Conformational isomerization kinetics of pent-1-en-4-yne with 3,330 cm⁻¹ of internal energy measured by dynamic rotational spectroscopy

Abstract

We demonstrate the application of molecular rotational spectroscopy to measure the conformation isomerization rate of vibrationally excited pent-1-en-4-yne (pentenyne). The rotational spectra of single quantum states of pentenyne are acquired by using a combination of IR–Fourier transform microwave double-resonance spectroscopy and high-resolution, single-photon IR spectroscopy. The quantum states probed in these experiments have energy eigenvalues of ≈3,330 cm⁻¹ and lie above the barrier to conformational isomerization. At this energy, the presence of intramolecular vibrational energy redistribution (IVR) is indicated through the extensive local perturbations found in the high-resolution rotation–vibration spectrum of the acetylenic C–H stretch normal-mode fundamental. The fact that the IVR process produces isomerization is deduced through a qualitatively different appearance of the excited-state rotational spectra compared with the pure rotational spectra of pentenyne. The rotational spectra of the vibrationally excited molecular eigenstates display coalescence between the characteristic rotational frequencies of the stable cis and skew conformations of the molecule. This coalescence is observed for quantum states prepared from laser excitation originating in the ground vibrational state of either of the two stable conformers. Experimental isomerization rates are extracted by using a three-state Bloch model of the dynamic rotational spectra that includes the effects of chemical exchange between the stable conformations. The time scale for the conformational isomerization rate of pentenyne at total energy of 3,330 cm⁻¹ is ≈25 ps and is 50 times slower than the microcanonical isomerization rate predicted by the statistical Rice–Ramsperger–Kassel–Marcus theory.

Rotational spectroscopy is one of the most powerful tools in physical chemistry for the determination of gas-phase molecular structure (1). The development of molecular beam spectrometers for rotational spectroscopy—most notably the Fourier-transform microwave (FTMW) spectroscopy technique introduced by Balle and Flygare (2)—has greatly expanded the range of chemical systems that can be studied. Rotational spectroscopy can be routinely used to study challenging structural problems such as the conformations of large molecules (3), weakly bound molecular complexes (4), and radicals (5). In addition, rotational spectroscopy of low-energy vibrational and torsional levels can be used to determine accurate potential energy surfaces for large amplitude motion in “floppy” systems (1). Here we demonstrate a fundamentally unique application of rotational spectroscopy. We show that isomerization kinetics can be determined from the rotational spectrum of a highly vibrationally excited molecule.

The physical basis of pure rotational spectroscopy is that when the total angular momentum is conserved, the rotational frequency is directly related to the principal moments of inertia of the molecule and, therefore, to its structure (1). In pure rotational spectroscopy, because the structure is static on the time scale of the rotational motion, the moments of inertia can be treated as constants. This simple model breaks down when a large polyatomic molecule is vibrationally excited. At high energy, vibrational energy can flow through the anharmonically coupled normal-mode vibrational states: a process called intramolecular vibrational energy redistribution (IVR) (6). For a molecule excited above the barrier to isomerization, intramolecular energy flow can ultimately lead to large-scale rearrangement of the molecular geometry. In both of these cases—simple IVR and isomerization—the molecular geometry is no longer static on the time scale of rotational motion. The time-dependent moments of inertia resulting from IVR and isomerization dynamics produce changes in the spectrum that are similar to well known effects in NMR spectroscopy (7, 8): motional narrowing is caused by IVR (9, 10) and, as demonstrated here, coalescence of the spectrum is produced by isomerization (11). The analogies between the new features in the rotational spectra of highly excited molecules and the dynamical effects in NMR spectra are the basis for calling this technique “dynamic rotational spectroscopy” (12).

Isomerization reactions are a fundamental class of chemical transformations and have drawn significant experimental and theoretical attention because of indications that the reaction rates are poorly predicted by statistical models of chemical kinetics [e.g., Rice–Ramsperger–Kassel–Marcus (RRKM) theory (13)] (14–16). Isomerization over low reaction barriers is believed to be especially susceptible to nonstatistical effects. One theoretical approach employed to gain a deeper understanding of isomerization dynamics has used extensions of random matrix theory (17) to make general arguments about the structure of the Hamiltonian for highly excited molecules (18). A second approach has examined the phase space structure of classical dynamics trajectories during isomerization (19–22). These studies have revealed the existence of bottlenecks to energy flow across the separatrix that divides the structurally localized regions of phase space. These low-dimensional models show that the full phase space cannot promote reaction. Instead, isomerization proceeds through restricted regions called “turnstiles.” Both of these theoretical approaches have pointed out that mode-specific dynamics can further inhibit the reaction rate.

Dynamic rotational spectroscopy provides a unique tool for measuring reaction rates for isolated molecules with precisely defined total internal energy. The technique directly measures the microcanonical reaction rate constant [k(E)] and, therefore, provides a stringent test of statistical reaction rate theories. Because the measurements take place in the cold, collision-free environment of a molecular beam, the reaction dynamics arise solely from the intramolecular processes. It has recently been shown (23) that conformational kinetics in room-temperature solutions can be obtained by using ultrafast 2D-infrared (IR) spectroscopy. Dynamic rotational spectroscopy and ultrafast 2D-IR spectroscopy provide complementary information about the isomerization dynamics that can be used to understand the interplay between intramolecular dynamics and solvation effects in thermal, condensed-phase systems (24, 25).

Results

Pent-1-en-4-yne (pentenyne) has two stable conformations: cis and skew. Interconversion occurs through rotation about the C–C single bond connecting the ethylenic and propargyl molecular subunits. The relaxed potential energy surface of pentenyne, calculated by using G98W (26) at the B3LYP/6-31G** level of theory, is shown in Fig. 1. At this level of theory, the cis conformer is 98 cm⁻¹ more stable than the doubly degenerate skew conformer. The calculated barrier to reaction for the cis conformer is 899 cm⁻¹ above the zero-point torsional energy level.

Fig. 1.

Potential energy surface (PES) of pentenyne for rotation about the conformational isomerization coordinate calculated at the B3LYP/6-31G** level of theory. Superimposed on the PES (lower trace) are the torsional energy levels (black lines) for 1D motion. The molecular structures correspond to the ground-state conformation. The red lines correspond to the probability distributions of the wavefunction at the relevant energy levels. The upper trace represents a model probability distribution for a quantum state near 3,330 cm⁻¹.

The probability distributions for the lowest energy one-dimensional (1D) torsional level of each conformer (27) are shown in Fig. 1. In these torsional levels, the probability distributions are peaked around a single molecular conformation. The pure rotational spectra of both conformers have been measured by using FTMW spectroscopy, and the measured rotational constants are in good agreement with predictions from the minimum energy structures in the ab initio calculations. From the pure rotational signal intensities, we estimate that 90% of the molecules in the free-jet expansion populate the ground vibration–torsion state of the cis conformer, and 10% are in the skew ground state.

The probability distribution for a molecular eigenstate at the excitation energy of the experiment (3,330 cm⁻¹) is also shown in Fig. 1. At this energy, IVR in the reaction coordinate mixes vibrational states with torsional wavefunctions that are localized around each conformational minimum. This interaction produces quantum states that can no longer be classified as belonging to either cis or skew conformations. Instead, the nuclear positions are delocalized through all torsional angles. This concept is analogous to the “resonance structures” used to describe the positions of electrons in polyatomic molecules. When two localized electron configurations can interact, the electronic wavefunctions have properties of both configurations, and the electrons are said to be “delocalized.”

The rotational spectra of the structurally mixed molecular quantum states produced by isomerization are qualitatively different from the pure rotational spectroscopy of localized torsional levels. The general appearance of the rotational spectra of quantum states of pentenyne with ≈3,330 cm⁻¹ of internal energy is shown in Fig. 2. The high-resolution IR spectrum of the acetylenic C–H stretch fundamental of pentenyne is shown on the left side of the figure. A list of the IR transition frequencies and intensities is provided in supporting information (SI) Table S1. IR-microwave double-resonance spectroscopy proves that the spectrum originates in the ground state of the more stable cis conformer. The spectra in the region of the 2₀₂-1₀₁ [standard asymmetric top notation (J_KaKc) (1), denoted as R(1) in Fig. 2] and 1₀₁-0₀₀ [R(0)] rotation–vibration transitions are shown. These spectra are “unfolded” from the frequency space into an energy space representation, to more clearly illustrate that the observed quantum states have a rotational energy separation that is characteristic of the cis geometry.

Fig. 2.

Isomerization-induced coalescence of the excited-state rotational spectrum of pentenyne. (Left) The eigenstate-resolved IR spectra of the 2₀₂-1₀₁ and 1₀₁-0₀₀ rotation–vibration transitions in the acetylenic C–H stretch fundamental of the cis conformer. These transitions are placed in an energy representation by adding the energy difference of the pure rotational levels (the 1₀₁-0₀₀ transition frequency, 6,368.39 MHz) to the 2₀₂-1₀₁ IR transition frequencies. The quantum state with the strongest transition in the 2₀₂-1₀₁ rotation–vibration spectrum (IR frequency 3,332.7859 cm⁻¹) is chosen as the reference energy (0 MHz). To facilitate comparison with measurements, the energy differences are plotted by using a frequency scale (MHz). The frequency for a J = 2–1 transition between the strongest IR transitions (red arrow) is 12,696.34 MHz, close to the 2₀₂-1₀₁ pure rotational frequency of the cis conformation of 12,671.75 MHz. (Right) After laser excitation of R(1) with a 600-MHz-bandwidth laser, the dynamic rotational spectrum of the laser-populated states is measured. The red (blue) reference line indicates the 2₀₂-1₀₁ pure rotational frequency for the skew (cis) conformer. The dynamic rotational spectrum displays coalescence between these characteristic rotational frequencies. Note that a few rotational transitions are observed near the cis pure rotational frequencies and are caused by the limited state mixing that leaves some laser-prepared quantum states with a large contribution from the acetylenic C–H stretch normal-mode vibration (a more complete description of this effect is given in Fig. S5 and SI Text).

The presence of IVR is indicated by the fragmentation of the IR spectrum into a set of transitions (6, 28). When IVR occurs, the rotation–vibration spectrum consists of transitions to all of the molecular eigenstates of the full molecular Hamiltonian that contain contributions from the acetylenic C–H stretch normal-mode via the state-mixing produced by vibrational anharmonicity and rotation–vibration interactions (e.g., Coriolis coupling). Analysis of the high-resolution rotation–vibration spectrum by standard methods gives an average time scale of 1,200 ps for IVR after coherent excitation of the acetylenic C–H stretch of cis-pentenyne (28). The 2₀₂ level depicted in Fig. 2 has an IVR time scale of 1,100 ps as shown in Fig. S1. This slow IVR time scale is characteristic of the acetylenic C–H stretch of planar conformations (29).

The rotational spectrum of the molecular eigenstates populated by laser excitation is shown to the right of the IR spectrum in Fig. 2. In this measurement, a pulsed IR laser is tuned to the R(1) rotation–vibration transition. This excitation populates quantum states with total angular momentum quantum number J = 2. The spectrum of all laser-populated molecular eigenstates in the region of the J = 2–1 rotational transition is shown. The important feature illustrated by this measurement is that the rotational spectrum of the structurally mixed quantum states peaks at a frequency between the characteristic rotational frequencies (Table S2 and SI Text) of the two stable conformations. For pentenyne at 3,330 cm⁻¹ of internal energy, the intramolecular isomerization dynamics cause coalescence of the rotational spectrum. The position of the intensity peak is approximately one-third of the way between the skew and cis pure rotational frequencies and corresponds to the weighted average rotational frequency of the vibrationally excited molecule. At excitation energies that are large compared with the conformer energy difference, twice as many quantum states are associated with the skew conformation because it has two equivalent minima in the potential.

The general appearance of the dynamic rotational spectrum of these highly excited states is independent of the initial conformational geometry, as illustrated in Fig. 3. Fig. 3A shows the cavity-FTMW spectrum in the 9- to 19-GHz frequency range obtained after excitation of the R(1) transition of the cis conformer (as in Fig. 2). The rotational spectrum of the laser-prepared quantum states displays peaks around 11 and 16.5 GHz. These frequencies (indicated by the vertical reference lines) fall between the J = 2–1 and J = 3–2 pure rotational frequencies of the skew and cis conformers and demonstrate coalescence after excitation of the cis conformer. Attribution of the spectral changes to isomerization is supported by measuring the rotational spectrum of quantum states prepared by exciting molecules in the ground torsional level of the skew conformation (Fig. 3B). In this case, the skew spectrum is only measured in the region of the J = 2–1 transition (10–13 GHz) because of sensitivity limits. This spectrum requires longer signal averaging because the skew conformer has significantly lower population in the molecular beam. The rotational spectrum of these quantum states is also peaked at a frequency between the characteristic conformer rotational frequencies and shows the same isomerization-induced coalescence behavior.

Fig. 3.

Rotational spectrum of laser-prepared quantum states of pentenyne in the region of the acetylenic C–H stretch fundamental. These spectra were recorded by fixing the IR laser on the R(1) transition, promoting population to the IR excited state, and subsequently scanning the microwave source. (A) Spectrum for excitation of the cis conformer. (A Inset) A small region of the actual experimental data, shown to illustrate the resolution and sensitivity. (B) Selective excitation of the skew conformer. The red dashed lines show the frequencies of the pure rotational transition of the J = 2–1 and J = 3–2 transitions for the cis (12,617 and 18,848 MHz, respectively) and skew (10,044 and 15,065 MHz, respectively) conformers. (C) Traces showing the IR action spectra recorded by monitoring the most intense excited-state peaks in A and B. The red traces are the expected action spectra calculated by using the cis (upper) and skew (lower) rotational constants. A +0.75 cm⁻¹ band origin shift is seen for the skew conformer.

The ability to selectively excite the cis or skew conformations comes from the high resolution of the laser excitation (0.02 cm⁻¹). Double-resonance measurements show a small frequency difference of ≈0.75 cm⁻¹ in the acetylenic C–H stretch fundamental frequency of the cis and skew conformers. The ability to selectively excite different conformations is verified in Fig. 3C. In this double-resonance measurement, the FTMW spectrometer frequency is set to one of the excited-state rotational frequencies, and the IR laser is scanned. The FTMW signal appears whenever the laser is resonant with a vibration–rotation transition that populates the FTMW-monitored excited state. The excited state can be reached through several transitions [e.g., R(1) and P(3)], and these frequencies are separated by known rotational energy-level spacings in the ground state. The IR laser signatures for excited-state transitions in Fig. 3 A and B are shown, along with the calculated signature for a transition originating from either the cis or skew ground state.

A second qualitative difference between dynamic rotational spectroscopy and pure rotational spectroscopy is that the rotational spectrum of a single, highly excited quantum state is composed of a large number of eigenstate-to-eigenstate rotational transitions. This effect is the rotational spectroscopy equivalent of the fragmentation of the IR spectrum by IVR. The rotational spectra of four individual quantum states extracted from the spectrum in Fig. 3A are shown in Fig. 4A. Lists of the transition frequencies and intensities for each eigenstate-resolved rotational spectrum are given in Fig. S2, Fig. S3, Table S3, and SI Text. Each single quantum state spectrum displays the coalescence profile with the intensity peaked between the pure rotational frequencies of the two localized conformations. Unlike the smooth line shape that would be observed in dynamic NMR spectroscopy, the dynamic rotational spectrum of an isolated molecule must be composed of a series of discrete transitions between the finite number of molecular eigenstates in the bound quantum system.

Fig. 4.

Rate determinations from the single quantum-state dynamic rotational spectra. (A) The four single-eigenstate rotational spectra extracted by triple-resonance techniques. These spectra were acquired by fixing the IR laser on the R(1) transition of the cis conformer to populate quantum states with J = 2. The cavity-FTMW spectrometer is tuned to each measured transition in Fig. 3A. A second microwave pulse is tuned to a known transition frequency for each eigenstate and interacts with the polarized sample. The destruction of the rotational coherence of the transition monitored by the cavity-FTMW spectrometer by this second pulse is used to confirm the assignment to each individual eigenstate. (B and C) Traces (J = 2 → 1; 10,000–14,000 MHz) (B) and (J = 3 ← 2; 14,000–18,000 MHz) (C) show the cumulative intensity analysis of the single-eigenstate spectra. The dashed blue lines are the single-eigenstate results; the solid red line is the average of the four composite eigenstate spectra; the black line represents the analytic line shape derived from a three-state Bloch model.

The isomerization rate is obtained from an analysis of the overall rotational line shape by using a three-state microwave spectroscopy Bloch model modified for chemical exchange (12). In this model, the cis and skew structures interact by coupling to above-barrier 1D torsional levels. These torsional levels have probability distributions that span both the cis and skew regions of the potential and provide the doorway for isomerization. The Bloch model produces a smooth line shape profile for the dynamic rotational spectrum, and it must be compared with the experimental spectrum, which consists of discrete eigenstate-resolved transitions. As illustrated in Fig. 4 B and C, we use the normalized cumulative spectral intensity of the experimental spectrum to smooth the quantum mechanical intensity fluctuations of the eigenstate-resolved spectrum (30, 31). The cumulative intensity profiles for the experimentally measured average dynamic rotational spectrum (red trace) and calculated coalescence line shape (black trace) of pentenyne at J = 2 → 1 and J = 3 ← 2 are shown in Fig. 4 B and C. The calculated line shape corresponds to an isomerization rate of 4.0 × 10¹⁰ s⁻¹ (isomerization lifetime of 25 ps). By comparing line shape predictions with the experimental spectra, we estimate that the errors on the experimental rate determination are <50% [k_iso = 4(2) × 10¹⁰ s⁻¹]. These error estimates are described more fully in SI Text and Fig. S4. The dashed blue lines in Fig. 4 B and C represent the cumulative intensity profiles of the individual eigenstate spectra. Fluctuations are observed about the average cumulative intensity distribution. These rate fluctuations are a key feature of quantum state-resolved chemical kinetics but are not analyzed here (32).

A more complete analysis of the average isomerization rate is shown in Fig. 5. The reaction rate for each conformer into the delocalized torsional levels is varied independently. The contours drawn on the plot give the conformational isomerization time scale that comes from the kinetic model underlying the Bloch analysis (12). The relationship between these rates and the overall isomerization reaction rate are described in more detail in SI Text. The surface shows a minimum error valley that approximately follows the contour of constant total isomerization rate (the separate forward and reverse reaction rates are not determinable after the onset of coalescence, but the total reaction rate is well defined). The isomerization rates are the same, within the estimated measurement error, for both the J = 2 → 1 and J = 3 ← 2 measurements.

Fig. 5.

Method for rate determination from dynamic rotational spectra for both the J = 2 → 1 (A) and J = 3 ← 2 (B) data on pentenyne. An error surface is generated by independently varying the localized–delocalized reaction rates. The rates are used in the three-state Bloch model to generate a cumulative intensity profile like those shown by the smooth, black curves in Fig. 4 B and C. The color-coding of the surface reflects the suitability of the calculated cumulative intensity to the measured cumulative intensity profile of the composite of the four single-eigenstate spectra (the red curves in Fig. 4 B and C). The error is the sum of the squares of the residuals between the simulated and experimental cumulative intensity distributions. The isomerization reaction rate is calculated by using the two rate constants and a reversible, first-order reaction scheme. This model is described further in SI Text. Contours that indicate the time scale for isomerization are shown in black and are labeled with the reaction time scale in picoseconds.

These measurements support the validity of using dynamic rotational spectroscopy to determine reaction rates. As indicated in Figs. 2 and 3, the separation of the characteristic cis and skew rotational frequencies increases linearly with the rotational angular momentum quantum number (Δ ν = 2,630 MHz for the J = 2 → 1 spectrum and 3,780 MHz for the J = 3 ← 2 spectrum). The intrinsic ability to vary the conformer frequency difference provides a check on the rate determination from line shape analysis. For pentenyne, the increased frequency difference, but constant isomerization rate, leads to a broader spectral distribution for the J = 3 ← 2 spectrum, as can be seen in Fig. 3 and gleaned from the cumulative intensity distributions in Fig. 4.

Discussion

Dynamic rotational spectroscopy directly measures the microcanonical rate constant [k(E, J)] for isomerization reactions. The measurement process is fundamentally different from most laser spectroscopy techniques in chemical dynamics (6, 23–25, 28). In laser spectroscopy, the dynamics of an initially prepared quantum state (such as a normal-mode vibration like the acetylenic C–H stretch) are detected. The subsequent reaction dynamics can be influenced by the IVR dynamics of this initial state. By contrast, dynamic rotational spectroscopy uses single quantum states of the molecule as the initial state for the rate determination. The dynamical information about the energy-resolved isomerization process is encoded in each eigenstate. In a dynamic view, these eigenstates reflect the nature of the system in the long time limit after coherent artifacts related to state-specific IVR of laser state preparation have dissipated.

The isomerization lifetime of pentenyne measured by dynamic rotational spectroscopy can be directly compared with the predictions of RRKM theory. We calculated the RRKM reaction rate by using the scaled harmonic frequencies of pentenyne obtained from electronic structure calculations with G98W at the B3LYP/6-31G** level of theory. These calculations were also used to estimate the conformer energy difference and reaction barrier (Fig. 1). Using the electronic structure parameters, RRKM theory predicts an isomerization time scale of ≈0.5 ps at an energy 3,330 cm⁻¹ above the cis ground vibrational state. This calculated rate is ≈50 times faster than the measured lifetime. This discrepancy cannot be removed through any reasonable adjustment of the normal-mode vibrational frequencies or the reaction barrier obtained from the electronic structure calculations.

In addition to its failure to calculate the low-barrier microcanonical isomerization rate of pentenyne, the statistical theory cannot account for a second feature of the reaction kinetics of pentenyne. The high-resolution IR spectrum of pentenyne (Fig. 2) has been used to determine the time scale for IVR after coherent excitation of the acetylenic C–H stretch of the cis conformer. From the spectra shown in Fig. 2, and the more complete dataset covering other rotation–vibration transitions, the time scale for intramolecular vibrational energy redistribution is ≈1,200 ps. The rotational contour of the acetylenic C–H stretch rotation–vibration spectrum shows that the coherent acetylenic C–H stretch excited state maintains the cis conformational geometry. Therefore, the IVR rate obtained from the rotation–vibration spectrum provides an upper limit to the conformational isomerization rate after coherent state preparation. For pentenyne, a dramatic slowing of the reaction rate from 25 to 1,200 ps occurs when coherent state preparation is used in the same energy region. The pentenyne conformational isomerization reaction demonstrates both mode-specific reaction kinetics and non-RRKM microcanonical reaction rates.

Materials and Methods

Pent-1-en-4-yne (98% purity) was obtained from GFS Chemicals and used without further purification. The high-resolution (6-MHz, 0.0002-cm⁻¹) single-photon IR measurements shown in Fig. 2 were performed with an electric-resonance optothermal spectrometer. The double-resonance capabilities of this spectrometer were used to provide rotational assignments (33). The rotational spectrum of laser-populated vibrational states, shown in Fig. 2, was measured by using a recently developed broadband FTMW spectrometer (34).

The quantum state-resolved dynamic rotational spectra in Figs. 3 and 4 were obtained by using IR-pulsed FTMW-MW triple-resonance spectroscopy (10). The gas sample used in these measurements was a mixture of 1% pentenyne in an 80/20% He/Ne inert gas mix. The gas sample was expanded into vacuum by using a pulsed valve (Series 9, 10 Hz, 0.8-mm orifice; General Valve). The FTMW spectrometer used in these measurements is a compact Balle–Flygare-type instrument based on the recent National Institute of Standards and Technology design (35).