## Theoretical Calculations Of Hydrogen Storage By Carbon Nanotubes

Theoretical models allow the study of ideal systems, and are useful tools for investigating complex systems. Models have been made of the physical adsorption of molecular hydrogen on carbon nanotubes, as well as chemical adsorption of both atomic and molecular hydrogen. These studies have investigated hydrogen binding sites on the carbon framework and adsorption isotherms and density profiles. Theoretical investigations can be extremely useful in developing an understanding of the elementary steps in the adsorption process, as well as in predicting an upper limit on the hydrogen storage capacity.

The theoretical calculations are generally classified into two groups according to how they treat time within their theoretical model. Statistical mechanics and molecular dynamics include time, and are used for looking at molecules in motion. Quantum mechanics (electronic structure methods) and molecular mechanics are time independent, and can perform accurate energy calculations on snapshots of the molecular system.

3.1. Statistical Mechanics (Monte Carlo) Methods

Most grand canonical Monte Carlo (GCMC) studies on hydrogen adsorption in carbon nanotubes have used a 12:6 Lennard-Jones [96-98] potential to model the classical molecular interactions. The study of hydrogen adsorption is complicated by a significant contribution from quantum effects [99, 100]. Darkrim and co-workers [99, 101] state that Chakravarty et al. [102] found that, for molecular hydrogen clusters at temperatures above 20 K, a full path-integral formalism was unnecessary, and the less computation-intensive Feynman-Hibbs [103] perturbative approach to modeling quantum effects seemed sufficient. Darkrim found that, for simulations of hydrogen adsorption on nanotubes, the Feynman-Hibbs-based corrections accounted for a 5% difference at room temperature and about 15-25% at 77 K. This clearly demonstrates that, while an uncorrected Lennard-Jones potential can be used successfully for qualitative work at room temperature and above, quantitative work requires treatment of the quantum effects. Another complicating factor now often included is a treatment of the quadrupole and dipole interactions [104, 105], and which were mandatory for the studies of charged systems [106]. A summary of the methods and some parameters can be found in Table 3.

All of the simulations reported so far have made two assumptions. First, they base their calculations on semiem-pirical potentials taken from work on planar graphite systems. Second, they assume that the same potentials can be used for both endohedral and exohedral adsorption. It is therefore reasonable to be skeptical of the quantitative accuracy of these studies until these two assumptions are shown to be reasonable enough to give good quantitative results.

### 3.2. Electronic Structure Method Studies

One approach to eliminate the two assumptions made for the statistical mechanic studies (vide supra) is to use electronic structure methods. These methods may also treat quantum effects and electron correlation directly or indirectly, and can lead to highly accurate results (±4 kJ • mol-1) [107]. The major difficulty of these methods is the computational effort required, with the more accurate methods scaling catastrophically with model size [e.g., O(N5-7)]. Additionally, processes involving hydrogen particularly require the more accurate methods to be used, due to the problem of basis-set incompletion [108]. However, these methods are ideal for studies of process mechanisms and thermodynamics. This makes these studies complementary to the statistical mechanics studies, as they can be used to refine the statistical mechanic models and parameters.

Recent advances in electronic structure methods have allowed larger systems to be studied more accurately than before. The development of several methods that use periodic boundary conditions has been very advantageous for indefinite systems such as graphite, crystals, and nanotubes. The pseudopotential plane-wave method [109] or the self-consistent charge-density-functional-based tight-binding method (SCC-DFTB) [110] are the main methods used to study nanotubes. These methods use either a local density approximation (LDA) [111] or a generalized gradient approximation (GGA) [112]. The GGA is generally more accurate than the LDA, but it has been criticized for overestimating repulsive interactions in systems characterized by weak bonding forces. Another new development

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