Stuart C. Althorpe

Observation and interpretation of a time-delayed mechanism in the hydrogen exchange reaction

Extensive theoretical and experimental studies have shown the hydrogen exchange reaction H + H2 → H2 + H to occur predominantly through a ‘direct recoil’ mechanism: the H–H bonds break and form concertedly while the system passes straight over a collinear transition state, with recoil from the collision causing the H2 product molecules to scatter backward. Theoretical predictions agree well with experimental observations of this scattering process. Indirect exchange mechanisms involving H3 intermediates have been suggested to occur as well, but these are difficult to test because bimolecular reactions cannot be studied by the femtosecond spectroscopies used to monitor unimolecular reactions. Moreover, full quantum simulations of the time evolution of bimolecular reactions have not been performed. For the isotopic variant of the hydrogen exchange reaction, H + D2 → HD + D, forward scattering features observed in the product angular distribution have been attributed to possible scattering resonances associated with a quasibound collision complex. Here we extend these measurements to a wide range of collision energies and interpret the results using a full time-dependent quantum simulation of the reaction, thus showing that two different reaction mechanisms modulate the measured product angular distribution features. One of the mechanisms is direct and leads to backward scattering, the other is indirect and leads to forward scattering after a delay of about 25 femtoseconds.

Quantum wavepacket method for state-to-state reactive cross sections

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 ...

Derivation of a true (t → 0+) quantum transition-state theory. I. Uniqueness and equivalence to ring-polymer molecular dynamics transition-state-theory

Surprisingly, there exists a quantum flux-side time-correlation function which has a non-zero t → 0+ limit and thus yields a rigorous quantum generalization of classical transition-state theory (TST). In this Part I of two articles, we introduce the new time-correlation function and derive its t → 0+ limit. The new ingredient is a generalized Kubo transform which allows the flux and side dividing surfaces to be the same function of path-integral space. Choosing this function to be a single point gives a t → 0+ limit which is identical to an expression introduced on heuristic grounds by Wigner in 1932; however, this expression does not give positive-definite quantum statistics, causing it to fail while still in the shallow-tunnelling regime. Positive-definite quantum statistics is obtained only if the dividing surface is invariant to imaginary-time translation, in which case the t → 0+ limit is identical to ring-polymer molecular dynamics (RPMD) TST. The RPMD-TST rate is not a strict upper bound to the exact quantum rate, but is a good approximation to one if real-time coherence effects are small. Part II will show that the RPMD-TST rate is equal to the exact quantum rate in the absence of recrossing.

Concerted hydrogen-bond breaking by quantum tunneling in the water hexamer prism

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 Rotational spectroscopy and accompanying theory uncover gearlike joint motion of a pair of water molecules in a cluster. [Also see Perspective by Clary] The nature of the intermolecular forces between water molecules is the same in small hydrogen-bonded clusters as in the bulk. The rotational spectra of the clusters therefore give insight into the intermolecular forces present in liquid water and ice. The water hexamer is the smallest water cluster to support low-energy structures with branched three-dimensional hydrogen-bond networks, rather than cyclic two-dimensional topologies. Here we report measurements of splitting patterns in rotational transitions of the water hexamer prism, and we used quantum simulations to show that they result from geared and antigeared rotations of a pair of water molecules. Unlike previously reported tunneling motions in water clusters, the geared motion involves the concerted breaking of two hydrogen bonds. Similar types of motion may be feasible in interfacial and confined water.

Calculation of the intermolecular bound states for water dimer

The intermolecular bound states of (H2O)2 are calculated using a simple approach previously found successful for (HF)2. The monomer bond lengths and bond angles are held fixed, and the angular part of the Hamiltonian is solved variationally at three values of the intermolecular radial coordinate. The results enable comparisons of the tunneling splittings obtained from three potential energy surfaces to be made with experiment. Estimates of three of the intermolecular vibrational frequencies are also obtained.

Calculation of the far-infrared spectra for (HF)2, (HCL)2 and (HBr)2

Abstract Calculations of far-infrared spectra for the weakly bound complexes (HF) 2 , (HCl) 2 and (HBr) 2 are reported. For (HF) 2 and (HCl) 2 , potentials fitted to ab initio data are used. Spectra are also predicted using simple classical potentials. The method involves diagonalising the Hamiltonian for the dimer with the monomer and intermolecular bond lengths held fixed. The approach gives very good agreement with energy levels calculated with the close coupling method, particularly for (HCl) 2 . Furthermore, the simple electrostatic potential gives a spectrum for (HCl) 2 that agrees surprisingly well with that obtained from the ab initio potential energy surface.

Ring-polymer instanton method for calculating tunneling splittings

The semiclassical instanton expression for the tunneling splitting between two symmetric wells is rederived, starting from the ring-polymer representation of the quantum partition function. This leads to simpler mathematics by replacing functional determinants with matrix determinants. By exploiting the simple Hückel-like structure of the matrices, we derive an expression for the instanton tunneling splitting in terms of a minimum on the potential surface of a linear polymer. The latter is a section cut out of a ring polymer, consisting of an infinite number of beads, which describes a periodic orbit on the inverted potential surface. The approach is straightforward to generalize to multiple dimensions, and we demonstrate that it is computationally practical by carrying out instanton calculations of tunneling splittings in HO(2) and malonaldehyde in full dimensionality.

Instanton calculations of tunneling splittings for water dimer and trimer.

We investigate the ability of the recently developed ring-polymer instanton (RPI) method [J. O. Richardson and S. C. Althorpe, J. Chem. Phys. 134, 054109 (2011)] to treat tunneling in water clusters. We show that the RPI method is easy to extend to treat tunneling between more than two minima, using elementary graph theory. Tests of the method on water dimer and trimer yield a set of instanton periodic orbits which correspond to all known tunneling pathways in these systems. Splitting patterns obtained from the orbits are in good overall agreement with experiment. The agreement is closer for the deuterated than for the protonated clusters, almost certainly because the main approximation in the calculations is neglect of anharmonicity perpendicular to the tunneling path. All the calculations were performed on a desktop computer, which suggests that similar calculations will be possible on much larger clusters.

State-to-state reactive scattering in six dimensions using reactant–product decoupling: OH + H2 → H2O + H (J = 0)

We extend to full dimensionality a recently developed wave packet method [M. T. Cvitaš and S. C. Althorpe, J. Phys. Chem. A 113, 4557 (2009)] for computing the state-to-state quantum dynamics of AB + CD → ABC + D reactions and also increase the computational efficiency of the method. This is done by introducing a new set of product coordinates, by applying the Crank-Nicholson approximation to the angular kinetic energy part of the split-operator propagator and by using a symmetry-adapted basis-to-grid transformation to evaluate integrals over the potential energy surface. The newly extended method is tested on the benchmark OH + H(2) → H(2)O + H reaction, where it allows us to obtain accurately converged state-to-state reaction probabilities (on the Wu-Schatz-Fang-Lendvay-Harding potential energy surface) with modest computational effort. These methodological advances will make possible efficient calculations of state-to-state differential cross sections on this system in the near future.

Time-dependent plane wave packet formulation of quantum scattering with application to H+D2→HD+D

We outline a new time-dependent wave packet formulation of quantum scattering theory. The theory obtains the differential cross section directly from the time-evolution of a plane wave packet, bypassing the usual S-matrix formulas. We introduce the theory for potential scattering, generalize it to reactive scattering, then explain how the theory was used recently to interpret the H+D2→HD (v=3 j=0)+D reaction in collaboration with experiment [S. C. Althorpe, F. Fernandez-Alonso, B. D. Bean, J. D. Ayers, A. E. Pomerantz, R. N. Zare, and E. Wrede, Nature (London) 416, 67 (2002)]. We also present new findings of quantum interference in the H+D2 reaction.