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, 17th, 2014
Chemical Reaction Process
Molecules
Electrons
(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
Nuclei
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
Quantum Tunneling
E>V
E<V
ZPE
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):
i

  x , y , t   Hˆ   x , y , t 
t
where x describes light quantum particles and y heavy “classical” particles.

Hˆ  
2
2m
x 
2

2

2M
2
y
 V x, y 
 i

  x , y , t   A  x , y , t  exp  S  x , y , t  


Trajectories follow reduced Newton’s Eqs and extra quantum correction :
py
 y    yV
y 
,
p
x
M
x 
px
m
,
 x  
p
x
V
U

D. Bohm. Phys. Rev., 85, 166-179, 1952
Linearized Quantum Force and Imaginary Time Dynamics
U is called quantum potential:
And fitting
A
A
U  

2
2 mA
xA
2
with a linear function of x.
Expectation values are computed by sums over the trajectories,
ˆ


t
 w x
t
, yt

q ,c
Boltzmann evolution/Imaginary Time Dynamics can be propagated
through imaginary time:


  x ,    Hˆ   x ,  

Cooling the system to temperature:
Reaction rate
  it  1 k B T
constant or ZPE
Where kB is the Boltzmann constant.
S. Garashchuk and V. Rassolov. J. Chem. Phys. 120, 1181-1190, 2004
Implementation




Quantum correction is
included for selected DoFs
The QT code is merged with
DFTB method
Electronic energy evaluation
is parallelized
Multiple independent subensembles 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 dotdash, 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
“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.
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.
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
k T  Q T  



C ff t  dt
k(T): rate constant
Q(T): quantum partition function of
reactants
Cff(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