Weak chemical interaction and van der Waals forces between graphene layers: A combined density functional and intermolecular perturbation theory approach

The interaction between graphene layers is analyzed using a local orbital occupancy approach and second-order perturbation theory. This perturbation theory yields the van der Waals forces which are calculated, within the local orbital approach, using an atom-atom interaction approximation and the local density of states of the graphene layers. Weak chemical interactions (one electron, Hartree, exchange) are calculated using an expansion in the interlayer orbital overlap. We find that the one-electron repulsion due to orthogonalization effects is much larger than the Hartree and exchange contributions. The sum of these contributions yields a net repulsive short-range energy that counteracts the attractive long-range van der Waals interaction. Our analysis of the van der Waals interaction highlights the importance of the $2s\to 3d$ atomic dipole transitions, which are responsible for more than half of the total van der Waals energy between two graphene layers. We obtain an interlayer equilibrium distance of $3.1–3.2\mathrm{\AA }$, with a binding energy of $60–72\mathrm{meV}$, in reasonable agreement with the experimental evidence for graphite.

Figure 1

Optimized (solid line) and standard (dashed line) fireball atomiclike orbitals (radial component) for carbon [see Eq. (31)]. (Left) $s$ orbital; (right) $p$ orbital.

Figure 2

One-electron, Hartree, and exchange-integral contributions (see text) to the interaction energy (per atom in the unit cell) for two graphene layers as a function of the graphene-graphene distance. Inset: decomposition of the one-electron term (see text).

Figure 3

Chemical (sum of different contributions shown in Fig. 2), van der Waals, and total interaction energies (per atom in the unit cell) for two graphene layers for the minimal basis set calculation.

Figure 4

Van der Waals and total interaction energy (per atom in the unit cell) when the dipole-dipole $s\to p^{*}$, $p\to s^{*}$, and $p\to d$ transitions are included in the calculation of the vdW energy, as discussed in the text. In this figure we have used the value $⟨p_z\mid z\mid d_{z^2}⟩=0.34\mathrm{\AA }$.