Assessing the polycyclic aromatic hydrocarbon anisotropic potential with application to the exfoliation energy of graphite.
ABSTRACT In this work we assess a recently published anisotropic potential for polycyclic aromatic hydrocarbon (PAH) molecules (J. Chem. Theory Comput. 2010, 6, 683-695). Comparison to recent high-level symmetry-adapted perturbation theory based on density functional theory (SAPT(DFT)) results for coronene (C(24)H(12)) demonstrate the transferability of the potential while highlighting some limitations with simple point charge descriptions of the electrostatic interaction. The potential is also shown to reproduce second virial coefficients of benzene (C(6)H(6)) with high accuracy, and this is enhanced by using a distributed multipole model for the electrostatic interaction. The graphene dimer interaction energy and the exfoliation energy of graphite have been estimated by extrapolation of PAH interaction energies. The contribution of nonlocal fluctuations in the π electron density in graphite have also been estimated which increases the exfoliation energy by 3.0 meV atom(-1) to 47.6 meV atom(-1), which compares well to recent theoretical and experimental results.
- [show abstract] [hide abstract]
ABSTRACT: The polymorphism of an industrial important pigment (PR179) was studied with a combination of standard crystal structure prediction and metadynamics. The former provided a starting set of candidate polymorphs whose structural and thermal stability were then probed by metadynamics. Moreover, metadynamics allowed for exploring the free energy surface to look for other possible polymorphs that were not included in the original set. Our calculations indicate that two structures have a high structural stability and are therefore good candidates to be found in experiments. The lower energy phase of the two indeed corresponds to the known polymorph, and we suggest that the other might be a metastable polymorph not yet experimentally discovered.The Journal of Physical Chemistry B 10/2008; 112(42):13231-7. · 3.61 Impact Factor
- Calculated ground-state structures of 13-molecule clusters of carbon dioxide, methane, benzene, cyclohexane, and napthalene. 3948-3961..
Assessing the PAHAP potential with application to the exfoliation energy of graphite
PreprintCambridge Centre for Computational Chemical EngineeringISSN 1473 – 4273
Assessing the PAHAP potential with application
to the exfoliation energy of graphite
Tim S. Totton1, Alston J. Misquitta2, Markus Kraft1
released: 22 August 2011
1Department of Chemical Engineering
University of Cambridge
New Museums Site
Cambridge, CB2 3RA
2Department of Physics
University of Cambridge
J J Thomson Avenue
Cambridge, CB3 0HE
Preprint No. 107
Keywords: PAH, anistropic potential, PAHAP
Computational Modelling Group
Department of Chemical Engineering and Biotechnology
University of Cambridge
New Museums Site
Cambridge CB2 3RA
World Wide Web:
+ 44 (0)1223 334796
In this work we assess a recently published anisotropic potential for polycyclic
aromatic hydrocarbon (PAH) molecules (J. Chem. Theory Comput. 2010, 6, 683-
695). Comparison to recent high-level SAPT(DFT) results for coronene (C24H12)
demonstrate the transferability of the potential whilst highlighting some limitations
with simple point charge descriptions of the electrostatic interaction. The potential
is also shown to reproduce second virial coefficients of benzene (C6H6) with high
accuracy and this is enhanced by using a distributed multipole model for the elec-
trostatic interaction. The graphene dimer interaction energy and the exfoliation en-
ergy of graphite have been estimated by extrapolation of PAH interaction energies.
The contribution of non-local fluctuations in the π electron density in graphite have
also been estimated which increases the exfoliation energy by 3.0meVatom−1to
47.6meVatom−1which compares well to recent theoretical and experimental re-
3Assessment of the PAHAP potential
3.1Comparing the PAHAP potential to SAPT(DFT) results for the coronene
dimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2Calculation of the second virial coefficients of benzene . . . . . . . . . .
3.3Binding energy of the graphene dimer . . . . . . . . . . . . . . . . . . .
3.4Exfoliation energy of graphite . . . . . . . . . . . . . . . . . . . . . . .
π-electron contribution to the dispersion . . . . . . . . . . . . . .
Polycyclic aromatic hydrocarbon (PAH) molecules are an important class of molecules
in the chemistry of hydrocarbon combustion. These molecules are found in a wide dis-
tribution of sizes in sooting flames [15–17, 25, 65, 85]. The examination of the result-
ing soot particles using high resolution TEM images reveals layered graphitic structures
[11, 34, 82] and has led to the conclusion that clustering of PAH molecules is responsible
for soot particle nucleation [27, 77, 92]. These findings have lead to the development of
predictive models to describe soot particle evolution in flame environments, such as en-
gines [8, 9, 56, 66], but there remain many unanswered questions concerning the precise
mechanisms of the nucleation and growth processes.
In order to access the structure and composition of nascent soot particles it is necessary
to understand the interactions of PAH clusters at a molecular level. At present the most
accurate means of doing this in a computationally feasible manner is by using atomistic
and they are often designed to be widely transferable between large classes of organic
molecules [2, 6, 12, 35–37, 81, 86–88, 90], and are typically simple in form, treating all
atom–atom interactions as isotropic. Often these potentials are augmented with a simple
point charge model to account for the electrostatic interaction explicitly.
In the crystallographic community, intermolecular potentials have been developed for
many years, notably by Williams [71, 86, 87] who used crystallographic data and heats
of sublimation of organic molecules to parameterise Buckingham potentials with an ad-
ditional point charge term. These potentials have been widely used in organic crystal
structure prediction [3, 14, 42, 58, 63, 95] and on work with molecular clusters [20, 75].
The 1967 Williams version (W67) has been used in the context of PAH interactions by
Miller  whilst the 1977 version (W77) has been fitted to a ‘Lennard-Jones plus point
charges’ form  (termed the LJ potential) which has proved popular due to its compu-
tational efficiency [28, 67, 77].
The most recent of Williams’ potentials, the W99 potential , was originally derived
from crystallographic data for hydrocarbon molecules but has since been extended to
oxyhydrocarbon molecules  and nitrogen-containing molecules  to yield a model
with 13 different atomic classes. For PAH molecules only the C(3) and H(1) classes are
considered resulting in a PAH interaction potential with just 6 parameters as well as
atomic point charges. In the crystallographic community Williams’ potentials have be-
come benchmarks , and in a recent work we have shown that the W99 potential gives
excellent agreement with high level quantum chemistry results for small stacked PAH
dimers . This agreement is due largely to the stacked dimers representing the low-
est energy conformations which correspond to the crystal structures used to parameterise
the potential. However, we also demonstrated that such potentials which use an isotropic
atom-atom description of the intermolecular interaction cannot accurately model PAH in-
teractions throughout conformational space. Consequently, if PAH interactions are to be
accurately modelled in situations where non-stacked conformations are expected, e.g. in
larger molecular clusters, a more complex atom-atom description is needed.
Until relatively recently, the paucity of experimental data and the computational limita-
tions of high-level quantum chemistry methods has precluded the development of com-
plex atom-atom potentials. The development of new computational quantum chemistry
methods which combine high accuracy with computational efficiency has enabled the
development of a new class of specific, high-accuracy intermolecular potentials. In par-
ticular, the development of symmetry-adapted perturbation theory based on Kohn-Sham
density functional theory orbitals (SAPT(DFT)) [29–32, 48–51] has led to a spate of new
potentials, in part due to its high accuracy and computational efficiency. Equally impor-
tantly, however, is the feature of perturbation theory that allows us to naturally partition
the total interaction energy into physical contributions: the exchange-repulsion, disper-
sion, polarisation and electrostatic interactions. This has enabled the parameterisation of
potentials capable of modelling important details of intermolecular interactions such as
atomic shape anisotropy, anisotropic polarizabilities and higher order dispersion contri-
butions [13, 53, 55, 61, 63, 78, 80]. In particular, this approach was successfully used in a
recent blind test of organic crystal structure prediction .
Podeszwa et al.  have used this methodology to develop an accurate intermolecular
potential for benzene using SAPT(DFT) interaction energies. The form of this potential
is a generalisation of the Buckingham potential, containing an exponential component
multiplied by a quadratic term and a more detailed dispersion model including C6, C8
and C10terms. A point charge electrostatic term was also included and this along with
the dispersion terms were multiplied by damping functions to attenuate the divergence
at short intermolecular separations. In addition to the usual atomic sites this potential
contained 13 off-atomic sites, resulting in a fit with a total of 92 parameters. The extra
sites were required to account for the shape-anisotropy of the atomic sites in benzene.
However, the large number of sites and parameters mean the potential is not obviously
transferable to larger PAH molecules, and thus while it acts as a benchmark in terms of
accuracy for benzene, it is not suitable for the study of the potential energy surface (PES)
of PAH clusters.
In recent work  we developed a transferable, anisotropic, atomistic potential to de-
scribe the interactions of PAH molecules. This anisotropic potential was based on the
methodology developed and successfully tested by Misquitta et al.  using a data set
that included a large number of conformations of the benzene dimer and additionally
dimers of larger PAH molecules (naphthalene, anthracene and pyrene ). The PAH
anisotropic potential (PAHAP) was designed to be simple enough to maintain transfer-
ability amongst planar pericondensed PAH molecules, while accurately reproducing the
potential was limited to three types of atom-atom interactions: carbon–carbon, carbon–
hydrogen and hydrogen–hydrogen. Each of these atom-atom terms is dependent on the
separation, Rab, and relative orientation, Ωab, with a functional form given by
which for simplicity we chose to be based on atom-centred point charges calculated using
the Merz-Singh-Kollman scheme .
The first term is a Born–Mayer term describing short-range interactions, the second is an
This form of the potential remedies two of the major deficiencies of traditional ‘exp-6’
potentials. Firstly, the short-range term now includes a shape-function, ρab, which models
the anisotropy of the interacting sites through a dependence on the relative orientation of
the two sites:
ρab(Ωab) = ρa(θa) + ρb(θb),
ρa(θa) = ρa
and similarly for ρb(θb). Here the angle θadefines the angle between the site–site vector
via shape-functions rather than additional sites, the resulting atom–atom potential is more
clearly transferable to other PAH molecules.
Secondly, the singularity in the dispersion term is removed by a Tang–Toennies damping
function , f6(Rab) for which the parameter β is calculated from the vertical ionisation
potential of benzene I according to β = 2(2I)1/2,
fn(Rab) = 1 − exp(−βRab)
The resulting potential has a total of 12 parameters for the short-range and dispersion
terms as well as the atom-centred point charges used in the electrostatic interaction.
SAPT(DFT) contributions up to second-order were used to fit the short-range term which
and exchange-induction energies. The isotropic C6dispersion model was calculated di-
rectly using the Williams-Stone-Misquitta method [45–47, 52, 91] and, due to a strong
linear correlation found between the model energies and the SAPT(DFT) dispersion en-
ergies, a simple scaling factor was introduced to match SAPT(DFT) energies .
The PAHAP potential has been carefully parameterised using very accurate ab initio data
from SAPT(DFT) and the WSM method, however two important short-comings remain:
(1) The potential could not be considered transferable because a new set of point charges
were needed for every type of PAH molecule, and (2) the potential was never validated
against data not used in the fitting process.
We have remedied the first deficiency in a recent paper  by developing an electro-
static model for PAH molecules based on a transferable set of atom-centred quadrupole
moments and point charges. In this paper we address the second deficiency by indepen-
dent assessment of the potential against experimental data and very recent ab initio data.
Firstly, we compare PAHAP and other potentials against recent high-level SAPT(DFT)
calculations of coronene (C24H12) interactions. We then compare the potential with others
against experimental data in the form of second virial coefficients of benzene. Finally, we
use PAHAP to estimate the interaction energy of the graphene dimer and the exfoliation
energy of graphite and compare results against experiment and recent high-level quantum
The geometries of all the molecules used in this work were optimised using density func-
tional theory (DFT) with the B3LYP functional and the 6-31G* basis set. DFT calcu-
lations were also used to calculate electrostatic-potential-fitted (ESP) point charges for
the electrostatic term using the Merz-Singh-Kollman scheme  with the PBE0 [1, 57]
functional based on the optimised atomic coordinates. The basis set used for calculating
charges depended on PAH size. The aug-cc-pVTZ was used for molecules up to the size
of coronene (C24H12) and for larger molecules the cc-pVTZ basis set was used due to nu-
merical instabilities encountered with the augmented version. All DFT calculations were
performed using the GAUSSIAN03 program .
For most of our calculations we have used ESP charges, but for some we have used a more
detailed distributed multipole description with multipoles up to rank 4 (hexadecapole) on
the carbon atoms and up to rank 1 (dipole) on the hydrogen atoms. These distributed
multipoles were calculated using the GDMA program [72, 73] from densities obtained
using the PBE0 functional. Where there is potential ambiguity we have explicitly stated
the kind of electrostatic model used with the PAHAP potential.
All evaluations of the PAHAP potential and other literature potentials were performed
using the ORIENT program . The local atomic axes used for the PAHAP potential,
along with the distributed multipoles for benzene and coronene in ORIENT format are
given in the Supporting Information. Also given are the geometries and ESP point charges
for all the PAH molecules.
3 Assessment of the PAHAP potential
3.1Comparing the PAHAP potential to SAPT(DFT) results for the
The PAHAP potential was originally fitted using SAPT(DFT) data for four small PAH
molecules (benzene, naphthalene, anthracene and pyrene dimers), and while the poten-
tial was shown to model the interactions of these molecules very well, the lack of inde-
pendent data to test the potential against weakened the claim of transferability. Recently
SAPT(DFT) interaction energies for the coronene (C24H12) dimer have been calculated in
a number of conformations  and provide the first opportunity to assess the transfer-
ability of PAHAP potential to larger PAH molecules.
The SAPT(DFT) calculations were performed by Podeszwa  using the aug-cc-pVDZ
calculations for the PAHAP potential. Figure 1 shows the four conformations for which
potential curves were calculated by varying the interplanar distance. Figure 2 compares
the PAHAP, W99 and LJ potentials with the SAPT(DFT) results. With one exception,
which we will come to later, these potentials were all used with an ESP point-charge
Three of the four conformations are very well matched by the PAHAP potential using the
ESP charge model. For the fourth, sandwich conformation, the match is not as good; the
well-depth is overestimated by around 10 kJ mol−1(∼ 15% of the interaction energy),
and the equilibrium separation is ∼ 0.1˚ A smaller. This discrepancy is mainly due to the
simplicity of the point-charge model. Also shown in figure 2 are interaction energies cal-
culated using PAHAP with a more detailed distributed multipole electrostatic model. The
agreement is clearly much better for the sandwich conformation, though this is accom-
panied with a slight degradation in accuracy at the other conformations. We expect some
systematic error when using the distributed multipoles in place of ESP charges because
the short-range terms in the PAHAP potential were obtained using electrostatic penetra-
tion energies calculated using ESP charges .
The improvement obtained with the distributed multipole model demonstrates the limita-
tion of the simple point charge model. Figure 3 shows sections of the PES for the coronene
dimer using the PAHAP potential with ESP charges and distributed multipoles. Here in-
teraction energies are plotted with the monomers kept parallel to each other and moved
around the xz plane. The twin minima in the potential wells show the two shifted graphite
positions, and the smaller hump connecting the two corresponds to the sandwich posi-
tion, identifying the conformation as a transition state between the two shifted graphite
minima. The SAPT(DFT) results suggest that the energy of this saddle point is too low in
the ESP case and that its location on the z axis should be further away from the repulsive
The PES generated using distributed multipoles resolves the finer details of the energy
landscape to a higher degree than the rather smoother PES generated using ESP charges.
This is clearly demonstrated for sandwich conformation with a more accurately located
saddle point. These results highlight the expected trade-off in accuracy when, in seeking
to generate a simple transferable potential, a simple point charge electrostatic model is
used. To maintain high accuracy in all conformations requires a more complex potential
form, but the success of the PAHAP potential in prediction of the low-energy minima of
the coronene dimer helps confirm its transferability to PAH interactions other than those
from which it was originally parameterised.
Also shown in Figure 2 are interaction energy curves obtained with the W99 and LJ po-
tentials. The W67 and W77 potentials are very similar to the LJ potential and are therefore
not shown. As has been previously noted  the W99 potential is very accurate for these
stacked conformations. However, we do not expect the W99 potential to fare as well in
the non-stacked conformations where it has been shown to underestimate the binding of
PAH molecules . By contrast, the LJ potential (and W67 and W77 potentials) overes-
timates the well depths for all the conformations by at least 20 kJ mol−1, and in the case
of the sandwich conformation by more than 30 kJ mol−1. This is a serious shortcoming
and potentially undermines work based on these potentials.
3.2 Calculation of the second virial coefficients of benzene
The second (pressure) virial coefficient is a common source of experimental data directly
related to the two-body interaction potential.
Figure 1: Structures of the coronene dimer: (a) graphite (d = 1.43˚ A), (b) shifted graphite
(d = 1.65˚ A), (c) crossed (twisted sandwich), and (d) sandwich.
An expression for the second virial coefficient, B(T), can be obtained from statistical
mechanics which depends upon the pair potential only, even if many-body terms occur in
the total energy. The variation of the second virial coefficient with temperature provides
a way of examining the potential energy surface. At low temperatures the second virial
coefficient is a measure of the volume of the potential well and at high temperatures it
is a measure of the average size of the repulsive core, although both the repulsive and
attractive parts of the potential surface contribute at each temperature. Since B(T) is
an integrated functional of the intermolecular potential, it is quite possible to obtain a
good second virial coefficient with a poor potential. We shall see an example of this. We
should therefore regard a good reproduction of B(T) as a necessary rather than sufficient
condition for an intermolecular potential.
the range 300-700 K using the ORIENT program. We have used the Gauss-Legendre inte-
gration scheme and have included the first-order quantum-correction [24, 33, 44], though
the latter does not make a significant contribution owing to the relatively large mass and
large moments of inertia of the benzene molecule; the effect being greatest at low temper-
ature but still relatively small (∼0.8%) at 200 K.
Figure 4 shows the second virial coefficients for the PAHAP potential with both ESP
point charges and a full distributed multipole (DMA) electrostatic term. Also included
are results obtained with the W67, W99 and LJ potentials and the benzene-specific po-
tential from Podeszwa et al. We have taken the experimental data from Bich et al. ,
Wormald et al.  and Francis et al.  We see that the PAHAP(ESP) potential results
in second virial coefficients that, while good at high temperatures, slightly underestimates
the experimental values at low temperatures. This perhaps indicates that on average the
potential slightly under-binds benzene. However, the PAHAP(DMA) potential is a clear
Podeszwa et al. potential results in equally good virial coefficients.
Eint (kJ mol-1)
Eint (kJ mol-1)
(b) Shifted graphite
Eint (kJ mol-1)
Eint (kJ mol-1)
Figure 2: Comparison of potentials with SAPT(DFT) for stacked coronene dimer con-
formations. All potentials use the ESP point charge model except the PAHAP
potential which is shown using both ESP charges and distributed multipoles
(DMA). The Williams’ 67 and 77 potentials (W67/W77) are not shown as they
result in interaction energies very similar to those from the LJ potential.
Figure 3: Potential energy surface for coronene dimer when one molecule is fixed and
the other is kept parallel and moved in the xz plane calculate using the PAHAP
potential with ESP charges, (a) and (b), and with distributed multipoles (DMA),
(c) and (d).
Podeszwa et al.
B(T) (cm3 mol-1)
Figure 4: Second virial coefficient of benzene. Experiment 1: ref. , Experiment 2: ref.
, Experiment 3: ref. , Podeszwa potential: ref. , W99 potential:
ref. , W67 potential: ref. , LJ potential: ref. 
Although the W99 potential performs well for stacked conformations, we know  that
this potential under-binds for other conformations, in particular the T-shaped conforma-
tion, and this seems to be reflected in systematic underestimation across the temperature
range. Interestingly, while the W67 and LJ potentials appear to be almost identical for
the coronene dimer, they result in quite different second virial coefficients for benzene.
The W67 potential matches the experimental data almost perfectly whereas the LJ po-
tential underestimates the data. However, as we have stated before, a good second virial
coefficients does not imply an accurate potential. In the case of the W67 potential this
is highlighted in the overestimation of the well depth for stacked conformations of the
coronene dimer (section 3.1), a trend also seen for benzene, which suggests other areas of
the PES must be underestimated to give an accurate average interaction.
3.3Binding energy of the graphene dimer
The interaction energy of stacked homo-molecular PAH dimers (per number of monomer
carbon atoms) increases with increased dimer mass, but tends towards an asymptote cor-
responding to the interaction energy of two infinite graphene sheets (Figure 6). This is an
important limit from the theoretical viewpoint, but also because PAH molecules in soot,
while not infinite in extent, can be very large, comprising hundreds of atoms .
the interaction energies for the stacked PAH dimers using a simple two-parameter model
b + nC
Figure 5: The pericondensed PAH molecules studied in this work: 1. Benzene (C6H6),
2. Naphthalene (C10H8), 3. Phenanthrene (C14H10), 4. Anthracene (C14H10),
5. Pyrene (C16H10), 6. Perylene (C20H12), 7. Benzo[g,h,i]perylene (C22H12),
8. Coronene (C24H12), 9. Bisanthene (C28H14), 10. Ovalene (C32H14), 11.
Hexabenzocoronene (C42H18), 12. Octabenzocoronene (C46H18), 13. Circum-
where nCis the number of carbon atoms per monomer and a and b are fitted parameters.
This satisfies the physical constraints of the system, being zero for nC= 0 and asymp-
totically constant for large nC. The interaction energy per carbon atom for the graphene
dimer is given by the limit as nC→ ∞, which is simply a.
Interaction energies for homomolecular PAH dimers were calculated for 13 different PAH
molecules (Figure 5) using the GMIN program . With this program, molecular cluster
geometry is optimised using the ‘basin hopping’ method [10, 41, 77, 84] which combines
local gradient-based minimisation with a Metropolis Monte Carlo scheme allowing dif-
ferent ‘basins’ on the PES to be located and ‘hopped’ between. This method provides an
efficient scheme to sample the potential energy landscape.
Weobtainanestimatefortheinteractionenergyofthegraphenedimerofa = 3.94kJmol−1
per carbon atom (40.9meVatom−1) and b = 7.59, using interaction energies calculated
with the PAHAP potential for dimers of PAH molecules shown in Figure 5. The rms de-
viation for the fit is 0.06kJmol−1per carbon atom (0.6meVatom−1). We excluded ben-
zene from the fit to Eqn. 5 as the minimum energy conformation of the benzene dimer
is not stacked (it is a tilted T-shaped structure). This result is similar to that found
by Podeszwa , who, using SAPT(DFT) calculations of interaction energies of three
PAH dimers (anthracene, pyrene and coronene) together with the extrapolation method
described above, obtained a graphene dimer interaction energy of 42.5meVatom−1.
There are, however, key differences in our approaches: we have used minimum energy
dimer conformations while Podeszwa constrained the dimer to be in the graphite con-
figuration and optimized interplanar separation only. By using SAPT(DFT) interaction
Eint/nC (kJ mol-1)
Figure 6: Interaction energies for PAH dimers using the PAHAP and W99 potentials.
The SAPT(DFT) energies used in Ref.  are included for comparison. These
energies were calculated using the full dimer-centered basis set plus mid-
bonds (DC+BS) approach and were used to estimate the interaction energy
of graphene (Table 1). A fit of the PAHAP results using Eqn. 5 is shown with
parameters a = 3.94kJmol−1and b = 7.59.
energies with a 1-dimensional fit Podeszwa minimizes the errors due to fitting, but con-
sequently, only a few small dimers can be used. This leads to an uncertainty in the ex-
trapolation using Eqn. 5. This uncertainty is largely removed in our approach, as with a
potential a much larger number of dimers can be sampled, thereby leading to a better con-
trol of extrapolation errors. The similarity of our results suggests that these differences
are probably inconsequential. Our results also serve as a validation of the extrapolation
formula (Eqn. 5).
3.4 Exfoliation energy of graphite
The exfoliation energy of graphite is the energy required for the uppermost graphene layer
to be removed from the graphite surface. This is different from the interaction energy of
two graphene sheets due to the interaction with graphene planes other than the nearest
The exfoliation energy must be differentiated from the binding energy and cleaving en-
ergy: the former is the interaction energy per carbon in the graphite crystal and the lat-
ter is the energy released per carbon upon cleaving a graphite crystal into two along a
graphene plane. Unfortunately, these energies are often undifferentiated in theoretical and
experimental papers, but because the dominant interaction is between nearest layers, the
differences in these three energies is expected to be small and is probably well within the