Simple Model for the Benzene Hexafluorobenzene Interaction - The

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Simple Model for the Benzene Hexafluorobenzene Interaction Andreas F. Tillack†,‡ and Bruce H. Robinson*,† †

Department of Chemistry, University of Washington, PO 371500, Seattle, Washington 98195, United States ABSTRACT: While the experimental intermolecular distance distribution functions of pure benzene and pure hexafluorobenzene are well described by transferable all-atom force fields, the interaction between the two molecules (in a 1:1 mixture) is not well simulated. We demonstrate that the parameters of the transferable force fields are adequate to describe the intermolecular distance distribution if the charges are replaced by a set of charges that are not located at the atoms. The simplest model that well describes the experimental distance distribution, between benzene and hexafluorobenzene, is that of a single ellipsoid for each molecule, representing the van der Waals interactions, and a set of three point charges (on the axis perpendicular to the arene plane) which give the same quadrupole moment as do the all atom charges from the transferable force fields.

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type of “charge separation” model. The particular model they chose made modest changes to g(r) from the standard OPLSAA model. The model of Hunter and Sanders showed that greater charge separation can make larger changes in g(r). This has led to a class of force fields in which charges are placed on either side of the atoms (particularly for arenes) to produce atom-centered quadrupolar terms.13−15 These two models bracket the experimental g(r). With the standard in-plane arrangement of charges the close π−π stacking is not well represented and a more off center interaction dominates nearest neighbor interactions. This experimental anomaly raises a very interesting question: Why does a standard transferable force field not work for this system? A search for an answer to this raises a second question: What is the simplest understanding of the interaction of aromatic compounds that explains both the homogeneous and the 1:1 heterogeneous mixtures of these molecules? The answer to this puzzle provides insight into new ways to think about modeling aromatic compounds in general.

INTRODUCTION Standard, nonpolarizable transferable all-atom force fields (such as OPLS-AA1 or GAFF2) generally give satisfactory accounting of condensed matter systems. For example when applied to benzene and hexafluorobenzene as pure liquids they reproduce many of the standard physical properties of liquids, including the center-to-center radial distribution function, g(r). However, when the same standard transferable force fields are used to describe the g(r) for the center-to-center distance between benzene and hexafluorobenzene, in a 1:1 m/m ratio, the discrepancy is quite striking.3−8 The distance distribution function, determined from neutron scattering data, predicts a large peak in g(r), around 3.7 Å, which is not reproduced by standard transferable force fields. To reproduce this peak two groups have chosen to model the van der Waals (vdW) force field (FF) interactions with a Williams model (rather than the traditional 6−12 Lennard-Jones, LJ, FF) with parameters tuned to reproduce the experimental data.9−11 In one case a different set of combining rules for the heterointeraction was also chosen. Additionally, in both studies the excess charges, located at the atom centers, were chosen to reproduce the experimental quadrupoles. Neither of these molecules has a monopole, dipole, or octupole, which leaves the quadrupole as the leading term in an expansion of the charge distribution around the center of the molecule. The potential part of the Hamiltonian consisted only of the van der Waals (vdW) and charge terms. As discussed by Elola and Ladanyi,9,10 the experimental peak at 3.7 Å in g(r) is expected as a liquid interaction analogue to the side−side stacking seen in the crystal structure of 1:1 benzene/ hexafluorobenzene. The close π−π stacking is not seen in the crystals of pure benzene or pure hexafluorobenzene.12 As Baker and Grant,11 following Hunter and Sanders,13 discuss: The side−side, π − π, interaction between arenes occurs in the 3.6 to 3.8 Å range, and it is not fully reproduced even in pure benzene when the charges are in the atomic plane. They suggested off plane charges be placed within the π cloud; as a © 2017 American Chemical Society

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METHODS Both molecular dynamics, MD, and Monte Carlo, MC, methods work quite well to simulate the radial distribution functions of condensed matter liquids.16 We have chosen to use MC with an isobaric, isothermal, NPT, ensemble at standard P and T (at 1 atm pressure and 293 K temperature) as the method for simulating the condensed forms of benzene, hexafluorobenzene, and the 1:1 mixture of the two. The allatom Hamiltonian includes the Lennard-Jones 6−12 potential between molecules in conjunction with the standard Lorentz− Berthelot combining rules; the charge interactions are simply Coulombic, using excess or partial electrostatic charges, which Received: March 9, 2017 Revised: May 25, 2017 Published: June 5, 2017 6184

DOI: 10.1021/acs.jpcb.7b02259 J. Phys. Chem. B 2017, 121, 6184−6188

Article

The Journal of Physical Chemistry B Table 1. All Atom Parameters for Benzene and Hexafluorobenzenea CC [Å] benzene (x = H) hexafluoro-benzene (X = F)

1.396 1.394

CX [Å] 1.09 1.34

rC,LJ[Å] εLJ [perg] −3

1.775 4.86 × 10 1.775 4.86 × 10−3

rX,LJ [Å] εLJ [perg] −3

1.21 1.08 × 10 1.425 4.24 × 10−3

⟨q⟩C [e] (DFT)

Qzz(DFT) [ϵÅ2]

Qzz(Exp) [ϵÅ2]

−0.0843 +0.0933

−1.067 +1.541

−2.08 +1.98

Bond distances (from the DFT simulation), the Lennard-Jones radii, rC,LJ, and energies, εLJ for the carbon; individual radii, rX,LJ, for each atom type (from OPLS-AA FF); the average charge, ⟨q⟩C, on the carbon atoms, and the quadrupoles, Qzz, from simulation, DFT, and experiment, Exp.21 a

are initially positioned on the atomic nuclei. The force field is nonpolarizable, and therefore, the high frequency dielectric is optionally included with the Coulombic interaction. Simulations of condensed molecules were performed with periodic boundary conditions, using a self-consistent reaction field.16−18 The molecular geometries and excess charges were obtained from the DFT optimized geometries using the hybrid B3LYP functional with a 6-31G(d) basis set using Gaussian 09D,19 with CHELPG20 partial charges placed at the nuclei. The vdW radii and LJ energies are based on the OPLS-AA force field.1 (See Table 1.) Similar results were obtained using the GAFF.2 A set of 216 molecules is initially placed at random orientations in a box large enough to allow free rotation of each molecule (about twice the equilibrium volume). As described in detail elsewhere,18,19 the initial placement is allowed to randomize for 2000 cycles during which only Lennard-Jones contributions are counted. During this randomization phase, the system already condenses to the equilibrium density under the NPT ensemble conditions. 120 000 cycles through the box are performed to ensure equilibrium conditions. The statistics needed to simulate g(r) and other properties are obtained from the last 60 000 cycles. The statistics represent the Boltzmann weighted averaged values over 8 independent replicates. As a test of the randomization, and as a way to ensure that the two components (in the 1:1 mixture of benzene and hexafluorobenzene) will not phase separate, the initial configurations were also started as fully separated components, each occupying half the volume. The results were identical whether the two components were initially placed randomly mixed or fully separated.

Figure 1. Radial distribution function, g(r), between the center of benzene and the center of hexafluorobenzene using the all atom, AA, force field for the 1:1 benzene/hexafluorobenzene liquid (given in Table 1). The blue dots represent the experimental g(r), reprinted with permission from JPC B 102, 10712 (1998) Figure 14a; Copyright 1998 American Chemical Society.8 All simulations use charges at the atom centers (contained in the arene plane): The charges, qH on hydrogen of benzene, and qF on fluorine of hexafluorobenzene, are given in Table 2.

interaction between benzene and hexafluorobenzene (as demonstrated by the height of the first peak in (r)). One can reproduce the first peak of g(r) with on-atom charges only by increasing those charges beyond what are suggested by transferable force fields, which include direct QM simulations, GAFF and OPLSAA force fields. To retain transferable charges, we now consider moving the charges to off-atom locations, while retaining the magnitude of the quadrupole moment. The on-atom charges are now replaced in such a way that the quadrupoles of benzene and hexafluorobenzene are maintained as given in Table 1. The replacement charges are defined as follows: The simplest quadrupole structure is that of a charge qQ placed a distance a on the Z axis, from the XY plane of the arene and a second charge, also qQ, placed at −a. At the center is a charge of −2qQ. This produces a quadrupole in the Z direction of Qzz = 2qQa2. Because the quadrupole is a traceless tensor, there is also a quadrupole in the x and y direction of 1 − 2 Q ZZ . We placed a charge qQ on both benzene and hexafluorobenzene, with the same magnitude but of opposite sign. And the distances, a, were adjusted for both so that the quadrupoles remained the same as the ones given by the DFT calculations, in Table 1. Qzz(x, DFT) = 2qQa2x , where x = H for benzene, and x = F for hexafluorobenzene. The distance for charge displacement for hexafluorobenzene is larger than for benzene because the quadrupole is larger: aF = 1.2aH For the case where qQ = −4096e for benzene (and qQ = +4096e for hexafluorobenzene) then aH = 0.011 Å. Note that the larger one chooses the magnitude of qQ, the closer the quadrupolar charges are moved to the center and the closer this

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RESULTS We now examine what steps must be taken to simulate the experimental g(r) spectra using the same AA LJ interactions and leaving the charges at the atoms, but changing the magnitude of the partial charges at the atoms. Figure 1 shows that the height of the first peak in g(r) increases as the magnitudes of the charges on the atoms increases. The smallest charges are given by QM simulations; larger are those from the OPLSAA force field; and larger still are the charges consistent with the experimental, gas phase quadrupole. The densities and the enthalpies of vaporization both increase with the increasing charges (Table 2). Reducing the dielectric has an effect similar to increasing the charges. Because these are all nonpolarizable force fields, the dielectric, ε, which would account for the linear polarizability of the medium, is optionally included (see Table 2) with the charge−charge interactions; assuming ε = n2, where n is the refractive index of the medium. Simulations using the NPT ensemble and MC methods with the OPLS-AA force field well reproduce the experimental g(r) for the pure liquids, benzene and hexafluorobenzene. However, to understand the interaction of benzene and hexafluorobenzene we now consider charges that are no longer placed on the atoms. As shown in Figure 1 and Table 2, the charges used in transferable force fields cannot reproduce the strength of the 6185

DOI: 10.1021/acs.jpcb.7b02259 J. Phys. Chem. B 2017, 121, 6184−6188

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The Journal of Physical Chemistry B

Table 2. Density and Enthalpy of Vaporization for the 1:1 Benzene and Hexafluorobenzene Mixture for Various Simulationsa method

refractive index (n)

experimentalg AAb (DFT)g AAb AAc(OPLSAA)g AAd (exp. Q)g AAd (exp. Q)g AAe(1.2 · expQ)g LoDf

1.5 1.5 1.0 1.0 1.5 1.0 1.0 1.5

charge qH [e]

charge qF [e]

density ρ [g/cc]

enthalpy ΔHvap [KJ/mol]

−0.0933 −0.0933 −0.130 −0.120 −0.120 −0.144

1.29 1.24 1.25 1.27 1.25 1.30 1.32 1.19

34.1 31.5 32.7 36.3 33.0 39.7 45.7 36.7

0.0843 0.0843 0.115 0.164 0.164 0.197

The charge on each hydrogen of benzene, qH, and each fluorine of hexafluorobenzene, qF, are adjustable parameters of the NPT AA simulations (Figure 1). The charges on the carbon atoms of benzene are −qH and of hexafluorobenzene are −qF. The refractive index, n, is used as a parameter in the Coulomb terms of the simulations. The density, ρ, and enthalpy of vaporization, ΔHvap, are those computed for the different charge conditions for the 1:1 mixture of benzene and hexafluorobenzene. bCharges determined from the DFT calculations, see Table 1. cCharges given in OPLSAA Force Field. dCharges that reproduce the experimental quadrupole moments, see Table 1. eCharges that give quadrupole moments 1.2 times the experimental values. fProperties derived from only four point charges placed on the z axis, normal to the arene plane, developed later. gRadial distribution function, g(r), shown in Figure 1. a

further decreased in magnitude. Table 2 gives the density and vaporization enthalpy for this system as well. When the g(r) for pure benzene and pure hexafluorobenzene are computed with this same quadrupole arrangement of charges, the changes are minimal from the original g(r) computed from the all atom charges placed on the nuclei.11,18 We have previously shown that the simulation of the vdW energies using the AA force fields can be well described by using a single ellipsoid.18,22,23 The features of the ellipsoid are found directly from those of the underlying AA force field (given in Table 3).18,23 We show that this substitution gives the

representation is to the analytic point quadrupole. In this case, because aH ∼ 0.01 Å this set of point charges should give a g(r) that is close to the pure quadrupole limit. On the other hand, for the case where qQ = −0.8 e, the distance aH = 0.816 Å for benzene, and aF = 0.982 Å for hexafluorobenzene the resulting g(r) is quite distinct from the pure quadrupole limit. Figure 2 shows the center-to-center g(r) for benzene to hexafluorobenzene for the AA force field for the 1:1 benzene/

Table 3. Semi-Axes of the Two Ellipsoidsa

benzene hexafluorobenzene

semi−axesLJ [Å]

wLJ [Å]

3.33; 3.27; 1.66 3.73; 3.79; 1.61

2.02 2.21

⎡ εLJ0⎣perg

( )⎤⎦ kJ mol

0.0834 (5.022) 0.0912 (5.492)

a

Single ellipsoids, replacing the AA LJ interactions, are given along with the adjusted semiaxis widths, wLJ[Å], and the Lennard-Jones 0 [perg], for benzene and hexafluorobenzene. The energies, εLJ intermolecular potential is a 6−12 type Lennard-Jones potential in which the widths and energies use the standard Lorentz−Berthelot combining rules. The energy is corrected by an interaction area that depends on the semi-axes and the particular pairwise orientation of the two ellipsoids.23 Figure 2. Radial distribution function, g(r), between the center of benzene and the center of hexafluorobenzene for the AA force field for the 1:1 benzene/hexafluorobenzene liquid. The dashed blue line is the experimental g(r).8 The fully atomistic g(r) is shown in red for the case where the charges (see Table 1) are in the arene plane. All other simulations use the charge of the quadrupolar term, qQ (shown in legend) on the Z axis to replace the partial charges on the nuclei. The values of the quadrupoles on benzene and hexafluorobenzene remain constant for all values of qQ.

same g(r) as the fully atomistic force field. The substitution of the single ellipsoid (called the LoD model) for the OPLS-AA FF for the Lennard-Jones interaction does not change the radial distribution function, g(r), or the enthalpy of vaporization, ΔHvap, markedly for both the case where the partial charges are on the nuclei and the case when the charges are replaced by charges at three specific places on the Z axis (perpendicular to the arene plane). Figure 3 compares the g(r) for AA and single ellipsoid when all the charges are kept the same, verifying that this replacement of the AA vdW FF with a single ellipsoid reproduces the same result as the AA force field. This same single ellipsoid (see Table 3) with the simplified quadrupolar set of charges, wherein q = −0.8e, is compared to the experimental g(r) with excellent agreement.

hexafluorobenzene mixture. This is compared with the experimental g(r) from Cabaco.8 The main interest is the peak (or lack thereof) in the 3.5 to 4.0 Å range. The AA force field predicts a g(r) less than 1, whereas the experimentally determined distribution function is larger than 3. In Figure 2, the particular charge arrangement for qQ = −0.8e has the closest agreement with the first experimental peak; and not only is the first peak in excellent agreement but the subsequent peaks also agree well with the experimental data. The peak in g(r) around 3.8 Å continues to increase as qQ is 6186

DOI: 10.1021/acs.jpcb.7b02259 J. Phys. Chem. B 2017, 121, 6184−6188

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The Journal of Physical Chemistry B

between the two different types of arenes; one could have used the experimental quadrupole values as well and obtained similar results as seen in Figure 1. Even modifying the charges on the atoms in the AA representation, to be consistent with the experimental quadrupoles, did not markedly change the g(r) as shown in Figure 1 for the AA model. Only after the linear polarizability was removed (by setting the refractive index to 1) was it possible in increase g(r). Fundamentally, there is no underlying, a priori, requirement that the DFT-based charges should reside on the nuclei. Indeed, there is considerable precedent for placing point charges in the π cloud of arene structures.13,14 Putting the excess (or deficit) charge at about 1 Å above and below the plane is no more or less correct than saying the charges should be placed on the nuclei: The parameters of the model (as with all transferable force fields) are tuned by agreement with data. One should note that these charges are excess charges, just as the charges that are positioned on the atoms of an AA force field are also excess charges. The nature of the substituents will induce either positive or negative deviations in the excess charges. For benzene it is simple to imagine that an extra electron of charge is contained in the π cloud. However, for hexafluorobenzene the π cloud must be envisioned as having a net deficit of an electronic charge due to the large electronegativity of fluorine pulling the electronic charge out of the core of the arene. The simplest model, we would suggest, is that benzene and hexafluorobenzene are individually well modeled by a single ellipsoid (whose structural parameters come directly from the underlying AA FF) and a single quadrupole (with a magnitude that can match either the theory or experiment, and optionally contain polarizability terms) described by single point charges. The ellipsoid we have chosen is in the family of the Gay-Berne type structures,23 and similar to that suggested by Golubkov and Ren.24 The placement of a quadrupolar set of charges, which is on the order of an electron at about one angstrom from the center of the plane, is about as simple a charge distribution as one can imagine (accomplishing results similar to a high order multipole expansion24,25) and yet it explains the structure of the delicately balanced benzene/hexafluorobenzene interaction while doing no harm to the structure of pure benzene or pure hexafluorobenzene; and predicts a reasonable enthalpy of vaporization (see Table 2). We are suggesting here also, as in water, that the search is for the optimal positions to locate the charges. Water models, in the SPC26 and TIPnP classes,27,28 have used 2, 3, 4, and even 5 center charge models, the positions of which have been a delicate balance between the vdW and the charge interactions.29 The simplest representation for the Lennard-Jones interactions in water is to use a single sphere. Here with the arenes, the simplest representation is with a single ellipsoid.

Figure 3. Center-to-center radial distribution function between benzene and hexafluorobenzene for the all atom force field (red dots) and the single ellipsoid approximation, LoD, (solid gold) with charges located on the nuclei. The single ellipsoid LJ approximation with the quadrupolar charge qQ = 0.8 ϵ along Z (as the quadrupole expansion, solid green) and the experimental radial distribution function (blue dots).8 The computational time for the LoD with all charges was about 3 fold longer than for the LoD with the four charges; and the computational time for the AA model relative to the LoD with four charges was 24 fold longer.

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DISCUSSION The forgoing results show that indeed the parameters of transferable AA force fields can be and should be used to define the LJ interactions; and no further modification is needed for the LJ interactions. However, explaining g(r) requires using onatom AA charges larger than those suggested by either DFT calculations or from standard transferable force fields, such as GAFF and OPLS-AA. As Elola and Ladanyi9,10 found, the partial charges for an on-atom, all atom model must be larger to reproduce the experimental g(r). As shown in Figure 1, just increasing (doubling) charges to match the experimental quadrupole moment only slightly increased the g(r). Once the linear polarization is off (setting n = 1) then charges, slightly in excess of those used to reproduce the gas phase quadrupole moment, are sufficient for a simulation to match g(r). When the on-atom charges are large enough to simulate g(r) well, the enthalpy of vaporization is larger than the optimal value determined from transferable force fields. (Both OPLSAA and GAFF were used.) In the case of arene structures, the nuclei are all in a single plane but the electronic charges are three-dimensional in nature. The π clouds, after all, represent distributed charges above and below the plane that contains all of the nuclei. Apparently the problem for transferable force fields arises when a two-dimensional arrangement of fractional/ excess charges is used to describe the fully three-dimensional nature of the arene. This answers the first question posed above: Transferable force fields can be used, but some modification of the placement of the all atom charges is necessary. Several other groups have considered strategies for off atom placement of charges. Perhaps the most systematic effort has been the extended electronic distribution, XED, force field.14 We have taken a simplified approach of trying to find a minimal set that is consistent with the charges used in transferable force fields, insightful, reproduces g(r), and gives reasonable values for the enthalpy of vaporization. In our simulation, the choice of using the DFT generated quadrupole was not necessary to understand the interaction

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AUTHOR INFORMATION

Corresponding Author

*[email protected]. ORCID

Bruce H. Robinson: 0000-0002-5579-953X Present Address ‡

Center for Computational Sciences, Oak Ridge National Laboratory, One Bethel Valley Road, P.O. Box 2008, MS-6008, Oak Ridge, TN 37831. 6187

DOI: 10.1021/acs.jpcb.7b02259 J. Phys. Chem. B 2017, 121, 6184−6188

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The Journal of Physical Chemistry B Notes

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The authors declare no competing financial interest. This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a nonexclusive, paidup, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

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ACKNOWLEDGMENTS The authors wish to thank Dr. Lutz Maibaum for helpful discussions throughout this project, and Dr. Bruce Eichinger for his careful readings of the manuscript and many suggestions along the way. We gratefully acknowledge the financial support of the National Science Foundation (DMR-1303080); and the Air Force Office of Scientific Research (FA9550-10-1-0558 and FA9550-15-1-0319). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Andreas F. Tillack has been partially supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Oak Ridge Leadership Computing facility under contract number DE-AC05-00OR22725.

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DOI: 10.1021/acs.jpcb.7b02259 J. Phys. Chem. B 2017, 121, 6184−6188