Srinivasan%20S.%20Iyengar

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Quantum wavepacket ab initio molecular dynamics
A computational approach for quantum dynamics in
large systems

• Srinivasan S. Iyengar
• Department of Chemistry and Department of
  Physics,
• Indiana University
• Group members contributing to this work
• Jacek Jakowski (post-doc),
• Isaiah Sumner (PhD student),
• Xiaohu Li (PhD student),
• Virginia Teige (BS, first year student)

Funding
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Predictive computations a few (grand) challenges

• Bio enzyme Lipoxygenase Fatty acid oxidation
• Rate determining step hydrogen abstraction from
  fatty acid
• KIE (kH/kD)81
• Deuterium only twice as heavy as Hydrogen
• generally expect kH/kD 3-8 !
• weak Temp. dependence of rate
• Nuclear quantum effects are critical
• Conduction across molecular wires
• Is the wire moving?
• Reactive over multiple sites
• Polarization due to electronic factor
• Polymer-electrolyte fuel cells
• Dynamics temperature effects

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Chemical Dynamics of electron-nuclear systems

• Our efforts approach for simultaneous dynamics
  of electrons and nuclei in large systems
• accurate quantum dynamical treatment of a few
  nuclei,
• bulk of nuclei treated classically to allow
  study of large (enzymes, for example) systems.
• Electronic structure simultaneously described
  evolves with nuclei
• Spectroscopic study of small ionic clusters
  including nuclear quantum effects
• Proton tunneling in biological enzymes ongoing
  effort

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Hydrogen tunneling in Soybean Lipoxygenase 1
Introduce Quantum Wavepacket Ab Initio Molecular
Dynamics
Catalyzes oxidation of unsaturated fat

• Expt Observations
• Rate determining step hydrogen abstraction from
  fatty acid
• Weak temperature dependence of k
• kH/kD 81
• Deuterium only twice as heavy as Hydrogen,
• generally expect kH/kD 3-8.
• Remarkable deviation

Quantum nuclei
The electrons and the other classical nuclei
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Quantum Wavepacket Ab Initio Molecular Dynamics
Distributed Approximating Functional (DAF)
approximation to free propagator
Ab Initio Molecular Dynamics (AIMD)
using Atom-centered Density Matrix
Propagation (ADMP) OR Born-Oppenheimer
Molecular Dynamics (BOMD)
References
S. S. Iyengar and J. Jakowski, J. Chem. Phys.
122 , 114105 (2005). Iyengar, TCA, In
Press. J. Jakowski, I. Sumner and S. S.
Iyengar, JCTC, In Press (Preprints
authors website.)
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1. DAF quantum dynamical propagation
Quantum Dynamics subsystem

• Quantum Evolution Linear combination of Hermite
  functions The Distributed Approximating
  Functional

is a banded, Toeplitz matrix
Time-evolution vibrationally non-adiabatic!!
(Dynamics is not stuck to the ground vibrational
state of the quantum particle.)
Linear computational scaling with grid basis
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Quantum dynamically averaged ab Initio Molecular
Dynamics

• Averaged BOMD Kohn Sham DFT for electrons,
  classical nucl. Propagation
• Approximate TISE for electrons
• Computationally expensive.
• Quantum averaged ADMP
• Classical dynamics of RC, P, through an
  adjustment of time-scales

acceleration of density matrix, P
Fictitious mass tensor of P
Force on P

• V(RC,P,RQMt) the potential that quantum
  wavepacket experiences

Schlegel et al. JCP, 114, 9758 (2001). Iyengar,
et. al. JCP, 115,10291 (2001).
Ref..
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Quantum Wavepacket Ab Initio Molecular Dynamics
The pieces of the puzzle
The Quantum nuclei
Distributed Approximating Functional (DAF)
approximation to free propagator
Simultaneous dynamics
Ab Initio Molecular Dynamics (AIMD)
using Atom-centered Density Matrix
Propagation (ADMP) OR Born-Oppenheimer
Molecular Dynamics (BOMD)
The electrons and the other classical nuclei
S. S. Iyengar and J. Jakowski, J. Chem. Phys. 122
, 114105 (2005) J. Jakowski, I. Sumner, S. S.
Iyengar, J. Chem. Theory and Comp. In Press
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So, How does it all work?

• A simple illustrative example dynamics of ClHCl-
  
• Chloride ions AIMD
• Shared proton DAF wavepacket propagation
• Electrons B3LYP/6-311G

• As Cl- ions move, the potential experienced by
  the quantum proton changes dramatically.
• The proton wavepacket splits and simply goes
  crazy!

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Spectroscopic Properties

• The time-correlation function formalism plays a
  central role in non-equilibrium statistical
  mechanics.
• When A and B are equivalent expressions, eq. (18)
  is an autocorrelation function.
• The Fourier Transform of the velocity
  autocorrelation function represents the
  vibrational density of states.

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Vibrational spectra including quantum dynamical
effects

• ClHCl- system large quantum effects from the
  proton
• Simple classical treatment of the proton
• Geometry optimization and frequency calculations
  Large errors
• Dimensionality of the proton is also important
• 1D, 2D and 3D treatment of the quntum proton
  provides different results.
• McCoy, Gerber, Ratner, Kawaguchi, Neumark

• In our case Use the wavepacket flux and
  classical nuclear velocities to obtain the
  vibrational spectra directly
• Includes quantum dynamical effects, temperature
  effects (through motion of classical nuclei) and
  electronic effects (DFT).

In good agreement with Kawaguchis IR spectra
J. Jakowski, I. Sumner and S. S. Iyengar, JCTC,
In Press (Preprints Iyengar Group website.)
References
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The Main Bottleneck The work around
Time-dependent Deterministic Sampling (TDDS)

• However, some regions are more important than
  others?
• Addressed through TDDS, on-the-fly

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The Main Bottleneck The quantum interaction
potential

1. Quantum Dynamics subsystem
2. AIMD subsystem (ADMP for example)

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Time-dependent deterministic sampling

• 1) Importance of each grid point (RQM) based on
  
• - large wavepacket density -? r
• - potential is low -
  V
• gradient of potential is high - ?V

2) So, the sampling function is
Ir , IV , IV --- adjust importance
of each component
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Time-dependent deterministic sampling

• 1) Importance of each grid point (RQM) based on
  
• - large wavepacket density -? r
• gradient of potential is high - ?V
• potential is low - V

2) Functional form
3) Grid point for potential evaluation are
determined by integrating Nw(x)
where
Ir , IV , IV --- adjust importance
of each component
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TDDS - Haar wavelet decomposition
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Generalization to multidimensions - Haar
wavelet decomposition
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TDDS/Haar How well does it work?
The error, when the potential is evaluated only
on a fraction of the points is really
negligble!!!
1 mEh 0.0006 kcal/mol 2.7 10-5 eV
Hence, PADDIS reproduces the energy
Computational gain three orders of magnitude!!
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TDDS/Haar Reproduces vibrational properties?
The error in the vibrational spectrum negligible
These spectra include quantum dynamical effects
of proton along with electronic effects!
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Hydrogen tunneling in biological enzymes The
case for Soybean Lipoxygenase 1

• Enzyme active site shown
• Catalyzes the oxidation of unsaturated fat!
• Rate determining step hydrogen abstraction

• Weak temperature dependence of k
• Hydrogen to deuterium KIE is 81
• Deuterium is only twice as larger as Hydrogen,
• Generally expect kH/kD 3-8.

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Soybean Lipoxygenase 1

• A slow time-scale process for AIMD
• Improved computational treatment through forced
  ADMP.
• The idea is the donor atom is pulled slowly
  along the reaction coordinate
• Bottomline Donor acceptor distance is not
  constant during the hydrogen transfer process.
• The donor-acceptor motion reduces barrier height

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Soybean Lipoxygenase 1 Proton nuclear
orbitals Look for the p and d type
functions!!
s-type
p-type
d-type
p-type
These states are all within 10 kcal/mol Eigenstat
es obtained from Arnoldi iterative procedure
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Reactant
Transition State

• Eigenstates obtained using
• Instantaneous electronic structure (DFT B3LYP)
• finite difference approximation to the proton
  Hamiltonian.
• Arnoldi iterative diagonalization of the
  resultant large (million by million) eigenvalue
  problem.

For Deuterium, the excited proton state
contributions are about 10 For hydrogen the
excited state contribution is about
3 Significant in an Marcus type setting.
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Transition state
classical
quantum
H
D
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Conclusions and Outlook

• Quantum Wavepacket ab initio molecular dynamics
  Seems Robust and Powerful
• Quantum dynamics efficient with DAF
• Vibrational non-adiabaticity for free
• AIMD efficient through ADMP or BOMD
• Potential is determined on-the-fly!
• Importance sampling extends the power of the
  approach
• In Progress
• QM/MM generalizations Enzymes
• generalizations to higher dimensions and more
  quantum particles Condensed phase
• Extended systems (Quantum Dynamical PBC) Fuel
  cells

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Additional slides
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Optimization of w(RQM) with respect to a,b,g
a1 (IY) b3 (IYP) g1 (IChi)
RMS error of intrepolation during a dynamics
within mikrohartrees
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Computational advantages to DAF quantum
propagation scheme

• Free Propagator
• is a banded, Toeplitz matrix
• Time-evolution vibrationally non-adiabatic!!
  (Dynamics is not stuck to the ground vibrational
  state of the quantum particle.)

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Quantum Wavepacket Ab Initio Molecular Dynamics
Working Equations
Quantum Dynamics subsystem
Trotter

• Coordinate representation
• The action of the free propagator on a Gaussian
  exactly known
• Expand the wavepacket as a linear combination of
  Hermite Functions
• Hermite Functions are derivatives of Gaussians
• Therefore, the action of free propagator on the
  Hermite can be obtained in closed form

• Coordinate representation for the free
  propagator. Known as the Distributed
  Approximating Functional (DAF) Hoffman and
  Kouri, c.a. 1992
• Wavepacket propagation on a grid

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Spreading transformation
We want to do potential evaluation for ? fraction
of grid

• Density from ?(x) may be larger than current grid
  density- exceeding density is spread over low
  density grid area
• for ?? 1 weighting ?(x) should tend to 1

Grid point for potential evaluation are
deteminned by integrating Nw(x)
Interpolation of potential

• Version of cubic spline interpolation
• based on on potentials and gradients
• easy to generalize in multidimensions
• general flexible form

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Another example Proton transfer in the phenol
amine system

• Shared proton DAF wavepacket propagation
• All other atoms ADMP
• Electrons B3LYP/6-31G
• C-C bond oscilates in phase with wavepacket

Wavepacket amplitude near amine
Scattering probability
References
S. S. Iyengar and J. Jakowski, J. Chem. Phys.
122 , 114105 (2005).
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Potential Adapted Dynamically Driven Importance
Sampling (PADDIS) basic ideas

• The following regions of the potential energy
  surface are important
• Regions with lower values of potential
• Thats probably where the WP likes to be
• Regions with large gradients of potential
• Tunneling may be important here
• Regions with large wavepacket density

Consequently, the PADDIS function is
The parameters provide flexibility