1. Ab Initio and Effective Fragment Potential Dynamics Heather M. Netzloff and Mark S. Gordon Iowa State University

2. Outline Computer simulations –Treatment of liquids Effective Fragment Potential Method Molecular Dynamics in GAMESS Applications Future Plans

3. Microscopic details Macroscopic properties Test theories and compare with experiments Simulate under extreme conditions unattainable to experimentalists Laws of Statistical Mechanics Computer simulation: Motivation

4. Computer simulations: Condensed phase Treatment of fluid-like states and solvation is essentially a many-particle problem… –Importance of accuracy and reliability of the method used Limited by the size of the system and computational costs GOAL: model CLUSTER to BULK behavior

5. Condensed phase For accuracy, preference = ab initio methods, BUT… –Computationally expensive –Only realistic for small systems Solvation Models: –Clusters: Ab initio, DFT, Effective Fragment Potential,… –Bulk: Continuum methods, TIPnP, SPC/E,… Limited number of methods which span the cluster  bulk gap well

6. The Effective Fragment Potential Method 1,2 Treatment of discrete solvent effects EFP1/HF: –Developed at Hartree Fock level of theory (DH(d,p)) –Reproduces RHF/ab initio results… E system = E ab initio + E interaction Standard Ab Initio Calculation Effective Fragment Potential Calculation 1 Day et. al. J. Chem. Phys. 105,1968(1996). 2 Gordon et. al. J. Phys. Chem A. 105,2(2001).

7. EFP: Menshutkin Reaction R 3 N + RX  R 4 N + X - –Reaction rate increases with polarity of solvent –Study process of solvation on ion formation EFP test (Simon Webb): NH 3 + CH 3 Br –Add EFP waters; AI solute/solvent (RHF/DZVP)

8. EFP: Water Clusters Monte Carlo Simulated Annealing with EFP (Paul Day, Grant Merrill, Ruth Pachter) –N = 6 - 20 Water hexamer –Re-optimization with HF, MP2 –CCSD(T) single points

9. Effect of “solvent” molecules (fragments) added as one-electron terms to ab initio Hamiltonian, H AR V = Fragment interaction potential Effective Fragment Potential Method Ab initio region EFP region

10. Effective Fragment Potential Method Interaction Potential Terms--EFP1/HF Electrostatics: Coulomb interactions –Multipolar expansion up to octupoles –Screening to account for overlapping charge densities –Expansion points: nuclear centers and bond midpoints Polarization: Dipole/induced dipole potential –Induced dipoles iterated to self-consistency –Centered on bonding and lone-pair localized molecular orbitals Exchange-repulsion/charge transfer: Remainder term –EFP1: exponential functions optimized by fitting procedure Limited to water –Located at fragment atom centers and center of mass

11. Effective Fragment Potential Method Solute explicitly treated with ab initio wavefunction of choice; remainder treated as effective fragments  QM/MM method Multipoles and polarizabilities –Determined from ab initio calculations on a single solvent molecule –Potential can be systematically improved! Exchange repulsion/charge transfer term –EFP1/HF: Currently fit to functional form  solvent specific –EFP2: generalize for any solvent NO FITS: Exchange repulsion from LMO intermolecular overlap and kinetic energy integrals NEW!! EFP1/DFT (B3LYP) All 3 terms are calculated independently from each other Only ONE-ELECTRON integrals are needed –MUCH less CPU intensive than quantum mechanics

12. EFP/Molecular Dynamics Do i = 1, number of simulation steps  Calculate PE and gradients  forces for particle at time t  Solve equations of motion for each particle to obtain KE at time t and new positions at time t + dt End do First test of EFP to reproduce bulk behavior with its implementation with Molecular Dynamics Basic MD Simulation loop: (user specifies time step size, dt, and length of simulation) MD is dependent upon the accuracy and reliability of the potential used to generate gradients/forces…

13. Molecular Dynamics Solve Newton’s equations of motion –Require both ENERGY and GRADIENT from PE routine –Integrate simultaneously for all atoms in the system to generate a trajectory for each atom –Perform integration in SMALL time steps (usually 1-10 fs) Time step size Energy conservation

14. Current MD Implementation in GAMESS EFP fragment-fragment interaction (EFP-EFP MD) –Classical interaction terms –EFP: EFP1/HF, EFP1/HF, EFP2 Ab initio interaction (AI MD) –Ab initio interaction –Basis set and level of theory of choice and availability in GAMESS

15. Current MD Implementation: EFP-EFP MD Integration EFP treats water as a rigid molecule –Center of mass motion: Translational motion  leapfrog integration algorithm Rotational motion  quaternions and modified leapfrog integration algorithm Water model parameters (calculated based on ab initio methods) are stored in GAMESS or given with input file –After each move all information must be rotated/translated along with the fragment

16. Current MD Implementation: EFP-EFP MD T 0 = desired/bath temperature T(t) = temperature obtained from translational or rotational kinetic energy at time t Ensembles: NVE (constant energy) NVT (constant temperature) : based on velocity scaling –Separate treatment for translational and rotational velocity components –Rescale by:

17. Periodic Boundary Conditions Minimum Image Convention Current MD Implementation: EFP-EFP MD Periodic boundary conditions –Introduced to minimize surface effects Minimum image convention –Restrict infinite number of terms in force calculation to only the closest N -1 periodic images –Based on center-of-mass (COM) positions

18. Current MD Implementation: AI MD Each atom is treated independently –Leapfrog integration algorithm NVE and NVT (velocity scaling) ensembles No periodic boundary conditions or minimum image convention currently implemented

19. EFP-EFP MD Applications Preparation of the system Heating: –NVE ensemble: Initial target temp = 50 K Initial quaternions chosen randomly Initial translational velocities sampled from Maxwell-Boltzmann distribution at 100 K; angular velocities start from zero –Use these coordinates to start simulation at 100 K –Same procedure to start 200 K and 300 K runs Equilibration--TARGET TEMP = 300 K: –Coordinates taken from 300 K heating run; initial translational velocities sampled from Maxwell-Boltzmann distribution at 600 K –NVT ensemble at 300K velocities rescaled only if they are outside 300 + 5 K (scaling factors not allowed to exceed 1.3) Production: –Initial coordinates, velocities, and quaternions from equilibration –NVE and/or NVT ensemble: measure observables of interest

20. Compare with other water potentials… Example: –SPC/E (Simple Extended Point Charge) 4 Point charges on O and H sites Reparameterized SPC model to include polarization Values for  O,  O, and q (point charge) are determined by parameterization with experimental density and E vap as targets EFP-EFP MD Applications +0.4238e -2.0q H 4 Berendsen et. al. J. Phys. Chem. 92,6269(1987).

21. EFP-EFP MD Applications Radial Distribution Function (RDF): –Measures the number of atoms a distance r from a given atom compared with the number at the same distance in an ideal gas at the same density –Gives information on how molecules pack in ‘shells’ of neighbors, as well as average structure –Can be measured spectroscopically (X-ray or neutron diffraction) –Test theoretical models versus experimental results 3 SITE-SITE radial distribution functions for water: –gOO(r), gOH(r), gHH(r)

22. gOO(r)--EFP1/HF gOO(r) r (Angstroms) Initial structure: 62 EFP waters, 26 ps equilibration Timestep size = 1 fs, Simulation = 5000 fs EFP1/HF--NVE EFP1/HF--NVT SPC/E--NVT Exp (THG) Exp (THG): X-ray; Sorenson et. al. J. Chem. Phys. 113,9149(2000).

23. gOO(r)--EFP1/DFT gOO(r) r (Angstroms) EFP/DFT--NVE EFP/DFT--NVT SPC/E--NVT Exp (THG) Initial structure: 62 EFP waters, 10 ps equilibration Timestep size = 1 fs, Simulation = 5000 fs

24. gOO(r) Error: 62 water analysis Error = gOO(exp)-gOO(X) r (Angstroms) X X X X EFP/DFT--NVE EFP1/HF--NVT SPC/E--NVT X Exp peak/valley locations

25. gOO(r)--EFP1/HF: 62 water analysis with Monte Carlo Initial structure: 62 EFP waters, 1.0 ps MD equilibration RDF measured with MC accepted structures (2 criteria for EFP MC) 6790 accepted (100,000) gOO(r) r (Angstroms) gOO(r) EFP1/HF--MC Exp (THG) Polynomial Fit (4th order) to EFP

26. gOO(r)--EFP1/HF: 512 water analysis gOO(r) r (Angstroms) X X X X X Initial structure: 512 EFP waters, 7.5 ps equilibration Timestep size = 1 fs, Simulation = 5000 fs r (Angstroms) Error = gOO(exp)-gOO(X) EFP1/HF--NVT SPC/E--NVT Exp (THG)

27. gOH(r): EFP1/HF, EFP1/DFT, SPC/E 62 waters EFP/DFT--NVE EFP/HF--NVT SPC/E--NVT Exp (ND) gOH(r) r (Angstroms) Exp (ND): Neutron Diffraction; Soper et. al.

28. gHH(r): EFP1/HF, EFP1/DFT, SPC/E 62 waters EFP/DFT--NVE EFP/HF--NVT SPC/E--NVT Exp (ND) gHH(r) r (Angstroms)

29. Timings: Energy + Gradient calculation Method* 20 water molecules 62 water molecules 122 water molecules 512 water molecules Ab initio**3.19 hrs--- ~157 yrs*** EFP23.3 sec26.1 sec95.3 sec26.8 min EFP1/HF0.2 sec2.6 sec5.1 sec97.8 sec SPC/E0.02 sec 0.1 sec0.7 sec * Run on 1200 MHz Athlon/Linux machine **Ab initio: DZP basis set, *** Assuming N 4 scaling

30. Future Plans Addition of minimum image convention to EFP2 implementation and AI MD –Coordinates, not only distances, must be manipulated Treatment of long range forces –First layer: Ewald sum method for charge-charge interaction –Second layer: Fast Multipole Method –Use continuum versus minimum image convention/periodic boundary conditions Utilization/development of parallel algorithms –Utilize parallel AI GAMESS code for calculation of AI energy and gradient –Parallelization of the EFP method

31. Acknowledgements Jon Sorenson, Grant Merrill, Mark Freitag $$$$$ DOE Computational Science Graduate Fellowship Thank You!