David E. Manolopoulos

An improved log derivative method for inelastic scattering

A new method for solving the close coupled equations of inelastic scattering is presented. The method is based on Johnson’s log derivative algorithm, and uses the same quadrature for the solution of the corresponding integral equations. However it differs from the original method in the use of a piecewise constant diagonal reference potential. This results in a reduction in matrix operations at subsequent energies, and an improved convergence of the solution with respect to the number of grid points. These advantages are clearly demonstrated when our method is applied to an atom–diatom rotational excitation problem.

Quantum statistics and classical mechanics: Real time correlation functions from ring polymer molecular dynamics

We propose an approximate method for calculating Kubo-transformed real-time correlation functions involving position-dependent operators, based on path integral (Parrinello-Rahman) molecular dynamics. The method gives the exact quantum mechanical correlation function at time zero, exactly satisfies the quantum mechanical detailed balance condition, and for correlation functions of the form C(Ax)(t) and C(xB)(t) it gives the exact result for a harmonic potential. It also works reasonably well at short times for more general potentials and correlation functions, as we illustrate with some example calculations. The method provides a consistent improvement over purely classical molecular dynamics that is most apparent in the low-temperature regime.

ABC: a quantum reactive scattering program

Abstract This article describes a quantum mechanical reactive scattering program for atom–diatom chemical reactions that we have written during the past several years. The program uses a coupled-channel hyperspherical coordinate method to solve the Schrodinger equation for the motion of the three nuclei on a single Born–Oppenheimer potential energy surface. It has been tested for all possible deuterium-substituted isotopomers of the H + H 2 , F + H 2 , and Cl + H 2 reactions, and tried and tested potential energy surfaces for these reactions are included within the program as Fortran subroutines.

A stable linear reference potential algorithm for solution of the quantum close‐coupled equations in molecular scattering theory

We show how the linear reference potential method for solution of the close‐coupled equations, which arise in inelastic scattering theory, can be reformulated in terms of an ‘‘imbedding‐type’’ propagator. Explicit expressions are given for the blocks of the propagator matrix in terms of Airy functions. By representing these functions in terms of moduli and phases, in both classically allowed and classically forbidden regions, one can evaluate the propagator without any numerical difficulty. The resulting algorithm is tested on a highly pathological problem—the rotationally inelastic scattering of a polar molecule by a spherical ion at extremely low kinetic energy—and found to be completely stable.

Ring-Polymer Molecular Dynamics: Quantum Effects in Chemical Dynamics from Classical Trajectories in an Extended Phase Space

This article reviews the ring-polymer molecular dynamics model for condensed-phase quantum dynamics. This model, which involves classical evolution in an extended ring-polymer phase space, provides a practical approach to approximating the effects of quantum fluctuations on the dynamics of condensed-phase systems. The review covers the theory, implementation, applications, and limitations of the approximation.

Competing quantum effects in the dynamics of a flexible water model

Numerous studies have identified large quantum mechanical effects in the dynamics of liquid water. In this paper, we suggest that these effects may have been overestimated due to the use of rigid water models and flexible models in which the intramolecular interactions were described using simple harmonic functions. To demonstrate this, we introduce a new simple point charge model for liquid water, q-TIP4P/F, in which the O-H stretches are described by Morse-type functions. We have parametrized this model to give the correct liquid structure, diffusion coefficient, and infrared absorption frequencies in quantum (path integral-based) simulations. The model also reproduces the experimental temperature variation of the liquid density and affords reasonable agreement with the experimental melting temperature of hexagonal ice at atmospheric pressure. By comparing classical and quantum simulations of the liquid, we find that quantum mechanical fluctuations increase the rates of translational diffusion and orientational relaxation in our model by a factor of around 1.15. This effect is much smaller than that observed in all previous simulations of empirical water models, which have found a quantum effect of at least 1.4 regardless of the quantum simulation method or the water model employed. The small quantum effect in our model is a result of two competing phenomena. Intermolecular zero point energy and tunneling effects destabilize the hydrogen-bonding network, leading to a less viscous liquid with a larger diffusion coefficient. However, this is offset by intramolecular zero point motion, which changes the average water monomer geometry resulting in a larger dipole moment, stronger intermolecular interactions, and a slower diffusion. We end by suggesting, on the basis of simulations of other potential energy models, that the small quantum effect we find in the diffusion coefficient is associated with the ability of our model to produce a single broad O-H stretching band in the infrared absorption spectrum.

Molecular graphs, point groups, and fullerenes

A general and convenient method is described by which one can obtain the point group, the 13C NMR pattern, and the number of IR‐ and Raman‐active vibrations of any given fullerene structure directly from its molecular graph. The method is based on certain ‘‘topological’’ atomic coordinates, the Cartesian components of which in a topological principal axis system are proportional to the components of three selected eigenvectors of the adjacency matrix of the graph. Its practical utility lies in combination with existing theoretical results, including analytical molecular‐orbital rules which predict two distinct families of closed‐shell fullerenes and a systematic search which generates all known Cn fullerene graphs. The overall strategy is illustrated in an experimentally relevant application to the 24 distinct isolated‐pentagon fullerene isomers of C84.

Chemical reaction rates from ring polymer molecular dynamics

We show how the ring-polymer molecular dynamics method can be adapted to calculate approximate Kubo-transformed flux-side correlation functions, and hence rate coefficients for condensed phase reactions. An application of the method to the standard model for a chemical reaction in solution--a quartic double-well potential linearly coupled to a bath of harmonic oscillators--is found to give results of comparable accuracy to those of the classical Wigner model and the centroid molecular dynamics method. However, since the present method does not require that one evaluate the Wigner transform of a thermal flux operator or that one perform a separate path integral calculation for each molecular dynamics time step, we believe it will prove easier to apply to more general problems than either of these alternative techniques. We also present a (logarithmic) discretization scheme for the Ohmic bath in the system-bath model that gives converged results with just nine bath modes--a surprisingly small number for a model of a condensed phase reaction. Finally, we present some calculations of the transmission through an Eckart barrier which show that the present method provides a satisfactory (although not perfect) description of the deep quantum tunneling regime. Part of the reason for the success of the method is that it gives the exact quantum-mechanical rate constant for the transmission through a parabolic barrier, as we demonstrate analytically in the Appendix.

Quantum mechanical angular distributions for the F+H2 reaction

Quantum mechanical integral and differential cross sections have been calculated for the title reaction at the three collision energies studied in the 1985 molecular beam experiment of Lee and co‐workers, using the new ab initio potential energy surface of Stark and Werner (preceding paper). Although the overall agreement between the calculated and experimental center‐of‐mass frame angular distributions is satisfactory, there are still some noticeable differences. In particular, the forward scattering of HF(v′=3) is more pronounced in the present calculations than it is in the experiment and the calculations also predict some forward scattering of HF(v′=2). A comparison with the quasiclassical trajectory results of Aoiz and co‐workers on the same potential energy surface shows that the forward scattering is largely a quantum mechanical effect in both cases, being dominated by high orbital angular momenta in the tunneling region where the combined centrifugal and potential energy barrier prevents classical...

An investigation of the F+H2 reaction based on a full ab initio description of the open-shell character of the F(2P) atom

Expanding on an earlier Communication [M. H. Alexander, H.-J. Werner, and D. E. Manolopoulos, J. Chem. Phys. 109, 5710 (1998)], we present here the full framework for the quantum treatment of reactions of the fluorine atom with molecular hydrogen. This involves four potential energy surfaces (PESs) and two, coordinate-dependent spin–orbit interaction terms, all of which were fitted to the results of ab initio calculations. Quantum scattering calculations, based on a time-independent method formulated in hyperspherical coordinates, were carried out to determine initial and final state-resolved reactive cross sections, for reaction of F in its ground (2P3/2) and excited (2P1/2) spin–orbit state with H2 in j=0 and j=2(pH2) and j=1(oH2). The overall reactivity of the excited state of F, which can occur only through nonadiabatic transitions, is found to be small, at most 25% of the reactivity of the ground spin–orbit state, which is adiabatically allowed. In addition, when compared with results of earlier calc...