Model Hamiltonians for Electron-Molecule Interactions

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Model Hamiltonians for Electron-Molecule
Interactions
K. D. Jordan
Electronically excited state of (H2O)45-
Dominant form of (H2O)13-
Anions 2007, Park City Utah
2
Support
Acknowledgements and Projects
National Science Foundation
Department of Energy
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.
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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 EAs 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
4
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
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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)
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Characterization of excess electron-water cluster
systems has proven especially challenging, both
experimentally and theoretically.

• 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?

Mass spectrum of the (H2O)n- clusters, from M.
Johnson
7
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)

• 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

Vibrational spectra of the (H2O)6-21- clusters
(Johnson et al.)
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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)
Photoelectron yield
Photoelectron spectra under different source
conditions. Colder clusters greater population
of structures with small VDEs. (from Neumark et
al.)
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Attributed to interior bound electron (Neumark et
al.)
Attributed to surface bound electron (Neumark et
al.)
Problems How can there be interior states for n
lt 10? Many more isomers than three are expected
Experimental vertical detachment energies of
(H2O)n- clusters. Data from the Bowen, Neumark,
and Johnson groups.
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What sort of theoretical method is needed to
address the questions posed by these recent
experiments?
To answer this, we need to consider the problem
of the binding of an excess electron to polar
molecules and their clusters

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

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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 cluster approaches. 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?
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Much prior work on model potentials for e-
(H2O)n Berne, Rossky, Nitzan, Landman, Borgis

• Models typically include
• Electrostatics e- - permanent charges on (H2O)
• Exchange/repulsion
• Polarization (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 of excess electron. Our approach - Drude
model of excess-electron molecule interactions.
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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
qrR/r3 r is the vector from the e- to the
oscillator
Drude model based on the Dang-Chang water model
q 0.52
H

M site 0.215 Å from O atom. Negative charge
(-1.04) plus Drude oscillator with q2/k a
1.444 Å3
O
H
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
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Hamiltonian (single Drude oscillator)
r - position of electron R - displacement of the
Drude oscillator
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Energies in eV All results are for the
MP2-optimized geometries.
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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.
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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.
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Recently Herbert and Head Gordon reported MP2
level electron binding energies (BE) of several
isomers of (H2O)20- and (H2O)24-.
Comparison of MP2 and Drude model electron
binding energies (meV) of selected isomers of
(H2O)20-
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)
Drude Model Drude Model 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)
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Comparison of MP2 and Drude model electron
binding energies (meV) of selected isomers of
(H2O)24-
Drude Model Drude Model 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
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What happens if the electrostatics (i.e., e-
interactions with both the permanent charges and
the induced dipoles) are reduced to zero?
21
Calculated photoelectron spectrum depends
sensitively on the neutral water model employed.
Best model
Position of major peak in expt. spectrum.
DPP2
DC
DPP1
Electron binding energy distributions of (H2O)6-
from the T 60K replica from parallel tempering
Monte Carlo simulations
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• Puzzle - reconciling the different conclusions of
  theoretical studies using local -a/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 aD
The -a/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.
Classical polarization potential
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This derivation shows that polarization models
with a/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.
Calculated electron binding energies (meV) of
selected (H2O)n- clusters
4a (dipole bound) 24b (cavity) 12a (network permeating) 24a (network permeating)
Drude model 304 839 27 331
adiabatic model (full) 302 1057 47 445
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-
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Comparison of Drude and adiabatic model
excitation energies (eV) of two (H2O)45- ions
(geometries from Turi and Rossky)
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
Blue 98, purple 60 of the charge density
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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
Solvated e- model shown above with R 8 Å.
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.
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• 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 a/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.
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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
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Comparison of ES polarization potentials from
adiabatic one-electron models and MP2 calculations
Damping employed in QDM
Damping employed in QDM
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.
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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
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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