Two most fundamental laws that govern the dynamical processes in the real world are Newton's equations in classical mechanics and the Schrödinger equation in quantum mechanics. For small atomic and molecular systems, it is possible to solve exactly Schrödinger equation to extract physical information, such as the scattering matrix in bimolecular reactions and resonances in unimolecular dissociation. This is one area in which we are actively involved in recent years. More specifically, we have developed world-leading Lanczos representation methods to deal with especially complex-forming reactions, with higher efficiency and more scalability. For more details, the reader is referred to our recent review articles in Phys Chem Comm 6, 884 (2004) and in Phys Chem Chem Phys. 6, 12 (2003). For large systems, while molecular dynamics (MD) simulation represents one of the best available methods, it can’t properly deal with the motions of light atoms such as proton transfer for which quantum effects play a crucial role. Our aim is to combine MD simulation with our quantum dynamical methods to deal with large biological systems in which quantum effects are also important. The following are two representative projects undertaken in Quantum & Molecular Dynamics Group.

The studies of unimolecular dissociation have been central to our understanding of chemical reaction dynamics. Quantum-mechanically a unimolecular dissociation is characterized by resonances, i.e., temporarily trapped meta-stable states. To fully describe a resonance, three quantities are needed, namely, resonance energies, resonance widths (decay rates) and the product state distributions arising from its decay. Though the phenomenon of resonances has long been recognized and qualitatively understood, the quantitative determination of resonances started to appear only during past two decades. In recent years quantum calculations based on iterative methods have become increasingly common due to better scaling properties than direct methods, and our group has been actively involved in developing two such iterative methods, namely, Chebyshev filter diagonalisation methods and Lanczos filter diagonalisation methods, in particular to deal with deep well systems like HO₂. For more details, the reader is referred to our two recent review articles in Phys Chem Comm 6, 12-20 (2003) and in Phys Chem Chem Phys. 6, 884-894 (2004) [the front-cover image on the right from CCMS is in association with this latter review article].

Parallel Computing for Non-Zero Angular Momentum:

While so far most of the resonance calculations performed focused on total angular momentum J = 0 cases, substantive progress has been made with non-zero J calculations very recently. It is well known that exact J > 0 calculations are essential in fully understanding reaction dynamics both in unimolecular dissociation and in bimolecular reactions. However, these J > 0 calculations are still very challenging even for triatomic reactions, especially when dealing with complex-forming systems. The major reason for this situation is the so called ‘angular momentum catastrophe’: many J > 0 calculations have to be performed, and the size of the Hamiltonian matrix increases linearly with J.

Recently, we have implemented parallel computing strategy in the two iterative filter diagonalisation methods and have for the first time reported the HO₂ unimolecular resonances for relatively high J values. The reasons for employing parallel computing are two-fold. On one hand, the cpu time required to resolve the resonance fine structures is substantial. On the other hand, the storage requirement of the potential matrix and overlapping integrals also increases linearly with J. The quantum J-specific unimolecular dissociation rates for HO₂ → H + O₂ have been compared with the results of Troe et al. from statistical adiabatic channel method / classical trajectory calculations (see Fig. 2). These parallel computations make it possible to compute long time and large amplitude motions with computational times and storage requirement comparable to J = 0 case.

Figure: Parallel computation of rates for J = 10 case in HO2 dissociation from both Lanczos and Chebyshev methods and compared with statistical/classical results of Troe. [Zhang & Smith, JCP 120, 9583 (2004)]

Product State Distributions from Resonance Decay:

In addition, we have developed very efficient low-storage methods to calculate the product state distributions for unimolecular dissociation. The product state distributions reflect scattering from the resonance into product states through the transition state region of the potential energy surface, and thus contain additional clues about the intra- and inter-molecular dynamics of the system. To specify product state distributions arising from resonance decay, the resonance eigenfunctions of the dissipative Hamiltonian must be calculated and analyzed for their amplitudes in different product channels. Through an elegant construction of the algorithm, both the complex eigenvalues and the product state distributions associated with each individual resonant state can be calculated from a single Lanczos iteration. Fig. 3

The research in this project is concerned with the detailed description of reactive molecular collision processes that are elementary to combustion and atmospheric chemistry. The most detailed description of a reactive scattering process is given by the scattering matrix, S matrix. For a reaction A+BC(v,j) AB(v',j')+C the scattering matrix elements,

 S^J_{v,j,lv',j',l'}(E), denote the probability amplitude for a reaction from an initial quantum state (v,j,l) of the reactants into a final quantum state (v',j',l') of the products at an energy E. Here v and v' are the vibrational state, j and j' the rotational state and l and l' the helicity quantum number of the reactants and the products respectively and J denotes the total angular momentum, the total rotation of the ABC molecule. The S matrix elements  characterize the complete scattering process. Only a few quantum dynamical studies that calculated the scattering matrix for the full range of quantum numbers involved have been reported to date. Most commonly, reactive scattering calculations are performed for J = 0 and results for J > 0 are obtained via approximation methods.

In this work we are developing new practical methods for calculating state-to-state information of reactive processes employing time-dependent and time-independent wave packet approaches. The iterative methods that we use are based on two well-known matrix recursions: namely the Lanczos and the Chebychev recursions.  These are very powerful and versatile methods and have been used successfully to treat reactive as well as photo-dissociation processes. The project involves the design and  coding of parallel strategies for the  implementation of these quantum dynamical methods, allowing us to exploit the throughput capabilities of the high performance cluster facility at the CCMS. In this way, the number of processors used will increase with the size of the problem but the actual wall-clock time for completion of the calculation remains the same.

In the Chebychev real wavepacket approach the solutions to the reactive scattering problem may be represented as evolving wavepackets in real time or in pseudo-time (the Chebychev order domain). The initial wavepacket at time t = 0 is set up for one single quantum state of the reactants in the reactant channel of the potential energy surface. The potential energy surface governs the dynamics of the reaction and is part of the Hamiltonian matrix. The wavepacket is then propagated in time utilising a three-term Chebychev matrix recursion. This procedure gives a physical picture of the reaction dynamics in real time.  The real wavepacket approach will yield results for one fixed quantum state of the reactants and for all product quantum states and a wide range of energies of interest.

In collaboration with the group of Prof. Gabriel G. Balint-Kurti at the University of Bristol (UK) our group has developed a state-of-the-art time-dependent wavepacket method and an associated computer code for the calculation of the full S matrix elements. This  method has been successfully applied for J = 0 to the reactions H + H₂(D₂) and

O(¹D) + H₂(HD, D₂). To facilitate the very difficult J > 0 calculations a parallel code has been developed employing the Messages Passing Interface (MPI).  Test calculations have been carried out for J = 0 - 26 for the H + H₂ system.

Within CCMS we have also implemented the pseudo-time version of the real Chebychev wavepacket approach, which converges the real part of the scattering wave directly through propagation in the Chebychev order domain. This latter code has thus far been implemented only for J=0 [J. Chem. Phys., 117, 5174-5182 (2002)].

The Real Wave packet Approach applied to the HO₂ System (J=0):
A serial version of the wave packet program is used to study the reaction
dynamics of the HO₂ system. We employ three different potential energy surfaces (DMBE4, Xu et. al and Troe's surface) to study the affects on the dynamics. We have calculated state-to-state probabilities for the DMBE4 surface for J=0 only.

The first figure shows a contour plot of the DMBE4 surface for a fixed angle. One can clearly see the deep well that corresponds to the HO₂ complex. The second figure shows 4 snapshots of the wave packet at different times of the propagation. The initial wave packet is located in the entrance channel at time zero and is very localized. With time it then moves into the interaction region and the product channel.
 

This project involves mixed quantum dynamics/molecular dynamics simulation of proton transfer processes in Green Fluorescent Protein (GFP). GFP is a unique fluorescent label and has found many biotechnological and cell biology applications, such as a reporter of gene expression, cell lineage, and protein trafficking and interactions. In GFP, the most important processes are the photo-absorption, the proton-transfer in the excited state and the green fluorescence, which involves enormous quantum photo-dynamics. Given the quantum effects involved and the complexity of GFP, full description of GFP should combine the quantum-dynamical model for one or several protons with MD simulations for all other atoms. We will implement this project from the simplified but quantum-dynamical model to the more complicated mixed quantum/classical method and some preliminary but encouraging results have been obtained recently.