Srinivasan S. Iyengar Department of Chemistry, Indiana University

1. 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 • Computational efficiency reqd.

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

4. [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) References… S. S. Iyengar and J. Jakowski, J. Chem. Phys. 122 , 114105 (2005). Quantum Wavepacket Ab Initio Molecular Dynamics

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 ofP acceleration of density matrix,P Force onP • 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. ADMP Spectrum!! 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

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 potentialexperienced 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 , 114105 (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.