Reactant-product Decoupling Research Articles

Reactant-Product Decoupling Technique Using the Intermediate Coordinate Method.

Although the reactant-product decoupling (RPD) technique was proposed over two decades ago, it remains an efficient approach for calculating product state-resolved information on some simple direct reactions using the quantum wave packet method. In the past, usually the RPD technique employed the collocation method to transform the wave function between reactant and product arrangements, which requires quite large computational efforts. In this work, the intermediate coordinate (IC) method is employed to realize the RPD technique. Numerical examples demonstrate that this new IC RPD (IRPD) technique has superior computational efficiency compared with the original method employing the collocation method. Especially, the new IRPD technique significantly saves disk space and computer memory. To illustrate the features of our new method, the total reaction probabilities of the H + H2, H + Br2, and F + H2 reactions with J = 0 and the differential cross sections of the H + H2 and F + H2 reactions at a series of collision energy are calculated and presented. With this efficient and effective new RPD technique, the Li + HF reaction, which involves sharp resonances with long-range wave functions in the van der Waals wells in both the reactant and product arrangements, is also calculated with several J at the product state-resolved level to reveal the ability of the RPD technique for describing resonance wave functions. With these numerical examples, it is found that, for the reaction with resonances, the RPD approach should be applied carefully. Otherwise, it is very possible that the resonances could disappear with the application of the RPD technique.

A Chebyshev method for state-to-state reactive scattering using reactant-product decoupling: OH + H2 → H2O + H

We extend a recently developed wave packet method for computing the state-to-state quantum dynamics of AB + CD → ABC + D reactions [M. T. Cvitaš and S. C. Althorpe, J. Phys. Chem. A 113, 4557 (2009)] to include the Chebyshev propagator. The method uses the further partitioned approach to reactant-product decoupling, which uses artificial decoupling potentials to partition the coordinate space of the reaction into separate reactant, product, and transition-state regions. Separate coordinates and basis sets can then be used that are best adapted to each region. We derive improved Chebyshev partitioning formulas which include Mandelshtam-and-Taylor-type decoupling potentials, and which are essential for the non-unitary discrete variable representations that must be used in 4-atom reactive scattering calculations. Numerical tests on the fully dimensional OH + H2 → H2O + H reaction for J = 0 show that the new version of the method is as efficient as the previously developed split-operator version. The advantages of the Chebyshev propagator (most notably the ease of parallelization for J > 0) can now be fully exploited in state-to-state reactive scattering calculations on 4-atom reactions.

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 wavepacket investigation of state-to-state reactive scattering of Cl with para-H2 including the open-shell character of the Cl atom

We describe a time-dependent wavepacket based method for the calculation of the state-to-state cross sections for the Cl+H(2) reaction including all couplings arising from the nonzero spin and electronic orbital angular momenta of the Cl atom. Reactant-product decoupling allows us to use a physically correct basis in both the reactant and the product arrangements. Our calculated results agree well with the experimental results of Yang and co-workers. We also describe a model with two coupled potential energy surfaces, which includes the spin-orbit coupling, which is responsible for the largest non-Born-Oppenheimer effects in the Cl+H(2) reaction but neglects the off-diagonal electronically diabatic coupling and all Coriolis couplings due to the electronic spin and orbital angular momenta. The comparison of the results of the full six-state and two-state models with an electronically adiabatic (one-state) description reveals that the latter describes well the reaction out of the ground spin-orbit state, while the two-state model, which is computationally much faster than the full six-state model, describes well the reaction from both the ground and excited spin-orbit states.

State-to-state reactive scattering using reactant–product decoupling

In this paper, we review a quantum wave packet method for computing state-to-state quantum dynamics of three-atom and four-atom bimolecular reactions using the reactant–product decoupling (RPD) approach. The long propagation times needed in order to extract state-to-state observables cover geometries at which the fragments are far apart on the reactant and product side of the reaction valley. To alleviate the coordinate problem, we partition the coordinate space into three regions, describing reactants, products and the interaction region each using a basis set optimized for the particular region. The wave packet is propagated simultaneously over all three regions, and it is transferred between the regions using a system of artificial reflecting and absorbing potentials. We review the accuracy and efficiency of the method and comment on its prospects for yielding accurate differential cross sections of four-atom reactions in the near future.

Quantum Wave Packet Method for State-to-State Reactive Scattering Calculations on AB + CD → ABC + D Reactions

We describe a quantum wave packet method for computing the state-to-state quantum dynamics of 4-atom AB + CD --> ABC + D reactions. The approach is an extension to 4-atom reactions of a version of the reactant-product decoupling (RPD) approach, applied previously to 3-atom reactions ( J. Chem. Phys. 2001, 114 , 1601 ). The approach partitions the coordinate space of the reaction into separate reagent, strong-interaction, and product regions, using a system of artificial absorbing and reflecting potentials. It employs a partitioned version of the split-operator propagator, which is more efficient than partitioning the (exact) time-dependent Schrodinger equation. The wave packet bounces off a reflecting potential in the entrance channel, which generates a source term; this is transformed efficiently from reagent to product Jacobi coordinates by exploiting some simple angular momentum properties. The efficiency and accuracy of the method is demonstrated by numerical tests on the benchmark OH + H(2) --> H(2)O + H reaction.

Time-dependent wave packet calculation for state-to-state reaction of Cl+H2 using the reactant-product decoupling approach

We present in this paper the application of the reactant-product decoupling (RPD) method [T. Peng and J. Z. H. Zhang, J. Chem. Phys. 105, 6072 (1996)] in a time-dependent wave packet calculation of the state-to-state reaction of Cl+H2→HCl+H on the G3 potential energy surface. In the RPD approach, the wave function is split into two components: the reactant ψR, which comprises the reagent and interaction regions, and the product ψP, which comprises the product region. The propagation of the reactant component ψR is separated (decoupled) from that of the product component ψP through the use of absorbing potential. The propagation ψP is entirely in the product space using the product Jacobi coordinates by using a coordinate transformation on the absorbed piece of wave function. The reaction probabilities from the ground state of H2 to specific rovibrational states of the product ClH are presented in detail. All calculations are done for total angular momentum J=0 on the G3 potential energy surface.

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 Schrödinger 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 (as was done in all previous applications of the RPD equations). We apply the method to the H+H2 reaction (for J=0–24), and obtain the first state-to-state differential cross sections to be calculated by an exact quantum wavepacket method. The method will allow state-to-state cross sections to be calculated for the same reactions for which wavepacket methods can currently calculate total cross sections.

Reactant–product decoupling approach to state-to-state reactive scattering H+DH

A state-to-state quantum reactive scattering calculation of the prototypical H+ DH reaction in three dimensions was studied using the reactant–product decoupling (RPD) method. This reaction or its reverse reaction had been widely studied with the implementation of different kinds of methods before [J.Z.H. Zhang, W.H. Miller, J. Chem. Phys. 91 (1989) 1528; D.S. Zhang, Q.G. Zhang, Y.C. Zhang, Chin. J. Atom. Mol. Phys. 13 (1996) 93; D.M. Charutz, I. Last, M. Baer, J. Chem. Phys. 106 (1997) 7654]. The main purpose of this paper is to explore further applicability of the RPD method for use as a general and efficient computational approach to study state-to-state quantum dynamics for polyatomic reactions. In this RPD method, the full time-dependent wave function is split into the reactant component and all product components. The calculation in different arrangements can be carried out using different components of the full wave function and the corresponding Jacobi coordinates, so a substantial amount of computational effort is saved. The state-to-state results for H+ DH on the LSTH potential energy surface for J=0 are given.

An application of distributed approximating functional-wavelets to reactive scattering

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 of previous TIWRPD calculations, demonstrating that the DGDW representation is an accurate and efficient representation for use in FFT wave-packet propagation methods, and that, more generally, the theory of wavelets and related techniques have great potential for the study of molecular dynamics.

Reactant–product decoupling approach to state-to-state dynamics calculation for bimolecular reaction and unimolecular fragmentation

The main purpose of the study is to explore a number of computational methods to be used in the RPD approach to state-to-state quantum dynamics calculation of polyatomic reactions. Specifically, a mixed time-dependent (TD) and energy-dependent (ED) approach to solve the RPD equations is investigated. In the mixed TD and ED approach, the reactant wavefunction is computed by the method of wavepacket propagation while the product wavefunction is calculated by energy-dependent methods. As a result, the reactant–product coordinate transformation only needs to be carried out for the number of energies at which the state-to-state S-matrix elements are needed, which is advantageous if state-to-state information at only a few energies are needed. Similar implementation of the mixed RPD approach is given for calculation of product state distribution in molecular photofragmentation dynamics.

Further partitioning of the reactant-product decoupling equations of state-to-state reactive scattering and their solution by the time-independent wave-packet method

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 method) by calculating state-to-state reaction probabilities and inelastic probabilities for the three-dimensional (J = 50) H+H2 reaction.

State-to-state reaction probabilities from the time-independent wavepacket reactant-product decoupling equations: application to the three-dimensional H + H 2 reaction (for J = 0)

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 + H 2 reaction (for J = 0).

Reactant-product decoupling approach to half-scattering problems: Photodissociation of H2O in three dimensions

In this paper, we present the RPD (reactant-product decoupling) approach to the calculation of final-state distribution in photodissociation of H2O in three-dimensional space. Although the RPD approach was recently developed for bimolecular state-to-state reactive scattering calculations, its application to photodissociation dynamics is very attractive. Specifically in photodissociation, the interaction (reactant) component wavefunction ψr (which in the present case of photodissociation is replaced by the interaction component ψint) is nonzero only in the strong interaction region, which greatly simplifies the numerical calculation for ψint in comparison to that for ψr in a full bimolecular reactive scattering calculation. In the following report, the time-dependent implementation of the RPD approach to the photodissociation of H2O in three dimensions is given and the calculated rovibrational state distributions of the OH fragment are presented.

A Chebyshev method for calculating state-to-state reaction probabilities from the time-independent wavepacket reactant-product decoupling equations

Recently, Peng and Zhang have introduced the reactant-product decoupling (RPD) equations. These are an exact formulation of quantum mechanical reactive-scattering, whereby the Schrödinger 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.

Reactant-product decoupling method for state-to-state reactive scattering: A case study for 3D H+H2 exchange reaction (J=0)

In this paper, we present theoretical and computational details of implementing the recently developed reactant-product decoupling (RPD) method (J. Chem. Phys. 105, 6072 (1996)) for state-to-state quantum reactive scattering calculations of the prototypical H + H2 reaction in three dimensions. The main purpose of this paper is to explore important features of the RPD scheme for use as a general and efficient computational approach to study state-to-state quantum dynamics for polyatomic reactions by using 3D H + H2 as an example. Specific computational techniques and numerical details are explicitly provided for efficient application of this method in the time-dependent (TD) implementation. Using the RPD method, the calculated state-to-state reaction probabilities for the 3D H + H2 reaction are in excellent agreement with those from the time-independent variational calculations, and the computational cost of the RPD method is significantly lower than other existing TD methods for state-to-state dynamics calculations.

Application of reactant–product decoupling method for state-to-state calculation of D+H2 reaction for J=0 and 1

The reactant–product decoupling (RPD) method (T. Peng and J. Z. H. Zhang, J. Chem. Phys., 1996, 105, 6072) provides a general and efficient approach to state-to-state quantum reactive scattering calculation for polyatomic reactions. The efficiency of the RPD method for reactive scattering is significantly enhanced by using a collocation-quadrature scheme to facilitate the coordinate transformation of the source term. This paper reports an application of the RPD method to the three-dimensional D+H 2 reaction for total angular momentum J=0 and 1. The state-to-state reaction probabilities from the present calculation are in excellent agreement with a previous time-independent variational calculation. The present result provides strong support for future application of the RPD method to more complex reaction systems.

Reactant–product decoupling approach to state-resolved reactive scattering Time-independent wavepacket formulation

S. C. Althorpe, D. J. Kouri, D. J. K. ,. D. K. H. J. Z. H. Z., D. K. Hoffman and J. Z. H. Zhang, J. Chem. Soc., Faraday Trans., 1997, 93, 703 DOI: 10.1039/A606300D