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The Hybrid Quantum Trajectory/Electronic Structure DFTB-based Approach to Molecular Dynamics Lei Wang Department of Chemistry and Biochemistry University of South Carolina James W. Mazzuca University of South Carolina Sophya Garashchuk University of South Carolina Jacek Jakowski NICS & UTK XSEDE14, Atlanta, GA July, 17 th, 2014

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Chemical Reaction Process Molecules Electrons Nuclei (a) Quantum method: explicitly solve Schrödinger equation for reactive system at low energy (b) Force field method: empirical parameters Density functional tight binding (DFTB): approximate quantum chemical method 1) cheap and accuracy 2) can do hundreds or a few thousand electrons Classical - Molecular Dynamics Light particles: Proton or Deuterium at low energy reactive process Include quantum effects for nuclei into MD: Zero Point Energy Tunneling

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Example of ZPE and Tunneling One-dimensional harmonic oscillator ZPE Quantum Tunneling E > V E < V

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Outline Bohmian dynamics and Linearized Quantum Force (LQF) method Implementation in real/imaginary time dynamics Collision of hydrogen with graphene sheet Proton transfer in soybean lipoxygenase-1

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Bohmian Dynamics: Mixed Quantum/Classical Trajectory Dynamics The polar form of time-dependent Schrödinger equation (TDSE): where x describes light quantum particles and y heavy “classical” particles. D. Bohm. Phys. Rev., 85, , 1952 Trajectories follow reduced Newton’s Eqs and extra quantum correction :

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Linearized Quantum Force and Imaginary Time Dynamics Expectation values are computed by sums over the trajectories, U is called quantum potential: And fitting with a linear function of x. S. Garashchuk and V. Rassolov. J. Chem. Phys. 120, , 2004 Boltzmann evolution/Imaginary Time Dynamics can be propagated through imaginary time: Cooling the system to temperature: Where k B is the Boltzmann constant. Reaction rate constant or ZPE

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Implementation Quantum correction is included for selected DoFs The QT code is merged with DFTB method Electronic energy evaluation is parallelized Multiple independent sub- ensembles represent nearly classical DoFs Fig 1: CPU time as a function of the number of cores trajectories were propagated for 25 steps

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Interaction of H+C 37 H 15 “A Quantum Trajectory-Electronic Structure Approach for Exploring Nuclear Effects in the Dynamics of Nanomaterials”, S. Garashchuk, J. Jakowski, L. Wang, B. Sumpter, J. Chem. Theory Comput., 9 (12), 5221–5235 (2013)

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DFTB Accuracy Test at Different Collision Sites Fig 2: The electronic potential energy curves for H+C 37 H 15 obtained with the DFT and with the DFTB at zero impact parameter with respect a) lattice-center, b) hexagon-center and c) bond-center geometries.

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Energy Transfer: Adsorbed Trajectories Fig 3: the collision energies E coll ={0.2,0.4,0.8} eV are plotted as a solid line, dash and dot- dash, respectively: (a) the kinetic energy of adsorbing hydrogen, (b) the kinetic energy of C 37 H 15, (c) the position of the colliding proton along the reactive coordinate z and (d) the potential energy of the system along the trajectories are shown as functions of time.

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Energy Transfer: Reflected Trajectories Fig 4: collision energies E coll ={ 0.05,1.2,1.6} eV are plotted as a solid line, dash and dot-dash, respectively: (a) the kinetic energy of reflected hydrogen, (b) the kinetic energy of C 37 H 15,(c) the z -coordinate of the colliding proton, and (d) the potential energy of the system along the trajectories are shown as functions of time.

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Movement of Hydrogen and Selected Carbons Fig 5: Positions of the colliding hydrogen and selected carbons along the z -axis for E coll ={ 0.05,0.8,1.6} eV represented as a solid line, dash and dot-dash, respectively: (a) the proton, (b) the central carbon and (c) the nearest-neighbor carbon. “Adsorption of a Hydrogen Atom on a Graphene Flake Examined with a Quantum Trajectory/Electronic Structure Dynamics”, L. Wang, J. Jakowski, S. Garashchuk, J. Phys. Chem. C, accepted.

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Three-dimensional proton ClassicalQuantum

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Three-dimensional Proton Dynamics Fig 6: Left: Adsorption probability; Right: Displacement of the central carbon. a) C 37 H 15 and b) C 87 H 23

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Adsorption Probability Averaged over Multiple Ensembles of Trajectories The ensembles are independent of each other Converged probabilities are obtained with 11 ensembles for the hydrogen and with 14 ensembles for the deuterium Fig 7: Adsorption probability of H on C 37 H 15 obtained with multiple ensembles: a) Hydrogen and b) Deuterium

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Conclusion Hybrid quantum/classical trajectory dynamics: reduced dimensionality quantum corrections on dynamics for light/heavy particles. It is suitable for up to 200 atoms. QTES-DFTB simulation of H+C 37 H 15 : the dominant QM effect is due to delocalization of initial wavepacket; neglect of nuclear effects can lead to an overestimation of adsorption.

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Biological Environment with Real & Imaginary Time Dynamics: Proton transfer in soybean lipoxygenase-1 Fig 8: DFTB potential energy as the hydrogen moves from carbon to oxygen. The effect of local substrate vibrations on the H/D primary kinetic isotope effect (KIE): QT = 51, QM = 49, Experimental value = 81 J. W. Mazzuca, S. Garashchuk, J. Jakowski. Chem. Phys. Lett. submitted

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Thermal Evolution of the Proton Wavefunction Real-time calculations are initialized using the trajectories in b)

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Rate Constant and Kinetic Isotopic Effect k(T) : rate constant Q(T) : quantum partition function of reactants C ff (t) : flux-flux correlation function

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Conclusion Substrate vibrations in SLO-1 active site increase the rate constant by 15%, and the kinetic isotope effect increases by 5-10%. The increase is moderate because the reaction is fast ~0.1ps.

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Acknowledgement Dr. Sophya Garashchuk Dr. Jacek Jakowski Dr. Vitaly Rassolov Dr. James Mazzuca Dr. David Dell’Angelo Bing Gu Brett Cagg Bryan Nichols

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