Quantum Wave-packet Ab Initio
molecular dynamics: An approach for
quantum dynamics in large systems
Srinivasan S. Iyengar
Department of Chemistry,
Indiana University
Computational Challenges for modelling complex
chemical processes



Complex interactions:
• Reactive over multiple sites
• Polarization and electronic factors
Ab initio, DFT at good level and
QM/MM
Nuclear quantization?
• Enzymes:
– SLO-1: KIE=81, weak T dep. of k
– DHFR
– ADH
• Condensed phase
• Materials, fuel cells

Computational efficiency reqd.
Quantum Wavepacket Ab Initio Molecular Dynamics
Full Electron-nuclear TDSE:
TDSCF separation:
i  ( R QM , R C , r; t )  Ĥ ( R QM , R C , r; t )
t
( R QM , R C , r; t )  1 ( RQM ; t )  2 ( RC ; t )  3 (r; t )  ei
i

 1 ( RQM ; t )  Hˆ 1 1 ( RQM ; t )
t
[Quantum-dynamical subsystem: (protons, excess electrons, etc..)]
i

 2 ( RC ; t )  Hˆ 2 2 ( RC ; t )
t
[Majority of the nuclei]
i

 3 ( r; t )  Hˆ 3 3 ( r; t )
t
[electrons]
Quantum Wavepacket Ab Initio Molecular Dynamics
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

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).
Quantum Wavepacket Ab Initio 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) [Pioneered by Hoffman
and Kouri, c.a. 1992]
Wavepacket propagation on a grid

Quantum Wavepacket Ab Initio Molecular
Dynamics: Working Equations
Ab Initio Molecular Dynamics (AIMD) subsystem:
•
BOMD: Kohn Sham DFT for electrons, classical nucl. Propagation
•
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



V(R C , P, R QM )
 1  P  P    μ 1/ 2
 1
P


R
V(R C , P, R QM )
1
1
R C
P
Force on P
V(RC,P,RQM;t) : the potential that quantum wavepacket experiences
Quantum Wavepacket Ab Initio Molecular Dynamics
 iHˆ t 
 1 ( RQM ; t )  exp 
  1 ( RQM ; t  0)
  
[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)
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
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 
Notice: all super- and sub-diagonals are the same.
Computational scaling: O(N) for large number of grid points

i
j
RQM
 RQM
2 ( t )

Some Advantages of ADMP
ADMP:
– Currently 3-4 times faster
than BO dynamics
– Computational scaling O(N)
– Hybrid functionals (more
accurate) : routine
– Good adiabatic control
ADMP Spectrum!!
Quantum Wavepacket Ab Initio Molecular Dynamics
 iHˆ t 
 1 ( RQM ; t )  exp 
  1 ( RQM ; t  0)
  
[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)
So, How does it all work?
•
Illustrative example: dynamics of ClHCl•
•
•
•
•
Chloride ions: AIMD (BOMD)
Shared proton: DAF wavepacket propagation
Electrons: B3LYP/6-311+G**
As Cl- ions move, the
potential experienced
by the “quantum”
proton changes
dramatically.
The wavepacket splits
spontaneously as is to
be expected from a
quantum dynamical
procedure.
Another example: Proton transfer 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).
The Main Bottleneck: The quantum potential
•
The potential involved in the wavepacket propagation is required at every
grid point!!

 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 
  
And the gradients are also required at these grid points!!
Expensive from an electronic structure perspective
The work around: Importance Sampling
Basic Ideas:
• Consider the phenol amine system
• The quantum nature of the proton: wavepacket on grid
• So we need the potential on grid
• Which grid points are most important?
This question: addressed by importance sampling approach,
“on-the-fly”
References…
J. Jakowski and S. S. Iyengar, In Preparation.
Basic ideas of importance sampling
Essentially: The following regions of the potential energy surface are important:
-Regions with large wavepacket density
-Regions with lower values of potential
-Regions with large gradients of potential
Consequently, the importance sampling function is:
( RQM ) 
f  ( RQM ; )  gV ( RQM ;  )
hV ( RQM ;  )
The parameters provide flexibility
Conclusions

Quantum Wavepacket ab initio molecular dynamics:
Robust and Powerful
 Quantum dynamics: efficient with DAF
 AIMD efficient through ADMP
 Importance sampling extends the power of the
approach

QM/MM generalizations are currently in progress, as
are generalizations to higher dimensions.