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Srinivasan S. Iyengar Department of Chemistry, Indiana University

Atom-centered Density Matrix Propagation (ADMP): Theory and Application to protonated water clusters and water/vacuum interfaces. Srinivasan S. Iyengar Department of Chemistry, Indiana University.

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Srinivasan S. Iyengar Department of Chemistry, Indiana University

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  1. Atom-centered Density Matrix Propagation (ADMP): Theory and Application to protonated water clusters and water/vacuum interfaces Srinivasan S. Iyengar Department of Chemistry, Indiana University

  2. This presentation is meant to be a quick outline of ADMP. You should read the related papers to get more complete understanding • Brief outline of ab initio molecular dynamics • Atom-centered Density Matrix Propagation (ADMP) • Results: • Novel findings for protonated water clusters • Preliminary results for ion-transport through biological channels • Nut-n-bolts issues

  3. Molecular dynamics in Chemistry • Molecular motion and structure determine properties: • Spectroscopic properties • Predicting Molecular Reactivity • Computationally molecular dynamicssimulates molecular motions: • determine properties from correlation functions • To Simulate molecular motions: • Need Energy of conformation • Forces to move nuclei: Simulate nuclear motion

  4. Methods for molecular dynamics on a single potential surface • Parameterized force fields (e.g. AMBER, CHARMM) • Energy, forces: parameters obtained from experiment • Molecules moved: 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 within some approximation for each nuclear structure • Nuclei are propagated using gradients (forces) • Works for bond-breaking but computationally expensive • Large reactive, polarizable systems: We need something like BO, but less expensive.

  5. Atom-centered Density Matrix Propagation (ADMP) : An Extended Lagrangian approach • Circumvent Computational Bottleneck of BO • Avoid repeated SCF for electronic SE • electronic structure, not converged,but propagated • “Simultaneous” propagation of electronic structure with nuclei: an adjustment of time-scales

  6. Lagrangian Constraint for N-representability of P: Idempotency and Particle number Nuclear KE “Fictitious” KE of P Energy functional Atom-centered Density Matrix Propagation (ADMP) • Construct a classical phase-space {{R,V,M},{P,W,m}} • TheLagrangian (= Kinetic minus Potential energy) • P: representedusing atom-centered gaussian basis sets

  7. acceleration of density matrix,P “Fictitious” mass ofP Force onP Euler-Lagrange equations of motion • Equations of motion for density matrix and nuclei • Classical dynamics in {{R,V,M},{P,W,m}}phase space • Solutions obtained using velocity Verlet integrator

  8. Direction of Increasing Frequency • Consequence of m: P changes slower with time: characteristic frequency adjusted • Consequence of m: P changes slower with time: characteristic frequency adjusted • Consequence of m: P changes slower with time: characteristic frequency adjusted • But Careful - too largem:non-physical • But Careful - too largem:non-physical • Appropriate m: approximate BO dynamics meffects an adjustment of time-scales: • Bounds for m: From a Hamiltonian formalism • m: also related to deviations from the BO surface

  9. Commutator of the electronic Hamiltonian and density matrix: boundedbymagnitude ofm “Physical” interpretation of m: • Magnitude of m: represents deviation from BO surface • macts as an “adiabatic control parameter”

  10. The Lagrangian • The Conjugate Hamiltonian (Legendre Transform) Bounds on the magnitude of m : • By controlling m:control deviations from BO surface and adiabaticity

  11. Hellman-Feynman contributions Pulay’s moving basis terms S=UTU, Cholesky or Löwdin Contributions due to [F,P]  0. Part of non-Hellman-Feynman Nuclear Forces: What Really makes it work

  12. Some Advantages of ADMP ADMP: • Currently 3-4 times faster than BO dynamics • Improvements will allow ADMP ~ 10 times faster • Computational scaling O(N) • Hybrid functionals (more accurate) : routine • Smaller m: Greater adiabatic control • QM/MM: localized bases: natural

  13. Born-Oppenheimer dynamics: Converged electronic states. Approx. 8-12 SCF cycles / nuclear config. dE/dR not same in both methods ADMP: Electronic statepropagatedclassically : no convergence reqd. 1 SCF cycle : for Fock matrix -> dE/dP Current: 3-4 times faster.  10 times Comparison with BO dynamics Reference… H. B. Schlegel, S. S. Iyengar, X. Li, J. M. Millam, G. A. Voth, G. E. Scuseria, M. J. Frisch, JCP, In Press.

  14. References… CP: R. Car, M. Parrinello, Phys. Rev. Lett. 55 (22), 2471 (1985). ADMP: H. B. Schlegel, J. Millam, S. S. Iyengar, G. A. Voth, A. D. Daniels, G. E. Scuseria, M. J. Frisch, JCP, 114, 9758 (2001). S. S. Iyengar, H. B. Schlegel, J. Millam, G. A. Voth, G. E. Scuseria, M. J. Frisch, JCP, 115,10291 (2001). Comparison with Car-Parrinello : Slide 0 • Atom-centered Density Matrix Propagation (ADMP) approach using Gaussian basis sets • Atom-centered Gaussian basis functions • Fewer basis functions for molecular systems • Electronic Density Matrix propagated • Asymptotic linear-scaling with system size • Car-Parrinello (CP) method • Orbitals expanded in plane waves • Occupied orbital coefficients propagated • O(N3) computational scaling

  15. ADMP: • Computational scaling O(N) • Hybrid functionals (more accurate) : routine • Smaller Greater adiabatic control: can use H2O • Properties independent of # References… Comparison with Car-Parrinello : Slide 1 Plane-wave CP: • Computational scaling O(N3) • Pure functionals (e.g. BLYP) Hybrid (B3LYP): expensive • Adiabatic control limited : larger : D2O for H2O • Properties depend on § § Scandolo and Tangney, JCP. 116, 14 (2002). # Schlegel, Iyengar, Li, Millam, Voth, Scuseria, Frisch, JCP, 117, 8694 (2002).

  16. ADMP: • Fewer basis fns. • QM/MM: localized bases: natural • Pseudopotentials not required for core • Important for metals e.g., redox species and enzyme active sites Comparison with Car-Parrinello : Slide 2 Plane-wave CP: • Larger no. of basis fns. • QM/MM: Plane-waves enter MM region • Pseudopotentials required for core

  17. Propagation of W Propagation of P: a time-reversible propagation scheme • Velocity Verlet propagation of P • Classical dynamics in {{R,V},{P,W}}phase space • Li and Li+1 obtained iteratively: • Conditions: Pi+12 = Pi+1 and WiPi + PiWi = Wi

  18. 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:

  19. Idempotency: To obtain Wi+1 • Given WiPi + PiWi = Wi, find appropriate Wi+1 • Guess: • Iterate Wi+1 to satisfy Wi+1Pi+1 + Pi+1Wi+1 = Wi+1

  20. Density Matrix Forces: • Use McWeeny Purified DM (3P2-2P3) in energy expression to obtain

  21. Hellman-Feynman contributions Pulay’s moving basis terms S=UTU, Cholesky or Löwdin Contributions due to [F,P]  0. Part of non-Hellman-Feynman Nuclear Forces: What Really makes it work

  22. Idempotency (N-Representibility of DM): • Given Pi2 = Pi, need Li to find idempotent Pi+1 • Solve iteratively: Pi+12 = Pi+1 • Given Pi, Pi+1, Wi, Wi+1/2, need Li+1 to find Wi+1 • Solve iteratively: Wi+1Pi+1 + Pi+1 Wi+1=Wi+1

  23. 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…

  24. Results • For Comparison with Born-Oppenheimer dynamics • Formaldehyde photo-dissociation • Glyoxal photo-dissociation • New Results for Protonated Water clusters • Protonated water wire • Ion transport through gramicidin ion channels

  25. 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, M. Okumura • Sum Frequency Generation (SFG) : Y. R. Shen, M. J. Schultz and coworkers • Variety of medium-sized protonated clusters using ADMP

  26. Protonated Water Clusters: Hopping via the Grotthuss mechanism True for 20, 30, 40, 50 and larger clusters…

  27. (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

  28. Larger Clusters and water/vacuum interfaces: Similar results

  29. Predicting New Chemistry: Theoretically A Quanlitative explanation to the remarkable Sum Frequency Generation (SFG) of Y. R. Shen, M. J. Schultz and coworkers

  30. 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

  31. 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

  32. 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) Experimental results seem to 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

  33. Protonated Water Cluster: Conceptual Reasons for “hopping” to surface Hydrophobic and hydrophillic regions: Directional hydrophobicity H3O+ has reduced density around Reduction of entropy of surrounding waters Is Hydronium hydrophobic ? H2O coordination 4 H3O+coordination =3

  34. Protonated Water Clusters: progress of the proton Most protonated water closer to the surface as simulation progresses 3 ang

  35. Protonated Water Cluster: Radial Distribution Functions • O*-O Radial Distribution function peaks: • BLYP : ~2.45 Angstrom and ~2.55 Angstrom • B3LYP : ~2.45 • Zundel [H5O2+]: ~2.45 • Eigen [H9O4+]: ~2.55 • BLYP : Zundel and Eigen • B3LYP: Zundel • BLYP : proton more delocalized

  36. Protonated Water Wire • Proton hopping across “water wire” • Model for proton transfer in: • ion channels • Enzymes • liquids • DFT - B3LYP / 6-31+G** / 300K / ~1 ps • Basis set / functional: good water-dimer properties

  37. Protonated Water Wire • Protonated Oxygen peak ~ 2.4 Angstrom • Non-protonated Oxygen peaks : spread (about 2.8 Ang.) • Results consistent with Brewer, Schmidt and Voth using EVB model

  38. Water wire to Ion Channels: QM/MM ADMP • Proton transport through ion-channel • QM/MM approach to AIMD

  39. QM/MM treatment of bio-systems (I) Unified treatment of the full system within ADMP

  40. ONIOM: Energy partitioning Link atom coordinates are expressed in terms of their neighbors: Link atoms factor out

  41. Preliminary results: Side-chain contributions to hop: B3LYP and BLYP: qualitatively different results

  42. Protonated Water Cluster v/s Protonated Water Wire • Cluster: Proton goes to surface • Wire: Proton tends to center • Why? • Cluster: • H3O+ coordination number 3 • Lone pair has reduced water density around • Wire: • 2 H-bonds at center: 1 H-bond at end • H3O+ lone pair has reduced density: center and edge • Reduced density not a factor: Number of H-bonds is

  43. 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**

  44. Glyoxal 3-body Synchronous photo-fragmentation

  45. What about BSSE? • Due to: • difference in instantaneous incompleteness in basis set. • Atom centered nature of basis set (not present in plane-waves). • Worst when neighbouring atoms leave completely (ie, total dissociation). • Present case: proton hopping, no complete dissociation (replaced by new proton). • Expected to be less. • Dominant sources of errors: • Off the BO surface • DFT functional

  46. What about BSSE? • Difference in completeness of basis set. • Worst when neighbouring atoms leave completely (ie, total dissociation). • Dynamics without total dissociation: • Effect expected to be less. • Dominant sources of errors: • DFT functional

  47. Chloride-Water Cluster • Conservation Properties : Fictitious KE = Change in Fict. KE ~ 0.0002% of total Energy

  48. Chloride-Water Clusters: Red-shifts • Bend: ~ 1600 cm-1, Stretch ~3400 & ~3600 cm-1 • Exptal. O-H Red Shift for Cl- (H2O)1 : • 3130 cm -1 Ar matrix : M. A. Johnson, Yale University • 3285 cm -1 CCl4 matrix : M. Okumura, CalTech • Critical to use hydrogens in these simulations DFT – B3LYP / 6-31G*

  49. Chloride-Water Cluster: Cl- (H2O)25 ADMP dynamics oscillates about the BO result.

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