Chapter - ADMP-Theory and Applications, PPT, Chemical Engineering Notes - Computer Science Engineering (CSE)

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

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Molecular dynamics on a single potential surface

• Parameterized force fields (e.g. AMBER, CHARMM)

  • Energy, forces: parameters obtained from experiment

  • Molecular motion: Newton’s 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.

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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

References

CP:   R. Car, M. Parrinello, Phys. Rev. Lett. 55 (22), 2471 (1985).

ADMP: Schlegel, et al. JCP, 114, 9758 (2001).   Iyengar, et al. JCP, 115,10291 (2001).   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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Atom-centered Density Matrix Propagation (ADMP)

• Construct a classical phase-space {{R,V,M},{P,W,μ}}

• 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,μ}} 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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μ effects an adjustment of time-scales:

• Consequence of μ : P changes slower with time: characteristic frequency adjusted

• But Careful - too large μ: non-physical

• Appropriate μ: approximate BO dynamics

• Bounds for μ: From a Hamiltonian formalism

• μ: also related to deviations from the BO surface

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“Physical” interpretation of μ : Bounds

• Commutator of the electronic Hamiltonian and density matrix: bounded by magnitude of μ

• Magnitude of μ : represents deviation from BO surface

• μ acts as an “adiabatic control parameter”

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Bounds on the magnitude of μ :

• The Lagrangian

• The Conjugate Hamiltonian (Legendre Transform)

Controlling μ: 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 -> dE/dP

  • Current: 3-4 times faster.

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Propagation of P: time-reversible propagation

• Velocity Verlet propagation of P

•  Propagation of W

• Classical dynamics in {{R,V},{P,W}} phase space

•   Λi and Λi+1 obtained iteratively:

  • Conditions: Pi+1 2 = Pi+1 and WiPi + PiWi = Wi   (next two slides)

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Idempotency (N-Representibility of DM):

• Given Pi2 = Pi, need Λi to find idempotent Pi+1

• Solve iteratively: Pi+12 = Pi+1

• Given Pi, Pi+1, Wi, Wi+1/2, need Λi+1 to find Wi+1

• Solve iteratively: Wi+1 Pi+1 + Pi+1  Wi+1 = Wi+1  

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Idempotency: To obtain Pi+1

• Given Pi2 = Pi, need to find indempotent Pi+1

• Guess: 

• Or guess: 

• Iterate Pi+1 to satisfy Pi+12 = Pi+1

• Rational for choice PiTPi + QiTQi above:

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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(Δt), W(Δt) : from analytical gradients and idempotency

• Similarly for R(Δt)

• 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/6-31+G** and BPBE/6-31+G**

• Castleman’s 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

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Spectroscopy: A recent quandry

Water Clusters: Important in Atmospheric Chemistry

Explains the experiments of M. A. Johnson

Bottom-right spectrum

From ADMP agrees well with expt: dynamical effects in IR spectroscopy

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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/6-31+G** and BPBE/6-31+G**

• Castleman’s 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 ?

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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

References

P. B. Miranda and Y. R. Shen, J. Phys. Chem. B, 103, 3292-3307 (1999).

M. J. Schultz, C. Schnitzer, D. Simonelli and S. Baldelli, Int. Rev. Phys. Chem. 19, 123-153 (2000)

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QM/MM treatment: ONIOM ADMP

Unified treatment of the full system within ADMP

(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: <J> = 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