Model Hamiltonians for Electron-Molecule Interactions

K. D. Jordan

Department of Chemistry

University of Pittsburgh

Pittsburgh, PA

Anions 2007, Park City Utah

Electronically excited state of (H2O)45-

Dominant form of (H2O)13-

National Science Foundation

Department of Energy

Acknowledgements and Projects

Excess electron/water clusters

A. DeFusco – Pitt.

F. Wang – Boston University

T. Sommerfeld – Southeastern Louisiana Univ.

K. Diri – Univ. Southern California

M.-K. Tsai – Brookhaven Natl. Lab.

M. A. Johnson - Yale

e- at TiO2/water interfaces

H. Petek, J. Zhao - Pitt.

History on evolution of my interest in e--molecule interactions

~1968 - Interest in photoelectron spectroscopy ever since hearing a seminar by Edgar Heilbronner when I was an undergrad at Northeastern

1970 - On to MIT to do Ph.D. studies with Bob Silbey

1971 - Meet Jack Simons who was an NSF Postdoc at MIT

1972 – Jack invites me to spend the summer at Utah

Worked on EOM theory for electron affinities

Realization that EA’s were not available for many molecules

1974 – Move to Engineering and Applied Science, Yale University

Interact with George Schultz, Paul Burrow, Arvid Herzenberg

Interest in temporary anions and start of long-time collaboration with Paul

Interest in dipole-bound anions prompted by question by Herzenberg

Spent summers in Utah, collaborating with Jack

Initial calculations on dipole-bound anions (LiH-, NaH-)

1976 - LiCl- (theory: Jordan and Luken; expt. Lineberger, et al.)

1979 - Paper exploring electron binding to quadrupole fields

1978 – Move to Univ. of Pittsburgh

Set up ETS experiment to study temporary anions (collab. with Paul)

Move into new research directions, setting aside dipole-bound anions

1993 – Papers with C.-J. Tsai on thermodynamics and stationary points of water clusters

1994 – Start of collaboration with Tim Zwier on benzene-(H2O)n clusters

1994 – Sabbatical at Univ. of Utah

Collaboration with Jack and Maciek Gutowski on the role of electron correlation in dipole bound anions

Begin thinking about how to do this via a model potential for e--water

1997 – Visiting Fellow, JILA (papers with Carl on (CS2)2-, etc.)

1999 – Start of Collaboration with Mark Johnson on (H2O)n- Clusters

2002 – Develop Drude model for e- - water with Feng (Seymour) Wang

2003 – Start of series of papers with Bob Compton and Kadir Diri on dipole-bound anions

2005 – Improvements to the Drude model with Thomas Sommerfeld

Begin work on local potential models

Eaq- is one of the most important species in chemistry and biology.

Yet the nature of this species has remained elusive

Clusters have proven especially useful for elucidating the nature of excess electrons and protons in water

The solvated electron has been known since 1863 – e- in liq. NH3 (Weyl, Ann. Phys.)

The hydrated electron (eaq-) was identified

in 1962 (Hart and Boag, JACS)

The prevailing view is that the hydrated electron is well described as an electron in a spherical cavity of radius ~ 2.4 Å

Lowest energy transition at 1.7 eV is essentially s → p

(H2O)n- clusters first observed (mass

spectroscopically) by Harberland in 1981

Expt. absorption spectra of e-

(aq) and selected (H2O)n-

clusters (Ayotte and Johnson, 1996)

• What is the origin of the magic numbers at n = 2, 6, 7, 11?

• Are the observed anions the most stable isomers?

• Role of the cluster temperature and of Ar atoms on the electron capture and dynamics?

• At what size cluster does the electron “prefer” to be in the interior?

• What is the mechanism of the electron binding?

Characterization of excess electron-water cluster systems has proven especially challenging, both experimentally and theoretically.

Mass spectrum of the (H2O)n-

clusters, from M. Johnson

Breakthrough: development of methods to measure the vibrational spectra of (H2O)n

- ions.

When combined with electronic structure calculations, have enabled the structures of the observed anions of the n ≤ 6 clusters to be established (Johnson et al., Science 2004)

Vibrational spectra of the (H2O)6-21─ clusters

(Johnson et al.)

Dominant isomer of the hexamer anion

7 OH groups pointing up

• results in a large dipole moment

• this is a very unstable arrangement for the neutral cluster

Time-resolved photoelectron spectroscopy studies (Neumark and Zewail groups, Science, 2004) have provided information on the dynamics of the larger (H2O)n

- clusters

Neumark et al. (Science, 2005) have shown that by using different source conditions, clusters with appreciably different electron binding energies can be prepared

Their interpretation:

strong binding = interior bound

weak binding = surface state

Called into question by:

Turi, Sheu, and Rossky (Science, 2005)

Sommerfeld + Jordan (JPC, 2005; JACS 2006)

Pho

toel

ectr

on y

ield

Photoelectron spectra under different source conditions. Colder clusters – greater population of structures with small VDEs. (from Neumark et al.)

Experimental vertical detachment energies

of (H2O)n- clusters. Data from the Bowen,

Neumark, and Johnson groups.

Attributed to surface bound electron (Neumark et al.)

Attributed to interior bound electron (Neumark et al.)

Problems:

How can there be interior states for n < ~ 10?

Many more isomers than three are expected

• Long believed that Koopmans’ theorem (which accounts for electrostatics, but not correlation) gives a good approximation of the e- binding. • But in 1990’s theoretical studies (Gutowski, Skurski, Jordan, Simons) provided evidence for large dispersion interactions between excess e- and electrons of the molecule/cluster.

e- + –

dispersion

To answer this, we need to consider the problem of the binding of an excess electron to polar molecules and their clusters

What sort of theoretical method is needed to address the questions posed by these recent experiments?

Examples: CH3CN, (HF)2

Binding Energy (cm-1)a

KT SCF MP2 CCSD(T) Expt.

CH3CN- 53 56 75 109 93-145b

(HF)2- 165 179 283 387 508c

aTheory results from Gutowski et al.bDefrancois et al. (1994, 1995); c Bowen et al. (1997)

High-order correlation effects are important, requiring coupled clusterapproaches.

Such calculations are restricted to small systems.

Cannot be used to address electron binding to (H2O)n, n ≥ 7.

The non-valence nature of the excess electron, suggests that a one-electron model potential may be applicable

but, how can we deal with the electron correlation problem?

Much prior work on model potentials for e- + (H2O)n Berne, Rossky, Nitzan, Landman, Borgis

Models typically includeElectrostatics [e- - permanent charges on (H2O)]Exchange/repulsionPolarization (e--water, water-water)

None of these models include explicitly dispersion interactions between the excess e- and the electrons of the water molecules

Cannot describe with C/R6 terms due to extended nature ofexcess electron.

Our approach - Drude model of excess-electron molecule

interactions.

Drude model for excess electron systems

+q -q Charges +q, -q coupled through a force constant k

R The position of the +q charge is kept fixed.

Polarizability = q2/k

An electron couples to the Drude oscillator via qr∙R/r3

r is the vector from the e- to the oscillator

O

H

H

M site: 0.215 Å from O atom. Negative charge (-1.04) plus Drude oscillator with q2/k = α = 1.444 Å3

q = 0.52

Drude model based on the Dang-Chang water model

{q = 0.52

We have recently found that it is essential to use more sophisticated models of neutral water, and many of the results presented have used such models

,el osc el oscH H H V

21

2je

j

r pex el

j

QH V

rV

2,3

(1 )el osc brqV e

r

r R

2 2 2 21 1( )

2 2osc

oo

H k X Y Zm

r - position of electronR - displacement of the Drude oscillator

repV

• Determined using procedure of Schnitker and Rossky

• Scaled so that the model potential KT energy reproduces ab initio KT result for (H2O)2

-

b

Damping coefficient scaled so that Drude model CI energy reproduces ab initio CCSD(T) result for (H2O)2

-

Hamiltonian (single Drude oscillator)

Energies in eV

All results are for the MP2-optimized geometries.

Contours enclosing 10 (innermost), 30, 50, 70, and 90% (outermost) of the total charge density of an excess e- bound to (H2O)2, and the AA form of (H2O)6.

(H2O)6-

(H2O)2-

As expected a large contraction of the charge density in going form the dimer to the hexamer.

But even for the hexamer the charge density within 2 Å of the H atoms is small.

The Drude model is also able to describe the electronically excited states of the excess electron/water cluster systems. Below the wavefunctions of the ground and excited sates of the (H2O)13

- cluster are shown

Even though the clusters are highly non-spherical, the low-lying excess electron states are s and p like.

Comparison of MP2 and Drude model electron binding energies (meV) of selected isomers of (H2O)20

-

Drude Model ab initioa

Isomer µ(D) ES PT2 Cl MP2

44 54 62 5.4 5 20 148 74

512 A 24.6 580 913 1078 1085

512 B 18.6 441 745 908 910

512 C 14.4 266 515 681 658

512 D 14.0 192 388 557 516

512 E 2.0 -4 -3 69 -28

512 F 0.04 13 127 398 229

aJ. M. Herbert and M. Head-Gordon, J. Phys. Chem. (2005)

Recently Herbert and Head Gordon reported MP2 level electron binding energies (BE) of several isomers of (H2O)20

- and (H2O)24-.

Good agreement for those clusters with large dipole moments

Ab initio MP2 BEs are significantly smaller than the Drude values for clusters with small dipole moments (i.e., for those systems in which high-order correlation effects are important)

Comparison of MP2 and Drude model electron binding energies (meV) of selected isomers of (H2O)24

-

Drude Model ab initioa

Isomer µ(D) ES PT2 Cl MP2

46 68 A 0.0 -5 -4 12 -32

B 0.0 -5 -4 638 576

41464 A 0.0 -5 -3 132 8

B 0.0 -5 -3 406 7

51262 A 0.26 -5 -4 44 -60

B 0.006 -5 -3 829 795

C 0.016 -4 59 236 130

414 A 0.0 -5 -4 76 27

aJ. M. Herbert and M. Head-Gordon, J. Phys. Chem. (2005)

Minus BE: unbound anion

What happens if the electrostatics (i.e., e- interactions with both the permanent charges and the induced dipoles) are reduced to zero?

Calculated photoelectron spectrum depends sensitively on the neutral water model employed.

Electron binding energy distributions of (H2O)6- from the T =

60K replica from parallel tempering Monte Carlo simulations

Position of major peak in expt. spectrum.

DC

DPP1

DPP2Best model

Puzzle - reconciling the different conclusions of theoretical studies using local -α/2r4 polarization potential and those that explicitly include correlation effects

• The former indicate that polarization effects are important

• The latter seem to indicate that dispersion effects are much more important than polarization effects for electron binding.

One can adiabatically separate the excess electron from the Drude oscillators to generate an effective potential for the excess electron

Assume there is a single Drude oscillator, with polarizabilty αD

Classical polarization potential

The -α/2r4 terms actually incorporate long-rage correlation effects

Much of the confusion is the result of semantics – i.e., polarization taking on different meanings in different communities.

This derivation shows that polarization models with α/2r4 term to actually recover some of the long-range correlation effects

So much of the confusion is semantics – polarization taking on different meanings in different communities.

4a(dipole

bound)

24b(cavity)

12a(network permeating)

24a(network permeating)

Drude model 304 839 27 331

adiabatic model (full)

302 1057 47 445

Calculated electron binding energies (meV) of selected (H2O)n- clusters

Overall, fairly good agreement with results from the one-electron polarization model and the many-body Drude model.

But required adoption a much stronger damping of the polarization [damping factor again fixed for (H2O)2

-]

W45- (int) W45-(surf)

Full Drude 1.7, 1.7, 1.8 0.8, 0.9, 1.1

adiabatic 1.8,1.8, 1.8 0.8, 0.9, 1.1

Comparison of Drude and adiabatic model excitation energies (eV) of two (H2O)45

- ions (geometries from Turi and Rossky)

Blue: 98%, purple: 60% of the charge density

A closer look at the results for selected (H2O)n- clusters

Electron binding energies (meV)

Drude

ES PT2 CI Adiab.

W4 -10 -54 (-3, -40) -86 -93

W7 -12 -90 (-4, -75) -681

W45 surf -776 -1226 (-152, -290) -1391 -1480

W45 int -592 -2525 (-1039, - 939) -2298 -2580

induction dispersion

For small clusters, the 2nd order induction contribution is much smaller than the 2nd dispersion contribution.

For large clusters induction and dispersion are of comparable importance.

Need to be careful in dissecting the interactions, when the 0th order wavefunction is not a good approximation.

Solvated e- model shown above with R = 8 Å.

Summary of work on (H2O)n- clusters:

A model using quantum Drude oscillators has been developed for describing the interactions of excess electrons with water molecules.

• Recovers the dominant electron correlation effects at a fraction of the cost of ab initio calculations

• Fast enough to be used in finite temperature simulations

• Inclusion of high-order correlation effects cause a sizable contraction of the charge distribution of the excess electron.

By use of an adiabatic approximation, we have derived a local polarization potential for the interaction of an excess electron with water clusters

• demonstrates that α/2r4 “polarization” potentials include long-range electron correlation effects as well as induction effects

• for most clusters the adiabatic model gives electron binding energies in fairly good agreement with the Drude model CI results

Preliminary results suggest that the excitation spectrum of an excess electron in water is well described by calculations using only two solvent shells.

In many cases the binding energies from the Turi-Borgis and our local potential are very close.

But there are some exceptions

Has led us to examine the model potentials more closely

Our potential is much more attractive near the H atoms.

But the Turi-Borgis potential has a more attractive long range tail (due to the 2.35 dipole moment in SPC/E water in contrast to 1.85 D in our model)

Or potential is less repulsive on the O end.

10% density, hexmer

10% density, dimer

Comparison of ES + polarization potentials from adiabatic one-electron models and MP2 calculations

The adiabatic ES + pol potential from the Drude model tracks the corresponding MP2 down to within 1.5 Å of the O atom, but this result is deceptive.

Both there are significant differences between the adiabatic model and the MP2 calculations for both the ES and polarization contributions starting near 2 Å from the O atom. (Polarization curves shown above)

Essential to greatly weaken the polarization potential at short r to get the correct e- binding energy for the dimer.

Damping employed in QDM

Damping employed in QDM

electrostatics

The Turi-Borgis potential is more attractive near the H atoms

Recovering charge-penetration?

This is partially compensated by the differences in the repulsive potentials

The Turi-Borgis potential has a much weaker polarization contribution than our local potential model or ab initio calculations

What are the take-away messages from the comparison of the two local potentials (and MP2 calculations)?

The Drude model (and corresponding local potential model) appears not to be sufficiently repulsive on O end of water, which would cause an overbinding of e- for network permeating and possibly also cavity bound states

The Turi-Borgis potential underestimates polarization, but has a more attractive long-range tail (on the H end)

Seem to balance out for dipole bound anions.

Not yet clear if this is the case for other types of anions

Projects underway

Improve the repulsive potential in the Drude and local potential models

Apply the improved model to Monte Carlo simulations of e- in bulk water

Extend to a flexible water model for calculating vibrational spectra