M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 50
Текст из файла (страница 50)
This is illustrated in Figure 7-4a.However, when the barrier of the reaction is low and the entropy variesrapidly in the region of the potential energy minimum, the transitionstate geometry differs from the transition structure (Figure 7-4b).As Williams stated [25]:The transition state is of strategic importancewithin the field of chemical reactivity.
Owing to itslocation in the region of the highest energy pointon the most accessible route between reactants andproducts it commands both the direction and therate of chemical change. Questions of selectivity(“Which way is it to the observed product?”) andefficiency (“How easy is it to get there?”) maybe answered by a knowledge of the structure andproperties of the transition state.The development of transition-state theory is due to Eyring andPolanyi [26], while the term transition state was first used by Evansand Polanyi [27].
Since then, it has been obvious that the properties of the region between the reactants and the products need to beFigure 7-4. Variation of ⌬G and E along a raction path, after Houk et al. [24],I, transition structure; II, transition state. (a) The transition state and the transition structure coincide; (b) The transition state and the transition structure differ.Adapted with permission.7.1. Potential Energy Surface319known in order to understand reaction mechanisms. However, the lifetime of the transition state is usually less than 10–12 s, and, therefore, for a long time this state could only be studied by theoreticalmethods. Only recently have experimental techniques become available that make the study of elementary reactions possible in realtime.
Direct measurements of the transition state have been carriedout using different sophisticated spectroscopic techniques (see, e.g.,References [28–30]). An example is the laser experiments that make itpossible to record snapshots of chemical reactions in the femtosecond(10–15 s) time scale, thus providing direct real-time observations of thetransition state [31–34].At the same time, with the ever increasing capabilities of computational techniques, it has become possible to calculate the details oftransition-state geometries and energetics with great precision.
Dueto the increasing reliability of quantum-chemical calculations on theone hand and to the possibility of real-time experimental observationof transition-state geometries on the other, the investigation of thestructure and dynamics of elementary chemical reactions has becomeone of the most exciting areas of modern chemical research. Nothingproves this better than the Nobel Prize for Chemistry in 1999, awardedto Ahmed Zewail “For his studies of the transition states of chemicalreactions using femtosecond spectroscopy.”7.1.2. Reaction CoordinateHow does symmetry come into the picture? It happens throughthe movement of the nuclei along the potential energy surface.
Asdiscussed in detail in Chapter 5, all possible internuclear motions of amolecule can be resolved into sets of special motions correspondingto the normal modes of the molecule. These normal modes alreadyhave a symmetry label since they belong to one of the irreduciblerepresentations of the molecular point group. The changing nuclearpositions during the course of a reaction are collectively described bythe term reaction coordinate. In simple cases, we may assume thatthe chemical reaction is dominated by one of the normal modes ofvibration, and thus this vibrational mode is the reaction coordinate.By selecting this coordinate, we may cut a slice through the potential energy hypersurface along this particular motion.
This was doneby Williams [35] in Figure 7-2 by cutting a slice of (a) in order to3207 Chemical Reactionsproduce (b). Figure 7-2b shows the reaction path along the reactioncoordinate. Points R and P are minima, corresponding to the initial(reactants) and final (products) stages of the reaction, while TS is thesaddle point corresponding to the transition structure and the energybarrier.The diagram in Figure 7-2b has several important features. First ofall, it represents only a slice of the potential energy hypersurface.
It isthe variation along one coordinate, and it is supposed that all the otherpossible motions of the nuclei, that is, all the other normal vibrations,are at their optimum value, so their energy is at minimum. Therefore,this reaction path can be taken as a minimum energy path. All otherpossible motions will be orthogonal to the reaction coordinate andwill not contribute to it. In other words, if we would try to leave thereaction path sideways, that is, along some other vibrational mode, theenergy would invariably increase.Figure 7-2a illustrates this point. The bold line shows the reactionpath.
It goes through a maximum point, which is the reaction barrier.The surface, however, rises on both sides of the reaction coordinate.Thus, with respect to the energy of the other vibrational coordinates,the reaction follows a minimum energy path indeed.7.1.3. Symmetry Rules for the Reaction CoordinateSymmetry rules to predict reaction mechanisms through the analysisof the reacton coordinate were first applied by Bader [36] (see, also,Reference [37]) and were further developed by Pearson [38].The energy variation along the reaction path can be characterizedin the following way. The energy of all vibrational modes except thereaction coordinate is minimal all along the path, i.e.,⭸E=0⭸Q iand⭸2 E>0⭸Q i2(7-1)where Qi is any coordinate (3N–7 for nonlinear molecules) except Qr ,the reaction coordinate. With respect to symmetry these vibrations areunrestricted.
(Of course, every normal mode must belong to one oranother irreducible representation of the molecular point group.)7.1. Potential Energy Surface321The energy variation of the reaction coordinate is different. At everypoint, except at the maximum and minimum values, it is nonzero,⭸E= 0⭸Q r(7-2)This is simply the slope of the curve on the potential energy diagram.At the minimum points (R and P in Figure 7-2a):⭸E= 0 and⭸Q r⭸2 E>0⭸Q r2(7-3)At the saddle point (TS in Figure 7-2a):⭸E= 0 and⭸Q r⭸2 E<0⭸Q r2(7-4)In order to predict reaction mechanisms and to estimate energybarriers, the energy can be expressed in terms of the reaction coordinate using second-order perturbation theory in such a way that theexpression contains symmetry-dependent terms (see References [39]and [40] for details).The expression of energy contains two different types of energyintegrals: ⭸E ⭸E ⌿iand⌿0 (7-5)⌿0⌿0 ⭸Q r ⭸Q r where ⌿0 and ⌿i are the wave functions of the ground state and anexcited state, respectively.
In the actual calculations, these wave functions are approximated by molecular orbitals, but their relationshipremains the same.Examine now the two energy integrals separately, bearing in mindwhat was said about the conditions necessary for an integral to havenonzero value (Chapter 4). The first integral contains only the groundstate wave function. It appears in the first-order perturbation energyterm that expresses the effect of changing the nuclear positions on the3227 Chemical Reactionsoriginal electron distribution. This integral will have nonzero valueonly if⌫⌿0 · ⌫⌿0 ⊂ ⌫ Qr(7-6)that is, if the direct product of the representation of ⌿0 with itself (afunction with the same symmetry) contains the representation of Qr .Concerning ⌿0 there are two possibilities: it can be degenerate ornondegenerate.
If ⌿0 is degenerate, the molecule will be unstable(this is the case of the Jahn–Teller effect, see Section 6.6) and itwill undergo a distortion that reduces the molecular symmetry anddestroys the degeneracy of ⌿0 . Consider now the case when ⌿0 isnondegenerate. We know that the direct product of two nondegenerate functions with the same symmetry always belongs to the totallysymmetric irreducible representation.
Therefore, Qr must also belongto the totally symmetric irreducible representation so that the integral will have a nonzero value. We can conclude that, except at amaximum or at a minimum, the reaction coordinate belongs to thetotally symmetric irreducible representation of the molecular pointgroup.The reaction coordinate is just one particular normal mode in thesimplest case. It must always be, however, a symmetric mode, and thisis so even if a more complicated nuclear motion is considered for thereaction coordinate. Such a motion can always be written as a sum ofnormal modes.
Of these modes, however, only those which are totallysymmetric will contribute to the reaction coordinate. The nonsymmetric modes may contribute only at the extremes of the potentialenergy function.The second integral in Eq. (7-5) appears in the second-order perturbation energy term, and it expresses the mixing in of the first excitedstate into the ground state during the reaction: ⭸E ⌿i .(7-7)⌿0 ⭸Q r This integral will be nonzero only if the direct product of the representations of the wave functions ⌿0 and ⌿i contains the representation towhich the reaction coordinate belongs,⌫⌿0 · ⌫⌿i ⊂ ⌫ Qr(7-8)7.1.
Potential Energy Surface323This expression contains important information regarding thesymmetry of the excited states as well. Only those excited states canparticipate in the reaction whose symmetry matches the symmetry ofboth the ground state and the reaction coordinate. We already knowthat Qr belongs to the totally symmetric irreducible representationexcept at maxima and minima.
This implies that only those excitedstates can participate in the reaction whose symmetry is the sameas that of the ground state. This information is instrumental in theconstruction of correlation diagrams, as will be seen later.The reaction coordinate can possess any symmetry at maxima andminima provided that the condition of Eq.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
