Precision test of statistical dynamics with state-to-state ultracold chemistry.

Chemical reactions represent a class of quantum problems that challenge both the current theoretical understanding and computational capabilities1. Reactions that occur at ultralow temperatures provide an ideal testing ground for quantum chemistry and scattering theories, because they can be experimentally studied with unprecedented control2, yet display dynamics that are highly complex3. Here we report the full product state distribution for the reaction 2KRb → K2 + Rb2. Ultracold preparation of the reactants allows us complete control over their initial quantum degrees of freedom, whereas state-resolved, coincident detection of both products enables the probability of scattering into each of the 57 allowed rotational state-pairs to be measured. Our results show an overall agreement with a state-counting model based on statistical theory4-6, but also reveal several deviating state-pairs. In particular, we observe a strong suppression of population in the state-pair closest to the exoergicity limit as a result of the long-range potential inhibiting the escape of products. The completeness of our measurements provides a benchmark for quantum dynamics calculations beyond the current state of the art.

A New Full-Dimensional Ab Initio Intermolecular Potential Energy Surface and Rovibrational Energies of the H2O-H2 Complex.

H2O-H2 is a prototypical five-atom van der Waals system, and the interaction between H2O and H2 plays an important role in many physical and chemical environments. However, previous full-dimensional intermolecular potential energy surfaces (IPESs) cannot accurately describe the H2O-H2 interaction in the repulsive or van der Waals minimum region. In this work, we constructed a full-dimensional IPES for the title system with a small root-mean-square error of 0.252 cm-1 by using the permutation invariant polynomial neural network method. The ab initio calculations were performed by employing the explicitly corrected coupled cluster [CCSD(T)-F12a] method with the augmented correlation-consistent polarized valence quintuple-ζ basis set. Based on the newly developed IPES, the bound states of the H2O-H2 complex were calculated within the rigid-rotor approximation. The transition frequencies and band origins agreed well with the experimental values [Weida, M. J.; Nesbitt, D. J. J. Chem. Phys. 1999, 110, 156-167] with errors less than 0.1 cm-1 for most transitions. Those results demonstrate the high accuracy of our new IPES, which would build a solid foundation for the collisional dynamics of H2O-H2 at low temperatures.

ABLRI: A program for calculating the long-range interaction energy between two monomers in their non-degenerate states.

An accurate description of the long-range (LR) interaction is essential for understanding the collision between cold or ultracold molecules. However, to our best knowledge, there lacks a general approach to construct the intermolecular potential energy surface (IPES) between two arbitrary molecules and/or atoms in the LR region. In this work, we derived analytical expressions of the LR interaction energy, using the multipole expansion of the electrostatic interaction Hamiltonian and the non-degenerate perturbation theory. To make these formulae practical, we also derived the independent Cartesian components of the electrostatic properties, including the multipole moments and polarizabilities, of the monomer for a given symmetry using the properties of these components and the group-theoretical methods. Based on these newly derived formulae, we developed a FORTRAN program, namely ABLRI, which is capable of calculating the interaction energy between two arbitrary monomers both in their non-degenerate electronic ground states at large separations. To test the reliability of this newly developed program, we constructed IPESs for the electronic ground state of H2O-H2 and O2-H systems in the LR region. The interaction energy computed by our program agreed well with the ab initio calculation, which shows the validity of this program.

Recent advances in quantum theory on ro-vibrationally inelastic scattering.

Molecular collisions are of fundamental importance in understanding intermolecular interaction and dynamics. Its importance is accentuated in cold and ultra-cold collisions because of the dominant quantum mechanical nature of the scattering. We review recent advances in the time-independent approach to quantum mechanical characterization of non-reactive scattering in tetratomic systems, which is ideally suited for large collisional de Broglie wavelengths characteristic in cold and ultracold conditions. We discuss quantum scattering algorithms between two diatoms and between a triatom and an atom and their implementation, as well as various approximate schemes. They not only enable the characterization of collision dynamics in realistic systems but also serve as benchmarks for developing more approximate methods.

Full-Dimensional Quantum Dynamics Studies of Ro-vibrationally Inelastic Scattering of H2O with Ar: A Benchmark Test of the Rigid-Rotor Approximation.

While the rigid-rotor (RR) approximation is usually considered to be accurate for describing pure rotationally inelastic scattering involving diatoms in their ground or low-lying vibrational states, its validity in scattering involving polyatomic molecules has not been fully examined. The existence of soft/anharmonic vibrational modes in polyatomic molecules could make rotational-vibrational energy transfer rather efficient, thus undermining the premise of the RR approximation. In this work, we conduct a benchmark test of the RR approximation in the rotationally inelastic scattering of the H2O(v2 = 0, 1) + Ar system by comparing with full-dimensional quantum scattering calculations. We demonstrate that the error in the RR rate coefficient for v2 = 0 is less than 5%, while it can reach up to 20% for some initial states within the v2 = 1 manifold. These results indicate that the RR approximation gradually deteriorates with increasing quantum number v2. Vibrational relaxation dynamics of this system was also studied, and it is found that transitions from initial states with a large rotational quantum number of projection on the a principal axis are more efficient. These results shed valuable light on ro-vibrationally inelastic scattering involving polyatomic molecules.

A time-dependent quantum approach to dissociative recombination, associative ionization, and Penning ionization.

A time-dependent wave packet method is introduced for calculating the integral and differential cross sections for the dissociative recombination (DR), associative ionization (AI), and Penning ionization (PI) processes. This method is demonstrated for DR/AI of the N + O ↔ NO+ + e- system and PI for the He* + Ar → He + Ar+ + e- system. Good agreement with previous theoretical and experimental results is obtained for these DR, AI, and PI processes. This method has the potential to provide a quantitative characterization of polyatomic ionization-involved processes on multidimensional potential energy surfaces.

ABC+D: A time-independent coupled-channel quantum dynamics program for elastic and ro-vibrational inelastic scattering between atoms and triatomic molecules in full dimensionality.

We discuss the details of a time-independent quantum mechanical method and its implementation for full-dimensional non-reactive scattering between a closed-shell triatomic molecule and a closed-shell atom. By solving the time-independent Schrödinger equation within the coupled-channel framework using a log-derivative method, the state-to-state scattering matrix (S-matrix) can be determined for inelastic scattering involving both the rotational and vibrational modes of the molecule. Various approximations are also implemented. The ABC+D code provides an important platform for understanding an array of physical phenomena involving collisions between atoms and molecules.

Full quantum scattering calculations of the line-shape parameters for the P and R branches of HF perturbed by Ar.

This work studied the rovibrational absorption spectral line-shape parameters of the P(1)-P(10) and R(0)-R(9) lines for Hydrogen fluoride perturbed by argon in the 0-0, 1-0, and 2-0 vibrational bands at 20-1000 K. A dataset of beyond-Voigt line-shape parameters (pressure broadening and shifting parameters, their speed dependencies, and the complex Dicke parameters) has been theoretically determined for the first time from generalized spectroscopic cross-section calculated by the full quantum scattering calculations. Then these parameters were employed to predict the line shape and asymmetry based on the partially-correlated speed-dependent hard-collision and the partially-correlated quadratic-speed-dependent hard-collision profiles. The effect of each parameter on the line shape and line asymmetry was further studied, which revealed that the beyond-Voigt effects were indispensable to accurately describe the line shape contour. Our results are in good agreement with the available experimental observations and provide a comprehensive set of theoretical references for further experimental measurements.

Full-dimensional quantum studies of vibrational energy transfer dynamics between H2O and Ar: theory assessing experiment.

We report the first full-dimensional quantum mechanical calculations of the ro-vibrational inelastic scattering dynamics between water molecules and argon atoms on an accurate potential energy surface, using a recently developed time-independent quantum method based on the close-coupling approach. The state-to-state integral cross-sections and rate coefficients show strong observance of gap laws. The calculated thermal rate coefficients for the relaxation of the stretching fundamental states of H2O are in good agreement with experimental values, while those for the bending overtone state are approximately five times smaller than the values extracted through a previous kinetic modeling of fluorescence decay data. Our state-specific quantum scattering results suggest the need to reassess the kinetic modeling of the experimental data. This work advanced our understanding of the quantum dynamics of vibrationally inelastic energy transfer processes involving polyatomic molecules.

Global Full-Dimensional Potential Energy Surface for the Reaction 23Na87Rb + 23Na87Rb → 23Na2 + 87Rb2 and the Formation Rate and Lifetime of the 23Na287Rb2 Collision Complex.

A full-dimensional global potential energy surface (PES) for the reaction 23Na87Rb + 23Na87Rb → 23Na2 + 87Rb2 was constructed based on high-level ab initio calculations. The short-range part was expressed as a permutation invariant polynomial-neural network (PIP-NN) fit of 22 003 ab initio points calculated using a coupled cluster method with the one-electron basis 5s5p5d2f plus effective core potentials and core polarizability potentials, while the long-range part was represented in an asymptotically correct form based on multipole expansion. The formation rate of the 23Na287Rb2 complex calculated using a quantum statistical method is in good agreement with experiment, while the estimated 19.20 µs lifetime of the complex from Rice-Ramsperger-Kassel-Marcus (RRKM) theory is significantly shorter than the measured millisecond decay rate, signaling either the inadequacy of RRKM theory or a yet unresolved loss mechanism.

Extended coupled-states approximation for full-dimensional quantum treatments of rovibrationally inelastic scattering between atoms and triatomic molecules.

While the rigorous time-independent close-coupling approach is ideally suited for cold and ultracold rovibrationally inelastic collision, its application beyond atom-diatom systems in full dimensionality is numerically expensive. Coupled-states (CS) approximation and its extensions are good choices to reduce the computational cost and have been successfully applied to diatom-diatom systems. In this work, we introduce the extended CS (ECS) approximation, in which one or a few nearest Coriolis coupled helicity channels are included. Its usefulness in atom-triatom systems is demonstrated for scattering of H2O with rare gas atoms. The results show that the ECS approximation, even when only the nearest neighbors are included, is generally much better than the CS approximation in describing scattering. At low collision energies, the ECS gradually converges to the exact results with the increasing number of Coriolis coupled helicity blocks. We further discuss three major factors that may lead to the failure of the CS approximation, namely, the reduced mass, collision energy, and triatomic rotational quantum number. It is illustrated that these factors could impact the relative importance of off-diagonal matrix elements in the Hamiltonian, thus influencing the coupling between different helicity channels.

Fully quantum calculations of the line shape parameters for 1-0 P(22) and P(31) lines of CO perturbed by He or Ar.

This work reports the full quantum calculations of the spectral line shape parameters for the P(22) line of 13CO and the P(31) line of 12CO in the fundamental band perturbed by He or Ar from 20 to 1000 K for the first time. The generalized spectroscopic cross sections of CO-He/Ar indicate that the Dicke narrowing effect competes with the pressure broadening effect. The pressure broadening can be explained by the dynamic behaviors of intermolecular collisions. The intermolecular inelastic collisions contribute more than 95% to the pressure broadening in both CO-He and CO-Ar systems at high temperatures. Regarding the state-to-state inelastic contributions to pressure broadening, the maximum contribution out of the final state of a given line is close to that out of the initial state. The Dicke narrowing effect influences the line shape profile significantly at high temperatures, which suggests that it is indispensable for reproducing the spectral line profile. With the Dicke narrowing effect, the calculated pressure-broadening coefficients and spectral intensity distribution are in good agreement with the available experimental observations.

Quantum Dynamics of Rotational Energy Transfer Processes for N2-HF and N2-DF Systems.

The rate coefficients of rotationally inelastic collision processes for N2-HF as well as N2-DF systems were calculated by applying the recently developed coupled-states approximation including the nearest neighbor Coriolis couplings approach, based on the full-dimensional ab initio intermolecular potential energy surface. It was found that the energy gap law governs these energy transfer processes. For rotational quenching of N2 (j1 = 2-10) by the ground rotational state of HF, j1 = 6 and 5 have the maximum quenching rate for ortho-N2 and para-N2, respectively. Quenching rate coefficients for initially excited HF and DF (j2 = 1) in collisions with N2 were also reported, where N2-DF has a larger quenching rate than N2-HF due to larger density of states of the N2-DF system.

A Time-Independent Quantum Approach to Ro-vibrationally Inelastic Scattering between Atoms and Triatomic Molecules.

A full-dimensional time-independent quantum mechanical theory for ro-vibrationally inelastic scattering of triatomic molecules with atoms is formulated. The Jacobi-Radau coordinate system used in the calculation allows not only a near perfect description of the vibrational problem but also the adaptation of the exchange symmetry for A2B type triatoms. The S-matrix elements are obtained by solving the close-coupling equations with contracted basis using the log-derivative method. This method is applied to the inelastic scattering of the water molecule by a chlorine atom, which sheds light on the energy gap law in energy transfer in atom-triatom collisions.

Full-Dimensional Global Potential Energy Surface for the KRb + KRb → K2Rb2* → K2 + Rb2 Reaction with Accurate Long-Range Interactions and Quantum Statistical Calculation of the Product State Distribution under Ultracold Conditions.

A full-dimensional global potential energy surface (PES) for the KRb + KRb → K2Rb2* → K2 + Rb2 reaction is reported based on high-level ab initio calculations. The short-range part of the PES is fit with the permutationally invariant polynomial-neural network method, while the long-range parts of the PES in both the reactant and product asymptotes are represented by an asymptotically correct form. The long- and short-range parts are connected with intermediate-range parts to make them smooth. Within a statistical quantum model, this PES reproduces both the measured loss rates of ultracold KRb molecules and the K2 and Rb2 product state distributions, underscoring the important role of tunneling in ultracold chemistry. The PES also correctly predicts the lifetime of the K2Rb2* intermediate complex within the Rice-Ramsperger-Kassel-Marcus limit. It thus provides a reliable platform for future dynamical studies of the prototypical reaction.

Quantum dynamics of the energy transfer for vibrationally excited HF (v = 7) colliding with D2 (v = 0): Theory assessing experiment.

It is still challenging to accurately qualify the rate coefficients for vibrationally excited molecules in experiment. In particular, for the energy transfer between HF (v = 7) and D2 (v = 0), which is a prototype for near resonant collisional transfer of vibrational excitation from one molecule to the other, the two available experimental results of rate coefficients contradict each other by a factor of nearly 20. In order to benchmark these data, in this work, the rate coefficients of vibration-vibration energy transfer processes of this system at temperatures ranging from 100 to 1500 K were calculated by employing the coupled-states approximation based on our recently developed full-dimensional ab initio intermolecular potential energy surface. The state-to-state rate coefficients were found to follow the general energy gap law. The calculated total vibration-vibration energy transfer rate coefficients decrease with the increase in the angular momentum of HF at most temperatures. The vibrational relaxation rate coefficient decreases monotonously with the temperature, and the calculated result of 8.1 × 10-11 cm3 mol-1 s-1 at room temperature is in very good agreement with the experimental value reported by Dzelzkalns and Kaufman [J. Chem. Phys. 77, 3508 (1982)].

Statistical quantum mechanical approach to diatom-diatom capture dynamics and application to ultracold KRb + KRb reaction.

A general and rigorous quantum method is proposed for studying capture dynamics between two diatomic molecules in full dimensionality. By solving the time-independent Schrödinger equation with proper boundary conditions, this method is ideally suited for studying quantum dynamics of cold and ultracold reactions. To illustrate its applicability, the capture dynamics between ultracold KRb molecules is characterized in full six dimensions for the first time using a first-principles based long-range interaction potential. The calculated capture rates for collisions involving distinguishable and indistinguishable 40K87Rb molecules are in good agreement with the experiment and exhibit clear Wigner threshold behaviors. Predictions for ultracold collisions between internally excited 40K87Rb suggest minor changes in the loss rate, consistent with experimental observations in similar systems.

Quantum dynamics of vibration-vibration energy transfer for vibrationally excited HF colliding with H2.

The rate constants for H2 -HF energy transfer processes, especially for those in vibrationally excited states, are very demanding in astrophysics and chemical laser engineering, especially for those in vibrationally excited states. Based on our recent potential energy surface, we used the coupled-states approximation including the nearest neighboring Coriolis couplings with energy-based local basis set to perform dynamics calculation for the H2 -HF energy transfer system. Rate constants for vibrational transitions (1; 3) → (0; 4), (1; 3) → (2; 2), and (0; 3) → (1; 2) were obtained. For state-to-state rate constants, transitions that have no internal angular momentum gap dominate at high temperatures. The vibrational-resolved rate constant for (1; 3) → (0; 4) initially decreases and then increases with the temperature, while those for (1; 3) → (2; 2), and (0; 3) → (1; 2) transitions monotonically increase. The calculated rate constants are in good agreement with the available experimental results. These dynamical data can be further applied to the numerical simulation of hydrogen fluoride chemical laser. © 2018 Wiley Periodicals, Inc.

A new full-dimensional ab initio intermolecular potential energy surface and vibrational states for (HF)2 and (DF)2.

We present a new full-dimensional ab initio potential energy surface (PES) of a hydrogen fluoride dimer [(HF)2] using the supermolecular approach. The calculations were performed at the coupled-cluster single double triple level, with an augmented correlation-consistent polarized valence quadruple-zeta basis set plus bond functions. The basis set superposition error was corrected by a full counterpoise procedure. With the exchange symmetry of the two HF molecules, the permutation invariant polynomial neural network approach was used to fit the hypersurface with a root-mean-square-error of 0.465 cm-1 for about 110 000 points. The ab initio noise of intermolecular potential in the long range was smoothed by the long-range coefficients method. The equilibrium configuration of the complex was found to be a Cs structure located at two equivalent minima with the well depth of 1573.495 cm-1. The eigenstates were calculated by employing a symmetry-adapted Lanczos propagation algorithm in the mixed radical discrete variable representation/angular finite basis representation. The tunneling splitting for the ground state of (HF)2 is 0.665 cm-1, agreeing well with experimental value of 0.65869 cm-1. Vibrational fundamentals are also very close to the observed values. The results of vibrational states calculations demonstrate the high accuracy of our new PES.

An improved coupled-states approximation including the nearest neighbor Coriolis couplings for diatom-diatom inelastic collision.

Solving the time-independent close coupling equations of a diatom-diatom inelastic collision system by using the rigorous close-coupling approach is numerically difficult because of its expensive matrix manipulation. The coupled-states approximation decouples the centrifugal matrix by neglecting the important Coriolis couplings completely. In this work, a new approximation method based on the coupled-states approximation is presented and applied to time-independent quantum dynamic calculations. This approach only considers the most important Coriolis coupling with the nearest neighbors and ignores weaker Coriolis couplings with farther K channels. As a result, it reduces the computational costs without a significant loss of accuracy. Numerical tests for para-H2+ortho-H2 and para-H2+HD inelastic collision were carried out and the results showed that the improved method dramatically reduces the errors due to the neglect of the Coriolis couplings in the coupled-states approximation. This strategy should be useful in quantum dynamics of other systems.