J . Phys. Chem. 1991, 95, 7605-7612
7605
A Simple Model for Predicting Structures of Gas-Phase van der Waals Dimers Containing a Rare-Gas Atom 2.Kisiel Institute of Physics, Polish Academy of Sciences, AI. Lotnikow 32/46, 02-668 Warszawa, Poland (Received: December 5, 1990)
A computationally simple model for predicting the preferred structures of gas-phase dimers containing a rare-gas atom is developed and tested. The structures are determined by minimizing the multicenter interaction energy arising from dispersive attraction counterbalanced by hard-sphere repulsion. In the absence of readily available good quality intramolecular atomic polarizability data, it is shown that the geometrical anisotropy of the dispersive interaction can be satisfactorily modeled by replacing atomic polarizabilitieswith cubes of covalent radii of the bound atoms. The model is found to accurately reproduce the structures of most known dimers containing a rare-gas atom and leads to a number of useful insights into the results for several dimers. The only significant failure of the model is for linear dimers and this is clearly ascribed to the deficiency in modeling the repulsive van der Waals shell and the necessity for including the inductive term. The ease of applicability makes the model an attractive add-on term for calculations on systems where anisotropy of the dispersive interaction may be important, in particular for other types of weakly bound complexes.
Introduction The recent rapid development of the molecular beam microwave and infrared spectroscopic methods has resulted in the determination of structures of a large number of weakly bound complexes.ls2 While with due care and computing effort such data can be rationalized and hopefully predicted at the ab initio level, there is still considerable scope for 'desktop" models that the experimenter can use interactively while in the process of conducting his investigation. The requirement for such models arises in particular a t the preparatory stage where it is necessary to predict the structure and the spectrum of a complex in order to expedite the spectroscopic experiment. The data processing stage also often requires the use of a model to select the most likely interpretation. For hydrogen-bonded complexes it has been established following various energy decomposition studies3s4that the geometrical anisotropy in the weak interaction is principally of electrostatic nature. On this basis, a simple electrostatic approach for modeling the angular geometry of hydrogen-bonded dimers has been proposed by Buckingham and Fowler5 and has enjoyed considerable ~ u c c e s s . ~ This - ~ model has also been successfully applied to van der Waals bonded dimers such as CO2+-HCCHBand, on addition of isotropically distributed dispersion, to N2-C029and N2-N20.10 For the much studied van der Waals dimers of the type where one member of the pair is a rare-gas atom there has hitherto been a lack of such a simple model. Structures of such complexes have been rationalized by arguments ranging from qualitative consideration of the likely magnitudes of London-type dispersion terms, through conclusion that the rare gas atom binds to the most electropositive site in the molecule," to consideration of electron donation by the rare-gas atom to lowest unfilled bonding orbital of the molecule,12including ( I ) Hobza, P.; Zahradnik, R. Chem. Rev. 1988, 88, 871. (2) Nesbitt, D. J. Chem. Rev. 1988, 88, 843. (3) Umeyama, H.; Morokuma, K.; Yamabe, S.J. Am. Chem. SOC.1977, 99, 330. (4) Hurst, G.J. B.; Fowler, P. W.; Stone, A. J.; Buckingham, A. D. Inr. J . Quantum Chem. 1986, 29, 1223. (5) (a) Buckingham. A. D.; Fowler, P. W. J. Chem. Phys. 1983,79,6426. (b) Buckingham, A. D.; Fowler, P. W. Con. J . Chem. 1985, 63, 2018. (6) Price, S. L.; Stone, A. J. J . Chem. Phys. 1987, 86, 2859. (7) Kisiel, Z.; Fowler, P. W.; Legon, A. C. J . Chem. Phys. 1990. 93,6249. ( 8 ) Pritchard, D. G.; Nandi, R. N.; Muenter, J. S.:Howard, B. J. J . Chem. Phys. 198%. 89, 1245. (9) Walsh, M. A.; Dyke, T. R.; Howard, B. J . J . Mol. Struct. 1988, 189, 111.
(IO) Randall, R. W.; Dyke, T. R.; Howard, B. J. Furuduy Discuss. Chem. SOC.1988,86, 21. ( I I ) Shea, J. A.; Campbell, E. J. J . Chem. Phys. 1983, 79, 4724. (12) Steed. J . M.;Dixon, T. A.; Klemperer, W. J . Chem. Phys. 1979, 70, 4095.
remnant hydrogen bonding.13 This state of affairs is not entirely satisfactory as in view of the relative simplicity of the isotropic rare-gas atom a single low-level unifying treatment could be anticipated. The geometries of the observed rare-gas dimers may be subdivided into four main categories as depicted in Figure 1 and the task of a model would be to satisfactorily explain the preference for any one of the four. The modeling of the skew-type structures, where the rare-gas atom fits into a cavity defined by atoms of the acceptor molecule appears to provide a particular challenge. A multisubstituted ethylene molecule, for example, will have four such sites, at the two ends and on both sides of the C - C bond. A recent systematic determination of the structures of dimers of Ar with a series of fluorosubstituted ethylenesi4 (Figure 2), has provided both an impetus and a guideline for developing a model. A requirement for an easily applicable model for rare-gas dimers also arises from the nature of the spectroscopic observables. The structure of the dimer is arrived a t from a fit to the observed rotational constants. However the addition of a point mass to a molecule leads, owing to the symmetry of the rotational ellipsoid, to more than one numerically indistinguishable solution. For example, for a complex between a rare-gas atom and a planar molecule of C, symmetry there are four identical solutions.i5 When straightforward elimination on grounds of unreasonably close Ar-atom distances is not possible, additional information has to be brought in. Isotopic substitution is frequently applied,i1*16q17 but may be less useful than is normally the case in view of the often significant vibration-rotation effects in weakly bound dimers. The usefulness of some other observables such as quadrupole coupling constants may also be curtailed for similar reasons. Another possibility, as has been done for Ar-PF3,l8 is to differentiate between the numerically identical solutions by means of ab initio calculations on both the dimer and the acceptor molecule. Since calculation of dimer properties would typically require a supermolecule Maller-Plesset-type configuration-exchange calculation, a full minimization is usually not practicable. (13) Cohen, R. C.; Busarow, K. L.; Laughlin, K. B.; Blake, G. A.; Havenith, M.; Lee. Y. T.; Saykally, R. J . J . Chem. Phys. 1988, 89, 4494. (14) Kisiel, 2.;Fowler, P. W.; Legon, A. C. J . Chem. Phys., in press. (15) Suenram, R. D.; Fraser, G. T.; Lovas, F. J. J . Chem. Phys. 1988,89,
6141. (16) Fraser, G. T.; Suenram, R. D.; Lovas, F. J. J . Chem. Phys. 1987,86,
3107. (17) Bohn, R. K.; Hillig, K. W., 11; Kuczkowski, R. L. J . Phys. Chem.
1989. 93. 3456. (18) (a) Hillig, K. W., II; Matos, J.; Scioly, A.; Kuczkowski, R. L. Chem. Phys. Lerf. 1987, 133, 359. (b) Taleb-Bendiab, A.; LaBarge, M. S.;Lohr, L. L.; Taylor, R. C.; Hillig, K. W., 11; Kuczkowski, R. L. J. Chem. Phys. 1989, 90. 6949.
0022-3654/91/2095-7605%02.50/0 0 1991 American Chemical Societv
I606 The Journal of Physical Chemistry, Vol. 95, No. 20, 1991 a)
c)
b)
Ar- CIF. H X I X = F .
Ar-OCS.COZ,CICN,
C1.81,CNI
CH3CI, PFl
Kisiel
d)
Ar-S03.EF3,fumn,
Ar-HZNCW, C H 2 C K N
oxirana. pyridma
CHZCHF
Figure 1. Four main types of structures for dimers containing a rare-gas atom and examples: (a) linear; (b) T-type.where , the rg-cm axis is nearly perpendicular to the highest symmetry axis of the acceptor molecule; (c) pyramidal, where the rg-cm axis is nearly coincident with either the highest symmetry axis or a distinguished inertial axis of the molecule; (d) skew, where the rg atom fits into a cavity defined by a chain of atoms in the molecule and the rg-cm axis is not directly related to any axis in the molecule.
(3 COS~O, - 1)) ~
Ai !
',
! \ ! '.
. (
g!
where ffa8, Aa,Or,... are components of the dipole-dipole, dipole-quadrupole, ... static polarizabilities for the two centers, the quantities U are set to the first ionization potentials, and the summation convention is employed. T4 etc. are elements of the T tensors which describe the geometric dependence of the energy V ~ Polarizabilities R,~~ and are given by Tap= ( ~ T ~ ~ ) - ~ V ,etc. can also be partitioned into bond polarizabilities such that a,, = a?3,a* = all = a 2 2 etc. for polarizability parallel and perpendicular to the bond direction, respectively. In this way dispersion interaction between a diatomic molecule, or a bond a, and a spherical atom b reduces toI9
\.
+ ~ , , - 7 ( 2 ~ 1cos3 / ea + ~ COS ea~ - 2 COS^ 3 6,)) 3
+ ...I (2)
where 6, is the angle between the bond axis and the direction of Rab. The energy for the whole rare-gas complex can be obtained by summing the pairwise contributions from eq 1 or eq 2 for all atoms or bonds of molecule a: #a
Udisp I
\
I
\
!
-kFi - 'k
L.-
F
\.
y H
k Figure 2. Experimentally determined structures for dimers between Ar
and fluorinated ethylenes." In the present work, a very simple minimizing model for predicting the most likely geometries of rare-gas dimers is developed and tested. The purpose of the model is to fulfill the above-mentioned requirements for rapid assessment of the geometrical anisotropy of the van der Waals interaction and to identify initial geometries for more rigorous calculations, while also providing some insight into the formative processes determining the structures of raregas-containing dimers. Model The attractive forces responsible for formation of dimers containing a rare-gas atom arise primarily from dispersive and inductive interactions. Of the two resulting contributions Ud@, and uindto the dimer energy, the dispersive term is usually much larger. By truncation of the treatment of interaction between a spherical atom and a linear polar m o l e ~ u l eat ' ~ the dipole polarizability a and the electric dipole moment p, it can be shown that for Ar-HF, Ar-HCI, and Ar-HBr, #jnd is 0.37, 0.053, and 0.022 of #disp. respectively. This estimate can only be regarded as approximate, since the molecular multipole expansion for the induction energy shows poor convergence at distances observed in the smaller dimers?O but the trend in energy partitioning identified here is confirmed in studies of hydrogen halide frequency shifts in inert-gas matrices.21 Induction is therefore likely to be of importance only for dimers of rare-gas atoms with molecules that carry a combination of high dipole moment and low polarizability and will be neglected. From the long-range theory of intermolecular forces, the dispersion energy between two centers a and b may be approximated byI9 ( 19) Buckingham, A. D. In Intermolecular Interactions-From Diatomics ro Biopolymers: Pullman, B., Ed.; Wiley: New York, 1978; Chapter I . (20) Hutson, J. M.; Howard, B. J. Mol. fhys. 1982. 45, 791.
(21) Barnes, A. J. In Vibrational Spectroscopy of Trapped Species; Hallam, H. E., Ed.; Wiley: New York, 1973; Chapter 4.
=
zu%p
(3)
The terms defined in eq 1 are particularly suited for use with multicenter distributed polarizabilities such as those defined by Stone,22 although these are as yet available on only a limited number of computational pa~kages.2~Some bond polarizabilities are available for use in terms in eq 2 but several would have to be assumed, and in any case additivity of the most generally quoted bond polarizabilities is poor.22924JS Since it is clear that without the use of carefully calculated multicenter polarizabilities a, A , ... considerable approximations are going to be made anyway, it is better to make the model as simple as possible, providing the primary objective of reliable structure prediction is retained. Further assumption of isotropic polarizabilities for atoms in molecule a and truncation at R6 reduces dispersion terms to the London dispersion formula (4)
Qualitative arguments a t this level have often been employed in discussing structures of rare-gas complexes. Unfortunately the usual atomic polarizabilities derived for bound atoms suffer from the same deficiencies as bond polarizabilities. Greater success at atomic partitioning of molecular polarizability has been achieved with the dipole-dipole interaction mode125*26 where the response of an atom to an electric field is modified by that of other atoms in the molecule. However the polarizabilities thus derived differ considerably from the standard additive and are available for only a limited number of atoms. The model also has its failures. In this light a viable alternative may be to replace the polarizabilities with quantities derived from some more easily available properties of bound atoms. Since the dispersion interaction arises from correlations between electron configurations, it may be expected to be extensive in the number of electrons available for correlation and hence, in the atomic multicenter approach, proportional to the total number of electrons on an atom, Z,. It should also be proportional to the "mobility" or ease of perturbing an electron which, for a covalently bound atom in a (22) Stone, A. J. Mol. Phys. 1985, 56, 1065. (23) Amos, R. D. CADPAC-The Cambridge Analytic Derivatives Package; SERC: Daresbury, U . K . (24) LeFevre. P. J. W. In Advances in Physical Organic Chemistry; Gold, V . , Ed.; Academic Press: New York. 1963; Vol. 3. (25) Applequist, J.; Carl, J. R.; Fung, K.K . J . Am. Chem. Soc. 1972.94, 2952. (26) Applequist, J. Arc. Chem. Res. 1977, I O . 79.
The Journal of Physical Chemistry, Vol. 95, No. 20, 1991 1607
Structures of Gas-Phase van der Waals Dimers
10
TABLE I: Summary of Atonic Parameters Assumed for Use in the Model (Eq 5) rvdW:A r,,bA r,3,au u , ~ a u u!au
I
1
:
H
t
m
1.2
B C
0.8 1
I.76 1.59
N 0
P S Br
Ne Ar
Kr I
1
I
lo
3
30
a / m 3 Figure 3. Comparison of ’r,
with atomic polarizabilities from the di-
poledipole interaction model.”
molecule, can be crudely approximated by the volume available for each electron or ‘/grrW3/Z. The product of these two factors suggests that polarizabilities d should be proportional to which conveniently carries the volume dimensions of polarizability. Gross proportionality of this type is confirmed by the plot in Figure 3, and thus the use of (roDv3)p may offer a general prescription for ordering bound atoms according to dis ersive response. On transition to atomic units it is noted that /3dJaUb/(Ua+ Ut,)= 1, and since this term shows only a relatively small variability, it is accordingly set to unity. It is proposed therefore that the geometry of a dimer between a rare-gas (rg) atom and a molecule a be identified and characterized by minimizing
where i runs over the atoms in a. The repulsive term Ere, = CfrPi(Rr,.!) simulates the effect of short range repulsive forces for which it is assumed that each atom is surrounded by a hard repulsive sphere defined by its van der Waals radius. For compatibility with numerical minimization it is convenient to assume a differentiable analytical form for this repulsive potential, one possible choice being6
h,(R,,) = &All + tanh KQ,, - R,, - c)/611
Xe
%
(6)
where Q.. is the sum of van der Waals radii on atoms i andj, and A, c, a d 6 are arbitrary parameters set to 50,0.6,and 0.1 atomic units respectively. The key atomic parameters required for the model are summarized in Table 1. Bondi’s2’ rather than Pauling’s2*van der Waals (vdW) radii are used since the former lead to more consistent effective radii for complexed rare-gas atoms. In addition, in view of experimental evidence from dimers of argon with C 0 2 , F2C0, CICN, and SO3 that an end-on approach to *-bonded atoms is much closer than suggested by their usual van der Waals radii, the repulsive shell is in such cases constructed from modified radii defined as follows. Electron withdrawal by multiple bonds is simulated by decreasing the hard-sphere radius for multiply bonded atoms by increments equal to the differences between singly bonded and corresponding multiply bonded covalent radii. At the same time a dummy repulsive center of radius equal to the mean of the standard vdW radii of atoms involved in the bond is placed at the midpoint of each multiple bond. The values of rm3 are compared in Table I with polarizabilities from both the dipole-dipole and the additive models. It appears (27) Bondi, A. J. Phys. Chem. 1964.68, 441. (28) Pauling, L. ” w e ofthe Chemical Bond Cornell University Press: Ithaca, NY, 1960. (29) Katelaar, J. A. A. Chemical Consfirurion;Elsevier: New York, 1953; p 90. (30) Hallett. A. C. H. In Argon, Helium and the Rare Gases; Cook, G . A., Ed.; Wiley Interscience: New York, 1961.
1.10
1.80 1.86 1.75 1.85 1.54 1.88 2.02 2.16
CI
.OH
0.77 0.74 0.74 0.72
1.56 1.47
F
0.3
0.37
1.04 0.99
1.14
0.34e 3.6 3.1 2.7 2.7 2.5 9.0 7.6 6.6
10.0
0.91
2.8
5.9 4.2 3.1 2.2
6.3 5.9 4.3 2.6
12.9 19.4
15.4 22.5 2.18 11.18
16.78 27.18
Reference 27. Reference 28. From dipoltdipole interaction model, ref 26. “From standard additive model, ref 29. CReplacedwith 0.505 for use in the model; see text. f R c d u d for atoms with multiple bonds and dummy repulsive centers placed at midpoints of such bonds; see text. #Reference 30.
c
l
E
- 100
V
\
w -200
“attr I
0
.
,
.
.
.
.
.
.
90
e
180
10
Figure 4. Energy decomposition of the experimentallydetermined potential for Ar + HCI” and comparison between the dispersion energy uw and E,t,r calculated with eq 5 for values of R , and 0 defining the po-
tential minimum umin. from Table 1 and also from Figure 3 that rGM3 may significantly underestimate the polarizability of hydrogen in relation to those of heavier elements. Since this is a crucial parameter a more accurate quantification of its value is essential. An opportunity for this is provided by the very accurate potential determined for Ar-HCI.” The minimum-energy path, u ~between , the Ar-HCI and the Ar-CIH minima on this two-dimensional potential surface is reproduced in Figure 4 as a function of the Ar-cm-CI (cm = center of mass) angle 0 (Rem not constant). The associated attractive energy uattrrhas also been calculated by using the procedure outlined in ref 31, as well as the inductive energy Uind, evaluated by using a three-center MP2 distributed multipole for HCIS2and 3 Uind
=
Fa = E ( - d T a -t- &Tap - Y&,
[email protected]) i= I
(7)
the summation for each field component F, being over the multipole centers. The difference between uattrand uindgives the dispersive energy udicp,also plotted in Figure 4. This is compared with Eattr calculated with eq 5 for the same values of R and 0 and scaled to coincide with the Ar-HCI minimum in Ud{,p. In order to reproduce the relative depth of the Ar-CIH minimum while using (rcov3)cI = 6.6 au, it was necessary to replace the corre(31) Hutson, J. M . J . Chem. Phys. 1988,89, 4550. (32) Stone, J. A.; Alderton, M . Mol. Phys. 1985, 56, 1047.
7608 The Journal of Physical Chemistry, Vol. 95, No. 20, 1991 sponding quantity for H with 0.505 au, and this value was used in all further calculations. The energy decomposition for Ar HCI shows that this system is not particularly amenable for structural prediction on only the dispersive basis as the reversed Ar-CIH linear geometry would be predicted to be the most stable. As expected, uindis much smaller than u but since its anisotropy is a sizable fraction of the anisotropy%umi, (as already noted in ref 20), it plays a deciding role in defining the deepest minimum to be at the Ar-HCI configuration. It is expected, however, that in molecules with more multielectron atoms the difference in energy between Uind and Udisp will increase sufficiently for the anisotropy in dispersion to dominate and be a reliable pointer to preferred structures on its own. It should be noted that both rW3 and polarizabilities from the dipoledipole model will, for Ar, favor van der Waals contact with fluorine over that with hydrogen, allowing the rationalization of the experimental structure for Ar-difluoroethylene in Figure 2. Additive polarizabilities on the other hand, when combined with van der Waals shells, will have a built-in bias for contact with hydrogen and in difluoroethylene will predict a preference for the FCCH rather than the FCF site by Ar. The dipoledipole model polarizabilities and r,’ also show the same trend with bond order, whereas additive polarizabilities increase with bond order. However, since the use of multiply bonded radii would be more appropriate to a bond-type multicenter formulation, and to keep the present model as simple as possible, only the single-bond covalent radii are adopted for use in eq 5 .
Kisiel
+
Performance of Model
To ascertain the performance of the model it is necessary to establish whether the minimized Eintin eq 5 has a positive correlation with the dimer binding energy and, most importantly, whether the associated geometry is a reliable indicator of dimer geometry. It appears that best understanding of the model will be obtained if its two components parts, E,,* and E.m, are tested separately. The calculation of Eat,,,and hence of dimer strength cannot be expected to be a strong point of the model but has a bearing on how well it will deal with competing geometries for a given dimer. The best generally available experimental indicator of binding energy is the Lennard-Jones well depth e derived from the centrifugal distortion constant DJ by means of the diatomic approximation33
0.0015
S
0
0.0010
\ L c U
0 W I
0.0005
0
0
200
100
E
/ cm-’
Figure 5. Correlation between (0) E,,,r and the pseudodiatomic Lennard-Jones well depth z and (@) E,,,, and the fitted well depth e(Oo) for dimers of HCI with Ne, Ar, Kr,and Xe.
0.003
i3 \
0.002
L c 4-
0
W I
0.001
0
E /cmi’ Figure 6. Correlation between E,,,, and the pseudodiatomic LennardJones well depth c for dimers of Ar with linear, T-type, and pyramidal geometries.
term. Thus, providing the repulsive exponent does not vary, the use of attractive energy instead of the total energy at minimum where B is the rotational constant for the complex, p, is the pseudodiatomic reduced mass, andf, is the force constant for the intermolecular stretch. For better comparability of values for e between dimers carrying the first three types of structures in Figure 1, the intermolecular center of mass separation R , is used in eq 10 rather than a separation evaluated following a one-dimensional vibration-rotation correction to B.33 An enhanced version of eq 8 derived for dimers with developed spatial geometry is also used” A test of the modeling of the attractive part of the interaction energy is therefore possible by comparing e from eqs 9-1 1 with E,,,, calculated for the observed dimer geometry. The problem of the unknown repulsive energy arises again, but it can be shown that for example, for the Lennard-Jones 12-6 potential, the repulsive term at minimum is a constant fraction (half) of the attractive term, or for the more general Maitland-Smith potentia135this is 6/n, where n is the exponent of R in the repulsive (33) Keenan, M. R.; Campbell, E. J.; Balle, T. J.; Buxton, L. W.; Minton, T. K.; Soper, P. D.; Flygare, W. H. J . Chem. Phys. 1980, 72, 3070. (34) Read, W . G.; Campbell, E. J.; Henderson, G. J . Chem. Phys. 1983, 78, 3501. (35) Hutson, J . M.; Howard. B. J . Mol. Phys. 1982, 45, 769. (36) Viswanathan, R.; Dyke, T. R. J . Chem. Phys. 1984,82, 1674.
(37) Dixon, T. A.; Joyner, C. H.; Baiocchi, F. A.; Klemperer, W. J . Chem. Phys. 1981, 74, 6539. (38) Novick, S.E.; Davies, P.; Harris, S.J.; Klemperer, W. J. Chem. Phys. 1973, 59, 2273. (39) DeLeon, R. L.; Mack, K. M.; Muenter, J. S.J . Chem. Phys. 1979, 71, 4487. (40) Shea, J. A.; Read, W. G.;Campbell, E. J. J . Chem. Phys. 1983,79, 2559. (41) Fraser, G.T.;Pine, A. S.;Suenram, R. D. J . Chem. Phys. 1988,88, 6157. (42) Keenan, M. R.; Wozniak, D. B.; Flygare, W. H. J . Chem. Phys. 1981, 75, 631. (43) Joyner, C. H.; Dixon, T. H.; Baiocchi, F. A.; Klemperer, W. J . Chem. Phys. 1981, 75, 5285. (44) Matsumoto, Y . ;Ohshima, Y.; Takami, M.; Kuchitsu, K. J . Chem. Phys. 1989, 90, 7017. (45) (a) Collins, R. A.; Legon, A. C.; Millen. D. J. THEOCHEM 1986, 135, 435. (b) Kisiel, Z.; Legon, A. C. Unpublished results. (46) (a) Kukolich, S.G.;Shea, J. A. J . Chem. Phys. 1982.77.5242. (b) Kukolich, S. G. J . Am. Chem. Soc. 1983, 105, 2207. (47) Klots, T. D.; Emilsson, T.; Ruoff, R. S.;Gutowsky, H. S.J . Phys. Chem. 1989, 93, 1255. (48) Harris, S.J.; Novick, S.E.; Klemperer, W. J . Chem. Phys. 1974.61, 193.
(49) Bowen, K. H.; Leopold, K. R.; Chance. K. V.; Klemperer, W. J . Chem. Phys. 1980, 73, 137. (50) (a) Campbell, E. J.; Read, W. G.; Shea, J . A. Mol. Phys. 1984.51, 1493. (b) Lovejoy, C. M.; Nesbitt, D. J . Chem. Phys. Leu. 1988, 147,490. (51) Balk T. J.; Campbell, E. J.; Keenan, M. R.; Flygare, W. H. J . Chem. _Phys. 1980. 72, 922. . (52) Keenan, M. R.; Buxton. L. W.; Campbell, E. J.; Balle, T. J.; Flygare, W. H. J . Chem. Phys. 1980, 73, 3523.
The Journal of Physical Chemistry, Vol. 95, No. 20, 1991 7609
Structures of Gas-Phase van der Waals Dimers
TABLE 11: Summary of Experimental Results a d of Derived Values for Pseudodiatomic Force Constant f, Well Depth c, 8d Attractive E E,, for V8riOUS " e m Containing a R8n-Crs Atom dimer B, MHz Dj, kHz h,N m-l R,, A c, cm-l -Eattr, au ref 3.977 58.7 0.000 778 36 . .. Ar-H2S 1725.45' 46.2 0.53 0.001 159 33 90.6 4.130 Ar-HBr 1106.67 12.40 0.76 ~~~
Ar-HF
Ar-H3'CI Ar-0,
Ar-OCS Ar-CO, Ar-CH,CI
Ar-CICN Ar-N20 Ar-BF3 Ar-F,CO Ar-PF3 Ar-oxirane Ar-furan Ar-pyridine Ar-pyrrole Ar-CIF Ar-S03 Ne-H35CI *'Kr-H"CI '29Xe-H35CI
m
~
3065.71 1678.5 1 183 1.29' 1368.49' 183 1.09' 1507.01' 1422.76' 1873.34' 1537.26 1523.52" 987.81' 1 7 1 6.1 4' 1361.45" 1189.78' 1355.39' 1327.11 1449.09
70.90 20.33 19.39 9.90 18.554 12.16 9.700 16.97 9.59 9.138 3.540 1 1.722 5.29 3.447 4.75 4.66 3.6
1.41 1.15 1.70 1.51 1.74 1.57 1.74 2.02 2.12 2.19 1.66 2.19 2.68 2.80 2.92 2.77 5.06
3.499 3.972 3.42 3.706 3.505 3.183 3.649 3.460 3.325 3.402 3.953 3.607 3.54 3.545 3.553 3.90 3.350
120.4 126.3 139.0 145.4 149.3 157.5 161.9 169.4 163.6 177.4 180.9 199.5 234.5 245.8 257.5 294.5 397.3
0.001 020 0.001 034 0.001 503 0.001 529 0.001 391 0.001 259 0.001 450 0.001 430 0.002 290 0.001 943 0.001 903 0.001 614 0.002 236 0.002 326 0.002 226 0.001 831 0.003 242
2732.18 1209.3 1 994.14
207.0 7.431 3.813
0.33 1.58 1.90
3.794 4.083 4.246
33.3 184.0 239.2
0.000 351 0.001 281 0.001 577
37 38 39 40 41 16 42 43 44
11 18 45 46 47 17 48 49 50
51 52
' ( B + c),I2.
should, under the conditions of one attractive term (or a constant ratio between attractive terms), lead only to a change in the gradient of a plot against c. The primary experimental data and the derived numerical results are summarized in Table 11. The experiment and the calculation are also compared graphically in Figures 5 and 6 in the form of plots of E,tt, against e for two series of rare-gas dimers. Figure 5 presents plots for one simple molecule, HCI, complexed to a series of rare-gas atoms. For these dimers, in addition to e from eqs 9-1 1, well depths e(Oo) from fitted V(R,B)potentials are also availableP5 The comparison shows that, a t least for this type of dimer, the pseudodiatomic e consistently underestimates the well depth and is nearer Doas, for example, for Ar-HCI,' Do= 1 1 7.8 cm-', De = 174.7 cm-l, and c = 126.3 cm-I. Both plots in Figure 5 show that changing the rare-gas atom is modeled satisfactorily enough, and the variation in binding energy is found to closely follow the changes in In Figure 6 a comparison for most of the known dimers of Ar with a diverse range of molecules is presented. Some dimers in which the established high degree of nonrigidity may affect D,, namely Ar-HCN,', Ar-H20,13 and Ar-HCCHM have been excluded, as well as A P N H , ~where ~ the combination of almost free rotation and inversion has not yet allowed the determination of minima in the angular geometry. Furthermore, dimers with skew-type structures have also been excluded as it is known from hydrogen-bonded dimers that the diatomic and related approximations perform poorly in cases where the mean interaction axis is not coincident with R,,.56 The point for Ar-C1F4* is added for comparison-in this dimer stronger bonding than that arising from only the dispersive interaction was stipulated, and this case will be discussed in more detail later. Figure 6 establishes a clear linear dependence between the quantities on both axes, which is especially evident if the point for Ar-BF3 is neglected. In view of uncertainties in the various assumptions made above the plot in Figure 6 shows that the multicenter approach adopted for modeling the dispersive interaction is performing better than has been anticipated*O and is a good prognostic of the model's selectivity. The deviation from the origin of the intercept in the fitted linear dependence in Figure 6 is probably due to the reasons discussed in connection with Figure 5 , namely the systematic underestimate
of dimer energy by the pseudodiatomic well depth e. The second and crucial test for the target application is whether the full model, with the added hard-shell repulsive term, reproduces the principal features of the experimental geometries. The results of applying eq 5 to known dimers are compared with experimentally determined structural parameters in Table 111 for linear, T-type, and pyramidal dimers, and in Table IV for skew-type dimers. The assumptions summarized in Table I have been rigidly adhered to, the only difference being that the van der Waals radius of 2 A has been used for argon, as the more usual radius of 1.88 A consistently underestimates center-of-mass separations. With the single exception of Ar-PF, (discussed below), the model structures in Tables 111 and IV are those of highest stability. It can be seen that the comparison with experiment is excellent for T-type, pyramidal, and skew-type geometries. The model not only correctly identifies the structural type but is also rather successful at reproducing the angular information. This is particularly so for the skew-type geometries where the model selects the correct cavity and reproduces the Ar-cavity atom distances to 0.1-0.2 A. For most dimers the difference in Ei,, calculated for the first and the second most strongly bound geometry is more than 10%. The lowest differences are for the skew-type dimers Ar-acrylonitrile (5.3%), Ar-formamide (7.3%), and Ar-difluoroethylene (8.8%), indicating that the discrimination margin below which all predicted geometries will have to be treated as equally probable may be below this level. The performance of the model for other rare gases is similar to that for Ar, as the behavior for rg-HCI shown in Figure 5 is typical, in that changing the type of rg changes the binding energy but does not alter the geometry. An interesting exception to this for dimers involving PF3 is discussed below. Some failures of the model are also apparent and offer an opportunity to obtain a more thorough understanding of its features. The center-of-mass separations R, are primarily a function of the assumed van der Waals shells and in some cases, such as Ar-SO3, point to considerable inadequacies in the repulsive shells. The inability of the model to resolve the two angular solutions for Ar-N20 is probably due to the same reason. There is also the class of linear dimers where the model fails for all members, predicting appreciably bent structures. These dimers are clearly established to have linear equilibrium geometries, and even though the average angles in the ground state are considerable and not far from those from the model, such angles arise entirely from the vibrational average for wave functions associated with highly anharmonic vibrations. Since it has already been shown for Ar + HCI that, even without induction, present multicenter modeling
arg/em.
(53) Lwpold. K. R.;Fraser, G. T.; Lin, F. J.; Nelson, D. D.; Klemperer, W . J . Chem. Phys. 198481, 4922. (54) Ohshima, Y.;lida, M.;Endo, Y . Chem. Phys. h i t . 1989, 161,202. (55) Nelson, D. D.; Fraser, G.T.; Peterson, K. 1.; Zhao, K.; Klemperer, W . ; Lovas, F. J.; Suenram, R. D. J . Chem. Phys. 1986,85, 5512. ( 5 6 ) Kisiel, Z.; Fowler, P.W.; Legon, A. C . J . Chem. Phys. 1990,93,3054. (57) Suenram, R. D.; Lovas, F. J. J . Chem. Phys. 1987, 87, 4441.
7610 The Journal of Physical Chemistry, Vol. 95, No. 20, 1991
Kisiel
TABLE III: Camp" of Observed and Predkted Geometries for h r , T-Type, and Pyramidd Dimers
obsd
R,, A
X
Ar-HF Ar-H3JCI Ar-HBr Ar-HCN Ar-CIF
3.499 3.972 4.130 4.343 3.90
H H H H CI
Ar-OCS Ar-C02 Ar-CICN Ar-N20 Ar-CH,CI Ar-PF, Ar-HCCH
3.706 3.505 3.649 3.460 3.783 3.953 4.04
0 0
Ar-H2S
3.977 3.42 3.402 3.607 3.54 3.545 3.553 3.350
S
dimer
8:
calcd
&.b deg
deg Linear
41 41.3 42.1 30.8 8.7
0 0 0 0 0
T-Type
74.17 90 82.40 82.6, 97.4 82.5, 97.5 70.3 90
N 0
CI P C
R,. A
4 deg
3.524 3.790 3.893 3.816 4.136
68.1 57.9 55.3 62.0 128.5
3.746 3.633 3.609 3.561 3.705 3.929 3.771
78.3 90 85.1 90.1 96.8 76.9F 76.9
Pyramidal
Ar-0,
Ar-F2C0 Ar-oxirane Ar-furan Ar-pyridine Ar-pyrrole Ar-SO,
100.8, 144.9
0 0 0
N N S
143.3 90 114.3 77.6 81.0 87.9 84.3 89.1
3.8 13 3.648 3.589 3.667 3.587 3.571 3.603 3.769
90 100.3 113.1, 73.3d 78.6: 101.4 86.5 84.5 90
0 2
O B = L(Ar-cm-X). bAveragevalue for 8 derived from (cos 8 ) or (cos28) from dipole moments and spin-spin or hyperfine coupling constants. CNotthe most strongly bound conformer; see text. dPreferred value; see text.
TABLE I V Comprrison of Obswved and Predicted Geometries for Skew-Type Mmers
dimer Ar- H 2NCH0
atoms defining acceptor cavity, XYZWP HNCO
Ar-H,CCHCN
HCCCN
Ar-H,CCHF
HCCF
Ar-H2CCF2
FC F
Ar-HFCCF2
FCCF
Ar-X obsdIs calcd obsd" calcd obsdI4 calcd obsdl' calcd obsd" calcd
3.24 3.30 3.22 3.30 3.49 3.30 3.55 3.55 3.53 3.54
Ar-cavity atom dist, A Ar-Z Ar-W
Ar-Y 3.78 3.62 3.91 3.73 3.94 3.75 3.56 3.66 3.75 3.82
3.93 3.78 4.18 3.90 3.95 3.88 3.55 3.55 3.71 3.81
Ar-P
3.56 3.66 3.74 3.64 3.46 3.54
3.77 3.83
3.45 3.54
deg3.625 3.549 3.653 3.516 3.620 3.571 3.514 3.592 3.546 3.634
42.3 58.9 36.9 54.1 48.2 50.6 73.0 65.2 60.6 58.8
2.44 3.04 2.19 2.85 2.70 2.76 3.36 3.26 3.09 3.11
'Angle between Ar-cm axis and plane of acceptor molecule. bPerpendiculardistance from Ar to molecular plane.
of dispersion predicts minima at linear configurations, it is the repulsive term that will now have to be considered in more detail. The hard-sphere assumptions of Table I are immediately seen to be inadequate, since for the first four linear dimers in Table 111 the vdW radius of the hydrogen would have to be reduced to ca 0.7 A in order for the experimental R,, to be reproduced. This is illustrated by the effective repulsive contour for HCI in Figure 7a, which was derived from the minimum-energy path R,(B) on the best experimental potential for Ar-HC13' by subtracting the effective vdW radius for Ar (here 2 A). The appropriate contour on the electron density diagram for HCls8 has a very similar shape. For HF, inspection of the published electron density plot59suggests that, in order to reproduce the experimental &,in A r H F , contact with Ar will have to be at the 0.02e electron density contour. The repulsive surface defined by this contour can be approximated along the lines adopted in ref 35 by R v d W ( e ) / A = 1.5 - 0.2511 - [1/2(1 + COS Q4]] - O.lg(0) (12)
e < 900 g(e) = cos2 e e > 900 g(e) =
o
(58) Cadc, P. E.; Bader, R. F. W.; Henneker, W. H.; Keaveny, 1. J. Chem. phys. i969, so, 53 I 3. ( 5 9 ) Bader. R. F. W.: Keavenv, 1.: Cadc. P.E. J. Chem. Phvs. 1967.47, 3381.
i
t
0
i
90
180
e
Figure 7. (a) Repulsive surface for HCI derived from the potential for Ar HCI," with characteristic distances marked in A; (b) Repulsive surface for H F derived by approximating the appropriate electron density contour; (c) Dispersive attractive energy E,,,, and inductive energy uid for Ar + HF along the contact with the repulsive surface for HF in (b).
+
The second term in eq 12 determines the transition between the hydrogen and the fluorine part of the contour and the third term simulates the flattening at the F end opposite the hydrogen. With this surface (Figure 7b) the model of eq 5 now yields minima in E,,,, at both linear configurations, as shown in Figure 7c. The inductive term has also been calculated with eq 6 and the distributed multipole for HF3* and, since for 0 = Oo the total at-
Structures of Gas-Phase van der Waals Dimers tractive term in the fitted potentia120is -286 cm-', E,, and qnd are seen to be nearly additive. The behavior previously noted for Ar + HCI is visible in that dispersion on its own favors the reversed linear configuration, and it is due to the large anisotropy in induction that the balance is tipped in favor of the Ar-HF minimum. Comparison between Figure 4 and Figure 7c shows that the anisotropy in dispersion increases with the size of the hydrogen halide, whereas induction decreases and in this light Ar + HBr would be of interest. This could be a turnover point where the experimentally observed structure would be predictable at only the dispersive level. The trend from Ar-HF to Ar-HCl shows that the two linear minima may also be much closer in depth, making the system an interesting candidate for infrared studies, and in fact a preliminary potential for Ar-HBr fitted by Hutson to only the molecular beam microwave data confirms this.60 The conclusion from the above is that the modeling of repulsion is probably a weaker link in the present model than the relatively crudely modeled dispersive attraction. This becomes less unexpected if we consider that the dispersive interaction with a leading Rb term varies much more sharply with distance than is the case for electrostatics where the expansion for the energy effectively begins with the dipole-dipole R3term and where for this reason the shape of the repulsive shell has not been found to be so critical. Unfortunately, even though distortion in the vdW shell such as the reduction in the effective vdW hydrogen radius in acidic H X molecules is in line with anticipations from chemical intuition and electron density plots, effects of this type are difficult to build into the model on the simple lines used here. In addition the inductive term may sometimes be of importance, and this appears to be due more to its anisotropy than to magnitude.
Selected Dimers In the preceding section, discussion of some discrepancies between the model and the experimental results has been omitted. For most of these cases, the model is found to usefully contribute to the interpretation of the results, and such dimers will now be considered in more detail. AI-PF~ Observed rotational constants lead to four numerically equivalent solutions in all of which the C3axis of PF3is at an angle of 70.3O to The authors select one of these, a structure where Ar is over a PFF face at 8, = 70.3O (Table 111), by eliminating two geometries on the basis of unreasonable Ar-molecule atom distances and carrying out an a b initio calculation to disqualify the third.18a The model predicts only one T-type structure with Op = 76.9O. This is very close to the preferred experimental geometry and demonstrates the usefulness of the model for differentiating between numerically equivalent solutions. However the model also predicts that a symmetric top pyramidal structure with Ar below the FFF face will have a 17% greater binding energy. Since the PFF-bound geometry is already observed experimentally, the FFF geometry is likely to be more weakly bound since only the most strongly bound isomer is normally observed owing to very sharp Boltzmann selection of dimer populations. For example, if dimer populations are not frozen out from some higher temperature, but can be described by an effective formation temperature comparable to the rotational temperature of, say, 2 K then the population ratio for two structural isomers with AE = 0.1 kJ mol-' = 8.3 cm-l will be 1 :400. There are grounds for suspecting that sizable and anisotropic inductive effects may be responsible for the failure of the model in correctly ordering the structural isomers in energy, as PF3 has a rather large molecular quadrupole 8 = 24.1 DA. Unfortunately, molecular multipole expansion fails to rationalize the measured induced dipoles,'8band satisfactory parametrization of electric charge distribution in PF3 is still to be derived. Interestingly, while a similar discrepancy between an experimental PFF-bonded configuration and the predicted FFF pyramidal configuration also takes place for KrPF3,Iabthe experiment and the model are in agreement for NePF3, which is found to be pyramidaL6I This lends further support for the balance tipping contribution from induction, as the field (60) Hutson. J. M. J. Chem. Phys. 1989, 91, 4455.
The Journal of Physical Chemistry, Vol. 95, No. 20, 1991 7611
c
-n n
'E 3 \
.c W
-100
I
0
/
I
I 1
90
180
e /@ Figure 8. Interaction potential for the dimer between Ar and acetylene calculated with eq 5.
around PF3 is much larger in the PFF direction than below the FFF Thus the large increase in polarizability from N e to Ar may be sufficient to shift the preference from the pyramidal to the PFF-bonded structure, as in the model the difference in Ehtbetween the two structures increases rather slowly in the series Ne, Ar, Kr. Ar-Oxirane. For this dimer the model predicted that Ar is at Bo = 77.9O, i.e. displaced toward the oxygen atom from the perpendicular to the COC plane a t the cm. This was in disagreement with the originally derived in which Ar is over the C-C bond at 0, = 113.1O. The discrepancy led to a reconsideration of the experimental results from which it turned out that a more satisfactory fit to the data is provided by a eo = 73.3', geometry with Ar displaced towards the consistent with the model. Ar-Furan. In the first communication regarding this dimer,& it was not possible to decide purely on the basis of the available evidence whether the Ar-cm axis is tilted toward or away from the oxygen atom. The tilt toward the oxygen atom was later confirmed by isotopic substitution!6b The model also predicts a tilt toward the oxygen, and a tilt toward a heteroatom is also apparent from results for Ar-pyrrole1' and A r - ~ y r i d i n e . ~This ~ leads to a simple rationalization that in a ring molecule consisting of atoms of comparable polarizability the rg atom will be displaced toward the smallest heavy atom in the ring. Ar-CH3Cl. The rotational constants for this dimer lead to a T-type structure with BCl = 8 2 S 0 or 97.5' (Table III), Le. with Ar displaced either toward the chlorine or toward the methyl group. The conclusion, reached on the basis of somewhat better consistency between Ar-CI distances for the 35Cland 37Clspecies of the dimer, was that the tilt toward the chlorine was to be preferred.I6 The model predicts the r e v e r s e A r displaced toward the carbon, which allows for vdW contact between Ar and both the methyl group and the chlorine atom. This also leads to the possibility of a nonzero 3-fold internal rotation barrier as implied by the experimental results, although the hard-sphere repulsion built into the model precludes an estimate of barrier height. In fact, further consideration of the experimental rotational constants6*shows that on subtraction of both harmonic vibrationrotation contributions and internal rotation contributions, these lead to the geometry with Ar tilted toward the methyl group. This suggests that further experimental data may be required to unambiguously fix the geometry of this dimer. Ar-HCCH. The conclusion from the latest R M W is that this dimer has a T-type geometry with an anomalously large center of mass separation, R,, = 4.04 A. In addition, a high degree of nonrigidity is suggested by the very large effective value (61) Hillig, K. W., 11; LaBarge, M.S.;Telab-Bendiab. A.; Kuczkowski, R. L. Chem. Phys. Lett. 1990, 171, 542. (62) Kisiel, 2. Unpublished results.
7612 The Journal of Physical Chemistry, Vol. 95, No. 20, 1991
of DIK = 21 MHz. The model was used to calculate the position of the minimum on the V(R,8) potential as shown in Figure 8, which confirms the general preference for a T-type geometry. The fine detail in the shape of the minimum is an artifact of the assumed van der Waals shell but the very shallow nature of the minimum up to R = 4 A, goes some way toward understanding the unusual experimental result. Ar-ClF. The experimental structure for this dimer has rather unexpectedly turned out to be linear with Ar-CIF atom ordering and a very short Ar-CI distance of 3.33 A.48Even if such close proximity is the result of appreciable repulsive contour flattening at the CI end of CIF, it is seen from Figure 6 that Eatt, underestimates the interaction energy by about The dipole and the quadrupole moment for CIF are each less than half of those for HF, and although uintis not particularly large, calculated with distributed multipolessb to be 32 cm-I, it goes some way toward making up the difference. The remaining energy may be provided by the higher order, R7and R8,dispersive terms, which will increase in importance on such close approach between Ar and Cl, and thus possibly no non-van der Waals type interactiona may have to be brought in. This is, however, the only dimer of those considered here for which the dispersive model a t the Rd level, even when augmented by induction, seems inadequate. A number of rg-containing dimers are known to be currently under studyt3 and it is tempting to use the model to predict their structures. Thus for Ar-CH,CHCI a skew-type geometry analogous to that for Ar-CH2CHF is predicted, with Ar over the cavity defined by HCCCl at an R,, of 3.63 A and at an out of plane angle of 34O. For Ar-ketene, the lowest energy structure is predicted to be planar with R , = 3.67 A and L(Ar-cm-0) = 87O. In addition, the barrier to rotation over the ketene plane is predicted to be very low, suggesting the possibility of internal rotation. For Ar-cyclopentadiene a pyramidal geometry is predicted with the Ar atom positioned almost exactly over the center of mass of cyclopentadiene. Finally, for Ar-CH3CN, an analogous situation to that for Ar-CH3CI is encountered where the model favors a T-type geometry with an Ar-cm-N angle of 92.4O over that with angle of 85.1' by about 10% in Eint. This prediction is, however, subject to increased uncertainty caused by omission of induction from the highly polar nitrile group and also by inaccuracies in representing the repulsive surface of this group.
Conclusions A simple model for dimer formation between a rare-gas atom (63) Schwendeman, R. H.; Nygaard, L. Microwave Spectroscopy Newsletrer; Michigan State University, 1990; No. XXXIII.
Kisiel and a molecule has been developed by seeking to reproduce the geometrical anisotropy of the dispersive interaction. The model is found to be successful for dimers that have T-type, pyramidal, and skew geometries. The model fails for linear dimers, most of which involve molecules with acidic hydrogens. The failure is identified to be due to the inadequacy of modeling the short-range repulsive potential with simple van der Waals shells, which is a t present the model's most serious deficiency. It is demonstrated that a significantly higher predictive accuracy, particularly in the more difficult cases, can be obtained if the repulsive potential can be based on parametrization of a calculated total electron density profile for the acceptor molecule. Another extension would be the inclusion of induction, which can readily be achieved with multiples from distributed-multipole analysis.32 The success of the model suggests that a simple generalization concerning the structures of rg-containing dimers is possible. This is that the preferred geometry will be that in which the raregas atom contacts the vdW spheres of the largest number of atoms, contacts with heavy atoms being preferred over those with hydrogen. An exception to the rule will be for cases where the vdW surface of the molecule is strongly distorted, and/or strong and anisotropic inductive effects are present. The sometimes quoted statement that the rg atom goes for the most electropositive site in the acceptor molecule" is seen to be true only in the phenomenological and not in the causative sense, as it is now reinterpreted to refer to the point of intersection of the resultant of the dispersive forces with the molecule. The present model may, with careful scaling, be combined with the electrostatic modelS to provide a more accurate description of the dispersive part of the interaction in dimers between two nonpolar molecules such as N2-C02 or C02-C2H4, which are increasingly the subject of high-resolution infrared studies. The ease of applicability may also make the model an attractive add-on term to the various 'desktop" molecular modeling packages for enhancing their reliability in predicting spatial configurations of molecules. A FORTRAN listing of the minimizing program using variable metric minimizationu is available from the author upon request.
Acknowledgment. The author is indebted to Patrick Fowler (Exeter) for introducing him to various aspects of multicenter modeling of intermolecular interactions and for helpful discussions. Financial support from the Research Project CPBP-01.12 and the Institute of Physics Research Project 14 are also acknowledged. (64) William, H. P.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W . T. Numerical Recipes; Cambridge University Press: New York, 1986.
