Research Article

Spectroscopic characterization of isomerization transition states

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Science  11 Dec 2015:
Vol. 350, Issue 6266, pp. 1338-1342
DOI: 10.1126/science.aac9668

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


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, ωeff. The method is applied to the cis-trans conformational change in the S1 state of C2H2 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 (25), 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 ωeff 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,Embedded Image (1)where ωeff is evaluated discretely for quantized systems. ωeff 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 ωeff(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 ωeff(n) all the way to the dissociation limit, but a linear extrapolation to ωeff = 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, ωeff(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, Embedded Image (2a)andEmbedded Image (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) = De[1 – exp(–ar)]2, where De is the dissociation energy, a is a length parameter, and r is the bond length displacement. In this case,Embedded Image (3a)andEmbedded Image (3b)where x is always negative. This linear decrease of ωeff with n reflects the migration of the Morse wave functions toward the softer outer turning point. When ωeff reaches zero at the dissociation limit, it becomes clear that the Morse and harmonic oscillators display very different dynamics.

Fig. 1 Effective frequency plots below their associated model potentials.

(A) Harmonic oscillator. (B) Morse oscillator. (C) Symmetric double minimum potential. (D) Asymmetric double minimum potential. In the top row, the quantized energy levels are marked with dashed lines. In the bottom row, the classical ωeff is shown as a solid line, with the quantum level spacings plotted as open circles versus Embedded Image (the midpoint energy for each interval). In (C), the upper and lower series of circles correspond to the vibrational level spacings and tunneling splittings, respectively. In (D), the ωeff curve and energy levels for the second minimum are shown in red, and the quantum level spacings are overlaid on the ωeff curves as triangles and squares.

Simple expressions for ωeff and ∂ωeff/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: ωeff 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 ωeff 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 ωeff 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 ωeff 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 SiC2 (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 (1720). For our purposes, it suffices that this dip in ωeff 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 ωeff as a function of energy, Embedded Image, defined as the midpoint energy for each vibrational interval:Embedded Image (4)where ω0 is the effective frequency at Embedded Image = 0 for the progression being analyzed, ETS 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 (2123), ETS = De, and ω0 = 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 ω0 at Embedded Image = 0 and ωeff = 0 at Embedded Image = ETS.

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 ETS. In such a case, ωeff falls instantly to zero and m = ∞.

Fig. 2 Relationship between potential shape and ωeff.

(A) Effective frequency curves. (B) The corresponding potentials as a function of the shape parameter m in Eq. 4.

Dissociation versus isomerization

The physical arguments presented here regarding the behavior of ωeff versus Embedded Image (and therefore Eq. 4) pertain only up to Embedded Image = ETS. Above that energy, ωeff 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 ωeff depends on the outer walls of the potential and is not described by Eq. 4. The mth root form of Eq. 4 suggests the presence of a branch point at ETS, 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/rn) 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)/2n. 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 = {1, 2, 3, 4, …}, the corresponding m values are Embedded Image. 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 Embedded Image and difference ωeff of adjacent level energies to obtain a set of (Embedded Image, ωeff) 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 (Embedded Image, ωeff) data reveals any dynamical trend in ωeff, whether constant, linear, or nonlinear. If a stationary point is present within the data range of Embedded Image, 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 ωeff equal to zero.

In principle, only this bare minimum of information is necessary to apply Eq. 4. A value for ETS 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 S1 state of C2H2. 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 C2H2, they exhibit many of the complications expected in larger molecules.

The examples of S0 HCN ↔ HNC and S1 C2H2

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 ν2 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 (2628). 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 S1 state of C2H2 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.

Fig. 3 Salient features of the cis-trans isomerization in S1 C2H2.

The barrier height and energy difference between the two conformers are shown, as well as the combination of trans normal modes (q3 and q6) that corresponds to the isomerization coordinate.

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 (3133). 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 ν2 (CC stretch) and ν3 (trans bend) (34), and the CH stretching modes ν1 and ν5 (35, 36). On the other hand, a large portion of thetrans vibrational manifold can only be understood within the framework of bending polyads Embedded Image (37), because of the Darling-Dennison and Coriolis interactions between the low-frequency ungerade bending modes, ν4 (torsion) and ν6 (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 3nB2 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 (Embedded Image) 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 ETS 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 ETS value (see supplementary text). A consistency check of the HCN-HNC analysis stems from another dynamical parameter that affects Embedded Image and the effective barrier height: the vibrational angular momentum, . The fitted ETS barrier heights are summarized in table S5 and, as expected, the barrier height increases approximately quadratically with .

Fig. 4 The Embedded Image ( = 0) effective frequency analysis for HCN and HNC.

Shown are experimental data points (blue), Dunham polynomial expansion predictions using only experimental data (green), and the assigned ab initio data points (red) (26) (see supplementary text for details). The fitted ETS parameters using Eq. 4 (blue) are compared with the ab initio barrier heights (red). A one-dimensional cut through the potential energy surface is shown as a red dashed line. The unusual shapes of the HNC potential and ωeff plot near 5000 cm–1 result from interaction with a low-lying excited diabatic electronic state (44).

Figure 5A shows plots of Embedded Image and Embedded Image for the 3n62 series of C2H2, where Eq. 4 can be seen to fit the observed data very well (see tables S1 and S2 for details of the fits). The 3n62 levels experience the effects of the barrier most strongly, whereas the 3n levels are completely uninfluenced by it, because a combination of q3 and q6 is required to access the transition state geometry. Both Embedded Image and Embedded Imagecan be obtained as a function of ν6 as well, reading the array of term values in table S2 horizontally rather than vertically. The same {Embedded Image, ωeff} data are obtained, but in different sets.

Fig. 5 The ωeff analysis for S1 C2H2.

The dip is expected near the ab initio value of 4979 cm–1. Data are from tables S1 to S3. (A) Experimental ωeff for the 3n62 levels, shown with fits to Eq. 4. Embedded Image is obtained directly from the progression of 3n62 levels, and Embedded Image is derived from the 3n62 and 3n61 levels at a given n3. (B) Experimental Embedded Image for the 3n62 levels as compared to Embedded Image for other progressions with varying quanta of ν6, shown with fits to Eq. 4. The 3n62 series has the sharpest dip, in close analogy to the K = = 0 series in the “Dixon dip” (12). (C) Embedded Image plots for the 3n, 3n42, and 3n62 progressions. The 3n42 series follows the normal behavior of the 3n levels, despite the isomerization dip observed in the 3n62 levels. This shows that the torsional mode ν4 is not involved in the isomerization process at these energies.

Reaction path analysis

Several possibilities arise when the ωeff 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 3n of trans-C2H2, show no sign of an isomerization dip. Here we see that surprisingly, the 3n42 levels exhibit the same Embedded Image as the 3n, 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 ν4 noted in (37). Furthermore, the very strong interactions between ν4 and ν6, 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 Embedded Image from the 3n42 levels. The absence of any such effects implies that the torsion is a spectator mode independent of the isomerization occurring in ν3 and ν6. 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 ωeff 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) + (ky2/2). Because the Hamiltonian is separable, E = Ex + Ey, and therefore Embedded Image does not depend on y, in the same way that Embedded Image does not depend on ν4 in trans-S1 C2H2. This means that the ωy of the transition state is unchanged from that of the minimum. We can then imagine a case where [∂2V(x, y)]/∂y2 varies with x. In such a case it is possible to extract a value for ωy of the transition state from the spacing of the Embedded Image curves, or by plotting Embedded Image directly, including data from above the saddle point energy (39). To give a specific example from a rotational degree of freedom, a quadratic fit of the -dependence of the 0ν20 HCN barrier height (table S5) yields (AEmbedded Image)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 ν2 progressions built upon excitation in ν1 and ν3 match well with the one-dimensional pseudopotentials Embedded Image from (4042) 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 ETS 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 ωeff is a multidimensional quantity, which could be treated by suitable multidimensional analysis. However, it is not yet clear how to define Embedded Image 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

Supplementary Text

Figs. S1 and S2

Tables S1 to S6

References (4560)

References and Notes

  1. 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, Embedded Image .
  2. Regarding ωeff versus Embedded Image as opposed to ωeff versus n: Using n simplifies derivations, but Embedded Image is more practical for actual use. Furthermore, plotting against Embedded Image gives more direct information about the potential energy surface.
  3. 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.
  4. 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.
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