Graphite unit cell optimization

The layers of materials such as graphite are held together purely by dispersion

interactions. Since the covalent bonds within the abplane are much stiffer than the van der Waals interactions in the cdirection, graphite displays a strongly anisotropic compressibility, which varies significantly with temperature. The thermal expansion of graphite was studied quite extensively in the late 1940s and early 1970s, both as a simple system of interest for testing physical models of heat capacity and lattice dynamics, and

also for the material’s importance to the nuclear industry.369–377 _{More recently, graphite} has received considerable attention as a material for benchmarking vdW inclusive DFT methodologies since the prediction of the interlayer binding energy (or the closely related exfoliation and cleavage energies) presents a particular challenge for many correlation methods. Graphitic nanomaterials were among the first examples of systems for which many-body vdW effects were realized to be qualitatively necessary.26

Dobson et al. have recently shown that the interaction between graphene sheets as they

are separated shows many-body effects beyond those captured by the random-phase approximation (RPA).117 _{In particular, they find that the long-distance}

graphene-graphene dispersion interaction is substantially less than the RPA prediction and is very sensitive to long-wavelength in-plane many-body modes that renormalize the velocity of the massless Dirac fermions in graphene.117 _{Since MBD is formally equivalent} (through the adiabatic connection fluctuation dissipation theorem) to the RPA,84,87,88 _this suggests that there may be many-body effects relevant to the properties of graphene and graphite that the DFT+MBD methodology will miss.

Since the c-axis of graphite expands considerably with temperature, it is important to

use low temperature experimental data as a reference for comparison to electronic

structure predictions of the unit cell size at 0 K. In 1970 Bailey and Yates378 _{examined the} anisotropic expansion of well-ordered pyrolytic graphite down to 30 K; they find that the linear coefficient of expansion in the c-direction isαk = (3.8±0.8)×10−6 K−1. Together

with the fact thatαk must go to zero as temperature approaches 0 K, this indicates that a

low temperature measurement of the c-axis spacing of graphite should be a good proxy for

the zero temperature spacing without the need for significant extrapolation. Baskin et al.

measured the lattice constants of single crystal graphite at 4.2±0.3 K and 78±0.3 K

using Cu Kα (λ= 1.5418 Å) X-ray radiation and determinedd(4.2K) = 3.3360(5) AA and d(78K) = 3.3378 Å witha= 2.4589(5) Å at both temperatures.379 Using the expansion

coefficients of Bailey and Yates, we performed a nonlinear extrapolation of the 78 K interlayer spacing and found that it is consistent with a value of 3.336 Å at zero

3.8. Results and Discussion temperature. Therefore, we take the 4.2 K measurement as our reference value.

c a

Figure 3.9: The unit cell of graphite, with the bonding atoms expanded outside the cell to show the AB stacking arrangement. The interlayer spacing isc/2and the C–C bond distance isa/√3.

In considering the interlayer binding distance and exfoliation energy of graphite, GGA functionals tend to predict very weak or no binding at all, while LDA functionals predict approximately the right binding distance with too shallow a potential energy surface (though LDA yields surprisingly good predictions considering that there are no terms in LDA that should account for vdW interactions).380 _{Almost all vdW inclusive treatments} improve upon these deficiencies, though several authors have shown that inclusion of self-consistent screening88,251 _{and many-body effects}86,88 _{can substantially renormalize the} cohesive energy of graphite, bringing the predicted values into closer agreement with experimental33,381,382 _{and quantum Monte Carlo estimates}383 _{than is found with nonlocal} correlation approaches.85,86,116,359,380,384

We optimized the unit cell of graphite (shown in Figure 3.9) using a 4×4×2 supercell with both PBE+MBD and PBE+TS.‡ _{Since the atoms reside in positions that are}

constrained by the symmetry of the P6₃/mmcspace-group and the hexagonal bonding

pattern of sp2 hybridized carbon, the primary quantity of interest in optimizing the

graphite unit cell is the interlayer spacing d=c/2. In Table 3.4 we compare our cell

parameters to those obtained by many different authors using a variety of functionals and ‡

Whenk-point integration becomes available we will check whether these results are converged since a

vdW correction schemes. We note that PBE is quite good at reproducing the C–C bond length of 1.42 Å and thus the a-axis length is well reproduced (results with the revPBE

functional are less compelling in this regard). As previously stated, in the absence of a dispersion correction PBE predicts much too large an interlayer spacing d∼3.9−4.9 Å.

The large variance in reported values is due to the very flat potential energy surface predicted by PBE. The revised GGAs, revPBE and PBEsol do better, but still have >3%

relative error in the interlayer spacing. The vdw-DF and vdW-DF2 nonlocal correlation methodologies tend to yield too large an interlayer spacing by ∼4−13%. The nonlocal correlation functional method VV10 and its revised variant both perform very well, giving relative errors of∼2% or less in the interlayer spacing. The outdated D2 method

performs reasonably well. Since the revisions to the D3 method included parameterizing

C₆ coefficients that distinguish between different hybridization states, we expect that

PBE+D3 would perform as well or better than PBE+D2. We note that although the effectively pairwise PBE+TS method does extremely well, the relative error is increased when the self-consistently screened variant is used. This is surprising since the SCS procedure makes such a drastic improvement to TS for predicting many other quantities. The vdw-WF-QHO method of Silvestrelli et al. is, like MBD, based on coupled QHOs,

but extracts its charge density partitioning using maximally localized Wannier functions rather that atomic Hirshfeld partitioning. Both vdW-WF-QHO and MBD perform extremely well relative to the experimental value and agree quite closely with the RPA result. Many more benchmarks of analytical MBD gradients for unit cell optimizations of dispersively dominated crystals should of course be performed, but we find this

preliminary result quite encouraging since graphite has proven challenging for many vdW correction methods.