The Hybrid Quantum Trajectory/Electronic Structure DFTB-based Approach toMolecular Dynamics

Lei WangDepartment of Chemistry and BiochemistryUniversity of South Carolina

James W. Mazzuca University of South CarolinaSophya Garashchuk University of South CarolinaJacek Jakowski NICS & UTK

XSEDE14, Atlanta, GAJuly, 17th, 2014

Chemical Reaction Process

Molecules

Electrons Nuclei

(a) Quantum method: explicitlysolve Schrödinger equation for reactive system at low energy(b) Force field method: empirical parameters

Density functional tight binding (DFTB): approximate quantum chemical method1) cheap and accuracy2) 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

Example of ZPE and Tunneling

One-dimensional harmonic oscillator

ZPE

Quantum Tunneling

E > V

E < V

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

Bohmian Dynamics: Mixed Quantum/Classical Trajectory Dynamics

The polar form of time-dependent Schrödinger equation (TDSE):

tyxHtyxt

i ,,ˆ,,

where x describes light quantum particles and y heavy “classical” particles.

yxVMm

H yx ,22

ˆ 22

22

tyxSityxAtyx ,,exp,,,,

D. Bohm. Phys. Rev., 85, 166-179, 1952

Trajectories follow reduced Newton’s Eqs and extra quantum correction :

UVpmpx

VpMp

y

xxx

xyyy

,

,

Linearized Quantum Force and Imaginary Time Dynamics

Expectation values are computed by sums over the trajectories,

cq

tttyxw

,

,ˆ

U is called quantum potential: AmA

U x2

2

2

And fitting with a linear function of x.AA

S. Garashchuk and V. Rassolov. J. Chem. Phys. 120, 1181-1190, 2004

,ˆ, xHx

Boltzmann evolution/Imaginary Time Dynamics can be propagated through imaginary time:

Cooling the system to temperature:Tkit B1

Where kB is the Boltzmann constant.

Reaction rate constant or ZPE

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. 4800 trajectories were propagated for 25 steps

Interaction of H+C37H15

“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)

DFTB Accuracy Test at Different Collision Sites

Fig 2: The electronic potential energy curves for H+C37H15 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.

Energy Transfer: Adsorbed Trajectories

Fig 3: the collision energies Ecoll ={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 C37H15, (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.

Energy Transfer: Reflected Trajectories

Fig 4: collision energies Ecoll ={ 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 C37H15,(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.

Movement of Hydrogen and Selected Carbons

Fig 5: Positions of the colliding hydrogen and selected carbons along the z-axis for Ecoll={ 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.

Three-dimensional proton

Classical Quantum

Three-dimensional Proton Dynamics

Fig 6: Left: Adsorption probability; Right: Displacement of the central carbon. a) C37H15 and b) C87H23

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 C37H15 obtained with multiple ensembles: a) Hydrogen and b) Deuterium

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+C37H15: the dominant QM effect is due to delocalization of initial wavepacket; neglect of nuclear effects can lead to an overestimation of adsorption.

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

Thermal Evolution of the Proton Wavefunction

Real-time calculations are initialized using the trajectories in b)

Rate Constant and Kinetic Isotopic Effect

dttCTQTk ff

k(T): rate constantQ(T): quantum partition function of reactantsCff(t): flux-flux correlation function

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.

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