The interlayer binding energy of graphite is obtained by a semiempirical method in which ab initio calculations based on the density functional theory (DFT) are supplemented with an empirical van der Waals (vdW) interaction. The local density approximation (LDA) and generalized gradient approximation (GGA) are used in the DFT calculations, and the damping (or interpolation) function used to combine these DFT results with an empirical vdW interaction is fitted to the observed interlayer spacing and c-axis elastic constant. The interlayer binding energies calculated in the LDA and GGA are quite different, but the combined results are nearly the same, which may be a necessary condition and provide reinforcements for validating the method. The present results are also consistent with those obtained by the empirical method based on the Lennard-Jones potential, and both are in reasonable agreement with the recent experimental data. These results indicate that, in contrast to the prevailing belief, the LDA underestimates the interlayer binding energy of graphite.

In this paper we have developed a theory of energetics for isolated single-wall carbon nanotubes ͑SWNTs͒ deformed in the radial direction, and applied this theory to investigate their deformation characteristics and stability under hydrostatic pressure. The starting point of the theory is the strain energy of SWNTs predicted by ab initio calculations based on the density functional theory ͑DFT͒, which shows the same behavior as that obtained for the continuum elastic shell model. We extend this result for inflated SWNTs with circular cross section to calculate the deformation energy of a deformed SWNT without performing further DFT calculations. This extension is then complemented by a van der Waals interaction, which is not fully taken into account in the DFT approximations currently in use but becomes important in highly deformed tubes. We find that the minimum pressure, P 1 , for the radial deformation to occur is proportional to the inverse cube of tube diameter, D, in agreement with the recent theoretical predictions as well as the classical theory of buckling. The radial deformation of SWNTs with D Ͻ 2.5 nm is found to be elastic up to very high pressure and they hardly collapse. On the other hand, SWNTs with D Ͼ 2.5 nm show a plastic deformation and collapse if the applied hydrostatic pressure exceeds a critical value, which is about 30-40% higher than P 1 and also varies as D −3 though approximately. These SWNTs with large D collapse when the cross-sectional area is about 60% reduced with respect to the circular one. It is also found that for SWNTs with D Ͼ 7.0 nm, the plastically deformed ͑collapsed͒ state is more stable than the inflated one. This critical value of D is somewhat larger than previously predicted.

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The constant-NVT Monte Carlo simulation is performed for model C60 molecules interacting via the Girifalco potential and a full free-energy analysis is made to predict the high-temperature phase diagram. The repulsive part of the C60 potential is very steep and the attractive part is relatively short-ranged. For such a system accurate computations of the virial pressure are difficult in simulations and it is argued that the discrepancies among the previous results for the phase diagram of C60 can partly be attributed to the uncertainties of the virial pressure involved in simulations. To avoid this difficulty we take the energy route to calculate equation of state (EOS), in which the absolute (Helmholtz) free energy is obtained by performing isochoric integration of the excess internal energy. A difficulty of the energy route in the high-temperature limit is resolved by the aid of an analytic method. The exact second and third virial coefficients are also used in the analysis of the fluid EOS. The pressure route is taken to calculate the EOS of the solid phase, in which the virial pressure is numerically more stable than in the fluid phase. The resulting high-temperature phase diagram of C60 is quite systematic and free from uncertainties, and the liquid–vapor critical point is found at Tc=1980 K and ρc=0.44 nm−3, whereas the triple point at Tt=1880 and ρt=0.74 nm−3, confirming the existence of a stable liquid phase over the range of ∼100 K.

We have developed a semiempirical method to obtain interlayer binding energy of graphite in the previous work ͓M. Hasegawa and K. Nishidate, Phys. Rev. B 70, 205431 ͑2004͔͒. In the present paper, we revisit this approach and develop an improved method, in which ab initio calculations based on the density functional theory ͑DFT͒ are also corrected through an empirical atom-atom van der Waals ͑vdW͒ interaction. The local density approximation ͑LDA͒ and generalized gradient approximation ͑GGA͒ are used in the DFT calculations. The parametrized damping function introduced to modify the asymptotic atom-atom vdW interaction is more flexible than the previous ones and covers a wider range of possibility in correcting for the approximate DFT calculations. The damping function is determined empirically by imposing the condition that the experimental interlayer spacing, in-plane lattice constant, and c-axis elastic constant are reproduced. We also require consistency between the LDA-and GGA-based methods ͑LDA+ vdW, GGA+ vdW͒ as the theoretically motivated necessary condition. The interlayer binding energy obtained by this method is 60.4 meV/atom at T = 0 K. The result of ϳ54 meV/atom at room temperature corrected by the thermal effect is consistent with the most recent experiment, 52± 5 eV/atom ͓R. Zacharia et al., Phys. Rev. B 69, 155406 ͑2004͔͒. The atom-atom vdW interaction obtained by the present semiempirical method favorably corrects for the overbinding and underbinding nature of the LDA and GGA, respectively, in the in-plane energetics of graphite. That interaction also provides a useful starting point for the studies of energetics of other graphitic systems such as fullerenes and carbon nanotubes.

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