A polarizable QM/MM model for the global (H2O)N

– potential surface

John M. Herbert

Department of Chemistry

Ohio State University

IMA Workshop“Chemical Dynamics: Challenges & Approaches”

Minneapolis, MNJanuary 12, 2009

Acknowledgements

Group members:Dr. Mary RohrdanzDr. Chris WilliamsLeif JacobsonAdrian LangeRyan RichardKatie Martins Mark Hilkert

CAREER

$$

B.B.G.2006

Dr. ChrisWilliams Leif

Jacobson

n -1/3

0.0 0.2 0.4 0.6 0.8

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

1.0

261115

2030

50100

200n

INeumark

JohnsonExperiments

– V

DE

(eV

)

Isomer I

VD

E /

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

(H2O)n– vertical electron binding energies (VEBEs)

n-1/3

Isomer I

?

VE

BE

/ eV

n—V

EB

E /

eV

n -1/3

0.0 0.2 0.4 0.6 0.8

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

1.0

261115

2030

50100

200n

INeumark

JohnsonExperiments

Simulation:

Internal

Surface

III

II

– V

DE

(eV

)(H2O)n

– vertical electron binding energies (VEBEs)

n-1/3

—V

EB

E /

eV

Simulations: Barnett, Landman, Jorter JCP 88, 4429 (1988) CPL 145, 382 (1988)

Theory (1980s):Surface to internal transition

occurs between n = 32and n = 64

?

Interior (cavity) states are stable only for T ≤ 100 K or n ≥ 200

Turi & Rossky, Science 309, 914 (2005)

simulated absorption spectra for (H2O)N–

Theory (21st century version)

Turi & Borgis, JCP 117, 6186 (2002)

expt.

J.V. Coe et al. Int. Rev. Phys. Chem. 27, 27 (2008)

V(anion)

V(neut)

VEBE

E(anion)

E(neut)

Importance of the neutral water potential for water cluster anions

Global minima

(H2O)20– isomers VEBE = 0.42 eV

E(anion) = 0.00 eVE(neut) = 0.45 eV

VEBE = 0.39 eV E(anion) = 0.01 eVE(neut) = 0.43 eV

VEBE = 0.72 eV E(anion) = 0.03 eVE(neut) = 0.78 eV

V(anion)

V(neut)

VEBE

E(anion)

E(neut)

Global minima

e– correlation is more important for cavity states

∆ = Ecorr(anion) - Ecorr(neutral)

(eV)

VEBE(eV)

correlation strength vs. e– binding motif

C.F. Williams & JMH,J. Phys. Chem. A 112, 6171 (2008)

∆ = Ecorr(anion) - Ecorr(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)

∆ = Ecorr(anion) - Ecorr(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)

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)

Electron–water pseudopotential

O

H H

1) Construct a repulsive effective core potential representing the H2O molecular orbitals:

(H2O)– wavefn.

nodeless pseudo-wavefn.

Electron–water pseudopotential

O

H H

1) Construct a repulsive effective core potential representing the H2O molecular orbitals:

(H2O)– wavefn.

nodeless pseudo-wavefn.

2) Use a density functional form for exchange attraction, e.g., the localdensity (electron gas) approximation:

3) In practice these two functionals are fit simultaneously

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 theCoulomb interaction can be expressed as

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)

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

Electron–multipole interactions

To avoid a “polarization catastrophe” at short range, we employ a dampedCoulomb interaction:

Recovering a pairwise polarization modelIn general within our model we have:

Imagine instead that each H2O has a single, isotropic polarizable dipole whosevalue is induced solely by qelec:

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 mayprevious simulations.

Fourier Grid Simulations

• Simultaneous solution of

where i = 1, ..., NMM.cI = vector of grid amplitudes for the wave function of

the Ith electronic stateH depends on the induced dipoles.

• Solution of the linear-response dipole equation is done via iterative matrixoperations. Dynamical propagation of the dipoles (i.e., an extended-Lagrangian approach) is another possibility.

• Solution of the Schrödinger equation is accomplished via Fourier gridmethod using a modified Davidson algorithm (periodically re-polarize thesubspace vectors)

• The method is fully variational provided that all polarization is done self-consistently

A few comments about guns

Vertical e– binding energies for (H2O)N–

Exchange/repulsion fit to (H2O)2– VEBE

34 clusters from N=2 to N=19 75 clusters from N=20 to N=35

Mod

el V

EB

E /

eV

Ab initio VEBE / eV

Non-polarizable model: Turi & Borgis, J. Chem. Phys. 117, 6186 (2002)

Vertical e– binding energies for (H2O)N–

Exchange/repulsion fit to entire database of VEBEs

Ab initio VEBE / eV

Mod

el V

EB

E /

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)

Relative isomer energies

Relative isomer energies

Relative isomer energies

electron–water polarization

(kcal/mol)

Analysis

∆ = Ecorr(anion) - Ecorr(neutral)

(eV)

VEBE (eV)

surface states, n = 2–24DFT 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)

∆ = Ecorr(anion) - Ecorr(neutral)

(eV)

VEBE (eV)

surface states, n = 2–24DFT 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)

∆ = Ecorr(anion) - Ecorr(neutral)

(eV)

VEBE (eV)

surface states, n = 18–22model 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)

∆ = Ecorr(anion) - Ecorr(neutral)

(eV)

VEBE (eV)

cavity states, n = 28–34model 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)

∆ = Ecorr(anion) - Ecorr(neutral)

(eV)

VEBE (eV)

cavity states, n = 14, 24DFT 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)

SOMO pair correlation energy / meV

0.1

0.2

0.3

0.4

0.5

0.6

cavity state, VEBE = 0.58 eV

frac

tion

of t

otal

pai

rs

1 3 5 7 9 11 13 15 17 19

surface state, VEBE = 0.87 eV

1 3 5 7 9 11 13 15 17 19

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)

Putting it all together:

water–water

e––waterelectrostatics

fit to exchange/repulsion