Research ArticleChemistry

Dynamical interference in the vibronic bond breaking reaction of HCO

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Science Advances  02 Jan 2019:
Vol. 5, no. 1, eaau0582
DOI: 10.1126/sciadv.aau0582


First-principles treatments of quantum molecular reaction dynamics have reached the level of quantitative accuracy even in cases with strong non–Born-Oppenheimer effects. This achievement permits the interpretation of puzzling experimental phenomena related to dynamics governed by multiple coupled potential energy surfaces. We present a combined experimental and theoretical study of the photodissociation of formyl radical (HCO). Oscillations observed in the distribution of product states are found to arise from the interference of matter waves—a manifestation analogous to Young’s double-slit experiment.


The wave nature of quantum particles gives rise to interference phenomena, which are analogs of the optical interference in Young’s double-slit experiment. When a particle reaches a destination via more than one path, the phase associated with each path is generally different. As the observable intensity is the squared sum of all contributing amplitudes, the phase differences lead to oscillatory patterns in intensity (1). While such wave-like behaviors are well known in physics, manifestations of interference in molecular reactions are rare, as they require distinct pathways with nearly equal probabilities. One such example is the inelastic scattering of NO by Ar (2, 3), where a Δj = even propensity can be attributed to the weakly broken symmetry of the diatom (4). Various forms of interferences have also been found in reactive scattering (57). More closely related to the current work is the photodissociation of water in its Embedded Image band, where nonadiabatic transitions between the Embedded Image and Embedded Image states of H2O near two conical intersections (HOH and HHO) provide distinct dissociation pathways leading to the same OH + H products, manifested by oscillations in the OH rotational state distribution (8). Here, we describe observations of interference in another chemical process, governed by Renner-Teller (RT) electronic state degeneracy at linearity, the photodissociation of HCO. In this case, photoexcitation from the ground electronic state (with a bent equilibrium geometry) to the lowest excited electronic state (with a linear equilibrium geometry) gives rise to a strong bending excitation. The two electronic states are degenerate at linearity, and once a nonadiabatic transition is made to the ground state, dissociation (to CO + H) occurs with considerable rotational excitation of CO. Moreover, the complex topography of the ground electronic state leads to interference between different paths to the same rotational product states, seen as oscillations in the CO rotational distribution.

The formyl radical (HCO) is an important combustion intermediate, which serves as the primary source of H in the chain initiation of hydrocarbon combustion. It also plays important roles in atmospheric and interstellar chemistry. The Embedded Image absorption band consists of a long progression of diffuse features, stemming from the bent-to-linear transition (9). Interestingly, the linear Ã2A′′ state and bent Embedded Image state become degenerate at linearity as an RT 2Π pair (10), which is subject to a type of non–Born-Oppenheimer coupling between the molecular rotational angular momentum (K) and the electronic angular momentum (Λ) (11). The potential energy surfaces (PESs) of these two states (12) are shown in Fig. 1, in which the seams of degeneracy for linear geometries are visible. For K > 0, this vibronic coupling facilitates rapid predissociation of the Ã2A′′ vibronic levels above the H + CO dissociation asymptote (13, 14). The CO product from HCO (Ã2A′′) photodissociation has been detected by Houston and coworkers and has been found to have rather high (up to j = 50) rotational excitation, while its vibrational excitation depends on the initial CO mode excitation in the parent molecule (1518). The product energy disposal can be rationalized by the anisotropy of the ground Embedded Image state PES, which has an equilibrium valence bond angle of θ = ~125°. The PES thus exerts a strong torque on the CO fragment as it emerges from the collinear degeneracy seam, as demonstrated by previous dynamical studies of the dissociation (1921).

Fig. 1 Surface plots of the coupled Embedded Image and Embedded Image states of the formyl radical relevant to photodissociation dynamics.

The CO bond distance is held fixed (rCO = 1.18 Å), while the plots illustrate the behavior of the PESs as a function of R and γ (Jacobi coordinates, angstroms, and degrees). Seams of degeneracy are noticeable for collinear geometries of HCO (0°) and HOC (180°). The region of the seam for collinear HCO (γ = 0) most relevant to the dynamics is highlighted by a red oval. Top and bottom panels on the right show rotated perspectives of the same plot.

We report here a new investigation of the dynamics and CO internal state distributions, recorded following photoexcitation of HCO at several wavelengths.

Fig. 2 Calculated absorption spectrum of the 202 state (the J′ label refers to the upper state in the transition) of HCO.

The extremely close agreement between calculated and recorded positions and widths is given in table S1. a.u., atomic units.


In Fig. 2, the calculated absorption spectrum is shown for the 202 state of HCO (Embedded Image), which has a substantial population (P = 0.18, assuming Boltzmann) at the estimated experimental temperature of 20 K. In agreement with experiments (9, 13), the spectrum is dominated by a pure bending progression, denoted as (0, Embedded Image, 0). Strictly speaking, because of the Coriolis coupling, K′ is not a good quantum number, but it can still be used to assign the peaks. It is apparent that the resonance width increases drastically with K′ because of increased RT coupling (10). As shown in table S1, the calculated transition frequencies and linewidths of the resonances are in excellent agreement with the existing experimental data, further validating the PESs.

In Fig. 3, the measured CO (v = 0) product rotational state distributions following excitation to the HCO (Ã2A′′) (0,101,0) and (0,121,0) states are compared with the calculated ones. The oscillatory structure in the experimental distributions noted previously by Houston and coworkers (1518) is unambiguously established and extended to the low-j region owing to the increased and uniform sensitivity of the high-n Rydberg-atom time-of-flight (HRTOF) technique and the CO background free product detection. The simulated product distributions obtained as thermal averages over initial rotational states (assuming 20 K beam temperature) are in very good agreement with the experiment. Any remaining differences between the measured and calculated distributions are most likely due to the strong sensitivity of the interference effect to subtleties of the PESs as well as to the experimental conditions. In fig. S7, the individual (initial rotational state) components are given. There, it can be seen that the rotational state distributions calculated from pure parent states are significantly more oscillatory than the experimental ones recorded at ~20 K, similar to those reported in earlier calculations (20). The less pronounced oscillation mainly reflects averaging over the initial rotational states but perhaps also contributions from broadening mechanisms (convolution of computed spectra with assumed linewidth parameters often leads to more similar comparisons).

Fig. 3 Comparison of calculated and measured CO rotational distributions.

The observed oscillatory and highly excited inverted product distributions are closely reproduced in the calculations. Simulated distributions for pure initial states (or lower beam temperatures) are even more oscillatory (see the Supplementary Materials).


Let us now focus on the oscillations of the CO rotational state distributions. To gain insight into the dynamics and possible interference effects, we analyze trajectories on the ground-state PES originating from linearity where the RT coupling takes place. For this purpose, the (0,121,0) state is chosen. In Fig. 4 (top panel), a few two-dimensional representative trajectories are shown on the Embedded Image state PES, in which the C-O distance is fixed at its equilibrium geometry (rCO = r0 = 1.18 Å). This is a reasonable approximation as CO vibration is nearly a spectator in the dissociation. To mimic the RT dissociation process, all trajectories were started at linearity with different R0 values and zero momentum in the R direction. The available kinetic energy is exclusively placed in the angular coordinate: Embedded Image. It is clear from the figure that typical trajectories are first attracted to the HCO well but are subsequently deflected by the isomerization barrier between HCO and HOC and then eventually dissociate. Other details of the trajectory calculations can be found in the Supplementary Materials.

Fig. 4 Insight into the origin of quantum interference in the reaction product distribution is obtained by a comparison of rigorous quantum scattering calculations with quasi-classical trajectories (for details of the calculations, see the main text and the Supplementary Materials).

Bottom: Interference fringes are seen in the quantum wave function as it leaves the molecular region, moving toward the asymptotic product channels. Top and middle: Illustration of how a rotational rainbow effect is predicted as classical trajectories with different phases contribute to the same CO product states labeled by j’s. Energies in the top panel are relative to the minimum on the ground electronic state. The square of the upper-state wave function provides the weight of the coordinate R (initially at R0). The Jacobi coordinates are used in which r and R denote the C-O distance and distance between H and the CO center of mass, while γ is the enclosing angle.

Although such an oversimplified classical model is not expected to generate quantitative results, the trajectories are useful nonetheless to shed light on the dissociation dynamics, particularly the interference effect, as explained below. It is interesting to note that various trajectories arrive at the same final rotational angular momentum, despite the fact that they sampled different parts of the PES and hence arrive out of phase. To better illustrate this effect, the excitation function, namely, the dependence of the final CO rotational quantum number on the initial R0 position, is plotted in Fig. 4 (middle panel). It is clear from the figure that the excitation function has a turning point near R0 ~ 3.6 atomic units, which is well within the range of the (0,121,0) wave function (as also shown in the same figure). The presence of this turning point Embedded Image suggests a singularity in the classical rotational excitation function. In this so-called rotational rainbow effect, trajectories with different initial positions lead to the same final rotational state (22). The quantum mechanical equivalent of the classical rainbow effect manifests in oscillations in the CO rotational state distribution, resulting from the constructive and destructive interference among the corresponding “quantum trajectories” because of the different dynamical phases associated with them (6). Although absent in the classical product rotational state distribution, the interference effect can be seen in the corresponding wave function in Fig. 4 (bottom panel), where waves emanating from different sources interfere as manifested by nodes. The interferences eventually lead to the oscillatory CO rotational state distribution in Fig. 3.

The interference effect reported here differs from that observed in the photodissociation of H2O(Embedded Image) (8) because there, the interference is between trajectories emanating from the HOH and HHO conical intersections. In the present case, on the other hand, the interference effect stems from the same source (RT coupling at linearity). Similar to the recently discussed scenario found in the H + D2 reactive scattering (6), these interference effects can all be traced back to the features on an adiabatic PES, which dictate the nuclear dynamics.

To summarize, we report quantum state-to-state dynamics in a prototypical system in which observed oscillations in the product-state distribution are attributed to an interference effect stemming from unique features on the ground-state PES of the system. As a result, these interference effects are exquisitely sensitive to the PES and thus can be used to probe dynamically important regions of the reactive PES.



The HRTOF technique (2327) was used to obtain the H + CO product center-of-mass (CM) translational energy and angular distributions. A pulsed HCO beam was generated by photolyzing a ~2% mixture of acrolein in Ar with the 193-nm radiation of an ArF excimer laser focused in front of the pulse nozzle. The HCO radicals produced were entrained in the molecular beam and subsequently cooled (to ~20 K) by supersonic expansion. The radical beam was collimated and then crossed with the visible photolysis radiation from a Nd:YAG pumped dye laser (defined here as the photolysis laser). This photolysis laser radiation was slightly focused and properly delayed with respect to the 193-nm radical production laser but preceded the H-atom product probe lasers; its polarization can be rotated by a Fresnel-Rhomb achromatic λ/2 plate for product angular distribution studies. Hydrogen atoms produced from photodissociation were first excited to the 22P level by a 121.6-nm Lyman-α (L-α) radiation generated by tripling a 364.7-nm dye-laser output in Kr (values given are vacuum wavelengths). The H atoms were further excited to a high-n Rydberg level by the 366.2-nm output of another dye laser (Rydberg probe laser). A small fraction of the radiatively metastable Rydberg atoms drifted with their nascent velocities to a microchannel plate detector that was perpendicular to the molecular beam, and were detected as ions after being field ionized in front of the detector. The nominal flight length was 37.1 cm. The accumulated H-atom TOF spectra represented 50,000 to 100,000 laser firings. Because the HRTOF tagging technique gives high resolution and uniform detection sensitivity for the kinetic energy of the H fragment and was rid of the CO background interference, the observed CO rotational-vibrational state distributions were well resolved and extended to the low-j states and can be more reliably interpreted than those reported previously (1518). It is clear from these experimental data that the CO rotational state distributions exhibit pronounced and reproducible oscillations in nascent product states, extending through the low-j excitation range (see Fig. 3). More details of the experiments and results are provided in the Supplementary Materials.


To understand these oscillatory features, we performed full-dimensional wave packet studies. While ab initio ground-state PESs of this system have been reported by several groups (2830), fully coupled PESs of the lowest few electronic states are rare (20, 31). The results reported here were obtained on the most recent coupled PESs determined from high-level ab initio calculations (12), proven to be considerably more accurate than the previous PESs (20). More details of these calculations are provided in the Supplementary Materials. Briefly, the RT coupling was treated using the theoretical approach of Petrongolo (32) and Goldfield et al. (19), as implemented in our recent work (33). The strong Coriolis coupling was included explicitly while neglecting spin-orbit coupling. The photoexcitation was simulated using the Condon approximation by exciting the thermally populated low-lying rotationally excited but vibrationally cold eigenfunctions (labeled Embedded Image) from the ground Embedded Image electronic state PES, up onto the Ã2A′′ state PES according to rigorous selection rules (the use of integer labels for J in this spin doublet system is due to the neglect of spin-orbit coupling) (34). They were then propagated on both coupled electronic states using the Chebyshev propagator (35) until reaching the H + CO dissociation asymptote, where the CO rotational-vibrational state distributions were determined by projecting the wave packet onto asymptotic product-state bases on the Embedded Image state PES (36).


Supplementary material for this article is available at

Section S1. Experimental

Section S2. Electronic structure and PES

Section S3. Quantum dynamics

Section S4. Classical trajectory calculations

Table S1. Comparison of the calculated and measured (5) positions and widths of resonances.

Fig. S1. Resonance enhanced multiphoton ionization spectrum of the HCO Embedded Image transition.

Fig. S2. H-atom photofragment yield spectrum of the HCO radical as a function of photolysis excitation wavelength around the Embedded Image transition.

Fig. S3. CM translational energy distributions P(ET,θ)’s and anisotropy parameter β of the H + CO products from photodissociation of the Embedded Image(0,121,0) vibronic state.

Fig. S4. CM translational energy distributions P(ET,θ)’s and anisotropy parameter β of the H + CO products from photodissociation of the Embedded Image(0,101,0) vibronic state.

Fig. S5. Internal energy and rotational state distributions of the CO (X1Σ+, υ = 0) product from photodissociation of the Embedded Image vibronic states.

Fig. S6. The convergence of the X and A states toward their collinear degeneracy is shown at the minimum of the seam [adapted from (12)].

Fig. S7. Simulated CO product distributions.

Fig. S8. Plot of typical trajectories shown over the contours of the ground-state PES in the plane with CO distance fixed.

References (3762)

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.


Acknowledgments: Funding: X.Z. acknowledges support from the National Natural Science Foundation of China (grant no. 11674003). D.X. was supported by the National Natural Science Foundation of China (grant nos. 21688102, 21733006, and 21590802) and the National Key Research and Development Program of China (grant no. 2017YFA0206500). J.Z. acknowledges support from the NSF (grant no. CHE-1566636). This research was also supported by the U.S. Department of Energy Office of Science, Office of Basic Energy Sciences (grant nos. DE-SC0010616 to R.D. and DE-SC0015997 to H.G.). Author contributions: All authors designed and conducted the research and participated in writing the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
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