1. Srinivasan S. Iyengar Department of Chemistry, Indiana University
Quantum Wave-packet Ab Initio molecular dynamics: An approach for quantum dynamics in large systems
Srinivasan S. Iyengar
Department of Chemistry,
Indiana University

2. 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
ADH: Thermophillic alcohol dehydrogenase
DHFR: Dihydroxyfolate reductase
SLO-1 Soybean lipoxygenase-1
Computational efficiency reqd.

3. Quantum Wavepacket Ab Initio Molecular Dynamics
Full Electron-nuclear TDSE:
TDSCF separation:
[electrons]
[Majority of the nuclei]
[Quantum-dynamical subsystem: (protons, excess electrons, etc..)]

4. Quantum Wavepacket Ab Initio Molecular Dynamics
S. S. Iyengar and J. Jakowski, J. Chem. Phys. 122 , (2005).
References…
[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)

5. 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) [Pioneered by Hoffman and Kouri, c.a. 1992]
Wavepacket propagation on a grid

6. 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
Force on P
V(RC,P,RQM;t) : the potential that quantum wavepacket experiences

7. 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)

8. Computational advantages to DAF quantum propagation scheme
Free Propagator:
is a banded, Toeplitz matrix:
Notice: all super- and sub-diagonals are the same.
Computational scaling: O(N) for large number of grid points

9. Some Advantages of ADMP
Currently 3-4 times faster than BO dynamics
Computational scaling O(N)
Hybrid functionals (more
accurate) : routine
Good adiabatic control
ADMP Spectrum!!

10. 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)

11. 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.

12. 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:
References…
S. S. Iyengar and J. Jakowski, J. Chem. Phys. 122 , (2005).

13. The Main Bottleneck: The quantum potential
The potential involved in the wavepacket propagation is required at every grid point!!
Trotter
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.

14. 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:
The parameters provide flexibility

15. 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.