J. Phys. Chem. 1984,88,4963-4910 at 4.2 and 1.9 K. k;::"' is a rate constant obtained experimentally in this work. k2;Eh"is a theoretical rate constant calculated here by a model of the quantum mechanical tunneling effect. The reaction H2 D H H D was studied in the gas phase in the temperature range 247-468 K.'* The rate constant without tunneling effect is given by the Arrhenius expression
+
-
+
-+
k = 3.63 X 1015T0.5exp(-9.4 X 103/RT)
(13)
+
The rate constant for the reaction H D + D H D2 is smaller than that for the reaction H2+ D H + HD,I9 but it is assumed H D2 is equal to k for H2 D here that k for H D D H HD. Thus the value represents an upper limit of the rate
+
+
+
-
(18) Ridley, B. A.; Schulz, W. R.; Le Roy, D. J. J. Chem. Phys. 1966,44, 3344. (19) Suplinskas, R. J. J . Chem. Phys. 1968, 49, 5046.
-
4963
+
constant for the H D + D H D2 reaction. In order to roughly estimate the order of magnitude, the rate constants at 4.2 and 1.9 K were calculated by eq 13, and denoted in Table I1 as k ~ ~ Since k::;?' is extremely small at 4.2 and 1.9 K, the thermally activated reaction for H D D H + D2 is completely suppressed at these temperatures. k2$ for V = 6.5 kcal mol-] is larger than k z y ' . But when the barrier height (VI) for reaction 8 is varied to 12 kcal mol-' considering the nonlinear arrangement of the HD-D system in crystalline HD, the value of k;$ is roughly consistent with the experimental value (k;:?').
+
-
Acknowledgment. This work was supported in part by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Science, and Culture. Registry No. H,, 1333-74-0; D,, 7782-39-0; HD, 13983-20-5; H, 12385-13-6; D, 16873-17-9.
Intermolecular Potential Calculations for Polycyclic Aromatic Hydrocarbons J. Houston Miller,*+ Department of Chemistry, George Washington University, Washington, D.C. 20052
W. Gary Mallard, and Kermit C. Smyth Center for Fire Research, National Bureau of Standards, Gaithersburg, Maryland 20899 (Received: March 29, 1984)
Methods of calculating the dispersive and overlap parts of the intermolecular potential for polycyclic aromatic hydrocarbons of Dshsymmetry are examined. A new, semiempirical method is utilized to generate the approximate angle-dependentpotentials. These potentials are added to the electrostatic potentials which arise from the permanent quadrupole moments, and the resulting total potential is used to predict the angle between the planes of the molecules in stable dimer configurations for the homologous series benzene, coronene, and circumcoronene. In addition, the corresponding well depths are evaluated for specific conditions.
I. Introduction Polycyclic aromatic hydrocarbons (PAH) are often proposed to be important building blocks and/or precursors in soot formation, and the mechanism of their production in hydrocarbon combustion has been studied for many years. Since the soot formation process is kinetically controlled and occurs rapidly (- 10 ms to reach diameters of 500-1000 A),l the possible chemical pathways to form soot may be severely limited. Although many mechanisms for soot formation have been proposed involving both radical and ionic intermediates, there is agreement on the general sequence of events leading to particle growth. Fuel molecules are broken down to form reactive intermediates which react further to form larger hydrocarbons ( 1 6 carbons). These may cyclize and lose hydrogen to form small aromatic species. Analogous chemical growth processes continue until the aromatic molecules become large enough that condensation processes can occur; i.e., the magnitude of attractive van der Waals forces will be sufficiently greater than the average kinetic energy of other molecules in the bath that a stable, bound dimer will be formed. An interesting question to examine from the point of view of soot formation is what is the minimum size required for PAH condensation to dominate the growth process. One approach to this problem utilizes homogeneous nucleation theory, in which the enhanced vapor pressure of the small droplet and supersaturation are taken into account explicitly. However, to apply this approach the supersaturation ratio as well as the surface tension must be known in order to predict the critical size necessary for condensation. Estimation of these quantities for small hydro+ NBS-NRC Postdoctoral Research Associate 1980-1982.
carbons leads to values for the critical radius as small as 2 &2,3 The prediction that molecules the size of C2H2would be the soot nuclei does not seem reasonable. In addition, details of orientation-dependent intermolecular potentials are lost in applying this calculated critical radius to planar hydrocarbon systems since spherical symmetry is assumed. The calculational approach adopted in this work for determining the probability of dimer formation as a function of temperature requires a knowledge of the interaction potential between pairs of molecules. In this paper potentials are examined, and a computational method suitable for large, nearly circular, aromatic molecules is formulated. In addition to simplifying the mathematics, the assumption of cylindrical symmetry for large aromatics is expected to be appropriate at high temperatures. Stein has shown that the PAH with the highest C/H ratio will be the most thermodynamically stable for any given number of carbon atoms: Thus, for a particular number of fused rings, the most nearly circularly symmetric PAH will be favored. The calculated intermolecular potentials can then be used to determine dimer concentrations as a function of molecular size and bath temperature by evaluating the second virial coefficient. The interaction of two molecules at medium- and long-range separations is a difficult problem which increases dramatically in complexity with molecular size. The application of modern (1) H. Gg. Wagner, "Seventeenth Symposium (International) on Combustion", The Combustion Institute, Pittsburgh, PA, 1979, p 3. (2) D. E.Jensen, Proc. R . SOC.London, Ser. A , 338, 315 (1974). (3) B. S. Haynes and H. Gg. Wagner, Prog. Energy Combust. Sci., 'I,229 (1981). (4) S. E. Stein, J . Phys. Chem., 82, 566 (1978).
This article not subject to U S . Copyright. Published 1984 by the American Chemical Society
~ ~ ' .
4964
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984
computational techniques, both perturbation and S C F ab initio methods, has resulted in quantitative descriptions of these interactions for a variety of system^.^ The perturbation approach considers the second-order effects which lead to van der Waals complex formation, hydrogen bonding, etc., as additions to the first-order exchange and Coulomb energies. In practice this method has been limited to small systems involving relatively few electrons6 The methodology usually adopted in the SCF ab initio calculations is the so-called “supermolecule” approach in which potential surfaces are calculated for the bimolecular complex and for the isolated molecules. The interaction potential is then the difference between these surfaces. This method has been applied to systems which are quite large (such as (c& , )2): and the results compare favorably with experiments if care is taken in the selection of the basis sets. However, calculations for these large systems are computationally intensive. For example, the calculation of a single point on the benzene-benzene surface consumed 140 min of CPU time on a Univac 1100/80 ~ o m p u t e r . ~Clearly, the construction of a complete potential surface and integration of a function which includes this surface over all space (as is required in the calculation of the second virial coefficient) would be an overwhelming task for benzene molecules and prohibitively time-consuming for larger aromatic systems. Fortunately, less rigorous approaches, which depend more on empirical input, have been applied to the calculation of intermolecular interactions of large systems; these methods will be examined in the following sections.
11. Nature of Intermolecular Forces There are three possible contributions to the medium- and long-range attractive forces between pairs of molecules: * electrostatic, inductive, and dispersive. Furthermore, the short-range repulsive interactions will be dominated by overlap and electrostatic forces. Herein we shall consider PAH which possess D6hsymmetry and, as such, do not have permanent dipole moments. However, the dimer of one of these species (benzene) has been shown to have a permanent dipole moment, indicating a ”TEE”-shaped g e ~ m e t r y . ~ Consideration J~ of only dispersive and overlap forces would have predicted that the planes of the molecules in the benzene dimer lie parallel to one another. The addition of the electrostatic forces from the quadrupole-quadrupole interaction to the intermolecular potential gives a predicted geometry in agreement with that observed experimentally.’ Except at short intermolecular separations quadrupole-induced forces between pairs of identical species are expected to be much less important than the permanent electrostatic, dispersive, and overlap forces.12 The calculation of the interaction potential of separated quadrupoles is straightforward and will be discussed in section V. Calculation of the dispersive and overlap potential for large molecules is somewhat more complicated. In the following sections several empirical methods for calculating potentials will be examined and a new hybrid technique which combines predictive abilities with calculational efficiency will be developed. 111. Methods of Calculation of the Dispersive and Overlap Potential Three methods which model the intermolecular potential will be considered. These are the atom-pair rnodel,l3 the Kihara core (5) H. Ratajczak and W. .I.Orville-Thomas, Eds., “Molecular Interactions”, Vols. 1-3, Wiley, New York, 1980. (6) H. Ratajczak and W. J. Orville-Thomas in ref 5, Vol. 1. (7) G . Karlstrom, P. Linse, A. Wallqvist, and B. Jonsson, J . Am. Cbem. SOC., 105, 3777 (1983). (8) G. C. Maitland, M. Rigby, E. B. Smith, W. A. Wakeham, “Intermolecular Forces”, Oxford University Press, New York, 1981. (9) K. C. Janda, J. C. Hemrninger, J. S. Winn, S. E. Novick, S . J. Harris, and W. Klemperer, J. Cbem. Phys., 63, 1419 (1975). (IO) J. M. Steed, T. A. Dixon, and W. Klemperer, J . Cbem. Pbys., 70, 4940 (1979). (11) T. B. MacRury, W. A. Steele, and B. J. Berne, J . Cbem. Pbys., 64, 1288 (1976). (12) J. S. Rowlinson, “Liquids and Liquid Mixtures”, Plenum Press, New York, 1969, p 236. (13) R. K. Boyd, C. A. Fyfe, and D. A. Wright, J. Pbys. Chem. Solids, 35, 1355 (1974).
Miller et al. I
!
, I
“FACE“
O,=B,=O $=anything
O,=@,=90
9=0
“TEE”
o,=o
@,=go $=arything
o,=o,=90 $=go ~
Figure 1. Illustration of the three angles required to specify the relative orientation of two disks for a few representative system geometries.
model,14 and the overlap r n ~ d e l . ” ~In~the ~ discussion of these methods the overall intermolecular dispersive and overlap potential takes the Lennard-Jones form:
Here t is the maximum well depth, is the radial distance at which the potential changes sign from positive to negative, and R is the separation of the molecular centers of mass. The Lennard-Jones potential does not explicitly contain the angular orientation of the molecules. It will be shown below that the angular dependence is contained within e, 0 , and n. In order to examine the effect of molecular orientation on the intermolecular potentials, it is useful to have a consistent nomenclature for orientation angles. Here the angles used follow the convention of Buckingham and other^'^^'^ for cylindrically symmetric colliders, such as disks. 0, and 0, are the angles that the vectors normal to the disks make with the line joining the centers of the disks, and 4 is the angle between the projections of these vectors onto a plane normal to this center line. Figure 1 illustrates these definitions. In the atom-pair model the potential of interaction between molecules is considered to be the sum of the interaction potentials of each of the atoms in one molecule with all of the atoms in the other. The magnitude of the atom-atom interactions is dependent on the atoms involved and is usually derived from heat of sublimation data and crystal packing distances by using a basis set of analogous molecular species. The ability of the atom-pair potential results to predict experimental dispersive intermolecular interactions is well established.18 Further, these results have been used to fit ab initio potential c a l ~ u l a t i o n s . ~The ~ ~ ~primary ~ difficulty with this approach is that the calculation of the potential is prohibitively time-consuming for large polyatomic molecules because the computation time at each point on the potential surface increases as the square of the number of atoms in the molecules. Due to this computational complexity, the method is unsuitable for studies which require construction of the entire interaction surface, such as evaluation of the second virial coefficient. In the Kihara model the molecule is taken to be any convex geometric solid which approximates the molecular shape.14 The intermolecular potential is then taken to be a function of p , the shortest distance between the surfaces of these cores. For example, benzene might be described as a regular hexagonal core. The intermolecular potential used in the Kihara description may take any reasonable form, but is usually similar to the Lennard-Jones potential. (14) T. Kihara, Reu. Mod. Phys., 25, 831 (1953). (15) B. J. Berne and P. Pechukas, J . Chem. Phys., 56, 4213 (1972). (16) A. D. Buckingham and J. A. Pople, Trans. Faraday Soc., 51, 1173 (1955). (17) G. A. Bottomley and T. H. Spurling, Aust. J. Cbem., 19,1331 (1966). (18) F. Mulder and C. Huiszoon, Mol. Pbys., 34, 1215 (1977). (19) T. Wasiutynski, A. van der Avoird, and R. Berns, J. Cbem. Pbys., 69, 5288 (1978).
Intermolecular Potential Calculations for PAH
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 4965
'
TABLE I: Atom-Pair Potential Parameters 'tJ
= -AIJ/rZf
BIJ
cc BENZENE CeH,
d=4.98a
CORONENE
exp(-CtJrIJ)
CH
HH
83630.0
125.0 8766.0
3.60
3.61
27.3 2654.0 3.74
A/(kcal mol-' A6) B/(kcal/mol)
568.0
C/A-'
parameters. The three molecules benzene, coronene, and circumcoronene (Figure 2) will be considered in these calculations because they are a homologous molecular series of nearly circular species, and if their shapes are assunkd to be disks, their relative bimolecular orientations may be completely specified by the four parameters R , S,, 02, and (Figure l).13 Features of the Model. First, the interaction potential at a particular system orientation (i.e., fixed 01, e2, and +) will be only a function of the intermolecular separation ( Y = f(R)). Specifically, in this work the interaction potential is written in the Lennard-Jones form (eq 1). Secondly, the long-range attractive part of the potential is taken to BB proportional to R" ( m = 6 in eq 1). While calculations were performed in which n of eq 1 was also held constant, no single value of n was found for which the overall potential accurately rep;esented the atom-pair potential well shapes for all system orientations. Thus, n is left as an adjustable parameter. Finally, the well depth ( e ) , the zero crossing point (a), as well as the power of the short-range dispersive repulsion (n) are each assumed to be functions of the orientation angles alone (el, 02, and +). Therefore, the computational effort can be divided into two parts: first, the calculation of V as a function of R at constant 01, 8,, and +; and second, the derivation of the functiopal relationship between the potential parameters (a, t, and n ) and the orientation angles (O1, 02, and 4). Computing V ( R ) at a Particular O,, 02, and +. In order to determine the functipnal relationship of u, e, and n on 01, 82, and +, the potential was calculated for a variety of system orientations for each of the three molecules. This was accomplished by applying the method of Boyd and c o - ~ o r k e r s in ' ~ which atom-pair potentials, U,of the Buckingham form
+
CIRCUMCORONENE Cdm
d=l4.528.
Figure 2. Homologous series of D6h polycyclic aromatic hydroearbons considered in this paper. The quoted diameters are the distances from a hydrogen to its counterpart on the opposite side of the molecule.
There are two difficulties in applying the Kihara model to the present calculations. First, the calculation of p for all possible relative molecular orientations is difficult for all but the simplest molecular geometries. Second, the well depth, e , does not have an explicit dependence on molecular orientation. In applications of the Kihara model the well depth is derived from experimental measures of the second virial coefficient and is, as such, an orientation-averaged value. The Kihara model cannot predict this quantity directly. In the overlap model the molecular shape is approximated by an e l l i p s ~ i d . 'The ~ electron density of the ellipsoid is dgscribed by Gaussian distributions, and the repulsive energy betwten two interacting ellipsoids is taken to be proportional to their overlap. Therefore, unlike the Kihara model, there is an explicit dependence of t on the relative molecular orientations, as well as a more complicated dependence for u. The exact forms for e and u as functions of the orientations have been given by MacRurV et al." In an application of the overlap model to benzene, the same well depth is predicted for the face-to-face and the edge-to-edge geometries." Larger interactions are expected in the face-to-face geometry since much more of the surface of the molecules is interacting than in the edge-to-edge geometry, when very little of the molecules' surfaces are adjacent to one another. Both the atom-pair model and an ab initio calculation' predict a difference in these two well depths for benzene dimers. An additional limitation of the overlap mode1 is that, as with the Kihara model, there is no facility to determine e and u without experimental input. Indeed, virial coefficient data for benzene are not sufficient to determine e, u, and K (a molecular shape parameter) exactly; the values of these parameters used in the example above resulted from consideration of crystal packing distances as well as heat of sublimation and virial coefficient data." Further, the need for experimental data limits the Kihara and overlap models to potentials of like molecules, whereas the atom-pair model conceivably could be extended to studies of mixed dimers. IV. Hybrid Scheme for the Calculation of the Dispersive and Overlap Potential In this and the following sections an approximation technique is developed in which the atom-pair potential is calculated for a number of relative molecular orientations. These potentials are then used to derive a functional relationship between the potential parameters of eq 1 ( e , u, n, and m ) and the molecular orientation
U(R) = -AR-6
+ B exp(-CR)
(2) are summed together to arrive at the intermolecular potential,
v;.
X
=
72 1=1
Y
1'1
UIJ(R,)
(3)
Here U , (=f(R,)) represents the interaction of atom i of one molecule with atom j on the other molecule. The numbers of atoms in the molecules are x and y (in this paper x = y ) . A , B, and C for the interactions of different atom pairs were taken from Boyd's work (listed in Table I) and were based on heat of sublimation data of Bromatic and aliphatic hydrocarbon ~tructures.'~Moleculat. geometries were simplified by the assumption that all C-C bond lengths were 1.4 A and all C-H bond lengths were 1.1 A. While the parameters of the atom-pair interactions are derived from data for pure compounds, their application is not restricted to interactions between identical molecules. Thus, the atom-pair potential calculations may be applied to any two interacting molecules. Calculations of the potential were carried out for a number of angles in 1.O-A radial separation steps. Seven to ten data points were collected for each molecular orientation spanning R's which produced highly repulsive to weakly attractive potentials. A nonlinear least-squares fitting algorithm20was used to fit the-data to eq 1 (with m = 6). The results of these calculations are tabulated in Table 11. Note that there is more than one entry in the table for certain combinations of 01, 02, and +. For example one of the "TEE" geometries of benzene was calculated with a single hydrogen closest to the plane of the second molecule while the other calculation was performed for a geometry in which two hydrogens were equally close. Averaging of these variations is (20) W. E. Wentworth, J . Chem. Educ., 42, 162 (1965).
4966 The Journal of Physical Chemistry, Vol. 88, No. 21, 1984
Miller et al.
TABLE 11: Lennard-Jones Parameters
angle, deg
benzene n
81
82
@
U
6
0 30 45 30 0 0 60 45 60 80 90 90 90
O 30 45 30 90 90 60 45 60 80 90 90 90
a 180 180 0 b b 180 0 0 180 90‘ Od Od
3.09 3.41 4.03 4.40 4.50 4.51 4.95 5.36 6.16 6.71 5.80 7.13 6.29
4.02 3.51 2.47 1.64 1.86 1.85 1.39 0.92 0.61 0.38 0.91 0.28 0.65
8.7 9.5 11.2 15.5 15.3 15.1 13.7 24.9 46.0 25.4 18.5 24.4 39.2
coronene n
ucale.ie
U
e
3.09 3.57 4.37 4.74 4.74 4.74 5.48 5.43 5.95 6.29 6.39 6.39 6.39
3.01 3.39 4.15 6.40 6.61 6.61 5.85 8.26 10.06 10.16 10.55 11.23 11.59
23.0 23.0 19.6 4.5 5.5 5.6 12.3 8.3 1.0 2.3 1.2 1.7 1.2
6.9 7.5 9.4 33.0 27.3 27.9 16.9 60.9 38.3 57.3 52.3 63.8 21.5
“calcd‘
U
3.01 3.48 4.26 7.06 7.06 7.06 6.02 8.74 10.03 10.64 11.11 11.11 11.11
2.98 3.33 4.21 8.45 8.52 8.52 5.83 ii.47 13.40 13.22 15.40 15.48 16.20
circumcoronene € n
0cale.i‘
58.7 61.7 54.8 6.5 9.8 9.8 41.9 2.9 1.6 4.9 1.5 1.1 0.8
2.98 3.45 5.99 9.33 9.33 9.33 8.00 11.97 13.98 14.52 15.68 15.68 15.68
6.2 6.5 8.2 51.7 40.8 40.8 9.7 55.2 126 82.6 90.2 93.8 98.9
““FACE”. *“TEE”. “PROP”. “EDGE”. e Section IV.
0
c c u
2
‘1.
0 0
v -
Y
W
2
4 6 SEPARATION (Angstroms) Figure 3. Atom-pair dispersive potentials for benzene.
a
2
6 10 SEPARATION (Angstroms) Figure 4. Atom-pair dispersive potentials for coronene.
14
required in making the approximation of cylindrical symmetry. Figures 3-5 show the potential curves for a few orientations for benzene, coronene, and circumcoronene, respectively. Functional Dependence of E , u, and n on (8,, B2, 4). Ideally, an explicit relationship for E , u, and n in terms of the orientation angles might be formulated. Although simple angular dependencies for E and n were not found in the analysis of the results presented in Table 11, functional relationships between e, the well depth, or n, the power of repulsion, and u, the zero crossing point, were derived: E = to exp[b(u - uo)] + c (4)
n = no exp[b’(u - uo)] + c’
(5)
For each of the three molecules, uo is assigned the value of u in the “FACE” orientation (the smallest value of cr), and a threeparameter nonlinear least-squares fit20 was performed on the appropriate data of Table I1 to find the parameters of eq 4 and 5. A curve illustrating the fit of the data by eq 4 is shown in Figure 6 . The results of the calculations are tabulated in Table 111. The remaining task in the calculation of the potentials is to define the functional relationship between the zero crossing point ( u ) and the orientation angles. No simple functional relationship between u and 4, 82, and C#J was found; however, it was possible to construct a u for each combination of 8,, B,, and C#J via numerical modeling. For this computation the basic premise of the Kihara model is employed: u is the separation of the centers of mass when the distance of closest approach of the disks is set equal to the distance defined by the face-to-face potential zero crossing point,
1
I
2
I
1
I
,
I
1
I
I
6 10 14 18 SEPARATION (Angstroms) Figure 5. Atom-pair dispersive potentials for circumcoronene.
I
uo. As has been pointed out, calculation of this u for three-dimensional molecules at various orientations is difficult. However, the disk approximation allows a simplificationto a two-dimensional
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 4967
Intermolecular Potential Calculations for PAH TABLE III: Least-Squares Parameters for Eq 7 and 8
n
t
molecule benzene coronene circumcoronene I
3.09 3.01 2.98 I
b
C
-0.55 -0.34 -0.19
-0.13
€0
00
4.22 26.2 71.7
I
I
A
-6.73
28.2 68.3
=
~,exp[b(a-a,)]
+
neighbor type
ei
A, A
-0.1 13 -0.110 -0.109 -0.110 -0.101 -0.122 -0.123
c
A
= 4.22 b = -.55 c
Q)I
2
1.98 -19.9 -59.0
A
A, B
E,
I I I I
c’
0.27 0.13 0.06
carbon type
BENZENE E
II
.oo
b‘
TABLE IV: Basis Set for Charge Density Calculations
\
\
-1
no 8.52
-0.125
= -.13
I
A A B
A, C B, B A, A
C
A, C
j
bo I I
-0.126 -0.151 -0.152 0.056 0.056 0.050 0.042
molecule phenanthrene phenanthrene naphthalene
anthracene benzene naphthalene anthracene phenanthrene phenanthrene phenanthrene anthracene naphthalene anthracene phenanthrene phenanthrene
1
I
TABLE V: Calculated Quadrupole Moments ( 0 )
I I I I
I I II
I
I
5
3
I
benzene coronene circumcoronene
I
7
n (Angstroms) Figure 6. Functional relationship (eq 4) between e, the well depth, and u, the zero crossing point, for benzene.
problem, which greatly facilitates computing u. The details of the transformation from a three-dimensional to a two-dimensional description are given in the Appendix. Table I1 includes the calculated u values for comparison of the hybrid method developed in this section with the atom-pair results.
V. Quadrupole-Quadrupole Interactions Since the molecules considered here have no permanent dipole moment, the first nonzero term in the electrostatic potential will be due to the quadrupole-quadrupole interaction. The magnitude of the interaction potential between the permanent quadrupole moments for axially symmetric molecules17 is given by
= ~ / Q ~ (1 R -- (5 ~ COS, e,) - (5 cos2 e,) (15 cos2 cos2 e,) 2[4 cos el cos 0, -I-sin el sin 0, cos
u,,
+
$I2] (6)
where Q is the molecular quadrupole moment. This potential assumes that the separation of charges within the molecule which is responsible for the quadrupole moment is small compared to the separation of the molecules ( R ) . Such an approximation is expected to break down for dimer geometries in which the intermolecular separation is small compared to the molecular radius. In these instances significant distortions of the molecular charge distributions will occur, and the use of eq 6 is questionable. Fortunately, there are only a few geometries (Le., the “FACE” geometries) for which the description of the intermolecular potential embodied in eq 6 is not appropriate. For cylindrically symmetric molecules Q is given by the quadrupole moment along the z axis of the molecule (for disks the z axis is normal to the plane of the disk): (7)
where ei is the charge density at atom i, ri is the distance from (21) P. T.
Eubank, AfChE J., 18, 454 (1972).
present work -28.4 -75.7 -171
10~~~/(c.m~) lit. method ref a 27 -29.0 b 26 -99.4 -215 b 26
“Experimental. bEstimated. the center of mass to the ith charge, and Zi is the distance parallel to the z axis of the ith charge. For disks this reduces to
Estimation of charge densities is possible assuming that reasonable values of localized charge can be derived from Huckel MO theory calculations for comparably bonded carbons in a basis set of polycyclic aromatic hydrocarbons. The group additivity method has been successfully applied to predicting thermodynamic quantities for PAH.4 This technique requires a basis set to estimate the charge densities. The molecules benzene, naphthalene, anthracene, and phenanthrene were chosen for this set because (1) they contain examples of the three types of PAH edge carbons (discussed below) and ( 2 ) the Huckel MO calculations for these species have been carried out.,, In this treatment there are four types of carbon atoms found in PAH: C(-H)(-C)Z (type A), C(-CH),(-C) (type B), C(-CH)(-C), (type C), and C(-C), (type D). Trends in charge densities have been noted by Hoffrnann.,, Specifically, charge densities for hydrogen in the absence of steric hindrance are close to +0.10 (*0.01). Carbons are negatively charged (-0.10 to -0.15) if they are bonded to hydrogens and slightly positive if bonded to other carbons (0 to +0.06). To examine these effects more precisely, carbon charge densities in the four basis set molecules were evaluated according to their carbon type designation and the carbon type. designation of their neighbors (Table IV). The group values used in the calculations were the numerical averages of the available examples in the basis set molecules. No examples of type D carbons are present in the basis set. A slightly positive charge (+0.003) was used for type D carbons on the basis of the discussion by Hoffmann.,, These approximate charge densities were applied to benzene, coronene, and circumcoronene. The resulting quadrupole moments (using eq 8) are listed in Table V. The literature presents a wide spread of possible values for the C-m2).23-29 quadrupole moment of benzene ((-10 to -40) X (22) R. Hoffmann, J . Chem. Phys., 39, 1397 (1963). (23) R. M. Hill and W. V. Smith, Phys. Reu., 82, 451 (1951).
Miller et al. I
I
1
1
-
I
1
I
\
I
I
I
I
1
I ,
BENZENE
1
I
1
I
I
I
I
I
I
h
I
I
I
I
1 1 1
. I
,l
BENZENE
i\ I
CORONENE
“PROP“
. . I
I
I
:I I
0
4
I
I
I
I
I
I
I
I
I
r
I
60
this simple approximation is used to estimate the expected moments for coronene (108 valence electrons) and circumcoronene (234 valence electrons), the resulting values are within 25% of those calculated with the present approach (Table V). These uncertainties are amplified in the calculation of the electrostatic potential because of the square dependence on the quadrupole moment. However, this effect is quantitatively important only at small intermolecular separations and does not significantly affect the positions and magnitudes of attractive potential minima which are important in the calculation of dimer configurations (this paper) and the virial coefficients (future work).
h(
1
I
Figure 8. Total potentials for the three molecules. Here 0, is fixed 0” and O2 is varied between 0” and 180” (6 = anything).
0
1
CIRCUMCORONENE I
90 120 150 ANGLE BETWEEN MOLECULAR PLANES (0,)
30
1
. .
.I.
1
1
8
1
1
12
1
1
1
16
1
20
S E PARATIO N (Angstrom q) Figure 7. Total potential curves for (a) benzene, (b) coronene, and (c) circurncoronene. Note the wholly repulsive potential curves in the “FACE” geometry for all three molecules.
Recently, the more accurate electric-field gradient jnduced birefringence measurements on gaseous benzene yielded a value of -(29.0 f 1.7) X lo4 C.mZ,27in excellent agreement with the value calculated in the present work. Unfortunately, the only polycyclic aromatic hydrocarbon for which there is a reliable experimental quadrupole moment is naphthalene,26so that direct comparison of the calculated moments of coronene and circumcoronene with experimental values is not possible. However, it is useful to comp?re the’ results of the present computation with the moments Calculated from another method which uses charge densities derived Trpm ab initio SCF MO calculations.z6 Chablo et al. fouqd for a limited set of aromatic molecules that the ratio of the calculated qpt-of-plane quadrupole moment to the number ’of moleculy valence electrons is -(0.92 f 0.05) X lo4 C.m2.26 If (24) A. Schweig, Mol. Phys., 14, 533 (1968). (25) R. L. Shoemaker and W. H. Flygare, J . Chem. Phys., 51, 2988 (1969). (26) A. Chablo, D. W. J. Cruickshank, A. Hinchliffe, and R. W. Mum, Chem. Phys. Lett., 78, 424 (1981). (27) M. R. Battaglia, A. D. Buckingham, and J. H. Williams, Chem. Phys. Lett.. 78. 411 (1981). (28) J . Vrbancich and G. L. D. Ritchie, J . Chem. SOC.,Faraday Trans. 2, 76, 648 (1980). (29) S. califano, R. Righini, and S. H. Walmsley, Chem. Phys. Lett., 64, 491 (197Q).
VI. Total Intermolecular Potentials The combined dispersive and overlap potential (eq 1) is added to the quadrupole potential (eq 6) to arrive at the overall bimolecular interaction potential for each of the three molecules benzene, coronene, and circumcoronene. These potentials as calculated for the “FACE”, “TEE”, and “PROP” geometries are shown in Figure 7. As has been found for benzenelo and ant h r a ~ e n e quadrupole-quadrupole ,~~ repulsion in the “FACE” geometry overwhelms an attractive contribution due to only the dispersive forces. An interesting application of the present model is the calculation of the most stable dimer geometry based on the potential energy. As mentioned earlier, the “TEE-shape geometry has been shown experimentally to be the most stable for benzene.I0 In a computational study of anthracene dimer structures, Morris30 showed that quadrupole-quadrupole repulsion made the “FACE” geometry unfavorable. Although a dimer structure could not be rigorously established in that work, it was shown that the long axes of the molecules were very nearly parallel and that the angle between the planes of the molecules was closer to 90” than Oo. These calculations support the results of an earlier spectroscopic study of anthracene dimers trapped in a rigid matrix, in which the ground-state dimer geometry was consistent with the long in-plane axes parallel and the short in-plane axes making an angle of 5 5 5” with one a n ~ t h e r . ~ ’ Within the disk approximation a calculation can be performed which will identify the most stable dimer configuration. Here, O1 is set to zero and B2 is allowed to vary from 0” to 180’. Figure 8 shows the minimum energy in the total intermolecular potential for the three molecules as a function of tI2. These calculations
*
(30) J. M. Morris, Mol. Phys., 28, 1167 (1974). (31) E. A. Chandross, J. Ferguson, and E. G. McRae, J. Chem. Phys., 45, 3546 (1966).
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 4969
Intermolecular Potential Calculations for PAH
I
,,(,,“’\b/“ I
ANGLE BETWEEN PLANES
I I
0”
~
45” 90”
10
5 10 15 SEPARATION ( A n g s t r o m s )
20
Figure 9. Contributions to the total potential from the quadrupole (eq 6 ) and dispersive potentials for the “FACE”(0, = O’), “TEE” (0, = go’), and an intermediate geometry (0, = Oo, = 45’) of circumcoronene
dimers. show that the most attractive well occurs for benzene at go’, coronene at 42O, and circumcoronene at 36’. Figure 9 shows the separate dispersive and quadrupole potentials for three values of O2 ( O O , 45O, and goa), Note that the quadrupole potential is important in destabilizing the face-to-face geometry (when u is small), but not so important for the “TEE”-shaped molecule (when u is relatively large). Since benzene is more nearly spherical than circumcoronene, the range of u for O2 = Oo to O2 = 90° is much less than in the larger molecule, and the short-range quadrupole potentials influence the overall potentials throughout this rotation. Another interesting point is that the binding energy in the dimer is much less than that which is indicated by London dispersive forces alone. For example, in circumcoronene 23 kcal/mol is calculated for the total potential vs. 65 kcal/mol from the dispersive part of the potential alone. It is important to realize that these calculations have been restricted to a specific set of initial angular conditions. Therefore, the dimer geometries and binding energies discussed above represent the most stable configurations for only these conditions. It will be necessary to extend the computations to a complete set of angular orientations in order to ascertain whether or not these geometries are in fact the most stable. It is useful to compare the predictions of this model with the more intensive S C F ab initio calculations on benzene dimers which were discussed in the Introduction.6 In the stable “TEE” geometry there is excellent agreement between the two models (2.27 kcal/mol binding energy [ref 61 vs. 2.41 kcal/mol [this work]). However, the comparison is less favorable for the “FACE” geometry. The ab initio calculation predicts a weakly attractive well for this orientation (-0.29 kcal/mol), whereas the present model predicts repulsion only. Reduction of the quadrupole moment would reduce the difference in these potential curves. However, the source of this discrepancy is more likely the application of the quadrupole-quadrupole potential (eq 6) when the electron clouds of the two molecules are close to one another and thus strongly interacting. These errors which occur at short intermolecular separations are less significant than the large number of more attractive wells which appear at larger intermolecular separations. In addition, the second virial coefficient is predominantly determined by long-range interactions. Consequently, the simple electrostatic potential has been retained in the calculation of the second virial coefficients for the bimolecular interactions of benzene, coronene, and circumcoronene. These values have
0’
{’ I 1 I
__
Figure 10. Diagram illustrating the transformation from three to two dimensions possible within the disk approximation. Here, N , and N2 are the normal vectors passing through the centers of the disks. Here = 8, cos $J where B,, B2, and @ are as shown in Figure 1.
been used with experimentally measured PAH concentrations to compute dimer concentration^.^^
VII. Conclusions Current methods of calculating the dispersive part of the intermolecular potential have been examined and are not satisfactory for describing the interactions of large polyatomic systems, for which the needed experimental data are not available. A hybrid scheme has been developed for a limited set of aromatic molecules which possess nearly circular symmetry. This method predicts the results of the computationally involved atom-pair potential method with a savings in computation time. High-order electrostatic effects, Le., quadrupole-quadrupole interactions, are considered through a group additivity method of calculating charge distributions. In agreement with the literature, the “TEE”-shaped (90’) benzene dimer is shown to be stable. The effect of quadrupole-quadrupole repulsion for coronene and circumcoronene also forces the dimer configuration away from the plane-parallel orientation to 42’ and 36’, respectively. The binding energies of these stable geometries of benzene, coronene, and circumcoronene are 2.4, 7.9, and 23 kcal/mol, respectively. The results of this paper provide the first step in the examination of the role of PAH condensation during soot formation. In contrast to classical nucleation theory this approach for treating condensation processes explicitly includes the important effects of ori(32) J. H. Miller, K. C. Smyth, and W. G. Mallard, “Twentieth Symposium (International) on Combustion”, The Combustion Institute, Pittsburgh, PA, in press.
J. Phys. Chem. 1984, 88, 4970-4973
4970
entation during the molecular collisions.
Appendix Transformationfrom Three to Two Dimensions within the Disk Approximation. While the calculation of u for the complex geometry of a molecule would involve very difficult three-dimensional geometric considerations, the disk approximation allows simplification to a two-dimensional problem. This transformation from three to two dimensions is illustrated as follows. The first disk is centered at the origin of a right-handed coordinate system ( a l , a2, as)and may rotate about the a l axis through an angle 0, (Figure loa). The center of the second disk is located at distance R along the a2axis and rotates through an angle O2 about an axis parallel to the a, axis and an angle 4 about the a2 axis. The range of 0, is 0-90°, of O2 is O1-9Oo, and of 4 is 0-180'. By requiring O2 > O,, one creates a nondegenerate set of orientations. Within these constraints, a point P on the edge of the second disk is defined which will always be the closest point to the first disk. The coordinates of P are ( r cos O2 sin 4, R - r sin 02, -r cos O2 cos 4) where r is the disk radius. (For example, the coordinates of P for the angle combination Or = 0, 02.= 90, 4 = anything "TEE" are [O,R-r,O].) This point P and the line joining the centers of the disks define a plane in a ] , a2,a3 space: a, cos 4
+ as sin 4 = 0
This plane intersects the plane of the first disk, whose equation is a2 cos O1 as sin 8, = 0
The two points where L crosses the edges of the first disk will be called E and E'; Q will be defined as the point of intersection of L and the line normal to L and passing through P (Figure lob). The distance of closest approach can now be found by consideration of the plane defined by P and the line EE'(Figure 1Oc). For convenience in nomenclature, an X Y coordinate system is superimposed on this plane such that the first disk (now the line segment E) is centered at the origin. The intermolecular axis, R , is coincident with the X axis. If Y'is an axis parallel to the Y axis and passing through the center for the second disk, then O2 is the angle that this disk (line segment) makes with Y'. Likewise, the first disk makes an angle $ with the Y axis and $ = 0, cos 4. The distance from P to the line s e g m e n t s represents the closest approach of the two disks and can be no less than the quantity go, defined as the disk separation distance in the face-to-face geometry (-3 A). This will be the perpenE'P.-A dicular distance, @, or one of the edge distances EP orcomputer algorithm was developed to determine which of EP, E'P, and represented the closest approach, and then set the length of that line segment equal to uo and evaluated R , the separation of the molecular centers of mass. The result depends on the value of r, the disk radius. This value was found by averaging the u values for all calculations with 8, = O2 = 90' (two "EDGE" geometries and the "PROP" geometry) and solving the equation
(E)
uav=
+
The equation of the line of intersection, L, is al
cos O1 sin4
-
a2
sin 0 cos 4
-
a3
-cos O1 cos 4
It should be pointed out that L always passes through the origin.
2r
+ uo
The resultant values of r are 1.65, 4.05, and 6.35 8,for benzene, coronene, and circumcoronene, respectively. In all cases this corresponds to a radial distance just outside the outermost carbons in the molecules. Registry No. Benzene, 71-43-2; coronene, 191-07-1; circumcoronene, 8486 1-6 1-0.
Ab Initio Studies of Some Hydrocarbon Complexes with Hydrogen Fluoride A. M. Sapse* Chemistry Department, City University of New York, Graduate Center and John Jay College, New York, New York 10019
and Duli C. Jain Natural Sciences Department, City University of New York, York College, Jamaica, New York 11451 (Received: April 10, 1984)
Complexes between hydrogen fluoride and the nonpolar molecules C2H2,CzH4,C2H6, and C6H6 have been studied at the Hartree-Fock level with the 6-31G* basis set. For comparison, the complex between hydrogen fluoride and the polar molecule CHsCN has also been studied. It is observed that the most stable structure of the *-bonded systems with H F is a "T" geometry for ethylene and acetylene, and the C,, structure for benzene.
Introduction Complexes between nonpolar molecules and hydrogen fluoride have formed the object of a series of experimental] and theoretical2 (1) (a) F. A. Baiocchi, T. A. Dixon, C. H. Joyner, and W. Klemperer, J . Phys. Chem., 74, 6544 (1981); (b) F. A. Biocchi, J. H. Williams, and W. Klemperer, J. Phys. Chem., 87,2079 (1983); (c) A. C. Legon, J. Phys. Chem., 87,2064 (1983); (d) J. A. Shea and W. H. Flygare, J. Chem. Phys., 76,4857 (1982); (e) W. G. Read and W. H. Flygare, J . Chem. Phys., 76,2238 (1982); (f) A. C. Legon, P. D. Aldrich, and W. H. Flygare, J . Phys. Chem., 75, 625 (1981).
0022-365418412088-4970$01.50/0
studies. These compounds are of a donor-acceptor nature, with binding energies ranging from less than 1 kcal/mol to a few kcal/mol. Since the molecules considered do not feature a dipole moment, the binding is due largely to polarization effects, quadrupole effects, charge transfer, and exchange terms. When (2) (a) P. Kollman, J. Am. Chem. Soc., 99,4875 (1977); (b) C. Petrongolo and J. Tomasi, Int. J . Quantum Chem., Quantum Biol. Symp., No.2, 181, (1975); (c) D. Bonchev and P. Cremaschi, Gazr. Chim. Ital., 104, 1195 (1974); (d) W. Jacubetz and P. Schuster, Tetrahedron, 27, 101 (1971); (e) A. M. Sapse, J . Chem. Phys., 78, 5733 (1983).
0 1984 American Chemical Society
