1. A polarizable QM/MM model for the global (H 2 O) N – potential surface John M. Herbert Department of Chemistry Ohio State University IMA Workshop “Chemical Dynamics: Challenges & Approaches” Minneapolis, MN January 12, 2009

2. Acknowledgements Group members: Dr. Mary Rohrdanz Dr. Chris Williams Leif Jacobson Adrian Lange Ryan Richard Katie Martins Mark Hilkert CAREER $$ B.B.G. 2006 Dr. Chris Williams Leif Jacobson

3. n -1/3 0.00.20.40.60.8 -3.5 -3.0 -2.5 -2.0 -1.5 -0.5 0.0 1.0 2611 15 20 30 50 100 200 n I Neumark Johnson Experiments – VDE (eV) Isomer I VDE / eV n Johnson: CPL 297, 90 (1998) JCP 110, 6268 (1999) Coe/Bowen: JCP 92, 3980 (1990) Neumark: Science 307, 93 (2005) Experiment: Abrupt changes at n = 11 and n = 25 followed by smooth (?) extrapolation – VDE / eV = –3.30 + 5.73 n –1/3 (H 2 O) n – vertical electron binding energies (VEBEs) n -1/3 Isomer I ? VEBE / eV n —VEBE / eV

4. n -1/3 0.00.20.40.60.8 -3.5 -3.0 -2.5 -2.0 -1.5 -0.5 0.0 1.0 2611 15 20 30 50 100 200 n I Neumark Johnson Experiments Simulation: Internal Surface III II – VDE (eV) (H 2 O) n – vertical electron binding energies (VEBEs) n -1/3 —VEBE / eV Simulations: Barnett, Landman, Jorter JCP 88, 4429 (1988) CPL 145, 382 (1988) Theory (1980s): Surface to internal transition occurs between n = 32 and n = 64 ?

5. Interior (cavity) states are stable only for T ≤ 100 K or n ≥ 200 Turi & Rossky, Science 309, 914 (2005) simulated absorption spectra for (H 2 O) N – Theory (21st century version) Turi & Borgis, JCP 117, 6186 (2002) expt. J.V. Coe et al. Int. Rev. Phys. Chem. 27, 27 (2008)

6. V(anion) V(neut) VEBE E(anion) E(neut) Importance of the neutral water potential for water cluster anions Global minima

7. (H 2 O) 20 – isomers VEBE = 0.42 eV E(anion) = 0.00 eV E(neut) = 0.45 eV VEBE = 0.39 eV E(anion) = 0.01 eV E(neut) = 0.43 eV VEBE = 0.72 eV E(anion) = 0.03 eV E(neut) = 0.78 eV V(anion) V(neut) VEBE E(anion) E(neut) Global minima

8. e – correlation is more important for cavity states ∆ = E corr (anion) - E corr (neutral) (eV) VEBE (eV) correlation strength vs. e – binding motif C.F. Williams & JMH, J. Phys. Chem. A 112, 6171 (2008)

9. ∆ = E corr (anion) - E corr (neutral) (eV) VEBE (eV) surface states correlation strength vs. e – binding motif e – correlation is more important for cavity states C.F. Williams & JMH, J. Phys. Chem. A 112, 6171 (2008)

10. ∆ = E corr (anion) - E corr (neutral) (eV) VEBE (eV) cavity states correlation strength vs. e – binding motif e – correlation is more important for cavity states C.F. Williams & JMH, J. Phys. Chem. A 112, 6171 (2008)

11. Motivation for the new model The electron–water interaction potential has been analyzed carefully, but almost always used in conjunction with simple, non- polarizable water models (e.g., Simple Point Charge model, SPC). –L. Turi & D. Borgis, J. Chem. Phys. 114, 7805 (2001); 117, 6186 (2002) A QM treatment of electron–water dispersion via QM Drude oscillators provides ab initio quality VEBEs, but requires expensive many-body QM –F. Wang, T. Sommerfeld, K. Jordan, e.g.: J. Chem. Phys. 116, 6973 (2002) J. Phys. Chem. A 109, 11531 (2005) How far can we get with one-electron QM, using a polarizable water model that performs well for neutral water clusters? –AMOEBA water model: P. Ren & J. Ponder, J. Phys. Chem. B 107, 5933 (2003)

12. Electron–water pseudopotential O H H 1) Construct a repulsive effective core potential representing the H 2 O molecular orbitals: (H 2 O) – wavefn. nodeless pseudo-wavefn.

13. Electron–water pseudopotential O H H 1) Construct a repulsive effective core potential representing the H 2 O molecular orbitals: (H 2 O) – wavefn. nodeless pseudo-wavefn. 2) Use a density functional form for exchange attraction, e.g., the local density (electron gas) approximation: 3) In practice these two functionals are fit simultaneously

14. AMOEBA electrostatics Define multipole polytensors and interaction polytensors where i and j index MM atomic sites and Then the double Taylor series that defines the multipole expansion of the Coulomb interaction can be expressed as

15. Polarization In AMOEBA, polarization is represented via a linear-response dipole at each MM site: The total electrostatic interaction, including polarization, is where * * P. Ren & J.W. Ponder, J. Phys. Chem. B 127, 5933 (2003)

16. Polarization work The electric field at MM site i is Some work is required to polarize the dipole in the presence of the field: So the total electrostatic interaction is really

17. Electron–multipole interactions To avoid a “polarization catastrophe” at short range, we employ a damped Coulomb interaction:

18. Recovering a pairwise polarization model In general within our model we have: Imagine instead that each H 2 O has a single, isotropic polarizable dipole whose value is induced solely by q elec : Then the electron–water polarization interaction is In practice we use an attenuated Coulomb potential, the effect of which can be mimicked by an offset in the electron–water distance: This is a standard ad hoc polarization potential that has been used in may previous simulations.

19. Fourier Grid Simulations Simultaneous solution of where i = 1,..., N MM. c I = vector of grid amplitudes for the wave function of the Ith electronic state H depends on the induced dipoles. Solution of the linear-response dipole equation is done via iterative matrix operations. Dynamical propagation of the dipoles (i.e., an extended- Lagrangian approach) is another possibility. Solution of the Schrödinger equation is accomplished via Fourier grid method using a modified Davidson algorithm (periodically re-polarize the subspace vectors) The method is fully variational provided that all polarization is done self-consistently

20. A few comments about guns

21. Vertical e – binding energies for (H 2 O) N – Exchange/repulsion fit to (H 2 O) 2 – VEBE 34 clusters from N=2 to N=19 75 clusters from N=20 to N=35 Model VEBE / eV Ab initio VEBE / eV Non-polarizable model: Turi & Borgis, J. Chem. Phys. 117, 6186 (2002)

22. Vertical e – binding energies for (H 2 O) N – Exchange/repulsion fit to entire database of VEBEs Ab initio VEBE / eV Model VEBE / eV 34 clusters from N=2 to N=19 75 clusters from N=20 to N=35 Non-polarizable model: Turi & Borgis, J. Chem. Phys. 117, 6186 (2002)

23. Relative isomer energies

26. electron–water polarization (kcal/mol) Analysis

28. ∆ = E corr (anion) - E corr (neutral) (eV) VEBE (eV) surface states, n = 2–24 DFT geometries correlation strength vs. e – binding motif e – correlation is more important for cavity states C.F. Williams & JMH, J. Phys. Chem. A 112, 6171 (2008)

29. ∆ = E corr (anion) - E corr (neutral) (eV) VEBE (eV) surface states, n = 2–24 DFT geometries correlation strength vs. e – binding motif e – correlation is more important for cavity states C.F. Williams & JMH, J. Phys. Chem. A 112, 6171 (2008)

30. ∆ = E corr (anion) - E corr (neutral) (eV) VEBE (eV) surface states, n = 18–22 model Hamiltonian geometries correlation strength vs. e – binding motif e – correlation is more important for cavity states C.F. Williams & JMH, J. Phys. Chem. A 112, 6171 (2008)

31. ∆ = E corr (anion) - E corr (neutral) (eV) VEBE (eV) cavity states, n = 28–34 model Hamiltonian geometries correlation strength vs. e – binding motif e – correlation is more important for cavity states C.F. Williams & JMH, J. Phys. Chem. A 112, 6171 (2008)

32. ∆ = E corr (anion) - E corr (neutral) (eV) VEBE (eV) cavity states, n = 14, 24 DFT geometries correlation strength vs. e – binding motif e – correlation is more important for cavity states C.F. Williams & JMH, J. Phys. Chem. A 112, 6171 (2008)

33. SOMO pair correlation energy / meV 0.1 0.2 0.3 0.4 0.5 0.6 cavity state, VEBE = 0.58 eV fraction of total pairs 135791113151719 surface state, VEBE = 0.87 eV 135791113151719 mainly just a bunch of weak interactions many stronger correlations Quantifying electron–water dispersion C.F. Williams & JMH, J. Phys. Chem. A 112, 6171 (2008)

34. Putting it all together: water– water e – –water electrostatics fit to exchange/ repulsion