## Shaking out details of transition states

Chemists liken reaction energetics to a landscape with hills and valleys. In this context, the transition state represents the highest barrier that reagents must pass over en route to forming products. Baraban *et al.* introduce a framework for extracting details about the transition state of rearrangement reactions directly from vibrational spectral data. They identified a characteristic pattern in the spacing between vibrational energy levels near the transition state, which revealed its energy as well as the specific motions involved in surmounting the barrier.

*Science*, this issue p. 1338

## Abstract

Transition state theory is central to our understanding of chemical reaction dynamics. We demonstrate a method for extracting transition state energies and properties from a characteristic pattern found in frequency-domain spectra of isomerizing systems. This pattern—a dip in the spacings of certain barrier-proximal vibrational levels—can be understood using the concept of effective frequency, ω^{eff}. The method is applied to the cis-trans conformational change in the S_{1} state of C_{2}H_{2} and the bond-breaking HCN-HNC isomerization. In both cases, the barrier heights derived from spectroscopic data agree extremely well with previous ab initio calculations. We also show that it is possible to distinguish between vibrational modes that are actively involved in the isomerization process and those that are passive bystanders.

The central concept of the transition state in chemical kinetics is familiar to all students of chemistry. Since its inception by Arrhenius (*1*) and later development into a full theory by Eyring, Wigner, Polanyi, and Evans (*2*–*5*), the idea that the thermal rate depends primarily on the highest point along the lowest-energy path from reactants to products has remained essentially unchanged. Most of chemical dynamics is now firmly based on this idea of the transition state, notwithstanding the emergence of unconventional reactions such as roaming (*6*, *7*), where a photodissociated atom wanders before abstracting from the parent fragment. Despite the clear importance of the transition state to the field of chemistry, direct experimental studies of the transition state and its properties are scarce (*8*).

Here, we report the observation of a vibrational pattern, a dip in the trend of quantum level spacings, which occurs at the energy of the saddle point. This phenomenon is expected to provide a generally applicable and accurate method for characterizing transition states. Only a subset of vibrational states exhibit a dip; these states contain excitation along the reaction coordinate and are barrier-proximal, meaning that they are more susceptible than other states to the effects of the isomerization barrier. Experimental evidence for this concept is drawn from our studies of two prototypical systems: the HCN ↔ HNC isomerization and the cis-trans conformational change in the first electronically excited singlet state of acetylene.

## Effective frequency and the isomerization dip

The effective frequency ω^{eff} is the central quantity in our model for the spectroscopic signature of isomerizing systems. In a one-dimensional system, the effective frequency is the derivative of the energy with respect to the quantum number *n*, (1)where ω^{eff} is evaluated discretely for quantized systems. ω^{eff} is a dynamic quantity that can change as excitation increases, unlike quantities such as harmonic frequency, ω, or fundamental frequency, ν, which are often listed as molecular constants. As such, it is a useful diagnostic of the behavior of the system.

Applications of effective frequency date back a long way. For example, the effective frequencies ω^{eff}(*n*) of a state of a diatomic molecule are its vibrational intervals, which decrease to zero at the dissociation limit. The sum of the effective frequencies is therefore the dissociation energy. In most cases it is not possible to observe ω^{eff}(*n*) all the way to the dissociation limit, but a linear extrapolation to ω^{eff} = 0 allows a very good estimate of the dissociation energy, notwithstanding nonlinearities in the trend of vibrational intervals near dissociation. This is the basis of the Birge-Sponer plot (*9*) where the area under a graph of the vibrational intervals, ω^{eff}(*n*), against *n* gives the dissociation energy. Leroy and Bernstein (*10*) have given a protocol for extrapolating the effective frequencies, which takes account of the exact long-range shape of the vibrational potential near dissociation. This procedure is found to give very accurate dissociation energies (*11*).

Effective frequencies also play a large part in our understanding of quasi-linear molecules. A quasi-linear molecule has a nonlinear equilibrium geometry but a comparatively small potential barrier to linearity. The pattern of the lowest vibrational levels is that of a bent molecule, but with increasing bending vibrational excitation, this changes smoothly into the pattern for a linear molecule, vibrating with large amplitude. Dixon (*12*) modeled a quasi-linear potential as a two-dimensional harmonic oscillator perturbed by a Gaussian hump at the linear configuration, and calculated its energy levels. These levels may be assigned vibrational (*v*) and angular momentum (*K*) quantum numbers (*13*). If the vibrational intervals (effective frequencies) for a given *K* value are plotted against *v*, they pass through a minimum at the energy of the potential barrier, thereby allowing determination of its value. The depth of this “Dixon dip” is greatest for *K* = 0 and decreases with increasing *K*. The reason is that the angular momentum results in a *K*-dependent centrifugal barrier at the linear configuration, which the molecule must avoid.

We now illustrate the concept of effective frequency in more detail, with the four types of potential shown in Fig. 1. For the harmonic oscillator,
(2a)and (2b)indicating that the dynamics of the system do not change as a function of energy. For a Morse oscillator, the potential is *V*(*r*) = *D*_{e}[1 – exp(–*ar*)]^{2}, where *D*_{e} is the dissociation energy, *a* is a length parameter, and *r* is the bond length displacement. In this case, (3a)and (3b)where *x* is always negative. This linear decrease of ω^{eff} with *n* reflects the migration of the Morse wave functions toward the softer outer turning point. When ω^{eff} reaches zero at the dissociation limit, it becomes clear that the Morse and harmonic oscillators display very different dynamics.

Simple expressions for ω^{eff} and ∂ω^{eff}*/*∂*n* cannot be derived for the other cases in Fig. 1, but these illustrate the most important feature even more clearly: The effective frequency goes to zero at the energy of each stationary point on the potential. Classically, this can be understood by imagining a ball released to roll on a double-minimum surface. If the ball starts on one side at exactly the height of a local maximum, it will reach that maximum with zero kinetic energy and stop. Because the ball never returns, the oscillation period is infinite and the frequency is therefore zero. We see immediately that this applies to the Morse oscillator as well: ω^{eff} reaches zero at the dissociation limit, which is a horizontal asymptote of *V*(*r*). It is clear that this phenomenon is quite general and that zeros or abrupt changes in ω^{eff} signal important changes in the dynamics of the system.

In their quantum and semiclassical analysis of highly excited states of HCP, Jacobson and Child (*14*) mentioned a dip in ω^{eff} as the signature of an approach to a saddle point. Because the HCP ↔ HPC potential energy surface exhibits some unusual features (HPC is a saddle point rather than a second minimum) and is not a true isomerization (*15*), the observed ω^{eff} trend was categorized as a peculiar “Dixon dip” rather than being recognized as the universal signature proposed here. Similarly, the onset of internal rotation in the ground state of SiC_{2} (*16*) is not an isomerization, although the ideas presented here are applicable to it. More generally, the behavior of systems as they encounter stationary points has been investigated from other perspectives as well (*17*–*20*). For our purposes, it suffices that this dip in ω^{eff} provides a marker of the chemically relevant transition state energy, as we demonstrate below.

## A model for measurement of the transition state energy

To determine the transition state energy, we propose the following semiempirical formula for ω^{eff} as a function of energy, , defined as the midpoint energy for each vibrational interval: (4)where ω_{0} is the effective frequency at = 0 for the progression being analyzed, *E*_{TS} is the energy of the transition state, and *m* is a parameter related to the barrier shape. For the Morse oscillator (Fig. 1B), *m* = 2 analytically (*21*–*23*), *E*_{TS} = *D*_{e}, and ω_{0} = ω, the harmonic frequency. Equation 4 can be regarded as a generalization of the Morse formula where *m* is allowed to take values greater than 2. The formula also satisfies the required physical boundary conditions of a limiting harmonic frequency ω_{0} at = 0 and ω^{eff} = 0 at = *E*_{TS}.

The dependence of the *m* parameter on potential shape is illustrated in Fig. 2 (see supplementary text for further details). The lower limit is *m* = 2, the Morse oscillator, where the asymptote is approached infinitesimally with *r*. The other limit is a truncated harmonic oscillator where the potential abruptly becomes constant at *E*_{TS}. In such a case, ω^{eff} falls instantly to zero and *m* = ∞.

## Dissociation versus isomerization

The physical arguments presented here regarding the behavior of ω^{eff} versus (and therefore Eq. 4) pertain only up to = *E*_{TS}. Above that energy, ω^{eff} can either remain at zero for an unbound system (as with the Morse oscillator) or rise again (in a bound system). The above-barrier behavior of ω^{eff} depends on the outer walls of the potential and is not described by Eq. 4. The *m*th root form of Eq. 4 suggests the presence of a branch point at *E*_{TS}, which separates the above-barrier and below-barrier eigenspectra into two distinct energy regions (*19*, *24*).

A semiclassical analysis of long-range interatomic potentials of the form *D* – (*C/r ^{n}*) was performed by LeRoy and Bernstein (

*10*) more than 40 years ago. They derived an expression that relates the change in energy per quantum number (i.e., the effective frequency) near the dissociation limit to a quantity proportional to [1 – (

*E/D*)]

^{(}

^{n}^{+2)/2}

*. This expression is clearly similar to our effective frequency formula, but the two models treat dynamically and mathematically distinct regimes. For inverse power-law potentials where*

^{n}*n*= {1, 2, 3, 4, …}, the corresponding

*m*values are . In the limit

*n*→ ∞, the effective

*m*value approaches 2 from below. In contrast, our model has a lower limit of

*m*= 2. In other words, these two similar effective frequency expressions treat essentially disjoint classes of potentials. The key difference is how the stationary point (or dissociation limit) is approached. For long-range potentials with inverse power-law forms, the stationary point at

*r*→ ∞ is approached only polynomially. Our treatment considers potentials where stationary points are local maxima and are therefore approached over a finite domain. The common system, the Morse potential, has a stationary point at

*r*→ ∞ but approaches it exponentially (i.e., faster than any power law) and is in some sense simultaneously long-range and local. Graphically, the dynamical distinction corresponds to positive curvature (LeRoy-Bernstein) versus negative curvature (our model) on a Birge-Sponer plot, with the linear plot of the Morse oscillator dividing the two regimes.

## Practical application

What is the best way to extract the desired saddle point energy from spectroscopically measured quantities? From the frequency-domain spectrum we measure the energies of a series of quantized vibrational levels, and take the average and difference ω^{eff} of adjacent level energies to obtain a set of (, ω^{eff}) data points (*25*). Equivalently, time domain spectroscopy may provide alternative or more direct ways to obtain vibrational periods or frequencies versus energy, especially for larger systems. A plot of these (, ω^{eff}) data reveals any dynamical trend in ω^{eff}, whether constant, linear, or nonlinear. If a stationary point is present within the data range of , the plot will dip to a minimum as the energy of the stationary point is approached, although in a bound quantum system there will never be a point with ω^{eff} equal to zero.

In principle, only this bare minimum of information is necessary to apply Eq. 4. A value for *E*_{TS} can then be obtained, which in favorable cases should have an uncertainty of 5 to 10% of the effective frequency, depending on the extent and quality of the input data. High resolution and detailed spectroscopic assignments are not a requirement, nor are ab initio calculations, although these can help to identify the active vibrations and the nature of the transition state. Even in larger molecules, where full spectroscopic and computational analyses are impractical, the problem must simplify to a very small number of vibrational modes that form the reaction coordinate and lead to the transition state. These active vibrations reveal themselves by their isomerization dips.

To demonstrate the capabilities of our method, we applied the model to two prototypical isomerizing molecules, HCN and the S_{1} state of C_{2}H_{2}. These systems have been spectroscopically characterized in great detail, such that we can apply the isomerization dip method and Eq. 4 to them with confidence. We emphasize that the levels of knowledge and quality of data available for these systems are not necessary in general for the application of our method. Furthermore, despite the small sizes of HCN and C_{2}H_{2}, they exhibit many of the complications expected in larger molecules.

## The examples of S_{0} HCN ↔ HNC and S_{1} C_{2}H_{2}

The potential surface for the electronic ground state of the [H,C,N] system has two minima: the linear HCN and HNC isomers, separated by approximately 5200 cm^{–1}. The reaction coordinate of the bond-breaking HCN-HNC isomerization corresponds mainly to the ν_{2} bending vibration, and the barrier to isomerization is nearly 17,000 cm^{–1} above the HCN minimum. Extensive experimental term values for both isomers are available up to 10,000 cm^{–1} above the HCN minimum (*26*). To continue the analysis up to and beyond the barrier energy, we used spectroscopically assigned ab initio eigenenergies (*26*–*28*). In (*26*), levels with high bending excitation were reported to deviate unexpectedly from effective Hamiltonian predictions, reflecting the presence of the double-well potential.

The S_{1} state of C_{2}H_{2} supports cis and trans conformers, with the cis conformer lying about 2672 cm^{–1} above the trans. As illustrated in Fig. 3, the transition state is planar and nearly half-linear (*29*). The bare saddle point energy is calculated to be 4979 cm^{–1} above the trans minimum, but with an uncertainty of hundreds of cm^{–1}, even for the most accurate calculations to date (*30*). A torsional isomerization path might have been expected on the basis of cis-trans isomerizations in other molecules, but this is not found here.

The height of the barrier relative to the fundamental frequencies leads us to expect at least some normal vibrational structure, even in the shallower cis well. Thus far, several cis vibrational levels have been identified, in reasonable agreement with ab initio calculations (*31*–*33*). In the trans well, almost all of the vibrational levels below the barrier have been assigned (*33*). Of the six trans conformer vibrational modes, four are fairly well behaved: the Franck-Condon active vibrations ν_{2} (CC stretch) and ν_{3} (trans bend) (*34*), and the CH stretching modes ν_{1} and ν_{5} (*35*, *36*). On the other hand, a large portion of thetrans vibrational manifold can only be understood within the framework of bending polyads (*37*), because of the Darling-Dennison and Coriolis interactions between the low-frequency ungerade bending modes, ν_{4} (torsion) and ν_{6} (cis bend).

Despite the success of the polyad model in reproducing the level structures associated with the bending vibrations, there are disturbing exceptions. As illustrated in figure 13 of (*38*), the series of 3^{n}B^{2} polyads exhibits a surprising trend, with the energy of the lowest member of the polyad decreasing rapidly relative to the energies of the other polyad members. Although inexplicable by conventional models, this occurrence turns out to be intimately related to the isomerization dynamics discussed here.

## Determination of the barrier height

We now apply the isomerization dip concept, and in particular Eq. 4, to the barrier proximal energy levels discussed above. Figure 4 shows the results of the pure bending () effective frequency analysis for HCN-HNC. The barrier heights for both wells are found to be within 1% of the ab initio values. To compare *E*_{TS} to calculated barrier heights, either the ab initio zero point energy must be subtracted from the calculated barrier height, or an effective zero point energy must be added to the fitted *E*_{TS} value (see supplementary text). A consistency check of the HCN-HNC analysis stems from another dynamical parameter that affects and the effective barrier height: the vibrational angular momentum, *ℓ*. The fitted *E*_{TS} barrier heights are summarized in table S5 and, as expected, the barrier height increases approximately quadratically with *ℓ*.

Figure 5A shows plots of and for the 3* ^{n}*6

^{2}series of C

_{2}H

_{2}, where Eq. 4 can be seen to fit the observed data very well (see tables S1 and S2 for details of the fits). The 3

*6*

^{n}^{2}levels experience the effects of the barrier most strongly, whereas the 3

*levels are completely uninfluenced by it, because a combination of*

^{n}*q*

_{3}and

*q*

_{6}is required to access the transition state geometry. Both and can be obtained as a function of ν

_{6}as well, reading the array of term values in table S2 horizontally rather than vertically. The same {, ω

^{eff}} data are obtained, but in different sets.

## Reaction path analysis

Several possibilities arise when the ω^{eff} analysis is extended to additional vibrational progressions. The first, shown in Fig. 5C, is that of differentiating between isomerization pathways. We have noted already that certain progressions, such as the 3* ^{n}* of

*trans*-C

_{2}H

_{2}, show no sign of an isomerization dip. Here we see that surprisingly, the 3

*4*

^{n}^{2}levels exhibit the same as the 3

*, from which we conclude that the torsional cis-trans isomerization pathway is closed at these energies. This observation is consistent with the harmonic behavior in ν*

^{n}_{4}noted in (

*37*). Furthermore, the very strong interactions between ν

_{4}and ν

_{6}, as well as the inevitable evolution of the torsion as the molecule straightens, would have led us to predict some kind of nonlinear behavior of the from the 3

*4*

^{n}^{2}levels. The absence of any such effects implies that the torsion is a spectator mode independent of the isomerization occurring in ν

_{3}and ν

_{6}. Unlike in many other molecules, torsion does not play a role in this cis-trans isomerization. It appears that the residual π bond, despite its incomplete complement of substituents, leads to the preference for in-plane isomerization (Fig. 3).

The clear distinction between spectator modes and isomerizing modes suggests that other properties of the saddle point may be obtainable from ω^{eff} analysis. Consider a separable system that consists of an asymmetric double minimum in *x* and a harmonic oscillator in *y*: *V*(*x*, *y*) = *V*(*x*) + (*ky*^{2}/2). Because the Hamiltonian is separable, *E* = *E _{x}* +

*E*, and therefore does not depend on

_{y}*y*, in the same way that does not depend on ν

_{4}in

*trans*-S

_{1}C

_{2}H

_{2}. This means that the ω

*of the transition state is unchanged from that of the minimum. We can then imagine a case where [∂*

_{y}^{2}

*V*(

*x*,

*y*)]/∂

*y*

^{2}varies with

*x*. In such a case it is possible to extract a value for ω

*of the transition state from the spacing of the curves, or by plotting directly, including data from above the saddle point energy (*

_{y}*39*). To give a specific example from a rotational degree of freedom, a quadratic fit of the

*ℓ*-dependence of the 0ν

_{2}0 HCN barrier height (table S5) yields (

*A*– )

_{TS}= 11.1 (±0.9) cm

^{–1}for the transition state, which agrees well with the ab initio value of 12.2 cm

^{–1}. These hitherto unmeasurable transition state rotational constants and frequencies, in addition to the saddle point energy, are critical inputs to the expression for the rate constant in transition state theory (

*2*).

The ideas about spectator modes articulated here are confirmed by analysis of the HCN-HNC stretching mode progressions; neither shows any dip up to 19,000 cm^{–1} above the HCN minimum. Furthermore, the shifts in the fitted barrier heights for ν_{2} progressions built upon excitation in ν_{1} and ν_{3} match well with the one-dimensional pseudopotentials from (*40*–*42*) and the stretching frequencies (table S4).

## Implications and outlook

The method described here provides qualitative and quantitative information about the isomerization mechanism and the transition state solely on the basis of experimental data. The reaction coordinate can be identified from the isomerization dips shown by active vibrations, and quantitative information about the energy, vibrational frequencies, and rotational constants of the transition state becomes available. It is especially promising that this analysis stems entirely from a small subset of the vibrational levels; in other words, a full vibrational analysis is not necessary. Special states exist that encode chemically important information, although it may not always be easy to recognize them.

The most exciting outcome of the excellent fits using Eq. 4 is the determination of the transition state energies. The uncertainties in *E*_{TS} are at least as good as what is currently available from theory. In many cases, experiments might only confirm theoretical predictions, but we expect them to sharpen our understanding, and the potential value of our method goes well beyond validating theory. For example, we envision that this approach is a step toward establishing kinetics on the same firm experimental foundation as that already enjoyed by thermochemistry, where precise information is broadly available. This could aid in modeling complex reaction networks as well as provide benchmarks for theoretical calculations.

The next stages in developing the concepts of isomerization dip and effective frequency will require more experimental data for model systems as well as refinement of the basic ideas. For example, it is clear that ω^{eff} is a multidimensional quantity, which could be treated by suitable multidimensional analysis. However, it is not yet clear how to define in many dimensions. On the experimental side, the MgNC ↔ MgCN isomerization (*43*) is similar to HCN ↔ HNC, although with a much lower barrier; with higher resolution than has been used so far, it should be possible to get detailed information on the isomerizing levels all the way to the barrier and beyond.

The most promising prospect is the application of this approach to larger molecular systems. For example, the principles presented in this work will provide a framework for experimental characterization of the transition state and detailed mechanism in other reactions, such as cis-trans isomerization, as epitomized by stilbene, and 1,2 hydrogen shifts, which are ubiquitous in organic chemistry. Remarkably little frequency-domain spectroscopy has been done at the high energies relevant to chemical kinetics. Partly this is because it was not clear previously what could be learned, and partly it is because the spectra rapidly become very complicated at such energies, so it is difficult to recognize the important patterns. In time it is likely that new classes of experiments will sample the information about isomerization that is encoded in time- or frequency-domain spectra of larger molecules. Our hope is that the concepts and examples described here will be viewed as templates for the characterization of isomerizing systems, thereby challenging and guiding spectroscopists to attack similar problems of chemical interest.

## Supplementary Materials

## References and Notes

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵The angular momentum quantum number
*K*plays two roles here: that of the asymmetric top rotational quantum number and the linear molecule vibrational angular momentum quantum number,_{}. - ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵Regarding ω
^{eff}versus as opposed to ω^{eff}versus*n*: Using*n*simplifies derivations, but is more practical for actual use. Furthermore, plotting against gives more direct information about the potential energy surface. - ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵In a real system, reaction path curvature could occur, such that the normal modes of the minima are not identical to those of the transition state. This would complicate matters, but the modes necessarily evolve smoothly from minimum to transition state.
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
**Acknowledgments:**We thank M. Joyeux for providing the HCN-HNC pseudopotentials. Supported by NSF Graduate Research Fellowship DGE 1144083 (P.B.C.), an Alexander von Humboldt Foundation Feodor Lynen fellowship for experienced researchers (G.C.M.), and U.S. Department of Energy grant DE-FG0287ER13671.