4746
J. Phys. Chem. 1989, 93, 4746-4751
Molecular Simllarity and Molecular Shape Changes along Reaction Paths: A Topological Analysis and Consequences on the Hammond Postulate Gustavo A. Arteca and Paul G. Mezey* Departments of Chemistry and Mathematics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 0 WO (Received: July 11, 1988; In Final Form: October 18, 1988)
The concept of similarity between formal nuclear configurations and the shape changes of molecular surfaces along reaction paths are discussed. A topological description of molecular surfaces allows one to quantify the relation between the changes in the nuclear geometry and in molecular shape. Molecular arrangements of common shape properties along reaction paths can be regarded as domains in configuration space where the topologically essential shape features of the molecular envelope are the same. The analysis of topological shape changes throws new light on some empirical relations among molecular structures, such as the Hammond postulate, which postulates that species with structural similarities are transformable into each other with a small energy variation. However, the concept of molecular similarity has no universally accepted interpretation, which restricts the usefulness of this postulate. In this study a topological concept of molecular similarity is investigated with respect to the Hammond postulate. The general method is illustrated by modeling the molecular surface by the van der Waals surface during a unimolecular reaction: the isomerization HNC HCN, a reaction of exceptional properties. The validity of the Hammond postulate for this reaction is discussed from a new perspective. A similar analysis is also provided for a bimolecular collisional reaction, N(4S) + 02(3z;) NO(*II,) + O(3P), on the doublet 2Af energy surface.
-
Introduction Structural changes in molecules undergoing chemical reactions can be described in several ways. A usual approach consists of following the changes in nuclear configurations along a path on the potential energy hypersurface. The shape of various regions of the potential energy surface provides the natural conditions for structural stability. According to the simplest, traditional approach, the structural changes are marked by the occurrence of critical points in the hypersurface, encountered along a selected path (“reaction path”’). Among various proposals for reaction coordinates (ref 2), the intrinsic reaction coordinate of FukuiZC and Tachibana and FukuiZjskis a natural choice. In all instances, nuclear arrangements can be characterized by a formal structure of a bonds-and-atoms framework; formal bond orders and atomic charges can be computed by population a n a l y s i ~ . ~ Alternatively, molecular structure can be defined from the properties of the total electronic d e n ~ i t y . ~Following this procedure, one can associate a molecular graph to every nuclear arrange~nent.~A molecular structure is then defined by the equivalence class of molecular graphs in configurational space.5 This latter approach emphasizes those characteristics of the molecule which allow one to reintroduce into quantum chemistry the conventional concepts of chemical bond, giving rise to a molecular skeleton. However, this approach has no precise relation to energy and stability, which are the very conditions for the ( I ) Eyring, H.; Polanyi, M. Z . Phys. Chem. B 1931, 12, 279. (2) (a) Eyring, H.; Eyring, E. Modern Chemical Kinetics; Reinhold: New York, 1963. (b) Marcus, R. A. J . Chem. Phys. 1968, 49, 2610, 2617. (c) Fukui, K. J . Phys. Chem. 1970, 74, 4161. (d) Sato, S.; Fukui, K. J. Am. Chem. SOC.1976.98.6395. ( e ) Peanon, R. G. Acc. Chem. Res. 1971,4,152. (f) Levine, R. D.; Hofacker, G. L. Chem. Phys. Lett. 1972, IS, 165. (g) Basilevsky, M. V. Mol. Phys. 1973, 26, 765. (h) Ratner, M. A. Ann. N.Y. Acad. Sci. 1973, 213, 31. (i) Pechukas, P. J. Chem. Phys. 1976, 64, 1516. G) Tachibana, A.; Fukui, K. Theor. Chim. Acta 1978,49,321. (k) Tachibana, A.; Fukui, K. Theor. Chim. Acta 1979, 51, 189. (I) Truhlar, D. G.;Garrett, B. C.; Hipes, P. G.; Kuppermann, A. J . Chem. Phys. 1984, 81, 3542. (m) Mezey, P. G. Potential Energy Hypersurfaces; Elsevier: Amsterdam, 1987. (3) (a) Mulliken, R. S. J . Chem. Phys. 1955, 23, 1833. (b) Huzinaga, S.; Narita, S. Isr. J . Chem. 1980, 19, 242. (c) Coulson, C. A,; Longuet-Higgins, H. C. Proc. R. SOC.London, Ser. A 1947, 191, 39. (d) Davidson, E. R. J. Chem. Phys. 1967, 46, 3320. (e) Reed, A. E.; Weinstock, R. B.; Weinhold, F. J . Chem. Phys. 1985, 83, 735. (4) (a) Bader, R. F. W.; Anderson, S. G.; Duke, A. J. J. Am. Chem. SOC. 1979, 101, 1389. (b) Bader, R. F. W.; Nguyen-Dang, T. T.; Tal, Y. J . Chem. Phys. 1979, 70, 4316. (c) Bader, R. F. W.; Nguyen-Dang, T. T. Adv. Quantum Chem. 1981, 14, 63. (d) Li, L.; Parr, R. G. J. Chem. Phys. 1986, 84, 1704. (e) Rychlewski, J.; Parr, R. G. J. Chem. Phys. 1986, 84, 1696. (5) Tal, Y.; Bader, R. F. W.; Nguyen-Dang, T. T.; Ojha, M.; Anderson, S. G. J. Chem. Phys. 1981, 7 4 , 5162.
0022-3654/89/2093-4746$01 .50/0
-
existence of chemical structures; furthermore, it does not address the three-dimensional aspect of the shape of the molecular body. Several applications suggest a three-dimensional closed surface, enclosing the atomic nuclei, as a suitable model to describe molecular shape. Examples of possible molecular surfaces commonly used are isodensity contour surfaces: surfaces with constant electrostatic potential,’ and van der Waals like surfaces.8 The shape description of these surfaces can be accomplished in several The relation between changes in nuclear geometry (Le., formal point distributions of nuclei) and molecular shape (Le., the shape of a 3D body) is an important aspect of molecular structural problems. For example, the interactions among biomolecules depend on the complementarity in shape between the molecular moieties, and it is of importance to know for which nuclear arrangements a molecule possesses a given molecular shape. Recently, we have developed several approaches to study the change of molecular surfaces along conformational rearrangements.lldJ2 (6) (a) Francl, M. M.; Hout, R. F.; Hehre, W. J. J. Am. Chem. Soc. 1984, 106, 563. (b) Purvis, G. D.; Culberson, C. Int. J. Quantum Chem., Quuntum Biol. Symp. 1986, 13, 261.
(7) (a) Tomasi, J. In Quantum Theory of Chemical Reacrions; Daudel, R., Pullman, A,, Salem, L., Veillard, A., Eds.; Reidel: Dordrecht, Holland, 1979; Vol. 1, and references quoted therein. (b) Politzer, P.; Truhlar, D. G., eds. Chemical Applications of Atomic and Molecular Electrostaiic Potentials; Plenum: New York, 1981. (c) Weinstein, H.; Osman, R.; Topiol, S.;Venanzi, C. A. In Quantitative Approaches to Drug Design; Dearden, J . C., Ed.; Elsevier: Amsterdam, 1983; Vol. 16. (d) Rein, R.; Rabinowitz, J. R.; Swissler, T. J. J . Theor. Biol. 1972, 34, 215. (e) Thornber, C. W. Chem. SOC.Rev. 1979, 8, 563. (f) Warshel, A. Acc. Chem. Res. 1981, 14, 284. (g) NBraySzab6, G.; Grofcsik, A,; K b a , K.; Kubinyi, M.; Martin, A. J. Comput. Chem. 1981,2,58. (h) Angyin, J.; Niray-Szab6, G. J . Theor. Biol. 1983,103,777. (i) Louie, A. H.; Somorjai, R. L. J . Theor. Biol. 1982, 98, 189. (8) (a) Langridge, R.; Ferrin, T. E.; Kuntz, I. D.; Connolly, M. L. Science 1981, 211, 661. (b) Pearl, L. H.; Honegger, A. J. Mol. Graphics 1983, I, 9. (c) Connolly, M. L. Science 1983, 221, 709. (9) (a) Carb6, R.; Arnau, M. In Medicinal Chemistry Advances; De las Heras, F. G., Vega, S., Eds.; Pergamon: Oxford, 1981. (b) Bowen-Jenkins, P. E.; Cooper, D. L.; Richards, W. G. J . Phys. Chem. 1985,89, 2195. ( c ) Richard, A. M.; Rabinowitz, J. R. Int. J . Quantum Chem. 1987, 31, 309. (IO) Aqvist, J.; Tapia, 0. J . Mol. Graphics 1987, 5, 30. (1 1) (a) Mezey, P. G. Int. J . Quantum Chem., Quantum Biol. Symp. 1986, 12, 113. (b) Mezey, P. (3. J . Comput. Chem. 1987, 8, 462. (c) Mezey, P. G. Int. J. Quantum Chem., Quantum Biol.Symp. 1987, 14, 127. (d) Arteca. G. A.; Mezey, P. G. Int. J. Quantum Chem., Quantum Biol. Symp. 1987.14, 133. ( e ) Arteca, G. A,; Jammal, V. B.; Mezey, P. G.; Yadav, J. S.; Hermsmeier, M. A.; Gund, T. M. J. Mol. Graphics 1988,6, 45. (f) Arteca, G. A.; Mezey, P. G.J . Comput. Chem. 1988, 9, 554. (g) Arteca, G. A.; Mezey, P. G. J. Mol. Struct. (THEOCHEM) 1988, 166, 1 I . (h) Arteca, G. A.; Jammal, V. B.; Mezey, P. G.J . Comput. Chem. 1988,9,608. (i) Mezey, P. G.J. Math. Chem. 1988, 2, 325.
0 1989 American Chemical Society
, n . . , . . n , . . n I .._.. Molecular aimiiariry ana iuoiecuiar anape Lnanges a,,
3 ,
,
I
.
,
Some of these ideas are extended in this work. A unimolecular reaction with a simple, one-dimensional (1 D) reaction barrier constitutes the most elementary model of a chemical reaction. In this case, three formal species are recognized along the reaction path: the reactant, the transition structure (“transition state” or “intermediate state”), and the product, in order of occurrence. How similar these species are to one another is an important question to address; however, there is no universally accepted interpretation of molecular similarity. The usual concept of these three structures is totally geometrical. (They correspond to critical points in a 1D barrier.) Consequently, they are associated with definite nuclear configurations. This concept is semiclassical and not quantum mechanical. It is customary to express similarity in terms of how close these critical points are along a formal reaction path. On the other hand, the shape of the fuzzy electronic density distribution provides a different model for the molecule and also for molecular similarity. It allows one to identify open sets in configuration space where the shape of the molecular surface is essentially the same. As a result, the geometrical view of molecular structures along a reaction path can be replaced by a topological one.*”’ There exist useful rules that relate the structures occurring along a reaction path. The Hammond postulate (HP) is one of such rules.13 It states that the interconversion of two species appearing consecutively in a reaction involves a small geometrical reordering of the nuclei, if the energy contents of the two species are similar. This implies that for simple, one-dimensional reaction barriers one should expect an early barrier for exothermic reactions, whereas for an endothermic reaction a late barrier should occur. The HP is usually interpreted in a so-called “quantitative” version:I4 for a large group of possible, one-dimensional reaction coordinates, the transition structure appears as a maximum occurring closer to the reactant (product) than to the product (reactant) for an exothermic (endothermic) reaction. An extensive discussion of the constraints on the shape of reaction barriers that satisfy the H P can be found in ref 15. These constraints are bounds to the internal forces and force constants associated with the reaction.15 In practice, most chemical reactions are found to follow the Hammond postulate, although exceptions are known. One of the purposes of this work is to provide a new perspective on the HP, by defining the species along the reaction coordinate in terms of the molecular shape rather than in terms of the nuclear geometry. As we shall show, this generalized description of the HP may lead to the reclassification of some simple reactions that formally violate the conventional HP. We show that reactions that violate the postulate in the conventional sense may, in fact, follow the main trend in satisfying the postulate, if the latter is stated in terms of molecular shapes. We also believe that this study will help to clarify the role of mathematical methods that are less common in the chemical literature and to show that topology, often perceived as an esoteric subject, has simple and useful applications in chemistry. Molecular Shape Characterization A molecular surface can be defined as an “isoproperty surface”, for example, as an isodensity surface or an isopotential surface, with respect to the “properties” of electronic density and electrostatic potential, respectively. For convenience in the topological analysis, it is advantageous to regard a molecular surface as a boundary G of a level set F of the physical functionflr) of interest, with r a position vector in 3-space, r E 3R, and a chosen function value a:
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989 4147
3R:flr) I a ) G(a,K) = (r E 3R:flr) = a ) F(a,K) = (r E
der, L. The Transition State; Royal Chemical Society Special Publications: London, 1962; Vol. 16. (c) Polanyi, J. C. J . Chem. Phys. 1959,31, 1338. (d) Mok, M. H.; Polanyi, J. C. J . Chem. Phys. 1969, 51, 1451. (14) (a) Miller, A. R. J . Am. Chem. SOC.1978, 100, 1984. (b) Agmon, N. J . Chem. SOC.,Faraday Trans. 2 1978, 74, 388. (c) Murdoch, J. R. J . Am. Chem. SOC.1983, 105, 2667. (15) Arteca, G . A.; Mezey, P. G. J . Comput. Chem. 1988, 9, 728.
(1b)
Here K represents the nuclear configuration: K is an element of the reduced nuclear configuration space M , K E M , a metric space.2m One may picture the level set F(a,K) as the collection of all points r, of the 3D space where the function valueflr) is equal to or greater than a chosen value a, for a given configuration K of the molecule. Our goal is to characterize the changes in shape of G(a,K) with changes in configuration K,in particular, when K moves along a given reaction path. The functionflr) can be, for instance, the electronic charge density. A number of topological methods have been proposed to characterize a three-dimensional molecular surface. The shape group method (SGM) is a symmetry-independent group theoretical technique for the description of the shape of various molecular surfaces.11J2a*bIt is based on the computation of a series of topological invariants, the shape groups, which are the homology groups of a series of truncated surfaces derived from the original surface G(a,K). Alternative techniques allow the description of three-dimensional shape in terms of graphs.i2a.q16 These methods can be implemented for both differentiable and nondifferentiable surfaces. For the purposes of this work it is sufficient to consider the simplest model for a molecular surface, the hard-sphere or van der Waals surface (VDWS).* This surface is built by assigning one sphere to every atom and considering their exterior, envelope surface. van der Waals surfaces are known to provide a reasonable approximation for some electronic density contour surfaces.” The VDWS is simple enough to allow one to perform analytically all the computations needed for its characterization. In our case G(a,K) will be a VDWS. As the value of constant a defining the level set for a VDWS surface is arbitrary for any given choice of van der Waals radii, one can drop it from the notation. The molecular surface will be denoted by G(K). An efficient method to characterize VDWSs is by means of their characteristic graphskk These graphs are denoted by g = g(G(K)), and they are determined by their vertices and edges, as follows:12c ( i ) Vertices of the Graph. Take the subsets Dk of the VDWS G = G ( K ) , Dk C G, formed by all the points on the VDWS (exterior surface) that belong to k, and only k, spheres simultaneously. The vertices of the graph, collected in the set Vg(G(K))], are the maximum connected components Dk(‘)of the subsets Dk of the VDWS G ( K ) : V [ g ( G ( K ) ) ]= (Dk(j)C G: i = 1, 2 , ..., nk; k = 1, 2,
..., m )
(2) In eq 2 the number m is the maximum number of spheres to which a point on G may belong. Notice that a set Dk of points belonging to k and only k spheres is formed by nk maximum (pathwise) connected components. Pictorially, one can visualize the spherical faces, circular arcs, and arc intersections on the VDWS as forming the vertices of a graph. (ii) Edges of the Graph. The edges, collected in the set E [ g ( G ( K ) ) ] are , defined by the set of vertex pairs that have a nonzero neighbor relation E [ g ( G ( K ) ) ]= ((Dk(i),D,O’)) E V X I/: N(Dk(’),D,O”) = 1) (3a) where the neighbor relation is given by 1, if clos [Dk(j)] n clos [DK(J)]# 0 and
N(Dk(i),D p ) = (12) (a) Mezey, P. G. J . Math. Chem. 1988, 2, 299. (b) Arteca, G. A.; Mezey, P. G., Int. J . Quantum Chem., Quantum Biol.Symp. 1988, 15, 33. (c) Arteca, G. A.; Mezey, P. G. Int. J . Quantum Chem. 1988, 34, 517. (13) (a) Hammond, G. S.J . Am. Chem. SOC.1955,77,334. (b) Melan-
(la)
1
lK(k) - K(kq1 = 1
0, otherwise
(3b)
Here the function K(k) is defined as K(k) = kiif k C 4, and K(k) = 3, if k 2 4; clos [Dk(i)]stands for the closure of set Dk(i)and (16) Harary, F.; Mezey, P. G. J . Math. Chem. 1988, 2, 377. (17) (a) Hout, R. F.; Hehre, W. J. J . Am. Chem. SOC.1983, 105, 3728. (b) Hout, R. F.; Pietro, W. J.; Hehre, W. J. J. Comput. Chem. 1983, 4, 276.
4748 The Journal of Physical Chemistry, Vol. 93, No. 12, 1989
0 for the empty set. Notice that the maximum connected components DLi), Dku),j # i, are never connected directly by an edge in the graph g ( G ( K ) ) . Only vertices corresponding to different k values can be connected. This corresponds to the pictorial notion that two spherical faces of a VDWS are "neighbors" if they share an arc or that two arcs are neighbors if they have at least a common boundary point. The graph g(G(K)),defined by the above relation, provides a simple, but detailed, shape characterization of a VDWS. It describes the details of the interpenetration pattern among atomic spheres and recognizes the existence of buried atoms as well as the occurrence of holes in the formal molecular body covered by the molecular surface. The information contained in the graph can be represented by its adjacency matrix. A configurational change leads in general to a different element K of configurational space M . If the change in the nuclear geometry introduces an essential modification in the shape of the molecular envelope surface, then there will be a change in the corresponding graph g(G(K)). Although the graphs describe only some particular shape features, this model is sufficient for detecting essential changes in the shape of a VDWS. Changes in the graph can be either in the number of vertices or in the edge connectivity. In this work we pay attention only to the changes in shape along a specified reaction path. Such paths can be viewed as a continuous assignment of numbers from the unit interval I to the points K of configurational space M . This assignment can be regarded as a parametrization of the path, the parameter taking the role of a reaction coordinate changing its value from 0 to 1 along the path. By use of the topological terminology, the path is as a mapping between topological spaces: P (Z,T) (M,T?, I = [0,1] (4)
-
where T and T' are usually taken as the respective metric topologies, although other choices are also possible. (For alternative, chemically motivated topologies for M , see ref 2m.) According to (4), the paths can be parametrized as P(t), by means of a single variable t,t E [0,1], and the shape of the molecular surface can be regarded as a function of t . In what follows we denote a molecular surface associated to a given point along the reaction path as G(P(t)). We use the following convention: G(P(0)) represents the molecular surface for the reactant, whereas C(P( 1)) stands for the molecular surface of the product. In the next section we discuss the changes in molecular shape, when following a reaction path for a unimolecular isomerization reaction.
-
Model Case: Reaction Path for the Isomerization HNC HCN In spite of its simplicity, the isomerization reaction H N C HCN provides a most interesting example to analyze the structural changes along a reaction path, from the point of view of the shape of a molecular surface. This reaction has been studied theoretically in detail in a classic paper by Pearson, Schaefer, and Wahlgren;18 we use their numerical results for the reaction path, obtained at a large scale CI computation (all single and double excitations, involving 11 735 configurations). For this reaction all the atomic coordinates along the rigorous minimum-energy path are available. Furthermore, the molecular structural changes for this reaction have been studied in terms of molecular graph^,^ making possible a comparison. The coordinate system used to follow the reaction is depicted in Figure 1. The plane spanned by the three atomic nuclei is denoted as the x-y plane, with its origin located at the center of mass of the N-C pair. Pearson, Schaefer, and Wahlgren18 have computed the reaction path, parametrized in terms of the angle 0, between the position vector of the H atom and the N-C direction, following the changes in the distances rl(@and rz(0). The change in the interatomic distance N-C during the reaction is very small. Notice that 6 = Oo corresponds to H C N (product), whereas 6 = 180° corresponds to H N C (reactant); consequently,
-
(18) Pearson, P. K.; Schaefer 111, H. F.; Wahlgren, U.J . Chem. Phys. 1975, 62, 350.
Arteca and Mezey
Y
c ,(
f
c.m. ..............................................
r
-
-
b
N
x
11,,
2
Figure 1. Coordinate system for the study of the isomerization reaction HNC HCN. Note that B = 0 corresponds to a configuration of HCN; hence, increasing angle 0 corresponds to progress along the reverse reaction HCN HNC.
-
-
reading the figures from left to right corresponds to the reverse HNC. reaction, H C N We have fitted the eight points available along the reaction path18 with a quartic spline (pair of sectionally defined fourthdegree polynomials), matched at 0 = 90'. This allows one to compute with reasonable accuracy the Cartesian coordinates of all the atoms during the reaction. The relevant coordinates are the x and y coordinates of the hydrogen atom (x(H) and y(H), respectively). Several choices of reaction coordinates are available. Internal coordinates, chosen according to chemical expectation or intuition, can describe the evolution of the reaction, but ultimately their choice is arbitrary. In order not to introduce artifacts by the choice of the reaction coordinate, in the general case we use a massweighted Cartesian system of coordinates, where the reaction coordinate is the arc length of the path P(t) represented in such a coordinate frame. In our case, due to the small change in the Cartesian coordinates of the nitrogen and carbon atoms, this arc length s ( t ) is given to a good approximation by
where x',(H) and y'&H) represent mass-weighed coordinates, that is, a Cartesian coordinate x(H) or y(H) multiplied by the square root of the mass of hydrogen. (Note that if the rearrangement affects only a single nucleus, then the mass weighing is not essential.) The subindex attached to the coordinates represents the progress along the reaction path as measured by parameter t ; so, for example, xb(H) corresponds to the mass-weighed x coordinate for the hydrogen atom in the starting configuration, whereas x:(H) is the x coordinate a t position P(t) along the reaction path. The arc length s(t) was computed by differentiating the function describing the x-y reaction path, composed by two matched sections of fourth-degree polynomials; the integration was performed numerically with the method of Romberg. Figure 2 shows the reaction barrier for the reaction as a function of the arc length s. Coordinates are measured in angstroms and masses in atomic units. In order to facilitate the comparison with ref 18, we have displayed the species along the path in a similar manner; notice that H C N is at the left-hand side (product) from where the computation of the arc length starts (Le., the parametrization corresponds to the reverse reaction). The reaction H N C HCN is exothermic. Nevertheless, the barrier is late for HNC, contrary to the Hammond postulate. The H N C geometry lies 2.5 units of arc length away from the transition structure, while the H C N geometry lies at only 2.0 units from it. Along the reaction path only three graphs g(G(K)) are found. The results are displayed in Figure 3. Observe that the essential shape features of the VDWSs for reactant and product are the same, as the respective graphs gRand gp are the same. For these two VDWSs one finds two Dz subsets (the boundaries between atomic spheres) and three D1 subsets (one associated to each atomic sphere). On the other hand, the VDWS of the transition structure (of geometry associated with the maximum
-
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989 4149
Molecular Similarity and Molecular Shape Changes n u)
a, a,
d L
L
a r
Y
al cn L
a,
c
W
0
1
2
3
A r c Length s
5
4
-2
(8)
-I
0
1
Mass-weighted x H coordinate
Figure 2. Reaction barrier for the HCN isomerization with the arc
length along the reaction path (eq 5 ) as reaction coordinate. The electronic energy is measured relative to HCN. The abscissa varies from HCN to HNC; masses are measured in atomic units to compute the arc length in mass-weighed coordinates. Shaded regions represent the uncertainty in the boundaries where the change in shape for the molecular surface occurs. This uncertainty is due to the range of possible choices of VDW radii for each atom.
2
(A)
Figure 4. Reaction barrier for the HCN isomerization along the massweighed x coordinate of the hydrogen atom as reaction coordinate. The abscissa varies from HCN to HNC. See text for the explanation about shaded regions.
n u)
a, L a,
c I
a
r
Y
3
L cn a,
c W
Figure 3. van der Waals graphs found along the reaction path for the
100
0
200
HCN isomerization. of the reaction barrier) has three subsets of type D2 and two subsets of type D3. The sets D2 correspond to three circular arcs on the VDWS, and the two sets D3 represent the only two points on the surface lying simultaneously on three atomic spheres (and found at the intersection of the three circular arcs). Accordingly, the transition structure corresponds to a molecular surface with a different shape, as described by the graph gT, where the three atomic spheres interpenetrate one another. Notice that there is no hole in the formal body enclosed by the surface. (Otherwise, no D3 sets would occur.) That is, the molecular surface for the transition structure is homeomorphic to a 2-sphere and not to a 2-torus. All the above conclusions remain valid for all choices of van der Waals radii quoted in the literature19for the given set of atoms. One can easily compute the first geometries along the reaction path at which “shape transitions””dJ2 gR gT and gT gR occur. The locations change in general with the sets of van der Waals atomic radii chosen. We have proceeded to compute these geometries for all the possible combinations of atomic radii for C, N, and H available in ref 19. After determining the geometries, the corresponding values of arc length on the reaction path were evaluated via eq 5 as described above. The results are shown in Figure 2. The shaded regions in the diagram show the uncertainty gT and g, gR due in the location of the shape changes gR to the changes in the values chosen for the radii. These regions represent a formal “boundary layer” among the shape regimes dominated by the shape characteristics of the reactant, product, and transition structure. The above conclusions may be viewed in a more general context. The geometrical notion of molecular structure can be replaced by a topological one, defined in terms of open sets in configuration space.2m If molecules are modeled with VDWSs, then a molecular structure corresponds to a domain of configurational space where the topological shape of the VDWS is invariant to geometrical changes within the domain. For instance, all the nuclear geom-
-
-
-
-
(19) (a) Bondi, A. J . Phys. Chem. 1964, 68, 441. (b) Hopfinger, A. Conformational Properties of Macromolecules; Academic: New York, 1973. (c) Gavezzotti, A. J . Am. Chem. SOC.1983, 105, 5220.
Angle
e
(degrees)
Figure 5. Reaction barrier for the HCN isomerization with the angle 6 as reaction coordinate. The abscissa varies from HCN to HNC. See text
for the explanation about shaded regions. etries lying between the reactant geometry and the first shaded region are topologically equivalent in the above sense. For all these geometries the VDWSs exhibit essentially the same shape. Accordingly, all these geometries represent the “reactant” and they belong to the “reactant domain in configurational space”. The analogous interpretation can be given to the “product domain” and “transition structure domain”. Figure 2 shows that the boundary between the reactant and transition structure regions lies only at approximately 1.O unit of arc length away from the reactant’s formal geometry. On the other hand, the boundary for the product and transition structure region is about 1.5 units of arc length away from the product’s formal geometry. In other words, the reactant (HNC) molecular surface may attain the shape of the transition structure molecular surface by a distortion of nuclear geometry smaller than that required for the product (HCN) molecular surface. It is noteworthy that this result is in agreement with the Hammond postulate, if the notion of molecular structure is understood as a topological concept, defined by the shapes of the van der Waals surfaces. We have studied other possible reaction coordinates in order to test alternative, simpler descriptions. Figure 4 shows the same reaction barrier as a function of the mass-weighted x coordinate for the hydrogen atom. As the corresponding y coordinate varies rather symmetrically about 90°, the x coordinate may serve as the most relevant Cartesian coordinate to follow the reaction. We show in Figure 4 the uncertainty in the boundaries of shape regions, due to the different possible choices of van der Waals radii, computed as describe above (shaded regions). As it is evident the conclusions are again the same, even though the transition structure geometry is more similar to that of the product than to that of the reactant, the essential shape features of the transition structure can be obtained from the reactant by a smaller distortion than from the product. Whereas the Hammond postulate does not hold if similarity is defined by the reaction coordinate, the
4750 The Journal of Physical Chemistry, Vol. 93, No. 12, 1989
same postulate holds if similarity is defined in terms of the molecular shape using a VDWS. In Figure 5 another alternative is considered, showing the reaction barrier and the shape regions in configurational space, using the angle 0 as a reaction coordinate. This is a representation of reaction coordinate usually followed in the literature. The qualitative conclusions are the same as above. The maximum corresponds to B = 74’ (measured from the product HCN). A distortion of about 35’ converts the shape of the reactant (HNC) molecular surface into that of the transition structure, whereas a distorsion of about 55’ is required for the product (HCN).
-
A Second Example: The Bimolecular Reaction N + O2 NO+O Simple collisional reactions are convenient illustrative examples of the concepts just discussed, since all atomic coordinates along the reaction path are available. The reaction considered here is
-
N(4S) + 02(32,-)
+ O(3P)
which is known to play an important role in upper atmospheric processes, where it is a source of infrared emission. From the theoretical point of view, this reaction is interesting since it provides an example of a simple bimolecular reaction on both doublet and quartet potential energy surfaces. The global minimum-energy paths on both surfaces have been reported in the literature.20 We use the results in ref 20a to perform the shape analysis along the reaction path on the doublet surface 2A’. This surface has been recognized to be the most important one in the reaction, clarifying some previous discrepancies in the literature.20~2’The doublet surface predicts a barrier height close to the experimental value and a bent ZA’ transition structure N00.20bThe accuracy of available computations is not sufficient to account for the observed vibrational relaxation spectra.22 On the global energy hypersurface 2Ar,the minimum-energy path for reaction 6 lies well away from the region where the stable species NOz occurs. Results in ref 20a for the path are based on an extensive MCSCF treatment, followed by an exhaustive valence CI. The roles of d functions and 2s core orbital excitations have also been estimated. With these results we have been able to perform the same analysis described for the unimolecular isomerization. The reaction path was described by mass-weighed Cartesian coordinates, and its arc length s was chosen as a reaction coordinate. By interpolation on the fitted path, we have estimated the Cartesian coordinates for any intermediate structure, from which the hard-sphere molecular surface can be built and analyzed. Although possessing many common features, this bimolecular reaction constitutes a noticeably different problem from the isomerization. The reactant and product states are rigorously defined only asymptotically. Accordingly, the so-called reaction path is only a section of the true path. We have followed the convention of ref 20a, considering the starting and end p i n t s of the path (the “reactants” and “products”) as the first and last computed points, respectively. Figure 6 displays the results obtained for reaction 6. The shaded regions represent boundary layers between conformational regimes with different molecular shape characteristics. As before, they are the result of the uncertainty in the choice of atomic radii. The reaction exhibits exactly the same three shapes found along the isomerization reaction path and characterized by graphs in Figure 3. The three corresponding molecular surfaces are topologically spherical, though the intermediate structure (TS) corresponds to a more compact arrangement of interpenetrating spheres. Notice that the starting point of the path is closer to the absolute maximum than to the end point. However, a smaller distortion is (20) (a) Benioff, P. A.; Das, G.; Wahl, A. C. J . Chem. Phys. 1977, 67, 2449. (b) Das, G.; Benioff, P. A. Chem. Phys. Lerr. 1980, 75, 519. (21) (a) Clyne, M. A.; Thrush, B. A. Proc. R. SOC.London, Ser. A 1961, 261, 259. (b) Wilson, C. W. J . Chem. Phys. 1975, 62,4842. (22) (a) Rahbee, A.; Gibson, J. J. J . Chem. Phys. 1981, 74, 5143. (b) H e m , R. R.; Sullivan, B. J.; Whitson, M. E. J . Chem. Phys. 1983, 79, 2221. (c) Winkler, I . C.; Stachnik, R. A.; Steinfeid, J. I.; Miller, S . M. J . Chem. Phys. 1986, 85. 890.
Arteca and Mezey -0 78
-0.82
0
2
4
6
8
10
A r c Length ( A ) Figure 6. Reaction barrier for the N + O2 NO + 0 bimolecular collisional reaction, with the arc length along the reaction path as reaction coordinate. Masses are measured in atomic units to compute the arc length in mass-weighed coordinates. Shaded regions represent the uncertainty in the boundaries where the change in shape for the molecular surface occurs. This uncertainty is due to the range of possible choices of VDW radii for each atom.
-
needed from the end-point geometry to reach a structure with the same shape as the TS. In ref 20 this reaction is regarded as possessing a single TS, even though two maxima are clearly seen as in Figure 6. Our results provide a criterion to establish that only one essential intermediate molecular shape occurs, common to both formal transition structures and the intermediate: the two maxima are “shapewise nondistinct”. This notion of “shapewise distinctness” of chemical structures along reaction paths gives a measure for structural similarity.
Further Comments and Conclusions Molecular structures can be defined as open sets in configurational space, corresponding to all the nuclear geometries that have a molecular surface with the same topological shape features. If the surface is chosen as a VDWS and its shape descriptor as the graph g(G(K)), then a molecular structure is represented by the equivalence class defined by a common graph g(G(K)) in the configuration space. This description is similar in spirit to the definition of molecular structure of Tal et al.5 In that case, structures are equivalence classes of the so-called “molecular graphs” which are constructed upon studying local and global properties of the total electronic density, and its derivatives, over the entire 3-space. However, those graphs bear no clear relationship with the shape of the 3D molecular “body”, neither are they related to a finite, closed molecular surface. All the nuclei appear as vertices of the graphs of Tal et aL5 In our case, some atoms may contribute no vertices if they are buried within the molecular surface. This is not an artificial feature of a VDWS. As a matter of fact, if the molecular surface is represented by a constant-density contour, a similar effect can take place: when the density value of the contour is low, then the shape of the surface is determined by the atoms in the peripheral regions of the molecule. The topological definition of molecular structures in terms of 3D molecular shape descriptors leads to new insights. The reinterpretation of the Hammond postulate according to similarity defined by shape descriptors may reclassify some reactions, showing a formal violation according to the standard formulation. In a previous s t ~ d y ,we ’ ~ have shown the conditions that a general family of one-dimensional potential energy functions must satisfy to follow the Hammond postulate. These conditions have been given in terms of ranges of values of internal forces and force constants. The present results provide another point of view: the relevant domains of the nuclear configurations are determined by the invariance of the topological shape of the molecular envelope. The method discussed in this work can be extended easily to molecular surfaces defined by conditions based on physical observables, for example, charge densities, instead of artificial constructions, such as VDWS’s. In these cases, the shape characterization can be accomplished by using some of the
J . Phys. Chem. 1989, 93, 4751-4756 methods described in ref 11. The shape groups, shape graphs, and other shape descriptors of charge densities, as functions of nuclear configurations, are associated with various domains of the configuration space.lldslZa,bIn the general case, the formal reaction path passes through several of these shape domains of the configuration space, and segments of the path can be characterized by the shape domains they belong. The order of occurrence and the relative lengths of these path segments can characterize the shape changes of charge density during the reaction. Nonetheless, van der Waals surfaces approximate surprisingly well some isodensity surfaces, and many of the shape changes found and analyzed here are expected to be found also for isodensity surfaces.
4751
The VDWSs can be characterized by alternative methods. A global descriptor, for example, the change of the molecular volume along a reaction path, may provide helpful information. A continuous change in volume provides a continuous characterization. Discrete characterizatons, such as that provided by the graphs g(G(K)):are advantageous for automated, computer-assisted shape comparison. Acknowledgment. The authors thank a referee for useful suggestions. This work was supported by a research grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada. Registry No. H C N , 74-90-8; 02,7782-44-7; N, 7727-37-9.
Uncatalyzed and V(V)-Catalyzed Reaction of Methylene Blue with Potassium Bromate in Aqueous Sulfuric Acid G. K. Muthakia Department of Chemistry, University of Nairobi, P. 0. Box 301 97, Nairobi, Kenya
and S. B. Jonnalagadda* Department of Chemistry, University of Zimbabwe, Box MP 167, Harare, Zimbabwe (Received: July 25, 1988; In Final Form: December 2, 1988)
The uncatalyzed and V(V)-catalyzed kinetics of reaction between methylene blue (phenothiazinium,3,7-bis(dimethylamino)-, chloride) and acidic bromate has been studied monitoring the absorbance of methylene blue (MB) at 665 nm. Both the reactions involved competitive and sequential steps, having slow reaction in the initial stages. A rapid reaction followed after an induction time. For the two reactions, the orders with respect to the reactants are the same: second order with H+,first order each with respect to bromate ion and MB. In addition, the catalyzed raction had first order dependence on catalyst concentration. In both the reactions HOBr is found to be the reaction intermediate. HOBr competed with bromate ion chloride. in the depletion of MB to give an intermediate, possibly phenothiazin-5-ium, 3-methylamino-7-dimethylamino-, The intermediate is further oxidized possibly by HOBr in fast step to final product, phenothiazin-5-ium, 3-amino-7-dimethylamino-, chloride. The stoichiometric ratios of MB to bromate are 2:3. The dual role of bromide ion as an inhibitor at low concentrations and as an autocatalyst at higher concentrations above a certain critical concentration in the reaction mechanism is discussed.
Introduction The chemistry of reactions involving acidic bromate ion has been extensively studied in the past decade due to its unique capability to generate complex temporal behavior in closed systems during oxidation of certain organic substrates. Elegant schemes and models were proposed for the metal ion catalyzed and uncatalyzed reactions leading to oscillatory phenomena.'-' A number of reactions have also been reported in the literature involving studies of various inorganic and organic reagents using acidic bromate as Indicator reactions with bromate ion in acid medium were also reported for kinetic determination of MO(VI),*OS(VIII),~V(V),1° etc. Methylene blue (phenothiazin-5-ium, 3,7-bis(dimethylamino)-, chloride) (MB), an intense blue dye, is known as a staining and sensitizing agent in biological reactions." MB is also used as a catalyst, as a polymerization inhibitor, and as a complexing agent in a number of studies.'* Burger and Field reported an uncatalyzed oscillatory reaction between MB and s ~ l f i d e . ' ~Literature survey shows no other kinetic studies were done using M B and any oxidizing agent. In pursuit of a selective indicator reaction for analysis of V(IV)/V(V), using acidic bromate as oxidant which is known to be catalyzed by V(V),I4 a number of organic reagents were scanned. During the preliminary investigations, MB was found *Towhom correspondence should be addressed. 0022-3654/89/2093-4751$01 S O / O
to satisfy certain criteria as substrate: a sharp absorption peak (A, 665 nm, In e 4.95) in the visible region, and a positive response to the presence of V(V). Further, when the kinetics were monitored, MB was found to have two depletion steps in reaction with acidic bromate: an initial slow step followed by a fast one after an induction time, It. Again the induction period was sensitive to vanadium(V) concentration. Hence, the kinetics of uncatalyzed and vanadium(V) ion catalyzed reactions between MB and potassium bromate in aqueous sulfuric acid were studied in detail. In this communication, we summarize the results of the (1) Noyes, R. M.; Field, R. J.; Koros, F. J . Am. Chem. SOC.1972, 94, 1394. (2) Jonnalagadda, S. B.; Srinivasulu, K. 2.Phys. Chem. (Leiprig) 1978, 259, 1191. (3) Herbine, P.; Field, R. J. J . Phys. Chem. 1980, 84, 1330. (4) Ruoff, P. J . Phys. Chem. 1984, 88, 2851. (5) Mottola, H. A.; Mark Jr., H. B. Anal. Chem. Rev. 1980, 52, 31R. (6) Jonnalagadda, S. B. Int. J . Chem. Kinet. 1984, 16, 1287. (7) Jonnalagadda, S. B.; Muthakia, G . K. J . Chem. SOC.,Perkin Trans. 2 1987, 1539. (8) Bartan, A. F. M.; Loo, B. J . Chem. SOC.A 1971, 19, 3032. (9) Rao, N . V.; Ramana, P. V. Microchim. Acra 1981, 11, 269. (10) Jonnalagadda, S. B. Anal. Chem. 1983, 55, 2253. (1 1) Santus, R.; Kohen, C.; Kohen, E.; Rcyftmann, J. P.; Morliere, P.; Dubertret, L.; Tocci, P. M. Photochem. Photobiol. 1983, 38, 71. (12) Cizek, Z.; Studlarova, V. Talanta 1984, 31, 547. (13) Burger, M.; Field, R. J. Nature 1984, 307, 720. (14) Otto, M.; Stach, J.; Kirmse. R. Anal. Chim. Acta 1983, 147, 277.
0 1989 American Chemical Society
