1. Srinivasan S. Iyengar Department of Chemistry and Department of Physics, Indiana University Atom-centered Density Matrix Propagation (ADMP): Theory and Applications

2. Iyengar Group, Indiana University  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 Outline

3. Iyengar Group, Indiana University 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.

4. Iyengar Group, Indiana University  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(N 3 ) 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 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). References… Extended Lagrangian dynamics

5. Iyengar Group, Indiana University Atom-centered Density Matrix Propagation (ADMP)  Construct a classical phase-space {{R,V,M},{P,W,  }}  The Lagrangian (= Kinetic minus Potential energy) Nuclear KE “Fictitious” KE of P Energy functional Lagrangian Constraint for N-representability of P : Idempotency and Particle number  P : represented using atom-centered gaussian basis sets

6. Iyengar Group, Indiana University Euler-Lagrange equations of motion for ADMP  Equations of motion for density matrix and nuclei  Classical dynamics in {{R,V,M},{P,W,  }} phase space  Next few slide: Forces, propagation equations, formal error analysis acceleration of density matrix, P Force on P “Fictitious” mass of P

7. Iyengar Group, Indiana University Nuclear Forces: What Really makes it work Pulay’s moving basis terms Hellman-Feynman contributions Contributions due to [F,P]  0. Part of non-Hellman-Feynman S=U T U, Cholesky or Löwdin

8. Iyengar Group, Indiana University Density Matrix Forces:  Use McWeeny Purified DM (3P 2 -2P 3 ) in energy expression to obtain

9. Iyengar Group, Indiana University  effects an adjustment of time-scales:  Bounds for  : From a Hamiltonian formalism also related to deviations from the BO surface   : also related to deviations from the BO surface  Consequence of  : P changes slower with time: characteristic frequency adjusted  But Careful - too large  : non-physical  Appropriate  : approximate BO dynamics  But Careful - too large  : non-physical  Consequence of  : P changes slower with time: characteristic frequency adjusted Direction of Increasing Frequency

10. Iyengar Group, Indiana University “Physical” interpretation of   Bounds  Commutator of the electronic Hamiltonian and density matrix: bounded by magnitude of   Magnitude of  : represents deviation from BO surface   acts as an “adiabatic control parameter” Iyengar et al. Israel J. Chem. 7, 191, (2002). Reference…

11. Iyengar Group, Indiana University Bounds on the magnitude of   The Conjugate Hamiltonian (Legendre Transform)  The Lagrangian Controlling   Deviations from BO surface and adiabaticity Iyengar et al. JCP. 115,10291 (2001). Reference…

12. Iyengar Group, Indiana University 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. References… Iyengar et al. Israel J. Chem. 7, 191, (2002). Schlegel et al. JCP 114, 8694 (2002). Iyengar and Frisch JCP 121, 5061 (2004).

13. Iyengar Group, Indiana University Propagation of Propagation of P: time-reversible propagation  Velocity Verlet propagation of P  Classical dynamics in {{R,V},{P,W}} phase space   i and  i+1 obtained iteratively: – Conditions: P i+1 2 = P i+1 and W i P i + P i W i = W i (next two slides)  Propagation of W

14. Iyengar Group, Indiana University Idempotency (N-Representibility of DM):  Given P i 2 = P i, need  i to find idempotent P i+1  Solve iteratively: P i+1 2 = P i+1  Given P i, P i+1, W i, W i+1/2, need  i+1 to find W i+1  Solve iteratively: W i+1 P i+1 + P i+1 W i+1 = W i+1

15. Iyengar Group, Indiana University Idempotency: To obtain P i+1  Given P i 2 = P i, need to find indempotent P i+1  Guess:  Or guess:  Iterate P i+1 to satisfy P i+1 2 = P i+1  Rational for choice P i TP i + Q i TQ i above:

16. Iyengar Group, Indiana University Idempotency: To obtain W i+1  Given W i P i + P i W i = W i, find appropriate W i+1  Guess:  Iterate W i+1 to satisfy W i+1 P i+1 + P i+1 W i+1 = W i+1

17. Iyengar Group, Indiana University How it all works …  Initial config.: R(0). Converged SCF: P(0)  Initial velocities V(0) and W(0) : flexible  P(  t), W(  t) : from analytical gradients and idempotency  Similarly for R(  t)  And the loop continues…

18. Iyengar Group, Indiana University 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 ADMP treatment of protonated water clusters: Iyengar, et al. JCP, 123, 084309 (2005). Iyengar et al. Int. J. Mass Spec. 241, 197 (2005). Iyengar JCP 123, 084310, (2005). References…

19. Iyengar Group, Indiana University Protonated Water Clusters: Hopping via the Grotthuss mechanism True for 20, 30, 40, 50 and larger clusters…

20. Iyengar Group, Indiana University (H 2 O) 20 H 3 O + : Magic number cluster  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  Hydronium goes to surface: 150K, 200K and 300K: B3LYP/6-31+G** and BPBE/6-31+G**

21. Iyengar Group, Indiana University (H 2 O) 20 H 3 O + : A recent spectroscopic quandry J.-W. Shin, N. I. Hammer, E. G. Diken et al., Science 304, 1137 2004. Theory Experiment

22. Iyengar Group, Indiana University 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

23. Iyengar Group, Indiana University ADMP Spectrum!! Iyengar et al. JCP, 123, 084309 (2005) Spectroscopy: A recent quandry

24. Iyengar Group, Indiana University (H 2 O) 20 H 3 O + : Magic number cluster  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  Hydronium goes to surface: 150K, 200K and 300K: B3LYP/6-31+G** and BPBE/6-31+G**

25. Iyengar Group, Indiana University Larger Clusters and water/vacuum interfaces: Similar results

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

27. Iyengar Group, Indiana University Protonated Water Cluster: Conceptual Reasons for “hopping” to surface H 3 O + has reduced density around Reduction of entropy of surrounding waters H 2 O coordination 4H 3 O + coordination =3 Is Hydronium hydrophobic ? Hydrophobic and hydrophillic regions: Directional hydrophobicity (it is amphiphilic)

28. Iyengar Group, Indiana University 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 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) References…

29. Iyengar Group, Indiana University 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).

30. Iyengar Group, Indiana University Side-chain contribute to hop “Eigen” like configuration possible using protein backbone B3LYP and BLYP: qualitatively different results

31. Iyengar Group, Indiana University  Photolysis at 29500 cm -1 : To S 1 state Returns to ground state vibrationally hot Product: rotationally cold, vibrationally excited H 2 And CO broad rotational distr: = 42. Very little vib. Excitation  H 2 CO  H 2 + CO: BO and ADMP at HF/3-21G, HF/6-31G** HCHO photodissociation

32. Iyengar Group, Indiana University Glyoxal 3-body Synchronous photo-fragmentation

33. Iyengar Group, Indiana UniversityConclusions  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

34. Iyengar Group, Indiana UniversityAcknowledgment  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)