https://www.pci.uni-heidelberg.de/tc/usr/mctdh/lit/rev.pdf

The multiconfiguration time-dependent Hartree (MCTDH) method: a highly efficient algorithm for propagating wavepackets

The theory of reactive (rearrangement) scattering for three atoms in three physical dimensions using adiabatically adjusting, principal axes hyperspherical (APH) coordinates is given. The relationships of the APH coordinates to Delves and Jacobi coordinates are given, and the kinetic energy operator is shown to be relatively simple. Procedures for solving the equations via either an exact coupled channel (CC) method or an optimum centrifugal sudden (CSAPH) approximation are given as well as procedures for applying scattering boundary conditions. Surface functions of two angles are obtained using a finite element method with an optimized, nonuniform mesh, and the CC equations are solved using the efficient VIVAS method. Sample CC results are given for the H3 system. The present approach has the advantages that all arrangements are treated fully equivalently; it is a principal axis system, so that both axes and internal coordinates swing smoothly with the reactions; it is directly applicable to both symmetric and unsymmetric systems and mass combinations and all total angular momenta; it gives convenient mappings for visualization of potential energy surfaces and wave functions; only regular radial solutions are required; all coordinate matching is by simple projection; and the expensive parts of the calculation are energy independent, so that, once they are done, the scattering matrices can be rapidly generated at the large numbers of energies needed to map out reactive thresholds and resonances. Accurate reactive scattering calculations are now possible for many chemically interesting reactions that were previously intractable.

Quantum reactive scattering in three dimensions using hyperspherical (APH) coordinates. Theory

A semiclassical theory of the width and shift of isolated infrared and Raman lines in the gas phase is developed within the impact approximation. A parabolic trajectory model determined by the isotropic part of the interaction potential allows a satisfactory treatment to be made of the close collisions leading to an analytical expression for the elastic collision cross section. A numerical test of this theory has been made for HCl-Ar by comparing the present results to those of previous infinite order treatments using numerical curved classical trajectories. Extension to the diatom-diatom collisions is then made by expressing the anisotropic potential using an atom-atom interaction model which takes both the long and short range contributions into account. Numerical applications have been performed for the Raman line widths of pure N2, CO2 and CO and for the infrared line widths of pure CO and of CO perturbed by N2 and CO2. A good quantitative agreement with experiments is obtained for all the considered cases and a correct variation of the broadening coefficient with the rotational quantum number is achieved in opposition to the previous results. A consistent variation of the line broadening with temperature is also obtained even for high rotational levels.

https://hal.archives-ouvertes.fr/jpa-00209180/document

Short range force effects in semiclassical molecular line broadening calculations

Extending our previous studies for the H2+OH reaction in five mathematical dimensions (5D) [J. Chem. Phys. 99, 5615 (1993); 100, 2697 (1994)], we present in this paper a full‐dimensional (6D) dynamics study for the title reaction. The 6D treatment uses the time‐dependent wave‐packet approach and employs discrete variable representations for three radial coordinates and coupled angular momentum basis functions for three angular coordinates. The present 6D study employs an energy projection method to extract reaction probabilities for a whole range of energies from a single wave‐packet propagation, while previous studies produced only energy‐averaged reaction probability from a single wave‐packet propagation. The application of the energy‐projection method allows us to efficiently map out the energy dependence of the reaction probability on a fine grid which revealed surprisingly sharp resonancelike features at low collision energies on the Schatz–Elgersma potential surface. Our calculation shows that the potential‐averaged 5D treatment can produce reaction probabilities essentially indistinguishable from the full‐dimensional result. We also report initial state‐selected reaction cross sections and rate constants which are in good agreement with our previous calculations. The effect of OH vibration on H2+OH reaction is examined in the present study and our calculation shows that the OH vibration can enhance the rate constant by about a factor of 1.7 in good agreement with the experimental estimate of about 1.5.

Full‐dimensional time‐dependent treatment for diatom–diatom reactions: The H2+OH reaction

Discrete variable representations and sudden models in quantum scattering theory

New coupled equations describing collisions of an atom and a diatomic molecule are derived in this paper. By utilizing a description of the collision in terms of rotating coordinates, all coupling in the z component of angular momentum is isolated into purely kinematic effects. By neglecting these couplings, one is led to approximate equations for which the jz component of angular momentum for the molecule is conserved. In addition, the scattering cross sections are formulated by neglecting the effect on the wavefunction of the rotation of the coordinate axes so that in place of Wigner rotation matrices dmmJ (Θ) appearing, one deals with simple Legendre polynomials and the orbital angular momentum l2 is approximated by l(l + 1) ℏ2. It is noted that the procedure involves no approximations so far as the potential matrix elements are concerned. Furthermore, the number of equations remaining coupled is drastically reduced and a completely quantum mechanical description of the dynamics of both internal states and relative motion is retained. The physical implications of the approximations are examined, and it is seen that the neglect of intermultiplet coupling gives rise to consideration of only transitions where both the orientation and magnitude of the rotor angular momentum change. Further, the neglect of transformation effects on the wavefunction is expected to be least accurate for the inelastic forward scattering and best for backward scattering and the j =0→0 elastic scattering. Finally, the present simplest version of the approximation obviously is not intended for treating processes dependent on mj transitions, e.g., NMR relaxation in He–H2. Next the formalism is applied in test calculations to He–H2 collisions using the Krauss‐Mies potential energy surface. Numerical results for elastic and inelastic integral and differential cross sections are compared with exact quantum mechanical close coupling solutions of the standard coupled channel equations. Over the energy range studied (from 0.1 eV up to 0.9 eV), agreement to within a few percent is obtained. Additional coupled states calculations are reported at 1.2 eV and computation times are compared against those required for a full close coupling solution. Calculations for the Roberts He–H2 surface are also reported to illustrate the independence of the approximations on the strength of the coupling (so long as the inelastic scattering is predominantly in the backward direction). The dramatic savings afforded by the present approach are such as to make possible fully converged calculations at collision energies typically studied in molecular beam experiments. Thus, for elastic and inelastic nonreactive collisions, involving a repulsive‐type interaction, the approach makes the a priori quantum mechanical description of the scattering of a diatom by an atom practical.

Quantum mechanical close coupling approach to molecular collisions. jz ‐conserving coupled states approximation

The elastic and inelastic collisions of an atom with a diatomic molecule are treated quantum mechanically in the body−fixed coordinate system. The body−fixed equations of motion are first compared with the usual spaced−fixed ’’close−coupling’’ equations and limiting cases are considered in which the two formalisms become equivalent. The recently developed ’’coupled−states’’ approximation in the body−fixed system is then described in which intermultiplet transitions are neglected and the eigenvalue of the orbital angular momentum operator l2 is approximated by h/l (l + 1). Numerically computed cross sections from this approximation are compared to those computed from the standard space−fixed close−coupling equations for the test system He−H2. Agreement to within a few percent is obtained for the integral as well as for the differential cross sections for elastic and for rotationally and vibrationally inelastic scattering in the energy range of 0.9 to 4.2 eV. A coupled−states large basis calculation (j = 0, 2, 4, 6, 8, 10 for n = 0 and j = 0, 2, 4, 6 for n = 1) at 4.2 eV is presented which demonstrates the enormous utility of the method.

Coupled-states approach for elastic and for rotationally and vibrationally inelastic atom-molecule collisions

The process of rotational excitation in the body-fixed frame is examined in detail. Physical arguments are presented to justify the coupled states approximation which involves a diagonalization of the body-fixed centrifugal potential. Cross sections for rotational transitions in HeHCN and in ArN2 collisions are reported and are compared to exact close coupling cross sections. The coupled states method is found to maintain its high accuracy for the extremely strong-coupling HeHCN system and also for the large Δj transitions in the ArN2 system even when many partial waves are required. General rules are given for the applicability of this approach in terms of the strength and the position of the anisotropy in the potential energy surface.

Validity of the coupled states approximation for molecular collisions

Converged coupled‐states integral cross sections were determined for the vibrational relaxation of the v=1 j=0 level of p‐H2 in collisions with 4He The collision energies ranged from 0005 to 04 eV The Gordon–Secrest (GS) potential was used with both a harmonic (HO) and rotating‐Morse oscillator (MO) description of the H2 molecule Additional calculations incorporated modifications in the long‐range and spherically symmetric (V0) parts of the purely repulsive GS surface Rotational coupling plays a major role in vibrational relaxation even at thermal energies Although the relative importance of individual vibration–rotation transitions is a sensitive function of the choice of intramolecular potential, the HO and MO total relaxation cross sections are nearly identical These total relaxation cross sections exhibit a power‐law dependence on the initial translational energy down to ∼01 eV above threshold; below which point a positive curvature appears for all the surfaces considered Rate constants for 

Paul McGuire

Papers

Quantum mechanical close coupling approach to molecular collisions. jz ‐conserving coupled states approximation

Coupled-states approach for elastic and for rotationally and vibrationally inelastic atom-molecule collisions

Validity of the coupled states approximation for molecular collisions

Cross sections and rate constants for low‐temperature 4He–H2 vibrational relaxation

A coupled-states approximation study of Li+-H2 collisions