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Hong Zhang a , Sean Smith a and Suresh Bhatia b

Mixed Quantum Dynamics/Molecular Dynamics Simulations of Transport Properties of Hydrogen/Deuterium in Nano-porous Materials. Hong Zhang a , Sean Smith a and Suresh Bhatia b. Centre for Computational Molecular Science a , Division of Chemical Engineering b

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Hong Zhang a , Sean Smith a and Suresh Bhatia b

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  1. Mixed Quantum Dynamics/Molecular Dynamics Simulations of Transport Properties of Hydrogen/Deuterium in Nano-porous Materials Hong Zhanga, Sean Smitha and Suresh Bhatiab Centre for Computational Molecular Sciencea, Division of Chemical Engineeringb University of Queensland, Brisbane QLD 4072 Australia 1 Introduction • Nano-porous materials are widely used as heterogeneous catalysts and in adsorption and membrane separation, as well as in sensing and energy storage. Examples include zeolites and other molecular sieves, activated carbon, and clays. • In many applications of nano-porous materials, the rate of molecular transport inside the pores plays a key role in the overall processes. A membrane or adsorption separation process exploits differences in rates of diffusion and/or adsorbed-phase concentrations to differentiate between the different molecular species and separate them. Molecular level modeling has come to be a powerful tool for gaining a better understanding of diffusion, surface interactions and dynamics in nano-porous materials. 2.3 Rotational effects and further approximations:  When T = 0, the wave function is a pure state, which is the ground vibrational state .For relatively low temperature T, only ground vibrational state will be involved, but different rotational states will have to be considered. In this case a Boltzman distribution can be assumed for the ro-vibrational states. • Molecular dynamics (MD) simulations have been employed to compute self- and/or transport- diffusion coefficients in nano-porous materials. However, for diffusion of light gases such as H2/D2, nuclear quantum effects might play a significant role, which has been indicated through both experimental evidence and approximate theoretical investigations.  For H2, ro-vibrational states with quantum numbers (ν= 0, j = 0, 1, …) are relatively easy to calculate. However, for larger systems such as H2O, CO2, ro-vibrational states with quantum numbers (ν= 0, J = 0, 1, …) are not easy to calculate albeit in our reach with parallel computing. Therefore approximations are unavoidable. One way is to use semi-classical Gaussian wavepacket, with its parameters fitted with the ground vibrational state wave function. Meanwhile, an additional width parameter in the Gaussian wave packet can be added to describe the rotational effects, instead of computing the mixed state wave function. The T-dependent width parameter should behave in such a way that when T, the packet narrows to a classical point, whereas asT0, the packet broadens to the width of the ground state vibrational wave function. Another way is to use J-shifting method. • In recent years, we have developed several quantum dynamics methods based upon Lanczos and Chebyshev iterations to solve large-scale quantum dynamics (QD) problem. These QD methods are very efficient and highly parallelizable, and are well suited to implement mixed QD/MD simulations in this project. 2 Methodology in QD/MD Simulation 2.1 Fundamental equations: Let’s start with the mixed quantum/classical equations for nuclear dynamics for guest molecules (the frame is treated as rigid model)  A further approximation is to replace the integral for V with its value at the average vibrational coordinates. This approximation is rationalized by the fact that the probability in these positions are the highest (assuming 100%), thus avoid the integration. 2.4 Hamiltonian for H2-zeolite and average potential term (translational motion of c.m.) (ro-vibrational motion) 2.2 Average potential term for translational mode: In Jacobi coordinates, the Hamiltonian for A-B pair is  Here atomic coordinates for frame have been fixed. Similarly, average potential term is For complex system two approximations for potential term have to be made, namely, pair-potential and additive potential, as in MD simulation. For diatomic systems such as H2, N2, O2, 2.5 Diffusion coefficients (or other quantities from MD simulations) 3 Comparison with Quantum Statistical Method Quantum statistical (QS) average potential term For tri-atomic systems such as H2O and CO2, From above Hamiltonian it can be seen that translational modes, vibrational modes and rotational modes are notoriously coupled, and any attempts to separate them will fail in rigorous sense. Here we first consider temperature T = 0 case, and ignore rotational effects. Zero-order vibrational wave function can be obtained by solving:  Both QM and QS methods have the same form for the average potential term. The first term (L-J) potential plays a dominant role whereas the other terms are correction terms.  However, the physical insight is very different in the two methods. In QS derivation, we suppose that the interaction between c. m. (i) and c. m. (j) can be expressed as L-J model and the centre of mass has a distribution g(u) in space. In QM method, QM wave function has been employed to compute the average QM potential. Later when consider T > 0 cases, a temperature dependent Gaussian wavepacket term will be added in the wave function to describe the rotational effects. Such treatment is accurate enough for transport phenomena. However, for spectroscopic properties, we will need to consider first- or second-order approximation. Potential term can be expressed as 4 Implementation and Future Work • A practical and reliable method has been proposed to incooperate quantum effects in MD simulations for transport of small molecules in nano-porous materials and the basic formulations have been derived for mixed QD/MD simulations. • Next we will combine our local QD codes with MD codes to implement the simulation package. • Comparison with quantum statistical method will also be made to test our method. V is a perturbation term. Thus the average potential for A-B pair is: We note that the first term in V is itself after integration, and the second and third term are constant with respect to RAB, which won’t contribute to the force for translational motion. Only the fourth term is important, which can be expanded at minimum energy point. We call it quantum mechanical correction term.

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