Role of Long-Range Intermolecular Forces in the Formation of

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Role of Long-Range Intermolecular Forces in the Formation of Inorganic Nanoparticle Clusters G. V. Gibbs,*,† T. D. Crawford,‡ A. F. Wallace,§ D. F. Cox,|| R. M. Parrish,^ E. G. Hohenstein,^ and C. D. Sherrill^ Departments of Geosciences, Materials Science and Engineering, and Mathematics, ‡Department of Chemistry, and Department of Chemical Engineering, Virginia Tech, Blacksburg, Virginia 24061, United States § Earth Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States ^ Center for Computational Molecular Science and Technology, School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332-0400, United States

)

†

ABSTRACT: An understanding of the role played by intermolecular forces in terms of the electron density distribution is fundamental to the understanding of the selfassembly of molecules in the formation of a molecular crystal. Using ab initio methods capable of describing both short-range intramolecular interactions and long-range London dispersion interactions arising from electron correlation, analyses of inorganic dimers of As4S4 and As4O6 molecules cut from the structures of realgar and arsenolite, respectively, reveal that the molecules adopt a configuration that closely matches that observed for the crystal. Decomposition of the interaction energies using symmetry-adapted perturbation theory reveals that both model dimers feature significant stabilization from electrostatic forces as anticipated by a Lewis acid/Lewis base picture of the interaction. London dispersion forces also contribute significantly to the interaction, although they play a greater role in the realgar structure near equilibrium than in arsenolite.

I. INTRODUCTION In an assessment of the role played by intermolecular forces in the self-assembly of nanoscopic entities into larger structures and materials, Bishop et al.1 concluded that “The formation of most assemblies can be modeled and justified only a posteriori, and there are few examples in which the course of nanoscale selfassembly was predicted a priori from a knowledge of individual interactions.” In this paper, we present a model based on directed, long-range intermolecular interactions associated with the formation of Lewis-acid
base-like complexes that not only may aid in the a priori prediction of how molecules self-assemble in the formation of molecular crystals but also may serve as a model for predicting the self-assembly of nanoparticles in the formation of oxide and sulfide Lewis acid
base complexes. The Lewis acid
base complex basis of our model has its origin in studies of the chemical reactivity of molecules undertaken by Bader et al.2 and Bader and MacDougall.3 They demonstrated that the initial approach of a reactant molecule in a chemical reaction is determined by the alignment of regions of locally concentrated and depleted electron density (ED) as defined by the topology of the Laplacian of the ED distribution, L(r) =
r2F(r). In a region where L(r) is positive, the ED is considered to be locally concentrated, defining a Lewis base region. In contrast, in regions where L(r) is negative, the ED is considered to be locally depleted, defining a Lewis acid region. In addition, it was discovered that the number and the spatial extent of the regions of concentrated and depleted ED in the valence shells of the atoms of the molecules as defined by L(r) are in close r 2011 American Chemical Society

agreement with the corresponding properties of the bonded and nonbonded electron pairs in the venerable VSEPR model for molecular geometry.4,5 This agreement between the bonded and nonbonded electron pairs in the VSEPR model and the regions of concentrated ED provides an additional, if somewhat more intuitive, motivation for considering the interactions in the framework of a Lewis acid
base complex model.6 We note that there are known cases for inorganic molecules such as ClF3 where dispersion is thought to play a more important role in the structuralization of a molecular crystal than the alignment of regions of charge concentration and depletion.7 The goal of our paper is to establish whether a connection exists between the alignment of the molecules displayed by some examples of inorganic molecular crystals and the long-range intermolecular Lewis acid/Lewis base (electrostatic) interactions that structuralize inorganic molecules as molecular crystals. It will also be established that the interactions that govern the alignment of the molecules can be explained and characterized in terms of the presence of a bond path (the path of maximum ED connecting a pair of bonded nuclei8) in the ED distribution while using the topology of L(r) in the same way in which Bader and MacDougall2 used it to determine the approach and the Special Issue: Richard F. W. Bader Festschrift Received: May 1, 2011 Revised: September 1, 2011 Published: September 22, 2011 12933

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The Journal of Physical Chemistry A alignment (i.e., the self-assembly) of a pair of molecules in a chemical reaction with the concomitant formation of a Lewis acid
base complex. An understanding of the connection between the ED distribution and intermolecular interactions is fundamental to the understanding of the self-assembly of nanoscopic entities as larger nanoparticles and materials. This is particularly true given that the ED distribution provides all observables for a material, regardless of its size, including its kinetic, potential, and total energy.9 We begin with a brief review of the topological properties of the ED distributions reported for several inorganic molecular crystals, properties that reveal that the structuralization of the molecules in a molecular crystal can be controlled by the intermolecular interactions embodied in Lewis acid
base directed bond paths.

II. EVIDENCE FOR DIRECTED LEWIS-ACID
BASE-LIKE INTERACTIONS IN THE SELF-ASSEMBLY OF INORGANIC MOLECULES II.a. Solid Molecular Chlorine. On the basis of a mapping of L(r) for solid molecular chlorine, Tsirelson et al.10 reported that the structuralization of the Cl2 molecules can be understood in terms of an alignment of Lewis acid
base regions connected by directed bond paths of maximum ED. Upon completion of a geometry optimization for a pair of Cl2 molecules at the secondorder Møller
Plesset (MP2) level of theory with a 6-311+ +G(d) basis set, they found that the Lewis acid
base regions of the molecules are aligned as observed in the molecular crystal with a binding energy for the resulting Cl2
Cl2 complex of 31.9 kJ/mol. Further, the geometry of the optimized pair of molecules was reported to have virtually the same structure as that adopted by the pair in the solid. On the basis of this evidence, it was concluded that each pair of molecules in the structure is aligned by directed Cl
Cl Lewis acid
base and base
base bond paths of ED. They also reported that attempts to derive the solid chlorine structure in terms of a pairwise Lennard-Jones (12
6) vdW central potential utterly failed even when quadrupole
quadrupole interaction terms were included in the model. On the other hand, they found that the alignment of the Cl2 molecules in the layers of solid chlorine can be predicted in terms of the topology of L(r) of an individual Cl2 molecule, as related to the alignment of the Lewis acid and base regions within the valence shell of the atom and the formation of a chlorine complex. In addition, the molecules are also aligned by Cl
Cl bond paths that are comparable to the O
O bond paths that are associated with the base
base interactions associated with the π-stacking interactions in DNA.11 II.b. Solid Molecular Tetrasulfur Tetranitride. In studies of the experimental and theoretical ED distribution for a molecular crystal consisting of S4N4 molecules, Scherer et al.12 and Tsirelson et al.13 found that the structure can likewise be understood in terms of the alignment of directed bond paths that connect the intermolecular Lewis acid
base regions on adjacent molecules. Scherer et al.12 considered the alignment of the acid
base regions to constitute “key-lock” mainstays in the structuralization of the molecular crystal. As the alignment of the molecules in the structure can be successfully explained in terms of the alignment of the Lewis acid
base regions connected by the directed bond paths of ED, it would appear that the self-assembly forces of the molecules can be ascribed to the stabilizing interactions between the acid and base regions of the molecules. It was asserted that the embodiment of the intermolecular directed bonded interactions

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in L(r) not only represents an important advance in our understanding of the self-assembly of molecules but also manifests how the molecules respond to one another and to their chemical environment. It was also asserted that the structuralization of the molecules in tetrasulfur tetranitride as well as that for solid chlorine can be taken as evidence that the alignment forces in these materials are encoded in the topology of the L(r). As the S4N4 and Cl2 molecules adopt quasi-close-packed molecular crystals and as the experimental intermolecular directed bond paths are comparable with those generated in procrystal ED distributions, Dunitz and Gavezzotti14 have questioned whether the presence of the bond paths between the acid and base regions in molecules serve as compelling evidence for the alignment of the intermolecular interactions. Further, if the cohesive intermolecular forces among the molecules are largely nondirectional, as expected for clusters whose interactions are dominated by London dispersion forces, then the molecules may be expected to adopt a close-packed structure as is often the case with the “bumps” on one molecule fitting into the “hollows” of adjacent molecules.15 However, as the binding energy obtained at the MP2 level for the Cl2
Cl2 complex is typical of a van der Waals (vdW) interaction10 and as the geometry of the complex matches the geometry of the dimers in the crystal, it appears that the crystal structure of Cl2 may be governed more by optimizing individual Cl2/Cl2 pair interactions than it is by maximizing the number of close vdW contacts. II.c. Solid Molecular Arsenolite, As2O3. Unlike solid chlorine and the terasulfide tetranitride molecular crystal, the tetrahedral As4O6 molecules in arsenolite are arranged in an open structure like the C atoms in diamond, occupying about one-third of the space that would be occupied by a close packed structure of molecules.16 Rather than the “bumps” of one molecule fitting into the “hollows” of the adjacent molecules in a close-packed arrangement, each As4O6 molecule is coordinated by four symmetry equivalent As4O6 molecules disposed at the corners of a tetrahedron. It has been shown that 24 directed As
O intermolecular bond paths radiate from each molecule and connect the L(r) acid
base regions of each of the four coordinated molecules, resulting in a cubic periodic molecular structure.17 The directed bond paths are indicated to serve as mainstays in the structuralization of the crystalline structure.17 The structures of the claudetite As2O3 polymorphs will not be considered in this study inasmuch as they consist of sheets of indeterminate extent rather than individual molecules of As2O3 composition. II.d. Solid Molecular Thioarsenides. In earlier work, the ED distributions and L(r) envelope isosurface maps were generated for the As4Sn (n = 3
5) thioarsenide molecular crystals alacr
anite (As8S9), pararealgar (AsS), uzonite (As4S5), realgar (AsS), βAsS, and α- and β-dimorphite (As4S3) and show that the bulk of the intermolecular directed As
S bond paths (∼70%) are aligned and connected with the Lewis acid
base regions on the adjacent As4Sn molecular complexes.18 Despite the preponderance of the directed nature of the Lewis acid
base As
S bond paths, the molecules in the thioarsenides each adopt a close-packed structure. On the basis of the directed As
S paths in the thioarsenides, a growth picture was proposed for the formation of thioarsenide complex aqueous species in sulfidic arsenic bearing water,18 on the basis of the arsenic speciation in sulfidic waters reported by Helz and Tossell.19 The question is, how might the aqueous thioarsenide molecules in the arsenic bearing sulfidic aqueous solution be pictured to react, self-assemble, and form complexes and thioarsenide nanoparticles and ultimately molecular crystals? 12934

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The Journal of Physical Chemistry A Given the evidence provided in the study of the thioarsenide molecules, it is anticipated that the surface of an aqueous thioarsenide species will be clad with Lewis acid and base regions as determined for each of the individual As4Sn thioarsenide molecules. Gibbs et al.18 anticipated that incipient directed intermolecular bond paths may be expected to potentially extend into space when an aqueous thioarsenide molecule is approached by another such molecule, like the burrs on a burdock (analogous to Velcro), catching onto either the acid or the base regions and forming Lewis acid
base directed bond paths that connect the acid and base regions of an approaching thioarsenide species with the concomitant formation of a complex. When in solution, the thioarsenide aqueous species will likely be coordinated by labile water molecules, and other aqueous species, attaching and detaching from the Lewis acid
base regions. When another aqueous thioarsenide molecule approaches, Lewis acid
base bond paths may be expected to develop, attract, and lock onto an aqueous thioarsenide molecule, ultimately eliminating all the other aqueous species with the loss of the labile water molecules so that thioarsenide complexes form an oriented attachment. It is noteworthy that the acid and base regions may also play a role in structuring the solvent around the growing particles through primarily electrostatic interactions with the solute. Further, the solvent structure may also influence the particle growth. If continued, this attachment process serves as a growth mechanism for a thioarsenide nanoparticle and may result in the structuralization of a molecular nanocrystal in which the number of longrange As
S Lewis acid
base intermolecular interactions is maximized. It is anticipated that As2O3 molecules like those in arsenolite will structuralize in a similar fashion.

III. COMPUTATIONAL METHODS Using the structure of realgar as a representative structure of the thioarsenides and arsenolite as test cases, we carried out three sets of computations for each mineral to develop a better understanding of the nature of long-range interactions and the mechanism of self-assembly for inorganic molecular crystals. First, the realgar and arsenolite structures were both simulated using density-functional theory (DFT) with the program package VASP,20
23 which makes use of periodic boundary conditions in conjunction with a plane-wave representation. The interaction between valence electrons and ionic cores was described by the projector augmented wave (PAW) method,24,25 using the generalized gradient approximation functional of Perdew, Burke, and Erzerhof.26,27 Energy cutoffs of 280 and 400 eV were used for realgar and arsenolite, respectively. For calculations involving the full crystalline structures, the unit-cell shape and atomic positions were geometry optimized within the observed crystalline symmetry. For calculations of the interactions within a dimer, the two molecules were confined to a fixed 25  15  15 Å3 cell to minimize the interaction with dimers in neighboring (periodic) cells, and the atomic positions optimized with no symmetry constraints. For both crystalline and dimer systems, the structures were optimized until all forces were less than 0.01 eV/Å using a (3  3  3) Monkhorst-Pack28 k-point sampling that was sufficient to converge the structures. Second, considering the well-known shortcomings of many density functionals for describing weak and long-range dispersion interactions accurately,29 we also carried out explicit manybody computations on an molecular (As4S4)2 dimer cut from the unit cell of realgar (Figure 1a) and on a (As4O6)2 dimer cut from

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Figure 1. (a) (As 4S4)2 dimer cut from the structure of realgar. (b) (As4O6)2 dimer cut from the structure of arsenolite. The silver spheres represent arsenic, yellow sulfur, and red oxygen atoms. The small black spheres represent the bond critical points for the intramolecular bonded interactions.

arsenolite (Figure 1b). Specifically, we optimized the structure of both dimers and As4S4 and As4O6 monomers using second-order Møller
Plesset perturbation theory30 (MP2) and the augmented correlation-consistent double-ζ (aug-cc-pVDZ) basis set of Dunning and co-workers31,32 with the Gaussian 09 quantum chemistry package.33 Total interaction energies were obtained using frozen core orbitals (1s2s2p3s3p for As, 1s2s for S, and 1s for O) and counterpoise corrections, the latter approximately accounting for basis-set superposition errors. The monomeric structures were confirmed to be minima on their respective potential energy hypersurfaces via harmonic vibrational analysis. The corresponding dimers were identified as minima (with the reasonable assumption that the monomer surfaces remain essentially unchanged) via direct computation of their interaction energy curves below. Third, to decompose the interaction energies for the two dimers into their essential components, we carried out a series of symmetry-adapted perturbation theory (SAPT) computations34 using a modified aug-cc-pVDZ basis set (denoted aug-ccpVDZ0 ), in which diffuse functions are removed from hydrogen atoms, and diffuse d-type functions are removed from the heavy atoms. The components of the interaction energy in the SAPT approach employed in this work depend on the total Hamiltonian H ¼ FA þ WA þ FB þ WB þ V where FA and FB denote the Fock operators of monomers A and B, respectively, and WA and WB are the corresponding fluctuation potentials. The potential, V, describes the interaction between the two fragments. In the MP2-like SAPT0 approximation used in this work, electron correlation effects are included in the interaction energies, but neglected on the individual monomers, yielding the following components Eint ¼ Eelst þ Eexch þ Eind þ Edisp where each contribution (electrostatic, exchange, induction, 12935

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Figure 2. (a) Several As4S4 molecules in the molecular crystal of realgar (defined in terms of the space group P21/c setting) including those in the unit cell. The b cell basis vector runs left to right, the c cell basis vector runs vertically, and the a cell basis vector runs perpendicular to b and c. The green spheres represent the arsenic atoms, and the yellow ones, sulfur atoms. The small black spheres represent the bond critical points along the intramolecular bond paths (silver colored) that connect the bonded atoms. (b) Copy of (a) with added intermolecular As
S bond paths (blue) connecting the arsenic and sulfur atoms and intermolecular As
As bond paths (green) that span the atoms of the adjacent As4S4 molecules. The S
S intermolecular bond paths are yellow paths, but they are difficult to see given that they are viewed nearly end on. The S
S bond paths in realgar are displayed much better in Figure 12 of Gibbs et al.18 where they are viewed perpendicular to the paths. The small black spheres represent the bond critical points.

and dispersion, respectively) is approximated as ð10Þ

Eelst ¼ Eelst

ð10Þ

Eexch ¼ Eexch ð20Þ

ð20Þ

Eind ¼ Eind, r þ Eexch
ind, r þ δHF ð20Þ

ð20Þ

Edisp ¼ Edisp þ Eexch
disp In the above equations, the superscript notation (mn) indicates that the interaction perturbation (V) is described to mth order and the monomer correlation (WA and WB) is included to nth order. The increment, δHF, provides an approximation of the magnitude of the perturbation, and thus a measure of the validity of the SAPT0 model. All SAPT0 computations were carried out using the PSI4 quantum chemical program package.35 It was found that the “-RI” auxiliary basis set for fitting the electrostatic and exchange terms is not sufficiently flexible to describe the core electrons and, subsequently, poorly describes the electrostatic interactions.

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Figure 3. L(r) isovalued envelopes generated for the molecules of the realgar structure. An indented spherical surface of L(r) =
12 au envelopes each arsenic site (green), a spherical L(r) =
12 au envelopes each sulfur site (yellow), a skullcap shaped isosurface L(r) = 0.001 au caps each arsenic site, and an L(r) = 0.001 au isosurface envelopes each sulfur site that extends about halfway along the intramolecular As
S bond paths. Each sulfur is partly enclosed by an earmuff shaped L(r) = 0.3 au isosurface, and an oblong shaped L(r) = 0.001 au isosurface occurs along the intermolecular As
As bond paths. The intramolecular bond paths are silver, the intermolecular As
S bond paths are blue, and the As
As intermolecular bond paths are green. As
S intermolecular bond paths connect the indented spherical envelopes that enclose the As sites (the Lewis acid L(r) =
12 au regions) with the earmuff shaped isosurface (Lewis base L(r) = 0.001 au region) that wrap about the sulfur sites and define the directed Lewis acid
base intermolecular bonded interactions that structuralize the molecules as a molecular crystal.

Consequently, the “-JKFIT” sets were used to fit the electrostatic and exchange terms in this study.

IV. RESULTS AND DISCUSSION An examination of the crystal structure of realgar36 together with an earlier examination of the calculated bond paths18 indicates that its As4S4 molecules are structuralized in a distorted cubic close packed array. Each unit cell contains four symmetry equivalent As4S4 molecules (Figure 2a). The average distance between the barycenters of adjacent molecules is 6.67 Å with the distances ranging between 5.17 and 8.54 Å. A total of 32 intermolecular bond paths branch from each As4S4 molecule connecting twelve nearest-neighbor equivalent molecules.18 Of these, 23 represent As
S, seven As
As, and two S
S intermolecular bonded interactions.18 Clearly, the As
S interactions comprise the bulk of the intermolecular interactions. The As
S bond lengths range between 3.42 and 3.77 Å, the As
As bond lengths range between 3.50 and 3.62 Å and the two S
S bond lengths are equivalent and identical in length: 3.72 (2) Å. The intermolecular As
S, As
As, and S
S bond paths and their bond critical points18 are also displayed in Figure 2b. L(r) isosurface envelope maps of the ED distribution are displayed in Figure 3. An examination of the map indicates that the intermolecular As
S bond paths connect regions of locally depleted ED on the arsenic atoms and regions of locally concentrated ED on the sulfur atoms, indicating a Lewis-acid base complexation of the structure. DFT/PBE computations using the VASP package were carried out for the realgar crystal. Although such calculations usually yield accurate predictions for many molecular and crystal properties, they are known to be less satisfactory for modeling molecular 12936

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Table 1. Realgar Dimer SAPT0/aug-cc-pVDZ0 Interaction Energy and Its Constituent Energy Components (Electrostatics, Exchange
Repulsion, Induction, and Dispersion) (kcal/mol) as a Function of Intermolecular Separationa Rcentroid (Å) ΔRMP2 (Å)

Figure 4. (a) MP2/aug-cc-pVDZ geometry optimized structure of the (As4S4)2 dimer. The green spheres represent arsenic, and the yellow ones represent sulfur. The intramolecular bond paths that connect the bonded atoms are displayed by silver rods. (b) Isosurface envelope of ED, F(r), calculated for the dimer at the F(r) = 0.009 au level. The envelope encloses all of the atoms of the dimer and contracts to constricted bond paths of ED between the As and S atoms, defining the intermolecular As
S bond paths (blue). (c) L(r) maps of the geometry optimized (As4S4)2 dimer. (See the legend of Figure 3 for a description of the map.)

structures where London dispersion forces play a significant role in the structuralization of the molecules.29 A geometry optimization of the structure of realgar reveals that this case is no exception with the resulting minimum-energy unit cell volume being substantially larger (∼30%) than the experimental one. Further, the intermolecular As
S and As
As bond lengths are substantially longer—r(As
S) = 4.59 Å, r(As
As) = 3.90 Å— than their experimental counterparts. Also, unlike the experimental structure, As
As bond lengths are shorter than the As
S bond lengths. However, the intramolecular As
S and As
As bond lengths are in close agreement with values observed for the As4S4 molecules in realgar. Given these inadequate results with DFT, a unit composed of a pair of adjacent As4S4 molecules from the realgar structure was geometry optimized at the MP2/aug-cc-pVDZ level, a method capable of capturing long-range London dispersion forces. If the two molecules attract one another and if the binding energy of the two is non-negligible, then an aggregate of molecules may be expected to self-assemble as a nanoparticle and ultimately form a molecular crystal. The MP2/aug-cc-pVDZ binding energy (with counterpoise correction) for the resulting (As4S4)2 complex (Figure 4a) was found to be significant at 53.0 kJ/mol. Furthermore, the orientation of the two molecules closely matches that observed for the crystal. The average intramolecular As
S (2.26 Å) and As
As (2.60 Å) bond lengths and the S
As
S

Eelst

Eexch

Eind

Edisp

4.456


1.00


300.56 605.38
105.86
96.44

4.706 4.956


0.75
0.50


164.99 331.28
91.30 179.07

5.206


0.25


51.39

5.456

0.00


29.52

ESAPT0 102.52


58.74
67.55
32.68
47.74

40.00 7.35

95.99


18.26
34.14


7.80

51.13


10.23
24.70
13.33

5.706

0.25


17.31

27.08


5.73
18.07
14.04

5.956

0.50


10.36

14.26


3.20
13.37
12.66

6.206

0.75


6.35

7.48


1.78


9.98
10.63

6.456

1.00


4.01

3.92


0.99


7.53


8.61

6.706 6.956

1.25 1.50


2.63
1.79

2.05 1.07


0.55
0.32


5.73
4.40


6.86
5.44

7.456

2.00


0.93

0.29


0.11


2.67


3.43

7.956

2.50


0.55

0.07


0.05


1.68


2.20

8.456

3.00


0.35

0.02


0.02


1.10


1.45

8.956

3.50


0.23

0.00


0.01


0.74


0.98

9.456

4.00


0.16

0.00


0.01


0.51


0.68

10.456

5.00


0.08

0.00

0.00


0.26


0.35

11.456 12.456

6.00 7.00


0.05
0.03

0.00 0.00

0.00 0.00


0.14
0.08


0.19
0.11

13.456

8.00


0.02

0.00

0.00


0.05


0.07

14.456

9.00


0.01

0.00

0.00


0.03


0.05

a

Also tabulated are the difference between each intermolecular distance considered and the MP2/aug-cc-pVDZ-optimized distance.

(94.5) and As
S
As angles (101.7) of the complex are comparable with the average values observed for the crystal, 2.24 Å, 2.57 Å, 94.8, and 101.3, respectively.36 The calculated intermolecular As
S bond lengths range between 3.29 and 3.31 Å with an average value of 3.30 Å whereas the experimental values are substantially longer and range between 3.42 and 3.84 Å with an average value of 3.60 Å. This result is anticipated given that each molecule in the structure of realgar is twelve-coordinated. It is well-known that the greater the coordination number of the molecule, the longer the expected intermolecular As
S distances and the greater the separation between the molecules. The ED and L(r) distributions calculated for the pair at the MP2/aug-cc-pVDZ level result in the features displayed in Figure 4b,c, respectively. The isosurface envelope of electron density, F(r), calculated at the F(r) = 0.009 au level (Figure 4b), displays contractions in the envelope between the pair of molecules. The ED distributed between the two molecules and passing through the contractions image the intermolecular As
S bond paths (blue lines) that connect the Lewis-acid regions of the pair of molecules as a (As4S4)2 complex. The L(r) envelopes for the pair display where the intermolecular As
S bond paths connect the Lewis acid regions of the arsenic atoms with the Lewis base regions on the sulfur atoms as observed in the crystal (Figure 4c).18 It is apparent that the L(r) distribution for the pair is comparable with those displayed by the dimers in the crystal, indicating that the bonded interactions among the molecules have little impact on the topology of L(r). Given that six intermolecular As
S bond paths span the intermolecular Lewis acid
base regions for the two molecules, the average interaction 12937

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The Journal of Physical Chemistry A energy per bond path is 8.8 kJ/mol, comparable to typical vdW binding energies.37 The dimensions of the (As4S4)2 complex of a bonded pair of molecules correspond to those of a nanoparticle. Further, on the basis of the binding energy, the (As4S4)2 complex may be considered to be a nanoparticle that is formed by the selfassembly two As4S4 molecules. If, as expected, the Lewis acid
base complexation of these molecules continues, it is anticipated that a nanoparticle of large dimensions will ultimately grow into a molecular crystal of realgar. To understand further the nature of the intermolecular interaction in the realgar dimer, we carried out SAPT0/aug-ccpVDZ0 computations as detailed above. Given that the (As4S4)2 dimer has no unique dissociation axis, we chose to determine the intermolecular potential curve along the axis connecting the monomer mass centroids, maintaining the Ci-symmetry structure of the dimer at each distance with frozen MP2/aug-cc-pVDZ monomer geometries at each step. The SAPT0 energy decomposition for the (As4S4)2 dimer may be found in Table 1 and is depicted graphically in Figure 5. The SAPT0 minimum occurs at ∼0.25 Å greater monomer centroid separation than the 5.46 Å distance predicted at the MP2/aug-cc-pVDZ level of theory. Near the minimum, electrostatic and dispersion interactions are similar in magnitude, with dispersion dominating the stabilizing

Figure 5. SAPT0/aug-cc-pVDZ0 interaction energy and its constituent energy components (electrostatics, exchange
repulsion, induction, and dispersion) for the realgar dimer as a function of intermolecular separation.

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energy components at long range and electrostatics dominating at short range. This result is somewhat surprising in that London dispersion forces, decaying as r
6, are inherently shorter range than electrostatic forces. As observed above, the As4O6 molecules in arsenolite adopt a much more open molecular structure than that of realgar, occupying one-third more space than would be occupied by a closed packed assembly of molecules (Figure 6). The experimental intramolecular As
O bond lengths of the molecules are each 1.79 Å, the intermolecular As
O bond lengths are each 3.05 Å, the O
As
O and As
O
As angles are 98.4 and 127.7, and the distance between the barycenters of the molecules is 4.80 Å.16 As in the case for the realgar structure, the DFT geometry optimization completed for the arsenolite structure resulted in a

Figure 7. L(r) isovalued envelopes for the arsenolite structure viewed down [110]. The blue line represent the intermolecular As
O bond paths that connect the red spherical isosurface (Lewis acid regions, L(r) =
22 au) enclosing each As site and the silver earmuff shaped isosurface (Lewis base region, L(r) = +12 au) capping each oxygen site. A silver skullcap shaped L(r) = 0.01 au isosurface caps each arsenic site and encloses each oxygen site.

Figure 6. Perspective drawings of the molecular structure of the As4O6 molecules in the molecular crystal arsenolite, As2O3, (a) viewed down [111] and (b) viewed down [111]. The silver spheres represent arsenic atoms and the red ones represent oxygen atoms. The gray lines represent the intramolecular bond paths between the bonded atoms of the molecules. 12938

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Table 2. Arsenolite Dimer SAPT0/aug-cc-pVDZ0 Interaction Energy and Its Constituent Components (Electrostatics, Exchange
Repulsion, Induction, and Dispersion) (kcal/mol) as a Function of Intermolecular Separationa Rcentroid (Å) ΔRMP2 (Å)

Eelst

Eexch

Eind

Edisp

ESAPT0

3.717


1.00


523.60 897.36
181.16
80.29

3.967


0.75


257.79 424.93


82.87
53.85

30.42

4.217


0.50


129.09 197.76


36.17
35.84


3.35

112.31

4.467


0.25


66.01

91.02

4.717

0.00


34.45

41.47


15.65
24.02
14.66
6.78
16.32
16.08

4.967

0.25


18.40

18.67


2.91
11.27
13.92

5.217 5.467

0.50 0.75


10.14
5.79

8.29 3.64


1.23
0.51


7.91
10.98
5.63
8.29

5.717

1.00


3.44

1.58


0.22


4.06


6.14

5.967

1.25


2.12

0.68


0.09


2.98


4.52

6.217

1.50


1.35

0.29


0.04


2.22


3.32

6.717

2.00


0.59

0.05


0.01


1.28


1.83

7.217

2.50


0.28

0.01

0.00


0.78


1.05

7.717

3.00


0.14

0.00

0.00


0.49


0.64

8.217 8.717

3.50 4.00


0.08
0.04

0.00 0.00

0.00 0.00


0.32
0.22


0.40
0.26

9.717

5.00


0.01

0.00

0.00


0.11


0.12

10.717

6.00


0.01

0.00

0.00


0.06


0.06

11.717

7.00

0.00

0.00

0.00


0.03


0.04

12.717

8.00

0.00

0.00

0.00


0.02


0.02

13.717

9.00

0.00

0.00

0.00


0.01


0.01

a

Also tabulated are the difference between each intermolecular distance considered and the MP2/aug-cc-pVDZ-optimized distance.

Figure 8. L(r) isovalued envelopes generated for the MP2/aug-ccpVDZ geometry optimized structure of the (As4O6)2 dimer. The intermolecular bond paths are blue. A description of the L(r) isosurfaces are given in the legend of Figure 7.

substantially larger unit cell volume (1617 Å3) than observed (1358 Å3). Further, DFT geometry optimization completed for a (As4O6)2 dimer cut from the arsenolite structure resulted in an intermolecular As
O separation of 3.23 Å, ∼6% larger than the experimental value given above. Isosurface envelopes of L(r) for the structure of arsenolite, viewed down [110], have been calculated in an earlier study17 and are displayed in Figure 7. The blue lines represent the intermolecular As
O bond paths that connect the red L(r) =
22 envelopes (Lewis acid regions) that center on the arsenic positions and earmuff shaped L(r) = 12 envelopes (Lewis base regions) that wraps around the oxygen atom positions.17 An (As4O6)2 dimer cut from the arsenolite structure was next optimized at the MP2/aug-cc-pVDZ level. The resulting dimer features an intermolecular As
O separation of 2.93 Å, an average intramolecular As
O separation of 1.82 Å, an As
O
As angle of 126, and an O
As
O angle of 101 and barycenter separation of 4.73 Å. The As
O separations and angles of the dimer and the barycenter separation are in reasonable agreement with values observed for the crystal. The resulting binding energy

Figure 9. SAPT0/aug-cc-pVDZ0 interaction energy and its constituent energy components (electrostatics, exchange
repulsion, induction, and dispersion) for the arsenolite dimer as a function of intermolecular separation.

(with counterpoise correction) for the dimer is 42.5 kJ/mol. L(r) isosurface envelopes for the dimer are displayed in Figure 8 where intermolecular bond paths connect the acid regions centered on the arsenic atoms and the base regions centered on the oxygen atoms. Given that six As
O bond paths span the acid and base regions of the pair, the average interaction energy per bonded interaction is 7.1 kJ/mol, slightly smaller than typical vdW binding energies. The nature of this interaction is elucidated by SAPT0 computations, analogous to those described above for realgar. Table 2 and Figure 9 provide the corresponding SAPT0/aug-cc-pVDZ0 12939

dx.doi.org/10.1021/jp204044k |J. Phys. Chem. A 2011, 115, 12933–12940

The Journal of Physical Chemistry A data for the arsenolite dimer, (As4O6)2. In this case, the SAPT0 minimum occurs close to the 4.72 Å minimum predicted by the MP2/aug-cc-pVDZ level of theory, where electrostatics dominate the stabilization of the complex. Induction is negligible until extremely close range, at which point the steric repulsions render the interaction unfavorable. At close range, electrostatics dominate for both complexes as may be expected for a Lewis acid
base type interaction. At longer distances, electrostatics and dispersion are roughly equal for the (As4O6)2 dimer, whereas dispersion dominates at long-range for (As4S4)2. Thus, in the structure of realgar, dispersion appears to play a larger role in the intermolecular interactions than in the more open structure of arsenolite.

V. SUMMARY AND CONCLUSIONS The calculations completed in this study show that a pair of molecules cut from the structure of an inorganic molecular crystal self-assemble and form an aligned Lewis acid
base complex. It is anticipated that if calculations were completed for a larger number of molecules that the molecules would align and selfassemble in the same way. The interactions that govern the alignment and self-assembly are indicated to be encoded in the Laplacian of the ED. The two molecular systems examined here, viz. dimers cut from the crystal structures of realgar (As4S4)2 and arsenolite (As4O6)2, are found to compare well with available experimental structural data when optimized at the MP2/aug-cc-pVDZ level of theory. In addition, decomposition of the dimer interaction energies using the SAPT0 approach indicates somewhat different sources of stabilization. In realgar, both electrostatic and dispersion interactions are comparable near the equilibrium structure, whereas in arsenolite, electrostatics dominate. However, in both cases, the intermolecular interaction features a significant electrostatic component, which is compatible with the view that these complexes can be understood in terms of favorable Lewis acid
base interactions between local regions of depleted or enhanced ED. ’ ACKNOWLEDGMENT This work was supported by the National Science Foundation and the U.S. Department of Energy through grants to G.V.G. (Grant No. EAR-0609885), T.D.C. (Grant No. CHE-1058420), C.D.S. (Grant No. CHE-1011360), and D.F.C. (Grant No. DEFG02-97ER14751) and a Multi-User Chemistry Research Instrumentation and Facility (CRIF:MU) award to T.D.C. (Grant No. CHE-0741927). ’ REFERENCES (1) Bishop, K. J. M.; Wilmer, C. E.; Soh, S.; Grzybowski, B. A. Small 2009, 5, 1600. (2) Bader, R. F. W.; MacDougall, P. J.; Lau, C. D. H. J. Am. Chem. Soc. 1984, 106, 1594. (3) Bader, R. F. W.; MacDougall, P. J. J. Am. Chem. Soc. 1985, 107, 6788. (4) Gillespie, R. J.; Nyholm, R. S. Quart. Rev. London 1957, 11, 339. (5) Macdougall, P. J.; Hall, M. B.; Bader, R. F. W.; Cheeseman, J. R. Can. J. Chem.-Rev. Can. Chim. 1989, 67, 1842. (6) Bader, R. F. W.; Gillespie, R. J.; Macdougall, P. J. J. Am. Chem. Soc. 1988, 110, 7329. (7) Shishkina, A. V.; Stash, A. I.; Civalleri, B.; Ellern, A.; Tsirelson, V. G. Mendeleev Commun. 2010, 20, 161.

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