What's in a Name-Transition
State
S. H. Bauer and C. F. Wilcox, Jr. Cornell University, Ithaca, NY 14853
The thought is father to the word. -an
old aphorism
However, the flip side to this is equally valid: In ancient cultures naming an object or a concept endowed it with a special reality. Wittgenstein once said, If a lian could talk we would not understand him.
D e ~ e t(tI )then commented,
.
On the contram.. if a lian could talk. that lion would have a mind so diKerent liom the general run of lion minds, that although we could understand him just fine, we would learn little about u r d i r m r y lions from h i m . Language plays a n enormous role in the structuring of the human mind Making Clear Semantic Distinctions
It is essential in the discioline of chemistm. as in all scientific dialogue, that term; be carefully d e k e d and that their implications be transmitted so that confusion is avoided with similar terms in general use. It is therefore regrettable that the widely used term transition state is consistently misapplied. Thoughtful students are initially troubled by use of the term state because the entity desienated is a structure, but they then learn by osmosiito tahe the necessary conceptual leap. Pedagogically, the confusion of state with structure (i.e., energy space with geometric space) is inappropriate and unnecessary.' Definitions For clarity, we prefer to use the following definitions. Molecular structure: A confirmration of atoms (eeometric) in 3D space with specified~onnectivitiesbetween the atoms such that the assembly has an identifiable (mean) center ofmass and moves as a unit in space. Structures are not rigid. Even a t the lowest temperature they have zeropoint oscillatoly motions (the atoms in their restless universe) due to the uncertainty principle, which does not
'In his first Daoer on reaction rate theorv 1J. Chem.'Phvs. 1935. 3. 1071 H. ~ v r i o~~~~ n b"sed the older term activatd comolex ,~~ liniroduced'b; tiarcellhi7 1915, an0 disc~sseotne c a c atton ~ of probabi tes lo; ~~~~
~
~~
~~~~~
.
~~~~~~
generating "act vated states" Inoeed, In tne c assical book, The The. ory 01 Reachon Rafes (1941).by Glasstone, Laldler, and Eyring transition state does not appear. On page 185 they introduce the term activated complex, and in Figure 53 they designate an "activated state". Laidler (1969)in Theories ofChernicalReaciionRatesprefers "activated complex" and argues that 'Transition state" is ambiguous, with which we heartily agree. About a decade ago, during a wnference on chemical kinetics, Eyring admitted that "transition state"was an unfortunate misnomer, and ascribed its origin to Wigner (1932). The term transition state theory was accepted by IUPAC (1979) based on decades of misuse by chemists. 2Somestructures are sufficiently labile that, even in the absence of external perturbations, atom connectivitiesshift spontaneously. 30fcourse,there are many intermediate atomicconfigurations,but not all of these are of interest-nlv the one that is assumed to reoresent the atom c conflg~rationat'the sadole pont aong tne m r i m~m-energypatn on tne potentla1 energy su~face.
allow a strict separation of geometric space from momentum soace.' However. because the models we use are based on m&roscopic conceits built around sharply localized geometries. we must be careful not to blur the distinction between geometric space and energy space. ~
~~
~~~~
Molecular states: A description of a molecnle is incomplete without some statement regarding its rotational and vibrational momentum. On the molecular scale. obsemable momenta are quantized. The allowed magnitudes are derived bv solvine (conceotuallv. if not actuallv) the ~ c h r o d i n i e requaiion for &tiunary states rori&nally called eigenstates). They are time-indupendent wave functions whosc squares are probability distributions for the electrons and nuclei that comprise the molecular unit. The critical parameter for each physically acceptable solution is a characteristic energy (an eieenvalue). In an assembly in thermal equilibrium~~molecu~es are partitioned among their characteristic molecular states according .to a temperature-dependent Boltzmann function. A New Approach ..
Although some chemists may find minor flaws in the above definitions, the clear distinctions so introduced focus on the core of thequestion posed in the title. When we tune in on discussions between chemists concerned with mechanisms and rates of molecular interconversions, it is immediately evident that their designation "transition state" usually refers to a configuration of atoms in geometric space. They propose that this configuration is the highest enerw structuri that develo~son the lowest electronic - energypath in the process of trkformation from reactant to p~oduct.~ We propose that for clarity the structure at the potential energy saddle point be called the critical transition structure (CTS). This structure is a unique (mean) configuration of atoms: In the random-walk approach (on the potential energy surface) toward the CTS, the electronic energy increases, and beyond the CTS that energy decreases. As developed below, the CTS is not rigid and is characterized by numerous states. Molecular and Internal Rotations and Vibrations The vibrational energy content of molecules, be they stable or in transition, is represented bv (3n - 6) modes. olus their respective coibinakons and overtones. All these are included in calculations of the corresponding partition functions. A fundamental postulate of chemical kinetic theory is that one may identify a reaction coordinate, that is, one mode, ora combination-ofmodes, that characterizes the chemical trnnsformation, leaving (3n - 7, modified modes to represent the dynamics oftthe less interesting changes in connectivities. All of these states (molecular rotations, internal rotations, and vibrations) plus the particular states of motion along the reaction cooidinate are incorporated in the computation of the partition function for the CTS. Indeed, in ab initio quantum mechanical computations of the stabilVolume 72 Number 1 January 1995
I?
In the language proposed here, T is the CTS. In this description,A and B represent the collection of species with total energies below the barrier height &M
< Va
that is, the potential energy barrier height (see Fig. 1). A* and B* have Ebw 2 V. but with the internal energy distributed so that the energy in the reaction coordinate is less than V. In T, E b is~ also greater than or equal to V., but with the energy in the reaction coordinate equal to or -ereater than V.. Reaction ~oorknate Thus. reactant s~ecies A* can have internal energies ibove the barrier, Figure 1. High barrier case. Empirically, the vibrational state density may be represented by but no reaction occurs until suffi& ~ ( E J= ela6-&) with a >> b (Robinson and Holbmok, Unimolecular Reactions; Wiley-inter. cient energy flows into the reaction coordinate to generate a Y species. science: London, 1972;p. €%). . For dissociations,the density of product states includes translations This is followed by sufficient inter= g r ~ ; where ~ , gr is a large number. nal enerm flow out of the reaction coordinate of that species to produce B*, which subsequently is deactiity of a CTS a correction is made for the heat capacity, vated to the product, B. Generally, in RRIiM it is assumed based on sums over these states. There is no logical justifithat the flow of energy out of T is fast compared to other cation for the oxymoron implied by the expression "the processes. states of the transition state". Reaction with a Modest Barrier Discussion of CTS Populations Consider typical reactions in which there is a modest Considering the present mindset, would this semantic barrier of, for example, 30 kcal mol-' and products that are switch serve a useful purpose? Foremost, it will eliminate considerably more stable than the starting material (as in the confusion due to discussing geometric entities using Fig. 1). For these, the population of the A* species is very the language of momentum. Also, it provides a basis for low due to its small Boltzmann factor. Thus, the producunderstanding the kinetics of rapid isomerizations (over tion of a Y species is small and essentially limited to the low barriers), which is a class of reactions that is badly tiny population of molecules that has accumulated energy re~resentedby the conventional RRKM (4) formulations. within a very narrow band just above V. ~ ; of e the more precise language advocated here frees our Under these circumstances the limited supply of Y speminds to focus on estimating magnitudes of obtainable cies is rapidly depleted due to the high density of B* states populations of CTSs and devising techniques for observing compared to that of the A* and Y states--a consequence of their spectroscopic signatures. the assumed much lower (electronic) energy of B relative As an illustration, a more detailed discussion of CTS to Aand Y. The net rate, on a quasiclassicalbasis, is deterpopulations follows. Recall that the conve.ntional (baremined by the rate of passage of critically excited molecules bones) RRKM description for the conversion through a configuration of "no return", that is, the short time required to traverse a distance 6 over the one-dimensional potential maximum. The reciprocal of this rate corresponds to a lifetime for the CTS,which is a fraction of is formulated as a sequence of four steps. (hlkT) < 1.4 x 10-13 s; the steady state population is very small. The relative population in the portion of phase space is given in the original Eyring derivation (3).
where A* and B* represent species that have been energized to levels a t (or above) the barrier energy; Y represents the subset of species that have sufficient momentum along the reaction coordinate to convert from A* to B*; and M stands for all energizing (and de-energizing) collision partners. 4R. D. Levine and R. 8. Bernstein in MolecuiarReactionDynamics (1987)explain, "the key assumption is that there exists a single configuration of no return, called the transition state".
14
Journal of Chemical Education
where mX is a reduced mass of the representative mass point for motion along the reaction coordinate, and the Qs' are partition functions. Let rn* = 50 Daltons, 6 = 0.5 A,and
for a large molecule. Then,
for a barrier of 30 kcal mol-' at 350 K. In systems represented by Figure 1, the residence time of the reacting moiety in the vibrational state that just skims the barrier maximum is about a single vibrational period due to the rapid rate of depletion into the high density of product states. Observine such T's is difficult but hs; been recently discussed and re. corded (4). However, the speties for which interesting data can be obtained are limited to very few atoms. In the concentrations of CTS's of most reactions are determined a steady state between a rate production with cOnversion to product. Direct Observation of a CTS: lsomerizations
-.
E
L'
-
.
Y--&!--L
Densities of Vib. States
d
C
Densities of Vib. States
Reaction Coord.
Figure 2. lsomerization over a low barrier. For states with total vibrational energy content Ex,f(E,;r) is that fraction wherein the amount E is localized in the reaction coordinate. Programs are available for calculating the fraction of A*molecules with any specified Ex> Va (at statistical equilibrium)that concurrently have any specified amount of energy in the reaction coordinate. For E > Va, the transition structure i.can exist as a well-defined species, which differs from both A*and B*in some respects. For any level of excitation EX,the steady-state population of the T structure is given by the number of states indicated by the shaded areas, weighted by the corresponding Boltzmann factors. The total i. population is detenined by an integral over Ex (range: Va to -). Approximately, for V. = 5000 cal, T = 350 K,
Although the direct observation of a CTS is difficult due to its low concentration, DroDer selection of A and B A d t h e reaction conditions that control the rates of formation and deactivation of T permit it to develop a measurable steady state concentration for molecules of substantial complexity For example, consider isomerizations over relatively low barriers, particularly when the two isomeric structures are of com~arableelectronic stability. For such cases one can i d e n t i f i ~ ~ sthat ' s are not confined to a narrow partition in phase space. Indeed, erecting such a partition at the top of the potential energy barrier is inconsistent with theoretical analyses that identify completely defined vibrational eigenstates (for the reaction coordinate) within the potential well, above the potential energy ma~imum.~ Conuersions with Low Barriers This ty e of conversion, that is, when low barriers (15 kcal mol- or lower) connect ground state species of comparable electronic stabilities, is illustrated in Figure 2. Under these conditions, the density of states of the A* and B* species are similar (or identical for degenerate rearrangements), and the generally assumed "no return" feature does not apply Furthermore, the low barrier implies that the concentrations of A* and B* species capable of producing r species are considerable. Under these circumstances it is feasible to develop a significant equilibrium population of the r species. The relative concentration of r to A is given approximately by
P
~.
For conversions represented by Figure 2, the residence time of a molecule in a single state, within the T collection of states, is about the same as in any A* or B* state, that is, 10-100 vibrational periods, assuming that all vibrational states are coupled about equally Lifetimes in specific states do not matter, as long as the rates of entrance and exit are sufficient to maintain significant steady state populations. Differences in Vibrational States Confusion of the statelstructure aspects of the l" species, rather than identifying it as a molecular species with a characteristic structure, has diverted kineticists from exploring techniques for investigating its properties. Significant differences between the vibrational states of A*. r, and B* are anticipated. Just as A* and B* have distinctive sets of (3n - 6) fundamental modes, so does T. It has its own set that consists of the large amplitude of motion in the reaction coordinate and the remaining (3n - 7) modified modes, which are modulated by coupling to the reaction coordinate. A preliminary analysis of the semihulvalene isomerization proved instructive. Application of the semiempirical AM1 program (5) led to a minimum-energy geometry and IR vibrational normal-mode frequencies that approximately reproduced the experimental values. The same prowam applied to the CTS gave interatomic distances and frequencies that appear reasonable. Of primary interest were the spectral shifts in the 5.5-6.5-pm region. (1770, 1769. and 1552 cm-' were disnlaced to 1626. 1603. and 1577'cm-I). Clearly, the large amplitude vibration within the reaction coordinate well above the potential maximum is coupled to the other (312- 7) molecular oscillators and affects their characteristic frequencies. A~
~
.
~
~
.
which for a bamer of 10 kcal mol-' at 350 K is about lo4. 'The distribution of these vibrational states is similar to those in a quartic potential well, that is. En = dl3where n is a quantum number.
Summaw Such information is of general chemical interest. The assignment ofspecificstructures to the components of chemiVolume 72 Number 1 January 1995
I!
cal systems is a basic requirement for quantitative descriptions of the reactants and their conversion to products. Knowledge of the spectra of CTS's will lead to a more detailed understanding of their properties than is generally possible from examination of product distributions. Additionally, locating various excitation levels for critical transition structures, even roughly (to a precision of about lo%), will provide valuable calibrating data for evaluating interaction parameters incorporated into empirical and semiempirical energy-minimization programs. Finally, a rational model for CTS's should supplant the confusing language chemists use during discussions of mechanisms.
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Journal of Chemical Education
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B m m : Boston, 1991.
2. ~designsteaUle~eoryfaunimunimleEulunimunimunimcticti~~dwelopedbyHiee,hp~ ger, Kaasel, end Ma-. See Bobinam, P J.; Halbalbk, K A Unimolrculorktions; Wk-Inter&-: landon. 1972.
3. Ev-, M.G:Polanyi, M. h a . F lass, 3,107
d Soo. IWS, 31,875; Eyling, H. J Chom.Fhye.
4. B m k , P. R Chom. Rau. 1988, 88, 401; Barnes, M. D.; Bmoks, P R: Curl,R F., Harknd,P.W.:Jobnson,B.RJ. C k m P h y s . 1889,96,3559;Neumarh,D.M.h~ Reu Phys. Chem 198e,43.153:Thiel.E.;hohagc,K H . Ckm.Phys.latt. 1889, 199.329.
5. Dewar,M.J.S.;ZoebiscbE.G.;Healy,E.F;Stewart, J.J.P.J.Am. C k m . S o e 1085, 107,3902;Stew* J. J. P. J. Compu&-dfdd Mol. Design 1980.4.1-105.
