Srinivasan S. Iyengar

1
Atom-centered Density Matrix Propagation (ADMP)
Theory and Applications

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

2
Outline

• Brief discussion of ab initio molecular dynamics
• Atom-centered Density Matrix Propagation (ADMP)
• Nut-n-bolts issues
• Some Results
• Novel findings for protonated water clusters
• QM/MM generalizations ion channels
• Gas phase reaction dynamics

3
Molecular dynamics on a single potential surface

• Parameterized force fields (e.g. AMBER, CHARMM)
• Energy, forces parameters obtained from
  experiment
• Molecular motion Newtons laws
• Works for large systems
• But hard to parameterize bond-breaking/formation
  (chemical reactions)
• Issues with polarization/charge
  transfer/dynamical effects
• Born-Oppenheimer (BO) Dynamics
• Solve electronic Schrödinger eqn (DFT/HF/post-HF)
  for each nuclear structure
• Nuclei propagated using gradients of energy
  (forces)
• Works for bond-breaking but computationally
  expensive
• Large reactive, polarizable systems Something
  like BO, but preferably less expensive.

4
Extended Lagrangian dynamics

• Circumvent Computational Bottleneck of BO
• Avoid repeated SCF electronic structure, not
  converged, but propagated
• Simultaneous propagation of electronic
  structure and nuclei adjustment of time-scales
• Car-Parrinello (CP) method
• Orbitals expanded in plane waves
• Occupied orbital coefficients propagated
• O(N3) computational scaling (traditionally)
• O(N) with more recent Wannier representations (?)
• Atom-centered Density Matrix Propagation (ADMP)
• Atom-centered Gaussian basis functions
• Electronic Density Matrix propagated
• Asymptotic linear-scaling with system size
• Allows the use of accurate hybrid density
  functionals
• suitable for clusters

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Atom-centered Density Matrix Propagation (ADMP)

• Construct a classical phase-space
  R,V,M,P,W,m
• The Lagrangian ( Kinetic minus Potential energy)

• P represented using atom-centered gaussian
  basis sets

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Euler-Lagrange equations of motion for ADMP

• Equations of motion for density matrix and nuclei

• Classical dynamics in R,V,M,P,W,m phase
  space
• Next few slide Forces, propagation equations,
  formal error analysis

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Nuclear Forces What Really makes it work
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Density Matrix Forces

• Use McWeeny Purified DM (3P2-2P3) in energy
  expression to obtain

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m effects an adjustment of time-scales

• Bounds for m From a Hamiltonian formalism
• m also related to deviations from the BO surface

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Physical interpretation of m Bounds

• Magnitude of m represents deviation from BO
  surface
• m acts as an adiabatic control parameter

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Bounds on the magnitude of m
Controlling m Deviations from BO surface and
adiabaticity
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Comparison with BO dynamics

• Born-Oppenheimer dynamics
• Converged electronic states.
• Approx. 8-12 SCF cycles / nuclear config.
• dE/dR not same in both methods

• ADMP
• Electronic state propagated classically no
  convergence reqd.
• 1 SCF cycle for Fock matrix -gt dE/dP
• Current 3-4 times faster.

References
Iyengar et al. Israel J. Chem. 7, 191, (2002).
Schlegel et al. JCP 114, 8694 (2002). Iyengar
and Frisch JCP 121, 5061 (2004).
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Propagation of P time-reversible propagation

• Velocity Verlet propagation of P

• Classical dynamics in R,V,P,W phase space
• Li and Li1 obtained iteratively
• Conditions Pi1 2 Pi1 and WiPi PiWi Wi
  (next two slides)

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Idempotency (N-Representibility of DM)

• Given Pi2 Pi, need Li to find idempotent Pi1
• Solve iteratively Pi12 Pi1
• Given Pi, Pi1, Wi, Wi1/2, need Li1 to find
  Wi1
• Solve iteratively Wi1 Pi1 Pi1 Wi1 Wi1
  

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Idempotency To obtain Pi1

• Given Pi2 Pi, need to find indempotent Pi1
• Guess
• Or guess
• Iterate Pi1 to satisfy Pi12 Pi1
• Rational for choice PiTPi QiTQi above

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Idempotency To obtain Wi1

• Given WiPi PiWi Wi, find appropriate Wi1
• Guess
• Iterate Wi1 to satisfy Wi1Pi1 Pi1Wi1
  Wi1

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How it all works

• Initial config. R(0). Converged SCF P(0)
• Initial velocities V(0) and W(0) flexible
• P(Dt), W(Dt) from analytical gradients and
  idempotency
• Similarly for R(Dt)
• And the loop continues

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Protonated Water Clusters

• Important systems for
• Ion transport in biological and condensed systems
• Enzyme kinetics
• Acidic water clusters Atmospheric interest
• Electrochemistry
• Experimental work
• Mass Spec. Castleman
• IR M. A. Johnson, Mike Duncan, M. Okumura
• Sum Frequency Generation (SFG) Y. R. Shen, M.
  J. Schultz and coworkers
• Lots of theory too Jordan, McCoy, Bowman, Klein,
  Singer (not exhaustive by any means..)
• Variety of medium-sized protonated clusters using
  ADMP

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Protonated Water Clusters Hopping via the
Grotthuss mechanism
True for 20, 30, 40, 50 and larger clusters
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(H2O)20H3O Magic number cluster

• Hydronium goes to surface 150K, 200K and 300K
  B3LYP/6-31G and BPBE/6-31G

• Castlemans experimental results
• 10 dangling hydrogens in cluster
• Found by absorption of trimethylamine (TMA)
• 10 dangling hydrogens consistent with our ADMP
  simulations
• But hydronium on the surface

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(H2O)20H3O A recent spectroscopic quandry
Theory
Experiment
J.-W. Shin, N. I. Hammer, E. G. Diken et al.,
Science 304, 1137 2004.
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Spectroscopy A recent quandry
Water Clusters Important in Atmospheric Chemistry
Bottom-right spectrum From ADMP agrees well with
expt dynamical effects in IR spectroscopy
Explains the experiments of M. A. Johnson
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Spectroscopy A recent quandry
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(H2O)20H3O Magic number cluster

• Hydronium goes to surface 150K, 200K and 300K
  B3LYP/6-31G and BPBE/6-31G

• Castlemans experimental results
• 10 dangling hydrogens in cluster
• Found by absorption of trimethylamine (TMA)
• 10 dangling hydrogens consistent with our ADMP
  simulations
• But hydronium on the surface

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Larger Clusters and water/vacuum interfaces
Similar results
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Predicting New Chemistry Theoretically
A Quanlitative explanation to the remarkable Sum
Frequency Generation (SFG) of Y. R. Shen, M. J.
Schultz and coworkers
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Protonated Water Cluster Conceptual Reasons for
hopping to surface
Hydrophobic and hydrophillic regions Directional
hydrophobicity (it is amphiphilic)

• H3O has reduced density around
• Reduction of entropy of surrounding waters

Is Hydronium hydrophobic ?
H2O coordination 4
H3O coordination 3
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Experimental results suggest this as well

• Y. R. Shen Sum Frequency Generation (SFG)
• IR for water/vapor interface shows dangling O-H
  bonds
• intensity substantially diminishes as acid conc.
  is increased
• Consistent with our results
• Hydronium on surface lone pair outwards, instead
  of dangling O-H
• acid concentration is higher on the surface
• Schultz and coworkers acidic moieties alter the
  structure of water/vapor interfaces

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QM/MM treatment ONIOM ADMP
Unified treatment of the full system within ADMP
(I)
(This talk will not overview the ONIOM scheme,
but the interested reader should look at the
reference below)
N. Rega, S. S. Iyengar, G. A. Voth, H. B.
Schlegel, T. Vreven and M. J. Frisch, J. Phys.
Chem. B 108 4210 (2004).
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Side-chain contribute to hop Eigen like
configuration possible using protein backbone
B3LYP and BLYP qualitatively different
results
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HCHO photodissociation

• Photolysis at 29500 cm-1 To S1 state
• Returns to ground state vibrationally hot
• Product rotationally cold, vibrationally excited
  H2
• And CO broad rotational distr ltJgt 42. Very
  little vib. Excitation
• H2CO ? H2 CO BO and ADMP at HF/3-21G,
  HF/6-31G

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Glyoxal 3-body Synchronous photo-fragmentation
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Conclusions

• ADMP powerful approach to ab initio molecular
  dynamics
• Linear scaling with system size
• Hybrid (more accurate) density functionals
• Smaller values for fictitious mass allow
• treatment of systems with hydrogens is easy (no
  deuteriums required)
• greater adiabatic control (closer to BO surface)
• Examples bear out the accuracy of the method

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Acknowledgment

• The work has enormously benefited from my former
  advisors and collaborators
• Greg Voth
• Berny Schlegel
• Gus Scuseria
• Mike Frisch
• At IU, people contributing to this work are
• Jacek Jakowski (post-doc)
• Isaiah Sumner (grad student)
• Xiaohu Li (grad student)
• Virginia E. Teige (Freshman)