J. Phys. Chem. 1995, 99, 4947-4954
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Two Symmetry Constraints on the Identity and Deformations of Chemical Species Paul G. Mezey Mathematical Chemistry Research Unit, Department of Chemistry and Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada, S7N OW0 Received: October 26, 1994; In Final Form: December 16, 1994@
Two point symmetry rules are proven, specifying constraints on the extent of deformations of nuclear configurations which preserve the identity of chemical species. These rules formalize some of the relations between nuclear point symmetry and the gradient field within relaxed cross sections of potential energy hypersurfaces. Relaxed cross sections have lower dimension than the potential energy hypersurface; consequently, their generation by relaxation algorithms is simpler than the computation of domains of a hypersurface of high dimensions. These rules have utility in the search for critical points within relaxed cross sections, relying on the easily calculable point symmetry properties of nuclear arrangements.
1. Introduction Most small deformations of molecules are not expected to lead to a change in chemical identity. Within a semiclassical model a chemical species, such as a stable conformer, is regarded as a dynamic, vibrating object that may take on a whole continuum of possible shapes,l without changing into a fundamentally different conformer. For an essential conformational change, involving a change of chemical identity from one stable conformer to another or from one molecule to another, usually a substantial deformation is required. Within the semiclassical model, if the electronic state is specified, the formal nuclear arrangement determines the electron distribution. Consequently, it is a common although not a fully rigorous approach to associate chemical identity with a set of nuclear arrangements. Which sets of nuclear arrangements can be regarded to “belong” to a given chemical species? Clearly, energy relations are important: different chemical species are separated by energy barriers. For a systematic treatment of the energy relations, one should consider the potential energy surface E(K) of the specified electronic state for the given set of nuclei (or, equivalently, for the given stoichiometric family of chemical species), where K denotes the nuclear configuration. Such potential surfaces E(K) are defined over the corresponding nuclear configuration space M , a metric space that is, however, never a vector space,2 a feature that is a source of many counterintuitive properties and occasional misconceptions. According to one proposal, chemical identity is associated with formal catchment region^^,^ of potential energy hypersurfaces,* relying on analogies with a mathematical model of “watersheds”, originally derived for geographical terrain^,^.^ and on the differential geometric model7-I0 of the intrinsic reaction coordinate. We adopt an earlier notation:2 K(A,i) is a critical point of the potential energy hypersurface E(&, where critical point index A (the number of negative eigenvalues of the Hessian matrix at critical point K ( l , i ) )expresses curvature information and i is a serial index. The catchment region C(A,i) is the collection of all nuclear configurations K E M , from where a formal, infinitely slow, vibrationless relaxation leads to the critical point K(A,i). In the above sense, a catchment region C(L,i) is the collection of all distorted nuclear arrangements K that are “K(il,i)-like” in a chemical context, since if these K @Abstract published in Advuncr ACS Ahsrrcrct.s, March 1, 1995.
arrangements are allowed to relax infinitely slowly and without vibration, then they change back to the configuration K(A,i). The catchment regions C(A,i) appear as the “natural” subsets of the nuclear configuration space M to represent chemical species, each such subset specifying the limits of deformations which preserve chemical identity in the above sense. Catchment regions C(A,i) of a given potential energy hypersurface E(K) of the given electronic state, usually the ground state, represent all chemical species of the stoichiometry associated with the configuration space M,3.4 including individual stable conformers of molecules, transition structures, and the stable relative arrangements of two or more molecules with a specified overall stoichiometry.* Clearly, there is a one-to-one correspondence between critical points K(A,i) and catchment regions C(A,i). Note that the above definition of chemical species is more restrictive (and arguably more precise) than the intuitive concept commonly used by chemists. In particular, we formally treat transition structures and the associated lower dimensional catchment regions as distinct “species”. For example, eclipsed ethane is a transition structure for an interconversion process between two staggered ethane minima, and according to the catchment region concept, we formally regard eclipsed and staggered ethanes to represent different chemical species, although ordinarily both structures are called ethane. The degree of distortion2 of any nuclear configuration K can be expressed in terms of the metric d of the space M,”-” with reference to any selected reference configuration KO, for example, to the critical point K(A,i)of a catchment region C(A,i):
This configuration space distance d(K,Ko) can also be regarded as a measure of dissimilarity2 between the two configurations K and KO. The point symmetry groups of the nuclear arrangements have important relations to the patterns of catchment regions and critical points of potential energy hypersurface^.'^ The point symmetry of a given nuclear arrangement is independent of the electronic state, that is, of the actual potential energy hypersurface. Consequently, many conclusions based on symmetry are universal for all energy hypersurfaces defined over a given nuclear configuration space M.I4 Use of some of the symmetry theorems requires only a simple test for the presence of some symmetry elements within a family of nuclear configurations, and conclusions on topological features of potential energy
0022-3654/95/2Q99-4947$09.Q0/0 0 1995 American .Chemical Society
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surfaces are obtained without any quantum chemical calculation.)“ (Note, however, that the determination of relaxed cross sections, vide infra, usually involves such calculations.) Here we list some relevant earlier results. (A) The catchment region point symmetry theoremI4 states the following: Within each catchment region the highest point symmetry occurs at the critical point. The term “higher point symmetry” refers to group-subgroup relations. If point symmetry group g’ is a subgroup of group g (g f g’), then g is of higher symmetry than g’. Each g is regarded as one of its own subgroups. One can rephrase this statement: no catchment region contains a nuclear configuration of a point symmetry group that is not a subgroup of the point symmetry group of the critical point configuration. (B) The vertical point symmetry theorem of nuclear configuration spaces gives a relation between the location and the point symmetries of nuclear configurations of critical points found on different potential surfaces corresponding to different electronic states. Divide the nuclear configuration space M into two parts, M I and Mr,by a hypersurface B that belongs to set M I(MI is a closed set). Let R’denote a family of some symmetry elements, R‘I,R‘z, ... R’,,, present for each nuclear configuration 6 along the boundary B:
R’= {R’l, R’?, ... R’,,} Choose some nuclear configuration K from set M I , and let R denote a family of some symmetry elements R I ,Rz, ... R, that are present at point K:
(3) The vertical point symmetry theorem of nuclear configuration spaces’“ states the following: (i) if no configuration along B possesses the family R of symmetry elements or (ii) if configuration K does not have all the symmetry elements of family R , then the family M Iof configurations must contain at least one critical point for the potential energy surface of each electronic state (of each possible overall electronic charge). (If condition (i) or (ii) is fulfilled, then K must be an interior point of M I ; that is, K cannot fall on the boundary B.) The proofs of these and some additional theorems, as well as several examples, can be found in ref 14. (C) The family of all point symmetry groups of all distorted nuclear configurations of a chemical species is an algebraic lattice. I This result follows from the catchment region point symmetry theorem. (D) Each point symmetry group g(K)present for any distorted nuclear configuration K of a (3N - 6 - A)-dimensional catchment region C(1.i) is also present at some configuration K‘ on a (3N - 6 - A - I)-dimensional sphere S(C(I,i),r)of center K(A,i) and radius r small enough so that S(C(A,i),r)falls within the catchment region C(A,i).I h The distortions are considered with reference to the equilibrium (critical point) nuclear configuration K(I,i), itself not regarded distorted. By testing the point symmetries only along a small sphere of distorted configurations around the critical point K(A,i),this theorem gives information on the extent of a catchment region C(A,i)by finding out all point symmetry groups which occur in C(1,i). Note that it is a much simpler task to find a critical
point of a potential surface and a small sphere around it than to find the boundaries of a catchment region. This theorem gives different information than the catchment region point symmetry theorem. It is not necessary for a catchment region C(A,i)to contain configurations having point symmetries corresponding to each subgroup of the group g(K(A,i))at the critical point K(2,i). (The same group g(K(A,i)) may be obtained by combining two different collections of its subgroups.) This theorem is the basis for the representation of all point symmetries of catchment regions by simple spherical maps.16 An interesting result is obtained by a homotopical transformation of these spheres.I6 By “blowing up” such a spherical map to “fill out” the catchment region C(A,i), one can reach the boundary hypersurface of C(A,i). The point symmetry group of the nuclear configuration at each point of the boundary must have a point symmetry group which contains the group of the corresponding point on the “blown up” sphere as subgroup. Consequently, the spherical symmetry map of the catchment region is “symmetry-dominated from both “inside” (by the critical point K(A,i))and “outside” (by the boundary of C(A,i)).16
2. Comments on Relaxed Cross Sections of Potential Energy Hypersurfaces and Configurational Spaces Not all internal coordinates of a molecule are equally important in some chemical processes. For example, the changes in C-H bond lengths in methyl groups are often unimportant and negligible in some chemical reactions. In such cases, the dimension of the subset of the configuration space used for the description of the problem can be reduced by taking relaxed cross sections. Relaxation is defined by having zero energy gradient (force) components locally orthogonal to the cross section. We shall assume that the potential energy hypersurface E(K) is everywhere twice continuously differentiable; if this condition is not fulfilled, then some of the treatments described in ref 2 can be applied. All energy gradients and steepest descent paths are determined in terms of mass-weighted coordinates or internal coordinates derived from them, in order to ensure that steepest descent paths correspond to relaxation paths, a necessary condition for the symmetry relations used.?.’4 We assume that the chemical problem involving N nuclei is confined to a (3N - 6)-dimensional subset A of the configuration space M,provided with a local coordinate system, where still all 3N - 6 of the internal coordinates are considered. We also assume that the first n of the internal coordinates (n 5 3N - 6) are the chemically important active coordinates for the chemical process, whereas the remaining n’ coordinates (n’ = 3N - 6 n) are regarded as the passive coordinate^.'^^'^ This does not mean that the passive coordinates cannot change their values in the process considered, but we usually expect that their changes are smaller than those of the active coordinates. However, all derivations in this paper are valid for any choice of the active coordinates; it is merely useful to designate as passive those coordinates which do not change much in the actual chemical process studied. In practice, this choice is dependent on the electronic state, that is, on the actual potential energy hypersurface E ( K ) . For any choice of the n active coordinates within set A, the values of the n’ passive coordinates are regarded as functions of the active coordinates, leading to the definition of relaxed cross sections:’.’“ An n-dimensional relaxed cross section C is characterized by the condition that at every point of C the gradient vector has vanishing components orthogoiial to C, if energy gradients are expressed with respect to the actual energy hypersurface
Two Symmetry Constraints
E(K). Such a set C is called an n-dimensional relaxed cross section of A according to the potential energy surface E(K). Note that for a poor choice of A, or for a poor choice of the n active coordinates, the relaxation may locate the surface C outside of A. Also note that for a given choice of the n active coordinates several relaxed cross sections may exist. For example, take two nearly parallel mountain chains, approximately aligned with the N-S direction in a geographical map, and declare the N-S and E-W displacements as the active and passive coordinates, respectively. Three relaxed cross sections are the line C along the valley bottom, and two lines, C' and C",along the two moutain ridges, defined by the condition that all forces are zero locally perpendicular to each of C, C , and C". Only the line C along the valley bottom has the property that each point of C is a minimum for displacements orthogonal to C the points of relaxed cross section C' are maxima for displacements locally orthogonal to C". A rope placed exactly along the ridge C' would not roll down on either side of the mountaiq chain; line C is, indeed, a relaxed cross section. If, however, the rope is subject to small E-W perturbations, then it may roll down to the lower, more stable relaxed cross section
C. Infinitely many relaxed cross sections pass through each critical point K(A,i) of a potential energy hypersurface (for an illustration of this property using the simplest, one-dimensional relaxed cross sections and for clarification of common misconceptions, see, for example, p 94 and Figures 11.9, 11.10 of ref 2). If the active coordinates are selected, starting from a critical point K(A,i), several different relaxed cross sections can be obtained. The following property is often overlooked in computational studies: simple energy minimization along the passive coordinates does not necessarily lead to a truly relaxed surface C, since along C the passive coordinates may also change. Vanishing energy derivatives along the passive coordinates imply a fully relaxed cross section C only if C is everywhere locally orthogonal to the subspace of the passive coordinate^,'^ a condition only seldom fulfilled. At each point K of C,the gradient vector is tangential to C; hence, the "gradient flux" is zero across C. The potential energy hypersurface E(K), restricted to C,is a function of the n active coordinates and is called an n-dimensional relaxed cross section of the original potential energy surface E(K). A relaxed cross section C, is determined by the jth potential energy hypersurface E,(K) (by the jth electronic state) according to which it is "relaxed". Whereas C, is a (possibly) curved surface that cuts across (some part of) the configuration space M , all the C, relaxed cross sections have projections to a common n-dimensional subspace Cnof the active coordinates. When set A is bounded, then the cross section C is also limited to a finite configurational domain that may have a boundary. If, however, the cross section C has no boundary within the interior of M , then its projection in Cnmay cut across the entire configuration space; in this case C and its projection in c" represent a complete relaxed cross section. A steepest descent path cannot leave a complete relaxed cross section. If, however, the cross section C is not complete, then it is possible for a steepest descent path to leave C, but only along a direction locally tangential to C at one of its boundary points. In the following we shall consider complete relaxed cross sections. Assume that a critical point K(1,i) of E(K) falls within a relaxed cross section C. The catchment region C"(A,i) of the relaxed cross section C is the collection of all the nuclear configurations of C from where steepest descent paths lead to the critical point K(A,i). In a complete relaxed cross section C no further conditions are needed: clearly, a steepest descent
J. Phys. Chem., Vol. 99, No. 14, 1995 4949 path from any point K of C must stay within C and must lead to a critical point within C. Complete relaxed cross sections C have many properties analogous to those of complete nuclear configuration spaces M . In particular, some of the theorems and rules listed in the previous section can be reformulated for relaxed cross sections of potential energy hypers~rfaces.'~.''For example, the relaxed cross section version of the catchment region point symmetry theorem states the following: Within each catchment region C"(A,i) of a relaxed cross section C the nuclear configuration corresponding to the critical point K(1, i ) has the highest point symmetry.
3. Two Symmetry Relations between Arbitrary Domains of Nuclear Configurations and Catchment Regions within Relaxed Cross Sections The main advantage of relaxed cross sections to the complete energy surface is their lower dimensions. A relaxed cross section can be obtained without the computation of steepest descent paths and algorithms following such paths. A relaxed cross section C can be found by moving an approximate cross section locally orthogonal to paths of steepest descent. These motions are continuous deformations of approximate cross sections which belong to a class of higher dimensional homotopies2 The process requires only the computation and minimization of selected gradient components locally orthogonal to the cross section, which implies considerable computational savings. These savings are realized at a price: it is possible to obtain a relaxed cross section without detecting any of the critical points which may belong to the cross section. However, the new symmetry rules described in this study provide altemative tools for critical point detection. These rules utilize the fact that it is simpler to determine the point symmetry of nuclear configurations than to calculate all the components of the gradient throughout a relaxed cross section. The motivation for the new symmetry rules is related to a rather general problem. Let us assume that a relaxed cross section is determined, and we are interested in the extent of those deformations within the cross section which preserve chemical identity. In Figure 1 two simple examples are shown where the C - 0 bond rotations are taken as the active coordinates. Without additional energy calculations, symmetry relations provide constraints on the extent of identity-preserving deformations. As we shall see later, in this example simple inspection can provide the same answers as the application of the rules; however, in higher dimensions, inspection is no longer reliable, yet the rules are applicable. In practice, the computation of relaxed cross sections can be accomplished using the analytic energy gradient methods available in most ab initio program packages. It is a much simpler task to determine and analyze a relaxed cross section of low dimension than to study a dimensionally unrestricted domain of the potential energy hypersurface for the same process. This is the motivation for the proposed symmetry rules, aiding the analysis of relaxed cross sections. If a relaxed cross section is specified and if the conditions of the rules are met, then constraints are obtained on the extent of deformations preserving chemical identity. Of course, these limitations on chemical identity preserving deformations remain valid within the full configuration space. Before introducing the main results of this report, we rephrase the rule on symmetry dominance in an alternative form: Within each catchment region Cn(l,i), the nuclear configurations of lower symmetry are "nested" between configurations of higher (or possibly equal) symmetry, located at the critical point K(A,i) and at the boundary hypersurface of C"(2,i).
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Figure 1. Two conformers of 2,4,6-trifluorophloroglucinol, examples for the application of rule 1.
With the exception of the vertical point symmetry theorem, the symmetry theorems and rules listed in the Introduction refer to subsets of the nuclear configuration space that are constrained by the catchment regions C(A,i). In the case of the catchment region point symmetry theorem, comparisons are made between all point symmetry groups within C(A,i) and the group at the critical point K(A,i), whereas the spherical symmetry map approach requires having the chosen sphere S(C(A,i),r)entirely contained within C(A,i). In both cases, some information concerning the location and extent of the actual catchment region C(A,i) is required. Such information, however, is not always readily available, and it is useful to explore altemative symmetry constraints not requiring knowledge of the catchment region boundaries. Rule 1. Consider a relaxed cross section C and two connected, open subsets A and A’ of C:
A,A’c C C M
(4)
We emphasize that there are no further restrictions on sets A and A‘; in particular, A and A’ are allowed but not required to intersect, and both may involve various parts of several catchment regions Cfi(A,i). Take two nuclear configurations K and K’, K # K’
(5 )
one from each of A and A‘, K E A
(6)
K‘ E A’
(7)
and
Assume that nuclear configuration K has a point symmetry element R not present anywhere else in A , and nuclear configuration K’ has a point symmetry element R‘ not present anywhere else in A’. Then K and K‘ are the critical points, hence the lowest eneigy points, of two dzflerent catchment regions of the relaxed cross section C: K = K(A,i) E c‘(A,i>
(8)
and K‘ = K(A’,i’) E C(A’,i’) where i
f
(9)
i’
The proof given below easily follows from the nile of conservation of point symmetry elements along steepest descent paths between critical point^'^.'^ and from the “zero gradient flux” property of relaxed cross sections.14.” The relaxed cross section C is a subset of M , C c M, hence the metric d of M is applicable within C, and balls of positive radii can be given for each point of C. Set A is an open set within the relaxed cross section C; hence there must exist an open ball B(K,r) of some positive radius r around point K E A such that B(K,r) c A . If K were not a critical point, then there would exist a path of steepest descent leading away from K. Since point symmetry elements are preserved along segments of paths of steepest descent containing no critical point (if steepest descent is defined in a mass-weighted coordinate system), consequently, the point symmetry element R must be present along a finite segment of the path leading away from K . Since C is a relaxed cross section and B(K,r) C A C C, this path must have a finite segment within B(K,r). However, according to the conditions of the rule to be proved, the point symmetry element R of K is not present anywhere else within B(K,r),a contradiction. Consequently, a point K with properties specified in the rule must be a critical point K = K(A,i) E c(A,i) for some catchment region of the relaxed cross section C. The same argument can be repeated for K’ E A’ and R’, resulting in the conclusion 6 = K ( 2 , i ‘ ) E Cn(A’,i’) C C. Since K f K’, and each catchment region has precisely one critical point, Cfi(A,i)and C‘(A’,i’)must be different, i i’ (Q.E.D.). Note that there is no restriction on the two critical point indices A and A‘; they may agree or differ. Similarly, the point symmetry elements R and R‘ may agree or differ. This rule provides some constraints on the extent of deformations preserving chemical identity: clearly, neither K nor K‘ can be deformed into the other without a change of chemical identity. Furthermore, if by the inexpensive determination of point symmetry elements within A and A’ such points K and K’
*
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are found, then the expensive energy minimizations of the more conventional critical point searches can be avoided. Rule 2. Consider a relaxed cross section C, nuclear configuration K , open set A , and symmetry element R fulfilling the same conditions as in rule 1:
KEAcCCM
(10)
Assume that nuclear configuration K has a point symmetry element R not present anywhere else in A; furthermore, consider another nuclear configuration K‘ with a point symmetry element R’ not present anywhere in A K (that is, in the set obtained from A by removing point K), where K # Kf
(11)
Then, K is a critical point, hence the lowest energy point of a catchment region of the relaxed cross section, and the nuclear configuration K‘ cannot be a distorted form of the chemical species of equilibrium configuration K. The proof relies on rule 1. Clearly, K is an equilibrium nuclear configuration, since it is a critical point K = K(A,i) E P ( l , i ) ,as follows from rule 1. Take the ball B(K,r) around K , as in the proof of rule 1. If nuclear configuration K‘, different from K , were to belong to the same catchment region Cn(A,i), then it could not be a critical point itself, and the unique steepest descent path from K f would have to lead to K , without encountering any other critical point. This path would necessarily have a segment within the open ball B(K,r), if it is to reach K . The conservation of point symmetry element R’ of K‘ along this path implies the presence of R‘ for all nuclear configurations of the segment of the path within B ( K , r ) K . However, according to the conditions of the rule to be proved, the point symmetry element R‘ of K‘ is not present anywhere within B ( K , r ) K , a contradiction. Consequently, nuclear configuration K‘ cannot belong to the catchment region of K , that is, K’ cannot be a distorted form of the chemical species of equilibrium configuration K = K(A,i) E Cn(l,i)(Q.E.D.). This result provides a different constraint of the extent of deformations that preserve chemical identity. If the distribution of symmetry element R in A and the lack of symmetry element R‘ in A K are confirmed, then symmetry considerations alone are sufficient to dismiss a nuclear arrangement K‘ as a distorted form of a given chemical species. Note that there is no restriction on the two point symmetry elements R and R‘; they may agree or differ. If, however, R and R‘ are different, then one of the conditions of rule 2 can be simplified and we may obtain a weaker but simpler rule by replacing the expression “the point symmetry element R’ not present anywhere in A K ” with “the point symmetry element R’ not present anywhere in A ”.
4. Examples The two rules are proposed for applications when (i) a relaxed cross section C has been determined and (ii) the point symmetries of the corresponding nuclear configurations are also specified. If the symmetry conditions of the rules are fulfilled, we conclude that certain configurations must be critical points and that limitations on the identity-preserving deformations apply. One should emphasize that any set of active coordinates can be used for generating relaxed cross sections. Specifically, there is no restriction that only chemically reasonable candidates for reaction coordinates can serve in the definition of relaxed cross sections. Whether the coordinates themselves are important or not as potential reaction coordinates is immaterial; what matters is the conclusion one obtains using these coordinates. In this
context, there is no such thing as a chemically dubious choice of active coordinates, only choices which do or don’t lead to an easy recognition of deformability properties within a subset of the full space. The point is that using low-dimensional relaxed cross sections based on judiciously selected active coordinates, one obtains conclusions on deformability constraints valid within the full space. In Figure 1, two nuclear configurations of the molecule 2,4,6trifluorophloroglucinol are shown, serving as simple, easily visualizable examples for the application of rule 1. The active coordinates are selected as the three-bond rotation angles along the carbon-oxygen bonds. These coordinates happen to be “reasonable choices” for conformational coordinates. Evidently, a nuclear configuration K I of c3h symmetry and a nuclear configuration K2 of C, symmetry, similar to those shown in Figure 1, must belong to the corresponding three-dimensional relaxed cross section C. Around K I and K2 there exist small, three-dimensional balls A I and A2, where within these balls only K I and K2, respectively, possess the point symmetry element u. The conditions of rule 1 are fulfilled. We conclude that K I and K2 are the critical points of two different catchment regions, and neither K I nor K2 is a distorted form of the chemical species (conformational catchment region) represented by the other configuration. This example is rather simple, and the conclusions obtained using rule 1 can also be obtained by simple, intuitive reasoning. Our next example is only slightly more complicated, primarily due to a more congested structure; however, it will also serve as an example for a generalization of rule 1. In Figure 2, three nuclear configurations of the halogenated isobutane HC(CHX2)3 are shown. In this family of molecules, the actual most stable nuclear geometry is dependent on the nature of the halogene X; nevertheless, a generalization of rule 1 provides a symmetry constraint common to all these molecules. The generalization is rather straightforward; the point symmetry elements R and R’ of rule 1 are replaced by a family R and a family R’ of point symmetry elements, respectively. By these replacements, a proof of identical structure applies for the generalized rule. The active coordinates are selected as the three bond rotation angles along the carbon-carbon bonds. These active coordinates are also “reasonable choices” for conformational coordinates. The nuclear configuration of structure A of C3,.symmetry is taken as K I , that of structure B (also of C3” symmetry) is taken as K2, and that of structure C of C3 symmetry is taken as K3. All three of these nuclear configurations must belong to the three-dimensional relaxed cross section C corresponding to the given choice of the active coordinates. Around both K I and K2 there exist small, three-dimensional balls A I and A2, where within these balls only K1 and K2, respectively, possess all three uL,point symmetry elements; that is, both nuclear configurations K I and K2 are entirely surrounded by others of lesser symmetry. If we take
then the conditions of the generalized rule 1 are fulfilled. Consequently, K1 and K2 must be critical points of two different catchment regions, and neither K I nor K2 can be a distorted form of the chemical species represented by the other configuration. It is also clear that structure C of symmetry C3, exhibiting partial hydrogen bonds, may undergo deformations within the same relaxed cross section C while preserving its C3 symmetry. This structure serves as an example where the conditions of rule 1 are not fulfilled; hence the rule cannot be applied. This does not exclude the possibility that for some choices of halogen X structure C is also a critical point of some catchment region.
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El
Figure 2. Three nuclear configurations of the halogenated isobutane HC(CHX& examples for the application of the generalized rule 1.
The two nuclear configurationsof 1,1,1,3-tetrachloropropane shown in Figure 3 serve as our first examples for the application of rule 2. We assume that we are interested primarily in two internal rotations, those about the two C-C bonds. The a and p rotation angles of the respective molecular fragments of fewer nuclei are defined in a counterclockwise sense. Both a and p are set equal to 0 in the C, nuclear configuration K I at the top of Figure 3. These a and p angles are taken as the active local coordinates, generating the corresponding two-dimensional relaxed cross section. In the lower part of Figure 3, a second C, nuclear configuration, K2, is shown. The first C, nuclear configuration K I can be converted into K2 along some (not necessarily gradient following) path entirely contained within this relaxed cross section. For example, a continuous increase of p to p' = p 180" can accomplish this transformation. Within the relaxed cross section, the nuclear configuration K1 is entirely surrounded by others of lesser symmetry: within a small ball A around nuclear configuration K I, only K I possesses the point symmetry element a (reflection plane). The nuclear configuration K2 also has a point symmetry element a; note, again, that this symmetry element is not present anywhere in the set Awl. The conditions of rule 2 are fulfilled. Consequently, nuclear configuration K I is a critical point, it has lower energy than any other configuration of the corresponding catchment region within the relaxed cross section, and K2 is not a distorted form of the chemical species (conformer) represented by its equilibrium structure K I at the top of Figure 3. In Figure 4, two nuclear configurations of the molecule 1,3diphenylpropane are shown, as the last example for rule 2. These
+
conformations serve as examples for a case that is already complex enough, where simple intuitive reasoning may miss the conclusions obtained when applying rule 2. Again, we assume that we are interested primarily in the two internal rotations about the C-C bonds marked in black, where a and p are the rotation angles of the respective smaller molecular fragments, defined in a counterclockwise sense, being equal to 0 in the C., nuclear configuration K1 shown on the left-hand side of Figure 4. We choose a and p as the active local coordinates and generate a corresponding two-dimensional relaxed cross section C. This choice of coordinates might appear odd. However, one should recall that any choice of the active coordinates corresponds to a valid cross section of the potential surface, and by a judicious choice of the active coordinates, the generation of the relaxed cross section and the application of the rule lead to information on the extent of deformations preserving chemical identity. Although the result is obtained within a relaxed cross section that is much easier to calculate than the full, (3N - 6)-dimensional potential surface, nevertheless, the conclusion on the extent of deformations is valid within the full configurational space. Note that there are four C-C bond rotations which dominate the conformational processes of this molecule. Consequently, we shall focus on these four coordinates. For simple visualization (only for visualization) we might assume that all other internal coordinates are fixed, and essentially we deal with a 4D conformational problem. A cross section C of active coordinates a and p involves only small energy variation within the local neighborhood of configuration K I , allowing the two phenyl rings to slide along each other, accompanied by slight
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1c,I
Figure 3. Two nuclear configurations of 1,1,1,3-tetrachloropropane,examples for the application of rule 2.
Figure 4. Two nuclear configurations of the 1,3-diphenylpropane molecule, examples for the application of rule 2.
twisting of the C-Phe bonds. We find that within this cross section C there exists a set A containing the C,s nuclear configuration KI, where only K Ihas the point symmetry element (T (reflection plane); that is, configuration K 1 is entirely surrounded by a set A of other configurations of lesser symmetry. Furthermore, the C2 nuclear configuration K2, shown on the right-hand side of Figure 4, has the point symmetry element C2 (2-fold axis), not present anywhere within Awl. Consequently, the conditions of rule 2 are fulfilled, and we conclude that nuclear configuration Kl is a critical point; hence it has the lowest energy within the corresponding catchment region of the relaxed cross section C, and nuclear configuration
Kz is not a distorted form of the chemical species (conformer) represented by the equilibrium structure K I on the left-hand side of Figure 4. Of course, since KI is a critical point, for the corresponding fixed pair of values of the two active coordinates a and p, there must exist infinitely many relaxed cross sections passing through K I. Some of these cross sections contain additional points which preserve the C, symmetry of K I , whereas some other cross sections do not. The fact that the cross section C we have obtained contains a domain A about K I that has no additional configuration with C , symmetry is both possible and the very source of the conclusion that K I is a critical point. It is also
4954 J. Phys. Chem., Vol. 99, No. 14, 1995
clear that in a conformational change within a relaxed cross section the two phenyl groups cannot maintain their local C , symmetry if this motion is to lead to the C2 nuclear configuration K2 depicted. Note that the rules do not state what type of critical point is found; this point can be a minimum or a saddle point of some index I . If A = 0, then the chemical species is a stable conformer; if, however, I = 1, then the formal chemical species is a transition structure, by analogy with the geographical example given in the previous section. Furthermore, by allowing local changes of configurations orthogonal to C, it is possible to reach other critical points, lower in energy than either K I or Kz. However, the important point is that as long as the relaxation starts from a configuration of the relaxed cross section C, the gradient remains tangential to C and the relaxation path must stay within the cross section. This holds even if some configuration change orthogonal to the cross section could lead to an energy lowering: the rope laid along the mountain ridge is relaxed, even if a locally orthogonal displacement of the rope could send it rolling down the slope. Note that relaxation is regarded in the strict sense: following a force. Specifically, in the context of the rules, we do not regard a motion relaxation if it involves motion along a zero component of the energy gradient. The rule is suitable to detect the fact that if the symmetry conditions are fulfilled within the cross section, then K I and, in the case of rule 1, also K2 are critical points, providing limits for deformations of the corresponding chemical species.
5. Conclusions and Comments Relaxed cross sections can be generated by gradient-following algorithms and by altemative techniques where only those gradient components are computed which are locally orthogonal to an approximate cross section. In the latter case, computational savings are achieved; however, the critical points falling on the cross section may remain undetected. Symmetry considerations, however, are useful for the detection of such critical points, as well as for detecting limits of deformations preserving chemical identity. The low dimension of cross sections, relative to the dimension of the full potential energy hypersurface, is an advantage that can be exploited. A special approach to the utilization of this feature, combined with a point symmetry analysis, is described in this report. Two simple rules are derived which give constraints on the extent of molecular deformations preserving chemical identity. Both rules rely on the easily calculable point symmetry groups of nuclear configurations; they are applicable within lower dimensional relaxed cross sections C of the full nuclear configuration space M . By contrast to earlier rules, these rules do not require a previous determination of any of the limit points of catchment regions of potential energy surfaces. Both rules and their proofs are phrased in the context of relaxed cross sections, and they apply to any relaxed cross section. However, one must keep in mind that if one declares all internal coordinates active coordinates, then the relaxed cross section C is the full configuration space M. Consequently, all results are also valid for M, for the ordinary catchment regions C(I,i),and for the corresponding subsets of M . Note, however, that for the full configuration space M a uniform “breathing” expansion of the molecule preserves all symmetry elements; hence within M the conditions of the rules are fulfilled only in trivial cases, such as the united atom configuration. Consequently, the rules are of little use if C = M . However, if the rules are used for relaxed cross sections of lower dimensions,
Mezey dim(C) .e dim(M), then the results are nontrivial and lead to conclusions valid within the full configuration space M , as shown by the examples. All the results presented here have been formulated for various point symmetry elements R and R‘. Note that analogous rules can be derived by replacing R and R‘ by framework group elements (R,P) and (R’,P), respectively, where P is the nuclear permutation operator formally “undoing” the effect of the point symmetry operator R. These operator pairs, as elements of the framework group, obey a multiplication rule (R,P) x (R’,P‘) = (R“,P“) = ( R x R’, P x P),and they, indeed, form a group. The actual pairing (R,P) may differ from nuclear configuration to nuclear configuration; the point symmetry operator R does not determine the permutation operator P in general, but there is a well-defined assignment for each nuclear configuration K . In either of the two altemative formulation^,'^.'^ framework groups provide more information on nuclear arrangements than ordinary point symmetry groups. Framework group elements (R,P)are also preserved along segments of steepest descent paths between critical points; hence the analogous rules 1 and 2 hold. Another generalization of these rules is obtained by applying them to the catchment regions of the “upside down” potential surfaces -E(K). Note that these formal catchment regions differ from the ordinary catchment regions in a nontrivial way; hence these versions of the rules provide new information. However, catchment regions on the “upside down” potential energy hypersurface -E(K) can no longer be associated with the concept of chemical species and with constraints on chemical identity preserving deformations. The rules and the outlined generalizations have utility in the search for critical points, on the basis of relaxed cross sections and easily calculable symmetry properties of nuclear arrangements.
Acknowledgment. This work was supported by an operating research grant from the Natural Sciences and Engineering Research Council of Canada and by the Computational Chemistry Laboratory of the Upjohn Company. References and Notes (1) Mezey, P. G. Shape in Chemisrty: An Introduction to Molecular Shape and Topology; VCH Publishers: New York, 1993. (2) Mezey, P. G. Potential Energy Hvpersurjaces; Elsevier: Amsterdam, 1987. (3) Mezey, P. G. Theor. Chim. Acta 1981, 58, 309. (4) Mezey, P. G. J . Chem. Phys. 1983, 78, 6182. ( 5 ) Cayley, C. A. Philos. Mag. 1859, 18, 264. (6) Maxwell, J. C. Philos. Mag. 1870, 40, 233. (7) Fukui, K. J . Phys. Chem. 1970, 74, 4161. (8) Tachibana, A.; Fukui, K. Theor. C h m Acta 1978, 49, 321. (9) Tachibana, A.; Fukui, K. Theor. Chim. Acra 1979, 51, 189. ( I O ) Tachibana, A.; Fukui, K. Theor. Chim. Acta 1979, 51, 275. ( I 1) Mezey, P. G. Int. J . Quantum Chem., Quantum. Biol. Symp. 1981, 8, 185. (12) Mezey, P. G. The Topological Model of Non-Rigid Molecules and Reaction Mechanisms. In Symmetries and Properties of Non-Rigid Molecules: A Comprehensive Survey; Maruani, J., Serre, J., Eds.; Elsevier Sci. Publ. Co.: Amsterdam, 1983; pp 335-353. (13) Mezey, P. G. Int. J . Quantum Chem. 1984, 26, 983. (14) Mezey, P. G. J . Am. Chem. Soc. 1990, 112, 3791. (15) Mezey, P. G. J . Math. Chem. 1990, 4, 377. (16) Mezev. P. G. Can. J . Chem. 1992. 70. 343 (suecial issue dedicated to Prof. S. Hizinaga). (17) Mezey. P. G. J . Math. Chem. 1993, 14, 79. (18) Pechukas, P. J . Chem. Phys. 1976, 64, 15 16. (19) Pople, J . A. J . Am. Chem. Soc. 1980, 102, 4615.
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