene shows the smallest affinity to all types of metal atoms while the binding energies for silicene and phosphorene or for their allotropies are notably higher. The lowest binding energy is detected for G-Mg complexes. Inspecting the different terms to the binding energy for the G-Mg complex, it was found that the dispersion and correlation are the most important binding contributions; electrostatic and exchange energies are very low in comparison to other systems suggesting mostly weak intermolecular interaction in this complex. The natural charge of the Mg atom is essentially 0 (Fig. 10) and the electronic configuration of Mg is 3S 2 , similar to that in the Mg atom.

Binding energies
On the other hand, haeckelite H shows much better affinity to the Mg atom, the most important binding contributions come from exchange and correlation energies suggesting chemical bonding between H and Mg, and dispersion becomes less important in this case. Mg natural charge in H5-Mg and H7-Mg complexes are between 0.9 and 1.0 while natural electron configuration of Mg atom becomes 3S 1 , implying total electron transfer from Mg to haeckelite. Such a big difference between two isomeric carbon NFs is due to higher aromatic stabilization of benzene fragments of graphene compared to haeckelites resulting in lower electron affinity of graphene NF compared to haeckelite ones. [12] In case of Li and Na, there is a complete electron transfer from the metal atom to the NF for all complexes. For haeckelite carbon allotropies, the binding energies of H complexes are higher compared to graphene. The difference is due to lower electrostatic repulsion and more negative Fig. 2 Topologies of studied NF complexes with Li, Na, and Mg exchange energies in case of haeckelite. Correlation and dispersion barely affect resulting binding energies. Similar trends for binding energies can be observed for silicon allotropies; however, binding energies are higher compared to carbon counterparts and increase in the order Mg < Na < Li. Higher binding energies for silicone haeckelite allotropy compared to silicene are due to better relation between electrostatic repulsion, exchange attraction, and better correlation stabilization for the formers. Dispersion does not play an important role there. The electrostatic repulsion in all silicone-containing NF-metal complexes decreases in the order Mg > Na > Li, thus explaining the relative stability of the complexes for different metals. Other contributions are less important. It is interesting to note (Fig. 10) that, for Mg complexes of silicone haeckelite allotropy and especially for HSi5-Mg one, the natural charge of Mg atom is 1.4-the second highest out of all complexes which corresponds to the natural electron configuration of Mg atom 3S 0.56 . In case of the Si-Mg complex, the natural charge at the Mg atom is only 0.7. This is due to almost 1 eV higher electron affinity of HSi compared to silicene [12].
The binding energy for complexes with phosphorene NFs and phosphorene allotropy follows the trend found for other complexes Li > Na > Mg for all except for HP7-Me systems where the highest binding energy was found for Mg, while Li and Na have almost the same binding energies. This anomaly can be explained by analyzing charges and geometry. As seen, the shortest Mg-P bond is about 0.5 Å shorter in HP7-Mg compared to P-Mg and HP5-Mg, and the second shortest Mg-P distance in HP7-Mg is 2.54 Å, still 0.4 Å shorter compared to the shortest Mg-P distance in P-Mg and HP5-Mg complexes. This suggests that in HP7-Mg complex Mg      Figures 11 and 12 show calculated activation energies for metal atoms diffusion along the NFs. In case of hexagonal allotropies: graphene, silicene, and phosporene (Fig. 11), the activation energies for hopping increase in this direction being the lowest for graphene and highest for phosphorene. As seen from Fig. 13, the activation energies of hopping generally increase with binding energies. The highest activation energies correspond to P-Li, having the highest binding energy. The lowest activation energy was calculated for G-Mg; this complex shows the lowest binding energy too. The highest activation energies are for Li while the smallest are for Mg, except for phosphorene where the lowest activation energy is for Na.

Activation energies for hopping
In case of haeckelite allotropies of carbon, silicon, and phosphorus, there are 3 different ways of hopping for metal atoms: heptagon-heptagon (7-7), pentagon-pentagon (5-5), and heptagon-pentagon (7-5 or 5-7). 5-5 transitions states (TSs) were detected only for Li atom and only for carbon and phosphorus allotropies. For other metals and systems, the structure of TS corresponds to a minimum. This is related to the larger sizes of Na and Mg atoms compared to Li and small size of the pentagon ring where Na and Mg atoms can form a complex with two pentagons at the same time as shown in Fig. 3. Hopping activation energies for carbon and silicon allotropies are close to these of graphene and silicene being in the range between 0.2 and 6 kcal/ mol, all easily  Binding vs activation energies of metal hopping for graphene, silicene, and phosphorene NFs assessable at room temperature. The activation energies vary greatly for the HP system from almost 17 kcal/mol for Na (5-7) to 0.5 kcal/mol for Li ; however, they are generally lower than those estimated for phosphorene.
In the estimation of average activation energies of metal atoms among all studied NF type, Mg shows the lowest average activation energy of 4.15 kcal/mol, Na: 5.70, and Li: the highest 6.13 kcal/mol.
As can be seen, the average activation energies of metal atom migrations do not vary much with metal nature; however, the average activation energies for different NF types do vary (Fig. 14). The lowest average activation energies were calculated for graphene NFs, while the highest one was detected for phosphorene. The average activation energies for haeckelites and its silicon analog differ by only 2 kcal/ mol from these for graphene and silicene, while HP allotropy shows noticeably lower hopping activation energies compared to phosphorene. A combination of graphene and Mg leads to the lowest activation energy of hopping while P-Li has the highest activation energy.

Conclusions
Complexes formed by Mg are quite different from these formed by Na and Li. In the latter case, there is always a complete electron transfer from a metal atom to a NF and the most important contribution to the binding energy comes from exchange and correlation stabilization. The binding of Mg to a NF depends strongly on the NF type, ranging from G-Mg complex where there is no charge transfer from Mg to NF with dispersion being the most important contribution to binding, to HP7-Mg complex showing the highest binding energy out of all studied complexes where Mg transfers almost 1.5 electrons to the NF. In this case, the most important contributions to the binding are from exchange and correlation. The Mg-NF distance is 1 Å shorter for the latter complex. Generally, the binding energies increase in the order Mg < Na < Li for all types of NFs with exchange and correlation contributing to binding, being the most important. Dispersion is only important for the G-Mg complex. Haeckelite allotropies for carbon and silicon show higher binding energies compared to graphene and silicene, unlike phosphorus allotropies where there is no clear difference between phosphorene and HP allotropy which could be related to weak p electron coupling between phosphorus atoms in these systems. The activation energies for hopping are easily achievable at room temperatures for carbon and silicon allotropies. However, they are significantly higher for phosphorene and HP allotropies achieving almost 18 kcal/mol. There is a reasonably good correlation between the activation and binding energies for graphene, silicene, and phosphorene NFs. Such correlation is not observed however for HC, HSi, and HP allotropies.
Author contribution Cesar Gabriel Vera de la Garza-a PhD studentperformed EDA analysis.
Wilmer Esteban Vallejo Narváez-a postdoctoral fellow-transition state search, and partial manuscript writing.
Luis Daniel Solís Rodríguez-an undergraduate student-routine optimizations of complexes.
Seguei Fomine-group leader-the manuscript idea and writing.

Conflict of interest
The authors declare no competing interests.