Srinivasan S. Iyengar Department of Chemistry and Department of Physics, Indiana University

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