Mixed Quantum/classical Theory Research Articles

Mixed quantum/classical theory (MQCT) approach to the dynamics of molecule-molecule collisions in complex systems.

We developed a general theoretical approach and a user-ready computer code that permit study of the dynamics of collisional energy transfer and ro-vibrational energy exchange in complex molecule-molecule collisions. The method is a mixture of classical and quantum mechanics. The internal ro-vibrational motion of collision partners is treated quantum mechanically using a time-dependent Schrödinger equation that captures many quantum phenomena including state quantization and zero-point energy, propensity and selection rules for state-to-state transitions, quantum symmetry and interference phenomena. A significant numerical speed up is obtained by describing the translational motion of collision partners classically, using the Ehrenfest mean-field trajectory approach. Within this framework a family of approximate methods for collision dynamics is developed. Several benchmark studies for diatomic and triatomic molecules, such as H2O and ND3 collided with He, H2 and D2, show that the results of MQCT are in good agreement with full-quantum calculations in a broad range of energies, especially at high collision energies where they become nearly identical to the full quantum results. Numerical efficiency of the method and massive parallelism of the MQCT code permit us to embrace some of the most complicated collisional systems ever studied, such as C6H6 + He, CH3COOH + He and H2O + H2O. Application of MQCT to the collisions of chiral molecules such as CH3CHCH2O + He, and to molecule-surface collisions is also possible and will be pursued in the future.

Symmetry Breaking: A Classic Example of Quantum Interference Captured by Mixed Quantum/Classical Theory.

The phenomena of propensity and inverse propensity are explored using time-dependent mixed quantum classical theory, MQCT, in which the rotational motion of the molecule is treated quantum mechanically, whereas the scattering process is described classically. Good agreement with the results of accurate full-quantum calculations is reported for a closed shell approximation to the NO + Ar system. It is shown that MQCT reproduces both phenomena in a broad range of the final states of the molecule and for various initial rotational states, offering a unique time-dependent insight. It permits seeing that both propensity and inverse propensity occur due to efficient depopulation of some states at the early postcollisional stage of the scattering process, when the molecule exists in a coherent superposition of many excited states that span a very broad range of angular momentum quantum numbers, populated by an efficient stepladder process of many consecutive transitions with small Δj.

MQCT 2024: A program for calculations of inelastic scattering of two molecules (new version announcement)

This is a revised and updated version of the package MQCT. The package includes routines for the calculations of ro-vibrational state-to-state transition cross sections for molecule + molecule collisions, using the mixed quantum/classical theory approach. One important addition to the package is an efficient and accurate adiabatic-trajectory approximation method (AT-MQCT), which enables calculations for larger and more complex molecules. An efficient parallel IO is implemented for state-to-state transition matrix, which now permits to handle very large matrixes, but also helps to truncate these matrixes by neglecting insignificant matrix elements and to balance the load on different CPUs during the matrix calculations and during the trajectory propagation. A stable recursive approach is added for computation of Wigner 3j-symbols, which expands the range of rotational states of molecules up to j∼ 100. Examples of input files are provided for all ten system types covered by the package, along with updated user manual. New version program summaryProgram title: MQCT 2024CPC Library link to program files:https://doi.org/10.17632/sg36r35njz.2Developer's repository link:https://github.com/MarquetteQuantum/MQCTCode Ocean capsule:https://codeocean.com/capsule/1260072/tree/v1Licensing provisions: GNU GPL v3.0Programming language: FORTRANJournal reference of previous version: Comput. Phys. Commun. 252 (2020) 107155Does the new version supersede the previous version?: YesReasons for the new version:1)Recently, we developed and tested the adiabatic-trajectory method [1–3] within MQCT framework, which appears to be both efficient and accurate, and is likely to become a “workhorse” for inelastic scattering calculations in molecule + molecule systems. In the new version of the package this method is made available as an option, AT-MQCT.2)Since MQCT trajectory calculations became more affordable with AT-MQCT, we started doing calculations with larger and larger rotational basis sets and quickly encountered several new problems. The major one was an extremely slow serial IO of the state-to-state transition matrix to a single file, which is now replaced by a parallel IO to multiple files written to a separate directory. Very large matrixes (up to 1010 non-zero matrix elements) are now handled efficiently.3)Another problem was an incorrect calculation of Wigner 3j-symbols for the rotational states around j∼ 40 and above. In the revised version a new stable subroutine is employed that uses a recursive method and works properly at least up to j=100.4)Several “bugs” were caught and fixed, with one major in the Monte-Carlo sampling of MQCT trajectories. Now users can add a batch of new trajectories to the previously sampled batch. In addition, users can monitor the convergence of Monte-Carlo sampling in detail. These options work for all levels of theory, including AT-MQCT. Several errors were found in the equations for calculations of state-to-state transition matrix using the expansion of potential energy surface, and in symmetrization of matrix elements for the case of identical collision partners. All these have been corrected.Summary of revisions: Several technical revisions concern the treatment of diatomic molecules in the code: A bug was found and fixed in the PES interface that converts the units of vibrational coordinate (the diatomic bond distance). Now the PES can be expressed in either Angstrom or Bohr, as specified in the input file. Integration of matrix elements over the vibrational coordinate is modified to use an equidistant grid by default (Gauss-Legendre method is still available as option). And the same for integration over Euler angles α and γ (for all system types). Integration over β uses Gauss-Legendre quadrature only. The format of ro-vibrational wavefunctions pre-computed externally and supplied as input is made more intuitive, as described in the manual. Finally, a new method for the sampling of orbital quantum number ℓ is implemented, which skips some values of ℓ and obtains results by interpolation of the data at the retained values of ℓ (for all system types, as described in the manual). Last version of the code is always available at https://github.com/MarquetteQuantum/MQCT.Supplementary material: Updated and expanded MQCT user guideNature of problem: Calculations of rotationally and vibrationally inelastic scattering of two molecules, with possible applications in astrophysics and atmospheric chemistrySolution method: Mixed quantum/classical theory (MQCT) approach References[1]B. Mandal, A. Semenov and D. Babikov, J. Phys. Chem. A 124 (2020) 9877-9888[2]B. Mandal, C. Joy, A. Semenov and D. Babikov, ACS Earth Space Chem. 6 (2022) 521-529[3]B. Mandal, C. Joy, D. Bostan, A. Eng and D. Babikov, J. Phys. Chem. Lett. 14 (2023) 817–824.

Rate coefficients for rotational state-to-state transitions in H2O + H2O collisions for cometary and planetary applications, as predicted by mixed quantum-classical theory

Aims. We present new calculations of collision cross sections for state-to-state transitions between the rotational states in an H2O + H2O system, which are used to generate a new database of collisional rate coefficients for cometary and planetary applications. Methods. Calculations were carried out using a mixed quantum-classical theory approach that is implemented in the code MQCT. The large basis set of rotational states used in these calculations permits us to predict thermally averaged cross sections for 441 transitions in para- and ortho-H2O in a broad range of temperatures. Results. It is found that all state-to-state transitions in the H2O + H2O system split into two well-defined groups, one with higher cross-section values and lower energy transfer, which corresponds to the dipole-dipole driven processes. The other group has smaller cross sections and higher energy transfer, driven by higher-order interaction terms. We present a detailed analysis of the theoretical error bars, and we symmetrized the state-to-state transition matrixes to ensure that excitation and quenching processes for each transition satisfy the principle of microscopic reversibility. We also compare our results with other data available from the literature for H2O + H2O collisions.

Mixed quantum/classical theory for rotational energy exchange in symmetric-top-rotor + linear-rotor collisions and a case study of the ND3 + D2 system.

The extension of mixed quantum/classical theory (MQCT) to describe collisional energy transfer is developed for a symmetric-top-rotor + linear-rotor system and is applied to ND3 + D2. State-to-state transition cross sections are computed in a broad energy range for all possible processes: when both ND3 and D2 molecules are excited or both are quenched, when one is excited while the other is quenched and vice versa, when the ND3 state changes its parity while D2 is excited or quenched, and when ND3 is excited or quenched while D2 remains in the same state, ground or excited. In all these processes the results of MQCT are found to approximately satisfy the principle of microscopic reversibility. For a set of sixteen state-to-state transitions available from the literature for a collision energy of 800 cm-1 the values of cross sections predicted by MQCT are within 8% of accurate full-quantum results. A useful time-dependent insight is obtained by monitoring the evolution of state populations along MQCT trajectories. It is shown that, if before the collision, D2 is in its ground state, the excitation of ND3 rotational states proceeds through a two-step mechanism in which the kinetic energy of molecule-molecule collision is first used to excite D2 and only then is transferred to the excited rotational states of ND3. It is found that both potential coupling and Coriolis coupling play important roles in ND3 + D2 collisions.

Adiabatic Trajectory Approximation within the Framework of Mixed Quantum/Classical Theory.

A hierarchy of approximate methods is proposed for solving the equations of motion within a framework of the mixed quantum/classical theory (MQCT) of inelastic molecular collisions. Of particular interest is a limiting case: the method in which the classical-like equations of motion for the translational degrees of freedom (scattering) are decoupled from the quantum-like equations for time evolution of the internal molecular states (rotational and vibrational). In practice, trajectories are pre-computed during the first step of calculations with driving forces determined solely by the potential energy surface of the entrance channel, which is an adiabatic trajectory approximation. Quantum state-to-state transition probabilities are computed in the second step of calculations with an expanded basis and very efficient step-size adjustment. Application of this method to H2O + H2 rotationally inelastic scattering shows a significant computational speedup by 2 orders of magnitude. The results of this approximate propagation scheme are still rather accurate, as demonstrated by benchmarking against more rigorous calculations in which the quantum and classical equations of motion are held coupled and against the full-quantum coupled-channel calculations from the literature. It is concluded that the AT-MQCT method (the adiabatic trajectory version of MQCT) represents a promising tool for the computational treatment of molecular collisions and energy exchange.

MQCT: User-ready program for calculations of inelastic scattering of two molecules

A program named MQCT is developed for calculations of rotationally and vibrationally inelastic scattering of molecules using the mixed quantum/classical theory approach. Calculations of collisions between two general asymmetric top rotors are now possible, which is a feature unavailable in other existing codes. Vibrational states of diatomic molecules can also be included in the basis set expansion, to carry out calculations of ro-vibrational excitation and quenching. Minimal input for the code assumes several defaults and is very simple, easy to set-up and run by non-experts. Multiple options, available for expert calculations, are listed in the Supplemental Information. The code is parallel and takes advantage of intrinsic massive parallelism of the mixed quantum/classical approach. A Monte-Carlo sampling procedure, implemented as option in the code, enables calculations for complicated systems with many internal states and large number of partial scattering waves. The coupled-states approximation is also implemented as an option. Integral and differential cross sections can be computed for the elastic channel. Rotational symmetry of each molecule, as well as permutation symmetry of two collision partners, are implemented. Potential energy surfaces for H2O + He, H2O + H2, and H2O + H2O are included in the code. Example input files are also provided for these systems. Program summaryProgram title: MQCTProgram files doi:http://dx.doi.org/10.17632/sg36r35njz.1Licensing provisions: GNU GPL v3.0Programming language: FORTRANSupplementary material: MQCT user guideNature of problem: Calculations of rotationally and vibrationally inelastic scattering of two molecules, with possible applications in astrophysics and atmospheric chemistry.Solution method: Mixed quantum/classical theory (MQCT) approachAdditional comments: This code can be used to simulate a collision of two asymmetric top rotors, such as H2O + H2O.

Understanding detailed balance for an electron-radiation system through mixed quantum-classical electrodynamics

We investigate detailed balance for a quantum system interacting with thermal radiation within mixed quantum-classical theory. For a two-level system coupled to classical radiation fields, three semiclassical methods are benchmarked: (1) Ehrenfest dynamics over-estimate the excited state population at equilibrium due to the failure of capturing vacuum fluctuations. (2) The coupled Maxwell-Bloch equations, which supplement Ehrenfest dynamics by damping at the full golden rule rate, under-estimate the excited state population due to double-counting of the self-interaction effect. (3) Ehrenfest+R dynamics recover detailed balance and the correct thermal equilibrium by enforcing the correct balance between the optical excitation and spontaneous emission of the quantum system. These results highlight the fact that, when properly designed, mixed quantum-classical electrodynamics can simulate thermal equilibrium in the field of nanoplasmonics.

Ehrenfest+R dynamics. I. A mixed quantum-classical electrodynamics simulation of spontaneous emission.

The dynamics of an electronic system interacting with an electromagnetic field is investigated within mixed quantum-classical theory. Beyond the classical path approximation (where we ignore all feedback from the electronic system on the photon field), we consider all electron-photon interactions explicitly according to Ehrenfest (i.e., mean-field) dynamics and a set of coupled Maxwell-Liouville equations. Because Ehrenfest dynamics cannot capture certain quantum features of the photon field correctly, we propose a new Ehrenfest+R method that can recover (by construction) spontaneous emission while also distinguishing between electromagnetic fluctuations and coherent emission.

Calculations of Differential Cross Sections Using Mixed Quantum/Classical Theory of Inelastic Scattering.

It is shown that the mixed quantum/classical theory (MQCT) for the description of molecular scattering is considerably improved by using integer values of orbital angular momentum l, just like in quantum theory, instead of treating it as a continuous classical variable related to the impact parameter. This conclusion is justified by the excellent accuracy of the modified theory for prediction of the differential cross sections, at various values of collision energy and in both forward and backward scattering regimes. This approach requires fewer trajectories, compared to the random Monte Carlo sampling, and the only convergence parameter is lmax (maximum orbital angular momentum) similar to Jmax in the full quantum theory (maximum total angular momentum). Calculations of differential and integral cross sections for elastic and inelastic channels are presented, and the role of the scattering phase is discussed. The low-energy range is analyzed in detail to obtain insight into how the mixed quantum/classical treatment works in the scattering regime dominated by resonances. The differential cross section for rotationally inelastic scattering, computed by MQCT approach, is presented for the first time.

Mixed quantum/classical theory for rotationally and vibrationally inelastic scattering of open-shell molecules and its application to the NH(X3Σ-) + He collisional system.

We developed the mixed quantum/classical theory (MQCT) for rotationally and vibrationally inelastic scattering of a diatomic molecule in a 2S+1Σ electronic state with a closed-shell atom. In this approach, the rotational (and vibrational) fine-structure states of the molecule are treated quantum-mechanically, whereas the relative motion of collisional partners is treated classically. This theory is benchmarked against full quantum (close-coupling) calculations for the NH(X3Σ-) + He system. Good agreement is found in all cases, and MQCT reproduces all main features of energy dependence of the cross section, except narrow scattering resonances at very low energies. Our method recovers propensity rules that govern values of close-coupling cross sections. Particularly, MQCT correctly predicts that the magnitudes of F-conserving transitions are significantly larger than those of F-changing ones. This new development makes MQCT a viable candidate for obtaining fine-structure resolved rotational rate coefficients at higher temperatures and for complex polyatomic systems where the standard full quantum treatment is computationally infeasible.

MQCT. I. Inelastic Scattering of Two Asymmetric-Top Rotors with Application to H2O + H2O.

A mixed quantum/classical theory (MQCT) for the inelastic collision of two asymmetric-top rotor molecules is developed. In this method, the quantum state-to-state transitions between the rotational states of molecules (internal) are treated quantum mechanically using the time-dependent Schrodinger equation, whereas their relative translational motion (responsible for scattering) is treated classically, using the average trajectory approach. Two versions of the formula for transition matrix elements are presented: a straightforward approach that uses numerical multidimensional quadrature over all the internal degrees of freedom and a more standard analytic approach that uses the expansion of the PES over the basis set of spherical harmonics. Adaptation to the case of identical molecules scattering is presented and is applied to the rotational excitation of two water molecules, H2O + H2O, using the PES from recent literature. Calculations of collisional excitation from the ground state of the system into a number of low-lying excited rotational states are carried out in a broad range of energies. Analysis of computed opacity functions shows a rather unusual scattering regime, dominated by a strong anisotropic long-range interaction (dipole-dipole). The coupled-states (CS) approximation is tested and found to agree semiquantitatively (within a factor of 2) with the fully coupled version of the method. Differential cross sections for the elastic scattering indicate a very narrow forward scattering peak.

Fokker-Planck quantum master equation for mixed quantum-semiclassical dynamics.

We revisit Caldeira-Leggett's quantum master equation representing mixed quantum-classical theory, but with limited applications. Proposed is a Fokker-Planck quantum master equation theory, with a generic bi-exponential correlation function description on semiclassical Brownian oscillators' environments. The new theory has caustic terms that bridge between the quantum description on primary systems and the semiclassical or quasi-classical description on environments. Various parametrization schemes, both analytical and numerical, for the generic bi-exponential environment bath correlation functions are proposed and scrutinized. The Fokker-Planck quantum master equation theory is of the same numerical cost as the original Caldeira-Leggett's approach but acquires a significantly broadened validity and accuracy range, as illustrated against the exact dynamics on model systems in quantum Brownian oscillators' environments, at moderately low temperatures.

Inelastic Scattering of Identical Molecules within Framework of the Mixed Quantum/Classical Theory: Application to Rotational Excitations in H2 + H2.

Theoretical foundation is laid out for description of permutation symmetry in the inelastic scattering processes that involve collisions of two identical molecules, within the framework of the mixed quantum/classical theory (MQCT). In this approach, the rotational (and vibrational) states of two molecules are treated quantum-mechanically, whereas their translational motion (responsible for scattering) is treated classically. This theory is applied to H2 + H2 system, and the state-to-state transition cross sections are compared versus those obtained from the full-quantum calculations and experimental results from the literature. Good agreement is found in all cases. It is also found that results of MQCT, where the Coriolis coupling is included classically, are somewhat closer to exact full-quantum results than results of the other approximate quantum methods, where those coupling terms are neglected. These new developments allow applications of MQCT to a broad variety of molecular systems and processes.

Mixed Quantum/Classical Theory for Molecule-Molecule Inelastic Scattering: Derivations of Equations and Application to N2 + H2 System.

The mixed quantum classical theory, MQCT, for inelastic scattering of two molecules is developed, in which the internal (rotational, vibrational) motion of both collision partners is treated with quantum mechanics, and the molecule-molecule scattering (translational motion) is described by classical trajectories. The resultant MQCT formalism includes a system of coupled differential equations for quantum probability amplitudes, and the classical equations of motion in the mean-field potential. Numerical tests of this theory are carried out for several most important rotational state-to-state transitions in the N2 + H2 system, in a broad range of collision energies. Besides scattering resonances (at low collision energies) excellent agreement with full-quantum results is obtained, including the excitation thresholds, the maxima of cross sections, and even some smaller features, such as slight oscillations of energy dependencies. Most importantly, at higher energies the results of MQCT are nearly identical to the full quantum results, which makes this approach a good alternative to the full-quantum calculations that become computationally expensive at higher collision energies and for heavier collision partners. Extensions of this theory to include vibrational transitions or general asymmetric-top rotor (polyatomic) molecules are relatively straightforward.

Mixed quantum/classical theory for inelastic scattering of asymmetric-top-rotor + atom in the body-fixed reference frame and application to the H₂O + He system.

The mixed quantum/classical theory (MQCT) for inelastic molecule-atom scattering developed recently [A. Semenov and D. Babikov, J. Chem. Phys. 139, 174108 (2013)] is extended to treat a general case of an asymmetric-top-rotor molecule in the body-fixed reference frame. This complements a similar theory formulated in the space-fixed reference-frame [M. Ivanov, M.-L. Dubernet, and D. Babikov, J. Chem. Phys. 140, 134301 (2014)]. Here, the goal was to develop an approximate computationally affordable treatment of the rotationally inelastic scattering and apply it to H2O + He. We found that MQCT is somewhat less accurate at lower scattering energies. For example, below E = 1000 cm(-1) the typical errors in the values of inelastic scattering cross sections are on the order of 10%. However, at higher scattering energies MQCT method appears to be rather accurate. Thus, at scattering energies above 2000 cm(-1) the errors are consistently in the range of 1%-2%, which is basically our convergence criterion with respect to the number of trajectories. At these conditions our MQCT method remains computationally affordable. We found that computational cost of the fully-coupled MQCT calculations scales as n(2), where n is the number of channels. This is more favorable than the full-quantum inelastic scattering calculations that scale as n(3). Our conclusion is that for complex systems (heavy collision partners with many internal states) and at higher scattering energies MQCT may offer significant computational advantages.

Rotational quenching of H2O by He: Mixed quantum/classical theory and comparison with quantum results

The mixed quantum/classical theory (MQCT) formulated in the space-fixed reference frame is used to compute quenching cross sections of several rotationally excited states of water molecule by impact of He atom in a broad range of collision energies, and is tested against the full-quantum calculations on the same potential energy surface. In current implementation of MQCT method, there are two major sources of errors: one affects results at energies below 10 cm(-1), while the other shows up at energies above 500 cm(-1). Namely, when the collision energy E is below the state-to-state transition energy ΔE the MQCT method becomes less accurate due to its intrinsic classical approximation, although employment of the average-velocity principle (scaling of collision energy in order to satisfy microscopic reversibility) helps dramatically. At higher energies, MQCT is expected to be accurate but in current implementation, in order to make calculations computationally affordable, we had to cut off the basis set size. This can be avoided by using a more efficient body-fixed formulation of MQCT. Overall, the errors of MQCT method are within 20% of the full-quantum results almost everywhere through four-orders-of-magnitude range of collision energies, except near resonances, where the errors are somewhat larger.

Mixed quantum/classical calculations of total and differential elastic and rotationally inelastic scattering cross sections for light and heavy reduced masses in a broad range of collision energies.

The mixed quantum/classical theory (MQCT) for rotationally inelastic scattering developed recently [A. Semenov and D. Babikov, J. Chem. Phys. 139, 174108 (2013)] is benchmarked against the full quantum calculations for two molecular systems: He + H2 and Na + N2. This allows testing new method in the cases of light and reasonably heavy reduced masses, for small and large rotational quanta, in a broad range of collision energies and rotational excitations. The resultant collision cross sections vary through ten-orders of magnitude range of values. Both inelastic and elastic channels are considered, as well as differential (over scattering angle) cross sections. In many cases results of the mixed quantum/classical method are hard to distinguish from the full quantum results. In less favorable cases (light masses, larger quanta, and small collision energies) some deviations are observed but, even in the worst cases, they are within 25% or so. The method is computationally cheap and particularly accurate at higher energies, heavier masses, and larger densities of states. At these conditions MQCT represents a useful alternative to the standard full-quantum scattering theory.

Efficient quantum-classical method for computing thermal rate constant of recombination: Application to ozone formation

Efficient method is proposed for computing thermal rate constant of recombination reaction that proceeds according to the energy transfer mechanism, when an energized molecule is formed from reactants first, and is stabilized later by collision with quencher. The mixed quantum-classical theory for the collisional energy transfer and the ro-vibrational energy flow [M. Ivanov and D. Babikov, J. Chem. Phys. 134, 144107 (2011)] is employed to treat the dynamics of molecule + quencher collision. Efficiency is achieved by sampling simultaneously (i) the thermal collision energy, (ii) the impact parameter, and (iii) the incident direction of quencher, as well as (iv) the rotational state of energized molecule. This approach is applied to calculate third-order rate constant of the recombination reaction that forms the (16)O(18)O(16)O isotopomer of ozone. Comparison of the predicted rate vs. experimental result is presented.

Forward–backward propagation in the mixed quantum–classical theory for the collisional energy transfer

A method of forward–backward propagation is incorporated into the mixed quantum–classical theory for calculations of the collisional energy transfer and ro-vibrational energy flow in a molecule+quencher encounter. This permits to avoid unphysical behavior of the energy transfer function in the range of large impact parameters. A case study is presented on stabilization of the metastable states of ozone in collisions with Ar bath gas.