J. Phys. Chem. 1995,99, 16909-16912
16909
Dispersion Interaction between Elongated Molecules Robert A. K r o m h o u t t and Bruno Linder**$ Chemical Physics Program, Florida State University, Tallahassee, Florida 32306-3006 Received: July 26, 1995@
As has been recognized for some time, the dispersion interaction of molecules, particularly large molecules, cannot be adequately represented by a simple London-type expression based on the total molecule polarizabilities. We investigate two models consisting of several anisotropic polarizabilities associated with molecular subgroups. In one model intramolecular interactions among the groups are included; in the other only intermolecular interactions are included, but different “effective” polarizabilities are used. These models are generalizations of models considered previously by others. We calculate the interaction energy as a function of molecular separation for several different relative orientations of the molecular axes of model PAA (p-azoxyanisole) molecules. We find that (1) at very short distances inhnolecular interaction is negligible; (2) at intermediate distances the distance dependence is approximately inverse fifth power; (3) the two models yield almost identical results over all but the very shortest distances; (4) at molecular separations greater than a few times the largest molecular dimension, the distance dependence is essentially the simple inverse sixth power London interaction based on total molecular polarizability. We compare these qualitative results with those found previously by others.
I. Introduction A number of authors have given expressions for the dispersion interaction between molecule^.'-^ Rabenold and Lindefla and MacRury and L i n d d have used the charge-density susceptibility in expressions that include effects of charge overlap that are exact except for neglect of intermolecular exchange and that take account of temporal and spatial correlations. The chargedensity susceptibility is the time-Fourier transform of the linear response function for the charge induced at one space-time point by a unit potential at another space-time point. MacRury and Linder paid particular attention to many-body effects and displayed forms that applied when the dipolar (polarizability and dipole-couplingtensors (T-tensor)) approximation was used. Rabenold and LindeflbvCmodeled parallel linear molecules as collections of identical Drude oscillators coupled by intramolecular and intermolecular T-tensors and showed that at small intermolecular separations the result was given to high accuracy by intermolecular pairwise interactions only, while at all distances the effects of terms beyond the second order in intermolecular interactions were negligible, as were effects of higher multipoles. Salem2 and Zwanzig3 investigated models of very long molecules consisting of isolated polarizabilities. Zwanzig included intramolecular T-tensors as well as intermolecular T-tensors, while Salem used “effective” polarizabilities and intermolecular T-tensors only. Both found that in the range of separation that is large compared to intermolecular polarizability separation and in the range that is small compared to molecular length, the second-order dispersion energy varied as the inverse fifth power of the molecular separation, in contrast to the simplest London approximation, which varies as the inverse sixth power. At separations that are large compared to the molecular length, one expects the interaction to reduce to the inverse sixth power behavior. In this paper, as an example of elongated molecules, we show two approximate representations of the dispersion interaction Physics Department, Emeritus.
* Chemistry Department. +
@
Abstract published in Advance
ACS Abstracts,
October 15, 1995.
0022-365419512099-16909$09.00/0
between two PAA @-azoxyanisole)molecules based on isolated molecular subgroup polarizabilities, one with and one without intramolecular interactions. Our model molecules are of finite length (length 5 times width), the polarizabilities are anisotropic, in contrast to previous calculations, and we calculate the interaction as a function of distance for a variety of relative orientations. We discuss and compare our results for this more general model with the findings of previous authors.
11. Method The formal expressions for the dispersion energy given by Linder et al?a35 have not been applied per se because of the very formidible problem of calculating the charge-density susceptibility of polyatomic molecules. The charge density susceptibility Po,(rl) Pn00.2) ho+E,-E,-ib
+
If one expresses the transition-charge densities pori in Maxwell multipoles, the dipole-dipole coefficient
is exactly the negative of the polarizability of the entire molecule. One can surmise that it would be possible to divide the molecule into regions (“basins” or molecular subgroups) between which there is negligible charge flow when a small potential is applied anywhere in the molecule. This would make feasible the subdivision of the susceptibility (or polarizability in the dipole-dipole approximation) into “effective” group susceptibilities (or polarizabilities). In this view, since the charge-density susceptibility which is being subdivided is characteristic of the whole molecule, no intergroup interaction would need to be included; however, for the dipole-dipole approximation to the group susceptibilities to be useful, the 0 1995 American Chemical Society
16910 J. Phys. Chem., Vol. 99, No. 46, 1995
Kromhout and Linder
TABLE 1: Parameters of the Models for PAAa group all all1 a2L a211 position H3c-o3.313 4.466 3.006 6.460 0.390
0 0 /o
7.759
9.496
7.037
13.920
0.174
1.943
3.230
1.762
4.734
0.000
7.759
9.496
7.037
13.920
-0.174
3.313
4.466
3.006
6.460
-0.390
-N=N-
-0-CH3
a Units of polarizabilities, a, are (molecular length3) x Positions are in fractions of molecular length measured from the center, and all are along the molecular axis. Conversion of polarizability in A3 to L3 is accomplished by multiplication by n/[6 x volume x (axial r a t i ~ )= ~ ]9.106 x
charge-density susceptibilities would have to be small for rI and r2 in different basins. Intergroup exchange would be included through the calculation of A second approach to a dipole-dipole approximation also uses the concept of division into basins between which charge flow is negligible. The calculation of the charge-density susceptibility of the molecular subgroups in isolation could be calculated. Then the zeroth-order approximation to the susceptibility, xo, might be taken to be the sum of the susceptibilities of the subgroups. The charge densities induced in one subgroup would interact with the other subgroups through the Coulomb interaction. Then it can be shown that the susceptibility of the whole molecule is given by a Fredholm integral equation in which xo appears both as the inhomogeneous term and in the kernel. Intramolecular exchange is inexact here. The dipoledipole approximation to this consists of subgroup polarizabilities coupled intramolecularly as well as intermolecularly through T-tensors. This second view is very closely related to the method of Applequid for the estimation of molecular polarizabilities from empirically derived atomic polarizabilities. It also corresponds to the dipole models used by Rabenold et al. and by Zwanzig, whereas the simpler model used by Salem more nearly corresponds to the effective group susceptibility model of the previous paragraph. To arrive at our two models for PAA, we began with empirical polarizabilities of molecules similar to the five chemically recognizable subgroups in PAA’ (see Table 1). These anisotropic polarizabilities were cylindrically averaged to produce components along and perpendicular to the molecular axis. For the effective polarizability model, with no intergroup intramolecular interactions, these polarizability components were scaled so that the sum over groups yielded the known parallel and perpendicular p~larizability~~ components of the whole PAA molecule. For the other model, the “interacting subgroup” model, we first set up the dipole stiffness matrix
x.
+ 9-
A = a-l(O)
(34
where A, a(O),and T a r e 15 x 15 matrices corresponding to one isolated molecule. This was inverted to produce -the generalized polarizability matrix,
B
~
-
These matrices are defined so that
A * p= E p = B-E
1
where p and E are 15-component column matrices representing the dipole moments and the electric field vectors at each group in the molecule. The overall polarizability components were then determined by summing all the elements of B in the rows corresponding to that component. The group polarizabilities were then scaled until the results were the empirical values for PAA, q = 4.553L3 and al = 2.185L3, where L is the molecular length, 22 A. The dipole approximation of MacRury and Linder5 for the dispersion interaction of two molecules becomes, for our models,
AE=
(&)hwdy tr,,{ln[l
+ a(-iy).H ln[l
-
+ a(-iy)-F]}
(5)
where 1 is the 3N-dimensional unit matrix, a(-iy) is a 3Ndimensional matrix with 3 x 3 submatrices which are the polarizabilities of the N subgroups on the two molecules, S i s a 3N-dimensional matrix whose 3 x 3 blocks are the ‘I*N(N1) dipole-coupling tensors comprising both the intermolecular tensors TF and intramolecular tensors T: (the subscripts refer to subgroups), F’ is the same but contains only the intramolecular couplings, and (-iy) is a point on the negative imaginary axis in the complex angular frequency plane. The lowest nonzero term from the expansion of the logarithm yields the simple London inverse sixth interaction. If we use the simplest (closure) approximation for the frequency dependence of the polarizabilities,
w2 w2+ y 2
a(-iy) = a(0)-
(6)
we can perform the integration over y. Neglecting the effects of retardation, we obtain
For model 1, the interacting group polarizability model, the interaction energy for a pair of molecules becomes, by expansion of eq 7,
where E = hii, is the mean excitation energy for all polarizabilities, and the subscript on a refers to the model. For model 2 the second term in the braces is missing, the interacting group polarizability matrix al(0) is replaced by the matrix of the effective polarizabilities, az(O),and the TZ components are set equal to 0. The values of the group polarizability components and the group separations used are given in Table 1 in units of molecular length. 111. Results
We have calculated the interaction energies for these two models for four different relative orientations of two molecules and for center of mass separations of up to two molecular lengths using 12 terms in the expansion of the energy, eq 8. The molecular length, L, here is taken to be 22 A and the diameter 4.4 A, corresponding to a spheroid volume of 230 A3, as suggested for PAA.8 We also give the asymptotic form for each case.9
Dispersion Interaction between Elongated Molecules
J. Phys. Chem., Vol. 99, No. 46, 1995 16911
PARALLEL CASE
45" CASE
101
I0-4
10-1
I0-4
102
I0 5
102
I0-5
10-3
10-6
103
10-6
EIE
E l i
E l i
E l i
104
I 0-7
104
107
10-5
10-8
105
108
10-9
106,
10-6
1.0
2.0
I
r/L
I
I
i
1.0
I I~
2.0
10.9
r/L
Figure 1. Dispersion interaction energy of parallel (012 = 0') molecules
Figure 2. Dispersion interaction energy for BIZ = 45'. Same symbols
divided by mean excitation energy vs intermolecular separation divided by molecule length. For 0.2 < r/L < 1.2 use left scale; for larger r use right scale as indicated by arrows. (1) Model 1, interacting group polarizabilities. (2) Model 2, effective group polarizabilities. (3) Line varying as inverse fifth power. (4) Simple London inverse sixth power, total molecule polarizabilities.
as in Figure 1.
Results for parallel ( 0 , 2 = 0 ' ) molecules and for 0 1 2 = 4 5 O , where 012is the angle between the molecular axes, with the separation, r, increasing parallel to its initial direction along the shortest line between centers, are shown in Figures 1 and 2 on a log-log plot. Because the values cover 6 orders of magnitude in the range I = 0.2L to r = 2.0L, the plots are divided into r 1.OL and r > 0.6L,with the left energy scale used for the smaller r and the right energy scale used for the larger r. Shown on the same plot are the curves for models 1 and 2, the simple London values? and a short dotted line corresponding to an inverse fifth power dependence. In Figure 3 are shown the model 1 results again for parallel and 45" plus two curves for 012 = 90', in which the highest curve is for the case where the molecules initially form a letter T and the separation is increased parallel to the stem of the T. Several characteristics should be noticed about these results: (1) The values from model 1 at small distances are nearly equal to the sum of the intermolecular dipolar pair interactions; that is, the effects of the T; were almost negligible, just as was found by Rabenold and Lindeil for parallel strings of harmonic oscillators. (2) A t distances smaller than the length of the molecule, the behavior is approximately inverse fifth power of the distance, corresponding to the results of Salem2 and Z w a n ~ i g . ~ (3) The differences between models 1 and 2 are only significant at very small separations. Further, in model 2 the second order (n = 2, pairwise additive term) contributes almost the entire value. See Table 2. (4)Each of the curves approaches the simple London inverse sixth power curve for r greater than 2L, as expected.
"INTERACTIVE GROUPS" 10.1
104
10-2
10-5
10-3
10-6
E l i
E l i
104
107
105
108
106
10-9
r/L
Figure 3. Dispersion interaction from model 1, interacting group polarizabilities. Same scales as Figure 1. (1) 8 1 2 = 45'. (2) Parallel molecules, 012 = 0'. (3) 012 = 90', crossed at center. (4)OI2 = 90', T configuration.
IV. Discussion The simple distributed polarizability models introduced here can be viewed as a logical step beyond the identical-sphericaloscillator model analyzed by Rabenold and Linder toward better
16912 .IPhys. . Chem., Vol. 99, No. 46, 1995
Kromhout and Linder
TABLE 2: Interaction Energy distance 0.2 0.6
1.o
2.0
large'
interaxial angle
0" 45" 0" 4 s 90°b 90"'
O0 45" 90°b 90"' 0" 45" 90°b 90"
0" 45" 90°b 90"'
model 1 -1.64 -4.43 -2.77 -2.54 -2.66 -2.78 -2.33 -2.32 -8.76 -2.63 -6.24 -6.16 -1.08 -6.25 -5.59 -5.23 -6.83 -7.62
x IO-' x 10-3 x
x 10-5 x 10-4 10-4 x x x x x IO-' x
IO-'
x 10-7 x
model 2 -1.9 x IO-* -5.02 10-3 -2.83 x 10-5 -2.54 x 10-5 -3.21 10-4 -2.87 x 10-4 -2.36 x -2.32 x -9.75 x 10-6 -2.68 x -6.47 x lo-' -6.27 x -1.16 x 10-7 -6.40 x -5.59 -5.23 -6.83 -7.62
model 1 pairwise additived -1.70 -4.69 -2.95 -2.65 -2.36 -2.75 -2.29 -2.32 -7.55 -2.61 -5.82 -5.91 -9.24 -6.22 -4.84 -4.82 -5.40 -4.77
x
x IO-'
10-5 x 10-5 x 10-4 10-4 x x x 10-6 x
x IO-' x
IO-*
x IO-' x lo-*
model 2 pairwise additived -1.89 -5.00 -2.83 -2.54 -3.21 -2.87 -2.36 -2.32 -9.74 -2.68 -6.47 -6.26 -1.16 -6.40 -5.59 -5.23 -6.83 -7.62
x IO-*
x 10-3 x 10-5
10-5 x 10-4 10-4 x x x 10-6 x x IO-' x lo-' x 10-7 x IO-*
simple London -8.73 -8.18 -1.20 -1.12 -1.46 -1.05 -5.59 -5.23 -6.83 -4.88 -8.73 -8.18 -1.07 -7.62 -5.59 -5.23 -6.83 -7.62
x x x x
IO-' lo-' 10-4 10-4
x lo-'
x IOW4 x 10-6 x x
x x lo-' x IO-' 10-7 x IO-'
a Energy in units of mean excitation, E. Distance is from molecule center to molecule center in units of molecular length. T-shaped orientation, distance parallel to base of T. Crossed at centers, distance along line of centers. Contains intermolecular, Tt-TJ,,couplings only. e Given here as IO6 times the coefficients of the inverse sixth power of the distance between molecular centers.
representations of the charge-density susceptibilities needed for dispersion interaction calculations. The ability of quantum chemists to calculate charge-density susceptibilities of small molecules may or may never be at hand, but the susceptibilities of molecules of interest (in liquid crystal and biophysical theories, for example) are not likely to be available in the near future. It might become feasible to calculate charge-density susceptibilities of a molecule large enough to divide into a pair of subgroups to test the feasibility of constructing the susceptibility of the large molecule from those of the subgroups. At this time we know of no such project in progress. It probably is feasible to calculate the polarizability of moderate-sized molecules and subgroups, which would give us better-founded subgroup polarizabilities to work with. Certainly it would be interesting to attempt to find empirical subgroup polarizabilities with which to represent the dipole approximation to the charge-density susceptibilities of large molecules. The problem here is to find data for determining the values. What we have shown in this paper is that dispersion interactions can be approximated with little difficulty from such models by using the effective polarizability model at all distances with the possible exception of very short distances; in the latter case pairwise additivity can be used to good approximation. At distances several times the longest molecular dimension the inverse sixth power approximation can be used based on the total molecule polarizability. We note that model 1 corresponds to the models of Zwanzig, of Yasuda, and of Rabenold and Linder, whereas model 2 corresponds to the model of Salem, who assumed pairwise additivity and effective polarizabilities. The fact that the "effective" and "interactive group" models yield very nearly the same results over the whole range of intermolecular separation and for all orientations, except where the molecules are nearly in contact, and the fact that the Ti.T;i terms give nearly all the result in the "effective" polarizability model are
very interesting from the users' point of view, as well as in reference to the remarks of Zwanzig in his analysis of Salem's paper. Not only are the "effective" polarizability model calculations much less complicated than those for the "interactive group" model they also suffer much less from round-off errors at the larger values of intermolecular separation, since it is essential to carry higher powers of F i n the latter case because the 9' contributions become more important as the separation becomes large. Finally, the question as to which model is actually more correct at very small separations seems moot. Certainly neither model is completely accurate at such small separations, but either should be better than the simple London inverse sixth formula. References and Notes (1) (a) Margenau, H.: Kestner, N. R. Theory of Intermolecular Forces; Pergamon: New York, 1969. (b) Mahanty, J.; Ninham, B. W. Dispersion Forces; Academic: New York, 1976. (c) Langbein, C. Theory of Van der Waals Attraction; Springer-Verlag: Berlin, 1974. (d) Hunt, K. L. C. J . Chem. Phys. 1990, 92, 1180. (e) Liang, Y. Q.;Hunt, K. L. C. J . Chem. Phys. 1993, 98, 4126. (f) Yasuda, Y. J . Phys. SOC. Jpn. 1969, 26, 163. (2) Salem, L. J . Chem. Phys. 1962, 37, 2100. (3) Zwanzig, R. J . Chem. Phys. 1963, 39, 2251. (4) (a) Linder, B.; Rabenold, D. A. Adv. Quantum Chem. 1972,6,203. (b) Rabenold, D. A.: Linder, B. J. Chem. Phys. 1975, 63, 4079. (c) Rabenold, D. A.; Linder, B. Int. J . Quantum Chem. 1977. 4, 443. (d) Rabenold, D. A.: Linder, B. J. Chem. Phys. 1977, 67, 1108. (5) MacRury, T. B.: Linder, B. J. Chem. Phys. 1973, 58, 5388. MacRury, T. B.; Linder, B. Ibid. 1973, 5398. (6) Applequist. J.; Carl, R. J.; Fung, K.-K. J. Am. Chem. Soc. 1972, 94, 2952. (7) (a) Applequist, J. J . Phys. Chem. 1993, 97, 6016. (b) LandoltBomstein, Zahlenwerte und Functionen; Springer: Berlin, 1951; Vol. 1, p 510. (c) Subramhanyam, H. S . : Prabha, C. S . : Krishnamurti, D. R. Mol. Cryst. Liq. Cryst. 1973, 28, 204. (8) Vieillard-Baron, J. J . Mol. Phys. 1974, 28, 809. (9) Eldredge, C. P.: Heath, H. T.; Linder, B.; Kromhout, R. A. J. Chem. Phys. 1990, 92, 6225. See also ref l a pp 37, 38.
JP952157X
