Effect of Li Termination on the Electronic and Hydrogen Storage Properties of Linear Carbon Chains: A TAO-DFT Study

Accurate prediction of the electronic and hydrogen storage properties of linear carbon chains (Cn) and Li-terminated linear carbon chains (Li2Cn), with n carbon atoms (n = 5–10), has been very challenging for traditional electronic structure methods, due to the presence of strong static correlation effects. To meet the challenge, we study these properties using our newly developed thermally-assisted-occupation density functional theory (TAO-DFT), a very efficient electronic structure method for the study of large systems with strong static correlation effects. Owing to the alteration of the reactivity of Cn and Li2Cn with n, odd-even oscillations in their electronic properties are found. In contrast to Cn, the binding energies of H2 molecules on Li2Cn are in (or close to) the ideal binding energy range (about 20 to 40 kJ/mol per H2). In addition, the H2 gravimetric storage capacities of Li2Cn are in the range of 10.7 to 17.9 wt%, satisfying the United States Department of Energy (USDOE) ultimate target of 7.5 wt%. On the basis of our results, Li2Cn can be high-capacity hydrogen storage materials that can uptake and release hydrogen at temperatures well above the easily achieved temperature of liquid nitrogen.

Nevertheless, these materials bind H 2 molecules very weakly (i.e., mainly governed by van der Waals (vdW) interactions), and hence, they perform properly only at very low temperatures.
For reversible hydrogen adsorption and desorption at ambient conditions (298 K and 1 bar), in addition to other thermodynamic considerations, the ideal binding energies of H 2 molecules on hydrogen storage materials should be in the range of about 20 to 40 kJ/mol per H 2 [6][7][8] . Consequently, various novel methods are being explored to increase the binding energies of H 2 molecules on high surface area materials to the aforementioned ideal range for ambient storage applications. To increase the H 2 adsorption binding energy, the surface of the adsorbent is generally modified with substitution doping, adatom adsorption, functionalization, etc. 2 . Among them, Li adsorption is especially attractive, because of its light weight with which a high gravimetric storage capacity could be easily achieved. Note also that Li-adsorbed carbon materials have been shown to possess relatively high gravimetric storage capacities with enhanced H 2 adsorption binding energies [9][10][11][12][13][14][15][16][17] , through a charge-transfer induced polarization mechanism 2, 18-20 .
Among carbon materials, linear carbon chains (C n ), consisting of n carbon atoms bonded with sp 1 hybridization (see Fig. 1(a), have recently attracted much attention owing to their unique electronic properties [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] . Note that C n may be considered for hydrogen storage applications due to their one-dimensional (1D) structures and the feasibility of synthesis of C n and their derivatives [24][25][26][27][28][29][30] . Recently, Pt-terminated linear carbon chains have been synthesized 28 . As mentioned above, due to a charge-transfer induced polarization mechanism 2, 18-20 , Li-terminated linear carbon chains (Li 2 C n ) can be good candidates for hydrogen storage materials (see Fig. 1(b-h). Because of the light elements (i.e., C and Li atoms) in Li 2 C n , high gravimetric storage capacities could be easily achieved. However, to the best of our knowledge, there has been no comprehensive study on the electronic and hydrogen storage properties of Li 2 C n in the literature, possibly due to the presence of strong static correlation effects in Li 2 C n (commonly occurring in 1D structures due to quantum confinement effects) 36 . Theoretically, the popular Kohn-Sham density functional theory (KS-DFT) 37 with conventional semilocal 38 , hybrid [39][40][41][42] , and double-hybrid [43][44][45][46] exchange-correlation (XC) density functionals can provide unreliable results for systems with strong static correlation effects 47 . For the accurate prediction of the properties of these systems, high-level ab initio multi-reference methods are typically needed 48 . Nonetheless, accurate multi-reference calculations are prohibitively expensive for large systems (especially for geometry optimization). To circumvent the formidable computational expense of high-level ab initio multi-reference methods, we have newly developed thermally-assisted-occupation density functional theory (TAO-DFT) [49][50][51] for the study of large ground-state systems (e.g., containing up to a few thousand electrons) with strong static correlation effects. In contrast to KS-DFT, TAO-DFT is a density functional theory with fractional orbital occupations, wherein strong static correlation is explicitly described by the entropy contribution (see Eq. (26) of ref. 49), a function of the fictitious temperature and orbital occupation numbers. Note that the entropy contribution is completely missing in KS-DFT. Interestingly, TAO-DFT is as efficient as KS-DFT for single-point energy and analytical nuclear gradient calculations, and is reduced to KS-DFT in the absence of strong static correlation effects. Therefore, TAO-DFT can treat both single-and multi-reference systems in a more balanced way than KS-DFT. Besides, existing XC density functionals in KS-DFT may also be adopted in TAO-DFT. Due to its computational efficiency and reasonable accuracy for large systems with strong static correlation, TAO-DFT has been successfully applied to the study of several strongly correlated electron systems at the nanoscale 17, 52-54 , which are typically regarded as "challenging systems" for traditional electronic structure methods (e.g., KS-DFT with conventional XC density functionals and single-reference ab initio methods) 47 . Accordingly, TAO-DFT can be an ideal theoretical method for studying the electronic properties of Li 2 C n . Besides, the orbital occupation numbers in TAO-DFT can be useful for examining the possible radical character of Li 2 C n . For the hydrogen storage properties, as the interaction between H 2 and Li 2 C n may involve dispersion (vdW) interactions, electrostatic interactions, and orbital interactions 3, 7, 55 , the inclusion of dispersion corrections 56,57 in TAO-DFT is important for properly describing noncovalent interactions. Therefore, in this work, we adopt TAO-DFT with dispersion corrections 50 to study the electronic and hydrogen storage properties of Li 2 C n with various chain lengths (n = 5-10). In addition, the electronic properties of Li 2 C n are also compared with those of C n to examine the role of Li termination.

Computational Details
All calculations are performed with a development version of Q-Chem 4.4 58 , using the 6-31G(d) basis set with the fine grid EML(75,302), consisting of 75 Euler-Maclaurin radial grid points and 302 Lebedev angular grid points. Results are computed using TAO-BLYP-D 50 (i.e., TAO-DFT with the dispersion-corrected BLYP-D XC density functional 56

Results and Discussion
Electronic Properties. To obtain the ground state of C n /Li 2 C n (n = 5-10), spin-unrestricted TAO-BLYP-D calculations are performed for the lowest singlet and triplet energies of C n /Li 2 C n on the respective geometries that were fully optimized at the same level of theory. The singlet-triplet energy (ST) gap of C n /Li 2 C n is calculated as (E T − E S ), the energy difference between the lowest triplet (T) and singlet (S) states of C n /Li 2 C n . As shown in Fig. 2(a), the ground states of C n and Li 2 C n are singlets for all the chain lengths investigated.
Because of the symmetry constraint, the spin-restricted and spin-unrestricted energies for the lowest singlet state of C n /Li 2 C n should be the same for the exact theory [49][50][51]59 . To assess the possible symmetry-breaking effects, we also perform spin-restricted TAO-BLYP-D calculations for the lowest singlet energies on the corresponding optimized geometries. The spin-restricted and spin-unrestricted TAO-BLYP-D energies for the lowest singlet state of C n /Li 2 C n are found to be essentially the same (within the numerical accuracy of our calculations), implying that essentially no unphysical symmetry-breaking effects occur in our spin-unrestricted TAO-BLYP-D calculations.
To assess the energetic stability of terminating Li atoms, the Li binding energy, E b (Li), on C n is computed using where E C n is the total energy of C n , E Li is the total energy of Li, and E Li C n 2 is the total energy of Li 2 C n . E b (Li) is subsequently corrected for the basis set superposition error (BSSE) using the counterpoise correction 60 , where the C n is considered as one fragment, and the 2 Li atoms are considered as the other fragment. As shown in Fig. 2(b), C n can strongly bind the Li atoms with the binding energy range of 258 to 357 kJ/mol per Li.
At the ground-state (i.e., the lowest singlet state) geometry of C n /Li 2 C n (with N electrons), the vertical ionization potential ( are obtained with multiple energy-difference calculations, with E N being the total energy of the N-electron system. For each n, Li 2 C n possesses the smaller IP v (see Fig. 3(a), EA v (see Fig. 3(b), and E g (see Fig. 4) values than C n . Note also that the IP v , EA v , and E g values of Li 2 C n are less sensitive to the chain length than those of C n .
To examine the possible radical character of C n /Li 2 C n , we calculate the symmetrized von Neumann entropy (e.g., see Eq. for the lowest singlet state of C n /Li 2 C n as a function of the chain length, using TAO-BLYP-D. Here, f i the occupation number of the i th orbital obtained with TAO-BLYP-D, which varies from 0 to 1, is approximately equal to the occupation number of the i th natural orbital [49][50][51]61 . For a system without strong static correlation ({f i } are close to either 0 or 1), S vN provides insignificant contributions, while for a system with strong static correlation ({f i } are fractional for active orbitals, and are close to either 0 or 1 for others), S vN increases with the number of active orbitals. As shown in Fig. 5, the S vN values of C n with even-number carbon atoms and Li 2 C n with odd-number carbon atoms are much larger than the S vN values of C n with odd-number carbon atoms and Li 2 C n with even-number carbon atoms, respectively.
To illustrate the causes of the odd-even oscillations in S vN , we plot the occupation numbers of the active orbitals for the lowest singlet states of C n (see Fig. 6(a) and Li 2 C n (see Fig. 6(b), calculated using TAO-BLYP-D. Here, the highest occupied molecular orbital (HOMO) is the (N/2) th orbital, and the lowest unoccupied molecular orbital (LUMO) is the (N/2 + 1) th orbital, with N being the number of electrons in C n /Li 2 C n . For brevity, HOMO,  HOMO − 1, HOMO − 2, and HOMO − 3, are denoted as H, H − 1, H − 2, and H − 3, respectively, while LUMO, LUMO + 1, LUMO + 2, and LUMO + 3, are denoted as L, L + 1, L + 2, and L + 3, respectively. As shown, C n with even-number carbon atoms and Li 2 C n with odd-number carbon atoms possess more pronounced diradical character than C n with odd-number carbon atoms and Li 2 C n with even-number carbon atoms, respectively.
On the basis of several measures (e.g., the smaller ST gap, smaller E g , larger S vN , and more pronounced diradical character), C n with even-number carbon atoms and Li 2 C n with odd-number carbon atoms should exhibit much stronger static correlation effects than C n with odd-number carbon atoms and Li 2 C n with even-number carbon atoms (i.e., possessing single-reference character), respectively. Note that KS-DFT with conventional XC density functionals can be unreliable for the properties of systems with strong static correlation effects, and accurate multi-reference calculations are prohibitively expensive for large systems (e.g., the longer C n and Li 2 C n ). In addition, due to the alteration of the reactivity of C n and Li 2 C n with n, it is highly desirable to adopt an electronic structure method that can provide a balanced performance for both single-and multi-reference systems, well justifying the use of TAO-DFT in this study.
Hydrogen Storage Properties. As pure carbon materials bind H 2 molecules very weakly (i.e., mainly governed by vdW interactions), they are unlikely to be promising hydrogen storage materials at ambient conditions 6 . Similarly, C n are not ideal for ambient storage applications, since the binding energies of H 2 molecules remain small. In addition, the number of H 2 molecules that can be adsorbed on C n is quite limited, due to the repulsive interaction between the adsorbed H 2 molecules at short distances 62 . Consequently, the more the adsorbed H 2 molecules, the less the average H 2 binding energy on C n . Therefore, C n cannot be high-capacity hydrogen storage materials at ambient conditions.
Here, we investigate the hydrogen storage properties of Li 2 C n (n = 5-10). As illustrated in Fig. 1(b-h), at the ground-state geometry of Li 2 C n , x H 2 molecules (x = 1-6) are initially placed on various possible sites around each Li atom, and the structures are subsequently optimized to obtain the most stable geometry. All the H 2 molecules  are found to be adsorbed molecularly to the Li atoms. The average H 2 binding energy, E b (H 2 ), on Li 2 C n is evaluated by where E H 2 is the total energy of H 2 , and − E x Li C 2 H n 2 2 is the total energy of Li 2 C n with x H 2 molecules adsorbed on each Li atom. Subsequently, E b (H 2 ) is corrected for BSSE using a standard counterpoise correction 60 . As shown in Fig. 7(a), E b (H 2 ) is in the range of 19 to 27 kJ/mol per H 2 for x = 1-4, in the range of 18 to 19 kJ/mol per H 2 for x = 5, and about 16 kJ/mol per H 2 for x = 6, falling in (or close to) the ideal binding energy range.
To assess if the binding energies of successive H 2 molecules are also in (or close to) the ideal binding energy range (i.e., not just the average H 2 binding energy), the binding energy of the y th H 2 molecule (y = 1-6), E b,y (H 2 ), on Li 2 C n is evaluated by Similarly, E b,y (H 2 ) is also corrected for BSSE using a standard counterpoise correction 60 . As shown in Fig. 7(b), E b,y (H 2 ) is in the range of 16 to 27 kJ/mol per H 2 for y = 1-4, in the range of 11 to 12 kJ/mol per H 2 for y = 5, and less than 5 kJ/mol per H 2 for y = 6. Therefore, while the first four H 2 molecules can be adsorbed on Li 2 C n in (or close to) the ideal binding energy range, the fifth and sixth H 2 molecules are only weakly adsorbed (i.e., appropriate only for storage at very low temperatures).
To assess the types of noncovalent interactions between H 2 and Li 2 C n , we compute the atomic charge on each Li atom for Li 2 C n (n = 5-10) with x H 2 molecules (x = 0-6) adsorbed on each Li atom (see Fig. 8), using the CHELPG (CHarges from ELectrostatic Potentials using a Grid based method) scheme 63 , in which atomic charges are fitted to reproduce the molecular electrostatic potential at a number of points around the molecule. For further clarification, we also plot the charge density isosurfaces of C 5 and Li 2 C 5 with x H 2 molecules (x = 0-6)  adsorbed on each Li atom (see Fig. 9). Similar charge density isosurfaces are also found for the longer Li 2 C n (n = 6-10) with the same number of adsorbed H 2 molecules. As the electronegativity of C is much higher than that of Li, the transfer of electronic charge in Li 2 C n is from Li to C n , resulting in a positive charge of 0.67-0.79 |e| on each Li atom in Li 2 C n . The positively charged Li atom can interact with more than one H 2 molecule, but the positive charge on Li decreases for the subsequent adsorption of H 2 molecules (up to x = 3). This type of adsorption can be attributed to the polarization of H 2 molecules by the positively charged Li atom (i.e., charge-induced dipole interaction) 2, 18-20 , leading to the enhanced H 2 binding energy and high hydrogen uptake in Li 2 C n . When the number of adsorbed H 2 molecules is large (e.g., x = 4-6), there is a significant overlap of the Li and H 2 charge densities, enhancing orbital interactions 3,7,55 . This suggests that orbital interactions should also be responsible for the H 2 binding energy, especially when a large number of H 2 molecules (e.g., x = 4-6) are adsorbed on the Li atom. In particular, due to the enhanced orbital interactions, when the fourth H 2 molecule is adsorbed on the Li atom, a small fraction of electronic charge is transferred from the Li atom to the adsorbed H 2 molecules, slightly increasing the positive charge on Li. Interestingly, there is no overlap between the charge density of the sixth H 2 molecule and the charge densities of other molecules, supporting that the sixth H 2 molecule is only weakly adsorbed (i.e., mainly governed by vdW interactions). Accordingly, the noncovalent interactions between H 2 and Li 2 C n should involve charge-induced dipole interactions, orbital interactions, and vdW interactions.
The desorption temperature, T D , of the adsorbed H 2 molecules is estimated using the van't Hoff equation 13,17,64,65 is the average H 2 binding energy (given by Eq. (3)), ΔS is the change in hydrogen entropy from gas to liquid phase (ΔS = 13.819R taken from ref. 66), p 0 is the standard atmospheric pressure (1 bar), p eq is the equilibrium pressure, k B is the Boltzmann constant, and R is the gas constant. As shown in Table 1, T D for Li 2 C n (n = 5-10) with x H 2 molecules (x = 1-4) adsorbed on each Li atom, is estimated using Eq. (5) at p eq = 1.5 bar (as adopted in ref. 6) and at p eq = 1 bar (the standard atmospheric pressure). As the E b (H 2 ) values are in the range of 19.47 to 26.53 kJ/mol per H 2 for x = 1-4, the corresponding T D values are in the range of 165 to 224 K at p eq = 1.5 bar, and in the range of 169 to 231 K at p eq = 1 bar, well above the easily achieved temperature of liquid nitrogen (i.e., 77 K). Therefore, Li 2 C n (n = 5-10) can be viable hydrogen storage materials that can uptake and release hydrogen at temperatures well above 77 K. Note that strictly, ΔS should be the change of total entropy before and after the hydrogenation. Therefore, the T D values given in Table 1 have to be taken as rough estimates for the desorption temperatures due to the definition of ΔS. For all metal-hydrogen systems, ΔS can be roughly estimated as the entropy change from molecular hydrogen gas to dissolved solid hydrogen 1 , that is 15.720R (taken Since Li 2 C n (n = 5-10) adsorb hydrogen as a form of molecule, ΔS should be smaller than 15.720R (as the entropy of adsorbed hydrogen should be positive yet nonvanishingly small). While there is no accurate estimate of the entropy of hydrogen in adsorbed state, it should be safe to assume that it is much less than that of the gas state 67 . Therefore, we estimate ΔS as the entropy change from molecular hydrogen gas to liquid hydrogen, as suggested by previous studies 13,17,64,65 . On the basis of Eq. (5), the larger the ΔS, the lower the T D values. However, even if the maximal ΔS (i.e., 15.720R) is adopted, the corresponding T D values will be only slightly lower (i.e., within 28 K for each case) than our reported T D values given in Table 1, being also well above 77 K. Accordingly, our comments remain the same even for the extreme case. As Li 2 C n (n = 5-10) can bind up to 8 H 2 molecules (i.e., each Li atom can bind up to 4 H 2 molecules) with the average and successive H 2 binding energies in (or close to) the ideal binding energy range, the corresponding H 2 gravimetric storage capacity, C g , is calculated using Here, M Li C n 2 is the mass of Li 2 C n , and M H 2 is the mass of H 2 . Note that C g (see Eq. (6)) is 17.9 wt% for n = 5, 15.8 wt% for n = 6, 14.1 wt% for n = 7, 12.8 wt% for n = 8, 11.7 wt% for n = 9, and 10.7 wt% for n = 10, satisfying the USDOE ultimate target of 7.5 wt%. Based on the observed trends for Li 2 C n , the maximum number of H 2 molecules that can be adsorbed on each Li atom with the average and successive H 2 binding energies in (or close to) the ideal binding energy range should be 4, regardless of the chain length. Therefore, the C g value of Li 2 C n should decrease as the chain length increases. Note, however, that the C g values obtained here may not be directly compared to the USDOE target value, which refers to the complete storage system (i.e., with the storage material, enclosing tank, insulation, piping, etc.) 5 . Nevertheless, since the C g values obtained here are much higher (especially for the shorter Li 2 C n ) than the USDOE ultimate target, the complete storage systems based on Li 2 C n are likely to be high-capacity hydrogen storage materials that can uptake and release hydrogen at temperatures well above the temperature of liquid nitrogen.

Conclusions
In conclusion, the search for ideal hydrogen storage materials has been extended to large systems with strong static correlation effects (i.e., those beyond the reach of traditional electronic structure methods), due to recent advances in TAO-DFT. In this work, we have studied the electronic properties (i.e., the Li binding energies, ST gaps, vertical ionization potentials, vertical electron affinities, fundamental gaps, symmetrized von Neumann entropy, and active orbital occupation numbers) and hydrogen storage properties (i.e., the average H 2 binding energies, successive H 2 binding energies, H 2 desorption temperatures, and H 2 gravimetric storage capacities) of Li 2 C n (n = 5-10) using TAO-DFT. As Li 2 C n with odd-number carbon atoms have been shown to possess pronounced diradical character, KS-DFT with conventional XC density functionals can be unreliable for studying the properties of these systems. In addition, accurate multi-reference calculations are prohibitively expensive for the longer Li 2 C n (especially for geometry optimization), and hence, the use of TAO-DFT in this study is well justified. On the basis of our results, Li 2 C n can bind up to 8 H 2 molecules (i.e., each Li atom can bind up to 4 H 2 molecules) with the average and successive H 2 binding energies in (or close to) the ideal range of about 20 to 40 kJ/mol per H 2 . Accordingly, the H 2 gravimetric storage capacities of Li 2 C n are in the range of 10.7 to 17.9 wt%, satisfying the USDOE ultimate target of 7.5 wt%. Consequently, Li 2 C n can be high-capacity hydrogen storage materials that can uptake and release hydrogen at temperatures well above the easily achieved temperature of liquid nitrogen.
For the practical realization of hydrogen storage based on Li 2 C n , Li 2 C n may be adopted as building blocks. For example, we may follow the proposal of Liu et al. 68 , and consider connecting Li-coated fullerenes with Li 2 C n , which could also serve as high-capacity hydrogen storage materials. A systematic study of the electronic and hydrogen storage properties of these systems is essential, and may be considered for future work. Since linear carbon chains 26,27 and Pt-terminated linear carbon chains 28 have been successfully synthesized, the realization of hydrogen storage materials based on Li 2 C n should be feasible, and is now open to experimentalists. In the future, we plan to examine how the electronic and hydrogen storage properties of linear carbon chains vary with different metal dopants (e.g., Na, Al, Ca, Ti, etc.).