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arXiv:cond-mat/0406549v1 [cond-mat.soft] 23 Jun 2004

APPLIED PHYSICS REPORT 2004–17

van der Waals interactions of the benzene dimer: towards treatment of

polycyclic aromatic hydrocarbon dimers

Svetla D. Chakarova∗and Elsebeth Schr¨ oder

Department of Applied Physics, Chalmers University of

Technology and G¨ oteborg University, SE–412 96 G¨ oteborg, Sweden

(Dated: June 21, 2004)

Although density functional theory (DFT) in principle includes even long-range interactions,

standard implementations employ local or semi-local approximations of the interaction energy

and fail at describing the van der Waals interactions. We show how to modify a recent density

functional that includes van der Waals interactions in planar systems [Phys. Rev. Lett. 91, 126402

(2003)] to also give an approximate interaction description of planar molecules. As a test case we

use this modified functional to calculate the binding distance and energy for benzene dimers, with

the perspective of treating also larger, flat molecules, such as the polycyclic aromatic hydrocarbons

(PAH).

Keywords: van der Waals Interactions; Density Functional Theory; Benzene.

I.INTRODUCTION

The benzene dimer is the prototype for aromatic interactions and has been studied extensively

both by theoretical [1, 2, 3, 4, 5, 6] and experimental [7, 8, 9] means. Describing benzene interactions

can be regarded as the first step toward describing interactions of polycyclic aromatic hydrocarbons

(PAHs). PAHs are planar molecules consisting of several aromatic rings, where the peripheral carbon

atoms are covalently bonded to hydrogen atoms. Both benzene and PAHs structurally resemble

graphite and exhibit very similar intra- and intermolecular bond lengths, particularly for (stacks of)

large PAHs. Like the interactions between the sheets in graphite, the interactions between parallel

benzene molecules or layers in PAH stacks are dominated by the weak, nonlocal van der Waals

(vdW) interaction. However, the description of the vdW interaction is absent in the traditional

implementations of density functional theory (DFT), implementations which have otherwise been

very successful in describing dense, hard materials on the atomic scale.

The past few years a number of publications [10, 11, 12, 13, 14, 15, 16, 17, 18] have addressed

the problem of consistently extending the common DFT implementations to also include the vdW

interaction. For the mutual interactions of planar sheets or surfaces a functional was obtained in

Refs. [13, 15]. For example, a relatively good estimate of the binding distance between two graphite

planes (graphene) was obtained with this planar functional [15], while the usual generalized gradient

approximations (GGA) fail to bind the graphene layers or do so at an unphysical binding distance and

energy. Since benzene and PAHs are similar to graphite, the correction to the regular calculations

will be significant also in dimers of these molecules.

In this work, we study dimers of benzene (C6H6) by modifying and approximating the planar

nonlocal density functional [13, 15] in order to treat molecules of finite size. We here concentrate on

dimers where the molecules are placed directly on top of each other (“AA stacking”). The method

we use is developed with the aim of being able to study also PAH molecules, and it is described

below. The results for some of the small PAH molecules naphthalene, anthracene, and pyrene

(C10H8, C14H10, and C16H10) will be reported in a forthcoming publication [19].

∗Corresponding author: Tel. +46 31 772 3632; Fax +46 31 772 8426; E-mail svetla@fy.chalmers.se

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II.METHOD

DFT in principle does include even long-range interactions, such as the vdW interactions, but

those are a part of the exchange-correlation functional Excwhich is approximated in the implementa-

tions of DFT. The most commonly used approximations to Excare the local density approximation

(LDA) and the generalized gradient approximation (GGA), both of which are local or semi-local

and do not include the vdW interaction. However, GGA (and to some extend LDA) has proven

very successful in describing the interactions within dense matter, e.g., within a molecule, and a

consistent correction scheme that includes the long-range intermolecular vdW interactions must not

change the short-range intramolecular interaction.

A. The graphite interactions

In the vdW correction scheme for planar sheets and surfaces, defined and described in Refs.

[13, 15, 16], the exchange energy from GGA is retained, and the correlation energy is determined as

a sum of the local correlation (obtained from LDA) and the nonlocal correlation Enl

described below. For the exchange part the GGA revPBE flavor of Zhang and Yang [20] is chosen

because it is fitted to exact exchange for atoms. Thus the (vdW-corrected) total energy is written

as

c, which will be

EvdW−DF= EGGA− EGGA,c+ ELDA,c+ Enl

c, (1)

where all energy terms are functionals of the electron charge density n(r), and EGGA,cand ELDA,care

the correlation from GGA and LDA, respectively. These energy terms are directly available from

our DFT calculations. Our DFT calculations are performed by the plane-wave pseudo-potential

based program Dacapo [21].

The nonlocal correlation, per area A, for a planar, translationally invariant system such as

graphite is given by [13]

Enl

c/A = − lim

L→∞

?∞

0

du

2π

?

d2k

(2π)2lnφ′(0)

φ′

0(0)

(2)

where φ(z) fulfills the differential equation (ǫkφ′)′= k2ǫ2

φ(L) = 0, and φ0(z) is the vacuum solution. The system is enclosed in a long box of length L

along the direction perpendicular to the graphite plane(s). ǫkis the dielectric function of graphite,

Fourier-transformed in the plane of the graphite sheet

kφ with boundary conditions φ(0) = 0 and

ǫk(z,iu) = 1 +

ω2

p(z)

⊥)2/3 + (k2+ q2

u2+ ν2

F(z)(k2+ q2

⊥)2/4. (3)

The plasmon frequency ωpis defined by ω2

n(z) is the planarly averaged electron density varying in the z-direction perpendicular to the plane,

and iu is the imaginary frequency. We use atomic units unless otherwise noted.

The parameter q⊥is introduced to compensate for the locality of ǫk in the z-direction. The

value of q⊥is materials dependent, and it is found from a separate set of GGA DFT calculations

with an applied, static electric field across the graphite layer.

In Figure 1(a) we show the interaction curve of graphene-graphenein the AA stacking according

to (1). These results of graphene in the AA stacking allow us to directly compare to our results of

benzene, and later PAHs, also in AA stacking. The binding separation 4.0˚ A of graphene (AA) is

larger than for graphene in the physically correct AB stacking (3.76˚ A), calculated within the same

vdW correction scheme [15]. We used the value q⊥= 0.756 in atomic units for a graphene sheet

[15].

p(z) = 4πn(z), the Fermi velocity by νF(z) = (3π2n(z))

1

3,

B. Extending the functional to treat planar molecules

It is a natural next step to modify Enl

large, but not infinitely extended molecules. The PAH molecules are pieces of graphene, passivated

c for planar, translational invariant sheets to treat planar,

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by hydrogen atoms at the broken carbon-carbon bonds.

appropriate approximations in order to apply the functional to planar, large, finite molecules.

The (semi-)local part of the total energy in (1), EGGA−EGGA,c+ELDA,c, can be determined

directly from a usual DFT implementation. Such calculations provide us, besides the energy terms,

with a three-dimensional electron density n(r). In the formalism for Enl

n(z) is assumed translationally invariant along the molecule.

n(z) = A−1

mol

?dxdy n(x,y,z) where Amol is the size of the molecular area that affects the vdW

interactions; this area must be estimated.

For carbon-dominated molecules such as benzene and PAH the dominating contribution to the

vdW correction comes from the carbon atoms. We thus assume that the dielectric function for the

molecules is well described by the functional form (3) with the graphene value q⊥= 0.756 a.u. By

comparing the dacapo-calculated static (u = 0) electric response of the isolated molecule with that

found from the model dielectric function (3), with Amolas a parameter, we can determine Amol. For

benzene we find that Amol= 51.7˚ A2reproduces the static response.

Given the model dielectric function (3) with the parameter Amol determined as described

above, the calculation of the Enl

c

contribution follows from (2). The physical analogy of using the

molecularly averaged n(z) in (2) is that of molecules moved together such that they each take up the

lateral area Amol. Enl

c

for molecules calculated this way therefore somewhat overcounts the vdW

interaction by including not only the physically correct interaction of one dimer molecule with its

partner, but also the interaction with the partner’s neighbors. Because the vdW attraction falls off

with distance this effect decreases as the molecule size increases. The other energy terms of (1) are

not affected.

We stress that Amolis larger than the area of the same amount of carbon atoms in a graphene

plane. In our procedure of finding Amol we compare the (static) response of a molecule isolated

in all three directions. We make sure that our Dacapo-calculated response really does come from

an isolated molecule by checking the convergence of the response with increasing lateral unit cell

size. The response of the low-density charge distribution region within the molecular plane is thus

included in our description, and therefore the vdW interaction of these sheets of close molecules is

different than the vdW interaction of two sheets of graphite, even though the molecules are mostly

composed of carbon atoms.

With the assumptions introduced above, necessary for applying the vdW planar functional to

finite molecules, the quality of the vdW interaction calculations increases with increasing molecular

size. We emphasize that in this sense, the benzene molecule is an extreme molecule, and we do not

expect full quantitative agreement of our results with full-powered quantum-chemistry calculations

[1, 6] or with results in our new (more costly) general-geometry formulation of the vdW-DF [18].

We would therefore like to introduce

c

in (2) the density profile

For the finite molecules we take

III. RESULTS

We here present the results of applying the modified vdW functional to the benzene dimer in

the AA stacking. The first step is to determine the structure of the isolated molecule, the static

response to an applied electric field, the (semi-)local energy contributions to the dimer binding,

and the electron density of the dimer. This step is carried out in a standard DFT program. The

second step is to calculate the nonlocal correlation energy contribution, Enl

discussed above.

The standard DFT calculations were performed with the plane-wave pseudopotential code

Dacapo [21] with periodic boundary conditions. We performed the self-consistent calculations (before

adding Enl

c) using the revPBE GGA functional. The choice of unit cell size involves a trade-off

between manageable calculations on the one hand, and large lateral and vertical separations between

periodically repeated images of the molecules (or dimers) on the other hand. When deciding on the

unit cell size we also benefit from the short-range nature of the traditional DFT-implementations.

In our calculations we used a hexagonal unit cell of size (17.112˚ A, 17.112˚ A, 26˚ A).

In the plane-wave code the number of plane waves is determined by a cutoff in the plane-

wave energy. A preliminary analysis showed a cutoff at 450 eV to be sufficient. However, as we

discuss below, there is a small energy difference between dimers within the same unit cell at very

large separation and two isolated molecules in two separate unit cells. This offset depends on the

plane-wave cutoff in a non-trivial way. We emphasize that the offset in the total energy difference

is unrelated to Enl

c.

c, with the approximations

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3456789

−0.05

0

0.05

separation [Å]

Energy [eV/unitcell]

(a)

3456789

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

separation [Å]

Energy [eV/dimer]

(b)

FIG. 1: Energy curves for graphene and benzene dimers in the AA stacking as a function of separation dis-

tance. Panel (a) shows the graphene-graphene interaction, panel (b) shows the benzene-benzene interaction.

The dashed lines are the self-consistently determined revPBE total energies, the dotted lines show the Enl

correction, and the solid lines show the EvdW−DF total energy curves obtained according to Eq. (1). The

circles on the EvdW−DF curves indicate the calculations performed.

c

A.Benzene-benzene interaction energy

The benzene dimer interaction is calculated by fixing the relative atomic positions within of

each of the two molecules of the dimer and varying the distance between them, while keeping the

size of the unit cell constant. Figure 1(b) shows the total-energy curves for benzene: the revPBE

curve (which shows no binding at all), the nonlocal correlation Enl

includes nonlocal correlation. The reference energy is the benzene dimer at large (11˚ A) separation.

We find in vdW-DF the binding energy 100 meV at the distance 4.1˚ A. These results are in

agreement with our results for the AA stacked graphene dimer as we expect the binding distance

there to be similar to that of the benzene dimer and PAH dimers in the AA stacking. Both the

benzene separation distance and interaction energy are in reasonable, although not perfect agreement

with experiments and other calculations. For instance, Refs. [2, 4, 5] report calculating binding

separations of around 3.8˚ A and binding energies in the range of 64–92 meV. As mentioned earlier,

we expect the quality of our results to improve for larger molecules.

As previously mentioned, no binding arises in the revPBE flavor of GGA. We note that GGA

Perdew-Wang 91 (PW91) [22] calculations (not shown) exhibit an unphysically small binding energy

of the benzene dimer (12 meV) at separation 4.7˚ A. However, this PW91 binding arises from the

unphysical interactions mediated by the exchange contribution in the manner previously documented

for graphite [15].

c, and the full EvdW−DF that

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B.Discussion of the energy reference level

During the course of this benzene and PAH study we encountered a technical problem in

determining an appropriate reference level for our (traditional) total energy calculations based on

GGA.

All dimer energies presented in this paper, whether revPBE or vdW-DF energies, are given

with reference to the energy of two molecules far apart. Thus the energies that we report, and which

we for now generally denote Ediff, are total energy differences given by

Ediff(d) = Etot(d) − Eref

(4)

where Etot(d) is the total energy of the dimer determined at separation d, and Eref is the energy

of two molecules far apart. This applies both to the benzene molecules and the graphite sheets.

The reference level Erefcan be calculated either by removing one molecule to find half the reference

energy, or by keeping both molecules within the unit cell and moving them “far apart” within the

box. In both cases, we make sure that the periodic cell is sufficiently large that the molecule does

not interact (electrostatically) with its periodic images. In our underlying GGA-DFT calculations

it turned out that these two seemingly equivalent ways of determining Eref lead to two slightly

different results. Numerically, the difference is more significant for the benzene molecules than for

graphene.

In Fig. 1(b) we report curves of interaction energies of dimers with increasing separation. The

natural choice of reference energy Erefis therefore the dimer with the two molecules “far apart”, by

which we mean 11–12˚ A separation in boxes of length of 24–26˚ A. However, the problem merits an

analysis.

In Figure 2 we show the difference in reference energies δEref — defined as the energy of a

pair of molecules “far apart” subtracted by the energy of two separate molecules — as a function

of the choice of DFT convergence parameters. In Dacapo, besides the plane-wave energy cutoff for

the wave functions, it is possible to set a different (larger) cutoff for the charge density, to enhance

the description of the charge density at the price of only a modest increase in the calculational size.

Most of the calculations in Figure 2 use a larger charge-density cutoff than the wave-function cutoff.

In order to analyze δEref also for large charge-density cutoff energies we carried out a number of

calculations in a unit cell with reduced lateral size (10˚ A).

A few immediate conclusions can be drawn on the basis of Figure 2. By comparing the single-

cutoff and the double-cutoff results at charge-density cutoff 500 eV and 550 eV we find that the

quality of the wave-function description does not affect δEref. We see this in Figure 2 by the collapsed

data points (indicated by arrows) for the revPBE total energy, but it is true also for the exchange

and the correlation parts of the energy, as well as for PW91 (not shown). This indicates that the

(noninteracting) kinetic part of the total energy, which is calculated from the wave functions, does

not contribute to δEref.

In contrast, δErefdepends non-trivially on the charge-density cutoff. It is not clear that a very

high cutoff value will yield good results, rather, δErefseems to oscillate with charge-density cutoff.

By comparing the results for the usual box size with the small-box results we further notice that

δErefdecreases with the amount of low-density (vacuum) region in the box. δErefthus seems to be

mainly affected by contributions from the low-density regions. A major part of δEref comes from

the exchange part of the total energy. Not shown in the figure are δEreffor the PW91 total energy

and for correlation (revPBE and PW91). These energies all give a finite, but smaller value of δEref.

Unfortunately the problems are worse exactly for the revPBE-GGA that enters our subsequent

vdW-DF calculations in (1).

Besides the ever-present risk of undetected programming errors, we speculate on three pos-

sible origins for the finite δEref that affects our underlying GGA-DFT calculations: (i) numerical

instabilities and formal inaccuracies arising from the implementation of the exchange-correlation

approximation; (ii) the mixing of different exchange-correlation approximations for creation of pseu-

dopotentials and use in calculations; and (iii) the use and/or implementation of ultrasoft pseudopo-

tentials. The possibility of formal problems in the exchange-correlation term borrows some support

from the work by Lacks and Gordon in 1993 [23]: PW91 exchange, and a number of other exchange

approximations, were shown to give values of |δEref| for noble gas dimers far exceeding those from

exact exchange. The deviation arises from contributions in the low-density, large-gradient regions

of space, which is precisely the regions that dominate our system. If this will turn out to be the

origin [24].

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-5

0

5

10

15

20

25

30

400 600800 100012001400 16001800 2000

δEref [meV]

Charge-density cutoff energy [eV]

FIG. 2: Difference in reference energies δEref as a function of charge-density cutoff energy. Diamonds are for

calculations with wave-function cutoff energy equal to the charge-density cutoff energy, empty circles have

wave-function cutoff 450 eV in the usual unit cell size and black circles have wave-function cutoff 450 eV in

a laterally smaller unit cell, as described in the text. Solid, dashed, and dash-dotted lines show δEref of the

revPBE total energy, of revPBE exchange, and of PW91 exchange.

of at least a part of the offset δEref, it is not too surprising that this does not seem to be

perceived as a general problem, since GGA and most present-day DFT codes have been implemented

and tested mostly on dense systems with less vacuum.

We might worry that using a mix of exchange-correlation approximations in the pseudo-

potential and in the energy calculations could lead to unwanted effects. Our calculations within

traditional DFT were carried out with Vanderbilt ultrasoft pseudopotentials [25], originally cre-

ated within the PW91 flavor of the GGA functional, and the energy and the charge density were

self-consistently determined within revPBE. However, we carried out a set of calculations for the

graphene-graphene system, where the offset is much less in size, that showed this mixing to not be

the cause of the offset. The calculations of δErefwere carried out with all combinations of pseudopo-

tentials (revPBE, PW91, PBE [26]), self-consistently determined charge density (revPBE, PW91,

PBE), and energy extracted (revPBE, PW91, PBE, RPBE [27], LDA, as well as all exchange and

correlation contributions). In all cases, the major part of the offset arises from the exchange part

of the energy, whereas the offset on the GGA and LDA correlation is generally almost an order of

magnitude smaller.

We cannot, within this study, test whether the use of ultrasoft pseudopotentials contributes

to δEref. This issue requires controlled comparison of results from different codes and choice of

pseudopotentials and will be addressed in a forthcoming investigation.

In addition to the tests mentioned above, we also tested that the exact position of the monomer

with respect to the Fast-Fourier-Transform grid does not matter on the scale of energy differences

considered here. Further, we speculated that δEref might be due to numerical errors arising when

very small charge-density values are left out in the calculation of the energies, but by changing the

limit of the smallest charge-density values included we found this to be irrelevant to δEref.

IV. CONCLUSIONS

We have presented a modification of an existing planar-system van-der-Waals DFT functional

to approximately treat also large but finite planar molecules. We use the modified functional for the

benzene dimer as an example, with the prospect of applying it to polycyclic aromatic hydrocarbon

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molecules. We have characterized a problem in traditional implementations of DFT for molecular

systems with large volumes of low charge density. This problem motivates further investigations.

The modified vdW functional presented is a step towards being able to treat sparse matter systems

consistently within DFT.

V.ACKNOWLEDGMENTS

We wish to thank Aaron Puzder for helpful discussions. This project was partly supported

by the Swedish foundation for strategic research (SSF) through consortium ATOMICS, by the EU

Human potential research training network ATOMCAD under contract number HPRN-CT-1999-

00048, and by the Swedish Research Council (VR). Allocation of computer resourses through the

Swedish National Allocation Committee (SNAC) is gratefully acknowledged.

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