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

TL;DR

This study uses the novel TAO-DFT method to accurately predict electronic and hydrogen storage properties of linear carbon chains and their lithium-terminated variants, revealing promising high-capacity hydrogen storage materials.

Contribution

The paper demonstrates the effectiveness of TAO-DFT in studying large, strongly correlated systems and provides new insights into the hydrogen storage potential of Li-terminated carbon chains.

Findings

Li2Cn chains have hydrogen binding energies near ideal ranges.

H2 storage capacities of Li2Cn reach 10.7 to 17.9 wt%.

Li2Cn can enable reversible hydrogen storage at near-ambient conditions.

Abstract

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 for reversible hydrogen uptake and release at near-ambient conditions.

Equations

$$E_b(\text{Li})=(E_{{\text{C}}_n}+2E_{\text{Li}}-E_{{\text{Li}}_2{\text{C}}_n})/2,$$

$$S_{\text{vN}}=-\frac{1}{2}\sum_{i=1}^{\infty}\bigg{\{}f_{i}\ \text{ln}(f_{i})+(1-f_{i})\ \text{ln}(1-f_{i})\bigg{\}}$$

$$E_b({\text{H}}_2)=(E_{{\text{Li}}_2{\text{C}}_n}+2x{E}_{{\text{H}}_2}-E_{{\text{Li}}_2{\text{C}}_n\text{-}2x{\text{H}}_2})/(2x),$$

$$E_{b,y}({\text{H}}_2)=(E_{{\text{Li}}_2{\text{C}}_n\text{-}2(y-1){\text{H}}_2}+2E_{{\text{H}}_2}-E_{{\text{Li}}_2{\text{C}}_n\text{-}2y{\text{H}}_2})/2.$$

$$C_g=\frac{8M_{{\text{H}}_2}}{M_{{\text{Li}}_2{\text{C}}_n}+8M_{{\text{H}}_2}}.$$

Full text

Introduction

Hydrogen (H2), as a pure energy carrier, has many attributes. Being light weight, it carries 142 MJ/kg of energy, which is approximately three times the energy content of gasoline, in terms of mass. Also, it is highly abundant on the earth in the form of water. More importantly, when hydrogen is burned with oxygen, it releases water vapor as the only effluent. Despite these advantages, there remain several problems to be clarified for the use of hydrogen. For example, hydrogen is highly flammable, and hence, if it comes in contact with the environment, it will burst. Another problem is related to its low energy content in terms of volume: it has only 0.0180 MJ/L, which is very low relative to gasoline (34.8 MJ/L). Moreover, over the past few years, the storage of hydrogen for onboard applications has been an active arena, which also requires a lightweight storage medium. Because of these reasons, storing a large amount of hydrogen reversibly in a small and lightweight container safely has been the biggest challenge in realizing a hydrogen-based economy Schlapbach2001 ; Jena2011 ; Park2012 ; Dalebrook2013 ; usdoe .

Over the years, the United States Department of Energy (USDOE) has monitored the research progress in the development of hydrogen storage materials for consumer vehicles. In 2015, the USDOE set the ultimate target of 7.5 wt% for the gravimetric storage capacities of onboard hydrogen storage materials for light-duty vehicles usdoe . As of now, there have been several methods for the storage of hydrogen Schlapbach2001 ; Jena2011 ; Park2012 ; Dalebrook2013 . The conventional methods for storing hydrogen are the high pressure method and the cryogenic method. In the high pressure method, one adopts carbon fiber reinforced tanks, which can withstand very high pressures (e.g., 350 to 700 bar), to store a large amount of completely recoverable hydrogen. In the cryogenic method, hydrogen is stored at very low temperatures (e.g., 20 K), typically requiring an expensive liquid helium refrigeration system. Both of these methods are not suitable for onboard automobile applications, because of the associated risk, high cost, and heavy weight. The storage of hydrogen in a metal hydride seems to be a convincing solution, but the irreversibility, slow kinetics, and high desorption temperature associated with this method are the problems yet to be overcome. Another promising solution is the storage of hydrogen in high surface area materials (e.g., graphene, carbon nanotubes, and metal-organic frameworks) through the adsorption-based methods. As high surface area materials could adsorb large amounts of hydrogen, the corresponding H2 gravimetric storage capacities could be rather high. Nevertheless, these materials bind H2 molecules very weakly (i.e., mainly governed by van der Waals (vdW) interactions), and hence, they perform properly only at low temperatures.

For reversible hydrogen adsorption and desorption at ambient conditions (298 K and 1 bar), the ideal binding energies of H2 molecules on hydrogen storage materials should be in the range of about 20 to 40 kJ/mol per H2 Bhatia2006 ; Lochan2006 ; Sumida2013 . Consequently, various novel methods are being explored to increase the binding energies of H2 molecules on high surface area materials to the aforementioned ideal range for ambient storage applications. To increase the H2 adsorption binding energy, the surface of the adsorbent is generally modified with substitution doping, adatom adsorption, functionalization, etc. Jena2011 . 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 H2 adsorption binding energies Chen1999 ; Deng2004 ; Deng2010 ; SeenithuraiLi ; Qiu2014 ; TAOH2S1 , through a charge-transfer induced polarization mechanism Niu1992 ; Niu1995 ; Froudakis2001 ; Jena2011 .

Among carbon materials, linear carbon chains (Cn), consisting of $n$ carbon atoms bonded with sp1 hybridization (see Figure 1(a)), have recently attracted much attention owing to their unique electronic properties Banhart2015 ; Cesari2016 ; Jin2009 ; Chuvilin2009 ; Kano2014 ; VanZee1988 ; Pan2003 ; Fan1989 ; Heimann1999 ; Horny2002 ; Belau2007 ; Lang1998 ; Pan2003 ; Souza2008 ; Li2009 ; Artyukhov2014 ; Banhart2015 ; Cesari2016 . Note that Cn may be considered for hydrogen storage applications due to their one-dimensional (1D) structures and the feasibility of synthesis of Cn and their derivatives Banhart2015 ; Cesari2016 ; Jin2009 ; Chuvilin2009 ; Kano2014 ; VanZee1988 ; Pan2003 . Recently, Pt-terminated linear carbon chains have been synthesized Kano2014 . As mentioned above, due to a charge-transfer induced polarization mechanism Niu1992 ; Niu1995 ; Froudakis2001 ; Jena2011 , Li-terminated linear carbon chains (Li2Cn) can be good candidates for hydrogen storage materials at near-ambient conditions (see Figure 1(b–h)). Because of the light elements (i.e., C and Li atoms) in Li2Cn, 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 Li2Cn in the literature, possibly due to the presence of strong static correlation effects in Li2Cn (commonly occurring in 1D structures due to quantum confinement effects Brus2014 ). Theoretically, the popular Kohn-Sham density functional theory (KS-DFT) Kohn1965 with conventional semilocal PBE , hybrid hybrid ; wM05-D ; LC-D3 ; SLC-D3 , and double-hybrid B2PLYP ; wB97X-2 ; PBE0-2 ; SCAN0-2 exchange-correlation (XC) density functionals can provide unreliable results for systems with strong static correlation effects Cohen2012 . For accurate prediction of the properties of these systems, high-level ab initio multi-reference methods are typically needed multi-reference . 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) ChaiTAO2012 ; ChaiTAO2014 ; ChaiTAO2017 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. ChaiTAO2012 ), 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 Wu2015 ; NK ; TAOH2S1 ; cycl , 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) Cohen2012 . Accordingly, TAO-DFT can be an ideal theoretical method for studying the electronic properties of Li2Cn. Besides, the orbital occupation numbers in TAO-DFT can be useful for examining the possible radical character of Li2Cn. For the hydrogen storage properties, as the interaction between H2 and Li2Cn may involve dispersion (vdW) interactions, electrostatic interactions, and orbital interactions Lochan2006 ; Park2012 ; Tsivion2014 , the inclusion of dispersion corrections BLYP-D ; Grimme2016 in TAO-DFT is important for properly describing noncovalent interactions. Therefore, in this work, we adopt TAO-DFT with dispersion corrections ChaiTAO2014 to study the electronic and hydrogen storage properties of Li2Cn with various chain lengths ($n$ = 5–10). In addition, the electronic properties of Li2Cn are also compared with those of Cn to examine the role of Li termination.

Computational Details

All calculations are performed with a development version of Q-Chem 4.4 Shao2015 , 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 ChaiTAO2014 (i.e., TAO-DFT with the dispersion-corrected BLYP-D XC density functional BLYP-D and the LDA $\theta$-dependent density functional $E_{\theta}^{\text{LDA}}$ (see Eq. (41) of Ref. ChaiTAO2012 )) with the fictitious temperature $\theta$ = 7 mhartree (as defined in Ref. ChaiTAO2012 ).

Results and Discussion

Electronic Properties

To obtain the ground state of Cn/Li2Cn ($n$ = 5–10), spin-unrestricted TAO-BLYP-D calculations are performed for the lowest singlet and triplet energies of Cn/Li2Cn on the respective geometries that were fully optimized at the same level of theory. The singlet-triplet energy (ST) gap of Cn/Li2Cn is calculated as $(E_{\text{T}}-E_{\text{S}})$, the energy difference between the lowest triplet (T) and singlet (S) states of Cn/Li2Cn. As shown in Figure 2, the ground states of Cn and Li2Cn 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 Cn/Li2Cn should be the same for the exact theory Rivero2013 ; ChaiTAO2012 ; ChaiTAO2014 ; ChaiTAO2017 . 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 Cn/Li2Cn 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, $E_{b}(\text{Li})$, on Cn is computed using

[TABLE]

where $E_{\text{C}_{n}}$ is the total energy of Cn, $E_{\text{Li}}$ is the total energy of Li, and $E_{\text{Li}_{2}\text{C}_{n}}$ is the total energy of Li2Cn. $E_{b}(\text{Li})$ is subsequently corrected for the basis set superposition error (BSSE) using the counterpoise correction Boys1970 , where the Cn is considered as one fragment, and the 2 Li atoms are considered as the other fragment. As shown in Figure 3, Cn 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 Cn/Li2Cn (with $N$ electrons), the vertical ionization potential ($\text{IP}_{v}={E}_{N-1}-{E}_{N}$), vertical electron affinity ($\text{EA}_{v}={E}_{N}-{E}_{N+1}$), and fundamental gap ($E_{g}=\text{IP}_{v}-\text{EA}_{v}={E}_{N+1}+{E}_{N-1}-2{E}_{N}$) are obtained with multiple energy-difference calculations, with ${E}_{N}$ being the total energy of the $N$-electron system. For each $n$, Li2Cn possesses the smaller $\text{IP}_{v}$ (see Figure 4), $\text{EA}_{v}$ (see Figure 5), and $E_{g}$ (see Figure 6) values than Cn. Note also that the $\text{IP}_{v}$, $\text{EA}_{v}$, and $E_{g}$ values of Li2Cn are less sensitive to the chain length than those of Cn.

To examine the possible radical character of Cn/Li2Cn, we calculate the symmetrized von Neumann entropy (e.g., see Eq. (9) of Ref. Rivero2013 )

[TABLE]

for the lowest singlet state of Cn/Li2Cn as a function of the chain length, using TAO-BLYP-D. Here, $f_{i}$ the occupation number of the $i^{\text{th}}$ orbital obtained with TAO-BLYP-D, which varies from 0 to 1, is approximately equal to the occupation number of the $i^{\text{th}}$ natural orbital ChaiTAO2012 ; ChaiTAO2014 ; ChaiTAO2017 ; noon . For a system without strong static correlation ($\{f_{i}\}$ are close to either 0 or 1), $S_{\text{vN}}$ 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), $S_{\text{vN}}$ increases with the number of active orbitals. As shown in Figure 7, the $S_{\text{vN}}$ values of Cn with even-number carbon atoms and Li2Cn with odd-number carbon atoms are much larger than the $S_{\text{vN}}$ values of Cn with odd-number carbon atoms and Li2Cn with even-number carbon atoms, respectively.

To illustrate the causes of the odd-even oscillations in $S_{\text{vN}}$, we plot the occupation numbers of the active orbitals for the lowest singlet states of Cn (see Figure 8) and Li2Cn (see Figure 9), calculated using TAO-BLYP-D. Here, the highest occupied molecular orbital (HOMO) is the ${(N/2)}^{\text{th}}$ orbital, and the lowest unoccupied molecular orbital (LUMO) is the ${(N/2+1)}^{\text{th}}$ orbital, with $N$ being the number of electrons in Cn/Li2Cn. 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, Cn with even-number carbon atoms and Li2Cn with odd-number carbon atoms possess more pronounced diradical character than Cn with odd-number carbon atoms and Li2Cn with even-number carbon atoms, respectively.

On the basis of several measures (e.g., the smaller ST gap, smaller $E_{g}$, larger $S_{\text{vN}}$, and more pronounced diradical character), Cn with even-number carbon atoms and Li2Cn with odd-number carbon atoms should exhibit much stronger static correlation effects than Cn with odd-number carbon atoms and Li2Cn 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 Cn and Li2Cn). In addition, due to the alteration of the reactivity of Cn and Li2Cn 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 H2 molecules very weakly (i.e., mainly governed by vdW interactions), they are unlikely to be promising hydrogen storage materials at ambient conditions Bhatia2006 . Similarly, Cn are not ideal for ambient storage applications, since the binding energies of H2 molecules remain small. In addition, the number of H2 molecules that can be adsorbed on Cn is quite limited, due to the repulsive interaction between the adsorbed H2 molecules at short distances Okamoto2001 . Consequently, the more the adsorbed H2 molecules, the less the average H2 binding energy on Cn. Therefore, Cn cannot be high-capacity hydrogen storage materials at ambient conditions.

Here, we investigate the hydrogen storage properties of Li2Cn ($n$ = 5–10). As illustrated in Figure 1(b–h), at the ground-state geometry of Li2Cn, $x$ H2 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 H2 molecules are found to be adsorbed molecularly to the Li atoms. The average H2 binding energy, $E_{b}(\text{H}_{2})$, on Li2Cn is evaluated by

[TABLE]

where $E_{\text{H}_{2}}$ is the total energy of H2, and $E_{\text{Li}_{2}\text{C}_{n}\text{-}2x\text{H}_{2}}$ is the total energy of Li2Cn with $x$ H2 molecules adsorbed on each Li atom. Subsequently, $E_{b}(\text{H}_{2})$ is corrected for BSSE using a standard counterpoise correction Boys1970 . As shown in Figure 10, $E_{b}(\text{H}_{2})$ is in the range of 19 to 27 kJ/mol per H2 for $x$ = 1–4, in the range of 18 to 19 kJ/mol per H2 for $x$ = 5, and about 16 kJ/mol per H2 for $x$ = 6, falling in (or close to) the ideal binding energy range.

To assess if the binding energies of successive H2 molecules are also in (or close to) the ideal binding energy range (i.e., not just the average H2 binding energy), the binding energy of the $y^{\text{th}}$ H2 molecule ($y$ = 1–6), $E_{b,y}(\text{H}_{2})$, on Li2Cn is evaluated by

[TABLE]

Similarly, $E_{b,y}(\text{H}_{2})$ is also corrected for BSSE using a standard counterpoise correction Boys1970 . As shown in Figure 11, $E_{b,y}(\text{H}_{2})$ is in the range of 16 to 27 kJ/mol per H2 for $y$ = 1–4, in the range of 11 to 12 kJ/mol per H2 for $y$ = 5, and less than 5 kJ/mol per H2 for $y$ = 6. Therefore, while the first four H2 molecules can be adsorbed on Li2Cn in (or close to) the ideal binding energy range, the fifth and sixth H2 molecules are only weakly adsorbed (i.e., not appropriate for ambient temperature storage).

As Li2Cn ($n$ = 5–10) can bind up to 8 H2 molecules (i.e., each Li atom can bind up to 4 H2 molecules) with the average and successive H2 binding energies in (or close to) the ideal binding energy range, the corresponding H2 gravimetric storage capacity, $C_{g}$, is calculated using

[TABLE]

Here, $M_{\text{Li}_{2}\text{C}_{n}}$ is the mass of Li2Cn, and $M_{\text{H}_{2}}$ is the mass of H2. Note that $C_{g}$ (see Eq. (5)) 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 Li2Cn, the maximum number of H2 molecules that can be adsorbed on each Li atom with the average and successive H2 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 Li2Cn 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.) usdoe . Nevertheless, since the $C_{g}$ values obtained here are much higher (especially for the shorter Li2Cn) than the USDOE ultimate target, the complete storage systems based on Li2Cn could serve as high-capacity hydrogen storage materials for reversible hydrogen uptake and release at near-ambient conditions.

Conclusions

In conclusion, the search for ideal hydrogen storage materials have 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 H2 binding energies, successive H2 binding energies, and H2 gravimetric storage capacities) of Li2Cn ($n$ = 5–10) using TAO-DFT. As Li2Cn 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 Li2Cn (especially for geometry optimization), and hence, the use of TAO-DFT in this study is well justified. On the basis of our results, Li2Cn can bind up to 8 H2 molecules (i.e., each Li atom can bind up to 4 H2 molecules) with the average and successive H2 binding energies in (or close to) the ideal range of about 20 to 40 kJ/mol per H2. Accordingly, the H2 gravimetric storage capacities of Li2Cn are in the range of 10.7 to 17.9 wt%, satisfying the USDOE ultimate target of 7.5 wt%. Consequently, Li2Cn can be high-capacity hydrogen storage materials at near-ambient conditions.

For the practical realization of hydrogen storage based on Li2Cn, Li2Cn may be adopted as building blocks. For example, we may follow the proposal of Liu et al. Liu2011a , and consider connecting Li-coated fullerenes with Li2Cn, 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 Jin2009 ; Chuvilin2009 and Pt-terminated linear carbon chains Kano2014 have been successfully synthesized, the realization of hydrogen storage materials based on Li2Cn should be feasible, and is now open to experimentalists.

Acknowledgements

This work was supported by the Ministry of Science and Technology of Taiwan (Grant No. MOST104-2628-M-002-011-MY3), National Taiwan University (Grant No. NTU-CDP-105R7818), the Center for Quantum Science and Engineering at NTU (Subproject Nos.: NTU-ERP-105R891401 and NTU-ERP-105R891403), and the National Center for Theoretical Sciences of Taiwan. S.S. would like to thank Kerwin Hui and Chih-Ying Lin for useful discussions.

Author Contributions

S.S. and J.-D.C. designed the project. S.S. performed the calculations. S.S. and J.-D.C. contributed to the data analysis and writing of the paper.

Additional Information

Competing financial interests: The authors declare no competing financial interests.