Rattling Motion of Alkali Metal Ions through the Cavities of Model

Article pubs.acs.org/JPCA

Rattling Motion of Alkali Metal Ions through the Cavities of Model Compounds of Graphyne and Graphdiyne S. Chandra Shekar and R. S. Swathi* School of Chemistry, Indian Institute of Science Education and Research, Thiruvananthapuram, Kerala 695016, India S Supporting Information *

ABSTRACT: We study the passage of the alkali metal ions (Li+, Na+, and K+) through some conjugated carbon-based ring systems, starting from C12 H 6 and C 24 H 12 (tribenzocyclyne; TBC), which serve as model compounds for graphyne to some of the higher analogues, C26H12, C28H12, and C30H12, the model systems for graphdiyne. The motion of the ions through cyclic carbon clusters, C12 and C14, is also investigated. The potential for the motion of the ions from one side of the ring to the other through the cavities of the molecules is a symmetric double well in most cases, while it is a rather flat potential in others, arising due to the free motion of the ions through the cavities. Electrostatic potential (ESP) analyses reveal that the ions bind to the ring systems at the most negative regions of ESP. The estimated energy barriers for the motion of Li+ through C12H6 and C24H12 are 4.7 and 4.3 kcal mol−1, respectively, and are comparable to the barrier for the classic case of umbrella-like inversion in ammonia. Transmission of Li+ through C26H12, C28H12, C30H12, C12, and C14 rings is barrierless. We predict that the rattling motion of Li+ through the model compounds of graphyne and graphdiyne should be experimentally observable. We also model the effectively one-dimensional motion of the ions through the rings using discrete variable representation (DVR) and calculate the energy levels of the complexes in the symmetric double well potentials. The molecular orbital analyses and the nuclear independent chemical shift (NICS) values for the rings suggest distinct trends based on the (4n + 2/4n) π electron count, leading us to propose two neutral complexes, (C12H6)Li2 and (C24H12)Li2, that are highly stable with binding energies of 400 and 356 kcal mol−1, respectively.

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INTRODUCTION Theoretical studies on the passage of atoms and ions through the rings of conjugated carbon-based molecules have been pursued with a lot of interest in the past,1,2 motivated largely by the classic case of ammonia inversion, the study of which eventually led to the development of ammonia maser.3 The potential for the passage of atoms and ions from one side of the ring to the other is a double well, analogous to that of the potential for ammonia inversion. Ammonia undergoes an inversion from one pyramidal geometry to the other via a planar geometry as the transition state. The two pyramidal geometries correspond to the minima in the double well, while the planar geometry represents the transition state. The energy barrier for the interconversion between the two pyramidal geometries is ∼5.7 kcal mol−1.4 For the case of passage of atoms and ions through rings, the minima in the double well correspond to the stable geometries of the atom or the ion on either side of the ring, and the transition state corresponds to the geometry in which the atom or the ion is in plane with the ring at its center. The mechanical motion of H+ and Li+ through benzene1,5 and cyclononatetraenyl anion,2 respectively, has been studied, and the complexes are suggested as molecular rattles. Experimental realization of such molecular devices is very challenging6 and is an area actively being pursued. Understanding the interaction of atoms, cations, anions, and lone pairs of electrons on atoms with the π clouds in © XXXX American Chemical Society

conjugated molecules is also important because of its fundamental role in various organic transformations.5,7−9 A lot of effort has also gone into studying the interaction of transition metal cations with the ring systems as they are vital in the formation of various organometallic compounds and their reactions.10,11 Recent studies considered the interaction of atoms, ions, and molecules with the rings of novel carbon-based systems like fullerenes, graphene, and carbon nanotubes and found that the interactions are fundamental for a variety of applications in gas sensors,12 hydrogen storage,13 lithium ion batteries,14 isotopic separation,15 and desalination of water.16 The motion of atoms and ions through the pentagonal and the hexagonal rings of fullerenes, graphene, and carbon nanotubes being an important step in the process has stimulated several theoretical studies.17,18 The tunneling motion of protons and hydrogen atoms through model systems, which could mimic portions of fullerenes and carbon nanotubes,1,19 is recently investigated. Diffusion of lithium through carbon nanotubes17 and other intercalation compounds20 is also studied. Storage of molecular hydrogen in these carbon-based structures in the presence of Special Issue: Structure and Dynamics: ESDMC, IACS-2013 Received: March 25, 2013 Revised: May 15, 2013

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dx.doi.org/10.1021/jp402896v | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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lithium is found to be highly efficient13 and is facilitated by efficient charge transfer from lithium to these structures.21 However, there are only few studies in the literature on the motion of other alkali metal ions like Na+ and K+ through the carbon networks.22−25 A more recent study14 considers the diffusion of the lithium ions through the hexagonal rings, Stone−Wales defects, and the vacancies created in the graphene lattice. The study was motivated by the necessity to increase the diffusion of lithium ions across the graphene layers in lithium ion batteries and found that it is feasible when there are vacancies in graphene that lead to the easy passage of the ions. Recent theoretical studies on the tunneling of helium atoms through graphene nanopores15,26−28 have found a difference in the tunneling probabilities of 3He and 4He across the pores, a potential method for isotopic separation using graphene. However, in the majority of these investigations, the lattice had to be modified by creating pores29 within the hexagonal networks because of the fact that the passage of the atoms and molecules through six membered rings is rather difficult with large barrier heights.18 The lattices based on graphyne30−32 and graphdiyne,33 new materials in the family of extended carbon-based systems involving networks of sp and sp2 hybridized carbon atoms, could be interesting candidates for such applications because of the bigger cavities that these structures possess enabling the facile passage of the atoms, ions, and small molecules through them. Unlike the case of graphene, they do not need an extra effort of creating pores in the lattice. Hence, in this article, we explore the possibility of using graphyne and graphdiyne as alternatives to graphene in such studies. Graphyne and graphdiyne have currently been drawing a lot of attention and are suggested to be potential candidates for various purposes.34 A very recent study based on density functional theory predicts remarkable hydrogen storage of graphyne on doping with lithium.35 Zhang et al.36 have studied the in-plane and the out-of-plane diffusion of lithium ions in graphyne, in which they find efficient lithium storage capacity of graphyne. They also suggest graphyne to be an interesting anode material in lithium ion batteries. However, a lot of studies on graphyne and graphdiyne are only in their infancy and the interaction of these structures with various atoms, ions, and molecules is an important area that needs to be investigated. Herein, we consider the possibility of the motion of the alkali metal ions (Li+, Na+, and K+) through the cavities of some conjugated carbon-based ring systems, starting from C12H6 and C24H12, which serve as model compounds for graphyne to some of the higher analogues, C26H12, C28H12, and C30H12, the model systems for graphdiyne. The experimental synthesis of these building blocks of graphyne and graphdiyne has been achieved only a few years ago.34,37 Complexes of the alkali metal ions with C24H12 (tribenzocyclyne, TBC) as the ligand are found in gas phase experiments.38 C12 and C14 cyclic carbon clusters, which are found recently in gas phase spectroscopic studies,39,40 are also considered for investigations. The electronic structure of carbon clusters of various sizes predicted interesting geometries for such clusters with bond length alternation.41−43 Studies of the transport properties of ions across the C14 rings are reported.44 C12 and C14 clusters serve as model systems for realistic pores in extended carbon-based network structures and are useful in estimating the effect of dispersion interactions that arise in such networks. We find that the Li+ ion can undergo an interesting rattling motion through the cavities of the conjugated molecules and that it should be possible to

experimentally observe this motion. The predicted energy barriers of 4.7 and 4.3 kcal mol−1 for the transmission of Li+ through C12H6 and C24H12 are comparable to the energy barrier in the classic case of ammonia inversion. Such low energy barriers for the passage of Li+ enable us to support an earlier proposal36 that the diffusion of Li+ through graphyne would be very facile, suggesting the use of graphyne as an electrode material in lithium ion batteries. However, our calculations for the motion of Li+ through C30H12, the model compound of graphdiyne, suggest that there is no energy barrier for the passage of the ion and hence graphdiyne could be an even better anode material, and this interesting possibility needs to be explored in experiments. To the best of our knowledge, there are no reports of the transmission of atoms or ions through graphdiyne, and we believe that our studies are an important step in this direction.

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METHODOLOGY All the geometry optimizations and single-point energy calculations reported in this article are performed using dispersion-including density functional methods with M062X, a medium-range functional, and ωB97XD, a long-range functional that is currently a widely used dispersion-corrected DFT (DFT-D) method. The above-mentioned functionals along with a variety of other interesting functionals45 have been extensively used in the past couple of years for estimating the interaction energies in various complexes and are found to give rise to accurate results.46,47 The calculations herein are performed at the triple-ζ (6-311G(d,p)) level using the Gaussian09 suite of programs.48 We consider the complexes of Li+, Na+, and K+ with the molecules, C12H6, C24H12, C26H12, C28H12, C30H12, C12, and C14. We first optimize the geometries of the molecules and calculate their energies (Emolecule). We also calculate the energies of the alkali metal ions (Eion). Subsequently, stable complexes of the alkali metal ions with each of the rings are obtained. The regions of highest negative potential on the rings are determined using the electrostatic potential (ESP) surfaces and contours and are compared with the locations of the ions in the stable complexes. We then consider the transmission of the ions through the cavities of the rings. We estimate the energy barriers for the motion of the ions across the rings using two methods. In the first method (Method 1), we consider the optimized structures of the molecules and keep the ions at various positions along an axis perpendicular to the molecular plane and passing through the location of the ion in the optimized geometry. Single-point energy calculations are performed for each of these configurations of the ions with respect to the molecules. The interaction energies between the molecule and the ion in the complexes are then estimated using E int = E(complex) − E(molecule) − E(ion)

(1)

From the resultant energy scans, we then find the minimum energy configurations for the ions near the rings as well as the transition states and estimate the energy barriers for the passage of the ions through the ring cavities using E b(1) = E int(transition state) − E int (minimum energy configuration)

(2)

In the second method (Method 2), we optimize the geometries of the transition states with the ions at the ring centers by constraining the symmetries of the complexes. Subsequently, B



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stable complexes thus obtained were found to be the highest for Li+ and the lowest for K+. From the optimized geometries of the complexes, it can be seen that the ions are located on axes passing through the centers of the rings for C12H6, C24H12, C30H12, C12, and C14 rings. However, in the case of complexes with C26H12 and C28H12, we find that the ions bind at an offcenter position from the geometric centers of the rings. This arises essentially due to the fact that the regions of highest negative electrostatic potential for these rings are at off-center positions, as revealed by our ESP analysis. As an illustration of this, in Figure 3, we show the ESP surfaces and the contours for the C24H12 molecule and its corresponding Li+ complex. In C24H12, the region of highest ESP is at the center of the molecule, and the binding of the ion at the same location can be seen from the ESP of the C24H12−Li+ complex. On the contrary, the ESP surfaces and the contours of C26H12 and C26H12−Li+ as shown in Figure 4 clearly reveal the binding of the ion at an off-center position. In order to analyze the rattling motion of the ions through the cavities, we consider the motion of Li+, Na+, and K+ across the rings along an axis passing through the location of the ions in the optimized geometries of the complexes. The interaction energies are then evaluated for the complexes of the ions with the molecules for various values of z at the M06-2X/6311G(d,p) and ωB97XD/6-311G(d,p) levels of calculation. However, in this article, we present the results obtained at the M06-2X/6-311G(d,p) level, while in the Supporting Information, results obtained at the ωB97XD/6-311G(d,p) level are presented. The interaction energies thus obtained from the single-point energy calculations (see Method 1 in Methodology section) for the motion of Li+ through graphyne- and graphdiyne-based model compounds, C12H6, C24H12, C26H12, C28H12, and C30H12 are shown in Figure 5. The potential for the motion of Li+ through C12H6 and C24H12 is a symmetric double well, with the minima occurring at stable geometries for Li+ on either side of the rings at distances of ±0.9 Å along the z-axis. The binding energies (−Eint) of the complexes are 42.9 and 63.2 kcal mol−1, respectively. The transition states correspond to the geometries in which the ion is in the center of the ring. The interaction energy differences between the transition states and the minima define the barrier heights and are denoted as E(1) b (see the Methodology section). The values + of E(1) b for the passage of Li through C12H6 and C24H12 are found to be 4.7 and 4.3 kcal mol−1, respectively. The barrier heights for the motion of the ion through the rings are rather small, meaning that the motion is facile. The cavity sizes for C12H6 and C24H12 are roughly the same and hence their barrier heights are similar. The difference in the binding energies of their complexes with Li+ could be attributed to the presence of dispersion interactions due to the additional benzene rings. The motion of Li+ through C26H12, C28H12, and C30H12 gives rise to a rather flat potential suggesting the free motion of the ion through the rings, which occurs essentially due to the fact that the cavity sizes of these molecules are larger than those of C12H6 and C24H12. There is no barrier for the passage of the ion from one side of the ring to the other through the center. It has recently been reported that the diffusion of Li+ ions across graphene can be facilitated by creating nanopores in the lattice.50 However, creating nanopores is often accompanied by the formation of defects. Hence, use of conjugated network structures like graphyne and graphdiyne with larger cavities is a promising alternative. Our prediction of the free motion of Li+ through the cavity of C30H12, the model compound for

we estimate the barriers as the energy differences between the optimized geometries of the transition states and those of the stable complexes as E b(2) = E(transition state) − E(stable complex)

(3)

All the computations reported herein have been corrected for the basis set superposition error (BSSE) using the counterpoise method. On estimating the potential energy for the motion of the ions through the rings, which is effectively a onedimensional motion, we use discrete variable representation (DVR), a grid-based numerical method49 to solve the effectively one-dimensional Schrödinger equation. We obtain the energy levels of the various complexes in the double well potentials to estimate the tunnelling splittings of the various levels. The theoretical nuclear independent chemical shift (NICS) values are calculated for the ring systems using the gauge-including atomic orbital (GIAO) method at the M062X/6-311G(d,p) level of calculation. Molecular orbital calculations and ESP analyses are also performed at the M062X/6-311G(d,p) level.

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RESULTS AND DISCUSSION Our major objective in this article is to analyze the energy barriers for the rattling motion of the alkali metal ions across graphyne and graphdiyne and compare them with those of graphene. Extended networks of C24H12 and C30H12 molecules give rise to graphyne and graphdiyne, respectively. Graphene could be thought of as an extended network of C6H6 molecules. Motion across the conjugated C12 and C14 rings is also considered. Figure 1 gives a schematic showing the effective

Figure 1. Schematic showing the effective pore sizes of C6H6, C24H12, C30H12, C12, and C14 molecules.

pore sizes of C6H6, C24H12, C30H12, C12, and C14. The complexes of Li+, Na+, and K+ with C12H6, C24H12, C26H12, C28H12, C30H12, C12, and C14 are investigated in this article. The choice of these ring systems enabled us to systematically investigate the effect of the cavity size on the transmission of the ions. The ring systems are first optimized to determine their stable geometries. Figure 2 shows the optimized geometries of the molecules. We then consider the complexes of Li+, Na+, and K+ with each of these rings and determine the optimized geometries of these complexes. The binding energies of the C



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Figure 2. Optimized geometries of the carbon-based ring systems. TBC and C12H6 are model compounds for graphyne. C26H12, C28H12, and C30H12 are model compounds for graphdiyne. C12 and C14 are cyclic carbon clusters.

Figure 5. Potential for the motion of Li+ through the cavities of C12H6, C24H12, C26H12, C28H12, and C30H12. Figure 3. ESP surfaces and contours for C24H12 and C24H12−Li+.

Figure 6. Potential for the motion of Li+ through the cavities of C12 and C14.

note that the binding energies of the complexes of Li+ with C12H6, C24H12, C26H12, C28H12, and C30H12 are ∼50.0 kcal mol−1, while those of Li+ with C12 and C14 are ∼20.0 kcal mol−1. Despite the fact that the cavities of C12H6, C24H12, and C12 contain the same number of rim carbon atoms (12), their binding energies with Li+ are different. The observed differences in binding energies arise due to varying dispersion interactions as well as effective cavity sizes. The binding energy of C24H12 is larger than that of C12H6 in spite of having same cavity size as C12H6 due to the dispersion interactions from additional benzene rings. The binding energy of C12 is lower than that of C12H6 because of an effective increase in the cavity size (see Figure 1).

Figure 4. ESP surfaces and contours for C26H12 and C26H12−Li+.

graphdiyne leads us to propose that out-of-plane diffusion of the lithium ion through graphdiyne can occur very easily and hence graphdiyne could be a better anode material than graphene or graphyne in lithium ion batteries. This essentially arises due to the fact that the effective cavity sizes of graphene, graphyne, and graphdiyne (see Figure 1) are different, and the bottleneck in the diffusion of Li+ is the passage through the cavity. The motion of Li+ through the carbon clusters, C12 and C14 (see Figure 6 for the plots of interaction energy), also occurs without an energy barrier. However, it is interesting to D



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Figure 7 shows the potential for the motion of Na+ through the cavities of C12H6, C24H12, C26H12, C28H12, and C30H12. The

Figure 8. Potential for the motion of Na+ through the cavities of C12 and C14. Figure 7. Potential for the motion of Na+ through the cavities of C12H6, C24H12, C26H12, C28H12, and C30H12.

complexes of Na+ with graphyne- and graphdiyne-based model compounds. The interaction energies for the motion of K+ through C12H6, C24H12, C26H12, C28H12, and C30H12 and that through C12 and C14 are shown in Figures 9 and 10, respectively. The

nature of the potential is a double well for motion through C12H6, C24H12, and C26H12, while it is flat for motion through C28H12 and C30H12. The cavity size of C30H12 is large enough to allow free motion of both Li+ and Na+ ions. However, C26H12 allows a free motion of the Li+ ion, while a finite energy barrier exists for the motion of Na+. The energy barriers for the transmission of Na+ through the rings are found to be 56.5, 56.1, and 10.8 kcal mol−1 for C12H6, C24H12, and C26H12, respectively. A significant decrease in barrier height with the increase in cavity size is clearly evident. The minima of the potential occur at distances of ±1.5, ±1.8, and ±1.2 Å, respectively, along the z-axis on either side of the rings. The binding energies for the complexes are ∼45.0 kcal mol−1 for most complexes, and the actual values can be found in Table 1. Table 1. Summary of the Results Obtained from the Evaluations of the Interaction Energies for the Complexes of Li+, Na+, and K+ with C12H6, C24H12, C26H12, C28H12, C30H12, C12, and C14 at the M06-2X/6-311G(d,p) Level of Calculation C12H6

C24H12

C26H12

C28H12

C30H12

C12

C14

17.2 14.3 9.6

22.2 17.8 13.9

5.0 41.5

3.8

1.2 1.5

1.2

Figure 9. Potential for the motion of K+ through the cavities of C12H6, C24H12, C26H12, C28H12, and C30H12.

potential is a double well for all cases, except for the passage of the ion through C30H12 in which we essentially find the free movement of the ion across the ring. We find large energy barriers for the passage of K+ through the cavities of most of the rings. The actual values of the barrier heights and the positions of the minima can be found in Table 1. The motion of K+

−1

Li+ Na+ K+

42.9 38.0 17.4

Li+ Na+ K+

4.7 56.5 175.9

Li+ Na+ K+

0.9 1.5 1.8

Binding Energies (kcal mol ) 63.2 56.4 59.1 54.7 45.5 48.7 49.9 49.8 35.4 37.4 38.5 41.7 Barrier Heights (kcal mol−1) 4.3 56.1 10.8 0.2 196.7 77.7 27.2 Positions of Minima (Å) 0.9 1.8 1.2 0.6 2.1 1.8 1.5 -

The interaction energies for the complexes of Na+ with C12 and C14 are shown in Figure 8. The potential is a neat double well for C12 and is flat for C14. The cavity size of C12 is such that it allows a free passage of Li+ but creates an energy barrier for the motion of Na+ (5.0 kcal mol−1). Such selectivity of the rings toward free passage of specific ions while not allowing the others can be interesting for a large number of applications. However, the binding energies of the complexes of Na+ with C12 and C14 are again smaller in comparison with those of the

Figure 10. Potential for the motion of K+ through the cavities of C12 and C14. E



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through the C14 ring is found to have a small energy barrier (3.8 kcal mol−1) in contrast to that of Li+ and Na+ for which the motion was found to be barrierless. The binding energies of the complexes of the alkali metal ions with the model compounds increase as one goes from C24H12 through C26H12 and C28H12 to C30H12 in most cases. The sequential addition of an extra acetylenic group introduces polarization in the system, thereby increasing the binding energies with the ions. However, in the case of the smallest Li+ ion, the cavity of C30H12 is rather large, and hence, the binding energy of this ring with the Li+ ion is the least. We have also performed geometry optimizations for each of the graphyne and graphdiyne-based complexes mentioned above by keeping the ions at a distance of 3.0 Å from the molecular plane along the z-axis, and the optimized geometries were found to be almost the same as the minima found using the single-point energy calculations. Further, the optimized geometries were found to be the true minima with no imaginary frequencies, in contrast to earlier studies that found the geometries to be saddle points.2 This is particularly important because in benzene−H+ complexes, it was found that the σ-complex is more stable than the π-complex, and hence, unless the proton approaches the benzene ring with enough energy along the z-axis, it was not possible to obtain the rattling motion.1,2 Interestingly, in our case, we find that the πcomplexes are very stable (binding energies of ∼50 kcal mol−1) and hence are more robust. For the motion of each of the ions through the rings, we find that the values of E(complex) at the minima are lower (Eint is negative) than the asymptotic energies, E(molecule) + E(ion), suggesting that the interaction between the molecule and the ion in the complex is favorable. The values of Eint at the transition state are also negative for all complexes involving Li+ and for most complexes involving Na+, suggesting that the transition state geometries are also favorable for these complexes. However, for most of the complexes of K+, the values of Eint are positive at the transition state, suggesting that the transition state geometries are unfavorable due to increased repulsion in the case of K+. This is also manifested in large barrier heights for the transmission of the K+ ion across the rings. We now determine the energy barriers for rattling using a different approach. Earlier, we have estimated the energy barriers from the single-point energy calculations for the complexes at various positions of the ions. We now use the optimized geometries of the stable complexes and the transition states to obtain the barriers. The geometries of the complexes with the ions in the centers of the rings were optimized by constraining the symmetries of the complexes. The optimizations are found to give rise to one imaginary frequency, confirming the geometries as the transition states. We then estimate the energy barriers (Eb(2); see the Methodology section) for the motion of the ions through the rings as the difference between the energies of the optimized geometries of the transition states and the complexes. The values of the binding energies, barrier heights, and the positions of the minima obtained using this approach are tabulated in Table 2. The barrier heights, E(1) and E(2) estimated using the two b b + methods, for the Li complexes are nearly the same. However, in the case of Na+ and K+ complexes, we notice a large (2) difference in the computed values of E(1) b and Eb . This can be rationalized as follows: the differences between the energies of the transition states and the stable complexes give rise to barrier

Table 2. Summary of the Results Obtained from the Optimized Geometries of the Minima and the Transition States for the Complexes of Li+, Na+, and K+ with C12H6, C24H12, C26H12, C28H12, C30H12, C12, and C14 at the M06-2X/ 6-311G(d,p) Level of Calculation C12H6 Li+ Na+ K+

42.9 38.0 17.4

Li+ Na+ K+

3.7 35.8 98.8

Li+ Na+ K+

0.9 1.5 1.8

C24H12

C26H12

C28H12

C30H12

Binding Energies (kcal mol−1) 64.2 67.5 66.3 58.9 46.3 48.8 49.9 50.3 35.7 37.5 38.6 52.4 Barrier Heights (kcal mol−1) 3.5 35.6 5.8 102.7 38.5 a 0.4 Positions of Minima (Å) 1.1 1.7 1.1 2.0 1.8 1.4 0.6

C12

C14

25.4 14.3 9.6

26.9 19.5 13.9

a a

a

1.1 1.5

1.4

a

Note that for the complexes of Na+ and K+ with C12 and those of K+ with C28H12 and C14, the transition state geometries could not be optimized.

heights. The energies of the stable complexes in the estimation (2) of E(1) b and Eb are roughly the same (see the binding energies in Tables 1 and 2), while the energies of the transition states were found to be rather different. A closer look at the optimized geometries of the transition states for the complexes of Li+ and K+ with C24H12 reveals a significant distortion of the ring system in the transition state for the relaxed geometry of C24H12−K+ complexes, while there is no distortion for the C24H12−Li+ complex (see Figure 11). In the transition state of

Figure 11. Schematic showing the effective cavity sizes for the transition state geometries of the C24H12−Li+ and the C24H12−K+ complexes.

C24H12−K+, the ring actually expands (effective size of 2.12 Å) in order to accommodate the larger K+ ion, while this is not needed in the case of Li+ ion (effective size of 1.85 Å). In Method 1, we actually do not allow for a change in the geometry of the ring with the approach of the ion and hence the difference in the computed values of the barrier heights. The rattling motion of alkali metal ions across the rings is also analyzed using the dispersion-corrected density functional, ωB97XD. The results of calculations performed at the ωB97XD/6-311G(d,p) level for all the complexes are presented in the Supporting Information (see Figures S1−S6 and Tables S1 and S2). The nature of the double well potentials, binding energies, and the positions of minima obtained using the two functionals, M06-2X and ωB97XD are similar. M06-2X is one F



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complexes. We solve the above equation numerically using the DVR method and determine the energy levels. DVR is a novel grid-based method for solving the quantum eigenvalue problems. We define a uniform grid along the z-axis in the interval (−12,12) Å. We choose 800 grid points within this interval so that the grid spacing is 0.03 Å. The values of V0 and d were taken from the computed interaction energies and geometries (from Table 1). With the above-mentioned parameters, the kinetic energy and the potential energy matrix elements are evaluated using the DVR formalism.49 In DVR, the kinetic energy matrix is a full matrix, while the potential energy matrix is a diagonal matrix. We diagonalize the resultant Hamiltonian matrix to obtain the energy levels of the complexes. The numerical calculations are performed using Mathematica 7.0.51 The vibrational energy levels for the complexes of Li+ with C12H6 and C24H12 and Na+ with C26H12 and C12 are shown in Figures 13−16, respectively. For E

among a class of Minnesota functionals that includes the middle-range interactions, while ωB97XD is a dispersioncorrected functional that includes long-range interactions. Our analysis using both the functionals shows that the motion of the ions, including the asymptotics, is captured well by both of them. This suggests that even a middle-range functional like M06-2X describes well the dispersion effects observed in our systems. An accurate description of the asymptotics is a challenging issue in the estimation of the interaction energies using various quantum chemical methods. For the systems under investigation, we find that dispersion-including DFT with both the functionals describes the asymptotics very accurately. For large separations between the molecule and the ion, we have been able to recover the asymptotic energies, i.e., we find that the numerical values of E(complex) approach E(molecule) + E(ion) (Eint ≈ 0). Having found that the motion of Li+, Na+, and K+ through some of the rings is facile, with low barrier heights, the rattling motion of these ions is probed in greater details. The complexes of Li+ with C12H6 and C24H12 and Na+ with C26H12 and C12 are considered for these investigations. We calculate the vibrational energy levels of the complexes. The potential for the motion of the ions was found to be a symmetric double well, and we model it by an analytical form of a quartic double well given by ⎡ 2z 2 z4 ⎤ V (z) = V0⎢ − 2 + 4 ⎥ ⎣ d d ⎦

Figure 13. Vibrational energy levels for the complex of Li+ with C12H6 along with the analytical form of the quartic double well potential.

(4)

where V0 is the barrier height and ±d corresponds to the positions of the minima of the well. Figure 12 shows a

Figure 14. Vibrational energy levels for the complex of Li+ with C24H12 along with the analytical form of the quartic double well potential.

> V0, the ions make free oscillations from one side of the ring to the other, while for E < V0, the complex is bound to be in either of the minima. For E ≪ V0 and close to the two minima, the

Figure 12. Comparison of the computed and the analytical potential well for the C24H12−Li+ complex.

comparison of the computed double well potential (data from Figure 5) and the analytical potential of eq 4. The analytical potential mimics the computed potential reasonably well in the regions of interest to us (−1.5 Å ≤ z ≤ 1.5 Å). Using the abovementioned form for the potential energy of motion of the ions through the rings, the Schrödinger equation for the onedimensional motion of the ions along the z-axis can be written as ⎡ ℏ2 d 2 ⎤ + V ( z ) ⎥ψ ( z ) = E ψ ( z ) ⎢− 2 ⎣ 2μ dz ⎦

(5)

Figure 15. Vibrational energy levels for the complex of Na+ with C26H12 along with the analytical form of the quartic double well potential.

where μ is the reduced mass of the complex. The solutions to the above equation give the vibrational energy levels of the G



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are (4n) π electron systems, while C26H12, C30H12, and C14 are (4n + 2) π electron systems. As we go from (4n) π electron systems to (4n + 2) π electron systems, the δε values increase due to the increase in the aromatic nature of the rings. Within the (4n) π electron systems, the δε for C28H12 is lower than that of C24H12 due to an increase in conjugation. Similar is the trend in the case of δε values of C26H12 and C30H12. The antiaromatic/aromatic nature of the molecules is further confirmed from the NICS values as shown in Table 5. C12H6, Table 5. Computed NICS Values of the Ring Systems

Figure 16. Vibrational energy levels for the complex of Na+ with C12 along with the analytical form of the quartic double well potential.

ions oscillate around the minima, and the energy levels are doubly degenerate. However, for E ≈ V0, we find that there is a splitting of energy levels due to tunneling of the ions from one side of the ring to the other. The potentials in each of the above-mentioned complexes support a varying number of bound states. The potentials for the complexes of Na+ support a large number of vibrational energy levels than those of Li+. This can be attributed to the higher reduced masses of their corresponding complexes. As an evidence for the double degeneracy of the energy levels near the minima and the tunneling splittings near the barrier top, in Table 3, we show

energy (kcal mol−1)

degeneracy

level

energy (kcal mol−1)

degeneracy

1 2 3 4 5 6 7 8 9 10

0.38184 1.12677 1.84094 2.51971 3.15579 3.73309 4.19094 4.27089 4.56866 4.80427

2 2 2 2 2 2 1 1 1 1

11 12 13 14 15 16 17 18 19 20

5.09522 5.39999 5.72413 6.06384 6.41807 6.78571 7.16593 7.55805 7.96147 8.37571

1 1 1 1 1 1 1 1 1 1

the lowest 20 energy levels for the complex of Li+ with C24H12. It should be possible to observe the rattling motion of the Li+ ion through the rings experimentally by a vibrational excitation to the 8th energy level. We now perform molecular orbital analysis of the rings. The computed values of HOMO−LUMO gaps (δε) for the ring systems are shown in Table 4. C12H6, C24H12, C28H12, and C12 Table 4. Results of Molecular Orbital Calculations on the Ring Systems molecule

HOMO−LUMO gap (eV)

C12H6 C24H12 C26H12 C28H12 C30H12 C12 C14 C12H62− C24H122−

4.70 5.40 5.84 5.19 5.64 4.69 5.76 5.30 3.22

NICS (ppm)

C12H6 C24H12 C26H12 C28H12 C30H12 C12 C14 C12H62− C24H122−

13.07 3.07 −0.65 2.18 0.04 31.04 −32.03 −18.86 −18.25

C24H12, C28H12, and C12, being (4n) π electron systems, show positive NICS values, confirming the antiaromatic nature of these molecules. C26H12 and C14 are (4n + 2) π electron systems, and hence, they show negative NICS values. The numerical values of NICS for C24H12, C26H12, C28H12, and C30H12 are found to be rather small, suggesting that they exhibit more of a nonaromatic character. It has to be noted that, in spite of having similar cavities, the NICS values of C12H6 and C24H12 are very different. This is due to the fact that antiaromatic rings when fused with aromatic rings (like the C12 framework with benzene rings in C24H12) tend to lose their antiaromaticity and become nonaromatic.52 C12 and C14 exhibit distinct antiaromatic and aromatic nature, respectively. On the basis of the molecular orbital analyses and the NICS values, we infer that C12H6 and C24H12 are antiaromatic and therefore could be made aromatic by introducing −2 charge. Thus, we consider C12H62− and C24H122− and their complexation with two Li+ ions. In fact, we find that (C12H6)Li2 and (C24H12)Li2 are highly stable, with binding energies of 400 and 356 kcal mol−1, respectively. It is interesting to note that the binding energy of the C24H12−Li+ complex is larger than that of the C 12H6−Li+ complex, while the binding energy of the (C12H6)Li2 complex is larger than that of the (C24H12)Li2 complex. C12H6 is antiaromatic, while C24H12 has nonaromatic character due to the presence of the benzene rings (see the NICS values in Table 5), indicating that C24H12 is more stable than C12H6. The HOMO−LUMO gaps for the two molecules (see Table 4) also suggest the higher stability of C24H12. Hence, the complex of C24H12 with Li+ has larger binding energy than that of C12H6. On introducing a negative charge of −2 units, the resultant C12H62− and C24H122− become aromatic (see the NICS values in Table 5). However, C12H62− is more stable than C24H122− (see the HOMO−LUMO gap values in Table 4), and hence, (C12H6)Li2 has larger binding energy than (C24H12)Li2. The Li+ ions are located at distances of 2.51 and 2.54 Å in (C12H6)Li2 and (C24H12)Li2, respectively. Figure 17 shows the optimized geometry of the (C24H12)Li2 complex. Such complexes are potential reagents in organometallic chemistry.

Table 3. Energy Eigenvalues Corresponding to the Vibrational Motion of Li+ through the Cavity of C24H12 level

molecule

H



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CONCLUSIONS Theoretical investigations on the complexes of alkali metal ions with some of the model compounds of graphyne and graphdiyne have led us to propose that the rattling motion of Li+ across the rings of C12H6, C24H12, C26H12, C28H12, and C30H12 occurs with energy barriers in the range 0−5 kcal mol−1. We believe that it should be possible to observe such rattling motion experimentally. In particular, the energy barrier for the tunneling of Li+ from one side of the TBC ring, a model compound for graphyne to the other is ∼4.0 kcal mol−1. Such low barrier height supports the earlier proposal that graphyne could be a potential anode material in lithium ion batteries. Further investigations on the complexes of the ions with the model compounds of graphdiyne reveal that the diffusion of Li+ through the cavity of C30H12 is a barrierless process and graphdiyne could be an even better material for use in lithium ion batteries. Our studies on the rattling motion across the C12 and C14 clusters are fundamental for the experimental investigations on motion across extended carbon networks with large pores. Studies of the interactions between alkali metal ions and a selection of model compounds covering important ligands in various organic and organometallic transformations can provide insights into the fundamental interactions governing these complexes. ASSOCIATED CONTENT

S Supporting Information *

Results of the calculations of interaction energies for various complexes performed at the ωB97XD/6-311G(d,p) level. This material is available free of charge via the Internet at http:// pubs.acs.org.

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REFERENCES

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Figure 17. Optimized geometry (two views) of the (C24H12)Li2 complex.

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Article

AUTHOR INFORMATION

Corresponding Author

*(R.S.S.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge Prof. Shridhar R. Gadre for an interesting suggestion to perform ESP analysis of the complexes investigated in this article and for his encouragement. We also thank IISER-TVM for computational facilities. S.C.S. thanks IISER-TVM for financial support. I



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Rattling Motion of Alkali Metal Ions through the Cavities of Model

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