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There are a variety of methods for including nuclear quantum effects in dynamics simulations by combining quantum Boltzmann statistics with classical dynamics. Among them are thermostatted ring-polymer molecular dynamics (TRPMD), centroid molecular dynamics (CMD), quasi-centroid molecular dynamics (QCMD), and the linearised semi-classical initial value representation (LSC-IVR). Here we make a systematic comparison of these methods by calculating the infrared spectrum of water in the gas phase, and in the liquid and ice phases (using the q-TIP4P/F model potential). Some of these results are taken from previous work, and some of them are new (including the LSC-IVR calculations for ice, and extensions of all the spectra into the near-infrared region dominated by overtone and combination bands). Our results suggest that QCMD is the best method for reproducing fundamental transitions in the spectrum, and that LSC-IVR gives the best overall description of the spectrum (albeit with large errors in the bend fundamental band caused by zero-point-energy leakage). The TRPMD method gives damped spectra that line up with the QCMD spectra, and is by far the cheapest method.

We describe improvements to the quasicentroid molecular dynamics (QCMD) path-integral method, which was developed recently for computing the infrared spectra of condensed-phase systems. The main development is an improved estimator for the intermolecular torque on the quasicentroid. When applied to qTIP4P/F liquid water and ice, the new estimator is found to remove an artificial 25 cm-1 red shift from the libration bands, to increase slightly the intensity of the OH stretch band in the liquid, and to reduce small errors noted previously in the QCMD radial distribution functions. We also modify the mass-scaling used in the adiabatic QCMD algorithm, which allows the molecular dynamics timestep to be quadrupled, thus reducing the expense of a QCMD calculation to twice that of Cartesian centroid molecular dynamics for qTIP4P/F liquid water at 300 K, and 8 times for ice at 150 K., The Journal of Chemical Physics, 157 (17), ISSN:0021-9606, ISSN:1089-7690

Classical molecular dynamics (MD) and imaginary-time path-integral dynamics methods underestimate the infrared absorption intensities of overtone and combination bands by typically an order of magnitude. Ple et al. [J. Chem. Phys. 155, 104108 (2021)] have shown that this is because such methods fail to describe the coupling of the centroid to the Matsubara dynamics of the fluctuation modes; classical first-order perturbation theory (PT) applied to the Matsubara dynamics is sufficient to recover most of the lost intensity in simple models and gives identical results to quantum (Rayleigh–Schrodinger) PT. Here, we show numerically that the results of this analysis can be used as post-processing correction factors, which can be applied to realistic (classical MD or path-integral dynamics) simulations of infrared spectra. We find that the correction factors recover most of the lost intensity in the overtone and combination bands of gas-phase water and ammonia and much of it for liquid water. We then re-derive and confirm the earlier PT analysis by applying canonical PT to Matsubara dynamics, which has the advantage of avoiding secular terms and gives a simple picture of the perturbed Matsubara dynamics in terms of action-angle variables. Collectively, these variables “Matsubara heat” the amplitudes of the overtone and combination vibrations of the centroid to what they would be in a classical system with the oscillators (of frequency Ωi) held at their quantum effective temperatures [of ℏΩi coth(βℏΩi/2)/2kB]. Numerical calculations show that a similar neglect of “Matsubara heating” causes path-integral methods to underestimate Fermi resonance splittings.

The energy landscapes for a discretized path integral representation of the water dimer, trimer and pentamer are characterized in terms of the localized (classical) and delocalized minima and transition states. The transition states are finite-temperature approximations to the exact instanton path, and they are typically used to calculate the tunneling splittings or reaction rates. The features of the path integral landscape are explored, thus elucidating procedures that could usefully be automated when searching for instantons in larger systems. Our work not only clarifies the role of minima and transition states in path integral calculations but also enables us to analyze the quantum-to-classical transition.

It was shown recently that smooth and continuous ‘Matsubara’ phase-space loops follow a quantum-Boltzmann-conserving classical dynamics when decoupled from non-smooth distributions, which was suggested as the reason that many dynamical observables appear to involve a mixture of classical dynamics and quantum Boltz- mann statistics. Here we derive a mean-field version of this ‘Matsubara dynamics’ which sufficiently mitigates its serious phase problem to permit numerical tests on a two-dimensional ‘champagne-bottle’ model of a rotating OH bond. The Matsubara- dynamics rovibrational spectra are found to converge towards close agreement with the exact quantum results at all temperatures tested (200–800 K), the only significant discrepancies being a temperature-independent 22 cm−1 blue-shift in the position of the vibrational peak, and a slight broadening in its lineshape. These results are compared with centroid molecular dynamics (CMD) to assess the importance of non- centroid fluctuations. Above 250 K, only the lowest-frequency non-centroid modes are needed to correct small CMD red-shifts in the vibrational peak; below 250 K, more non-centroid modes are needed to correct large CMD red-shifts and broaden- ing. The transition between these ‘shallow curvature’ and ‘deep curvature’ regimes happens when imaginary-time Feynman paths become able to lower their actions by cutting through the curved potential surface, giving rise to artificial instantons in CMD., G.T. acknowledges a University of Cambridge Vice-Chancellor’s award and support from St. Catharine’s College, Cambridge. S.C.A. acknowledges funding from the UK Science and Engineering Research Council.

We report rovibrationally selected differential cross sections (DCSs) of the benchmark reaction H + D2 → HD(v' = 3, j' = 4-10) + D at a collision energy of 3.26 eV, which exceeds the conical intersection of the H3 potential energy surface at 2.74 eV. We use the PHOTOLOC technique in which a fluorine excimer laser at 157.64 nm photodissociates hydrogen bromide (HBr) molecules to generate fast H atoms and the HD product is detected in a state-specific manner by resonance-enhanced multiphoton ionization. Fully converged quantum wave packet calculations were performed for this reaction at this high collision energy without inclusion of the geometric phase (GP) effect, which takes into account coupling to the first excited state of the H3 potential energy surface. Multimodal structures can be observed in most of the DCSs up to j' = 10, which is predicted by theory and also well-reproduced by experiment. The theoretically calculated DCSs are in good overall agreement with the experimental measurements, which indicates that the GP effect is not large enough that its existence can be verified experimentally at this collision energy.

Matsubara dynamics is the quantum-Boltzmann-conserving classical dynamics which remains when real-time coherences are taken out of the exact quantum Liouvillian [T. J. H. Hele et al., J. Chem. Phys. 142, 134103 (2015)]; because of a phase-term, it cannot be used as a practical method without further approximation. Recently, Smith et al. [J. Chem. Phys. 142, 244112 (2015)] developed a "planetary" model dynamics which conserves the Feynman-Kleinert (FK) approximation to the quantum-Boltzmann distribution. Here, we show that for moderately anharmonic potentials, the planetary dynamics gives a good approximation to Matsubara trajectories on the FK potential surface by decoupling the centroid trajectory from the locally harmonic Matsubara fluctuations, which reduce to a single phase-less fluctuation particle (the "planet"). We also show that the FK effective frequency can be approximated by a direct integral over these fluctuations, obviating the need to solve iterative equations. This modification, together with use of thermostatted ring-polymer molecular dynamics, allows us to test the planetary model on water (gas-phase, liquid, and ice) using the q-TIP4P/F potential surface. The "planetary" fluctuations give a poor approximation to the rotational/librational bands in the infrared spectrum, but a good approximation to the bend and stretch bands, where the fluctuation lineshape is found to be motionally narrowed by the vibrations of the centroid.

Modern computational methods have become so powerful for predicting the outcome for the H + H 2 → H 2 + H bimolecular exchange reaction that it might seem further experiments are not needed. Nevertheless, experiments have led the way to cause theorists to look more deeply into this simplest of all chemical reactions. The findings are less simple.

We investigate the calculation of approximate non-equilibrium quantum time correlation functions (TCFs) using two popular path-integral-based molecular dynamics methods, ring-polymer molecular dynamics (RPMD) and centroid molecular dynamics (CMD). It is shown that for the cases of a sudden vertical excitation and an initial momentum impulse, both RPMD and CMD yield non-equilibrium TCFs for linear operators that are exact for high temperatures, in the $\textit{t}$ = 0 limit, and for harmonic potentials; the subset of these conditions that are preserved for non-equilibrium TCFs of non-linear operators is also discussed. Furthermore, it is shown that for these non-equilibrium initial conditions, both methods retain the connection to Matsubara dynamics that has previously been established for equilibrium initial conditions. Comparison of non-equilibrium TCFs from RPMD and CMD to Matsubara dynamics at short times reveals the orders in time to which the methods agree. Specifically, for the position-autocorrelation function associated with sudden vertical excitation, RPMD and CMD agree with Matsubara dynamics up to O($\textit{t}$$^{4}$) and O($\textit{t}$$^{1}$), respectively; for the position-autocorrelation function associated with an initial momentum impulse, RPMD and CMD agree with Matsubara dynamics up to O($\textit{t}$$^{5}$) and O($\textit{t}$$^{2}$), respectively. Numerical tests using model potentials for a wide range of non-equilibrium initial conditions show that RPMD and CMD yield non-equilibrium TCFs with an accuracy that is comparable to that for equilibrium TCFs. RPMD is also used to investigate excited-state proton transfer in a system-bath model, and it is compared to numerically exact calculations performed using a recently developed version of the Liouville space hierarchical equation of motion approach; again, similar accuracy is observed for non-equilibrium and equilibrium initial conditions.

The instanton method obtains approximate tunneling rates from the minimum-action path (known as the instanton) linking reactants to the products at a given temperature. An efficient way to find the instanton is to search for saddle-points on the ring-polymer potential surface, which is obtained by expressing the quantum Boltzmann operator as a discrete path-integral. Here we report a practical implementation of this ring-polymer form of instanton theory into the Molpro electronic-structure package, which allows the rates to be computed on-the-fly, without the need for a fitted analytic potential-energy surface. As a test case, we compute tunneling rates for the benchmark H + CH

Quantum reaction rates for bimolecular gas-phase reactions can be computed efficiently and to a realistic degree of approximation by applying ring-polymer molecular dynamics within a free-energy interpolation scheme [R. Collepardo-Guevara, Y.V. Suleimanov, and D.E. Manolopoulos, J. Chem. Phys. 130, 174713 (2009)]. Here, we present modifications to this scheme which simplify the implementation of the method, and have the advantage of yielding directly the free-energy as a function of the interpolation coordinate. We also take the opportunity to verify the benchmark results obtained for the H + H2 and Cl + HCl reactions by Collepardo-Guevara et al., obtaining excellent agreement for H + H2 and reasonable agreement for Cl + HCl.

We report an improved method for the calculation of tunneling splittings between degenerate configurations in molecules and clusters using path-integral molecular dynamics (PIMD). Starting from an expression involving a ratio of thermodynamic density matrices at the bottom of the symmetric wells, we use thermodynamic integration with molecular dynamics simulations and a Langevin thermostat to compute the splittings stochastically. The thermodynamic integration is performed by sampling along the semiclassical instanton path, which provides an efficient reaction coordinate as well as being physically well-motivated. This approach allows us to carry out PIMD calculations of the multi-well tunneling splitting pattern in the water dimer and to refine previous PIMD calculations for one-dimensional models and malonaldehyde. The large (acceptor) splitting in the water dimer agrees to within 20% of benchmark variational results, and the smaller splittings agree to within 10%.

International audience; Recent calculations on the hydrogen-exchange reaction [Bouakline et al., J. Chem. Phys. 128, 124322 (2008)], have found strong geometric phase (GP) effects in the state-to-state differential cross sections (DCS), at energies above the energetic minimum of the conical intersection (CI) seam, which cancel out in the integral cross sections (ICS). In this article, we explain the origin of this cancellation and make other predictions about the nature of the reaction mechanisms at these high energies by carrying out quasiclassical trajectory (QCT) calculations. Detailed comparisons are made with the quantum results by splitting the quantum and the QCT cross sections into contributions from reaction paths that wind in different senses around the CI and that scatter the products in the nearside and farside directions. Reaction paths that traverse one transition state (1-TS) scatter their products in just the nearside direction, whereas paths that traverse two transition states (2-TS) scatter in both the nearside and farside directions. However, the nearside 2-TS products scatter into a different region of angular phase-space than the 1-TS products, which explains why the GP effects cancel out in the ICS. Analysis of the QCT results also suggests that two separate reaction mechanisms may be responsible for the 2-TS scattering at high energies.

We have measured differential cross sections (DCSs) for the vibrationally inelastic scattering process H + o -D 2 ( v = 0, j = 0,2) → H + o -D 2 ( v′ = 1–4, j′ even). Several different collision energies and nearly the entire range of populated product quantum states are studied. The products are dominantly forward-scattered in all cases. This behavior is the opposite of what is predicted by the conventional textbook mechanism, in which collisions at small impact parameters compress the bond and cause the products to recoil in the backward direction. Recent quasiclassical trajectory (QCT) calculations examining only the o -D 2 ( v′ = 3, j′ ) products suggest that vibrationally inelastic scattering is the result of a frustrated reaction in which the D—D bond is stretched, but not broken, during the collision. These QCT calculations provide a qualitative explanation for the observed forward-scattering, but they do not agree with experiments at the lowest values of j′ . The present work shows that quantum mechanical calculations agree closely with experiments and expands upon previous results to show that forward-scattering is universally observed in vibrationally inelastic H + D 2 collisions over a broad range of conditions.

It is well known that path-integral methods can be used to calculate the energy splitting between the ground and the first excited state. Here we show that this approach can be generalized to give the splitting patterns between all the lowest energy levels from different symmetry blocks that lie below the first-excited totally symmetric state. We demonstrate this property numerically for some two-dimensional models. The approach is likely to be useful for computing rovibrational energy levels and tunnelling splittings in floppy molecules and gas-phase clusters.

We illustrate how path-integral molecular dynamics can be used to calculate ground-state tunnelling splittings in molecules or clusters. The method obtains the splittings from ratios of density matrix elements between the degenerate wells connected by the tunnelling. We propose a simple thermodynamic integration scheme for evaluating these elements. Numerical tests on fully dimensional malonaldehyde yield tunnelling splittings in good overall agreement with the results of diffusion Monte Carlo calculations.

Gear-like rotation by a wobbly water duo The molecules in liquid water move about constantly, but on average they cling to each other through hydrogen bonds, like dancers who keep switching partners. Richardson et al. uncovered a fresh twist in this molecular dance (see the Perspective by Clary). Studying clusters of six molecules each—essentially the smallest three-dimensional water droplets—they observed coupled motion of two different molecules in the cluster. The process breaks two different hydrogen bonds concurrently in a pattern akin to rotating gears. Science , this issue p. 1310 ; see also p. 1267

The recently developed ring-polymer instanton (RPI) method [J. Chem. Phys. 2011, 134, 054109] is an efficient technique for calculating approximate tunneling splittings in high- dimensional molecular systems. The key step is locating the instanton tunneling path at zero temperature. Here, we show that techniques previously designed for locating instantons in finite-temperature rate calculations can be adapted to the RPI method, where they become extremely efficient, reducing the number of potential energy calls by 2 orders of magnitude. We investigate one technique that employs variable time steps to minimize the action integral, and two that employ equally spaced position steps to minimize the abbreviated (i.e., Jacobi) action integral, using respectively the nudged elastic band (NEB) and string methods. We recommend use of the latter because it is parameter-free, but all three methods give comparable efficiency savings. Having located the instanton pathway, we then interpolate the instanton path onto a fine grid of imaginary time points, allowing us to compute the fluctuation prefactor. The crucial modification needed to the original finite-temperature algorithms is to allow the end points of the zero-temperature instanton path to describe overall rotations, which is done using a standard quaternion algorithm. These approaches will allow the RPI Method to be combined effectively with expensive potential energy surfaces or on-the-fly electronic structure methods.

In a previous article [T. J. H. Hele and S. C. Althorpe, J. Chem. Phys. 138, 084108 (2013)], we showed that the t → 0+ limit of ring-polymer molecular dynamics (RPMD) rate-theory is also the t → 0+ limit of a new type of quantum flux-side time-correlation function, in which the dividing surfaces are invariant to imaginary-time translation; in other words, that RPMD transition-state theory (RMPD-TST) is a t → 0+ quantum transition-state theory (QTST). Recently, Jang and Voth [J. Chem. Phys. 144, 084110 (2016)] rederived this quantum t → 0+ limit and claimed that it gives instead the centroid-density approximation. Here we show that the t → 0+ limit derived by Jang and Voth is in fact RPMD-TST.

The scattering dynamics of the state-to-state reaction H + D2 (v(i) = 0, j(i) = 0, m(i) = 0) --> HD (v(f) = 3, j(f) = 0, m(f) = 0) + D is investigated, where vi, ji, mi and vf, jf, mf are initial and final vibrational, rotational, and helicity quantum numbers, respectively. We use accurate quantum scattering matrix elements for total energies in the range 1.52-2.50 eV (calculated stepwise in 0.01 eV increments). The theoretical tools used are a nearside-farside (NF) analysis of the partial wave series (PWS) for the scattering amplitude, together with NF local angular momentum (LAM) theory. We find that the backward scattering, which is the energy-domain analog of the time-direct reaction mechanism, is N dominated, whereas the forward scattering (time-delayed analog) is a result of NF interference between the more slowly varying N and F subamplitudes. The LAM analysis reveals the existence of a "trench-ridge" structure. We also resum the PWS up to three times prior to making the NF decomposition. We show that such resummations usually provide an improved physical interpretation of the NF differential cross sections (DCSs) and NF LAMs. We analyze two resummed scattering amplitudes in more detail, where particular values of the resummation parameters give rise to unexpected unphysical behavior in the N and F DCSs over a small angular range. We analyze the cause of this unphysical behavior and describe viable workarounds to the problem. The energy-domain calculations in this paper complement the time-domain results reported earlier by Monks, P. D. D.; Connor, J. N. L.; Althorpe, S. C. J. Phys. Chem. A 2006, 110, 741.

We report quantum wave packet calculations on the H′ + HBr ( v = 0, j = 0) → H 2 ( v ′, j ′) + Br reaction over collision energies in the range 2.1–2.5 eV. The product rotational distributions show good agreement with previous experimental results and quasi-classical trajectory predictions. We report the first quantum differential cross sections for this system, which show strong forward scattering into rotationally excited products. Cuts through the wave packet show evidence of an additional rapid reaction mechanism, in which the system first visits the H′Br + H exchange arrangement.

Quantum transition-state theory (QTST) and free-energy instanton theory (FEIT) are two closely related methods for estimating the quantum rate coefficient from the free-energy at the reaction barrier. In calculations on one-dimensional models, FEIT typically gives closer agreement than QTST with the exact quantum results at all temperatures below the crossover to deep tunneling, suggesting that FEIT is a better approximation than QTST in this regime. Here we show that this simple trend does not hold for systems of greater dimensionality. We report tests on several collinear and three-dimensional reactions, in which QTST outperforms FEIT over a range of temperatures below crossover, which can extend down to half the crossover temperature (below which FEIT outperforms QTST). This suggests that QTST-based methods such as ring-polymer molecular dynamics (RPMD) may often give closer agreement with the exact quantum results than FEIT.

We recently obtained a quantum-Boltzmann-conserving classical dynamics by making a single change to the derivation of the “Classical Wigner” approximation. Here, we show that the further approximation of this “Matsubara dynamics” gives rise to two popular heuristic methods for treating quantum Boltzmann time-correlation functions: centroid molecular dynamics (CMD) and ring-polymer molecular dynamics (RPMD). We show that CMD is a mean-field approximation to Matsubara dynamics, obtained by discarding (classical) fluctuations around the centroid, and that RPMD is the result of discarding a term in the Matsubara Liouvillian which shifts the frequencies of these fluctuations. These findings are consistent with previous numerical results and give explicit formulae for the terms that CMD and RPMD leave out.

We show that a single change in the derivation of the linearized semiclassical-initial value representation (LSC-IVR or classical Wigner approximation) results in a classical dynamics which conserves the quantum Boltzmann distribution. We rederive the (standard) LSC-IVR approach by writing the (exact) quantum time-correlation function in terms of the normal modes of a free ring-polymer (i.e. a discrete imaginary-time Feynman path), taking the limit that the number of polymer beads $N \to \infty$, such that the lowest normal-mode frequencies take their Matsubara values. The change we propose is to truncate the quantum Liouvillian, not explicitly in powers of $\hbar^2$ at $\hbar^0$ (which gives back the standard LSC-IVR approximation), but in the normal-mode derivatives corresponding to the lowest Matsubara frequencies. The resulting Matsubara dynamics is inherently classical (since all terms $\mathcal{O}\left(\hbar^{2}\right)$ disappear from the Matsubara Liouvillian in the limit $N \to \infty$), and conserves the quantum Boltzmann distribution because the Matsubara Hamiltonian is symmetric with respect to imaginary-time translation. Numerical tests show that the Matsubara approximation to the quantum time-correlation function converges with respect to the number of modes, and gives better agreement than LSC- IVR with the exact quantum result. Matsubara dynamics is too computationally expensive to be applied to complex systems, but its further approximation may lead to practical methods.

The plane wave packet (PWP) and nearside–farside approaches are combined, in order to obtain time-dependent differential cross-sections (DCSs) describing the scattering of a plane wave packet over the full θ = 0 → 360° sphere. This produces a better separation of different reaction mechanisms than previous θ = 0 → 180° PWP calculations. Application to the He + H 2 + reaction separates clearly the direct and time-delayed mechanisms, showing that the latter (caused by decay of HeH 2 + scattering resonances) does not begin until the HeH 2 + complex has rotated by over half a revolution, and then continues until the HeH 2 + has described almost an entire revolution. The resulting nearside–farside interference from this decay dominates the (energy-domain) DCS.

The crossing of two electronic potential surfaces (a conical intersection) should result in geometric phase effects even for molecular processes confined to the lower surface. However, recent quantum simulations of the hydrogen exchange reaction (H + H 2 → H 2 + H) have predicted a cancellation in such effects when product distributions are integrated over all scattering angles. We used a simple topological argument to extract reaction paths with different senses from a nuclear wave function that encircles a conical intersection. In the hydrogen-exchange reaction, these senses correspond to paths that cross one or two transition states. These two sets of paths scatter their products into different regions of space, which causes the cancellation in geometric phase effects. The analysis should generalize to other direct reactions.

An experimental and theoretical investigation of the collision energy dependence of the HD(nu' = 2,j') rotational product state distribution for the H + D2 reaction in the collision energy range of Ecol = 1.30-1.89 eV has been carried out. Theoretical results based on time-dependent and time-independent quantum mechanical methods agree nearly perfectly with each other, and the agreement with the experiment is good at low collision energies and very good at high collision energies. This behavior is in marked contrast to a previous report on the HD(nu' = 3,j') product state rotational distribution [Pomerantz et al., J. Chem. Phys. 120, 3244 (2004)] where a systematic difference between experiment and theory was observed, especially at the highest collision energies. The reason for this different behavior is not yet understood. In addition, this study employs Doppler-free spectroscopy to resolve an ambiguity in the E, F-X resonantly enhanced multiphoton ionization transition originating from the HD(nu' = 2,j' = 1) state, which is found to be caused by an accidental blending with the transition coming from the HD(nu' = 1,j' = 14) state.

We present experimental rotational distributions for the reaction H + D2 --HD(nu' = 3,j') + D at eight different collision energies between 1.49 and 1.85 eV. We combine a previous measurement of the state-resolved excitation function for this reaction [Ayers et al., J. Chem. Phys. 119, 4662 (2003)] with the current data to produce a map of the relative reactive cross section as a function of both collision energy and rotational quantum number (an E-j' plot). To compare with the experimental data, we also present E-j' plots resulting from both time-dependent and time-independent quantum mechanical calculations carried out on the BKMP2 surface. The two calculations agree well with each other, but they produce rotational distributions significantly colder than the experiment, with the difference being more pronounced at higher collision energies. Disagreement between theory and experiment might be regarded as surprising considering the simplicity of this system; potential causes of this discrepancy are discussed.

The hydrogen exchange reaction has a conical intersection and is therefore a prototype for studying geometric phase effects in reaction dynamics. We compute wave packets in hyperspherical coordinates for the reaction H + H 2 ( v 0 =0, j 0 =0) → H 2 + H, over the collision energy range E coll =0.75−2.02 eV, using the plane wave packet method. Geometric phase effects are not included. The wave packets visualise the quantum dynamics around the conical intersection and relate it to the centre-of-mass scattering angle. The wave packet fails to encircle the conical intersection, making slightly less than half a revolution. This provides a physical picture for why geometric phase effects in this reaction are expected to be reduced.

We develop a very simple, previously neglected, idea for enhancing the power of quantum wave packet calculations to interpret chemical reactions. This is that, once the energy-domain wave function is calculated, an infinite variety of time-dependent wave packets may be recovered from it, simply by multiplying by an energy filter and taking the Fourier transform. Each packet corresponds to a different initial wave packet along the scattering coordinate. We show that, when incorporated into the plane wave packet method [J. Chem. Phys. 2002, 117, 4623], the filters can isolate features in the differential cross section and generate separate wave packets visualizing the dynamics of each feature. The filters also enable one to focus the wave packets, so as to minimize artifacts caused by spreading. We demonstrate these ideas on the F + HD reaction, by isolating the low energy ridge from the differential cross section and generating a focused plane wave packet that visualizes its dynamics. We find that the ridge is produced by a "Catherine wheel" rotation and decay of the FHD complex through about 180°.

The recently developed plane wave packet method is applied to the F + HD reaction on the Stark–Werner potential surface, yielding converged differential cross-sections (d.c.s.) over the collision energy range E =0–0.15 eV. A wave packet ‘movie’ is obtained showing how the HF product scatters into space. Two direct and two time-delayed mechanisms are observed. One of the latter corresponds to a recently observed Feshbach resonance. The movie is related to the energy-dependent d.c.s. by a time-dependent d.c.s., which reveals quantum interference between the direct and time-delayed mechanisms.

We present a 3D quantum wavepacket method for calculating state-to-state reactive cross sections for the A+BC→AC+B reaction. The method avoids the coordinate problem (of A+BC arrangements being difficult to represent by AC+B coordinates, and vice versa) by solving the reactant-product decoupling (RPD) equations [T. Peng and J. Z. H. Zhang, J. Chem. Phys. 105, 6072 (1996)] in their further partitioned form [S. C. Althorpe, D. J. Kouri, and D. K. Hoffman, J. Chem. Phys. 107, 7816 (1997)]. These equations decouple the nuclear dynamics Schrodinger equation into separate reactant, strong-interaction, and product regions, permitting different coordinates to be used in each region. We solve the equations using A+BC Jacobi coordinates in the reactant region, and AC+B Jacobi coordinates in the strong-interaction and product regions. In test calculations on the J=0 H+H2 reaction, we show that this partitioning of coordinate systems is much more efficient than using A+BC coordinates in the strong-interaction region ...

We discuss the application of photoelectron angular distributions as a probe of time-evolving rovibrational wavepackets. A physical picture is drawn of the mechanism through which rotation–vibration coupling mechanisms are reflected in the time-dependent photoionisation asymmetry parameters. The effects of complicating features that may obscure the interpretation of time-resolved photoelectron spectroscopies, including the field intensity and the inherent complexity of ionisation processes are examined.

We study numerically the ability of time-resolved photoelectron angular distributions to provide useful information regarding the alignment of wave packets and hence a new view on excited state dynamics. The calculations employ a recently developed theory of pump–probe photoelectron spectroscopy [J. Chem. Phys. 107, 7859 (1997)] which treats both laser pulses nonperturbatively. Taking the NO system as a prototypical example, we first describe the alignment dynamics in the course of the perpendicular Π→Σ transition. The observation of alignment perpendicular to the field polarization which converts upon turn-off of the pulse to alignment parallel to the field is explained in terms of the phase relation between the wave packet components. Considering next the ionization stage, we find only weak dependence of the photoelectron angular distribution on the field intensity in the range typically employed in gas-phase femtosecond experiments. Our results illustrate the utility of time-resolved photoelectron angu...

The Navier-Stokes equations with both periodic and non-slip boundary conditions are solved using a new class of wavelets based on distributed approximating functionals (DAFs). Extremely high accuracy is found in our wavelet-DAF integration of the analytically solvable Taylor problem, using 32 grid points in each of the two spatial dimensions, for Reynolds numbers from Re = 20 to Re = ∞. The present approach is then applied to the lid-driven cavity problem with standard non-slip boundary conditions. Physically reasonable solutions are obtained for Reynolds numbers as high as 3200, using 63 grid points in each spatial dimension. Our results indicate that wavelet methods are readily applicable to those dynamical problems for which the existence of possible singularities demands highly accurate solution methods.

The time-independent wave packet reactant−product decoupling (TIW−RPD) method is a new method for calculating state-to-state reaction probabilities, which we recently developed by extending the original reactant−product decoupling method of Peng and Zhang (Peng, T.; Zharg, J. Z. H. J. Chem. Phys. 1996, 105, 6072; Zhu, W.; Peng, T.; Zhang, J. Z. H. J. Chem. Phys. 1997, 106, 1742). In the TIW−RPD method, the nuclear dynamics Schroedinger equation is partitioned into a set of completely decoupled equations, each of which describes the nuclear dynamics in either the reactant channel, one of the product channels, or the strong-interaction region. In this paper we apply the TIW−RPD method to the (three-dimensional) Li + HF → LiF + H (J = 0) reaction. We also describe an improvement to the reactant channel part of the method. The state-to-state reaction probabilities for Li + HF converge very well with respect to the size of the strong-interaction region, demonstrating that the TIW−RPD method is robust enough to...

A newly developed distributed approximating functional (DAF)-wavelet, the Dirichlet–Gabor DAF-wavelet (DGDW), is applied in a calculation of the state-to-state reaction probabilities for the three-dimensional (3-D) (J=0)H+H2 reaction, using the time-independent wave-packet reactant-product decoupling (TIWRPD) method. The DGDWs are reconstructed from a rigorous mathematical sampling theorem, and are shown to be DAF-wavelet generalizations of both the sine discrete variable representation (sinc-DVR) and the Fourier distributed approximating functionals (DAFs). An important feature of the generalized sinc-DVR representation is that the grid points are distributed at equally spaced intervals and the kinetic energy matrix has a banded, Toeplitz structure. Test calculations show that, in accordance with mathematical sampling theory, the DAF-windowed sinc-DVR converges much more rapidly and to higher accuracy with bandwidth, 2W+1. The results of the H+H2 calculation are in very close agreement with the results o...

We obtain a shallow-tunnelling correction factor for use with Wigner-Eyring transition- state theory (TST). Our starting point is quantum transition state theory (QTST), which approximates the accurate quantum rate as the instantaneous flux through a delocalised transition-state ensemble of ring-polymers. Expanding the ring-polymer potential to second order gives the well-known Wigner tunnelling- factor which diverges at the cross-over temperature between deep and shallow tunnelling. Here, we show how to remove this divergence by integrating numerically over the two softest ring-polymer normal modes. This results in a modified Wigner correction factor involving a one-dimensional integral evaluated along a straight line on the potential energy surface. Comparisons with accurate quantum calculations indicate that the newly derived correction factor gives realistic estimates of quantum rate coefficients in the shallow- tunnelling regime.

The reactant‐product decoupling (RPD) equations are a rigorous formulation of state‐to‐state reactive scattering recently introduced by Peng and Zhang. For an N‐arrangement reaction there are a total of N RPD equations, each of which describes the dynamics in just one region of coordinate space. One of the regions (the r‐region) encloses the reactant channel and the strong interaction region; each of the other N−1 regions encloses one of the product channels. In this paper we develop a suggestion later made by Kouri and co‐workers: that the original RPD equations can be further partitioned into a set of new RPD equations, in which the original r‐region is now partitioned into three regions—two enclosing the reactant channel, and one enclosing the strong interaction region. After introducing the new RPD equations, we derive the time-independent wave‐packet (TIW) form of the equations, and show how to solve them using an extended version of the Chebyshev propagator. We test the new RPD equations (and the me...

The reactant-product decoupling (RPD) equations are a rigorous formulation of quantum reactive scattering recently introduced by Peng and Zhang. Solving the RPD equations yields state-to-state reaction probabilities from a set of decoupled wavepacket propagations, each of which is confined to an arrangement of the reaction. Previously, we derived the time-independent wavepacket version of the RPD equations, and developed an efficient Chebyshev-based propagator for solving them. In this Letter we apply our method to 3D reactive scattering, presenting state-to-state reaction probabilities for the H + H2 reaction (for J = 0).

Recently, Peng and Zhang have introduced the reactant-product decoupling (RPD) equations. These are an exact formulation of quantum mechanical reactive-scattering, whereby the Schrodinger equation is partitioned into a set of uncoupled equations, each of which describes the dynamics in one arrangement of the reaction. In this paper we derive an efficient method for solving the RPD equations which is based on the Chebyshev propagator. The derivation makes use of the recently derived time-independent wavepacket version of the RPD equations. We test the method by applying it to the collinear H+H2 reaction.