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
Funding:
Group members contributing to this work:
Jacek Jakowski (post-doc),
Isaiah Sumner (PhD student),
Xiaohu Li (PhD student),
Virginia Teige (BS, first year student)
Iyengar Group, Indiana University
Predictive computations: a few (grand) challenges
Lipoxygenase: enzyme

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?
Ion (proton) channels




Reactive over multiple sites
Polarization due to electronic factor
Polymer-electrolyte fuel cells
Dynamics & temperature effects
Iyengar Group, Indiana University
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
Iyengar Group, Indiana University
Hydrogen tunneling in Soybean Lipoxygenase 1: Introduce
Quantum Wavepacket Ab Initio Molecular Dynamics
Catalyzes oxidation of unsaturated fat
Expt Observations
“Quantum” nuclei

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
i

 1 ( RQM ; t )  Hˆ 1 1 ( RQM ; t )
t
i

 2 ( RC ; t )  Hˆ 2 2 ( RC ; t )
t
The electrons and the
“other” classical nuclei

i  3 ( r; t )  Hˆ 3 3 ( r; t )
t
Iyengar Group, Indiana University
Quantum Wavepacket Ab Initio Molecular Dynamics
The “Quantum” nuclei
i

 1 ( RQM ; t )  Hˆ 1 1 ( RQM ; t )
t

 iHˆ t 
 1 ( RQM ; t )  exp 
  1 ( RQM ; t  0)
  
[Distributed Approximating Functional
(DAF) approximation to free propagator]


i  2 ( RC ; t )  Hˆ 2 2 ( RC ; t )
t
The electrons and the
“other” classical nuclei

i  3 ( r; t )  Hˆ 3 3 ( r; t )
t
References…


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.)
Iyengar Group, Indiana University
1. DAF quantum dynamical propagation
Quantum Dynamics subsystem:
•
 iHˆ t 
 1 ( RQM ; t )  exp 
  1 ( RQM ; t  0)



Quantum Evolution: Linear combination of Hermite functions: The
“Distributed Approximating Functional”
i
j
2
M /2



(
R

R
)
iKt

 j
QM
QM
i
RQM exp 
 RQM   C2 n exp 
H 2n
2
2 ( t )
  
n 0




i
j
RQM
 RQM
2 ( t )




is a banded, Toeplitz matrix
a
b

.
.

0
0

b . .
a b
b a b
b a
b
0 . .
0 0
0

.
b .

a b
b a 
Time-evolution: vibrationally
non-adiabatic!! (Dynamics is
not stuck to the ground
vibrational state of the
quantum particle.)
Linear computational scaling with grid basis
2.
Iyengar Group, Indiana University
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
“Fictitious”
mass tensor of P
acceleration of
density matrix, P
d 2P
dt 2

d 2R C
M
dt 2
•
 μ

1 / 2


 1/ 2
V(R C , P, R QM )
 1  P  P    μ
 1
P


R
1
V(R C , P, R QM )
1
R C
P
Force on P
V(RC,P,RQM;t) : the potential that quantum wavepacket experiences
Ref.. Schlegel et al. JCP, 114, 9758 (2001).
Iyengar, et. al. JCP, 115,10291 (2001).
Iyengar Group, Indiana University
Quantum Wavepacket Ab Initio
Molecular Dynamics: The pieces of the puzzle
The “Quantum”
nuclei
 iHˆ t 
 1 ( RQM ; t )  exp 
  1 ( RQM ; t  0)
  
[Distributed Approximating Functional
(DAF) approximation to free propagator]

Simultaneous dynamics
Ab Initio Molecular Dynamics (AIMD) using:
The electrons and
the “other”
classical nuclei
Atom-centered Density Matrix Propagation
(ADMP)
OR
Born-Oppenheimer Molecular Dynamics
(BOMD)
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?
•
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 Properties

The time-correlation function formalism
plays a central role in non-equilibrium
statistical mechanics.
C (t )  A(0) B(t )   d A( ;0) B( , t )

(18)
When A and B are equivalent expressions,
eq. (18) is an autocorrelation function.
 The Fourier Transform
of the velocity

i t
autocorrelation
the
C ( )  function
e v(0)v(t ) represents
(19)

vibrational density
of states.
Vibrational spectra
including quantum dynamical effects
Iyengar Group, Indiana University


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).
J  x, t   



i *

    *  Im  *
2m
m

In good agreement with
 i

 p
J(t )  J  Re   (t )
  (t )  
~ " v"
Kawaguchi’s IR spectra
m

 m
References…
J. Jakowski, I. Sumner and S. S. Iyengar, JCTC, In Press
(Preprints: Iyengar Group website.)
Iyengar Group, Indiana University
The Main Bottleneck: The work around:
Time-dependent Deterministic Sampling (TDDS)
•
Consider the phenol amine system
Need the quantum mechanical
Energy at all these grid points!!
•
•
However, some regions are more important than others?
Addressed through TDDS, “on-the-fly”
Iyengar Group, Indiana University
The Main Bottleneck: The quantum interaction potential
1. Quantum Dynamics subsystem:
 iHˆ t 
 1 ( RQM ; t )  exp 
  1 ( RQM ; t  0)
  
 iHˆ t 
 iVt   iKt   iVt 
3
exp 
  exp   exp   exp    O(t )
 2      2 
  
2. AIMD subsystem (ADMP for example)

1/ 2
V ( RC , PC , RQM )
d 2 PC 1 / 2




 1  [PC  PC    ]
1
dt 2
PC
The Interaction Potential:
A major computational
bottleneck
V ( RC , PC , RQM )
d 2 RC
M
  1
1
dt 2
PC
•
The potential for wavepacket propagation is required at every grid point!!
•
And the gradients are also required at these grid points!!
•
Expensive from an electronic structure perspective
Iyengar Group, Indiana University
Time-dependent deterministic sampling
1) Importance of each grid point (RQM) based on:
- large wavepacket density
- 
- potential is low
- V
- gradient of potential is high - V
2) So, the sampling function is:
 ( RQM ) 
[   1 / I  ]  [V '1 / IV ' ]
[V  1 / IV ]
I , IV , IV’ --- adjust importance
of each component
Iyengar Group, Indiana University
TDDS - Haar wavelet decomposition
Gen 2 i 1
 ( RQM )   ci , j i , j ( RQM )
i
1
 ( x)  
0
0  x 1
otherwise
j
 i , j ( x)   (2i x  j )
Iyengar Group, Indiana University
Generalization to multidimensions - Haar wavelet decomposition
Gen 2 i 1
 ( RQM )   ci , j i , j ( RQM )
i
1
 ( x)  
0
0  x 1
otherwise
j
 i , j ( x)   (2i x  j )
Iyengar Group, Indiana University
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
gain three orders of magnitude!!
Iyengar Group, Indiana University
TDDS/Haar: Reproduces vibrational properties?
The error in the vibrational spectrum: negligible
These spectra include quantum dynamical effects of
proton along with electronic effects!
Iyengar Group, Indiana University
Hydrogen tunneling in biological
enzymes: The case for Soybean Lipoxygenase 1
Lipoxygenase: enzyme





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.
Iyengar Group, Indiana University
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
“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
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.
Iyengar Group, Indiana University
Transition state
quantum
H
D
classical
Iyengar Group, Indiana University
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
Iyengar Group, Indiana University
Additional slides
Iyengar Group, Indiana University
Optimization of ‘(RQM) with respect to a,b,g
a1 (IY
b3 (IYP)
g1 (IChi
RMS error of intrepolation during a dynamics within mikrohartrees
Iyengar Group, Indiana University
Computational advantages to DAF
quantum
propagation scheme
•
Free Propagator:
i
QM
R
i
j
 ( RQM
 RQM
)2  M / 2  ( 0) 2 n 1 1 n 1
1
 iKt  j
1 / 2




exp 
exp 

(
2

)
H 2n
 RQM 


 ( t )
4
n!
2
 (0)
2 ( t )
  

 n 0

i
j
RQM
 RQM
2 ( t )

is a banded, Toeplitz matrix:
a
b

.
.

0
0

•
b . .
a b
b a b
b a
b
0 . .
0 0
0

.
b .

a b
b a 
Time-evolution: vibrationally non-adiabatic!! (Dynamics is not stuck
to the ground vibrational state of the quantum particle.)
Group, Indiana University
Quantum Wavepacket Ab InitioIyengar
Molecular
Dynamics: Working Equations
Quantum Dynamics subsystem:

 iHˆ t 
 1 ( RQM ; t )  exp 
  1 ( RQM ; t  0)



Trotter
 iHˆ t 
 iVt   iKt   iVt 
3
exp 

exp

  exp   exp    O(t )
 2      2 
  
Coordinate representation:
•
•
•
•
i
QM
R
•
•
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:
i
j
 ( RQM
 RQM
)2  M / 2  ( 0) 2 n 1 1 n 1
1
 iKt  j
1 / 2




exp 
exp 

(
2

)
H 2n
 RQM 


 ( t )
4
n!
2
 (0)
2 ( t )
  

 n 0

i
j
RQM
 RQM
2 ( t )

Coordinate representation for the free propagator. Known as the
Distributed Approximating Functional (DAF) [Hoffman and Kouri, c.a.
1992]
Wavepacket propagation on a grid
Spreading transformation
Iyengar Group, Indiana University
We want to do potential evaluation for η fraction of grid
-Density from ω(x) may be larger than current grid densityexceeding density is spread over low density grid area
- for η 1 weighting ω(x) should tend to 1
 ' ( RQM )   ( RQM )  U ( ( RQM ))
Grid point for potential
evaluation are deteminned
by integrating [N*(x)]
Interpolation of potential
Version of cubic spline interpolation
- based on on potentials and gradients
- easy to generalize in multidimensions
- general flexible form
Group, Indiana University
Another example: Proton transferIyengar
in the
phenol amine system
•
•
•
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

Scattering probability:  1 (0) G ( E ) 1 (t )
References…
S. S. Iyengar and J. Jakowski, J. Chem. Phys. 122 , 114105 (2005).
Iyengar Group, Indiana University
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
-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:
( RQM ) 
f  ( RQM ;a )  gV ( RQM ; b )
hV ( RQM ; g )
The parameters provide flexibility