srinivasan s. iyengar department of chemistry and department of physics, indiana university group...
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Srinivasan S. IyengarDepartment 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)
Quantum wavepacket ab initio molecular dynamics: A computational approach for
quantum dynamics in large systems
Funding:
Iyengar Group, Indiana University
Predictive computations: a few (grand) challengesPredictive 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
Lipoxygenase: enzyme
Ion (proton) channels
Iyengar Group, Indiana University
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
Chemical Dynamics of electron-nuclear systemsChemical Dynamics of electron-nuclear systems
Iyengar Group, Indiana University
Hydrogen tunneling in Soybean Lipoxygenase 1: Introduce Hydrogen tunneling in Soybean Lipoxygenase 1: Introduce Quantum Wavepacket Ab Initio Molecular DynamicsQuantum Wavepacket Ab Initio Molecular Dynamics
Expt ObservationsExpt 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
);(ˆ);( 333 trHtrt
i
);(ˆ);( 222 tRHtRt
i CC
The electrons and the
“other” classical nuclei
Catalyzes oxidation of unsaturated fat
);(ˆ);( 111 tRHtRt
i QMQM
Iyengar Group, Indiana University
Quantum Wavepacket Ab Initio Molecular Quantum Wavepacket Ab Initio Molecular DynamicsDynamics
Ab Initio Molecular Dynamics (AIMD) using:
Atom-centered Density Matrix Propagation(ADMP)
OR
Born-Oppenheimer Molecular Dynamics(BOMD)
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: author’s website.)
References…
)0;(ˆ
exp);( 11
tR
tHitR QMQM
[Distributed Approximating Functional (DAF) approximation to free propagator]
);(ˆ);( 111 tRHtRt
i QMQM
The “Quantum” nuclei
);(ˆ);( 333 trHtrt
i
);(ˆ);( 222 tRHtRt
i CC
The electrons and the
“other” classical nuclei
Iyengar Group, Indiana University
1. 1. DAF DAF quantum dynamical quantum dynamical propagatpropagationion
• Quantum Evolution: Linear combination of Hermite functions: The “Distributed Approximating Functional”
)0;(ˆ
exp);( 11
tR
tHitR QMQM
Quantum Dynamics subsystem:Quantum Dynamics subsystem:
is a banded, Toeplitz matrix
ab
bab
babbab
bab
ba
.
.
.00
0
..
0
00.
.
. Time-evolution: vibrationally non-adiabatic!! (Dynamics is not stuck to the ground vibrational state of the quantum particle.)
Linear computational scaling with grid basis
)(222
22/
02 )(2
)(expexp
t
RR
n
jQM
iQM
M
nn
jQM
iQM
jQM
iQMH
t
RRCR
iKtR
Iyengar Group, Indiana University
• 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
1
PC
QMC12
C2
R
)RP,,V(RRM
dt
d
acceleration of density matrix, P
Force on P
“Fictitious” mass tensor of P
PPP
)RP,,V(RP1
R
QMC12
2
dt
d 2/1μ 2/1μ
2.2. Quantum dynamically Quantum dynamically averaged ab Initio Molecular Dynamicsaveraged ab Initio Molecular Dynamics
• V(RC,P,RQM;t) : the potential that quantum wavepacket experiences
Schlegel et al. JCP, 114, 9758 (2001). Iyengar, et. al. JCP, 115,10291 (2001).Ref..
Iyengar Group, Indiana UniversityQuantum Wavepacket Ab Initio Quantum Wavepacket Ab Initio Molecular Dynamics: Molecular Dynamics: The pieces of the puzzleThe pieces of the puzzle
)0;(ˆ
exp);( 11
tR
tHitR QMQM
[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)
The “Quantum” nuclei
The electrons and the “other” classical nuclei
Simultaneous dynamics
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
Iyengar Group, Indiana University
So, How does it all work?So, How does it all work?
• A simple illustrative example: dynamics of ClHCl- • Chloride ions: AIMD • Shared proton: DAF wavepacket propagation• Electrons: B3LYP/6-311+G**
• As Cl- ions move, the potential experienced by the “quantum” proton changes dramatically.
• The proton wavepacket splits and simply goes crazy!
Iyengar Group, Indiana University
Spectroscopic PropertiesSpectroscopic 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.
)18(),()0;()()0()( tBAdtBAtC
)19()()0()(
tvveCti
Iyengar Group, Indiana UniversityVibrational spectra Vibrational spectra including quantum dynamical effectsincluding 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 Kawaguchi’s IR spectra
***
mm
itxJ Im
2,
""~)()(Re)( vm
pt
m
itJt
J
J. Jakowski, I. Sumner and S. S. Iyengar, JCTC, In Press (Preprints: Iyengar Group website.)References…
Iyengar Group, Indiana University
• Consider the phenol amine system
The Main Bottleneck: The work around: Time-dependent Deterministic Sampling (TDDS)
Need the quantum mechanicalEnergy at all these grid points!!
• However, some regions are more important than others?
• Addressed through Addressed through TDDS, TDDS, “on-the-fly”“on-the-fly”
Iyengar Group, Indiana University
1. Quantum Dynamics subsystem:
2. AIMD subsystem (ADMP for example)
The Main Bottleneck: The quantum interaction potentialThe Main Bottleneck: The quantum interaction potential
)(2
expexp2
expˆ
exp 3tOiVtiKtiVttHi
][),,(
112/1
2
22/1
CCC
QMCCC PPP
RPRV
dt
Pd
112
2 ),,(
C
QMCCC
P
RPRV
dt
RdM
)0;(ˆ
exp);( 11
tR
tHitR QMQM
• The potential for wavepacket propagation is required at every grid point!!The potential for wavepacket propagation is required at every grid point!!
• And the gradients are also required at these grid points!!And the gradients are also required at these grid points!!• Expensive from an electronic structure perspectiveExpensive from an electronic structure perspective
The Interaction Potential:A major computational bottleneck
Iyengar Group, Indiana University
]/1[
]/1'[]/1[)( '
V
VQM IV
IVIR
1) Importance of each grid point (RQM) based on: - large wavepacket density - - potential is low - V
- gradient of potential is high - V
Time-dependent deterministicTime-dependent deterministic sampling sampling
I , IV , IV’ --- adjust importance of each component
2) So, the sampling function is:
Iyengar Group, Indiana UniversityTDDS - Haar wavelet decomposition TDDS - Haar wavelet decomposition
)()(12
,, QM
Gen
i
i
jjijiQM RcR
otherwise
xx
0
101)( )2()(, jxx i
ji
Iyengar Group, Indiana University
Generalization to multidimensions - Haar wavelet decomposition Generalization to multidimensions - Haar wavelet decomposition
)()(12
,, QM
Gen
i
i
jjijiQM RcR
otherwise
xx
0
101)( )2()(, jxx i
ji
Iyengar Group, Indiana University
TDDS/Haar: How well does it work?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 Eh = 0.0006 kcal/mol = 2.7 * 10-5 eV
Hence, PADDIS reproduces the energy: Computational Computational gain three orders of magnitude!!gain three orders of magnitude!!
Iyengar Group, Indiana University
TDDS/Haar: Reproduces vibrational properties?TDDS/Haar: Reproduces vibrational properties?
These spectra include quantum dynamical effects of proton along with electronic effects!
The error in the vibrational spectrum: negligible
Iyengar Group, Indiana University
Hydrogen tunneling in biological Hydrogen tunneling in biological enzymes: The case for Soybean Lipoxygenase 1enzymes: The case for Soybean Lipoxygenase 1
Lipoxygenase: enzyme
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.
Enzyme active site shown Catalyzes the oxidation of
unsaturated fat! Rate determining step:
hydrogen abstraction
Iyengar Group, Indiana University
Soybean Lipoxygenase 1:Soybean Lipoxygenase 1:
Lipoxygenase: enzyme 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
Iyengar Group, Indiana University
Soybean Lipoxygenase 1: Proton nuclear Soybean Lipoxygenase 1: Proton nuclear “orbitals”: Look for the “p” and “d” type functions!!“orbitals”: Look for the “p” and “d” type functions!!
s-type
p-type
p-type d-type
These states are all within 10 kcal/mol
Eigenstates obtained from Arnoldi iterative
procedure
Iyengar Group, Indiana University
ReactantReactant
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.
Transition StateTransition 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.
Iyengar Group, Indiana University
Transition stateTransition state
quantumquantum classicalclassicalH
D
Iyengar Group, Indiana University
Conclusions and OutlookConclusions 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
Iyengar Group, Indiana University
Additional slidesAdditional slides
Iyengar Group, Indiana University
(IY(IYP)(IChi
Optimization of Optimization of ‘(R‘(RQMQM) with respect to ) with respect to
RMS error of intrepolation during a dynamics within mikrohartrees
Iyengar Group, Indiana University
• 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.)
Computational advantages to Computational advantages to DAF DAF quantum quantum propagatpropagation schemeion scheme
ab
bab
babbab
bab
ba
.
.
.00
0
..
0
00.
.
.
)(22
2/1!
141
2/
0
12
)()0(
2
2
)2()(2
)(exp
)0(
1exp
t
RR
nnn
M
n
n
t
jQM
iQMj
QMiQM
jQM
iQMH
t
RRR
iKtR
Iyengar Group, Indiana University
• Coordinate representation for the free propagator. Known as the Distributed Approximating Functional (DAF) [Hoffman and Kouri, c.a. 1992]
• Wavepacket propagation on a grid
Quantum Wavepacket Ab Initio Molecular Quantum Wavepacket Ab Initio Molecular Dynamics: Working EquationsDynamics: Working Equations
)0;(ˆ
exp);( 11
tR
tHitR QMQM
)(
2expexp
2exp
ˆexp 3tO
iVtiKtiVttHi
)(22
2/1!
141
2/
0
12
)()0(
2
2
)2()(2
)(exp
)0(
1exp
t
RR
nnn
M
n
n
t
jQM
iQMj
QMiQM
jQM
iQMH
t
RRR
iKtR
Trotter
Quantum Dynamics subsystem:Quantum Dynamics subsystem:
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:
Iyengar Group, Indiana UniversitySpreading transformation Spreading transformation
-Density from ω(x) may be larger than current grid density- exceeding density is spread over low density grid area- for η 1 weighting ω(x) should tend to 1
We want to do potential evaluation for η fraction of grid
))(()()(' QMQMQM RURR
Grid point for potential evaluation are deteminned by integrating [N*(x)]
Interpolation of potentialInterpolation of potentialVersion of cubic spline interpolation- based on on potentials and gradients - easy to generalize in multidimensions- general flexible form
Iyengar Group, Indiana UniversityAnother example: Proton transfer in the Another example: Proton transfer in the phenol amine systemphenol amine system
S. S. Iyengar and J. Jakowski, J. Chem. Phys. 122 , 114105 (2005). References…
• Shared proton: DAF wavepacket propagation • All other atoms: ADMP• Electrons: B3LYP/6-31+G**
• C-C bond oscilates in phase with wavepacket
Wavepacket amplitude near amine )()()0( 11 tEG Scattering probability:
Iyengar Group, Indiana UniversityPotential Adapted Dynamically Driven Importance Sampling (PADDIS) : basic ideas : basic ideas
);(
);( );()(
QMV
QMVQMQM Rh
RgRfR
The following regions of the potential energy surface are important:
-Regions with lower values of potential-That’s 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
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