10892
J. Phys. Chem. 1996, 100, 10892-10911
Identifying and Analyzing Intermolecular Bonding Interactions in van der Waals Molecules† Richard G. A. Bone*,‡ and Richard F. W. Bader Department of Chemistry, McMaster UniVersity, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4M1 ReceiVed: NoVember 28, 1995; In Final Form: April 10, 1996X
Atomic interaction lines and (3,-1) critical points were located in the intermoleclar regions of a total of 36 configurations of 11 van der Waals dimers and a trimer, none of which contain a hydrogen bond. The complexes comprised combinations of five moieties: argon atoms and the molecules C2H2, CO2, OCS, and SO2. Atomic interaction lines in the intermolecular region denote the dominant atom-atom interactions underlying the weak interactions. The set of complexes allows us to draw conclusions about the role of the intermolecular charge density in binding. We find that the values of the charge density and its principal axes of curvature at intermolecular critical points in these systems are an order of magnitude less than those found in hydrogen-bonded systems and that the (3,-1) critical points display the hallmarks of “closed-shell” interactions. The strong correlation between the value of F at a bond critical point and the binding energy of the complex that has been found previously for hydrogen-bonded systems is harder to expose with the van der Waals molecules presented here. For example, comparison must be restricted to structures in which the same pairs of atoms are interacting. We find that “conflict structures” may be associated with both equilibrium and transition state conformations but that significant nonrigidity is usually a consequence. We are able to make some predictions concerning the likely occurrence of intermolecular critical points in systems larger than those described here.
Introduction The concept of a chemical bond has allowed chemists to both rationalize and predict the structures of countless molecular species.1 It is infrequent for there to be any doubt about whether two atoms are actually bonded to one another or not, but when such cases arise, it is necessary to resort to theoretical methods. Currently, the longer range forces between distinct, “separated” molecules are the subject of extensive study. Opportunities for isolating clusters of formally discrete molecules were at one time scarce, but the advent of molecular beam technologies has enabled experimentalists to create and study weakly bound complexes of “closed-shell” molecules relatively easily.2,3 Hydrogen-bonded and “van der Waals” molecules are important because they offer a window on the pair-potentials which govern fluid behavior.4 Their structures and dynamics have traditionally and successfully been understood with the long-range theories of intermolecular forces5,6 in which it is assumed that each monomer retains its structural and chemical integrity. Nevertheless, with the exception of hydrogen bonds, the question of whether the interactions between atoms on separate monomers could be called chemical bonds has not been answered in the general case. The classification of bonding interactions is a diverse subject, but most models have invoked the sharing of electrons between atoms as a crucial consideration. By contrast, the theory of atoms in molecules (AIM)7 offers a self-consistent way of partitioning any system whose (ab initio) wave function can be computed or whose electron density distribution may be measured into its atomic fragments, on the basis of a consideration of the gradient vector field of its charge density, F.8 From a topological analysis, features such as critical points and † This work formed the basis for an oral presentation at the 208th ACS Meeting, August 1994, Washington, DC. ‡ Current address: Terrapin Technologies Inc., 750-H Gateway Blvd., South San Francisco, CA 94080-7020. X Abstract published in AdVance ACS Abstracts, June 1, 1996.
S0022-3654(95)03512-X CCC: $12.00
paths of maximum electron density (“atomic interaction lines”9) between nuclei and/or critical points emerge.10 In this way, AIM leads to a “molecular graph” which is a representation of the bonding interactions and which may be viewed as a description of “structure”. AIM is universally applicable11 and has been immensely successful in rationalizing the chemical structures of single isolated molecules (including those with challenging bonding environments, such as the boranes12). In the overwhelming majority of cases, atomic interaction lines and associated critical points are found between pairs of atoms which would typically be called “chemically bound”. Well-ingrained concepts of chemical intuition have thus found a mathematical underpinning in the topology of the charge density. But AIM offers qualitative interpretations in cases where the existence of a bonding interaction is the subject of doubt. Crystal structures often provide the strongest experimental evidence for significant bonding interactions where, traditionally, short contact separations have proved to be useful indicators. Atomic interaction lines have been located in the solid state,13 for both metallic structures14 and molecular crystals.15,16 In such cases, AIM is able to highlight the dominant atom-atom interactions which contribute to weak binding or to the composite structure. For example, the structure of crystalline Cl2 displays a number of atomic interaction lines between Cl atoms which are on different molecules15 but whose contact separation is somewhat smaller than the sum of van der Waals radii. The presence of the principal intermolecular interaction lines and the directional property they impart on the interaction account for the layered structure of solid chlorine.15 Price et al.17 in a recent analysis of Cl‚‚‚Cl interactions conclude that there is little evidence for preferred orientations being attributable to charge transfer. They argue that the apparent need for a specific Cl‚‚‚Cl attractive force can be avoided by using an anisotropic model of the Cl atoms. © 1996 American Chemical Society
Intermolecular Bonding in van der Waals Molecules Similar arguments justify an electrostatic model of the hydrogen bond.18 Nevertheless, the hydrogen bond is also manifested in the charge density by an atomic interaction line between the proton and the acceptor atom in a number of systems.19-22 Consequently, we expect that other long-range atom-atom interactions, although less pronounced, may find similar expression. Cioslowski et al. looked at other weakly bound complexes including benzene-tetracyanoethylene and Ne@C6023 and found a number of conformationally dependent intermolecular critical points. They also looked at long-range intramolecular interactions23-26 in sterically crowded molecules. They focused primarily on relationships between pairs of hydrogens,25 but also found atomic interaction lines between O-O, F-F, H-N, and H-F pairs,24 which would usually be described as “nonbonded” but which have short contact distances. Therefore, there is considerable precedent for finding longrange bonding interactions between atoms other than hydrogen, not only within single molecules but in complexes and extended systems. In justifying our work, we argue that a large number of important complexes do not contain H-bonding species, yet may contain equally interesting bonding topologies. It is not always possible, a priori, to predict just which pair of atoms in a complex will be joined by atomic interaction lines. So, we broaden the range of systems studied and further investigate the possible trends. We are searching for more examples of bonding interactions in environments typically thought of as “nonbonding”, in the sense that the forces between the monomers (or noble gas atoms) within a van der Waals complex have come to be regarded as weak and nondirectional and such that individual pairwise interactions have been considered to have no bonding character. In this paper, many configurations of 11 van der Waals complexes containing argon, CO2, C2H2, OCS, and SO2 are analyzed and intermolecular bonding interactions are found in a number of interesting and surprising circumstances. We are careful to make no use of the word “prediction” in this context. Rather, we are searching for patterns in systems which have already been described by calculation and are reasonably wellunderstood experimentally. We aim to elucidate any distinguishing features of such systems. For example, can stability be related to the number of intermolecular atomic interaction lines? Do we find that atomic interaction lines are prevalent between certain pairs of elements, and are they quantified in comparable ways? From this we hope to be able to generalize and perhaps to extrapolate to other complexes whose structures are known but whose bonding topology remains unknown, and even to complexes whose structures are as yet undetermined. Theory In this section we present a brief description of the theory of atoms in molecules7 that is necessary to understand the results for van der Waals systems which we present in the main body of the paper. Characteristics of Critical Points. Trajectories of the gradient vector field of the charge density, ∇F, are our device for exploring the three-dimensional topology of F;27 critical points, at which ∇F vanishes, are our key to understanding its structure. Each critical point is quantified by its three principal curvatures (eigenvalues of the Hessian matrix), λ1-3, and characterized by the ordered pair of integers, (r,ω), designating respectively its rank (number of nonzero eigenvalues) and signature (number of positive eigenvalues minus the number of negative eigenvalues). The positions of nuclei coincide with maxima, i.e., (3,-3) critical points in F. Two paths originate and ascend in opposite
J. Phys. Chem., Vol. 100, No. 26, 1996 10893 directions from (3,-1) critical points. These paths typically terminate at nearby nuclei and together comprise an “atomic interaction line”, a line of maximum electron density. In an equilibrium structure, such an atomic interaction line is referred to as a “bond path” and the (3,-1) critical point is termed a “bond critical point” (BCP).28 Their presence meets the necessary and sufficient conditions that the two atoms are bonded to one another;7 the network of bond paths defines the molecular graph. The fact that bond paths coincide with traditional chemical bonds in almost all cases lends weight to chemical interpretations of the charge density. Pairs of atoms which are linked by bond paths are thus readily identified to be those whose interaction dominates the forces which hold the structure together. The value of F at a BCP, Fb, gives a loose indication of bond strength.7 Boyd and Choi found linear relationships between the value of F at the H-bond critical point and the H-bond strength21 as well as the intermolecular bond length22 for a set of complexes between organic nitriles and hydrogen halides. Infinitely many trajectories descend from a BCP and define an interatomic surface (which in general is not planar), such that the condition ∇F‚n ) 0 is obeyed everywhere in it (n is a vector normal to the surface at a given point). The two negative eigenvalues, λ1,2, quantify curvatures in the interatomic surface, “transverse” to the atomic interaction line, whereas the positive λ3 denotes the “longitudinal” curvature along the atomic interaction line itself. Consequently, a pair of atoms which share an interatomic surface are also linked by an atomic interaction line which intersects that surface perpendicularly at a (3,-1) critical point. The molecular graph may be regarded as a set of primary attractors (the nuclei) linked by secondary attractors (the BCPs). Whereas the location and number of these features are dictated by the physics of the system, occasions demand the existence of “ring” (or (3,+1)) critical points for which one curvature is negative and two positive. Two oppositely oriented trajectories descend perpendicular to the plane of the ring. (Additionally, “cage” (or (3,+3)) critical points which are minima found in cavities surrounded by three or more rings, may also arise but are not encountered in this study.) It was pointed out by Collard and Hall29 that the number of critical points of all types is governed by a fundamental theorem of topology, the Poincare´Hopf relationship,
n-b+r-c)1
(1)
where n is the number of nuclei and b, r, and c are the numbers of bond, ring, and cage critical points, respectively. Sometimes, as a result of this counting rule (in concert with point-group symmetry, if present), it becomes inevitable that certain critical points must exist. In particular, the action of bringing two molecules together to form a complex demands the existence of at least one new “intermolecular” BCP. Furthermore, for any “intramolecular” rearrangement in which one net extra BCP arises, a corresponding ring critical point must also be created. Conversely, atomic interaction lines in formally “intermolecular regions” persist as a complex is split into infinitely separated monomer units, whereas atomic interaction lines between sterically hindered atoms (in intramolecular environments) may change in nature (and may cease to exist or may be replaced by others) under certain conformational changes.24,25 It should be stressed, though, that the topology of F is not dictated by geometry alone; given only a set of primary attractors (nuclei), there is no requirement that any scalar field should adopt the form that F does. In fact there is a homeomorphism between F and the virial field.30 A molecular graph is mirrored
10894 J. Phys. Chem., Vol. 100, No. 26, 1996
Bone and Bader
by a corresponding virial graph in such a way that lines of maximum density linking bonded nuclei are matched by lines of maximally negative potential energy density. Laplacians. The nature of BCPs in H-bonded, van der Waals, and ionic species needs to be reconciled with that of their counterparts in “regular” molecules. Bader and Esse´n made the first comprehensive study of the observed categories of BCPs28 and uncovered the distinction between “shared” and “closed-shell” interactions. They concluded that the hallmark of “shared” (i.e., covalent) interactions is a high value of the charge density at the BCP, Fb (>10-1 au). In addition, the curvatures of the charge density are usually large. The Laplacian, ∇2Fb, may be positive or negative and is usually of the same order of magnitude as F. A negative Laplacian denotes a perpendicular contraction of charge toward the atomic interaction line which is more significant than the longitudinal depletion of charge away from the BCP, toward the nuclei. This is in accord with one’s intuitive understanding of the nature of chemical bonding in regular molecules. A positive Laplacian may occur for polar covalent bonds, e.g., the double bonds between C and O or S. By contrast, in hydrogen-bonded systems, noble-gas dimers, and also ionic systems, Fb is quite small (∼10-2 au or less and ∼10-3 au in van der Waals complexes7,19,21,25) and the Laplacian is positive. These two observations appear to have no exceptions and are indicative of a “closed-shell” interaction. Subsequent comprehensive studies of H-bonded systems by Carroll et al. verified this description and added the observation that Fb was only very slightly greater than the sum of unperturbed monomer densities, ∑Fb, at that point. We conclude that in the weaker interactions between closed-shell molecules the longitudinal curvature is such as to squeeze electron density away from the BCP much more dramatically than the concomitant radial compression of charge toward it. There is an energetic interpretation of these observations which derives from the local statement of the virial theorem,
p2 2 ∇ F(r) ) 2G(r) + V(r) 4m
(2)
which relates the Laplacian to the values of the potential and kinetic energy densities, V(r) and G(r), respectively. Regions where the Laplacian is negative are such that the potential energy is dominant (in magnitude), whereas regions where it is positive are such that the kinetic energy dominates. This corresponds to the idea that normal molecules are bound together as a consequence of increased electronic potential energy in the interatomic binding region. Conversely, interactions between closed-shell species suffer from Pauli exclusion of electrons from those very regions, and instead the potential energy is separately increased within one or both of the atomic basins. Cremer and co-workers31 have advocated use of the total energy density, H(r), to complement the Laplacian in the analysis of bond types. H(r) is given by
H(r) ) G(r) + V(r)
(3)
and its value at a BCP, denoted by Hb, is negative for “shared interactions” and positive for “closed-shell interactions”.32 Unlike the Laplacian, whose sign is determined by the local virial expression, eq 2, the sign of Hb is determined by the energy density itself and therefore is negative for all interactions which result from an accumulation of electron density at the BCP. Therefore, terms such as “covalent” and its converse, “nonbonding”, which prevail in the chemical literature, find both qualitative and numerical expression within AIM as “shared”
and “closed-shell” interactions, respectively.28 Hence, we are able to accommodate intermolecular bond paths between atoms in van der Waals molecules in a more general classification scheme. We can acknowledge that they denote significant atom-atom interactions while differentiating them from the intramolecular bonds which define the structure of the individual monomers. Chemical Change. The proximities of partner monomers in van der Waals molecules are comparable to those of the precursors of chemical reactions. Previous investigation of chemical reactivity demonstrated that the morphogenesis of the charge density as species approached one another could be modeled by catastrophe theory.29,33 Both “bifurcation catastrophies” and “conflict structures” signify a transition from one bonding re´gime to another. Catastrophes are symptomatic of intermediate molecular separation and herald the coalescence of a ring critical point with one of its neighboring BCPs in a singularity, i.e., where the electron density is locally very flat. In this case, the axis of soft curvature itself may be indicative of structural instability. From the study of many examples, it has emerged that the geometry at which catastrophes in the charge density are observed does not necessarily correspond exactly to transition states on potential energy surfaces. Conflict structures arise at larger separations and indicate that an infinitesimal change in geometry may cause a bond path to switch from one nucleus to another; at the conflict structure itself, it is usual for the “atomic interaction line” to connect a nucleus with an orthogonally disposed BCP instead of another nucleus. Again, at these geometries, the electron density is usually very flat in the region of the catastrophe. Hitherto it has been held that a conflict structure corresponds to an unstable or transition state structure, possibly because it is energetically favorable for a “bond path” to relax to either of two nuclei. This may well be the case for “regular” molecules, but in van der Waals complexes, where the forces involved are qualitatively different, we will show that this may not hold true in all circumstances. Description of Calculations Choice of Systems. We selected 11 “van der Waals” molecules as candidates for the discovery of atomic interaction lines in unusual circumstances. None of them contain “traditional” hydrogen bonds, but all are typified by “closed-shell” interactions and structures whose form is dominated by electrostatic and/or dispersion forces. This set of molecules is useful because almost all have an equilibrium conformation which is well-understood both theoretically, from prior published (ab initio) calculations, and experimentally from recent molecular beam measurements. But in addition, many of the molecules are interesting because they permit isomerization (and in several cases more than one structure has been observed). This justifies the examination of points on disparate regions of their PESs, e.g., structures which are transition states and local minima. All of the systems are sufficiently “small” that a reasonable level of theory may be used. All our calculations were carried out at the second-order Møller-Plesset (MP2) level34 with basis sets of at least “triple-ζ” in quality. In many cases, fully optimized geometries were employed. The details of calculation for each of the systems are listed in the Appendix. Our belief is that the quality of each wave function is such that calculational artifacts are effectively removed. Indeed, the levels of theory employed in this study are such that agreement with experiment is as good as can reasonably be expected at this time. Certainly we do not attach any significance to small geometric discrepancies which are attributable solely to aspects of the ab initio calculation, namely, basis set and level of theory.
Intermolecular Bonding in van der Waals Molecules
J. Phys. Chem., Vol. 100, No. 26, 1996 10895
TABLE 1: Bond Critical Point Data (au) for Monomers C2H2, CO2, OCS, and SO2 at the MP2 Level of Theory: Only the Respective Biggest Basis Sets Used for Each Are Shown (See Appendix) bond C2H2 (TZ2P+p/df) C-H CtC CO2 (TZ2P+d) CdO OCS (TZ2P) OdC CdS SO2 (TZ2P+df) SdO
Fb
32Fb
λ1 (|)
λ2 (⊥)
λ3
Hb
0.2936 0.4111
-1.1962 -1.3493
-0.8456 -0.7014
-0.8456 -0.7014
0.4950 0.0534
-0.3385 -0.6289
0.0 0.0
0.4475
0.3082
-1.1578
-1.1578
2.6235
-0.7994
0.0
0.4493 0.2320
0.5302 0.2069
-1.2119 -0.2487
-1.2119 -0.2487
2.9317 0.7071
-0.8007 -0.2699
0.0 0.0
0.3019
0.8544
-0.5822
-0.5197
1.9563
-0.3950
0.1202
We note that all previous work with the theory of atoms in molecules on weakly bound systems has employed only selfconsistent-field wave functions,19 often with quite small basis sets.23-26 The fact that we have used exclusively a correlated method gives us confidence that our conclusions are not likely to be prone to the artifacts which may arise with inadequate levels of theory. Although an earlier study showed that electron correlation did not qualitatively affect the bonding topology within a number of small molecules (and had a minimal impact on the quantitative description of BCPs,35 our belief is that the known effect of correlation on the equilibrium structures and geometries of van der Waals systems justifies its incorporation here. Procedures. The charge density obtained is a result of a “supermolecule” calculation on each molecular conformation.36 In this study, none of the calculations were explicitly corrected for “basis set superposition error” (BSSE) because we do not anticipate significant changes in the charge density itself, due to BSSE. Furthermore, there remains significant controversy over the matter of how to correct for BSSE in a correlated calculation.37 We also point out that in each of the prior ab initio investigations on the systems studied here, it was concluded that the level of theory attained was high enough to mean that BSSE would not qualitatively alter the structural picture, i.e., the topology of the potential energy surface. At the MP2 level of theory, it is not sufficient to carry out a single-point energy calculation. To obtain the “perturbed” charge density, part of a gradient calculation must be computed.38 Consequently the “orbitals” (and corresponding occupation numbers) we use are obtained by diagonalization of the perturbed one-particle density matrix. The number of these is equal to the dimension of the one-particle basis set (cf. an SCF calculation where we do not use the “virtual” orbitals on account of their zero-occupancy). Therefore, charge-density analysis is rather time consuming for these systems. The locations of the critical points of the charge density were obtained with the program SADDLE, a constituent of the wave function analysis package, AIMPAC.39 Charge-density and gradient vector plots were obtained with the program MORPHY.40 Some of our basis sets contain f-orbitals. Their importance in obtaining a good description of both monomer properties41 and dimer geometries has been demonstrated elsewhere.42 Consequently, all of the programs within AIMPAC, as well as MORPHY, were extended to handle f-basis functions. The first through fourth derivatives of these functions are available as supporting information. Results and Analysis We begin with a brief description of the bonding topology in each of the monomer molecules, before describing the argon complexes and then the clusters of C2H2, CO2, OCS, and SO2. In each of the plots of the charge density, nuclei are represented by b and the symbols 9 and 2 represent the locations of bond and ring critical points, respectively. Atomic interaction lines
are shown as bold lines, whereas charge density contours and intersections of interatomic surfaces with the plane of the plot are regular lines. Nuclei in the plane of the plot are shown in bold type, whereas those which are out of plane are projected onto the plot and shown in open face. Where there are symmetrically inequivalent forms of the same element the element symbol is followed by a number. (i) The Monomers, C2H2, CO2, OCS, and SO2. BCP data for each isolated monomer are presented in Table 1. These molecules all exhibit bonds typical of shared interactions: the values of Fb are high and the Laplacians span a range of values. The CtC and C-H bonds in C2H2 are shown to be “shared” interactions, with negative Laplacians, but the double bonds in CO2, OCS, and SO2 are all indicative of “shared polar” interactions.28 Notable is the shallowness of the curvature along the CtC bond in C2H2, which indicates a uniformity to the density along the bond axis between the two atoms. Earlier studies had located a spurious “maximum” in the charge density at the midpoint of the acetylene CtC bond,7 but this is found to disappear at higher levels of theory and is not observed in our calculations. The values of Hb are all negative, so their shared character is confirmed.32 (ii) The Ar2 System. The existence of a BCP between two noble gas atoms has already been reported.28 Long regarded as prototypical of van der Waals interactions, Ar2 (1) is included in this study for completeness and for comparison with other complexes. In Table 2, we see that the description of its BCP by the quantities Fb, ∇2F, and Hb places its bonding type firmly within the “nonbonding” category. (iii) Ar-C2H2. The argon-acetylene van der Waals complex, 2, is regarded as a fundamental test case in the modeling of intermolecular forces. It is very weakly bound, and for a long time there was considerable debate as to the value of the intermolecular distance.43 Structural uncertainties have recently been resolved with the understanding that the system is very floppy.44 Several attempts have been made to describe the variation in binding energy as the argon atom traverse a path around the acetylene.45,46 The picture here derives from the ab initio studies of Bone47 in which it was shown that the global minimum on the potential energy surface is planar and nonsymmetric (2b), though barely distinguishable energetically from the “T”-shaped structure (2a). A planar transition state (2c) is encountered between the global minimum and a linear local minimum (2d). Plots of the electron density, atomic interaction lines, and interatomic surfaces are shown in Figure 1. BCP data for the atomic interaction line joining the argon atom to its neighbor are presented in Table 2. It is clear that the largest curvature is along the atomic interaction line in each case, as is typical for “closed-shell” interactions. Of the remaining curvatures, it is the one in the plane of the complex which is shallowest, demonstrating a slight tendency for the electron density to accumulate not only in the region of the BCP but along the contact surface between the argon atom and the C2H2 molecule. Note that this accords with the observation
10896 J. Phys. Chem., Vol. 100, No. 26, 1996
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TABLE 2: Bond Critical Point Data (au) for Different Conformations of Various Ar Complexes: In Each Case the Atomic Interaction Line Is between Ar and Atom “X”, in Which Case 9 Denotes a BCP on Another (Orthogonal) Interaction Line and b Denotes a Ring Critical Point; Binding Energies, De, Are Expressed as Magnitudes; the Larger the Value, the More Strongly Bound with Respect to Isolated Monomers; Values of De Come from ab initio Calculations in References Described in the Appendix and Are Not Corrected for Basis Set Superposition Error De/cm-1 X 1 2 a b c d 3 a b c a b c 4 a b c d e 5 a b
Ar2 D∞h Ar-C2H2 C2V Cs Cs C∞V Ar-CO2 (TZ2P) C2V Cs C∞V (TZ2P+df) C2V Cs C∞V Ar-OCS Cs Cs C∞V Cs C∞V Ar-SO2 Cs C2V
c C2V d Cs e Cs
RXY/Å
RAr•/Å RX•/Åd ∑Fb/10-3a Fb/10-3 32Fb/10-2 λ1/10-3 (|)b λ2/10-3 (⊥)c λ3/10-2 Hb/10-4
2.9318
2.8887
1.2194
-1.8560
-1.8560
1.5906
1.9007 1.7769 1.1603 1.0953
3.1401 3.3674 3.7833 4.2284
3.0163 3.2700 3.8543 4.3644
1.0614 1.1776 1.6433 1.8149
-1.3022 -1.4656 -3.0479 -3.7758
-1.7233 -1.9602 -3.1728 -3.7758
1.3640 7.0092 0.3234 1.5202 7.2575 0.3374 2.2653 10.5102 0.0410 2.5700 10.7042 0.0000
C O O
3.5336 1.8799 1.6537 3.5291 1.8729 1.6564 3.4539 1.8454 1.6086
3.1053 3.0186 3.3052
3.1081 2.9282 3.1518
1.3417 1.3054 1.5376
-0.6692 -2.0008 -2.2854
-2.0815 -2.0356 -2.2854
1.6168 8.7107 2.1103 1.7090 9.0889 0.0174 1.9833 10.8103 0.0000
236 160 172
C O O
3.4463 1.8397 1.6066 3.4548 1.8364 1.6195 3.4142 1.8273 1.5870
3.7365 3.5433 3.5464
3.7776 3.4338 3.4186
1.6504 1.5792 1.7074
-0.8430 -2.4463 -2.5248
-2.6391 -2.4646 -2.5248
1.9986 10.2221 2.1306 2.0702 10.8195 0.0075 2.2124 11.8740 0.0000
303 191 192 177 212
C O O S S
3.5883 3.4622 3.4398 3.9459 3.7187
1.8768 1.8472 1.8376 1.9114 1.8577
1.7115 1.6154 1.6022 2.0364 1.8610
3.2566 3.3129 3.4034 2.8488 3.4206
3.2391 3.2063 3.2888 2.7975 3.4383
1.3582 1.5302 1.6188 1.0093 1.4154
-0.3688 -2.3185 -2.4163 -1.6435 -2.0280
-2.0891 -2.3119 -2.4163 -1.5607 -2.0280
1.6040 8.5899 4.6650 1.9933 10.7004 0.0028 2.1020 11.3256 0.0000 1.3298 6.8313 0.0530 1.8210 9.2283 0.0000
306 254
S O Se S S O
3.7183 3.6381 4.1447 3.9116 3.7228 3.4135
1.8492 1.8949 1.9508 1.8931 1.8424 1.8159
1.8676 1.7465 2.1939 2.0186 1.8808 1.5965
3.7235 2.8978 2.5363 2.9904 3.8057 3.8447
3.7422 2.8350 2.4768 3.0318 3.7921 3.6854
1.5071 1.1483 0.9998 1.1154 1.5003 1.7201
-2.1126 -1.4524 -1.5234 -1.9922 -1.5970 -2.5667
-0.9682 -1.8858 1.4783 -1.8949 -2.3614 -2.6972
1.8152 9.4158 1.1819 1.4821 7.8231 0.2984 1.0043 6.7836 5.7935g 1.5041 7.7495 0.0514 1.8961 11.4847 0.4787 2.2464 9.2701 0.0508
133
Ar 3.7744 1.8872 1.8872
216 221 186 196
9 C H H
3.7928f 3.6502 2.9800 2.8929
190 122 125
162 233 207
1.8921 1.8744 1.8237 1.7976
8.2792 0.0
a The sum of the values of the monomer densities at points corresponding to the position of the BCP. b The “parallel” curvature is that in the plane of the complex (when planar), orthogonal to the atomic interaction line. c The “perpendicular” curvature is that perpendicular to the plane of the complex (when planar). d Note that sum of two bond distances does not necessarily equal the internuclear distance if the bond path is curved. e The ring critical point is regarded as being between Ar and S. λ is assumed to be perpendicular to the ring plane. f Ar-C distance is 3.8406 Å. 1 g Calculated using the ratio of the two “uphill” curvatures: i.e., λ /λ - 1, assuming that λ > λ . 2 3 2 3
that the argon atom can slide very easily in this lateral direction. There is a connection between the shallowest direction of the electron density and the ease of motion of the atom in that direction. It is also noteworthy that the longitudinal curvature along the CtC bond is shallow. In moving from structures 2a through 2d, we observe regular trends in F, each of the Hessian eigenvalues, and consequently ∇2F. But not only do these trends fail to parallel the variation of the binding energy (which does not decrease smoothly from 2a through 2d) we see that, roughly speaking, the critical points with higher values of F correspond to higher energy conformations. There are clearly subtle effects at work. It may not be reasonable to compare all the configurations in the same way. In the two highest energy conformations, the atomic interaction line links argon with a hydrogen. In the global minimum, 2b, it links argon to a carbon; and the C2V symmetry transition state, 2a, is a conflict structure in which argon is connected to the BCP between the two carbons. (It is interesting to note that Cremer and Kraka obtained a different picture for the H2O-O2 complex,32 where the oxygen atom of the water molecule was linked to both the other oxygen atoms by atomic interaction lines, rather than to the bond midpoint. The absence of a ring critical point in their diagram suggests that the electron density must be very flat, perhaps too flat to reliably locate such a subtle feature.) The existence of a conflict structure in the Ar-C2H2 C2V transition state, 2a, upholds the picture that the potential energy surface is very flat in the region of C2V symmetry (where the argon interacts strongly with both carbon atoms and their electron-rich triple bond) and that “stability” is achieved by a lateral motion of the argon atom toward either carbon. In the
C2V conformation, the picture corresponds with “approach” of the argon atom toward the acetylene molecule along a line which is contained in the CtC interatomic surface. (We speculate that, if the argon atom were pushed closer (along the same line) toward the acetylene molecule, the single intermolecular interaction line would bifurcate into a pair which link the argon atom with each of the carbons and a ring critical point would form in between them.) By contrast, the point at which the argon atom crosses the contact surface between C and H does not correspond energetically to a transition state,47 even though another conflict structure will arise at that point. However, the amplitude of motion that the argon atom must undergo before this conflict structure is reached is quite large. The argon atom can enjoy effectively unhindered motion within the combined basin of the two carbon atoms. The energy profile for this motion was given in ref 47; the point at which the argon atom crosses either C-H surface corresponds with a steep rise in the intermolecular potential. If we are safe in restricting comparison to the pair of Ar-H bonded structures (2c and 2d) and, separately, the other two structures, then we do indeed find the relationship that higher values of F and λ3 correspond to tighter binding. We note that the H-bonding relationship discovered by Boyd and Choi21 was strictly for nitriles; that is, in each of their systems, the bond studied was linear and between the same pair of elements in similar alignments. It is quite possible that anisotropy of the density around given atoms at the level of resolution which we are considering may mask real correlations between values of the density and binding energies. We expect such effects to be more marked for the heavier elements. On the other hand, there are other properties which correlate in a parallel fashion, for
Intermolecular Bonding in van der Waals Molecules
J. Phys. Chem., Vol. 100, No. 26, 1996 10897
Figure 1. Four stationary points on the Ar-C2H2 PES: F, atomic interaction lines, and interatomic surfaces. (a) Structure 2a (C2V transition state). (b) Structure 2b (Cs global minimum). (c) Structure 2c (Cs transition state). (d) Structure 2d (C∞V local minimum).
example, the Ar-X distance and the respective distances from Ar and X to the BCP, just as Fb was found to correlate with H-bonding distance.22 (iv) Ar-CO2. Little is known of Ar-CO2, 3, except that its equilibrium configuration has C2V symmetry,48 3a. Calculations47 suggest that there is a subsidiary linear minimum and an “angular” transition state which separates it from the global minimum. For this complex we are able to compare results with two basis sets. The data (at fully optimized geometries) show that the effect of basis set is minimal with respect to the qualitative behavior of the eigenvalues at the BCPs. We note that the charge density is very uniform around the O atom, from the transition state to the linear configuration. This leads to eigenvalues of the Hessian, λ1 and λ2, having very similar magnitudes. On the other hand, values of the charge density display different trends with respect to binding energy with the two basis sets. This demonstrates the high sensitivity of these features to the basis sets employed. In neither case, however, can we see a real correlation between Fb and binding energy. In each case, it is easier to suggest that intermolecular separation (or the atomic separations from the BCPs) is a real factor in dictating structural stability.
There is less evidence for nonrigidity in Ar-CO2 than in ArC2H2; we can rationalize that with the observation that the symmetric configuration of Ar-CO2 is not a “conflict” structure, even though its in-plane curvature of the density orthogonal to the interaction line is smaller than that in Ar-C2H2. The charge density (Figure 2) shows clearly that argon is bound to carbon in the global minimum structure, 3a. The effect of the approach of argon has been to squash the C-O interatomic contact surfaces. As the argon atom moves tangentially about the CO2 molecule, it will cross one of these contact surfaces (forming a conflict structure), at which point the atomic interaction line will “hop” to the nearer oxygen atom. The argon atom will remain “bound” to that atom until it reaches linearity, 3c. The transition state on this path, 3b, clearly does not correspond to the point at which the bond switches. But note that the argon atom need not move very far at all before its atomic interaction line will encounter either of the two C-O interatomic surfaces. It is essentially trapped in a very anharmonic well; the switch of the atomic interaction line from C to O implies a reorganization of the charge density which may be unfavorable. (v) Ar-OCS. The argon-OCS dimer, 4, is similar to ArCO2 in that it is observed to have a fairly rigid “T”-shaped equilibrium configuration, 4a.49 Ab initio calculations have
10898 J. Phys. Chem., Vol. 100, No. 26, 1996
Figure 2. Three stationary points on the Ar-CO2 PES: F, atomic interaction lines, and interatomic surfaces. (a) Structure 3a (C2V global minimum). (b) Structure 3b (Cs transition state). (c) Structure 3c (C∞V local minimum).
shown that the full pathway of the argon atom about the OCS molecule includes four further stationary points: two linear
Bone and Bader minima, 4c and 4e, in which argon is adjacent to the oxygen and sulfur atoms, respectively; and two planar transition states, 4b and 4d, one between each of the two respective linear structures and the global minimum.42 The potential energy surface is very shallow around the oxygen so that the linear structure is unlikely to be “trapped”. Although we do not expect the intermolecular geometries to change dramatically (probably by no more than 0.05 Å) on optimization, because these calculations were performed at intermolecular separations optimized with a smaller basis set, we are unable to draw quantitative conclusions. Nevertheless, we do find that a higher value of F at the BCP is associated with stronger binding in the complex, at least where BCPs between constant pairs of elements are concerned. It is also clear (from Table 2) that the change in the curvatures of the electron density at each critical point is larger for motions of the argon around the sulfur than for the oxygen and that the form of the binding of the argon to the carbon is very similar to that in Ar-CO2. Once again, the bonding re´gimes are defined by the relationship of the argon atom to the interatomic surfaces of the partner monomer. The angular region over which the argon is bound to the carbon (just as with Ar-CO2) is small, but the presence of the atomic interaction line (Figure 3) stabilizes this structure with respect to lateral motions, even though the in-plane curvature of the electron density is shallowest near the minimum. The energy profile for motions of Ar around OCS, via conformations 4a through 4e, was shown in ref 42. We see again the presence of steep rises in the intermolecular potential at regions where an atomic interaction line is expected to switch from one atom to another. It is becoming clear that motions in which the argon atom will cross an interatomic surface (i.e., pass through a conflict structure and thereby switch the allegiance of its interaction line) are energetically unfavorable, even though the crossings themselves are not at the highest points of energetic barriers. (vi) Ar-SO2. The argon-sulfur dioxide dimer 5 is probably the most interesting of these argon-containing systems because of the presence of so many stationary points on a potential energy surface with only 3 (intermolecular) degrees of freedom. Early spectroscopic measurements demonstrated that this system is floppy enough to exhibit interconversion tunneling.50 Its equilibrium configuration has Cs symmetry with the argon atom astride the SO2 molecule, 5a, but the dynamics are such that the argon may reach the other side of the SO2 molecule by going over a low barrier at a planar structure with C2V symmetry, 5b. The actual conformation of the global minimum was rationalized to be one in which the argon atom could remain close to all three of the atoms of SO2.42 Extensive ab initio calculations have demonstrated the existence of a point of bifurcation on the tunneling path and three further geometrically distinct stationary points, 5c, 5d, and 5e, all of which are planar.42 The equilibrium structure has a single intermolecular interaction line between argon and sulfur (Figure 4). Although the argon atom is clearly poised to move closer to the plane of the SO2 molecule, i.e., to decrease the angle it makes with the sulfur and the symmetry axis of the SO2 molecule, and eventually forms bonding interactions with the oxygens, such a motion would move it too far from the sulfur atom to benefit from the favorable van der Waals interaction. The curvature in the density orthogonal to such a motion is actually shallower than that along the tunneling coordinate, a fact which might suggest that more than one large-amplitude motion should be considered when modeling this complex. By contrast, the ab initio
Intermolecular Bonding in van der Waals Molecules
J. Phys. Chem., Vol. 100, No. 26, 1996 10899
Figure 3. Five stationary points on the Ar-OCS PES: F, atomic interaction lines, and interatomic surfaces. (a) Structure 4a (Cs global minimum). (b) Structure 4b (Cs O-bonded transition state). (c) Structure 4c (C∞V O-bonded local minimum). (d) Structure 4d (Cs S-bonded transition state). (e) Structure 4e (C∞V S-bonded local minimum).
calculations showed that the two respective vibrational frequencies corresponding to these motions were of similar magnitudes.42
In the planar transition state, 5b (where there is no longer a bond between argon and sulfur), the presence, now, of two bonds (by symmetry), from the argon to both the oxygens, demands
10900 J. Phys. Chem., Vol. 100, No. 26, 1996
Bone and Bader
Figure 4. Five stationary points on the Ar-SO2 PES: F, atomic interaction lines, and interatomic surfaces. (a) Structure 5a (Cs global minimum). (b) Structure 5b (C2V O-bonded transition state). (c) Structure 5c (C2V S-bonded maximum). (d) Structure 5d (Cs S-bonded transition state). (e) Structure 5e (Cs O-bonded maximum).
the formation of an intermolecular “ring” critical point. Its origin is rather interesting: it is very close to the triple point of
intersection of the two O-S interatomic surfaces with the Ar-O surfaces. We speculate that at larger separations, the argon-
Intermolecular Bonding in van der Waals Molecules oxygen atomic (3,-1) critical points coalesce with the ring critical point. The data for the BCPs between Ar and O in this conformation show that the out-of-plane curvature is shallowest. Consequently, the energy changes along the tunneling coordinate can be expected to be rather anharmonic. The other three conformations of Ar-SO2 all contain ordinary atomic interaction lines in which the argon atom is “bonded” to its nearest neighbor atom of SO2, but again, although these structures are all very close to being stationary points (either transition states or maxima), none of them correspond to conformations in which the bonds switch from atom to atom. We note in passing that the anisotropy of the charge density about the oxygen appears to be higher here than in OCS or CO2. The trends in the topological descriptors may be masked by the fact that, for two of the structures, the geometries at which the calculations were performed are not fully optimized (see the Appendix), but there is a limited indication that the strength of binding can be correlated with the magnitude of Fb. For structures 5c and 5d, both of which contain Ar-S interactions, the larger Fb goes with the stronger binding; structure 5a does not fit in with this trend, and we must argue that this is because the argon atom approaches the sulfur from a different “face”. Such problems of interpretation did not plague the earlier study of H-bonded nitriles, in which there was less variation in overall configuration and angle of approach between hydrogen and acceptor. To resolve this potential uncertainty of interpretation, a number of points were computed at 10° increments in θ along the tunneling coordinate. In each case, all degrees of freedom, except for motion along the tunneling coordinate itself, were fully optimized. Analysis of the wave functions is presented in Figure 5. The ring critical point persists until the argon atom is more than 40° out of the plane of the SO2 molecule, after which point the ring critical point coalesces with the two neighboring (3,-1) critical points in a catastrophe. The two atomic interaction lines from argon to the oxygen atoms become a single line to the sulfur atom. If we consider the first five points on the curve, we see that the value of Fb (Figure 5c) increases steadily as the binding becomes stronger, whereas the Ar-O distance contracts. In fact, all of the topological parameters vary smoothly over this range of conformations. For the interactions with the sulfur atom, the trends are not in line. The energy continues to decrease to the global minimum (at θ ) 63°) before rising steadily toward the other C2V stationary point, 5c (θ ) 180°). By contrast, Fb increases to a maximum at θ ≈ 100° before falling again. The Laplacian value at the Ar-S critical point (Figure 5d) and Hb (Figure 5f) have parallel trends. Although Fb does not faithfully follow the binding energy, it does, however, approximately follow the variation in the Ar-S distance, which itself will measure the degree of penetration. This observation illustrates the danger in attempting to correlate Fb with binding energy when there are other interactions which contribute significantly to binding over and above the pairwise interaction(s) highlighted by the atomic interaction line(s). In this case, Fb correlates well with the Ar-X distance for both the Ar-O and Ar-S interactions. It correlates well with the binding energy in the case of Ar-O interactions because the Ar-S distance remains large enough not to interfere. In the region where pairwise interactions between argon and all three other atoms influence the binding energy strongly, the ability of a single pairwise interaction to describe this accurately is diminished. As the argon atom moves to configurations beyond θ ≈ 100°, further away from the oxygens, we observe the steady decrease in binding energy paralleled by a decrease in Fb.
J. Phys. Chem., Vol. 100, No. 26, 1996 10901 (vii) Generalization of Features Found in Argon Complexes. The idea that closed-shell interactions are characterized by a small value of Fb and small positive values of ∇2Fb and Hb is upheld for all of these argon complexes. Atomic interaction lines behave in ways very similar to those found in bound molecules, even though the properties of the interactions are very different. We therefore find that the theory of atoms in molecules has successfully identified the major intermolecular atom-atom interactions in noble-gas complexes. There is an approximate transferability of argon-X interaction features. For example, the values of Fb in all the bond critical points involving argon fall within a very tight range, and all are comparable with those in Ar2. Correlations which may exist between Fb and other physical quantities of interest are hard to discern, because there may often be a number of inseparable factors involved. For a given atom, X, there can be an approximate link between Fb at the Ar-X BCP and the binding energy but only if the binding is dominated by that single pairwise interaction at the configurations considered. It may be the case that increasing the number of atomic interaction lines of a given type leads to an increased binding. For example, in the ring system in Ar-SO2 (structure 5b), the presence of two Ar-O interactions leads to a stronger interaction than does the single Ar-O bond in the Cs structure, 5e, even though the value of Fb is greater in the latter. In fact, Fb appears to be most closely aligned with the degree of penetration of the electron cloud on Ar by the neighboring atom; that is, the greater the penetration, the greater the value of Fb. This data is obtained from the RAr• distances in the tables. Assuming a fixed, transferable, nonbonded (“van der Waals”) radius for argon (say to the F ) 0.001 au contour, i.e., 1.99 Å for Ar here), then the degree of penetration of the argon atom is simply (1.99 - RAr•). Thus, the smaller the value of RAr•, the greater the penetration. For example, in the linear ArC2H2 structure, the Ar-• distance is shortest for any of the complexes and the value of Fb is highest. Similar analysis was found to be valuable in the study of hydrogen-bonded systems by Carroll and Bader,19 but insufficient data on any given pair of elements and the use of several different basis sets prevent our study from drawing quantitative conclusions on the argon complexes. Carroll and Bader also noted that in hydrogen-bonding systems, the difference between Fb and the sums of the respective values at the corresponding coordinates in the isolated monomers, ∑Fb, is typically less than 10-3 au for the “intermolecular” BCPs.19 From Table 2, it is seen that the difference in the two quantities for the argon complexes is even smaller, having its greatest value as 1.6 × 10-4 au for conformation 5e of Ar-SO2; most differences are less than 10-4 au. This can be taken to be further evidence for the weakness of the interaction: there is virtually no enhancement of the electron density in the bonding region with respect to the unperturbed monomers. This explains why the value of Fb (or ∑Fb) can be quite successfully correlated with the degree of penetration of the interacting atoms. To analyze the true extent of the perturbation of the monomers, atomic integrations were carried out for all symmetry-unique atoms in each conformation of these argon complexes using the integration routines of the AIMPAC suite of programs.39 In this way, the total net charge on the argon atom and that on its partner monomer could be computed, by summation. Detailed study of these results will be presented in a forthcoming publication; for now it is sufficient to know that charge transfer is essentially negligible in these systems. The numerical error in individual integrations as is suggested
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Bone and Bader
Figure 5. Variations along the tunneling coordinate of Ar-SO2. (θ is the angle between the Ar-S vector and the principal axis of SO2 such that θ ) 0° corresponds to structure 5b, with the argon atom in contact with the oxygens.) (a) Binding energy of the complex. (b)-(f) Variation of critical point data. The two re´gimes represent the distinct regions of bonding between argon and each of oxygen and sulfur. The solid curves represent the properties of the bond critical points; the dashed curve in the oxygen region represents the property of the associated ring critical point. (b) Distance between argon and the atom to which it is connected by an atomic interaction line. (c) Fb. (d) ∇2Fb. (e) λ3 (curvature along the atomic interaction line for the BCPs; curvature orthogonal to the ring plane for the ring critical points). (f) Hb.
by computing net charges on individual monomers ranges from -5.9 × 10-4 au for OCS to +1.2 × 10-4 au for SO2. (For argon itself, the integration could be performed to an accuracy of 10-6 au.) The net charges found for the monomers in the complexes lie in the following ranges (in units of 10-4 au): C2H2 (-20.1, 36.8); CO2 (-13.7, -3.8); OCS (-14.3, +19.0); SO2 (+2.6, +20.4); and Ar (-40.0, +16.6). These are, on average, only 10-3 au, which is quite close to the level of precision of the atomic integrations anyway. These values are an order of magnitude smaller than the effects of charge transfer found for hydrogen-bonded complexes.19 (vii) Acetylene Dimer and Trimer. The structures of homogeneous acetylene complexes are dominated by two types of quadrupole-quadrupole interactions,51 giving rise to station-
ary points in which monomers are in a perpendicular (“T”shaped) or less favorable “slipped-parallel” arrangement. Acetylene dimer (6) exhibits both of these,52,53 a C2V global minimum (6a) and a C2h transition state (6b). The cyclic global minimum of acetylene trimer (7) is a compromised structure in which the angle between the monomers deviates from 90° in order to allow three pairwise interactions.51,52 The perpendicular conformation of the dimer 6a is another “conflict” structure, according to the topology of its charge density (Figure 6). In this case there is an interaction line between the hydrogen of one monomer and the BCP at the midpoint of the CtC bond of the other. Although this corresponds to a minimum on the potential energy surface, we note that the system is very floppy.52-54 A “gearwheel” motion
Intermolecular Bonding in van der Waals Molecules
Figure 6. Representative stationary points on the PES of homogeneous acetylene clusters, F, atomic interaction lines, and interatomic surfaces. (a) (C2H2)2: structure 6a (C2V global minimum). (b) (C2H2)2: structure 6b (C2h transition state). (c) (C2H2)3: structure 7 (C3h global minimum); interatomic surfaces are omitted for clarity.
J. Phys. Chem., Vol. 100, No. 26, 1996 10903 of the monomers leads to the slipped-parallel configuration, 6b, in which the atomic interaction line now links a pair of carbons. Neither of the two stationary points correspond to a situation in which an atomic interaction line switches from one atom to another, though the energy profile for the motion,53,54 being approximately sinusoidal, demonstrates again that the switching occurs at points where the energy rises steeply. We note that at the C2V configuration the shallowest curvature orthogonal to the intermolecular interaction line is in the plane of the complex, i.e., in the direction of the gearwheel motion. To see what has happened at an intermediate configuration, we can turn to the trimer, 7. At this stage, the atomic interaction line joins hydrogen to a carbon, the pair of atoms which share a contact surface. A further switch will occur once two carbons on separate monomers share a contact surface. We can also envisage a scenario in which the acetylenes are in a slippedparallel configuration but more extended than the one shown, and hence the atomic interaction line links a pair of hydrogens. Table 3 shows that the characteristics of the charge densities for the two extremes of bonding are quantitatively different. In each case, curvatures in the plane of the complex are smaller than that perpendicular to the plane, but for the interactions involving hydrogen, all the curvatures are much larger in magnitude and the ellipticities smaller than the respective quantities for C‚‚‚C interactions. (viii) Acetylene-CO2. By contrast with (C2H2)2, the heterogeneous dimer, C2H2-CO2 (8), is a fairly rigid system with only one observed form, a planar C2V symmetry equilibrium structure.55 Ab initio calculations identified a further subsidiary minimum at linearity and a transition state corresponding to opposed out-of-plane internal rotation of the monomers.53 The topology of the charge density (Figure 7) shows that the equilibrium conformation 8a is a conflict structure. In the case of (C2H2)2 and Ar-C2H2, conflict structures arose from the approach of a single atom or molecule along an axis contained in the plane of an interatomic contact surface and corresponded ultimately to rather floppy configurations. Here, in a similar way, the approach of the carbon atom of CO2 along the plane of the interatomic surface of C2H2 leads to a conflict structure, but it corresponds to a relatively stable conformation. A demonstration of the shallowness of the intermolecular charge density is provided by the failure of the gradient algorithm to locate a stable downhill trajectory from the BCP. We note that the bonding situation observed at equilibrium is preserved during out-of-plane rotations of either monomer. Thus, the C2V transition state, 8b, in which the monomer axes are mutually orthogonal also has a single BCP between the C of CO2 and the midpoint of the CtC bond. The linear configuration of this complex, 8c (not shown), offers no new special insight in that there is simply an atomic interaction line linking H of C2H2 with the O of CO2, the only two atoms which are in close contact. (ix) (CO2)2. The (CO2)2 system is analogous to (C2H2)2 in that there are two stationary points dictated by quadrupolequadrupole interactions.53,56,57 But in this case, the shape of the monomer is such that in the C2h symmetry arrangement, 9a, the center-of-mass to center-of-mass separation of the monomers can be less than that in the “T”-shaped configuration, rendering it more stable.53 In addition we investigated configurations in which the monomer axes were fixed parallel to one another, and a lateral motion between the global minimum and a D2h symmetry structure was carried out. The C2V symmetry structure 9b presents no surprises in that there is an interaction line between an oxygen on one monomer and the carbon on the other (Figure 8). On the other hand, the C2h structure 9a displays a single atomic interaction line in the
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Bone and Bader
TABLE 3: Bond Critical Point Data (au) for Various Conformations of van der Waals Complexes Containing C2H2, CO2, OCS, and SO2: In Each Case the Atomic Interaction Line Is between the Pair of Atoms Which Constitutes the “Bond”, in Which Case 9 denotes a BCP on Another (Orthogonal) Interaction Line and b Denotes a Ring Critical Point; Binding Energies, De, Are Expressed as Magnitudes; the Larger the Value, the More Strongly Bound with Respect to Isolated Monomers; Values of De Come from ab initio Calculations in References Described in the Appendix and Are Not Corrected for Basis Set Superposition Error structure 6 a b 7 8 a b c 9 a b 10 a b c d 11 a b c d e f
(C2H2)2 C2V C2h (C2H2)3 C3h C2H2-CO2 C2V C2V C∞V (CO2)2 C2h C2V (OCS)2 C2h C2h Cs C∞V (SO2)2 Cs Ci C2 C2h C2V C2V
“bond”
Fb/10-3
32Fb/10-2
λ1/10-3 (|)a
λ2/10-3 (⊥)a
551 454
H-9 C-C
7.1570 4.5213
2.1815 1.7308
-4.6204 -0.8155
-5.9800 -3.1759
1613
H-C b
7.5209 1.7792
2.3422 0.7305
-4.9981 4.1691
704 398 416
C-9 C-9c H-O
5.9168 4.5056 7.4302
2.1801 1.6140 3.3319
445 392
O-O O-C
6.2931 5.4400
625 508 509 416
C-S b O-O C-S S-O
1102 1111 971 786 658 542
O-S O-O O-O O-O S-O S-O
De/cm-1
λ3/10-2
Hb/10-4
3.2415 2.1299
8.8508 11.1019
0.2942 2.8947
-6.3722 4.1691
3.4793 -1.0335
9.3084 5.3233
0.2749 0.0000
-0.3437 -2.1754 -2.4254
-3.9811 -1.0743 -2.4254
2.6126 1.9390 4.9044
11.9509 9.4006 15.4720
10.5816 1.0250 0.0000
2.9667 2.7902
-2.6832 -1.5960
-5.1298 -4.5002
3.7480 3.3998
17.6853 16.9763
0.9118 1.8197
4.2508 4.2418 5.4809 4.6169 6.4302
1.4176 1.4089 2.6273 1.5421 2.9726
-0.1755 0.1005 -1.6402 -0.9076 -4.6708
-2.3210 1.6363 -4.3707 -2.6176 -4.6708
1.6673 -2.2738 3.2283 1.8946 3.9067
8.5450 8.8145 16.3619 8.6134 7.4314
12.2278 15.2816 1.6645 1.8842 0.0000
7.6756 6.6257 6.8080 6.7786 4.3452 4.0455
3.0673 2.8752 2.9211 2.8613 1.5965 1.4919
-5.0393 -0.8702 -1.2549 -1.6033 -2.4174 -2.1961
-3.0263 -4.8702 -4.9965 -5.0579 -3.2004 -2.7993
3.8739 3.4492 3.5462 3.5274 2.1583 1.9915
15.6874 16.6530 16.7625 15.8011 9.9755 9.4596
0.6652 4.5970 2.9816 2.1546 0.3239 0.2747
a The “parallel” curvature is that in the plane of the complex (when planar), orthogonal to the atomic interaction line. b The “perpendicular” curvature is that perpendicular to the plane of the complex (when planar). c The plane of the complex is taken to be that containing the C2H2 molecule and the intermolecular BCP.
intermolecular region between a pair of oxygens. It seems to be counter-intuitive that the two electronegative atoms are linked in this way, but this pair of atoms share an atomic surface, and therefore there must be an interaction line between them. Moving the monomers laterally soon creates a repulsive configuration. The values of the various descriptors of the bond critical points at each intermediate configuration are shown in Table 4. In the region between equilibrium and θ ) 65°, Figure 8c(i) a catastrophe occurs, in which the single O-O interatomic interaction line splits into a pair enclosing a new ring critical point. This has occurred because of a singularity in the shallow intermolecular charge density; note the very high value of the ellipticity for this (3,-1) critical point. The reverse motion corresponds to the ring critical point coalescing with the two adjacent (3,-1) critical points to form the single O-O interaction line. While similar catastrophe events are well-known in ring systems, the interesting thing about this situation is that the “ring” is formed in this case by a pair of atoms. So, we have the topological equivalent of a “double bond” in the charge density, in which a pair of atomic interaction lines link the same pair of atoms. This topology is rather short-lived and only persists until the interaction lines may switch to the neighboring carbons, just before θ ) 75°, Figure 8(ii). The switch in these interactions occurs when each oxygen atom begins to share more of its surface with the carbon on the adjacent monomer than with the symmetrically equivalent oxygen. As the monomers are forced to move further, another switch occurs so that the interaction lines now join the closest pairs of oxygens, Figure 8c(iii), θ ) 80°. This switchover point, occurring between θ ) 75° and θ ) 80°, corresponds to the position at which the C-O interatomic surfaces of the two monomers cross one another. Even though this is not a stationary point on the potential energy surface, there are still two intermolecular BCPs. The existence of the ring critical point at the center of symmetry is required by topology. The
ring critical point persists until the D2h symmetry conformation, Figure 8c, iv, as is required by the overall topology. Consistent with the previous example of C2H2-CO2, the intermolecular electron density is very shallow in the plane of the complex. Although “double bonds” of the sort seen here have not been found in “normal” molecules, they are not prohibited by the theory of atoms in molecules. The occurrence here is almost certainly an unusual consequence of very flat intermolecular charge densities. (x) (OCS)2. The (OCS)2 complex 10 has features in common with that of (CO2)2 except that it boasts twice as many stationary points in which the monomers are parallel, due to the lack of a center of symmetry of the OCS monomer. Experimental measurements have detected the global minimum structure, a C2h symmetry complex in which the sulfurs are in “internal” positions.58 Ab initio calculations located two further planar stationary points and a linear local minimum.59 No “T”-shaped structures have been considered. We restrict attention to the planar structures (Figure 9). Here, we see that the equilibrium configuration 10a is similar to the intermediate conformations of slipped-parallel (CO2)2. There are two atomic interaction lines in the intermolecular region, related by symmetry and linking each carbon to the sulfur on the partner monomer. The ring critical point occurs at the center of symmetry, in this case where the C-S interatomic surfaces align with one another. The density is so flat in the intermolecular region that, once again, the gradient algorithm failed to make reasonable paths off the two saddles in the plane of the intermolecular surface. The presence of a ring critical point, here, in an equilibrium configuration contrasts with the existence of ring critical points in nonequilibrium configurations of (CO2)2, above. Therefore, we cannot exclusively attribute the presence of ring critical points to either stability or instability. The other C2h symmetry structure 10b is related to the equilibrium configuration by a lateral shift in the monomers
Intermolecular Bonding in van der Waals Molecules
J. Phys. Chem., Vol. 100, No. 26, 1996 10905 BCPs and the ring) are much smaller than the corresponding values in the other configurations. (xi) (SO2)2. The last complex we studied is the complicated (SO2)2 system, 11. Long the subject of spectroscopic debate, the most recent measurements deduced a Cs symmetry structure similar in form to that of the water dimer and also undergoing a complicated internal rotation.60 Ab initio calculations found the Cs symmetry structure to be the global minimum and identified five other stationary points.61 The global minimum structure 11a displays a single atomic interaction line in between the internal oxygen of the monomer in the symmetry plane and the sulfur of the partner monomer (Figure 10). The principal tunneling motion in this complex involves a gear wheel-like rotation of the Cs structure through the Ci transition state,61 11b. The curvatures of the electron density which correspond with this are the shallowest for both configurations involved. The next two structures ascending in energy, 11b and 11c, are of Ci and C2 symmetry, respectively. The bonding features in both these structures are very similar, comprising a single interaction line connecting the oxygens, through the center of symmetry (or symmetry axis). This feature is repeated in the planar C2h symmetry structure 11d (Figure 10). We note the very similar values of Fb and the critical point curvatures at each of these three critical points. The binding energies of these conformations are somewhat different from one another, so we have an indication that the overall binding is determined by factors beyond the specific O-O interaction. The highest energy C2V symmetry structures, 11e and 11f, have ring critical points between S-O interaction lines, similar to the arrangement in the planar transition state of Ar-SO2. We note, though, that even the presence of two atomic interaction lines is not sufficient to confer overall stability on the complex. On the other hand, we note that for the S-O interactions in 11e and 11f and the global minimum, 11a, the larger the value of Fb at any given critical point, the greater the overall binding energy. Discussion and Conclusions
Figure 7. Two of the stationary points on the C2H2-CO2 PES: F, atomic interaction lines, and interatomic surfaces. (a) Structure 8a (C2V global minimum). (b) Structure 8b (C2V torsional transition state).
which takes the sulfurs to “external” positions (and the oxygens to corresponding “internal” positions). The bonding topology here is very similar to that encountered in the equilibrium configuration of (CO2)2, viz., a single atomic interaction line linking the adjacent oxygens. The Cs symmetry structure 10c (obtained by rotating one of the OCS monomers through 180° in the plane) has no rotational symmetry and so contains a single interaction line between sulfur and carbon, whose atomic surfaces are in contact. There is a linear configuration 10d (not shown); it contains the by now expected atomic interaction line between the sulfur of one monomer and the adjacent oxygen of the other. As with (CO2)2, the curvatures of the electron density in the intermolecular region (along a notional line of contact of the monomers) are very flat. Although the most stable structure, 10a, appears to benefit from a pair of intermolecular atomic interaction lines, an inspection of the in-plane curvatures and high value of the ellipticity shows that the structure is very close to a “bifurcation catastrophe”. Motions of the two OCS monomers parallel to one another can be expected to be facile, the result of which would probably be that the critical points coalesce to a BCP on a single S-S interaction. Note that the in-plane curvatures of the electron density in 8a (at both the
We have characterized a diverse collection of van der Waals complexes with the theory of atoms in molecules. In technical aspects, the main difference between this and previous work of a similar nature is that we have used large basis sets and a correlated level of theory which together were known to reproduce key monomer properties satisfactorily. Using the usual combination of gradient-based geometry optimization, guesswork, and good fortune, prior ab initio calculations had determined optimized structures for each complex which were, where data existed, in good overall agreement with those experimentally observed. But the theory of atoms in molecules7 has provided a description of these structures which goes beyond geometry: it has identified the key atom-atom interactions which allow chemical interpretations to be offered. We have therefore uncovered the existence of “van der Waals bonds”, features of F which are true counterparts of hydrogen bonds, but between pairs of heavy atoms. It is fair to say that the existence of very flat surfaces of electron density bodes ill for detailed interpretations in many cases, not merely because the precision possible in the calculation of characteristic quantities is limited but because such regions herald onset of facile structural change. For example, in (OCS)2, the intermolecular charge density is sufficiently flat in most conformations that the difference between one bonding scenario and a similar alternative is often slight. But, in general, we find that the topological features in
10906 J. Phys. Chem., Vol. 100, No. 26, 1996
Bone and Bader
Figure 8. (CO)2)2: (a) Structure 9a (C2h global minimum). (b) Structure 9b (C2V transition state). (c) The progression of the critical points as the monomer pair move past one another: (i) θ ) 70°; (ii) θ ) 75°; (iii) θ ) 80°; (iv) θ ) 90° (D2h symmetry configuration).
the charge density of these complexes are no different from those encountered in “regular” molecules, in the sense that there are bond and ring critical points and catastrophe and conflict
structures. The sole exception is the “O-O double bond” in the nonequilibrium conformation of (CO2)2. We do find that atomic interaction lines in the intermolecular region are quite
Intermolecular Bonding in van der Waals Molecules
J. Phys. Chem., Vol. 100, No. 26, 1996 10907
TABLE 4: Critical Point Data (au) for the Intermolecular Bond and Ring Critical Points in Various Conformations of (CO2)2 in Which the Monomer Axes Are Parallel: Angle θ Is the Angle Between a Line Joining the Carbon Atoms on the Two Monomers and a Monomer Principal Axis; The Pair of Atoms Joined by the Intermolecular Atomic Interaction Line Are Denoted by “Bond”; Where Symmetrically Related Pairs Are Present, Only One Pair Is Given θ deg
a
De/cm-1 a
58 60 65 70 75 80 85 90
445 442 372 219 10 -206 -369 -448
70 75 80 85 90
219 10 -206 -369 -448
“bond” O1-O1 O1-O1 O1-O1 O1-O1 O1-C O1-O2 O1-O2 O1-O2
Fb/10-3
32Fb/10-2
6.2931 6.2986 6.1102 5.8816 5.8598 5.9869 6.1389 6.2998
2.9667 2.9692 2.8720 2.7392 2.7193 2.8318 2.9407 3.0320
5.8788 5.6836 5.5411 5.4535 5.5319
2.7415 2.6253 2.5425 2.4948 2.5271
λ1/10-3 (|) BCP -5.1298 -5.1168 -4.8928 -4.6787 -4.7045 -4.8302 -4.9944 -5.1551 Ring Point -4.6594 -4.4854 -4.3720 -4.3082 -4.3897
λ2/10-3 (⊥)
λ3/10-2
Hb/10-3
-2.6832 -2.3158 -0.9930 -0.1539 -0.8474 -1.4412 -2.1618 -2.4311
3.7480 3.7124 3.4606 3.2224 3.2745 3.4589 3.6464 3.7906
1.7685 1.7768 1.7576 1.6791 1.6395 1.7477 1.8079 1.8486
0.9118 1.2095 3.9273 29.4059 4.5516 2.3515 1.3104 1.1205
0.0788 0.6600 0.7916 0.7016 0.6209
3.1995 3.0079 2.9006 2.8554 2.9040
1.6951 1.6050 1.5153 1.4508 1.4499
39.6028 3.5574 2.6642 3.0698 3.6771
A negative De denotes a repulsive configuration.
highly kinked, presumably a consequence of the substantial range of F values which is encountered. Individual monomers retain their structural integrity to a high degree because charge transfer between them is negligible, and although we have not presented this information here, we observe that their geometry and the nature of their intramolecular BCPs change hardly at all on complex formation. Therefore, just as was found previously with hydrogen bonds, absence of charge transfer is not incompatible with the existence of “directed” interactions. Furthermore, the electron density at the “closed-shell” BCPs found here is virtually the same as the sum of respective values for the isolated monomers at the corresponding point; the difference between these two quantities is an order of magnitude less than the difference found with hydrogen-bonded systems. We have confirmed that “intermolecular” BCPs occurring in van der Waals complexes display all the hallmarks of closedshell interactions. They can be characterized by values of Fb which are 100 times less than those found for normal shared interactions and around 10 times smaller than values found in hydrogen bonds and by positive values of both the Laplacian and Hb. The range of values of Fb is in line with the observed range of strengths of binding of these species. We find no evidence that specific values of parameters such as Fb, ∇2Fb, or Hb will give, per se, a detailed indication of strength of binding, but this is partly because it is difficult to make comparisons between conformations in which individual atom-atom interactions may differ and between calculations in which different basis sets have been employed. There is, though, strong correlation between the value of Fb and the degree of penetration of adjacent atoms, often expressible as the atomatom distance itself. Beyond that, any correlation between Fb and binding energy is less obvious. This is partly because it is difficult to find large amounts of data which may be fairly compared and partly because the binding energy of these complexes is affected by many factors. An inspection of the charge density identifies clearly the dominant atom-atom interactions but does not quantify the binding energy. In a typical (empirical) model of binding, it is usually found that the actual binding energy represents a sum of many pairwise contributions (irrespective of the pair of atoms which are linked via an atomic interaction line). Only if one of those contributions is much stronger than the others (over a particular range of geometries) can a property such as Fb between such a pair of atoms be used to give a reliable estimate of binding as a function of conformation. In van der Waals molecules, all the interactions, including the dominant ones, are weak, and it is
often not possible to isolate a pair of atoms whose interaction can be said to dominate the binding energy. By contrast, in the hydrogen-bonded systems for which there has been success in correlating hydrogen-bond energy with Fb,21 the binding is dominated by a single pronounced interaction. The simplicity of the systems studied and their close homology21 allowed very clear demonstration of this effect. One of our initial aims, to generalize our results into a simple set of observations governing pairwise atom-atom interactions (similar to the hydrogen-bonding rules proposed by Legon and Millen62), has not proved easy. What this paper shows is that even a noble gas atom may form a weak directional interaction with elements whose electronegativities are both high (O and S) and low (H and C). The diversity of form within our examples is too great to permit reduction to a small number of descriptive statements. The most general observation we can make is that, as with any other systems, the location of atomic interaction lines is determined by those atoms which share interatomic surfaces and therefore that atoms which are in close proximity are highly likely to display a bonding interaction. We have shown that traditional indicators of structural stability (with respect to conformational changes) do not hold as convincingly for these systems. For example, conflict structures may correspond to both minimum energy conformations (e.g., C2V (C2H2)2) and transition states (C2V Ar-C2H2), though we tentatively suggest that the potential energy surface will usually be very flat in the region of such structures. We find ring critical points in both equilibrium (C2h (OCS)2 and C3h (C2H2)3), transition state (C2V Ar-SO2), and unbound (slipped conformations of (CO2)2) structures. The assessment of stability in each case derives from studying the charge density. In weakly bound systems, the internal intramolecular features of the partner monomers are relatively unperturbed, but the close proximity of other monomers forces the external form of their respective charge densities to be distorted. The switch of an intermolecular atomic interaction line from one nucleus to another often requires that line to lie momentarily inside the atomic surface which separates the two atoms; this is the basis of the “conflict” mechanism of structural change. Such a point does not usually correspond with a stationary point on the potential energy surfacesnot even a transition statesbut what appears to be the case is that the energy of binding rises quite steeply in the neighborhood of such regions and so they effectively hinder motion, e.g., in Ar-C2H2, Ar-CO2, and ArOCS. We conclude, then, that large-amplitude motions which demand that an atomic interaction line pass through an
10908 J. Phys. Chem., Vol. 100, No. 26, 1996
Figure 9. Three of the stationary points on the (OCS)2 PES: F, atomic interaction lines, and interatomic surfaces. (a) Structure 10a (C2h global minimum). (b) Structure 10b (C2h local minimum). (c) Structure 10c (Cs transition state).
interatomic surface in this manner are rather unfavorable in that energetic barriers to such motions are steep, even if the curvature
Bone and Bader of the charge density in the direction of such a motion at the nearest critical point is shallow. The steric and long-range interactions studied by Cioslowski and co-workers23-25 are typified by the presence of both “ring” and “bond” critical points in close proximity (with a propensity for bifurcation catastrophes to occur). This fact allowed them to characterize the interactions by the nature of the ellipticity of the bond critical points and the separation between adjacent bond and ring points. They noted that the maximum curvature perpendicular to the atomic interaction line could be either parallel or perpendicular to the nearby ring surface and identified the former scenario with complexes and the latter with steric interactions. We regard that this is obvious in the context of the systems they studied: the steric interactions that they monitored all introduced rings into the bonding topology. Certainly, in any “regular” molecule, the introduction of a new “bond”, without introduction of a new atom, demands the creation of a new “ring” point, by the Poincare´-Hopf relationship. So it is possible to say that, in the absence of cage topologies, any intramolecular long-range atomic interaction line (i.e., a “closed-shell” interaction) must have a ring critical point associated with it. But steric interactions, while important, do not have sufficient flexibility or variety to demonstrate the full range of the effects in long-range interactions which should be searched for. The limitation of their analysis to ring-like critical points is not general enough for our purposes. We have shown that the majority of complexes do not contain any (“intermolecular”) ring critical points at all. Cioslowski and Mixon pursued their research further25 by suggesting that the interpretation of atomic interaction lines should now be tempered, viz., “interaction lines delineate major (not necessarily bonding!) interactions present”. In more lengthy discussion, they argue that the properties of Fb and its Laplacian may allow one to distinguish between “strong” bonds and “significant nonbonding repulsive” interactions and therefore that “bond critical points are not necessarily indicative of attractive bonding interactions”. We object to the term “repulsive” (and its counterpart, “attractive”) interaction, at least in equilibrium conformations where there are no net forces on the nuclei. In any case, it has always been acknowledged that the occurrence of atomic interaction lines is not confined to structures whose energy is stationary with respect to nuclear degrees of freedom. (For example, in the unbound D2h conformation of (CO2)2 there are two intermolecular atomic interaction lines and a ring critical point.) Consequently, we do not see the need to make qualifying statements about the interpretation of atomic interaction lines. It is much more useful to define interactions as being either “closed shell” or “shared”, and in either case, the atomic interaction lines identify the dominant interactions. In fact, the theory of atoms in molecules is quite clear about when to expect an atomic interaction line: it is when a pair of atoms share an atomic surface. Therefore, one inevitably expects that mere proximity of atoms is a strongly influencing factor. As we have shown, the properties of critical points of F in the intermolecular region are adequately summarized by the triplet: Fb ≈ 10-3 au; Hb > 0; ∇2Fb > 0. The idea that the theory of atoms in molecules is a model with general applicability has been upheld in this case. It has enabled us to see that directional pairwise interactions may be found in almost any circumstance. In the present context, there are ramifications for the choice of representation of intermolecular forces. Nevertheless, an envisaged practical difficulty of using models of specific pairwise attractive forces vs anisotropic models is that over the range of the potential energy surface (i.e., the interaction geometries sampled) the former may change usually
Intermolecular Bonding in van der Waals Molecules
J. Phys. Chem., Vol. 100, No. 26, 1996 10909
Figure 10. Four of the stationary points on the (SO2)2 PES: F, atomic interaction lines, and interatomic surfaces. (a) Structure 11a (Cs global minimum). (b) Structure 11d (C2h maximum). (c) Structure 11e (C2V maximum): (i) xz-plane; (ii) yz-plane. (d) Structure 11f (C2V maximum).
discontinuously, such as when an atomic interaction line flips from one nucleus to another. In a subsequent publication we
will investigate the utility of the Laplacian of the charge density for further rationalizing the structure of these species.
10910 J. Phys. Chem., Vol. 100, No. 26, 1996 Some Predictions We conclude with some brief predictions on systems for which we were unable to carry out calculations (at equivalent levels of theory) due to time and resources. We suggest the probable occurrence of bond (and ring) critical points in three more van der Waals complexes. Ar-C6H6. In the C6V symmetry conformation (see, for example, ref 62) we expect a single interaction line which links the argon atom to the ring critical point at the center of the benzene ring. Slight movements of the argon atom from the symmetry axis will cause the interaction line to jump to either the midpoint of a C-C bond or a carbon atom, dependent upon proximity. In a planar C2V configuration in which the argon-ring midpoint axis bisects a C-C bond, we expect a pair of BCPs between Ar and the two protons which are nearest to it and a ring critical point inside the C-C-H-Ar-H ring. (C6H6)2. In a D6h symmetry sandwich structure in which the two benzene rings adopt a “π-π stacking” arrangement, although probably not the global minimum,64 we expect to see BCPs between each of the eclipsed pairs of carbon atoms. In addition there will be ring critical points in the four-membered rings formed by adjacent pairs of such carbons. The occurrence of analogous BCPs between similarly related hydrogens will depend on their actual proximity. There will be a cage critical point at the center of symmetry of the complex. SO2-C2H2. A complex whose existence has recently been reported65 is found to have a C2V symmetry configuration. We expect to find an atomic interaction line between the sulfur atom and the midpoint of the CtC bond. This will be a conflict structure and will likely exhibit large-amplitude motion in which the atomic interaction line will switch alternately to the two carbons. Acknowledgment. R.G.A.B. acknowledges SERC (now EPSRC), U.K., for a 1992-4 NATO postdoctoral Fellowship. Computer facilities at McMaster University were used for this work. Encouraging comments and advice on scientific content from Dr. P. L. A. Popelier and Dr. T. A. Keith were most useful. Dr. P. L. A. Popelier is also thanked for access to a copy of MORPHY with which to generate custom plots. Appendix: Details of ab Initio Calculation Geometries and Basis Sets Used in the Calculations. The MP2/TZ2P basis sets used on C2H2, CO2, OCS, and SO2 are those of Dunning and Huzinaga.66,67 Calculations with these bases on dimers containing those monomers are described in references 51, 53, 59, and 42, respectively. The additional set of p/df polarization functions used on C2H2 and the d-set on CO2 are described in ref 47. The additional sets of d/f polarization functions used on OCS and SO2 are detailed in ref 42. The MP2/MC+sp3d and MP2/MC+sp3d2f basis sets used on argon are also described in ref 42. For the monomers, all wave functions were analyzed and geometries optimized at their respective levels of theory. For the complexes, where possible, the geometry at which the wave function has been analyzed is that optimized at the level of theory in question. (C2H2)2. For both (i.e., C2V and C2h) structures the geometry is that optimized at the MP2/TZ2P level of theory.53 (C2H2)3. For the C3h structure studied here, the MP2/TZ2P geometry is that used in ref 51, i.e., intermolecular parameters optimized at the MP2/DZP level,67,68 but the MP2/TZ2Poptimized monomer bond lengths.
Bone and Bader C2H2-CO2. For all three (i.e., both C2V and the C∞V) structures the geometry is that optimized at the MP2/TZ2P level of theory.53 (CO2)2. For both (i.e., C2V and C2h) structures the geometry is that optimized at the MP2/TZ2P level of theory.53 (OCS)2. For the C2h (SCO-OCS), Cs, and linear structures, the MP2/TZ2P geometries are those used in ref 59, i.e., the MP2/DZP-optimized intermolecular parameters, but the MP2/ TZ2P-optimized monomer bond lengths. The wave function for the global minimum (C2h OCS-SCO) structure was computed at a geometry in which the center-of-mass to centerof-mass vector between the monomers is at the MP2/DZPoptimized value and makes an angle of 92.5° with the monomer axes, corresponding closely with the experimental structure;58 see ref 59. (SO2)2. The MP2/TZ2P wave functions were calculated at MP2/TZ2P-optimized monomer bond lengths. For the two C2V structures and the C2h structure, the intermolecular parameters are those optimized at the MP2/DZP level.61 For the C2 and Ci structures the intermolecular parameters are those optimized at the MP2/MC(A) level.61 For the Cs structure, the intermolecular parameters were chosen to most closely reproduce the experimental rotational constants, as described in ref 61. Ar-C2H2. All geometries are those optimized at the MP2 level of theory with the larger basis set on argon (including two f-shells) and the TZ2P+p/df basis on C2H2. Geometries are given in ref 47. Ar-CO2. All geometries are those optimized at the MP2 level of theory with each of the two basis sets considered, viz., the smaller argon basis set with TZ2P on CO2 and the larger basis set on argon (with the two f-shells) and the TZ2P+d basis on CO2. Geometries are quoted in ref 47. Ar-OCS and Ar-SO2. The geometries of the two C2V symmetry structures and the global minimum of Ar-SO2 were fully optimized. For all other structures, the OCS and SO2 bond lengths were set at the MP2/TZ2P+d/f level of theory; intermolecular parameters were set at those optimized at the MP2/TZ2P level with the smaller basis set on argon. Details are found in ref 42. Supporting Information Available: Derivation of expressions for analytical partial derivatives (up to fourth-order in Cartesian coordinates) of the set of 10 Gaussian basis functions of f-symmetry (7 pages). Ordering information is given on any current masthead page. References and Notes (1) Pauling, L. The Nature of the Chemical Bond, 3rd Ed.; Cornell Univ. Press: Ithaca, 1960. (2) Miller, R. E. J. Phys. Chem. 1986, 90, 3301. (3) Nesbit, D. J. Chem. ReV. 1988, 88, 843. (4) Maitland, G. C.; Rigby, M.; Smith, E. B.; Wakeham, W. A. Intermolecular Forces; Oxford University Press: Oxford, 1981. (5) Buckingham, A. D. AdV. Chem. Phys. 1967, 12, 107. (6) Buckingham, A. D.; Fowler, P. W.; Hutson, J. M. Chem. ReV. 1988, 88, 963. (7) Bader, R. F. W. Atoms in Molecules-A Quantum Theory; International Series of Monographs on Chemistry, No. 22; Oxford University Press: Oxford, 1990. (8) Bader, R. F. W.; Beddall, P. M. J. Chem. Phys. 1972, 56, 3320. (9) Runtz, G. R.; Bader, R. F. W.; Messer, R. Can. J. Chem. 1977, 55, 3040. (10) Bader, R. F. W.; Ngyen-Dang, T. T.; Tal, Y. J. Chem. Phys. 1979, 70, 4316. Bader, R. F. W.; Ngyen-Dang, T. T.; Tal, Y. Rep. Progr. Phys. 1981, 44, 893. (11) Eberhart, M. E.; Donovan, M. M.; Maclaren, J. M.; Clougherty, D. P. Progr. Surf. Sci. 1991, 36, 1. (12) Bader, R. F. W.; Legare, D. A. Can. J. Chem. 1992, 70, 657. (13) Zou, P.-F.; Bader, R. F. W. Acta Crystallogr. 1994, A50, 714. (14) Eberhart, M. E.; Donovan, M. M.; Maclaren, J. M.; Clougherty, D. P. Prog. Surf. Sci. 1991, 36, 1. Eberhart, M. E.; Clougherty, D. P.; Maclaren, J. M. J. Mater. Res. 1993, 8, 438.
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