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The dissociative single ionization of the LiH molecule using a two-color UV-UV pump-probe scheme is simulated in ab initio calculations demonstrating how the dynamics initiated on intermediate Rydberg states of diatomic molecules can be imaged by such experiments. The theoretical treatment combines nuclear motion with highly correlated descriptions of electronic continua and bound electronic states. Nuclear dynamics on Rydberg states are only weakly reflected by changes in photoelectron energies ejected by time-delayed ionizing pulses if the Rydberg potential curves parallel those of the ion states being produced. However, coincidence measurements guided by knowledge of the photoionization amplitudes as a function of internuclear distance can still reveal intermediate-state dynamics in pump-probe experiments.

We present an experimental and theoretical energy- and angle-resolved investigation on the non-dissociative photoionization dynamics of near-resonant, one-color, two-photon, single valence ionization of neutral O2 molecules. Using 9.3 eV femtosecond pulses produced via high harmonic generation and a 3-D momentum imaging spectrometer, we detect the photoelectrons and O2 + cations produced from one-color, two-photon ionization in coincidence. The measured and calculated photoelectron angular distributions show agreement, which indicates that a superposition of two intermediate electronic states is dominantly involved and that wavepacket motion on those near-resonantly populated intermediate states does not play a significant role in the measured two-photon ionization dynamics. Here, we find greater utility in the diabatic representation compared to the adiabatic representation, where invoking a single valence-character diabat is sufficient to describe the underlying two-photon ionization mechanism.

Filming atomic motion within molecules is an active pursuit of molecular physics and quantum chemistry. A promising method is laser-induced Coulomb Explosion Imaging (CEI) where a laser pulse rapidly ionizes many electrons from a molecule, causing the remaining ions to undergo Coulomb repulsion. The ion momenta are used to reconstruct the molecular geometry which is tracked over time (i.e., filmed) by ionizing at an adjustable delay with respect to the start of interatomic motion. Results are distorted, however, by ultrafast motion during the ionizing pulse. We studied this effect in water and filmed the rapid "slingshot" motion that enhances ionization and distorts CEI results. Our investigation uncovered both the geometry and mechanism of the enhancement which may inform CEI experiments in many other polyatomic molecules.

Nonlinear spectroscopies can disentangle spectra that are congested due to inhomogeneous broadening. In conjunction with theoretical calculations, attosecond extreme ultraviolet (XUV) four-wave-mixing (FWM) spectroscopy is utilized here to probe the dynamics of autoionizing inner valence excited Rydberg states of the polyatomic molecule, CO2. This tabletop nonlinear technique employs a short attosecond XUV pulse train and two noncollinear, few-cycle near-infrared pulses to generate background-free XUV wave-mixing signals. FWM emission is observed from the n=5-7 states of the Henning sharp ndσg Rydberg series that converges to the ionic Bςu+2 state. However, these transient emission signals decay with lifetimes of 33 ± 6, 53 ± 2, and 94 ± 2 fs, respectively, which calculations show are consistent with the lifetimes of the short-lived n=6-8 members of the nsσg character Henning diffuse Rydberg series. The oscillator strengths of transitions between states involved in all possible resonant FWM processes are calculated, verifying that the nonlinear spectra are dominated by pathways described by an initial excitation to the diffuse nsσg Rydberg series and emission from the sharp ndσg Rydberg series. The results substantiate not only that attosecond XUV FWM spectroscopy produces rigorous and meaningful measurements of ultrafast dynamics in polyatomic systems, but also that nonlinear spectroscopic techniques are versatile tools to selectively probe dynamics that are otherwise difficult to access.

The double photoionization of a molecule by one photon ejects two electrons and typically creates an unstable dication. Observing the subsequent fragmentation products in coincidence can reveal a surprisingly detailed picture of the dynamics. Determining the time evolution and quantum mechanical states involved leads to deeper understanding of molecular dynamics. Here in a combined experimental and theoretical study, we unambiguously separate the sequential breakup via D+ + OD+ intermediates, from other processes leading to the same D+ + D+ + O final products of double ionization of water by a single photon. Moreover, we experimentally identify, separate, and follow step by step, two pathways involving the b 1Σ+ and a 1Δ electronic states of the intermediate OD+ ion. Our classical trajectory calculations on the relevant potential energy surfaces reproduce well the measured data and, combined with the experiment, enable the determination of the internal energy and angular momentum distribution of the OD+ intermediate.

Löwdin's symmetry dilemma is an ubiquitous issue in approximate quantum chemistry. In the context of Hartree-Fock (HF) theory, the use of Slater determinants with some imposed constraints to preserve symmetries of the exact problem may lead to physically unreasonable potential energy surfaces. On the other hand, lifting these constraints leads to the so-called broken symmetry solutions that usually provide better energetics, at the cost of losing information about good quantum numbers that describe the state of the system. This behavior has previously been extensively studied in the context of bond dissociation. This paper studies the behavior of different classes of HF spin polarized solutions (restricted, unrestricted, and generalized) in the context of ionization by strong static electric fields. We find that, for simple two electron systems, unrestricted Hartree-Fock (UHF) is able to provide a qualitatively good description of states involved during the ionization process (neutral, singly ionized, and doubly ionized states), whereas RHF fails to describe the singly ionized state. For more complex systems, even though UHF is able to capture some of the expected characteristics of the ionized states, it is constrained to a single Ms (diabatic) manifold in the energy surface as a function of field intensity. In this case, a better qualitative picture can be painted by using generalized Hartree-Fock as it is able to explore different spin manifolds and follow the lowest solution due to lack of collinearity constraints on the spin quantization axis.

Multiple Rydberg series converging to the O2+c4Σ-u state, accessed by 20-25 eV extreme ultraviolet (XUV) light, serve as important model systems for the competition between nuclear dissociation and electronic autoionization. The dynamics of the lowest member of these series, the 3sσg state around 21 eV, has been challenging to study owing to its ultra-short lifetime (

Multiple Rydberg series converging to the O2+c4Σ-u state, accessed by 20-25 eV extreme ultraviolet (XUV) light, serve as important model systems for the competition between nuclear dissociation and electronic autoionization. The dynamics of the lowest member of these series, the 3sσg state around 21 eV, has been challenging to study owing to its ultra-short lifetime (

We demonstrate a theoretical treatment of dissociative single ionization of the LiH molecule using two-color UV-UV pulse sequences that makes use of a highly correlated description of both the ionization continuum and target molecular ion and neutral states to which it is coupled. The present results emphasize how the details of the ionization process at various internuclear distances combine to form a lens through which such experiments image the dynamics of intermediate electronic states populated by the pump pulse. While ionization yields (dissociative and nondissociative) provide information about the amplitudes and phases that build up the molecular wave packet in the neutral states, molecular frame photoelectron angular distributions exhibit the changing character of those states, i.e., from ionic to covalent. In addition, the time-dependent mean kinetic energy of the wave packet on neutral states is clearly mapped onto the kinetic energy release of the atomic fragments produced by the probe ionization pulse.

We report experimental results on three-dimensional momentum-imaging measurements of anions generated via dissociative electron attachment to gaseous formamide. From the momentum images, we analyze the angular and kinetic-energy distributions for NH2-, O-, and H- fragments and discuss the possible electron attachment and dissociation mechanisms for multiple resonances for two ranges of incident electron energies, from 5.3 to 6.8 eV and from 10.0 to 11.5 eV. Ab initio theoretical results for the angular distributions of the NH2- anion for ∼6-eV incident electrons, when compared with the experimental results, strongly suggest that one of the two resonances producing this fragment is a A′′2 Feshbach resonance.

We present state-selective measurements on the NH2+ + H+ and NH+ + H+ + H dissociation channels following single-photon double ionization at 61.5 eV of neutral NH3, where the two photoelectrons and two cations are measured in coincidence using 3D momentum imaging. Three dication electronic states are identified to contribute to the NH2+ + H+ dissociation channel, where the excitation in one of the three states undergoes intersystem crossing prior to dissociation, producing a cold NH2+ fragment. In contrast, the other two states directly dissociate, producing a ro-vibrationally excited NH2+ fragment with roughly 1 eV of internal energy. The NH+ + H+ + H channel is fed by direct dissociation from three intermediate dication states, one of which is shared with the NH2+ + H+ channel. We find evidence of autoionization contributing to each of the double ionization channels. The distributions of the relative emission angle between the two photoelectrons, as well as the relative angle between the recoil axis of the molecular breakup and the polarization vector of the ionizing field, are also presented to provide insight on both the photoionization and photodissociation mechanisms for the different dication states.

We present an experimental and theoretical energy- and angle-resolved study on the photoionization dynamics of nonresonant one-color two-photon single-valence ionization of neutral N2 molecules. Using 9.3-eV photons produced via high-order harmonic generation and a three-dimensional momentum imaging spectrometer, we detect the photoelectrons and ions produced from one-color two-photon ionization in coincidence. Photoionization of N2 populates the X ?g+2, A ?u2, and B ?u+2 ionic states of N2+, where the photoelectron angular distributions associated with the X ?g+2 and A ?u2 states both vary with changes in photoelectron kinetic energy of only a few hundred meV. We attribute the rapid evolution in the photoelectron angular distributions to the excitation and decay of dipole-forbidden autoionizing resonances that belong to series of different symmetries, all of which are members of the Hopfield series, and compete with the direct two-photon single ionization.

We investigate the two-photon double ionization of atomic magnesium induced by ultrashort pulses. Though the initial and final state symmetries are comparable to the same process in helium, in stark contrast the range of photon energies for which nonsequential ionization is the only open pathway is narrow (less than 1 eV) in magnesium. Thus several sequential ionization pathways feature heavily in these processes. Nonetheless, it is found that for pulse durations between 0.25 and 2.0 fs, the joint angular dependence of the ejected electrons can depend sensitively on pulse length, varying between the strictly back-to-back ejection characteristic of nonsequential ionization to other distributions. The significance of excited-state correlating configurations in representing the initial state of magnesium is discussed in the light of their consequences for the resulting angular distributions at photon energies where sequential ionization can access intermediate states that lie nearby in energy, particularly for longer pulses.

We report measurements on the H++H+ fragmentation channel following direct single-photon double ionization of neutral NH3 at 61.5 eV, where the two photoelectrons and two protons are measured in coincidence using three-dimensional (3D) momentum imaging. We identify four dication electronic states that contribute to H++H+ dissociation, based on our multireference configuration-interaction calculations of the dication potential energy surfaces. The extracted branching ratios between these four dication electronic states are presented. Of the four dication electronic states, three dissociate in a concerted process, while the fourth undergoes a sequential fragmentation mechanism. We find evidence that the neutral NH fragment or intermediate NH+ ion is markedly rovibrationally excited. We also identify differences in the relative emission angle between the two photoelectrons as a function of their energy sharing for the four different dication states, which bare some similarities to previous observations made on atomic targets.

The simple single-channel static-exchange approximation completely ignores correlation between the continuum and molecular ion electrons. In molecular systems with symmetry equivalent atoms, the single-channel approximation can seriously fail in core ionization when using delocalized orbitals to represent the core hole states. We present cross sections and molecular frame photoelectron angular distributions with both localized and delocalized core orbitals in CF4 F (1s) ionization. We show that only a full coupled-channel calculation can recover an accurate description of the physics of inner-shell photoionization when using delocalized orbitals, whereas nearly the same result can be obtained from independent single-channel static-exchange calculations when localized core orbitals are used. A grid-based variational method described here makes such single-channel calculations possible on larger systems without local-exchange approximations. Illustrative calculations on the core ionization of SF6 are presented to illustrate the power of the grid-based method.

We report recent results of mass-resolved anion fragment momentum imaging experiments to investigate dissociative electron attachment to formic acid, for incident energies between 5 eV and 9 eV. A remarkable site-selectivity is found for a resonance at 8.5 eV by comparing anion fragment yields for two deuterated isotopologues of formic acid. This results in an H- fragment from the O-H bond of the transient anion, with negligible contribution from C-H break. In contrast, a lower-energy resonance at 7.1 eV dissociates by C-H or O-H break to produce H- and the neutral radicals HOCO or HCOO.

We publish three Roadmaps on photonic, electronic and atomic collision physics in order to celebrate the 60th anniversary of the ICPEAC conference. In Roadmap I, we focus on the light-matter interaction. In this area, studies of ultrafast electronic and molecular dynamics have been rapidly growing, with the advent of new light sources such as attosecond lasers and x-ray free electron lasers. In parallel, experiments with established synchrotron radiation sources and femtosecond lasers using cutting-edge detection schemes are revealing new scientific insights that have never been exploited. Relevant theories are also being rapidly developed. Target samples for photon-impact experiments are expanding from atoms and small molecules to complex systems such as biomolecules, fullerene, clusters and solids. This Roadmap aims to look back along the road, explaining the development of these fields, and look forward, collecting contributions from twenty leading groups from the field.

We investigated the fragmentation dynamics of highly excited states of molecular oxygen using femtosecond transient photoelectron spectroscopy. An extreme ultraviolet (XUV) pulse populates the autoionizing Rydberg series converging to O2+cςu-4, and a femtosecond near-infrared (IR) pulse was used to photoionize these states as they dissociate. Monitoring the differential photoelectron spectra as a function of time delay allowed us to obtain the relaxation lifetimes of these Rydberg states. We observed a photoelectron signal corresponding to the formation of a 4p excited atomic oxygen fragment, which is not an expected dissociation product of the (O2+cςu-4)nlσg Rydberg series. Analysis of the time-dependent photoelectron spectra and photoionization calculations indicate that this fragment results from a previously unexplored (O2+Πg4)4p repulsive state and that, contrary to expectations, this multielectron excitation pathway presents a substantial cross section. Our study demonstrates that two-color time-resolved differential photoelectron spectroscopy is an excellent tool to study the fragmentation dynamics of such multielectron excited states, which are not easily probed by other means.

The electronic and nuclear dynamics in methanol, following 156 nm photoexcitation, are investigated by combining a detailed analysis of time-resolved photoelectron spectroscopy experiments with electronic structure calculations. The photoexcitation pump pulse is followed by a delayed 260 nm photoionization probe pulse to produce photoelectrons that are analyzed by velocity map imaging. The yields of mass-resolved ions, measured with similar experimental conditions, are found to exhibit the same time-dependence as specific photoelectron spectral features. Energy-resolved signal onset and decay times are extracted from the measured photoelectron spectra to achieve high temporal resolution, beyond the 20 fs pump and probe pulse durations. When combined with ab initio calculations of selected cuts through the excited state potential energy surfaces, this information allows the dynamics of the transient excited molecule, which exhibits multiple nuclear and electronic degrees of freedom, to be tracked on its intrinsic few-femtosecond time scale. Within 15 fs of photoexcitation, we observe nuclear motion on the initially bound photoexcited 21A″ (S2) electronic state, through a conical intersection with the 11A' (S3) state, which reveals paths to photodissociation following C-O stretch and C-O-H angle opening.

By decomposing the initial-state wave function into its unique natural orbital expansion, as defined in the 1950s by Löwdin and used in modern studies of entanglement, we analyze the role of electron correlation in the initial state of an atom or molecule in determining the angular distribution of one-photon double ionization. Final-state correlation of the two ejected electrons is treated completely in numerically accurate calculations as the initial states of He, H-, and H2 are built up from correlating configurations in strict order of decreasing natural orbital occupations. In the two-electron atoms it is found that the initial-state correlation plays a sometimes modest but generally measurable role. In striking contrast, for H2 a large number of correlating configurations in the ground state is often necessary to produce angular distributions even approximately resembling the correct ones. One-photon double photoionization of oriented H2 is found to be particularly sensitive to left-right correlation along the bond.

The measurement of the triple differential cross section in the body frame for double photoionization of a molecule can be made in principle by detecting the ionic fragments and the two photoelectrons in coincidence - but only if the dynamics and geometry of dissociation of the doubly charged molecular ion are known. A classical trajectory study of the nine lowest states of the water dication is presented using high quality ab initio potential-energy surfaces. Sampling from a semiclassical initial distribution of positions and momenta is used to approximate ionization from the Frank-Condon region of the ground vibrational state of neutral H2O. Excellent agreement in comparison with preliminary experimental momentum imaging measurements of double photoionization of water show that eight dication states can be unambiguously identified in the experiment with the aid of theory. The theoretical trajectory results allow body frame measurements of double photoionization to yield all eight states even though the usual assumption of direct dissociation, the "axial recoil" approximation, breaks down for three of the dication electronic states seen in the experiment.

We present a simple two-dimensional model of the indirect dissociative recombination process. The model has one electronic and one nuclear degree of freedom and it can be solved to high precision, without making any physically motivated approximations, by employing the exterior complex scaling method together with the finite-elements method and discrete variable representation. The approach is applied to solve a model for dissociative recombination of H2+ in the singlet ungerade channels, and the results serve as a benchmark to test validity of several physical approximations commonly used in the computational modeling of dissociative recombination for real molecular targets. The second, approximate, set of calculations employs a combination of multichannel quantum defect theory and frame transformation into a basis of Siegert pseudostates. The cross sections computed with the two methods are compared in detail for collision energies from 0 to 2 eV.

We have verified a mechanism for Raman excitation of atoms through continuum levels previously obtained by quantum optimal control using the multiconfigurational time-dependent Hartree-Fock (MCTDHF) method. For the optimal control, which requires running multiple propagations to determine the optimal pulse sequence, we used the computationally inexpensive time-dependent configuration interaction singles (TDCIS) method. TDCIS captures all of the necessary correlation of the desired processes but assumes that ionization pathways reached via double excitations are not present. MCTDHF includes these pathways and all multiparticle correlations in a set of time-dependent orbitals. The mechanism that was determined to be optimal in the Raman excitation of the Ne 1s22s22p53p1 valence state via the metastable 1s22s12p63p1 resonance state involves a sequential resonance-valence excitation. First, a long pump pulse excites the core-hole state, and then a shorter Stokes pulse transfers the population to the valence state. This process represents the first step in a multidimensional x-ray spectroscopy scheme that will provide a local probe of valence electronic correlations. Although at the optimal pulse intensities at the TDCIS level of theory the MCTDHF method predicts multiple ionization or excitation ionization of the atom, at slightly lower intensities (reduced by a factor of about 4) the TDCIS mechanism is shown to hold qualitatively. Quantitatively, the MCTDHF populations are reduced from the TDCIS calculations by a factor of 4.

The method of complex basis functions is applied to molecular resonances at correlated levels of theory. Møller-Plesset perturbation theory at second order and equation-of-motion electron attachment coupled-cluster singles and doubles (EOM-EA-CCSD) methods based on a non-Hermitian self-consistent-field reference are used to compute accurate Siegert energies for shape resonances in small molecules including N2-, CO-, CO2-, and CH2O-. Analytic continuation of complex ?-trajectories is used to compute Siegert energies, and the ?-trajectories of energy differences are found to yield more consistent results than those of total energies. The ability of such methods to accurately compute complex potential energy surfaces is investigated, and the possibility of using EOM-EA-CCSD for Feshbach resonances is explored in the context of e-helium scattering.

We used extreme ultraviolet (XUV) transient absorption spectroscopy to study the autoionizing Rydberg states of oxygen in an electronically- and vibrationally-resolved fashion. XUV pulse initiates molecular polarization and near-infrared pulse perturbs its evolution. Transient absorption spectra show positive optical-density (OD) change in the case of nsσg and ndπg autoionizing states of oxygen and negative OD change for ndσg states. Multiconfiguration time-dependent Hartree-Fock (MCTDHF) calculations are used to simulate the transient absorption and the resulting spectra and temporal evolution agree with experimental observations. We model the effect of near-infrared perturbation on molecular polarization and find that the laser-induced phase-shift model agrees with the experimental and MCTDHF results, while the laser-induced attenuation model does not. We relate the electronic-state-symmetry-dependent sign of the OD change to the Fano parameters of the static absorption line shapes.

The computation of Siegert energies by analytic continuation of bound state energies has recently been applied to shape resonances in polyatomic molecules by several authors. We critically evaluate a recently proposed analytic continuation method based on low order (type III) Padé approximants as well as an analytic continuation method based on high order (type II) Padé approximants. We compare three classes of stabilizing potentials: Coulomb potentials, Gaussian potentials, and attenuated Coulomb potentials. These methods are applied to a model potential where the correct answer is known exactly and to the Πg2 shape resonance of N2- which has been studied extensively by other methods. Both the choice of stabilizing potential and method of analytic continuation prove to be important to the accuracy of the results. We conclude that an attenuated Coulomb potential is the most effective of the three for bound state analytic continuation methods. With the proper potential, such methods show promise for algorithmic determination of the positions and widths of molecular shape resonances.

We present an experimental and theoretical energy- and angle-resolved investigation on the non-dissociative photoionization dynamics of near-resonant, one-color, two-photon, single valence ionization of neutral O2 molecules. Using 9.3 eV femtosecond pulses produced via high harmonic generation and a 3-D momentum imaging spectrometer, we detect the photoelectrons and O 2 + cations produced from one-color, two-photon ionization in coincidence. The measured and calculated photoelectron angular distributions show agreement, which indicates that a superposition of two intermediate electronic states is dominantly involved and that wavepacket motion on those near-resonantly populated intermediate states does not play a significant role in the measured two-photon ionization dynamics. Here, we find greater utility in the diabatic representation compared to the adiabatic representation, where invoking a single valence-character diabat is sufficient to describe the underlying two-photon ionization mechanism. [ABSTRACT FROM AUTHOR]

This work describes the implementation and applications of non-Hermitian self-consistent field (NH-SCF) theory with complex basis functions for the ab initio computation of positions and widths of shape resonances in molecules. We utilize both the restricted open-shell and the previously unexplored spin-unrestricted variants to compute Siegert energies of several anionic shape resonances in small diatomic and polyatomic molecules including carbon tetrafluoride which has been the subject of several recent experimental studies. The computation of general molecular properties from a non-Hermitian wavefunction is discussed, and a density-based analysis is applied to the (2)B1 shape resonance in formaldehyde. Spin-unrestricted NH-SCF is used to compute a complex potential energy surface for the carbon monoxide anion which correctly describes dissociation.

Attosecond transient absorption spectra near the energies of autoionizing states are analyzed in terms of the photon coupling mechanisms to other states. In a recent experiment, the autoionization lifetimes of highly excited states of xenon were determined and compared to a simple expression based on a model of how quantum coherence determines the decay of a metastable state in the transient absorption spectrum. Here it is shown that this procedure for extracting lifetimes is more general and can be used in cases involving either resonant or nonresonant coupling of the attosecond-probed autoionizing state to either continua or discrete states by a time-delayed near infrared (NIR) pulse. The fits of theoretically simulated absorption signals for the 6p resonance in xenon (lifetime = 21.1 fs) to this expression yield the correct decay constant for all the coupling mechanisms considered, properly recovering the time signature of twice the autoionization lifetime due to the coherent nature of the transient absorption experiment. To distinguish between these two coupling cases, the characteristic dependencies of the transient absorption signals on both the photon energy and time delay are investigated. Additional oscillations versus delay-time in the measured spectrum are shown and quantum beat analysis is used to pinpoint the major photon-coupling mechanism induced by the NIR pulse in the current xenon experiment: the NIR pulse resonantly couples the attosecond-probed state, 6p, to an intermediate 8s (at 22.563 eV), and this 8s state is also coupled to a neighboring state (at 20.808 eV).

Attosecond transient absorption spectroscopy of highly excited states allows investigating the dynamics of both the decay and the light induced coupling between excited states. We present a rigorous quantum mechanical treatment that provides a generalized discussion of the on- and off-resonant decay dynamics, of resonant and non-resonant couplings to either discrete or continua states and the temporal evolution of the quantum beating of the excited wavepackets. The comparison of the model to experimental results obtained in xenon proves its general applicability. © OSA 2015.

The method of complex basis functions for computing positions and widths of molecular resonances is revisited. An open-ended and efficient implementation is described. The basis set requirements of the complex basis are investigated within the computationally inexpensive static-exchange approximation, and the results of this investigation lead to a hierarchy of basis sets for complex basis function calculations on small molecules. These basis sets are then applied in static-exchange calculations on some larger molecules with multiple low energy shape resonances: carbon tetrafluoride, benzene, pyridine, pyrimidine, pyrazine, and s-triazine. The results indicate that more sophisticated methods using complex basis functions are worth pursuing in the search for accurate and computationally feasible methods for computing resonance energies in molecular systems.

The decay of highly excited states of xenon after absorption of extreme ultraviolet light is directly tracked via attosecond transient absorption spectroscopy using a time-delayed near-infrared perturbing pulse. The lifetimes of the autoionizing 5s5p66p and 5s5p67p channels are determined to be (21.9 ± 1.3) fs and (48.4 ± 5.0) fs, respectively. The observed values support lifetime estimates obtained by traditional linewidth measurements. The experiment additionally obtains the temporal evolution of the decay as a function of energy detuning from the resonance center, and a quantum mechanical formalism is introduced that correctly accounts for the observed energy dependence. © 2014 American Physical Society.