Theoretical Investigation of Solar Energy
Conversion and Water Oxidation Catalysis
by
MASSACHUSEMS INSTITTE
OF TEC OLGYT
Lee-Ping Wang
JUN 0 7 2011
Submitted to the Department of Chemistry
in partial fulfillment of the requirements for the degree of
LIBRARIES
Doctor of Philosophy in Chemistry
ARCHIVES
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2011
@ Massachusetts Institute of Technology 2011. All rights reserved.
A uthor ................
...............
uep tment of Chemistry
May 20, 2011
Certified by......
Troy Van Voorhis
Associate Professor
Thesis Supervisor
Accepted by ...
Robert W. Field
Chairman, Department Committee on Graduate Theses
2
This doctoral thesis has been examined by a Committee of the
Department of Chemistry as follows:
Professor Robert J. Silbey...
Chairman, Thegis Committee
Professor of Chemistry
Professor Troy Van Voorhis.
Thesis Supervisor
Associate Professor of Chemistry
Professor Robert G. Griffin
Member, Thesis Committee
Professor of Chemistry
I
Theoretical Investigation of Solar Energy Conversion and
Water Oxidation Catalysis
by
Lee-Ping Wang
Submitted to the Department of Chemistry
on May 20, 2011, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Chemistry
Abstract
Solar energy conversion and water oxidation catalysis are two great scientific and engineering challenges that will play pivotal roles in a future sustainable energy economy.
In this work, I apply electronic structure theory and molecular dynamics simulation
methods to understand some of the detailed mechanisms behind solar energy conversion and water oxidation catalysis. I will present a detailed atomistic picture of charge
separation processes at a donor-acceptor interface between two organic semiconductor
materials in an organic photovoltaic device. This will be followed by an investigation of the water oxidation mechanism of a homogeneous ruthenium water-splitting
catalyst and a heterogeneous cobalt phosphate water-splitting catalyst. I will also
introduce several advances in theoretical methods that allow us to more accurately
compute observables that are critically relevant to the operation and performance of
these materials and devices.
Thesis Supervisor: Troy Van Voorhis
Title: Associate Professor
6
Acknowledgments
To Professor Troy Van Voorhis, my research advisor, thank you for being an excellent
research mentor beyond compare. Over the course of my graduate education, Troy
has played the central role in my growth as a scientist, helping me to hone my critical
thinking, strengthen my perserverance, and spark my creativity. Troy is a brilliant
inventor of new ideas, yet he also cherishes and supports his students' independent
ideas. He is a leader and a motivator by example and by encouragement. And perhaps
most importantly, I know Troy cares about his students from the bottom of his heart.
I am honored to have had the opportunity to learn from him.
I would like to thank my thesis committee, Professors Robert G. Griffin and
Robert J. Silbey, for always being available to share their breadth of experience and
guide me on my pursuit of scientific research and an academic career. I would also
like to thank the wonderful teachers that have kindled my interest in science from
elementary school all the way through graduate school: Mr. John McClain, Ms.
Tanya Phillips, Professor Yuen-Ron Shen, Professor Dung-Hai Lee, Professor John
Neu, Professor Joel Fajans, Professor Robert Jaffe, Professor Hsiang Wu-Yi. I would
like to thank Professor Luke Lee for introducing me to the world of scientific research,
and the teachers at the Lawrence Hall of Science for introducing me to the world of
science education. To all of the faculty in the MIT Department of Chemistry, it has
been my privilege to work alongside the best chemists in the world.
My officemates in MIT Theoretical Chemistry, past and present, have long been
my friends and companions, my teachers, my supporters, and my helpful critics. They
are: Hang Chen, Jiahao Chen, Xin Chen, Chiao-Lun Cheng, Seth Difley, Jeremy
Evans, John Harkless, Eric Hontz, Ben Kaduk, Tim Kowalczyk, Tamar Mentzel,
Steve Presse, Xiaogeng Song, Laken Top, Bas Vlaming, Oleg Vydrov, Jianlan Wu,
Qin Wu, Sina Yeganeh, Shane Yost, Shen Young, and Eric Zimanyi. Thank you for
making "The Zoo" such a nice place to work.
I will remember Li Miao not only as the most efficient administrative assistant
in my memory, but also as a reliable source of fun conversation and good advice, a
confidant and a friend.
To the folks in the Chemistry Education Office Melinda -
especially Susan, Mary, and
thank you for all of your kindness and assistance, and for remembering
all of our faces as hurry our way through these short years.
To my good friends, at MIT, in California and elsewhere in the world: Thank you
for sharing my success with me -
you're the source of my happiness. I look forward
to many good times in the days to come. Go Bears!
To Yan-Ping, my younger brother and best friend: You have been at least as much
an inspiration to me as I have been to you.
Above all, I owe my success to the love of my mother and father, Ivy Kuo and
Chi-Yuen Wang. Amid all of the turbulence and uncertainty of the past five years,
through all of the ups and downs, your unyielding support and confidence in me
has been the single invariant, the only conserved quantity. No words can express
my gratitude for the love that can only come from Baba and Mama. This thesis is
dedicated to you.
Contents
1 Introduction
21
1.1
Chemistry and Renewable Energy . . . . . . . . . . . . . . . . . . . .
21
1.2
Basic Chemistry and its Application to the Energy Problem .....
23
1.2.1
Catalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
1.2.2
Light energy conversion
. . . . . . . . . . . . . . . . . . . . .
26
1.2.3
Electrochemistry . . . . . . . . . . . . . . . . . . . . . . . . .
29
1.2.4
Biology
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
Theoretical Chemistry: Foundations and Applications . . . . . . . . .
33
1.3.1
Quantum Chemistry and Electronic Structure Theory . ....
34
1.3.2
Statistical Thermodynamics and Molecular Dynamics Simulation 41
1.3
1.4
Exposition: Theoretical Investigation of Solar Energy Conversion and
Water Oxidation Mechanism . . . . . . . . . . . . . . . . . . . . . . .
1.4.1
QM/MM Investigation of charge separation in an organic donor/acceptor
interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
45
45
1.4.2
Mechanistic investigation of a ruthenium water oxidation catalyst 47
1.4.3
Improved computation of standard reduction potentials
1.4.4
Mechanistic investigation of a cobalt water oxidation catalyst
1.4.5
Force Matching Method for Improved Classical Molecular Models 56
. . . .
51
54
QM/MM Study of Exciton Dissociation and Charge Separation in
Organic Photovoltaic Devices
61
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
2.2
T heory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
2.3
2.4
3
2.2.1
Constrained DFT as a Route to CT Excited States . . . . . .
63
2.2.2
QM/MM Modeling of Disordered Systems . . . . . . . . . . .
64
2.2.3
Simulation Procedure . . . . . . . . . . . . . . . . . . . . . . .
65
2.2.4
Interface Structure . . . . . . . . . . . . . . . . . . . . . . . .
66
2.2.5
Density Functional Calculations . . . . . . . . . . . . . . . . .
66
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
67
2.3.1
Bulk Materials
. . . . . . . . . .. ..... .. ... ... ..
67
2.3.2
Organic-Organic Interface . . . . . . . . . . . . . . . . . . . .
69
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
Acid-Base Mechanism for Ruthenium Water 0 xidation Catalysis
75
3.1
Introduction . . . . . . . . . . . . . . . . . . . .
3.2
Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . .
79
3.3
3.4
3.5
.
.
.
.
.
.
.
.
.
.
.
75
.
3.2.1
General
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
3.2.2
Redox Potentials . . . . . . . . . . . . . . . . . . . . . . . . .
79
3.2.3
Reaction Path Investigations . . . . . . . . . . . . . . . . . . .
81
Single-Center Catalyst . . . . . . . . . . . . . . . . . . . . . . . . . .
82
3.3.1
Energetics and Redox Potentials . . . . . . . . . . . . . . . . .
84
3.3.2
Oxygen-Oxygen Bonding . . . . . . . . . . . . . . . . . . . . .
85
3.3.3
Dioxygen Displacement . . . . . . . . . . . . . . . . . . . . . .
88
Two-Center Catalyst . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
bond formation . . . . . . . . . . . . . . . . . . . . . .
93
3.4.1
Mechanism
3.4.2
0-0
3.4.3
Deprotonation of the hydroperoxo group and dioxygen release
Discussion and Conclusions
. . . . . . . . . . .
4 A Polarizable QM/MM Explicit Solvent Model for Computational
Electrochemistry in Water
101
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
101
4.2
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
105
4.2.1
Implicit Solvent Calculation of Standard Reduction Potentials
106
4.2.2
4.3
4.4
4.5
4.6
5
QM/MM Calculation of Standard Reduction Potentials . . . . 107
Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.3.1
Implicit Solvent Model . . . . . . . . . . . . . . . . . . . . . . 108
4.3.2
QM/MM Model . . . . . . . . . . . . . . . . . . . . . . . . . . 109
R esults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.4.1
Standard Reduction Potentials in Implicit Solvent Model . . . 111
4.4.2
Standard Reduction Potentials in QM/MM Model . . . . . . . 111
D iscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.5.1
Atomic Radii in ISM . . . . . . . . . . . . . . . . . . . . . . . 113
4.5.2
Solute Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.5.3
Solute-Solvent Correlation . . . . . . . . . . . . . . . . . . . . 115
4.5.4
Hydrogen bonding effects . . . . . . . . . . . . . . . . . . . . . 116
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Direct-Coupling 02 Bond Forming Pathway in Cobalt Oxide Water
Oxidation Catalysts
119
6 Hybrid ensembles for improved force matching
129
7 Conclusions
139
A Supporting Information for Chapter 2: Acid-Base Mechanism for
Ruthenium Water Oxidation Catalysis
143
B Supporting Information for: Direct-Coupling 02 Bond Forming Pathway in Cobalt Oxide Water Oxidation Catalysts
149
B.1 M ethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
149
C Supporting Information for: Hybrid Ensembles for Improved Force
Matching
165
12
List of Figures
1-1
Photograph of mountaintop removal coal mining, in which entire mountaintops are dynamited and pulverized in order to mine the underlying coal
seam s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-1
Illustration of the QM/MM method.
22
Left: Disordered cell of the
H2Pc/PTCBI system described by MM. Center: Selection of a H2 Pc
and PTCBI pair at the interface for calculation of the CT state energy.
Right: Density-of-states plot obtained by repeating the calculation over
different snapshots of a MM trajectory. . . . . . . . . . . . . . . . . .
2-2
62
Calculated absorption spectrum (dashed) and experimental spectrum
(solid) of PTCBI (top, blue) and H2 Pc (bottom, red).
The calcu-
lated spectra contain 750 calculated energies sampled from 15 molcules
each over 50 snapshots, each given a Gaussian distribution with width
1.7 nm.
The inserted molecules show the attachment/detachment
(blue/orange) densities of the lowest excited state of PTCBI and H2 Pc. 68
2-3
Full calculated spectra of all relevant energy states: bulk absorption
(left axis) of H2 Pc (red) and PTCBI (blue), CT density of states (black,
right axis), and the location of the average bulk band offset (brown).
Each data point is given a Gaussians distribution with a width of 1.7
nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
2-4
Plot of the distance dependence of the PTCBI/H 2 Pc CT state binding
energies. The coordinate R is a linear combination of intermolecular
distances. Each different color/shape combination represents distinct
dimer pairs in the simulation cell. . . . . . . . . . . . . . . . . . . . .
2-5
71
Plot of the average IP and EA of H 2 Pc (red) and PTCBI (blue) crystal
planes as a function of their distance from the interface. Each point
has a standard deviation of about 50 meV. . . . . . . . . . . . . . . .
72
3-1
Schematic representations of acid-base and direct coupling pictures. ....
77
3-2
Complete catalytic cycle for the single-center water oxidation catalyst. Lowestenergy spin states are denoted in subscripts.
Bracketed species are not
experimentally observed. The Ruv species is sufficiently high in energy
and unstable that we denote it using "2T", where "T" stands for "transient". Experimental values for redox potentials are reported in parentheses.
For brevity, numerical labels are used to denote the various intermediates
throughout the paper. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-3
83
Starting and ending points of a geometry optimization starting at Ruv(O)
and ending at Ru"'(OOH). The color scheme is: Ru (green), C (teal), N
(blue), 0 (red), H (white). The reacting oxygen atoms are connected by
a dotted line. Selected Mulliken populations are labeled. Charges on the
water hydrogen atoms are summed into the oxygen atoms.
3-4
. . . . . . . .
86
Isosurface plots of the HOMO and LUMO in 2T + H2 0. These two orbitals
combine to create the 0-0
o- bond. Note the Ru-O antibonding r*
character of the LUMO; 0-O bonding simultaneously weakens the Ru-O
bond. The orientation of the H2 0 molecule may significantly affect O-0
bonding, due to the importance of HOMO-LUMO overlap.
3-5
. . . . . . . .
88
Reaction diagram for oxygen displacement comparing the geometries and
energetics of the starting, ending, and transition state configurations. 3-D
plots of the system are shown at the initial, final, and transition states. Key
distance measurements (H20-Ru and 00-Ru) are indicated. . . . . . .
89
3-6
Dibenzofuran (DBF) bridged two-center ruthenium catalyst.
3-7
Proposed catalytic mechanism for the DBF-bridged two-center catalyst.
. . . . . . .
Lowest-energy spin states are shown. . . . . . . . . . . . . . . . . . . . .
3-8
91
92
Initial (6T, left) and final (7, right) configurations for the O-O bonding step. The RuV side (left side of the molecule) is probed with water
molecules. Key hydrogen bonds are indicated with dotted lines. . . . . . .
3-9
93
Attempted deprotonation of the hydroperoxo group in the monomeric analogue, [Ru"'(tpy)(bpy)(OOH)j
2+.
. . . . . . . . . . . . . . . . . . . . .
94
3-10 Marcus energy diagram showing the reactant [(O)RuiVRuII(OOH)] 4 + and
product [(HO)Ru"RuIV (00)]4+ configurations in the final PCET step.
4-1
.
96
Illustration of a QM/MM simulation of a [Fe(H 2 0) 6 ] 3+ complex showing
QM solute (green, Fe; red, 0; white, H) and superimposed positions of MM
solvent molecules (red, 0; black, H). The geometry of the solute is held
fixed. The positions of MM atoms is an approximate representation of the
probability distribution of nuclear positions, and the short-range solvent
structure (interpreted as a consequence of solute-solvent hydrogen bonding)
is visible.
4-2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
105
Calculated values of EO plotted against experiment. Gray: ISM calculations
of EO for molecules chosen from Ref. 16 (organic molecules, triangles; sandwich complexes, diamonds; octahedral complexes, circles).
Blue crosses:
ISM calculations performed on aqueous transition metal complexes. Red
crosses: QM/MM calculations performed on aqueous transition metal complexes.
4-3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
111
Mean and standard deviation of differences between calculated EO values
and experimental measurements. Functionals are ordered by mean error,
and a correlation between mean error and fraction of Hartree-Fock exchange
is observed..........
..................................
113
5-1
Outer ring: Proposed mechanism of water oxidation catalysis; intermediates are labeled in bold face, with net charge and lowest-energy spin
state in subscript (AF denotes an open-shell electronic state with antiferromagnetic coupling). Standard reduction potentials (Eo) and free
energy barriers for the first half of the cycle, computed by QM/MM, are
given. Terminal aqua ligands and the bottom half of the compound are
omitted for clarity. Center: MM dynamics trajectory of model catalyst
structure (held static) with superimposed positions of nearby explicit
solvent water molecules (oxygen: red spheres, hydrogen: black dots),
highlighting the strong hydrogen-bonding effects.
5-2
. . . . . . . . . . .
120
Energetically disfavored structures and pathways. 2A: An alternative
protonation state of 2, containing a Cov bis-hydroxo; E0 (labeled)
is significantly higher than that of the 1/2 couple. 3A, 3B, 3C: Alternative 0-0
bonding pathways with reaction energies and activa-
tion barriers labeled; the computations for 3B included extra water
molecules or phosphate anions as proton acceptors (not shown). 3B
was ruled out because Cov was a prerequisite to reactivity. . . . . . .
5-3
DFT energies and (S2 ) as a function of 0-0
122
distance for the doubly
oxidized model catalyst, 3. Left inset: HOMO at roo =3.0 A. Right
inset: o-bonding orbital at roo =1.4
5-4
A. . . . . . . . . . . . . . . . . . 123
QM/MM free energy profile as a function of 0-0
distance for CoIV2 Co1" 2 0
2
(3-+ 4). Simulation snapshots are shown for the free energy minima
at roo =3.0 A and roo =1.4
6-1
A, indicated by triangles.
. . . . . . . .
126
(a) QM and MM< potential energy curves. (b) QM and MM> potential
energy curves. The error region and thermally accessible regions are labeled
using arrows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
131
6-2
Optimized MM potentials from force matching, starting from the initial
guess (black line) and fitting to the quantum PES (black dots). Initial
guesses are MM< (a) and MM> (b); optimized MM potentials using X2M
(red), X2M (green), and X2 (blue) are shown. . . . . . . . . . . . . . . .
6-3
134
Radial distribution functions of water for (a) O--O distances, (b) 0-H
distances, and (c) H-H distances. Thirty AIMD RDFs of length 60 ps
were combined to produce the reference curve. . . . . . . . . . . . . . . .
Si
136
Correlation diagram of calculated and experimental redox potentials for
various organic molecules and metallocenes as reported in Baik and Friesner
[16]. No linear fit was performed. R2
S2
=
0.98, MAE
=
0.150V .......
144
Correlation diagram and linear fit of calculated and experimental redox
potentials for various Ru-0H2 coordination complexes. R 2 - 0.55, MAE
=
0.110V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
S1
Gas-phase geometry optimized model catalyst (1 from Figure 1), with
four intramolecular distances labeled (Co-p-oxo, Co-OH, Co-Aq, CoCo). The Co-O and Co-Co distances are in agreement with previously
published XAS data. . . . . . . . . . . . . . . . . . . . . . . . . . . .
S2
151
Orbital diagram for twice-oxidized model catalyst in open position (3,
on left) and bonded position (4, on right). Only orbitals with L0.3
occupancy on the oxo groups are shown. The formation of a 0- bond is
evident, as well as occupation of all 7r and -r* orbitals. Some orbitals
(antibonding partners of 7r on bottom left and o- on bottom right) are
not shown, because they are high in the virtual-orbital manifold and
have very diffuse character . . . . . . . . . . . . . . . . . . . . . . . .
S3
152
MM dynamics trajectory of model catalyst 1 (freely moving) with superimposed positions of solute hydrogens (black dots) and nearby explicit solvent water oxygens (red spheres).
A high solute degree of
freedom is evident. The hydrogen-bonding effects are not as clearly
visible as in Figure 1, due to free rotation of the hydrogens . . . . . .
153
S4
Spin density plot of twice-oxidized model catalyst in open position
(3).
Note the similarity between the spin density and the HOMO
in S2, left column; this indicates that the electronic ground state is
a broken-symmetry singlet with metal d, character. [24] There is no
corresponding spin-density plot for the ground state of 4, because it is
a closed-shell singlet.
Si
. . . . . . . . . . . . . . . . . . . . . . . . . .
154
Evolution of parameters during force matching for MMo (blue), MM 5 o
(green), and MM 80 (red) . . . . . . . . . . . . . . . . . . . . . . . . . .
165
S2
Comparison of gun using MMo, MMso, and the SPC/E initial guess. . . .
166
S3
Comparison of gHH using MMO, MM 80 , and MMioo; the latter is fitted to
an AIMD trajectory. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
166
List of Tables
2.1
Calculated transport properties for H2 Pc and PTCBI. Experimental
values, taken from Refs. 222, 320, and 227, are given in parentheses.
All values are reported in eV; computed values have a statistical uncertainty of le 0.07 eV. . . . . . . . . . . . . . . . . . . . . . . . . . .
67
5.1
Standard reduction potentials (Eo) computed by QM/MM. . . . . . .
125
S1
Redox potentials of relevant half-reactions vs. NHE. All potentials are standardized to pH 0 by applying a -0.059V/pH correction. For compounds 1
through 19, see Baik and Friesner [16]. . . . . . . . . . . . . . . . . . . .
S2
145
Redox potentials of relevant half-reactions vs. NHE. All potentials are standardized to pH 0 by applying a -0.059V/pH correction. For compounds 1
through 19, see Dovletoglou and Meyer [92]. For compounds 20 through 22,
see Masllorens and coworkers [179].
S1
. . . . . . . . . . . . . . . . . . . .
147
DFT+PCM computations of redox potentials for PCET from each of
the three Co"'(OH) groups of 2, indicating no energetic difference for
creating a cofacial oxo group (on the top face) compared to a distal
oxo group (on the bottom face).
The DFT+PCM calculations are
intended for comparing cofacial and distal deprotonations only, not for
quantitative predictions of EO.
S2
S3
. . . . . . . . . . . . . . . . . . . . .
151
XYZ coordinates for model catalyst 1, also shown in S1. (E = -1796.176824084
a.u.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
155
. . . . . . . . .
156
XYZ coordinates for 2. (E = -1795.499445861
a.u.)
S4
XYZ coordinates for 3. (E = -1794.7933790035
a.u.) . . . . . . . . .
157
S5
XYZ coordinates for 4. (E
a.u.) . . . . . . . . .
158
S6
Initial XYZ coordinates for water addition to the once-oxidized equiv-
=
-1794.8369508192
alent of 4. (E = -1870.6779521845
S7
a.u.) . . . . . . . . . . . . . . . .
Transition state XYZ coordinates for water addition to the once-oxidized
equivalent of 4. (E = -1870.6671448537
S8
159
a.u.)
. . . . . . . . . . . . .
160
Final XYZ coordinates for water addition to the once-oxidized equivalent of 4; this is similar to 5, but with one additional electron and
proton. (E
S9
-1870.6927839533
a.u.) . . . . . . . . . . . . . . . . . .
Initial XYZ coordinates for water addition to 5. (E
161
-1946.477303537
a.u.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S10 Transition state XYZ coordinates for water addition to 5.
162
(E =
-1946.470836279 a.u.) . . . . . . . . . . . . . . . . . . . . . . . . . .
163
S11 Final XYZ coordinates for water addition to 5; this is identical to 6+
02. (E
=
-1946.498637907
a.u.)
. . . . . . . . . . . . . . . . . . . .
164
Sl
Force matching for the helium dimer, starting from MM<.
. . . . . .
166
S2
Force matching for the helium dimer, starting from MM>.
. . . . . .
167
Chapter 1
Introduction
1.1
Chemistry and Renewable Energy
The search for renewable energy conversion and energy storage technologies is a central scientific and engineering problem of our time. The reliance of modern civilization
on fossil fuels as its main energy source is a practice that cannot be sustained in coming century; for many reasons, this problem is greatly serious and urgent. On the
one hand, fossil fuel resources are non-renewable and becoming increasingly scarce,
and the absolute requirement of developed and developing nations for a secure and
steady stream of energy is a primary compelling factor in international tensions and
global conflicts. In addition, the extraction and consumption of fossil fuels comes
with great collateral environmental impact. The irreversible release of carbon dioxide
into the atmosphere due to the consumption of fossil fuels is expected to cause major
disruptions in global climate over the next several decades via the greenhouse effect;
however, one need look no further than the Deepwater Horizon oil spill or the destruction of the Appalachian Mountains in mountaintop removal coal mining to witness
the more immediate and obvious environmental impacts involved in harvesting these
fuels. Thus, renewable alternatives to fossil fuels are desperately needed for the sake
of a stable, peaceful, and sustainable future civilization.
Yet, currently there is no single alternative source of energy that rivals fossil fuels
in terms of usability and convenience. Alternative energy sources including sunlight
Figure 1-1: Photograph of mountaintop removal coal mining, in which entire mountaintops
are dynamited and pulverized in order to mine the underlying coal seams.
and solar-derived kinetic energy (this includes wind, rivers, and tides) cannot be
harnessed directly in most modern applications, and the wide spatial and temporal
variations in their availability is at odds with the requirements of civilization. Thus,
alternative energy sources must be converted into usable forms and stored in order to
match the variations in supply and demand. The need for improved energy conversion and storage technologies presents a great challenge and opportunity for science
and engineering; indeed, most energy-related research fits into this overall framework,
including solar electricity and thermal generation, fuel-forming and fuel cell catalysis, hydrogen storage, wind and hydroelectric turbine design, biomass and biofuel
generation, thermoelectric power, and electricity distribution just to name a few.
The fundamental challenges of energy conversion and storage can be understood
thoroughly in terms of chemistry, and this sets the stage for chemistry to make great
contributions to these important problems. In this rest of this Chapter, I will first
attempt to frame these fundamental challenges using basic concepts of chemistry,
drawing specifically on our knowledge of catalysis and excited-state processes.
I
will illustrate how these concepts are united in biology by the incredibly efficient
photosynthetic machinery that performs solar energy conversion and storage for all
living things on Earth. I will then give an introduction to the methods of theoretical
and computational chemistry, and the ways in which these methods can be applied to
improve our understanding of both natural and synthetic energy conversion systems;
the unifying theme is that theoretical and computational chemistry has the ability to
provide richly detailed pictures of chemical processes on the smallest and most rapid
timescales, granting insight that can explain and validate an experimental result and
perhaps assist in guiding the next experiment. This will be followed by an exposition
of my thesis project, which encompasses several studies of energy conversion-related
chemistry that touches on both the method development and application aspects of
theoretical and computational chemistry. In rough chronological order, the topics of
my research are:
* A hybrid quantum/classical (QM/MM) model of organic semiconductor materials and organic photovoltaics
9 A mechanistic study of a molecular ruthenium water oxidation catalyst
e An explicit solvent method which provides a detailed treatment of solute-solvent
hydrogen bonding in aqueous solvents, and its application to the computation
of standard reduction potentials
e A mechanistic study of a cobalt phosphate water oxidation catalyst, which
makes use of the above explicit solvent model
e A force-matching method for systematically increasing the accuracy of molecular models by fitting empirical potentials to high-level calculations
1.2
Basic Chemistry and its Application to the Energy Problem
The energy that enters or exits a chemical reaction is principally determined by the
difference in chemical bond energies.
Consider, for example, a chemical fuel such
as hydrogen. In the combustion reaction of hydrogen with oxygen, the two H-H
single bonds and an O=O double bond in the reactants are traded for four 0-H
single bonds in the products. The potential energy difference in the chemical bonds
is released as kinetic energy.
Similarly, the reverse' reaction (known as water splitting), given by:
2 H2 0 -- + 2H2+02
AE = 4.92 eV,
converts four O-H single bonds into two H-H single bonds and an O=O double
bond. This reaction is thermodynamically uphill, and the energy that we use to drive
the reaction is stored as potential energy in the chemical bonds of the products. Thus
chemical bonds are a type of energy currency [163], and the large amounts of energy
contained in chemical bonds imparts a high energy density to chemical fuels.
Hydrogen is a carbon-neutral fuel, and its energy density by mass is the highest
known for any chemical fuel. 1 Thus, it would be wonderful if alternative energy
sources could be easily channeled into the water splitting reaction to produce hydrogen
fuel from water. This is much easier said than done. Our everyday experience is
that visible photons (most of which contain 1-3 eV of energy) do not split water
spontaneously, even though it would only take two or three photons to provide enough
energy to carry out one equivalent of the reaction.
Obviously, something else is
required to carry out the reaction in addition to the energy supply. This missing
piece is a reaction pathway or mechanism.
1.2.1
Catalysis
All chemists know that the reaction pathway is an indispensable part of any chemical
reaction. To carry out a desired reaction, one must gain an understanding of the
reaction pathway and control it. For instance, the most trivial pathway would be to
dissociate the reactant molecules into individual atoms by breaking all of the chemical
bonds, and then reconnecting the atoms to make the products. Chemical reactions
are almost never carried out this way, because the individual atoms are much higher
'However, gaseous hydrogen has an extremely low energy density by volume. High-density hydrogen storage represents a major unmet requirement for a hydrogen economy in which chemistry
can provide significant contributions, but we won't go into the details here.
in energy than the molecular reactants or products (in the water splitting reaction,
the energy of the isolated atoms is 19.12 eV higher than the reactants and 14.20 eV
higher than the products). Even if complete dissociation does not occur, at some
point a chemical bond must be broken. This is the origin of the activation energy
or kinetic barrier, and we can see that a kinetically viable reaction pathway must
avoid regions of excessively high energy where many chemical bonds are broken. This
provides the basis for the fascinating field of catalysis.
Generally, all catalysis can be understood in terms of providing low-barrier reaction pathways, but there is an incredibly rich amount of detail in the myriad ways
by which each catalyst lowers the barrier for its corresponding chemical reaction. For
instance, in a simple atom-transfer reaction A-H + B -+ A + H-B the catalyst may
bind to the reactants such that they are brought close together (i.e. H close to B),
thus reducing the entropic penalty involved [210]. The catalyst may reduce the barrier height by bringing the energy of the transition state A ... H ... B closer to that
of the reactant [293], and this can be accomplished either by stabilizing the transition state or by destabilizing the reactant. 2 Alternatively, the catalyst may create an
entirely new reaction intermediate by first accepting H from A and then transferring
it to B. [8] Historically, the mechanisms for most catalysts were elucidated after the
discovery of the catalyst itself, but significant advances have been made in catalyst
design, including the design of new biological enzymes from the ground up [240].
The underlying reasons for precisely how a catalyst modifies the energy landscape
of a reaction pathway can be found in the details of its electronic structure, and often
the best catalysts represent a precise balancing of different interaction strengths;
for example, a catalyst for the reaction AB -+ A + B must bind to AB strongly
enough to destabilize it relative to the transition state, but at the same time it
must not bind A and B too strongly in order to release them [101]. The correlations
between different interactions can lead to the fine-tuning of a catalyst by modifying its
electronic structure once the basic mechanism is known [205]. By now, I have already
2
1n the latter case the binding energy of the catalyst must exceed the destabilization energy of
the reactant or the binding will not occur in the first place - such an energy decomposition may not
be unique.
mentioned several areas in which theory and computation can be highly helpful in
understanding catalysis and reaction pathways in general; this includes (1) performing
molecular simulations for observing the reaction mechanism [82,109,159,195,215], (2)
investigating the electronic structure to understand how the catalyst modulates the
relative energies of intermediates [111], (3) building simple models for reaction rates
that depend only on a few key observables derived from the electronic structure [143],
and (4) optimizing the catalyst by fine-tuning these observables [114,206]. This way,
theoretical and computational chemistry can become a greatly helpful companion to
experimental chemistry in providing both explanatory and predictive power.
1.2.2
Light energy conversion
Solar energy is the most abundant alternative energy source, and a means for channeling the energy of solar radiation into chemical bonds is vital. Conceptually, the conversion of solar energy into chemical energy can also fit into the framework of reaction
pathways (in which the photon participates as a reactant), though here the language
of quantum mechanics is the most intuitive. The first step of any light-molecule interaction is absorption of the photon and excitation of the molecule's quantum electronic
state. Here we shall concern ourselves mainly with the lowest excited state, which
can be understood in terms of independent electrons (i.e. molecular-orbital (MO)
theory) as the promotion of a single electron from the highest occupied molecular
orbital (HOMO) to the lowest unoccupied molecular orbital (LUMO). We know that
water does not interact strongly with visible light; this is because water does not
possess an electronic quantum state at visible energies, for the lowest excited state is
at ~ 7 eV. [91]
As a first step towards solar energy conversion, we need to absorb light using
molecules with excited states in the appropriate energy regime. In this area, chemistry provides us with an wide range of dyes and pigments (also known as sensitizers)
which have excellent absorption properties; most organic dyes and pigments have an
extended conjugated electronic structure which brings the HOMO-LUMO excitation
(usually a ir -+ 7r* transition) to within the visible region, while metal-containing dyes
and pigments are colored by virtue of their d -± d transitions. [153] These molecules
have strong absorptivity or extinction coefficients because electromagnetic waves couple strongly to the large transition dipole moment associated with the excitation. The
energy of the HOMO-LUMO transition and the transition dipole moment are clearly
important properties of any light-absorbing molecule, but as we shall soon see, there
are many other challenges on the long pathway of solar energy conversion.
After light absorption occurs, the excited state is very short-lived, and any process that might harness the energy of this excited state must do so very rapidly.
Most excited states simply relax back to the ground state by emission or coupling
to vibrational modes in the surrounding environment. One way to utilize vibrational
relaxation in energy conversion is to convert light into heat, large amounts of which
can then be used to perform mechanical work or drive chemical reactions. Several
thermochemical cycles have been developed to drive water splitting at very high temperatures, [260,261,269] which can be achieved by solar furnaces; this is a promising
route for industrial-scale energy conversion, but we will not go into the details here.
Alternatively, there are some very interesting pathways by which excited states
can bring about chemical change, and these pathways can be very useful for energy conversion purposes. Processes that proceed via electronic excited states are
advantageous in that they do not require prodigious amounts of heat, enabling solar energy conversion at ambient conditions. For example, an excited state localized
on one molecule can move to another molecule by a process known as energy transfer, [36, 87, 116] or it can cause the excited electron to travel to another molecule
by charge transfer. [18,113,174,181] Energy transfer and electron transfer processes
form the basis of operation of molecular electronics, including several different types
of photovoltaics [64, 112,118,218,220, 255, 273]. Light can also directly cause some
chemical reactions to occur, since an electronic excitation causes a change in the
electronic potential energy surface and allows atoms to rearrange. [279]
The ability of molecules to undergo processes such as energy transfer and charge
transfer after absorbing a photon represents another important step toward solar
energy conversion. Charge transfer is especially important in this regard, because
it provides a pathway for splitting the short-lived excited state into free carriers
(electrons and holes); if these electrons and holes could be separated and collected at
electrodes, their energy can be used to perform electrical work or harnessed using electrochemistry to carry out chemical reactions. An improved understanding of charge
transfer processes would help us to design molecules or materials that easily separate
charge and prevent the reverse process of electron-hole recombination, leading us to
more efficient photovoltaic devices. Efficient energy transfer is also an essential component in most systems that harness light to produce electricity, since large areas of
absorptive material are needed to gather light energy yet charge separation can only
occur in special environments with offsets in energy levels. A widely used paradigm
in biological and synthetic photovoltaic systems is that of an antenna, where a highly
absorptive material is combined with an efficient network that transfers the excitation
energy to smaller regions that promote charge separation. [116]
Theory and computation are capable of providing much insight into light energy
conversion. The physics of energy transfer can be understood in terms of F6rster
resonant energy transfer, Dexter energy transfer, and coherent energy transfer. [181]
These mechanisms of energy transfer are operative in different distance regimes and
chemical environments. In particular, coherent energy transfer is considered to be
the most efficient and most rapid for energy conversion, and recent research indicates
that a moderate amount of quantum coherence, coupled to thermal fluctuations of
the environment, may be responsible for the exceedingly efficient energy transfer processes in biological systems. [49,56,299] Electron transfer between molecules can be
understood in the framework of Marcus theory, [18,174] which states that the electron
transfer process is driven by environmental fluctuations that bring the energies of the
electron donor and acceptor into degeneracy. The key parameters which determine
energy transfer and electron transfer rates in these models can be accessed by computational techniques, including electronic structure theory and molecular dynamics
simulations.
1.2.3
Electrochemistry
Electrochemistry is defined as the electron transfer between a conducting electrode
and a molecule or ion in solution. It is both an indispensible analytical tool in chemistry and a key step in the energy conversion pathway. In the framework of solar
energy conversion, electrochemistry comes into play after a photogenerated excited
state separates into charge carriers, and it remains to convert that electrical energy
into chemical energy [163]. The concepts of electron transfer and oxidation/reduction
half reactions will help us understand the kinetics and thermodynamics of electrochemical reactions, respectively.
The conductive properties of metals are imparted by a highly delocalized continuum of electrons. These electrons reside within continuum of quantum energy levels
called a band. Because the empty levels (the conduction band) are in direct contact
with the filled levels (the valence band), any metal at nonzero temperature has a
thermal population of electrons in the conduction band, giving rise to high electron
mobilities. The energy of the highest occupied electrons, called the Fermi energy,
can be adjusted by changing the electric potential of the metal electrode. When the
electrode is brought into contact with a solute, electron transfer may occur from/to
the solute depending on the relative position of the Fermi level and the energy levels
of the solute's oxidized/reduced states.
Using the language of electrochemistry, the water splitting reaction can thus be
decomposed into water oxidation and proton reduction half-reactions (here I write
them in the reduction direction, according to standard):
4e- + 4H +02 -+ 24H
2 0
2 H+ + 2 e- -+ H2
EO = 1.23 V vs. NHE,
EO = 0 V vs. NHE.
The standardreduction potential of a half-reaction is given by the potential difference
required to drive that half-reaction coupled to a "standard" half-reaction; the NHE
reference half-reaction is just the proton reduction reaction given in the second line.
We can see that the energy input for the overall reaction is 1.23 eV per electron, for
a total of 4.92 eV in one equivalent of the total reaction.
The value of 1.23 V is a thermodynamic potential; that is to say, it reflects the total
per-electron energy difference in the overall reaction. To actually drive the reaction,
a much higher potential needs to be applied, often in the neighborhood of > 1.7
V. [65,80,168] This quantity is known as the overpotential and indicates the large
energy barriers that are present in the pathway that connects reactant and product.
A detailed breakdown of the mechanism reveals that the transfer of four electrons
in water splitting can give four very different reduction potentials, yet the overall
potential required to drive the reaction must necessarily be the largest one. [282]
Thus, in multielectron oxidation-reduction reactions, the energetics of intermediate
oxidation states is important to consider. This provides some insight into why the
water oxidation reaction is so difficult to manage; the simultaneous leveling of four
redox potentials must be a challenging task, especially given that they often correlate
with one another.
Perhaps one of the most interesting aspects of the water oxidation mechanism is
the formation of an oxygen-oxygen bond. The 0-0 bond is a relatively difficult
bond to form in chemistry, as it necessarily requires coupling between two highly
electronegative oxygen atoms. Since oxygen atoms in molecules are relatively electronrich, they are unlikely to form bonds with other electron-rich species. For the 0-0
bond forming reaction to proceed in such asymmetric fashion (in what is called an
acid-base mechanism), one oxygen moiety must be made sufficiently electron-poor in
order for it to be act as a Lewis acid, and the other, relatively electron-rich oxygen
moiety would play the role of the Lewis base. [29] Thus, a good water-oxidation
catalyst must enhance the Lewis acidity of one water molecule and the Lewis basicity
of the other. [186] Here in electrochemistry, we can again see the fine balance that a
good catalyst must always achieve; positive charge must be focused sufficiently so in
order to strip low-energy electrons from water, but not overly so as to avoid excessive
energy penalties. A catalyst with strong oxidizing power may rapidly strip electrons
away from water molecules, but in doing so may create strongly bound oxo groups
which cannot be released to complete catalytic turnover.
An alternative pathway to 0-0 bond formation, which may be applied in synthetic water oxidation environments, is to form oxygen radicals. Because oxygen
radicals have unpaired electrons, it is possible to form an 0-0 single bond by a
mechanism known as mdical coupling. [29] There are challenges associated with performing water oxidation by this mechanism - namely, the need to form two oxygen
radicals in close proximity to one another - but this challenge can be tackled in chemistry by the design of redox catalysts with two metal centers. Indeed, much work has
been done to study the radical coupling mechanism, including the design of many
catalysts containing two metal centers such as the ones in Refs. 287 and 250.
1.2.4
Biology
We can apply our understanding of solar energy conversion and fuel-forming chemistry
to understand the most successful energy conversion process on Earth - photosynthesis
(in fact, our fossil fuels are nothing more than photosynthetic products preserved since
prehistoric times!) In green plants, the first step in photosynthesis begins with the
absorption of light by the photosystem II protein, located on the thylakoid membrane
inside the chloroplast of a plant cell. The many chlorophyll molecules in Photosystem
II are all capable of absorbing light, and their excitation energy is channeled by
exceedingly efficient and rapid energy transfer processes to a key chlorophyll molecule
in the protein's reaction center. [232] The reaction-center chlorophyll then undergoes
an electron-transfer process, transferring the energetic electron in several steps to
reduce a quinone (Q) to hydroquinone (QH 2 ). The holes that result from charge
separation is transferred to a cluster of manganese, calcium, and oxygen atoms called
the oxygen-evolving center (OEC). [97, 315] At the OEC, the water oxidation halfreaction occurs, in which four electrons total are stripped from two water molecules.
The products of this reaction are four protons, which are released into the solvent,
and molecular oxygen. [185,186] The total reaction is:
2 H2 0+ 2Q -+ 2QH2 +0
2
AE = 3.60 eV,
and the corresponding oxidation/reduction half reactions are:
4e- + 4H+ +02 -+
2H+ +2e- + Q -
21120
QH2
E0
1.23 V vs. NHE,
E0 = 0.34 V vs. NHE.
Thus, electrical energy is converted to chemical energy, in the ratio of 0.90 eV per
electron. The quinone carries the reduced electron further along the electron transport
chain where ultimately CO 2 reduction is achieved.
Photosystem II is an exceedingly efficient energy conversion system, and the OEC
is a very efficient water oxidation catalyst. The overall energy conversion efficiency of
Photosystem II, that is the ratio of incident solar energy to stored chemical energy,
is 16%. By comparison, the efficiency of converting solar energy to separated charges
is 20%. The ability of Photosystem II to convert 20% of the incident solar energy
to electrical energy suggests performance comparable to a good silicon solar cell, but
even more impressive is its ability to convert three fourths of this electrical energy
into chemical energy. One fourth of the energy is lost is due to the overpotential of the
electrochemical reactions, and almost all of it can be assigned to the water oxidation
half-reaction. This gives us an estimate of the overpotential of water oxidation at the
OEC, which is approximately 300 mV. [81] This is an impressively low overpotential
for the highly difficult water oxidation reaction, and this combined with the turnover
rate of 500-1000 cycles/s establishes the water oxidation efficiency of the OEC as
unmatched under ambient conditions.
Numerous experimental and computational studies have been performed on Photosystem II and the OEC in particular in order to understand the water oxidation
mechanism. [63,122,188,201] If we focus specifically on the electrochemical step, the
OEC must be oxidized four times to perform one catalytic turnover. Typically, oxidizing a molecule more than once is highly difficult, as the accumulation of positive
charge is energetically unfavorable. Here the process may be facilitated by protoncoupled electron transfer, in which protons are released into the solvent along with
the oxidation event in sequential or concerted fashion. [192] This prevents the ener-
getically unfavorable buildup of positive charge. Another factor that facilitates this
process is the delocalization of positive charge; since the OEC contains more than
one metal atom, positive charge may delocalize across the entire cluster, once again
avoiding the energetic penalties associated with "focusing" the positive charge on
a single point. However, the OEC does not have an overly extended structure, as
a highly delocalized positive charge affords no additional oxidizing power over bulk
materials.
Theoretical studies have indicated that the 0-0 bonding step in the OEC proceeds via the acid-base mechanism. While one water molecule is bound to the cluster
and is oxidized to a metal oxo group, the other water molecule is simultaneously preoriented in order to facilitate 0-0 bond formation and made more basic by nearby
proton-accepting groups on the protein. [258] The fine-tuned electrostatic control over
both reactant water molecules in the OEC may be one of the keys to its high efficiency,
and perhaps synthetic catalysts can also benefit from controlling the orientation and
electronic character of nearby solvent molecules.
1.3
Theoretical Chemistry: Foundations and Applications
The research content of this thesis is based on theoretical and computational chemistry. During the course of my thesis project, I have applied established methods in
theoretical chemistry to understand the mechanism of electrochemical water oxidation. My earlier investigations indicated several ways in which the existing theoretical
methods can be improved, so a significant portion of my project was dedicated to the
re-examination of established assumptions and the improvement of existing theoretical methods.
Since my thesis project involves theoretical chemistry on both the
method development and application sides, here I shall give an introduction to the
theoretical methods in this work and briefly explain how these methods can be applied
to our problems of interest.
In the broadest sense of the word, theoretical chemistry is the application of
physics to explain and predict chemical phenomena. Here, I will not attempt to
provide a survey of the entire field, and instead I shall focus on the small subset of
theoretical chemistry that is most relevant to the current work.
1.3.1
Quantum Chemistry and Electronic Structure Theory
A proper description of molecules must necessarily describe the quantum mechanical
behavior of the electrons. A full nonrelativistic description of a molecule's stationary
quantum state is available from the solution of the time-independent Schr6dinger
equation:
HIT) = E|W),
11
HW(r, R) = (Tn+ Te + Vn, + Ve + Ve)(r, R)= ET[(r, R).
(1.2)
or, in the position representation,
Here r and R represent the electronic and nuclear coordinates, respectively.
All terms in the Hamiltonian operator H can be explicitly written down; they
amount to the nuclear and electronic kinetic energies, and the Coulomb interactions
between all nuclei and electrons. It remains to solve this linear partial differential
equation for the eigenfunctions (quantum states) |W') and eigenvalues (energies) Ej.
Yet, the Schr6dinger equation is not analytically solvable for any system more complex than a single electron interacting with a single nucleus, and finding the solution
numerically requires an exponentially scaling computational cost. [267] Thus, approximate solutions must be found. In fact, a large part of a theoretical chemist's training
consists of finding the most appropriate approximations for treating a specific chemical problem.
Often we apply the Born-Oppenheimer approximation, which separates nuclear
and electronic degrees of freedom and decomposes the Schr6dinger equation into an
electronic part and a nuclear part. The electronic part is given by:
He(r; R)" 'e(r; R) = Ee (R) Te(r; R).
(1.3)
This equation resembles the full Schr6dinger equation, but the key difference is that
We represents only the electronic part of the quantum state. The nuclei enter into
the electronic Hamiltonian as fixed potential wells, and the nuclear kinetic energy
operators are removed. Thus, the electronic quantum state and electronic energy
depend parametricallyon the nuclear coordinates. This electronic energy then enters
into the nuclear Schr6dinger equation as a potential energy surface:
(Tn + Ee(R))'Fn(R) = En'W(R).
(1.4)
The Born-Oppenheimer approximation allows us to treat the vast majority of problems in quantum chemistry by solving the electronic Schrbdinger equation with a set of
fixed nuclear coordinates. In certain situations the Born-Oppenheimer approximation
does break down (for example, when two potential energy surfaces become degenerate
or during proton-transfer events with significant nuclear quantum effects [119, 161]),
but for most situations it is appropriate. From here, I will use |T) to denote the
electronic quantum state
IIe),
because the treatment of nuclear quantum effects is
not a part of this thesis.
After separating the electronic and nuclear degrees of freedom, solving the electronic Schradinger equation is still a major challenge. An approach to find the ground
quantum state is to apply the variationalprinciple,which states that the ground state
is that which minimizes the energy expectation value, given by:
Eo = (To IH1I'o)
(TOIT)
(1.5)
The wavefunction contains the positions of every electron in the system; for manyelectron systems, the true wavefunction is a vastly complicated function of a huge
number of variables.
The formulation of a good approximate functional form for
the wavefunction, or variational ansatz, for the variational problem is an active and
ongoing field of research in physics and chemistry; we will only consider a small subset
of wavefunctions most familiar to quantum chemists. First, to get a better grasp of
this problem, we can consider a very basic wavefunction ansatz by forming a direct
product of N single-electron solutions to Equation 1.3:
I(Xi, x2 , ...
, XN)H
= Xi(X1)Xj(X2) ... Xk(XN),
(1-6)
Here Xi(xa) is a wavefunction for a single electron, called a spin orbital,which depends
only on a single electron's spatial and spin degrees of freedom. Note that a product
wavefunction can only fully solve Schr6dinger's equation if the Hamiltonian doesn't
contain any electron-electron interactions; here we are only forming a variational
ansatz, not searching for the full solution.
This product wavefunction WH is called a Hartree product. The Hartree product
does not solve the electronic Schrddinger equation for two reasons; for one, it neglects
exchange antisymmetry, which is the property that the wavefunction must change sign
when any two fermion labels are exchanged. The wavefunction also does not include
electron correlation,which is the tendency for electrons to avoid one another due to
their mutual repulsion. The exchange antisymmetry can be included by building a
determinant from the one-electron orbitals:
X1 (x1)
Xi(x2)
...
Xi(XN)
Xk(X1)
Xk(X2)
...
Xk(XN)
This wavefunction is called a Slater determinant. While the Slater determinant does
not contain any electron correlation, it does possess the property of exchange antisymmetry.
Note that the Slater determinant can only exactly solve a Schr6dinger equation
with no electron-electron interactions. But what happens if we use it as a variational
ansatz for a fully interacting problem? The result is that we obtain a set of singleelectron equations where each individual electron's potential energy function contains
the mean field of the other electrons, including the effects of exchange antisymmetry.
Thus, electron-electron interactions are captured in an average sense, though electron
correlation is still missing. In a wide class of problems, the Slater determinant is
a remarkably accurate wavefunction ansatz, although the contribution of electron
correlation is still required in order to obtain the high levels of accuracy required by
chemical problems.
Once we have a wavefunction ansatz, the task is to variationally minimize the
energy. Any variational wavefunction ansatz must contain variational parameters,
and the Slater determinant is no different. The N one-electron orbitals in the Slater
determinant are built from linear combinations of X spatial basis functions, where
X > N -
these can either be plane waves or spatially localized functions called
atomic orbitals:
#o.
Xi=
(1.8)
P
By varying the linear combination coefficients, the one-electron orbitals are adjusted
and the energy expectation value is minimized to obtain an variational ground-state
wavefunction. The variational minimization of the energy of a Slater determinant is
known as Hartree-Focktheory.
Hartree-Fock theory is a rigorous treatment of independent electrons and thus
forms the foundation of a large class of quantum chemical methods that provide a
treatment of electron correlation. Generally these methods capture electron correlation effects either by expanding the flexibility of the wavefunction ansatz or using
perturbation theory. One such class of methods known as configuration interaction
(CI) allows the wavefunction to be built from several Slater determinants (where
Hartree-Fock uses only one). If the wavefunction ansatz contains all C(X, N) possible Slater determinants that can be built from a set of basis functions, it forms a
complete forms a complete space of antisymmetric N-variable functions in that basis
set. Variationally minimizing this wavefunction provides a full treatment of electron
correlation in what is called full configuration interaction (CI). Full CI is impossibly
expensive for all but the smallest systems, and almost all CI methods will cleverly
choose a small subset of configurations. A wavefunction that contains significant
contributions from a small number of configurations is said to contain static correlation, while those that contain small contributions from many configurations is said to
contain dynamic correlation. Many methods exist for capturing dynamic correlation
effects without explicitly constructing the different configurations, including perturbation theory and coupled cluster theory. In general, wavefunction-based treatments
of electron correlation involve a large computational cost and scales unfavorably with
the number of electrons in the system; an alternative treatment of electron correlation
is density functional theory, which I briefly describe below.
Today, density functional theory (DFT) is perhaps the most widely used treatment of electronic structure. Conceptually, DFT is fundamentally different from the
wavefunction-based methods described above. The Hohenberg-Kohn theorem [127]
provides the mathematical rigor of DFT; it proves the existence of a one-to-one correspondence between the quantum mechanical Hamiltonian and the ground-state electron density. Consequently, the ground-state electron density uniquely determines
the ground-state wavefunction. This is an important and nontrivial statement. Consider that the electron density is merely a population map of the wavefunction's
intricately interacting electrons, a statistical representation of the wavefunction in
three-dimensional space; the existence of a one-to-one mapping between the density
and the wavefunction proves that the wavefunction contains no more information
about the system than the ground-state density, and all of its additional complexity
originates not from the specific characteristics of the system but from universal laws
of Nature. [212]
The existence of this one-to-one correspondence has great implications, perhaps
the most important of which is the existence of an exact density functional for the
electronic energy, E[p], which provides an exact mapping between the ground-state
electron density and the ground-state energy of the quantum mechanical system. The
Hohenberg-Kohn theorem also provides a variationalmethod for electron densities,
stating that the exact density functional assumes its minimum for the correct groundstate density. Thus, the solution of the Schrbdinger equation can be recast in terms of
variationally minimizing the density functional. This statement carries the promise
of solving the electronic structure problem at a greatly reduced computational cost,
as it may be possible for one to evaluate the energy directly from the density without
having to involve any wavefunctions.
The greatest challenge of DFT is that the exact density functional is not known
(although its existence is proven by the Hohenberg-Kohn theorem). Furthermore, not
enough is known about the exact density functional in order to design systematically
improvable approximations. However, significant strides have enabled the development of highly accurate density functional approximations, perhaps the greatest of
which is the Kohn-Sham (KS) method which I shall describe below.
The Kohn-Sham (KS) method [154] provides a foundation for highly accurate density functional approximations. The basic problem can be understood by performing
a basic decomposition of the electronic energy into kinetic and potential energy contributions:
E[p] = T[p] + V[p].
(1.9)
Most density functional approximations to the electronic kinetic energy T[p] are wildly
inaccurate. This is because the electronic kinetic energy is largely a property of individual electrons. In wavefunction-based methods, the kinetic energy is easily evaluated (it is a sum of one-electron operators), but this becomes a difficult problem
in DFT because the electronic coordinates are lost. The KS method treats the electronic kinetic energy by introducingfictitious electronic coordinates. This is done by
introducing a Slater determinant of one-electron orbitals (the Kohn-Sham orbitals)
which shares the same electron density as the true system. There is no rigorous
correspondence between the Kohn-Sham wavefunction and the true electronic wavefunction; nevertheless, applying the kinetic energy operator to the KS wavefunction
gives us a highly accurate estimate of the electronic kinetic energy. Intuitively, this
method works so well because most of the electrons' kinetic energy originates from
their independent movement.
With the KS method, one now has a highly effective energy partitioning with
which to design accurate density functional approximations. Within this framework,
the total electronic energy is given by
E[p]
=
T.[{oi}] +
2
J
p('~'r dr dr' +
|fr - r'|
v(r)p(r) dr + V.'[p].
(1.10)
where the first three terms (the electronic kinetic energy, the classical electron selfrepulsion, and the interaction between the electron density and the external potential)
account for most of the electronic energy. The fourth term, known as the exchangecorrelation functional, contains all of the remaining nontrivial effects and remains
to be approximated. The literature abounds with approximations to the exchangecorrelation functionals, and the most popular general-purpose density functionals such
as B3LYP [25] and PBE [216] achieve remarkably high accuracy for many types of
calculations. However, every density functional approximation has a different strategy
and makes a different set of assumptions, and much care must be taken in choosing
the most appropriate approximation for each specific problem of interest.
Over the last few decades, countless contributions have been made to electronic
structure theory, and it would be impossible to list them all in this thesis. Some of
the most contributions in electronic structure theory that have been very helpful to
my research projects, include but are not limited to:
" The construction of charge-transfer electronic states via application of constrained density functional theory. [301,302]
* The treatment of excited electronic states, either via direct construction of the
wavefunction [155] or by measuring the response of the wavefunction to an
oscillating electric field. [93]
" The computation of couplings between different electronic states in DFT [88,
303,304] and using these couplings to predict Marcus electron transfer rates [89]
or to construct multiconfigurational wavefunctions. [138,300]
o The approximate treatment of solute-solvent interactions using implicit and
explicit solvent models. [16,48,152]
1.3.2
Statistical Thermodynamics and Molecular Dynamics
Simulation
Electronic structure methods may be well equipped to describe the energies of fixed
and isolated molecules, but to gain a complete picture of a chemical reaction one
needs a different set of tools designed for exploring the space of configurations and
momenta, also known as phase space. As molecules explore the phase space, their
properties no longer correspond to a single configuration, but rather to an ensemble
average over the entire configuration space. 3
In a canonical ensemble (i.e. constant volume, temperature, and number of particles), the probability density P of encountering a particular configuration r is proportional to its Boltzmann factor:
P(r) = -e-E(r)(
Z
Here, Z is the partition function which is the integral of e-jE(r) over all space. Thus,
the thermally averaged observable of any molecular system at a constant temperature
can in principle be obtained by calculating the property at every point in configuration
space and then performing a weighted average over the Boltzmann distribution using
the integral:
(A)
j
P(r)A(r) dr.
(1.12)
JR3N
Clearly, evaluating this integral directly is not practical for computational applications. For one, a numerical integral over the 3N-dimensional configuration space
requires an exponentially increasing number of points with the number of atoms.
Furthermore, most points in configuration space are high in energy, and their contribution to the average quantity is expected to be infinitesimal given that their
3
In this thesis we are mainly interested in molecular properties that depend on the positions of
the atoms rather than their momenta, so the momentum degrees of freedom are integrated out.
Boltzmann factors are exponentially small. Fortunately, many thermal averages can
be computed accurately by importance sampling; that is, the integral is evaluated
at points chosen by their thermal probability using techniques like the Metropolis
algorithm. This gives us a protocol for computing thermal averages of computable
single-point properties: one samples the configuration space using the Metropolis
algorithm, computes the property at each sampled point, and takes an unweighted
average over the samples. [9,102]
As a simple example, if we wish to compute a thermally broadened absorption
spectrum, we would first use a molecular dynamics or Monte Carlo simulation to
sample the configuration space of our system. We then compute the excitation energies at every sampled configuration, or at a fixed interval. The level of theory
used in computing the spectrum need not be identical to that used to sample the
configuration space, although it is desirable that the two methods are of comparable
accuracy. Finally, we make a histogram from the many peaks, or give each peak a
small Gaussian width, to obtain our final thermally broadened spectrum. There are
two things to note. First, we typically obtain converged measurements long before
complete sampling of the configuration space; the length of the trajectory needed
to obtain convergence often depends on the particular observable of interest. Sampling of the entire configuration space would suffice for computing every observable
imaginable, but for most observables this is not necessary. Second, the Boltzmann
factors are not needed in the average because the importance sampling is implicitly
performed by the simulation. The computation of thermal averages is an important
method for several of the studies in this thesis, not least because the method for
computing a standard reduction potential is closely related to computing thermally
averaged ionization potentials and electron affinities.
Only a subset of interesting properties are computable using thermal averages
of configuration space, and perhaps the most important exception is the free energy.
From statistical mechanics, the free energy is given by the log of the partition function,
F = kbT log Z,
(1.13)
and such a quantity is, by its very nature, not a thermal average. Rather, the free
energy is really a measure of the amount of available phase space for a particular
system. It is nearly impossible to simulate an absolute free energy directly. However,
derivatives of the free energy are thermal averages and clever integration schemes can
be employed to find free energy differences between properly defined states.
Molecular dynamics (MD) is a simulation method that can be used to compute
many of the interesting equilibrium properties that I described above. Essentially,
MD is a computer simulation of molecular motion generated from integrating the
equation of motion for the nuclei. Many different methods fall under the framework
of MD, and they are mainly differentiated by (1) the theory used to determine the
potential energy and forces, and (2) the theory used to generate the nuclear motions
from these forces. The simplest method for obtaining nuclear motion is using the
Born-Oppenheimer approximation. Within the Born-Oppenheimer approximation,
differentiating the electronic energy with respect to the nuclear coordinates gives us
a set of forces acting on the nuclei:
F = -VREe(R) = ma.
(1.14)
In a Born-Oppenheimer molecular dynamics (BOMD) simulation, these forces are
used to move the nuclear positions according to Newton's equations of motion. [20]
This is an acceptable approximation as long as nuclear quantum effects are negligible, and we expect this approximation to hold for our studies. However, in certain
situations (i.e. proton tunneling or degeneracies in the electronic states) the BornOppenheimer approximation breaks down. There are several treatments for situations where quantum nuclear effects are important, including quantum wavepacket
propagation methods [305,309], path-integral molecular dynamics [50,177,278], and
ring-polymer molecular dynamics [73] to name a few.
The other ingredient to running a MD simulation is a method to compute the energy and the force. If quantum methods are used to compute the energy, the resulting
simulation is known as ab initio molecular dynamics (AIMD). Generally, AIMD sim-
ulations are computationally very costly as they require one quantum calculation per
simulation time step (~ 1 fs), and these simulations rarely exceed 100 ps (100,000 calculations) in length. However, one notes that in AIMD all of the quantum mechanical
behavior is encoded in the potential energy surface (PES), and the nuclei undergo
purely classical motion on this complex surface. Thus, it is possible to approximate
the quantum PES with a simple function of the nuclear coordinates. Such a function
is known as an empirical potential or a force field; for example, a simple force field for
diatomic hydrogen would be a harmonic oscillator with an equilibrium length of 74
pm and a frequency of 1.3x 1014 Hz. MD simulations employing force fields are known
as molecular mechanics (MM), and their computational costs are several magnitudes
lower compared to AIMD; this allows one to access much larger molecules and longer
simulations, and so MM is highly popular in biomolecular simulations. [9,102]
The key assumption of MM is obvious; the molecular behavior in any MM simulation depends completely on the functional form and parameters of the force field.
The MM simulation only corresponds to reality when the force field is an accurate
description of the true PES, and so an accurate choice of functional form and parameters is essential. The design of accurate force fields is great challenge in the field of
molecular simulation, and there are many possible strategies for force field design, including fitting to experimental data or higher-level quantum calculations. A portion
of my thesis research is dedicated to this important problem.
Many applications of theoretical chemistry require a simulation that can provide
a highly accurate description of the reactants and also capture the effects of the
surrounding environment. In these situations, the environment is expected to affect
the reaction energetics or kinetics in a significant way without directly participating
in the chemical transformation. For these applications a precise description of the
reactants is required, but it may still be possible to treat the environment and the
environmental effects in a relatively approximate way. One treatment is to divide
the system into quantum mechanical and molecular mechanical subsystems; this is
known as a hybrid quantum/classical (QM/MM) framework [3]. As long as the QM
and MM subsystems are not strongly coupled, their interactions can be described
using a relatively simple QM/MM interaction Hamiltonian. Molecular dynamics in
general, and specifically the QM/MM framework, are essential tools for several studies
in this thesis.
1.4
Exposition:
Theoretical Investigation of So-
lar Energy Conversion and Water Oxidation
Mechanism
The previous two sections provides us with a framework for introducing my work on
investigating solar energy conversion and water oxidation mechanism using theoretical
chemistry methods. My work is focused on synthetic, rather than biological, systems
due to their relative simplicity and more immediate relevance to alternative energy
applications.
1.4.1
QM/MM Investigation of charge separation in an organic donor/acceptor interface
An organic semiconductor is, relatively speaking, a new concept. One usually thinks of
semiconductors as being an inorganic solid-state material with a medium sized band
gap, or HOMO-LUMO gap in the language of chemistry. Organic semiconductor
molecules have the requisite electronic properties; usually they contain extended grelectron systems that reduces the size of their HOMO-LUMO gap, making them
interesting candidates for semiconductor applications.
The most interesting aspects about organic semiconductors are their differences
from inorganic semiconductors. Organic semiconductors are flexible materials, and
the molecules are soluble in organic solvents; this opens the door for many novel
applications. For example, one could imagine flexible display screens, or solar cells
being manufactured by a printing press.
The promise of organic semiconductors for renewable energy applications is great,
but even the most modern materials face serious limitations. The power conversion
efficiency of even the best organic solar cells is less than half that of conventional
silicon cells. To make matters worse, organic semiconductors are much more prone
to photodegradation compared to their inorganic counterparts, and this shortens the
lifetime of the device. Perhaps if simulations were able to gain some insight into
the microscopic processes that enhance or limit solar cell efficiencies - for example,
charge recombination, exciton diffusion, or even photodegradation, theory could prove
a helpful guide to experiment.
As the first step toward this goal, we developed a new method for simulating
organic semiconductor materials, as they are dramatically different on the molecular
scale compared to their inorganic counterparts. London dispersion is the main intermolecular interaction, giving rise to highly disordered structures. Disorder leads to
localization of charge carriers and excitations, which must be described by hoppingtype models. These highly disordered systems lead to a wide dynamic range of energy
levels, couplings, and other parameters.
We judged a QM/MM method to be an ideal framework for describing these disordered organic semiconductors. For one, the localized nature of the excited state
requires only one (or perhaps two) molecules to be treated using the full quantum
theory, while the others can be described using a much simpler classical model. Ensemble averages of properties can be found by averaging the computed property over
a molecular dynamics trajectory. The QM/MM interaction Hamiltonian is expected
to be a good approximation to the true Hamiltonian so long as there are no strong
electronic interactions between the QM and MM regions; this is expected to hold in
these weakly charged, dispersion-bound systems.
We applied the QM/MM method to simulate an organic photovoltaic device consisting of a donor/acceptor interface between two organic semiconductor molecules,
H2 Pc and PTCBI. Using the constrained-DFT method [301], we computed the energies of charge-transfer (CT) states at the interface and compared it to the energies of
the free carriers.
Our study had two key findings. The first finding was that the CT states had
a very broad energy distribution that depended on molecular orientation as well as
distance from the interface. This implies that CT states have alternate mechanisms for
lowering their energy, rather than simply returning to the interface and recombining;
in other words, the thermal fluctuations allow them to dissociate yet still decrease in
energy.
Our other key finding was that CT states were globally unbound with respect to
the infinitely separated free carriers. This was quite a strange result, given that CT
states are nominally held together with a Coulomb binding energy. Our question was
answered when we computed the energy of free carriers at the interface, and found
their energies to be higher than in the bulk. This is known as a band bending effect,
and it was caused by a decreased dielectric screening at the interface, which in turn
was caused by irregularities in the molecular polarizabilities and crystal packings.
The band bending effect raised the energy of interfacial free carriers, allowing CT
states to remain locally bound even as they were globally unbound.
1.4.2
Mechanistic investigation of a ruthenium water oxidation catalyst
The earliest molecular synthetic water oxidation catalyst is widely cited to be the
ruthenium "blue dimer", discovered in 1985. [108] The blue dimer contains two octahedrally coordinated ruthenium centers, bridged by an oxygen atom. On each ruthenium center, four coordination sites are occupied by bidentate bipyridine ligands, and
the remaining coordination site is bound to water. The blue dimer catalyzes water
oxidation at an extremely acidic pH range and has a very low turnover number; nevertheless, it has generated an intense amount of interest due to its ability to catalyze
this extraordinarily difficult reaction.
Several aspects of the water oxidation mechanism by the blue dimer are known.
For example, the resting state of the catalyst contains two Rur1 centers and two
coordinating water molecules (denoted here by (H2 0)Ru"'ORum'(OH 2 ). Under the
experimental conditions, three oxidation event proceeds with the concurrent loss of
four protons to the solvent, until the mixed-valence electronic state (O)RuIVORuv(O)
is reached. The fourth oxidation event is the most thermodynamically demanding and
rapidly leads to an intermediate containing an 0-0 bond, (HOO)RuIII(O)Ruv(O).
It is conjectured that the reaction proceeds through a short-lived (O)RuvORuv(O)
oxidation state, and that the 0-0
bond forms by an acid-base mechanism in which
a water molecule attacks one of the electron-poor Ruv(O) sites. [168]
The blue dimer coordinates to two water molecules, and this had led people to
think that two metal centers was required for water oxidation. In the following two
decades, it was widely believed that the presence of two metal centers implied some
type of direct coupling mechanism. Thus, a number of molecular water oxidation
catalysts containing two metal centers were developed [250,287,291,292]; their performance, so far, has been mixed.
Recently, there have been several surprising discoveries of molecular water oxidation catalysts containing only one metal center. [66,178,183,277,327] Both ruthenium and iridium one-center catalysts have been discovered, and the general catalyst
structure is largely the same; the metal center is generally coordinated to pyridine-like
complexes, leaving one coordination site open for water. The ability of one-center catalysts to oxidize water suggests that simpler mechanisms may be possible than were
thought previously. Several experimental mechanistic studies have been performed
on one particular single-center ruthenium catalyst [58,65,294]; the experimental evidence points toward an acid-base mechanism, which is plausible given that only one
metal center participates in the reaction. However, questions remain concerning the
mechanism. Can the direct-coupling mechanism be ruled out? What is the electronic
structure of the species immediately prior to forming the 0-0
bond? What are some
of the possible side reactions that lead to catalyst inactivation? Does the atomistic
picture provided by theoretical calculations agree with the experimental result?
I will attempt to provide some insight into the water oxidation mechanism by
a single-center ruthenium catalyst. Using density functional theory combined with
an implicit solvent model, we computed the one-electron reduction potentials corresponding to each of the four oxidation events that take place over the course of
the catalytic cycle. The catalyst starts in a Ru" (OH 2 ) resting state. The first two
oxidation events are proton-coupled and do not change the charge of the catalyst
molecule, leading to RuIV(O); here our computed values are in good agreement with
experimental measurements [66]. The third oxidation event is the most interesting
one, for it corresponds to the oxidation of Ruiv(O) to Ruv(O). This oxidation is exceptionally demanding because there are no more protons that can leave in order to
balance the charge. The Ruv(O) species is not experimentally observed; if it were to
occur at all, it would react extremely rapidly. One of the goals of our study is to use
theoretical calculations to compute the reduction potential of the RuIv/RuV couple,
in order to assess the validity of invoking Ruv(O) in the mechanism.
We computed the redox potential of the RuJV/RuV couple to be 1.98 V v. NHE,
which was 330 mV higher than the potential of the experimental catalytic wave. That
is to say, our computation suggests that the Ruv(O) species is much more energetically demanding than experimentalists claim. There are three possible solutions to
this disagreement. One possibility was that our theoretical method was not able to
accurately describe the energy of the RuV(O) oxidation state. Another explanation
was that the oxidation of Ruiv(O) to Ruv(O) could be thermally activated (i.e. the
330 mV is part of the activation energy), and the product is consumed so rapidly that
the reaction is driven forward by Le Chatelier's principle. The third explanation was
that the Ruv(O) species does not actually exist even as a transient intermediate, and
we proposed an atom-coupled 0-0 bonding pathway which avoids the formation of
Ruv(0).
While the atom-coupled pathway was potentially the most interesting, it also implied a dependence of the catalytic potential on pH, which was not experimentally
observed. [58] The accuracy of our model could have been tested if there were a wealth
of experimental data on RuIv/Ruv redox couples that we could use to calibrate our
theory (we did not locate such a dataset, likely because any Ruv-containing species is
likely to be highly reactive). Thus, we concluded that an activation barrier may exist
at the experimental potentials for Ruv(O) oxidation, but also that our model may
have been insufficiently accurate for the computation of the redox potential. Nev-
ertheless, we went forward with the mechanistic investigation, keeping the potential
flaw of the model in mind for a later project.
Assuming that the Ruv(O) species is experimentally accessible, we wished to investigate its reactivity with water. Our finding was that Ruv(O) immediately reacted
with surrounding water molecules to produce an 0-0 bonded species, Ru"'(OOH),
in a classic nucleophilic attack mechanism. Interestingly, the reaction could only proceed if some other water molecules were present nearby to accept the proton that
is released; this showed that 0-0
bonding in the acid-base mechanism is strongly
proton-coupled, and lends credence to the hypothesis that this step can be kinetically
enhanced if the basicity of spectator water molecules could be locally increased. By
visualizing the electronic structure of the catalyst and attacking water, we found that
the 0-0
bond is formed from combining the HOMO of the water molecule with the
LUMO of the Ruv(O). The LUMO has a nodal surface on the Ru-O bond, so as
the 0-0
bond is formed the Ru-0 bond is weakened. This is expected to facilitate
the release of molecular oxygen at the end of the catalytic cycle.
From Ru"(OOH), one more oxidation is required to extract all four electrons in
one catalytic turnover. We computed the last reduction potential, bringing us to
RuIV(OO), and we found it to be much lower than that of the previous Ruiv/Ruv
couple. One proton is released to the solvent and the Ru-0 bond weakens even
further. Finally, adding a single water molecule displaces the bound molecular oxygen
and returns us to the resting state; this final step occurs in SN2 fashion with a small
activation barrier.
We also investigated the mechanism of a two metal-center catalyst to see whether
it could also catalyze water oxidation at a single metal center. We found this indeed to
be the case; by applying the same mechanism as above, we were able to carry out the
entire water oxidation cycle on a single catalytic subunit. By contrast, our attempts to
locate a direct-coupling mechanism failed completely, as a strong Coulombic repulsion
between the catalytic subunits prevented them from coming together.
Our investigation provided a clear atomistic picture of water oxidation via the
acid-base mechanism in a single-center catalyst and a two-center catalyst. With the
results of this study, we wanted to apply our methodology to somewhat more complex water oxidation systems with improved efficiencies and lower overpotentials. We
also began to suspect the implicit solvent methodology for computing redox potentials in aqueous solvent, especially in situations where there are strongly localized
electrostatic interactions from hydrogen bonds that are not well described by the
homogeneous implicit solvents. This motivated the next two projects for improved
computation of reduction potentials and the mechanistic investigation of a cobalt
phosphate water oxidation catalyst, described below.
1.4.3
Improved computation of standard reduction potentials
In the last section, where I described a mechanistic investigation of a ruthenium
water oxidation catalyst, the standard reduction potentials corresponding to the four
oxidation events in the catalytic cycle were computed using a combination of density
functional theory (DFT) and an implicit solvent model (ISM). The results of our
study led us to question the appropriateness of applying the approximations of ISM
to the water oxidation problem. More specifically, we wanted to test the validity
of the ISM approximation for computing standard reduction potentials in hydrogen
bonded species.
ISMs have been in development since the introduction of the Born solvation model
in the 1920s [34], and they are commonly employed as a companion to quantum
chemical calculations as a simple description of a solvent environment. The most
important part of an implicit solvent model, not least for the purpose of computing
standard reduction potentials, is its description of the solvent's dielectric screening.
Both polar and nonpolar solvents can provide dielectric screening in that the electron
density on the solvent molecules are able to polarize in response to the solute electric
field. This electronic polarization happens on a much faster timescale than the nuclear
dynamics. Polar solvent molecules have the additional ability to reorient themselves in
response to the solute electric field, providing additional screening via their molecular
dipole moments. Both of these physical effects contribute to the solvent's dielectric
constant, though the electronic and orientational components can be separated using
rapidly oscillating electric fields.
Most ISMs are comprised of two simple quantities; the dielectric constant of the
solvent and the solute cavity. The solute molecule is placed inside the solute cavity,
and the solvent is described by assigning a homogeneous dielectric constant to the
space outside of the cavity. The quantum chemistry computation provides the electron
density of the solute, and the implicit solvent provides a polarization response to the
solute's electrostatic potential. Because the electronic structure of the solute and the
dielectric response of the implicit solvent influence one another, the two quantities
are solved self-consistently in a fashion highly analogous to an ordinary (solvent-free)
electronic structure calculation. For these reasons, ISMs have become highly popular
as simple and cost-effective descriptions of the solvent. [16,48,152,241,296,297]
One would intuitively expect ISMs to perform well in situations where the solvent/solute interaction is indeed well characterized by a homogeneous dielectric response. Indeed, the performance of ISMs is best for nonpolar organic solvents with low
dielectric constants. Water, on the other hand, is on the complete opposite end of the
spectrum; its very structure is maintained by strong electrostatic interactions between
solvent molecules. Water possesses the ability to form hydrogen bonds, a uniquely
strong intermolecular interaction which blurs the boundary between an electrostatic
dipole-dipole interaction and a covalent bond. If water forms hydrogen bonds with
the solute, one would expect the implicit solvent model to be a poor description.
Indeed, reports in the literature have confirmed that ISMs perform exceptionally
poorly for predicting the redox potentials of transition metal aqua complexes. [60,
165,280] This was the expected finding, because these aqua complexes are among the
most obvious candidates for having strong solute-solvent hydrogen bonding effects
(here we define the "solute" as the metal ion plus its six aqua ligands). Since the
solute-solvent interaction for a water oxidation catalyst is expected to be quite strong
indeed (especially the interaction between water and a Ruv(O) group preceding the
reaction), we judged that future studies on water oxidation catalysts would absolutely
require an improved description of the solvent.
Since our physical intuition suggests an important role for short-range, spatially
inhomogeneous solute-solvent interactions via hydrogen bonding, we decided that
an explicit solvent model employing a hybrid quantum-classical (QM/MM) method
would be able to provide an improved description of the system.
In this type of
simulation, both solute and solvent molecules are explicitly included, but the levels
of theory used to treat the solute and solvent are different. The solute is treated
using electronic structure methods as before. The solvent is treated using a classical molecular mechanics (MM) model, in which atoms are essentially represented
using partial charges and hard spheres connected by harmonic springs; our model
is augmented using mobile partial charges to represent electronic polarization. The
classical approximation greatly reduces the computational cost for explicit solvent
simulations, as an electronic structure calculation on a full solute+solvent system
can be prohibitively expensive. The classical and quantum regions of the simulation
interact with each other through a QM/MM interaction Hamiltonian HQM/MM.
With any simulation it is important to keep track of assumptions and approximations. In a QM/MM explicit solvent simulation, the key assumption here is that
the solute-solvent interaction is well described using the QM/MM interaction Hamiltonian HQM/MM.
The form of HQM/MM is really quite simple, typically consisting
of an electrostatic interaction between the solute electron density and solvent partial
charges, and van der Waals repulsion between the QM and MM atomic centers. If the
true solute-solvent interaction is beyond the description of this model - for example,
if there is any solute-solvent chemical bonding taking place - the approximation is no
longer valid. Nevertheless, our expectation is that an explicit model of water, with
its discrete point charges and heterogeneous electrostatic interactions at short range,
should provide a more complete description of hydrogen bonding than an implicit
solvent model.
Using the explicit solvent model, we computed the standard reduction potential of
nine transition metal complexes in water; this set included seven hexaaqua complexes
which ISMs failed to describe plus the Co(NH 3 )6 and Fe(CN) 6 ions. Each computation
involved averaging over hundreds of electron detachment
/
attachment calculations,
which were performed on configurations extracted from QM/MM molecular dynamics
trajectories. Roughly, each explicit solvent computation was 10' times more expensive
than the ISM computation, largely due to the need to sample the solvent configuration
space. Our results strongly indicated that the explicit solvent model was able to
predict more accurate redox potentials compared to ISMs, which failed completely
for strongly hydrogen bonded systems. With these results in hand, we proceeded to
study a cobalt phosphate water oxidation catalyst discovered in the Nocera lab.
1.4.4
Mechanistic investigation of a cobalt water oxidation
catalyst
In 2008, a cobalt phosphate water oxidation catalyst was discovered in the Nocera lab
which had wide-ranging impact in the water splitting community. [140] This catalyst
was incredibly simple to synthesize, requiring no more than a solution of cobalt ions
and phosphate buffer in an electrochemical cell. At moderate oxidizing potentials
(P 200 meV higher than the thermodynamic potential of water splitting), a thin film
was observed to form on the anode. This thin film proved to be a highly efficient
water oxidation catalyst, with the ability to sustain high catalytic current densities
for long periods of time at a low overpotential
(a
400 meV, only 100 meV higher
than the estimated overpotential of the OEC in Photosystem II.) This system was
able to split water at room temperature and at neutral pH conditions, making it more
versatile compared to commercial and industrial electrolyzers which require extreme
pH ranges or very high temperatures. Moreover, this system makes use of cobalt,
which is a relatively abundant and inexpensive metal compared to the iridium and
ruthenium oxides that are common in industrial applications.
Given all of its ease of synthesis and use, the catalyst has been quite challenging
to characterize. It appears only to form as an amorphous film with no long-range
ordering; structural models of the short-range order have been proposed. [142] Extensive mechanistic studies have determined that the main rate-determing step is purely
chemical in nature [170,265,266] (i.e. does not involve electron or proton transfer),
but the exact nature of 0-0 bonding is unknown. Some evidence of the metal oxidation state has been reported; electron paramagnetic resonance studies suggest the
presence of Co'v. [182]
Compared to the ruthenium catalyst study, the cobalt phosphate catalyst is a
much more challenging system to study theoretically. For one, there is no single
established structure; in fact there may be more than one catalytically active motif
in the amorphous film. Thus, one must come up with structural models that are
consistent with the existing experimental measurements. The experimental knowledge of the mechanism is relatively incomplete, meaning that there are many more
candidate mechanisms to assess. At the same time, the increased challenge of this
system provides more opportunities for theory to make an interesting and meaningful
contribution.
Our initial model of the catalyst was a cubane, consisting of a Co4"O 4 core decorated with aqua and hydroxo ligands in the resting state. The molecule is electrically
neutral. We assumed that oxidation events were coupled to proton release, given
the abundance of potentially acidic protons on the molecule surface; thus, oxidation
events did not alter the charge on the compound. Taking cues from our previous
study on ruthenium catalysts, we investigated a wide range of 0-0 bonding possibilities ranging from nucleophilic attack of water on terminal ligands and corner oxo
groups, to direct coupling between various groups in the compound. Our strongest
finding was a direct-coupling pathway between two terminal Corv(O) groups across a
cube face. This transformation was energetically favorable and proceeded with a low
activation barrier, and furthermore it occurred after only two out of a total four oxidation events. Subsequent oxidation events were shown to weaken the Co-O bonds,
helping to release the dioxygen product.
We further tested the plausibility of the mechanism by computing the standard
reduction potentials associated with these two oxidation events; this was done using the QM/MM explicit solvent model, described in the previous section. We found
that the redox potentials were within the experimentally accessible range, and impor-
tantly, that deprotonation from the hydroxo groups was favored over deprotonation
from the aqua groups -
the computed redox potential was much higher when the
deprotonation occurred at the aqua groups. Taken together, our theoretical investigation predicts a thermodynamically accessible pathway for 0-0 bond formation
through a direct-coupling mechanism between two terminal metal oxo groups; we
hope that this finding will be helpful in helping experimentalists to focus their mechanistic studies. In the meantime, our collaborators in the Nocera lab have updated
their model of the catalyst to a cobaltate-like structure. Recently, we have applied
our methodology to the study of this new model, with some interesting new findings.
The QM/MM explicit solvent model was instrumental in obtaining meaningful
results for our model catalyst structures due to the strong hydrogen bonding effects
with the solvent. However, this resulted in extremely costly simulations; the QM/MM
simulations of the cubane took three months on 400 CPUs, and the simulations on
the larger updated model were more costly still. Thus, we were prompted by this and
other studies to develop new theoretical methods for accelerating the computation
of the potential energy surface and rapidly sampling the configuration space. My
main method development project, which is based on the force matching method for
automatic parameterization of molecular mechanics (MM) force fields, is motivated
by these applications. In the long term, we hope that the force matching method will
be a critical component of the computational infrastructure for calculating rapid yet
accurate ensemble averages of observables in complex systems with explicit solvent
environments.
1.4.5
Force Matching Method for Improved Classical Molecular Models
If one were to imagine a ranking of theoretical chemistry methods for estimating the
ground state energies and forces of a quantum mechanical system roughly by their
accuracy
/
computational cost, the full numerical solution of the time-independent
Schr6dinger equation (i.e. full CI) would naturally be at the top. Full CI is not
practical for all but the smallest systems, so we are forced to make approximations;
approximate wavefunction-based methods include Hartree-Fock, many-body perturbation theory, coupled-cluster theory and others. [267] Density functional theory also
lives in this region; although the Hohenberg-Kohn theorem proposes a direct route
from the electron density to the energy, in practice the most accurate approximations to the energy are obtained by reintroducing wavefunctions using the Kohn-Shan
method. All of the methods described above are often called ab initio calculations because the computation evalutes the potential energy surface by solving the electronic
structure problem and assumes no additional information about the system beyond
the nuclear positions and number of electrons. In general, ab initio techniques contain
no system-dependent parameters, which makes them highly flexible and potentially
highly accurate; the associated computational cost, however, is relatively high due to
the need to solve the electronic structure problem.
Molecular mechanics (MM), on the other hand, is a method in which the potential
energy function is completely specified for the system of interest. [102] This potential
energy function is often called the force field. In contrast to ab initio calculations, the
force field contains explicit knowledge of the system; for example, the connectivity
of atoms in molecules usually needs to be specified before the simulation begins.
The force field contains not only the bonding topology of the system, but it also
contains terms corresponding to different types of intramolecular and intermolecular
interactions; these are usually simple analytic functions with fixed parameters. Since
the potential energy function is completely specified, MM simulations do not require
a solution of the electronic structure problem and consequently the computation of
energies and forces is several orders of magnitude faster. However, the accuracy of
the simulation depends entirely on the force field; a force field can be more accurate
than DFT or it can give completely unreasonable results, depending on how closely it
reproduces the true PES. Force field development strategies can fall into two categories
- fitting to experimental measurements or fitting to ab initio calculations.
While some research groups are specialized in force field development, many more
are interested in running simulations if accurate parameters are accessible. To this
end, a number of highly transferable parameter libraries (also called force fields) are
routinely used for a diverse range of simulations, including CHARMM [42], AMBER
[55], and AMOEBA [221] for biomolecules, OPLS-AA [136,137] for general simulations
of liquid systems, and MM3 [10] for hydrocarbons. Importantly, these force fields
are designed to be highly transferable across the different systems in their general
class. Transferability implies uniformity of the functional forms and parameters in
different systems; since a biomolecular force field only includes one set of amino acid
parameters, the same parameters are used for all proteins. Intuitively we know that
force fields are not fully transferable; assuming identical interactions for different
molecules is an approximate treatment at best. An attractive alternative would be to
develop a highly transferable method for creating a force field specifically tailored to
the system of interest; this forms the basic motivation of the force matching method.
The concept of fitting the parameters in MM by matching its forces to those
derived from ab initio calculations is known as the force matching method. Conceptually, one would compute two sets of forces for the a set of molecular configurations once with the MM force field and once with the ab initio method - and then tune the
force field parameters to bring the forces into close agreement. Originally developed
in 1994 for simulations of solids, different incarnations of the force matching method
have been developed to generate force fields for a diverse range of systems ranging
from gas-phase reaction dynamics to biomolecules. The fitting of approximate forces
to higher-level reference forces is conceptually very simple; it is simply a least-squares
fitting problem. However, successfully applying the force matching method reveals a
series of difficult challenges, which may have limited its popularity. I will list some of
the limitations here:
" The accuracy of the force field cannot, in general, exceed the reference ab initio
method that it is fitted to.
" The functional form cannot be overly flexible; too many parameters may lead
to overfitting of the forces and cause linear dependencies to appear in the parameters, leading to instability.
* The functional form must be sufficiently descriptive; an insufficiently flexible
force field may result in some terms being "borrowed" to fit interactions present
in the ab initio method but not described by the force field.
" The configuration space must be sampled in such a way that fitting to the
reference data produces a force field that performs similarly to the reference
method in molecular dynamics simulations.
These are all problems that have appeared time and again in my work on force
field development. A careful choice of ab initio method to generate reference data
is required to successfully apply the force matching method, as is a careful choice of
fitting functional forms and parameters; these considerations need to be taken for each
system of interest. However, proper sampling of training data from the configuration
space is still a major challenge for any system. I developed two complementary
methods - hybrid ensembles and the weighted histogram analysis method (WHAM)
- which can be generally applied to the sampling problem. These methods have
been greatly helpful in obtaining force fields which consistently sample the region of
configuration space that they were fitted to, increasing the applicability of the force
matching method.
60
Chapter 2
QM/MM Study of Exciton
Dissociation and Charge
Separation in Organic Photovoltaic
Devices
2.1
Introduction
The study of organic semiconductors (OSCs) presents a new landscape upon which
chemists, materials scientists, and many others are rapidly expanding the scope of
semiconducting materials with a palette of small molecules and polymers. [223] These
are typically r-conjugated systems with low-lying excited states and the ability to
transport both holes and electrons. [70] In the context of renewable energy, significant
progress has been made in the field of organic photovoltaic (OPV) applications. [100,
219, 270] These devices derive their functionality from at least one donor-acceptor
interface between two OSC materials; this is where excitons are converted into charge
transfer (CT) states or vice versa, caused by the offset in the HOMO and LUMO
energy levels of the different molecules [37].
The main intermolecular interaction in OSC devices is the weak van der Waals
force, which allows for larger thermal fluctuations and greater disorder, especially
at a donor-acceptor interface. These effects can lead to Anderson localization [13]
of carrier states on a few molecules, or even a single molecule. In this picture, the
key quantities are the energies and couplings betwee the different localized electronic
states, and an array of computational tools exist that give us access this information.
Individual chromophores can be described using electronic structure methods including density functional theory (DFT) [84,148] or wavefunction-based methods. [249]
Implicit solvent models (ISMs) can be invoked to provide an average description of
the dielectric response of the environment. [199]
650
DOS
Figure 2-1: Illustration of the QM/MM method. Left: Disordered cell of the
H2Pc/PTCBI system described by MM. Center: Selection of a H2 Pc and PTCBI
pair at the interface for calculation of the CT state energy. Right: Density-of-states
plot obtained by repeating the calculation over different snapshots of a MM trajectory.
Based on the localization of carrier states and short-range disorder, one expects a
combined quantum mechanical/molecular mechanical (QM/MM) [3] scheme to give
a faithful description of the interface. By explicitly including intermolecular interactions in the QM/MM potential, the effect of neighboring molecules on optical and
transport properties can be better described. Even for an amorphous material with
nanometer-scale disorder, the localized nature of the electronic states means that the
QM region need only include one or two monomers. This makes QM/MM calculations vastly faster than a full DFT calculation on the same system. Indeed, we have
successfully used QM/MM to model a crystalline OSC (Alqa) [89], yielding quantitative predictions for both transport and optical properties. Here, we will apply
these methods to study a functioning OPV device formed from the interface between
two well-studied organic molecules: metal-free phthalocyanine (H2 Pc) and 3,4,9,10
perylenetetracarboxylic bisbenzimidazole (PTCBI).
At every stage of our simulation, we observe the crucial influence of the disordered
environment, on the distribution of electronic states, and on the CT binding energies
in particular. We also observe band bending effects at the interface, leading to shifted
values in the donor ionization potential (IP) and acceptor electron affinity (EA).
Importantly, the combination of band bending effects and fluctuations in CT binding
energies make it possible for relaxed CT states to dissociate into free carriers with no
barrier. This finding improves our understanding of exciton dissocation and carrier
generation mechanisms in OPVs, which is a subject of much current interest [62,158,
326].
This chapter will begin with a short description of the computational algorithms
used, including constrained DFT which is used to generate the charge transfer (CT)
state, and the nature of the QM/MM interaction. Next, we present the results of
calculations on the bulk H2 Pc and PTCBI materials and compare our computed
observables with experimentally measured properties. This is followed by a simulation
of the organic donor-acceptor interface, and we discuss the effects of the interface on
excitons and free carriers. We focus specifically on the dissociation mechanism of the
CT state. Finally, we summarize the implications of our results for OPVs and discuss
the insight gained on the carrier generation mechanism.
2.2
2.2.1
Theory
Constrained DFT as a Route to CT Excited States
In what follows, we employ constrained DFT (CDFT) to define the charge-transfer
electronic states which are critical intermediate states in OPV devices.
Within CDFT, a diabatic ET state such as D+A- is defined to be the lowest
energy state such that the partial charges on the atoms that compose the acceptor
(A) sum to -1.
If the overall system is neutral, the donor (D) atom charges will
then sum to +1, reflecting the hole left behind by the transferred electron. Other
diabatic states, such as A A- or D+D are defined analogously. The constraint is
enforced by applying a local chemical potential bias, V, between donor and acceptor.
By varying the strength of this potential, one can control the charge on the acceptor.
In CDFT, V is determined self-consistently with the charge density to determine the
lowest energy state that satisfies the charge constraint exactly [301].
Here, we note in passing that the electronic coupling matrix elements VRP can
be computed in CDFT directly from a combination of the overlap between electronic
states and the matrix elements of the chemical potential between them [306]. Recent
applications have shown that CDFT can provide accurate predictions of Marcus electron transfer parameters within this framework [283,303]. In this study, however, we
will compare energetics throughout.
2.2.2
QM/MM Modeling of Disordered Systems
QM/MM techniques have a long history of application to systems where a small active
site is surrounded by a much larger environment. The simulation cell is thus divided
into classical (MM) and quantum (QM) regions, which interact via the Hamiltonian.
[3] The important quantum phenomena are contained within the molecules of the
QM region, while the surrounding MM region provides essential corrections (e.g.
local electric fields) from the heterogeneous environment.
The total energy of the QM/MM simulation is given by:
(2.1)
E = EQM(rQ) + EMM(rM) + EQM/MM(rQ, rM).
Here, EQM and EMM are the energies of the QM and MM regions, which depend on
the coordinates of the QM (rQ) and MM (rm) nuclei respectively. EQM/MM represents
the interaction energy between the regions and consists of electrostatic interactions
plus an empirical vdW repulsion:
EQM/MM
,IEM
-
iEM jEQ
rr')dr'
r
-
VvdW(ri,
+
iEM
jCQ
rj). (2.2)
In addition to simple electrostatic effects, the QM/MM treatment must account
for the fact that the OSC environment is polarizable. We do this by introducing
fictitous "Drude" charges, qi , which are harmonically bound to MM atoms [157]:
di
EDrude
.
-
- ri
2
. .A
-
r2|
. .d
-
(2-3)
d2,I
Electronic polarization is modeled by allowing the Drude particles to respond instantaneously to local electric fields, so that for any given nuclear configuration {r},
EDrude
is minimized with respect to the positions of the Drude particles. The posi-
tions of the Drude particles must be determined self-consistently with the QM charge
density, because the electric field experienced by the Drude particles depends on the
QM charges and vice versa. For Alq 3 , the empirical parameters contained in the FF
terms above were fitted to reproduce available experimental and ab initio data as
detailed in Ref. 89.
2.2.3
Simulation Procedure
We begin our QM/MM calculations by obtaining thermally sampled configurations
of the OSC material. A pure MM trajectory with a large number of molecules is
simulated at constant temperature, from which we extract a large number of snapshots
at fixed time intervals for QM/MM calculations.
Each snapshot contains a disordered supercell of many molecules (Figure 2.1, left
pane). Because electronic states in OSCs are largely localized, molecules distant from
the active region mostly contribute long-range effects that are well described by MM.
We invoke QM/MM by selecting a small subset of the snapshot (Figure 2.1, center
pane), which by physical intuition we assume to contain the interesting quantum
effects. The quantum region is intentionally chosen as small as possible to minimize
the spurrious charge delocalization problems that plague DFT. [303] Finally, the QM
molecules are embedded in the MM simulation and treated using a full QM electronic
structure calculation (Figure 2.1, right pane); the interaction with the MM system
is replaced by the QM/MM Hamiltonian (Equation 2.1). The MM atomic charges
appear as external point charges in the QM calculation, which affect the QM electronic
density.
Our study can be divided into 1) calculations performed on bulk H2 Pc and PTCBI
systems and 2) calculations performed on the H2Pc/PTCBI interface; this allows us
to benchmark our calculations by comparing to experimental measurements on single
crystals and also examine effects of the interface by comparing bulk and interface
calculations.
2.2.4
Interface Structure
For construction of the MM systems we started with a pure 14x 7x5 (3x14x5) supercell of the experimental cyrstal structure for a total of 490 (420) PTCBI (H2 Pc)
molecules. The H2 Pc/PTCBI interface was constructed by aligning the (001) and
(010) crystal faces of the H 2Pc and PTCBI supercells along z; periodic boundary
conditions were applied along X and
y
(i.e. perpendicular to the interface), and the
system was relaxed under constant pressure (1 bar and 300K) for 1 ns. All three
systems were evolved under NVT dynamics for 5 ns at 300 K. The final 4 ns of the
constant-volume dynamics were sampled at 40 ps intervals to obtain 100 snapshots
for QM/MM calculations; the 40 ps time interval was chosen to minimize correlations
between snapshots.
2.2.5
Density Functional Calculations
All of the QM/MM calculations were done using the CHARMM [42]-Q-Chem [251]
interface [298], and all pure MM calculations were run in Gromacs 4.0 [125].
All
quantum calculations were performed with Q-Chem 3.2 using the PBEO functional
and 6-31G* basis set. All of the singlet excited state calculations used linear-response
time dependent density functional theory (TDDFT) [93] on one molecule. The charge
transfer states were obtained using constrained DFT [301] on two molecules with an
extra electron placed on PTCBI and one electron removed from H2 Pc. The PBEO
functional was chosen because it offered the best compromise between accurate pre-
diction of the singlet energy and the band offset, as the singlet energies increased and
the band offset decreased with respect to the fraction of exact Hartree-Fock exchange
for various functionals tested (PBEO contains 25% exact exchange).
2.3
Results and Discussion
2.3.1
Bulk Materials
Table 2.1: Calculated transport properties for H2Pc and PTCBI. Experimental values,
taken from Refs. 222, 320, and 227, are given in parentheses. All values are reported
in eV; computed values have a statistical uncertainty of le 0.07 eV.
Material
H2 Pc
PTCBI
IP
4.74 (5.2)
5.53 (6.2)
EA
2.43 (3.0)
3.20 (3.6)
TG
2.31 (2.2)
2.33 (2.6)
Band Offset
1.54(1.6)
We started by computing the band offset and Frenkel exciton energies of bulk
H2Pc and PTCBI, given in 2.1. To obtain the IP and EA values, we collected
data from three different monomers in twenty distinct snapshots; the transport gap
(TG) for a single material was given by Ep - EEA and the bulk band offset by
E
=
E pC
-
EPCBI.
Here we note that the TGs and band offset, the more
critical quantities for device performance, are in good agreement with experimental
values despite larger errors in the IPs and EAs themselves. This is because there are
roughly equal shifts in the IP and EA when increasing the basis set (+0.2 eV with
6-311G*) and when placing a molecule in the electrostatic environment of the crystal
(-0.3 eV).
Turning our attention to optical properties, we note that most OSC materials
have a broad absorption in the solid phase due to many different effects such as
heterogeneous broadening, coupling between excited states, and vibronic transitions.
Here we focus on heterogeneous broadening only. We computed the excited state
energies of individual molecules in different snapshots and plotted the resulting absorption spectra along with the experimental spectra [1] in Figure 2. Both of the
a -Experiment
0-
450
Calculated Singletl
500
550
650
600
Energy (nm)
700
750
800
500
550
600
650
Energy (nm)
700
750
800
CU
450
Figure 2-2: Calculated absorption spectrum (dashed) and experimental spectrum
(solid) of PTCBI (top, blue) and H2 Pc (bottom, red). The calculated spectra contain 750 calculated energies sampled from 15 molcules each over 50 snapshots, each
given a Gaussian distribution with width 1.7 nm. The inserted molecules show the
attachment/detachment (blue/orange) densities of the lowest excited state of PTCBI
and H2 Pc.
absorption features are in roughly the right spectral region, but we note that with
only heterogeneous broadening the calculated lineshapes are not nearly as broad as
the experimental results. It thus appears that Franck-Condon (FC) and/or HerzbergTeller (HT) effects play a significant role in determining OSC absorption spectra, even
in disordered environments [244,245].
Looking at PTCBI in particular, our calculated spectrum is also missing a peak
at around 660 nm. This peak is also absent with the higher-accuracy RI-CC2 [121]
method in Turbomole [5] with the larger TZVP basis, which predicts only one bright
peak at ~525 nm. We suspect the missing peak is a HT effect; specifically, with either
PBEO or RI-CC2, there is a "dark" state in the 600-700 nm range with an oscillator
strength that is essentially zero. This sets up a classic situation where these dark
excitons can borrow intensity from the bright state via vibronic coupling.
To better picture these excitons, the attachment-detachment plots [124] of both
molecules are shown alongside the spectra in Figure 2. Both of the molecules have a
strong transition dipole in the plane of the molecules; for PTCBI it points along the
long molecular axis. The strength and alignment of the transition dipole moments
suggest that exciton-exciton coupling in the solid phase could also have a significant
effect on the lineshapes [69, 256] of these crystalline materials, although we expect
such effects to diminish in more realistic, disordered systems. In summary, our current implementation of the QM/MM model can reproduce the band offset accurately
and obtain a qualitative picture of the excitonic levels, but obtaining a more accurate spectrum would require combining all of the above physical effects with the
heterogeneous broadening presented here.
2.3.2
Organic-Organic Interface
Next, we examine the CT state at the donor-acceptor interface. The CT state is
highly susceptible to changes in the electrostatic environment and its energy depends
on proximity to the interface. We sampled five crystallographically distinct nearestneighbor CT pairs at the interface over twenty snapshots and plotted their density
of states alongside the absorption spectra in Figure 4. By comparing the energy
levels, we see that a singlet exciton in either material is able to transfer its energy
into an interfacial CT state, which can then separate into isolated charges; thus,
our calculations correctly reproduce the experimental observation that PTCBI/H 2 Pc
forms a functional photovoltaic device [4]. Not surprisingly, the CT states have a
broader energy distribution (FWHM ~220 meV) than excitonic states; this is in part
due to the distribution of CT pairs, most notably the variation in the donor-acceptor
distance between different pairs.
We performed further analysis on the distance dependence of the CT binding
energy (BE, given by
EBE =(E1HPc -
EcBI)
ECT). Using the procedure provided
in Ref. 171, we fit the inverse of the BE to a linear combination of intermolecular
distances; our results are shown in Figure 5. The BE has a clear R-
decay as
a function of distance, arising from the Coulomb interaction between the electron
on the acceptor and the hole on the donor; however, this trend was not observed
when center-of-mass or closest contact distances were used, highlighting the important
orientational dependence for these planar molecules. The overall fit is good, with a
correlation of 0.85 between the data and R-; there is also a clear scatter of 0.1 eV on
top of the Coulombic decay which we attribute to thermal fluctuations. From moment
to moment, the CT energy of a given dimer will fluctuate by a few kT. Thus, at any
instant there can easily be a more distant CT pair that has a lower energy than a
compact pair due to random fluctuations in molecular orientation. These variations
are expected to aid the initial charge separation at the organic-organic interface.
-
PTCBI
H2PC
---
CT States
-
Band Offset
C
(I)
00
550
600
650
700
750
800
850
Energy (nm)
Figure 2-3: Full calculated spectra of all relevant energy states: bulk absorption (left
axis) of H2Pc (red) and PTCBI (blue), CT density of states (black, right axis), and
the location of the average bulk band offset (brown). Each data point is given a
Gaussians distribution with a width of 1.7 nm.
Perhaps surprisingly, the average energy of the CT states
( 1.6 eV) is higher than
the bulk band offset (1. 5 eV), giving an apparent CT binding energy of ~::-0. 1 eV; that
is to say, the CT states seem to be unbound! We found that this can be explained
by the significant contribution of interface effects to the band offset. In Figure 6 we
plot the IP and EA of H2Pc/PTCBI vs. the distance from the interface, each point
corresponds to an average over four monomers each using twenty snapshots. The EA
of PTCBI (IP
of H2Pc) decreases (increases) as one moves toward the interface by
0.1 (0.15) eV, such that the band offset at the interface is 0.25 eV larger the bulk
value and giving an average CT binding energy of ~04.15 eV. Thus the CT states
are locally bound; the energy of the electron-hole pair at the interface is more stable
than a single electron plus an single hole at the same sites. At the same time, the
CT states are globally unbound; the electron and hole gain energy by migrating away
10.15
-
-
0.10 0.05-
002
4
6
R(A)
8
10
Figure 2-4: Plot of the distance dependence of the PTCBI/H 2Pc CT state binding
energies. The coordinate R is a linear combination of intermolecular distances. Each
different color/shape combination represents distinct dimer pairs in the simulation
cell.
from the interface.
The 'gap bending' effect at the OSC donor-acceptor interface has been previously
calculated in different systems and with different models [167,284,285].
In those
cases, the effect was caused by an interfacial dipole that shifted the electron and
hole levels asymmetrically. Our calculations did not find a significant dipole at the
H2Pc/PTCBI interface; instead, the gap bending appears to be due to differences in
the polarizability and crystal packing. The interface has a stabilizing (destabilizing)
effect on carriers in H2Pc (PTCBI) because PTCBI has a higher dielectric constant
than H2Pc, and the relatively sparse packing introduces an overall destabilizing effect;
our QM/MM simulations with a polarizable MM model were uniquely able to capture
these effects [89].
There is much discussion in the literature on understanding the origins of the high
internal quantum efficiency in OPVs and why the separation of a CT state appears
to be essentially barrierless [214]. One prominent view is that the excess energy from
exciton dissociation creates a "hot" CT state with sufficient kinetic energy to break
free of the binding energy before thermal relaxation takes place [62,326].
On the
other hand, there is also evidence that thermally relaxed CT states are separating
into free charges [158,325]. Our work indicates the latter model to be more accurate,
(IP)- --
5
-- --
-- -
5.0
S(IP)
4.5-
I
4 0
.
l(EA)
3.0
3.02.5
2.01
__
-20
_
-15
_
_
-10
__-
_
-5
0
Distance
_
_
5
_
_
10
_
_
15
20
(A)
Figure 2-5: Plot of the average IP and EA of H2 Pc (red) and PTCBI (blue) crystal
planes as a function of their distance from the interface. Each point has a standard
deviation of about 50 meV.
and suggests that thermally relaxed CT states can break up easily due to competition
between the decreased dielectric screening at the interface and the Coulomb attraction, the first increasing and the second decreasing the CT energy. For our current
H2Pc/PTCBI model system, the decrease in dielectric screening is larger than the
Coulomb attraction, and thus there is little to no energy barrier for CT separation.
Future studies spanning a broad range of molecules and interfaces would be useful
for testing the generality of these results.
2.4
Conclusion
In this study we used a QM/MM model to investigate the H2 Pc/PTCBI donoracceptor interface and obtained thermal distributions of the exciton, IP, EA, and CT
energies. We obtained an accurate prediction of the band offset for this interface
system and a qualitative description of the excitonic states. We also find a strong
dependence of the BE on the relative orientation of the molecules forming the CT
pair. We addressed two effects on the CT state energy that depend on proximity
to the interface: the electrostatic changes at the interface cause the band offset to
increase by 0.25 eV, and the CT binding energy is strongest at the interface with a
typical value of 0.15 eV. The competition between two effects create a situation where
thermally relaxed CT states at the interface can easily separate into free carriers. In
our model system, charge separation is downhill by about 0.1 eV.
The basis of our investigation is a synthesis of tools that together provide an
efficient means of computing accurate electronic properties in a large, disordered
system. The primary approximation is the assumption that the electronic states are
localized, which is a necessary component for the QM/MM decomposition. This is
not generally valid in inorganic devices but is thought to hold quite generally in OSCs,
where Anderson localization dominates the energy landscape.
The current set of calculations have helped to clarify the molecular processes at
an organic-organic interface, but there are a number of effects that still need to be included in the calculations. First, in order to get a more accurate absorption spectrum,
the Franck-Condon factors, Herzberg-Teller corrections, and exciton coupling should
be included; any of these effects can shift the peaks as well as significantly broaden
them. Second, the effect of vacancy and substitutional defects on the performance of
OSC devices needs to be studied, since realistic interfaces in thin films are less ideal
than the one presented in this study. Finally, it would be fruitful to extend this study
to different OSC molecules, in order to aid in our understanding of how the interface
affects the band offset in a variety of situations.
74
Chapter 3
Acid-Base Mechanism for
Ruthenium Water Oxidation
Catalysis
3.1
Introduction
The sunlight-driven splitting of water into hydrogen and oxygen is a promising longterm source of clean energy with the potential to mitigate the modern crises of fossil
fuel dependence and global warming. [163] Conceptually, this process may be accomplished by a photoelectrochemical cell in which a photovoltaic converts energy from
sunlight into an electric current, which is used to drive the electrolysis of water.
2H+ + 2e- -+
H2
2H20 -+ 2 H++ 2e + 02
E0 = 0.00 V vs. NHE
E0
1.23 V vs. NHE
Within this picture, the water oxidation half reaction is especially challenging. The
overall reaction is a four-electron oxidation, coupled to the transfer of four protons
and the formation of an 0-0 bond. The complexity of this reaction is illustrated
by the prohibitive overpotentials associated with performing this reaction via any of
several stepwise fashions [198]:
2H20 -+ 3H+ + 3e- + HO
E
2 H20 -+ 2H+ + 2e- + H202
E
H2 0--+
H++ e + OH'
=
1.66 V vs. NHE
1.76 V vs. NHE
E= 2.72 V vs. NHE
Thus, the optimal pathway is likely to be highly concerted, and special catalysts
are needed to perform this reaction at potentials accessible using photovoltaics that
capture visible light.
Nature catalyzes water oxidation using a Mn 4 cluster embedded within the oxygenevolving complex (OEC) of photosystem II. [173,307] This remarkable metalloprotein
system oxidizes water at efficiencies and rates that far exceed those of any synthetic
catalyst. [242] There have been many efforts to resolve the structure of the OEC in
plants using X-ray studies, [19, 97,225,316] as well as experimental [6, 39,122,130,
184, 185, 189, 202,237, 242,286,295,314] and computational [239,253,258] efforts to
understand the mechanism of water splitting within the OEC. The essential steps of
the water oxidation mechanism can also be studied rigorously and comprehensively by
focusing on much simpler synthetic systems, which do not have the extensive protein
scaffolding that often complicates efforts to understand the natural system.
Toward this end, a number of synthetic catalysts for water oxidation have been
synthesized and characterized, ranging from single-center transition metal complexes,
[66,183,277] dimers, [108,250,287,327] and four-center clusters [38,246] to amorphous
materials [140] and periodic metal-oxide systems. [105,328] Well-characterized families
of catalysts, e.g. the two-center ruthenium catalysts based upon the famous blue
dimer, [108] have demonstrated that catalyst performance can depend on chemical and
structural variation; however, a method for systematic catalyst optimization remains
elusive, highlighting the need for mechanistic understanding. Recently, theoretical
studies using density functional theory (DFT) [23,24, 35, 168,198, 277, 281, 310-312]
have been helpful in the effort to understand the water oxidation mechanism by
providing highly detailed geometrical and energetic data at the atomistic level.
H
0
0
Acid-Base (AB)
Mechanism
~
Direct Coupling (DC)
Mechanism
Figure 3-1: Schematic representations of acid-base and direct coupling pictures.
There are currently two dominant pictures for the water oxidation mechanism,
[29,238] known as acid-base (AB) and direct coupling (DC) (3-1). In the AB picture,
an oxygen nucleophile (e.g. water or hydroxide) attacks a metal-bound, electrophilic
oxo group. From an orbital perspective, an orbital of a character (HOMO) approaches
the M-0 -7r* orbital (LUMO). The combination leads to the formation of an 0-0
o-
bond while breaking one of the M-O 7r bonds, representing a formal 2e- reduction
of the metal center. The AB picture is thought to describe the 0-0
bond forming
chemistry in the oxygen-evolving complex of photosystem II. [213]
In the DC picture, 0-0
oxo groups. The 0-0
bonding occurs between two metal-bound radicaloid
bonding orbital arises from combining two singly occupied
orbitals of M-0 -r* character. While there is no biological precedent for the DC
picture, it has been widely implicated in the design of two-center water oxidation
catalysts, many of which use cofacial or conformationally locked geometries to encourage direct coupling. [238,327] Muckerman and coworkers [198] have performed
DFT studies on a two-center catalyst developed in the Tanaka lab [287] and found
stable intermediates containing direct 0-0
coupling. Muckerman's study contains
many interesting insights into the proposed catalyst intermediates, but no energetics
were provided to assess the viability of the overall cycle.
Recently, the AB picture has received increasing attention as a possible operative mechanism in synthetic water-splitting catalysts. [29,312] The AB picture was
suggested early on by Hurst as a possible pathway for the blue dimer, [128] and this
has been supported recently by experimental studies which suggest nucleophilic attack of water as a key step. [168,308] More recently, Yang and Baik have proposed
that the AB picture is operative for the two-center Ru-Hbpp water-oxidation catalyst
developed in the Llobet lab [250] based on a detailed computational investigation of
both pathways, but the results have been somewhat controversial as more recent isotopic labeling experiments [238] and calculations [35] appear to favor the DC picture
instead. In addition, the recent emergence of single-center catalysts in the Thummel, Bernhard, Sakai, and Meyer labs [65-67,183, 277,318] suggest that the entire
catalytic cycle can be accomplished at a single metal center. Recent experimental
studies of single-center catalysts on oxide surfaces [65] and in multi-component lightdriven water oxidation systems [94] provide further support for the viability of the AB
picture, and work has begun on optimizing these catalysts by adjusting the electronic
parameters of the ligands [294].
The purpose of this paper is to assess the viability of the AB picture as a mechanism for water oxidation. We first conduct a thorough mechanistic investigation
of a recently-published single-center catalyst; [66] our findings strongly indicate that
the catalyst operates via the AB picture, providing verification for Concepci6n and
Meyer's original proposal [66,67] with some slight modifications. We then present a
water oxidation cycle for a two-center catalyst, [29] which had been designed in accordance with the DC picture. We find that the water oxidation mechanism proceeds
via an AB picture highly similar to that of the single-center catalyst.
We conclude with several proposals and suggestions for both the theoretical and
experimental wings of the catalyst development community. From a theoretical standpoint, our methodology demonstrates the importance of using explicit solvent when
investigating reaction kinetics and barriers, a consideration that has often been overlooked in previous computational work. In terms of catalyst design, our conclusions
strengthen the AB mechanistic hypothesis and suggest that catalysts optimized for
the AB pathway should be a fruitful avenue of study.
3.2
3.2.1
Computational Details
General
The calculations in this paper can be roughly divided into two categories; the determination of energies for intermediates, and the investigation of reaction pathways.
The determination of energetics required single-point energy calculations in the solvent phase, geometry optimizations, and frequency calculations. Reaction pathway
investigations involved constrained geometry optimizations, transition state searches,
and molecular mechanics (MM) simulations for thermodynamic sampling.
All DFT calculations were carried out using the B3LYP hybrid exchange-correlation
functional. [25] Geometry optimizations, frequency calculations, and transition state
searches were performed in the gas phase using the 6-31G* basis set; [120] Ru atoms
were represented using the Los Alamos LANL2DZ effective core potential. [123] In
all cases, the crystal-field-derived spin states were computationally verified to be the
lowest-energy configurations; all states were modeled using the unrestricted-spin formalism and a broken symmetry guess. We employed constrained DFT [301] to obtain
the symmetric mixed-valence electronic states in the two-center catalyst, where conventional DFT suffers systematic errors due to electron self-interaction. [217] Singlepoint energies in the solvent phase were computed at the gas-phase optimized geometries using the expanded TZVP [247] basis and the Stuttgart/Dresden ECP [14] for
Ru atoms. The solvating effect of water was represented using the C-PCM polarizable
continuum model. [21] Solvated energies were calculated using GAUSSIAN 03 [103]
and all other electronic structure calculations were performed in Q-Chem 3.1. [252]
The MM simulations, described below, were performed in GROMACS. [85]
3.2.2
Redox Potentials
We calculated redox potentials for oxidative couples in the catalytic cycle using the
polarizable continuum model for solvent effects; [16,296] our approach is also highly
similar to the method proposed by Norskov et al. [207] for determining reaction free
energies in solid-state catalysts. The redox potentials are related to the standard
reduction free energy in solution by
-FE* = AG oE
(3.1)
where AG oE is the free energy change associated with reduction in a solvent environment at standard conditions and F is the Faraday constant. AGoEA is the difference
between several energy components computed for the reduced and oxidized species.
AGo
= AG
FA
(3.2)
+ AAGsiv
AG*oFA = AESCF + AH T - TAS(g)
(3.3)
ESCF is the gas-phase electronic energy, HT is the gas-phase enthalpy correction
(including the vibrational zero-point energy), -TS is the entropic contribution to
the Gibbs free energy, and AG,,,1 is the free energy of solvation.
A large portion of oxidation events are characterized by proton-coupled electron
transfer (PCET), where electron transfer (ET) is accompanied by proton transfer
(PT) to the solvent. Thus, there are many redox potential calculations for which we
must include the standard free energy of a proton in solution, GoHq). (Ref. 187)
GoHj
(aq) =
H
(a)
+ AGH±
SONv
(3.4)
We adopt a widely accepted value for the free energy of solvation of a proton, AG H,
-265.9 kcal/mol. [274] The gas-phase Gibbs free energy is a small correction, given
by GoH, =
kT + PV - TS = -6.3 kcal/mol, and yields the final value of
GH
-11.803 eV.
Thus far, E* has been referenced to vacuum. We obtain potentials referenced to
NHE (Ea, = -4.24 V [145]) via:
E*=-F
x
(AGoEA + AGxo, -
F(g)
nH+X G oHj) - 4.24V
-aq
(3.5)
The measured redox potential E1 at experimental pH conditions is related to
standard conditions (pH 0) using the Nernst equation:
RT
E =i E* - -
F
ln(10) x
nH
-
ne
x pH
(3.6)
where ne and nH+ are the number of electrons and protons involved in the redox
reaction. At experimental conditions (the experiments using Cerv are performed at
pH 1), 59 mV is subtracted from the standard potential for each proton transferred.
Using this method, we first calculated the redox potentials for a set of wellcharacterized organic molecules and metallocenes; [16] our method reproduced the
experimental values well, with a mean absolute error (MAE) of 0.15 V (Figure Si).
Following this, we computed redox potentials for a large number of ruthenium transition metal complexes with experimentally determined redox potentials. [92,179] We
found a systematic deviation between calculated and experimental redox potentials;
our model underestimated small values of E* and overestimated large values. This
suggests that the redox potentials are largely governed by explicit solvent effects,
coordinating anions, and/or electronic effects [92] that are beyond the descriptive
capability of our model. However, a linear fit (Figure S2) demonstrates strong correlation between theory and experiment.
E* ic = aE,, + b where E,,
Thus, the final potentials are given by
is the raw calculated potential, and a = 0.528 and
b = 0.623 are fitting parameters; redox potentials for the catalyst itself are not included in the fit. The MAE of our predictions using the linear regression is 0.11 V,
and should be considered as the typical uncertainty in subsequent values of Eg1 c in
this report.
3.2.3
Reaction Path Investigations
A key concern surrounding the proposed mechanism is whether it makes predictions
in agreement with the experimentally observed intermediates and measured kinetics.
We address this question by investigating the reactivity of catalyst intermediates
before and after oxidation events. Many of these reactions involve solvent water
molecules as reactants or proton acceptors; to address the role of solvent, we developed
a procedure to study catalytic steps that incorporates explicit, thermally sampled
solvent configurations.
We first performed exploratory sampling of the solvent configuration space, in
which the catalyst interacts with solvent through a simple MM force field using OPLSAA [137] Lennard-Jones interactions and Mulliken partial charges. The geometry of
the catalyst was fixed at the gas-phase optimized configuration for the duration of the
simulation. Numerous snapshots of the catalyst and its solvation environment (~800
water molecules, modeled with the SPC/E [27] force field) were sampled from an
NPT [28,208] equilibrated simulation at fixed time intervals. From these snapshots,
the catalyst is extracted along with several nearby water molecules (the water nearest
the oxo group, and its three nearest neighbors) for DFT calculations. The sampled
configurations were fully relaxed to the quantum energy surface using a geometry
optimization. Oxidation events were simulated by removing an electron from the
system at the relaxed configuration. Following oxidation, the geometries were reoptimized, and the optimized configurations and energies were analyzed to determine
whether spontaneous reactions occurred. Reaction barriers for non-spontaneous steps
were determined using a transition state search.
3.3
Single-Center Catalyst
We begin with the water oxidation mechanism of the single-center catalyst, [Ru"(tpy)(bpm)(OH 2 )]2 +
(Ref. 66) The coordination environment of the central Ru atom is pseudo-octahedral,
and catalysis occurs at the water-bound site. Kinetic studies in Ref. 66 have shown
that the rate of water oxidation is first-order in the catalyst concentration, indicating
that no more than one metal center is involved in the rate-determining step, and
perhaps the entire catalytic cycle. Concepci6n and Meyer have proposed that the
water oxidation mechanism for this catalyst proceeds via the AB picture, in which a
water molecule attacks an electrophilic oxo group bound to the highly oxidized Ruv.
The purpose of this theoretical study is to explore the water oxidation mechanism
0
(q=+2, S=O)
e-, H+
(q=+2,S=/2)
4 (q=+2, S=1)
AG* = 26.28 kcal/mol
AG* = -15.01 kcal/mol
Ea= 4.4 kcal/mol
0.o
N
E*= 1.14V (>1.20V)
RL
N
I
e-, H+
AG* = 30.21 kcal/m ol
E= 1.31V (<1.20V1)
AG*= 29.98 kcal/mol
e-, H+
Eo =1.30V (<1.65V)
N
,oN
N .,N
AG 0 = -6.46 kcal/mol
AGI = 1.7 kcal/mol
AG*= 45.66 kcal/mol
E= 1.98V (1.65V)
H2 0
(q=+2,
IN
N
2 (q=+2,
S=1)
v N N
N
N
N
2Tq=+7 I
Figure 3-2: Complete catalytic cycle for the single-center water oxidation catalyst. Lowestenergy spin states are denoted in subscripts. Bracketed species are not experimentally
observed. The RuV species is sufficiently high in energy and unstable that we denote it
using "2T", where "T" stands for "transient". Experimental values for redox potentials
are reported in parentheses. For brevity, numerical labels are used to denote the various
intermediates throughout the paper.
and the intermediates involved. Here, we present a complete picture of the catalytic
cycle, including calculations of all reactants, products, intermediates, energetics, and
reaction barriers.
3.3.1
Energetics and Redox Potentials
Four oxidation events are required for the production of a single oxygen molecule.
The various intermediates in the catalytic cycle are labeled in 3-2, beginning with
the aquo complex Ru"(OH 2 ), labeled as 0. Experimental measurements and their
interpretation are as follows. A two-electron wave is observed at 1.20 V vs. NHE,
taking the aquo complex 0 directly to the oxo species RuIv(O) (2). The hydroxo
species 1 is not experimentally observed, indicating that it disproportionates to 0
and 2; thus the redox potentials of the Ru"/Ru'r and RuIII/Ru'v couples average
out to 1.20 V, with the former being higher. Both oxidation events are pH-dependent
PCET steps. At 1.65 V, a pH-independent one-electron wave appears as a prefeature
of the catalytic wave, and is assigned to the oxidation of 2 to Ruv(O) (2T). 2T
contains the highly oxidized Ruv, which undergoes nucleophilic attack by a solvent
water molecule, forming an oxygen-oxygen bond and losing a proton to create the
hydroperoxo complex Ru"'(OOH) (3). This complex is oxidized a fourth and final
time to yield the peroxo complex RuW(OO) (4); this fourth couple has an unknown
potential that is bounded from above by the catalytic wave. The peroxo complex 4
returns to the rest state 0 via displacement of the peroxo group by a solvent water
molecule, releasing oxygen.
The four calculated standard redox potentials in this catalytic cycle are shown
in 3-2. For the Rur/Ru" redox couple (0-+1), our calculated value of 1.14 V is a
slight underestimate compared to the experimentally reported lower bound of 1.20
V. We calculated the potential for the RuIII/RuIv couple (1-+2) to be 1.31 V, which
is an overestimate, given that the experimentally reported upper bound is 1.20 V.
Unfortunately, our protocol gives the incorrect ordering of the redox potential and
does not reproduce the experimentally reported disproportionation of 1. However,
we note that our theoretical predictions agree with experiment to within the margins
of error (110 mV).
The calculated redox potential for the RuIv/Ruv couple (2-+2T) is very high
at 1.98 V - this potential lies 330 mV above the 1.65 V prefeature, which is where
Concepci6n and Meyer observe the redox couple. This redox couple is a necessary
part of the mechanism, because the experimental observation of a pH-independent
catalytic wave [65] dictates that the highest redox potential must not be coupled to
PT. This statistically significant discrepancy between computation and experimental
observation persists through our explorations of different density functionals, basis
sets, and electronic states; more comprehensive simulation techniques, such as fully
explicit solvent models or the inclusion of counterions, may be essential for determining the redox potentials of these highly oxidized species. The disagreement between
theory and experiment may be exacerbated by the fact that 2T is the only species
in the entire cycle with a +3 charge. Alternatively, the oxidation of 2 at 1.65 V
may be possible as an activated process, but reaching 2T would require crossing an
additional kinetic barrier. The calculated potential for the final redox couple (3
-+
4) is 1.30 V, below the experimental upper bound of 1.65 V.
Oxygen-Oxygen Bonding
3.3.2
The 0-0
bonding step is the primary focus of our study. This step is the most
chemically interesting, because it corresponds directly to the AB picture where a
water nucleophile attacks an electrophilic metal oxo group. In preliminary investigations, we tested a number of alternatives to the proposal in Ref. 66. We found
no stable 0-0
bonded structure involving Rui(O); this indicates that higher ox-
idation states are essential for 0-0 bonding. We also found that any bimolecular
0-0
bonded structure Ruiv/v(O)-(0)Rulv/v is at least 100 kcal/mol higher in en-
ergy than the two monomers, which is likely a consequence of electrostatic repulsion
between these positively charged species. Despite our likely overestimation of the
repulsion (counterions were omitted from the calculation), our results largely rule out
a bimolecular non-rate-determining step, which would have been difficult to detect
by experimentally measuring the overall rate.
After eliminating the possibility of bimolecular direct coupling, we examined the
addition of water to Ruv(O) (2T-+3). The kinetics surrounding water addition is
1D-159
-OW
-0
20
(b)
"
J1B
0.92
9
o.441
-. 398
Figure 3-3: Starting and ending points of a geometry optimization starting at RuV(O)
and ending at Ru"'(OOH). The color scheme is: Ru (green), C (teal), N (blue), 0 (red),
H (white). The reacting oxygen atoms are connected by a dotted line. Selected Mulliken
populations are labeled. Charges on the water hydrogen atoms are summed into the oxygen
atoms.
instrumental for validating the proposed mechanism; for example, finding a kinetic
barrier to water addition would predict the observation of 2T in the reaction mixture,
contradicting the experimental observations and bringing the proposed mechanism
into question. We used explicit solvent simulations as detailed in the Computational
Methods section to model this critical step.
40 configurations of 2 + 4H 2 0 were sampled, fully relaxed, and then oxidized.
After oxidation of the system to 2T + 4H 2 0 and subsequent relaxation, we observed
that 3 out of 40 configurations spontaneously underwent the reaction 2T+ 4H 2 0 -+
3 + 2H 2 0 + H3 0+. These reactions proceeded by nucleophilic attack of a water
molecule on the metal oxo group, forming an 0-0
bond, coupled with PT to a
neighboring water molecule. The Mulliken charge analysis clearly indicates that a
unit of positive charge is transferred to the solvent after PT (3-3). The distribu-
tion of energies energies shows that the 3 O-O bonding configurations had initial
energies within 5 kcal/mol of the mean, and lower final energies than the other 37
configurations by z 18 kcal/mol.
Our calculations confirm that the bonding step is thermodynamically favorable;
solvent-phase calculations give a driving force of AG = -6.5 kcal/mol at experimental
conditions. The reaction pathway of the 0-0
bonding step (2T-+3), characterized
by nucleophilic attack of a water molecule followed by PT, is consistent with the AB
picture for water oxidation.
We estimated the free energy of activation for 2T-+3 using transition state theory. Using 3/40 as the reaction probability allows us to estimate the free energy of
activation using AG
-
-kTln(P.x)
e 1.7 kcal/mol. Although our sample size is
small, it is clear that 2T is a transient species that reacts readily with water at room
temperature (kT = 0.6 kcal/mol). Our calculation of a small free energy barrier may
help to explain the difficulty in detecting 2T in experiments.
The remaining 37 out of 40 configurations relaxed to a variety of final geometries;
they are listed here in order of increasing energy. In one case, 0-0
bonding was not
followed by PT. In another case, a solvent water molecule attacked a carbon atom
in the 3 position of the tpy ligand, followed by PT to the solvent water molecules to
form 3-hydroxyl-tpy. Water addition to the ligand has been previously observed in
the blue dimer; [51] this is reminiscent of a ligand-based water oxidation hypothesis
proposed by Yamada and Hurst, [52,308] but may also be an indicator of eventual
catalyst degradation. 24 cases resulted in an O-0 approach distance ranging from
1.9
A to
2.2
A, with
no covalent bonding. In 11 cases, the water molecules migrated
far away from the oxo group and no reactivity occurred after oxidation. The variety
of final states for this step is indicative of the high reactivity of the Ruv center.
We examined bond lengths and molecular orbitals through the course of 0-0
bonding to verify consistency with the AB picture. We found that oxidation from
RuIv to Ruv (2-+2T) caused the Ru-O bond to shorten from 1.78
A to
1.71 A, and
reduced the Mulliken spin populations on Ru and 0 from 1.0/1.0 to 0.5/0.5. This
is consistent with removing an electron from a Ru-O lr* orbital and increasing the
LUMO
HOMO
'-
7
Figure 3-4: Isosurface plots of the HOMO and LUMO in 2T + H2 0. These two orbitals
combine to create the 0-0 a bond. Note the Ru-O antibonding 7r* character of the
LUMO; 0-0 bonding simultaneously weakens the Ru-0 bond. The orientation of the
H20 molecule may significantly affect 0-0 bonding, due to the importance of HOMOLUMO overlap.
bond order to 2.5. The subsequent nucleophilic attack of water and 0-0
(2T-+3) resulted in an 0-0
bond with length 1.37
A, characteristic
of a bond order
of 1; the Ru-0 bond order decreased as the bond lengthened from 1.71
A.
bonding
A to
1.94
Once again, this demonstrates consistency with the AB picture, for the 0-0
o bonding orbital is predicted to have Ru-0 antibonding 7r* character (3-4). 3-4
shows that the water 1bi molecular orbital must be oriented towards the Ru-0 axis
for bonding to occur; it may be possible to design catalysts which coordinate nearby
water molecules to facilitate this process.
3.3.3
The 0-0
Dioxygen Displacement
bonding step takes the catalytic cycle to the hydroperoxo complex Ru"'(OOH)
(3). From here, we have a fourth and final proton-coupled redox event that takes us
to the peroxo complex Ruiv(OO) (3-+4). We performed an investigation of this
reaction path using the explicit solvent method. We found that for all 40 sampled
configurations, the oxidized RuIV(OOH) complex deprotonated immediately to yield
4. The Ru--O bond lengthens even more to 2.13
1.23
A; this indicates
an increase in 0-0
A while the O-0
bond shortens to
bond order and an accompanying decrease
in the Ru-0 bond order, in accordance with the AB picture. Unlike the 0-0
bond-
ing step, the solvent water molecules were predisposed toward PT, due to hydrogen
bonding with the hydroperoxo group. Our results show that the fourth oxidation
very likely follows a PCET pathway, similar to the first two oxidations. The absence
of a kinetic barrier for this step predicts that 3 is not an observed intermediate, in
agreement with experimental observations.
E,=4.38 kcal/mol to
55
-1
-15
Reaction Coordinate
Figure 3-5: Reaction diagram for oxygen displacement comparing the geometries and energetics of the starting, ending, and transition state configurations. 3-D plots of the system
are shown at the initial, final, and transition states. Key distance measurements (H20-Ru
and 00-Ru) are indicated.
Finally, we examined the reaction barrier for the oxygen displacement step 4
+
H20 -+ 0 + 02.
Energy minima were found for the endpoints of the oxygen
displacement reaction, 4 + 3H 2 0
-+
0 + 02 + 2H 2 0. We found the energy minimum
for 4 to be a triplet six-coordinate structure, rather than the seven-coordinate singlet
proposed in Refs. 66 and 67. Solvent-phase energy calculations of the endpoints give
a significant forward driving force of AG = -15.0 kcal/mol. The geometries of the
endpoints are shown in 3-5.
The transition state for 4 -> 0 was found with a small barrier of 4.4 kcal/mol
(3-5). In this geometry, the Ru-00 distance is 2.87 A, and the Ru-OH2 distance
is 3.33
A. The 00-Ru-OH 2 angle
is 52.2 degrees, and it is nearly exactly bisected
by the vertical axis running through the octahedral coordination site. We can infer
from this investigation that the oxygen displacement step, the only step that is not
triggered by oxidation, has a significant downhill driving force and a small enough
activation barrier that it proceeds under experimental conditions at a limited rate.
The presence of an activation barrier is in agreement with the observed appearance
of the intermediate 4 in the reaction mixture; Concepci6n and Meyer report that this
is the rate-determining step. [67] Following replacement of dioxygen by water, the
catalyst returns to the rest state 0 and the catalytic cycle is completed.
3.4
Two-Center Catalyst
Following our investigation of the water oxidation mechanism in the single-center
catalyst, we turn our attention to a two-center catalyst. We have investigated the
reaction mechanism of the dibenzofuran (DBF) bridged two-center Run complex
[(bpy) (H20)Run (tpy-DBF-tpy)Ru" (OH2)(bpy)] 4+, shown in 3-6. Recent experiments
[29] have shown that the above complex undergoes reversible redox reactions to reach
the stable oxidation state of [(O)RuIvRuiv(0)]4+ (inert ligands and linker have been
omitted for clarity); further oxidation shows some evidence of water oxidation catalysis, but at present the experimental performance of this catalyst is uncertain.
14+
Figure 3-6: Dibenzofuran (DBF) bridged two-center ruthenium catalyst.
3.4.1
Mechanism
In a previous account, [29] we probed the possibility for a DC mechanism in the DBFbridged two-center complex. We found a large steric and electrostatic barrier (>50
kcal/mol) against bringing the two subunits close together, and concluded that direct
coupling was unlikely. Here, we focus instead on the viability of the AB picture for
this two-center system. Our results are summarized in 3-7, and involve intermediates
highly similiar to those of the single-center mechanism. The computations involved
consist mainly of geometry optimizations1 ; redox potential calculations were omitted
due to the lack of detailed experimental redox data, and explicit solvent sampling was
omitted due to the prohibitive computational cost of electronic structure calculations
on this much larger system.
The initial state is the singly oxidized species [(HO)Ru"Rur(OH 2 )]4 + (5). Three
proton-coupled redox events give rise to the [(O)RuIvRuiv(o)] 4 + state (6), containing two ruthenium oxo groups. A fourth redox event yields the high-energy Ruv_
'There is experimental evidence for weak electronic coupling between the two Ru(OH 2) subunits,
which is plausible considering the long distance between the two metal centers. Our computational
results also support this conclusion; the high-spin configuration and the low-spin broken symmetry
(BS) configuration are isoenergetic as long as the number of unpaired electrons in each subunit is
kept consistent. This allowed us to simulate the whole complex in a high-spin state instead of the
much harder to converge BS state.
0
/
0
OH
RuH"
/
\ RuIv
-
OH
Ru1
H20
H20
5 (q=+4, S=1/2)
\ Ru"1
E =5 kcal/mol
(q=+4, S=3/2)
(PCET)
DBF
AE = -17 kcal/mol
E =12 kcal/mol
3e-,3H+
AE =-0.4 kcal/mol
OH
0
0
0
RuIv
0
Ru1"
Ru11
RuIv
D=
7(q=+4, S=3/2)
(rapid)
/ 6 (q=+4., S=-2)
0
RuIV
0
RuV
(q=+5, S=3/2)
Figure 3-7: Proposed catalytic mechanism for the DBF-bridged two-center catalyst.
Lowest-energy spin states are shown.
containing intermediate [(O)RuIVRuv (O)]+
(6T), which undergoes nucleophilic at-
4
tack by water and deprotonates to give [(O)RuivRuIII(OOH)] + (7).
There is a
marked similarity between this transformation and the 0-0 bond forming step in the
single-center case (2-+[2T]-+3). The Rurr(OOH) group in 7 is deprotonated via an
intramolecular PCET event; this results in the peroxo species [(HO)Rur1IRurv(0O)]4+
(8). The Ruiv(00) peroxo is replaced by water in a substitution reaction to yield 5,
releasing dioxygen and completing the catalytic cycle. Here we observe a similarity
between this step and the turnover step in the single-center case (4-W).
3.4.2
O-O bond formation
(a)
(b)
Figure 3-8: Initial (6T, left) and final (7, right) configurations for the O-O bonding step.
The Ruv side (left side of the molecule) is probed with water molecules. Key hydrogen
bonds are indicated with dotted lines.
We found that no Ruv(O)-OH2 bond can be formed at the 6 state despite an
exhaustive search of bonding possibilities; this is not surprising considering the singlecenter catalyst 2 cannot undergo 0-0 bonding at the same oxidation state. Thus,
we investigated the 0-0
bonding chemistry of the RuV-containing intermediate
6T, analogous to 2T. Conventional DFT predicts that electrons are equally delocalized across this symmetric mixed-valence species in a formal [(O)Ru" 5 Ru4"(O)]+
oxidation state, in which Ruv is absent and reactivity is decreased.
Given the
weak electronic coupling between the metal centers, this result is clearly unphysical, and likely a manifestation of the electron self-repulsion present in DFT. [217,
303] Thus, constrained DFT [301] was used to obtain the proper electronic state of
[(O)RuIvRuv(O)] 5 +. The molecule was partitioned into three regions - the two subunits plus the linker - for constrained DFT; the constraint regions were chosen to
keep the cDFT potential from disrupting extended ir-systems.
We found that the RuV(O) oxo group in 6T undergoes nucleophilic attack by a
water molecule, accompanied by concerted PT to solvent water molecules to form 7.
The reaction is spontaneous; i.e. no energy barrier is observed, similar to what we
found for the reaction (2T-+3) in the single-center case. This conclusion is based on
the following results: first, the system [(O)RuivRuv(0)-OH 2 ]5 + (6T + H2 0) has
an energy minimum in the Ru(0)-OH2 coordinate where the distance is approximately 2.0 A (3-8a). Three additional water molecules are manually placed nearby
as hydrogen-bond acceptors. Geometry optimization starting from this configuration leads to the product configuration, [(O)RuIvRur1(OOH)] 4 + (7), where an 0-0
bond has formed and a proton has been transferred to an acceptor water molecule
(3-8b). From the Mulliken charge distribution of the product configuration, and judging by the essential nature of PT in this process, it is clear that the 0-0
bonding
mechanism follows the same AB picture as in the single-center case.
3.4.3
Deprotonation of the hydroperoxo group and dioxygen
release
Figure 3-9: Attempted deprotonation of the hydroperoxo group in the monomeric analogue,
[Ru"'(tpy)(bpy)(OOH)] 2+ .
We investigated the deprotonation of the Rul"(OOH) group in 7 by first examin-
ing unassisted PT from the monomeric analogue, [Ru(tpy)(bpy)(OOH)]2 +, to three
solvent water molecules (3-9). Using constrained geometry optimizations, we lengthened the Ru(OO-H) bond from 1.068 A to 1.52 A. The H-OH 2 distance from the
terminal proton to the nearest water molecule concurrently decreased from 1.43
A to
1.06 A, giving a small indication of PT. However, we found that the energy steadily
increased to a net penalty of 4.84 kcal/mol, and no stable PT configuration was found.
We inferred from this result that the Ru"'(OOH) group in 7 will not spontaneously
lose a proton to solvent.
We found that deprotonation can occur via an alternative pathway involving the
second Ru center, characterized by a combined intramolecular PCET process. To
investigate this pathway, we fully optimized the geometry of [(O)RuIvRunI(OOH)] 4+,
then initiated ET using CDFT to create the [(O)RurrIRuiv(OOH)] 4 + electronic state.
Subsequent relaxation of the geometry leads to the [(HO)Ruuu1Rulv(O)]
4
+,
provided
that a water bridge is included to facilitate PT. 3-10 (top) displays the reactant and
product structures of this reaction.
We did not investigate the sequential pathways by which PT and ET may occur.
However, if a concerted pathway is assumed, which is reasonable because it has the
lowest activation energy, [77, 254] we could estimate the barrier height using the
driving force AG and reorganization energies A [303] using the Marcus theory formula
E, = (AG + A)2 /4A. 3-10 (bottom) shows the energetic relationships using a single
combined PT and ET coordinate. We found a small driving force (AG = -0.4
kcal/mol). Our calculated values of A are slightly different for reactant and product;
AR = 50.9 kcal/mol and Ap = 46.1 kcal/mol. We chose to use the average of the two
A values which yields an activation energy of 11.9 kcal/mol.
In the single-center case, we note that the deprotonation of 3 was accomplished
by a fourth redox event.
In the case of 7, the second Ru center effectively acts
as an oxidant in allowing the final deprotonation to occur; Ru"'1(OOH) is oxidized
to RuW (OO), and Ruiv(O) is reduced to Rul"(OH). The small driving force and
relatively large barrier against PT here suggests that the final deprotonation step in
the two-center catalyst may be very slow.
50.5 kcal/mol 146.1
0.0 kcal/mol
kcal/mol
-0.4 kcal/mol
Figure 3-10: Marcus energy diagram showing the reactant [(0)RuVRuIII(OOH)] 4 + and
product [(HO)RuIIIRuIW(00)14+ configurations in the final PCET step.
The release of 02 from the RuIv(00) group in 8 is accomplished by a ligand
substitution with water, similar to the single-center case. The overall reaction (8-+5)
is downhill by 17.8 kcal/mol. We found the transition state with a barrier of 5.9
kcal/mol; this value is consistent with the barrier for dioxygen displacement in the
reaction (4-+0).
3.5
Discussion and Conclusions
The design of efficient and robust water oxidation catalysts has much to benefit from
an improved understanding of the mechanism. [1911 Until recently, there has been an
abundance of cofacial two-center Ru systems in attempts to pursue the rational design
of water oxidation catalysts, which are largely inspired by the blue dimer. While many
of these systems had been designed with the DC picture in mind, [250,287] there is
also evidence that catalysis proceeds via the AB picture at one metal center, with the
other metal center playing an assisting role. [168,308] These two-center systems are
capable of water oxidation, but syntheses of these catalysts are highly difficult, and
a strategy for optimizing catalyst figures of merit (overpotentials, turnover numbers,
and operating conditions) remains elusive.
The recent discovery of single-center catalysts [66,183,327] introduced a new perspective on catalytic water oxidation.
The single-center catalysts strengthen the
possibility of an AB picture where a water nucleophile attacks a single metal oxo
group to form the 0-0 bond. Following formation of the 0-0
bond, the water
nucleophile loses a proton to the solvent, lifting the requirement for a second metal
center to assist in deprotonation.
In this work, we have presented a complete picture of the water oxidation mechanism in the single-center catalyst [Ru(tpy)(bpm)(OH 2 )]2 +. Our calculations clearly
support the AB picture as a viable pathway for 0-0
bond formation, providing
important validation for the mechanistic proposal put forth in Ref. 66. The oxidation
of Ruiv(O) to Ruv(O) is energetically demanding, but the subsequent nucleophilic
attack of water on Ruv(0) occurs nearly spontaneously, forming an 0-0
out going through a large kinetic barrier. The steps following 0-0
bond with-
bonding proceed
spontaneously to release molecular oxygen and complete the catalytic cycle.
We have also found a water oxidation mechanism that proceeds via the AB picture
in the two-center DBF-bridged catalyst, [(bpy)(H 2 0)Runl(tpy-DBF-tpy)Ru"(OH 2 )(bpy)]4 +
There exists a significant similarity between the O-0 bonding steps of the two-center
catalyst and the one-center catalyst. For both catalysts, 0-0
bonding occurs via
an acid-base reaction initiated by oxidation of one Ru center to Ruv, followed by
nucleophilic attack of water and PT to the solvent. We found that the second metal
center does not play an assistive role in the 0-0
bonding step, as has been pro-
posed in DFT studies of other two-center catalysts; [310,312] however, it assists in
deprotonation of the hydroperoxo group after 0-0
bonding is complete via a PCET
mechanism.
There are several obvious directions for extending this work. In terms of theoretical methods, in the future we would like to improve the redox potential estimates
using more advanced solvation models; the determination of solvation energies using
explicit solvent free energy perturbation (FEP) is a promising, if costly, option. [176]
Our methodology for investigating 0-0
bonding could be further refined; we used
an MM simulation to sample the solvent configuration space, but a QM/MM simulation may be more appropriate. Furthermore, one may obtain more accurate estimates
of reaction probabilities and discover alternative pathways by using a larger number
of snapshots. The theoretical exploration of alternative reactions (e.g. catalyst poisoning and degradation), especially at the highly reactive Ruv oxidation state, may
also prove useful in identifying factors that limit catalyst turnover. Notably, we have
shown that the inclusion of explicit solvent is indispensible when investigating key
steps of the catalytic cycle, and we hope that future theoretical studies will take this
important consideration into account.
On the more experimental side, we note that while the AB picture is by no means
universal, our results suggest that the mechanism is at least transferable across several
systems. Thus, an obvious future direction entails applying the same AB mechanism
to other catalyst families, including the array of Ru-center catalysts discovered in
Thummel's group [277] and the Ir-center catalysts discovered in Bernhard's group.
[183] One would also like to confirm the viability of the AB mechanism in more wellknown systems such as the blue dimer and (ultimately) the OEC. The mechanism of
these catalysts (AB versus DC) is likely determined by the ability of the highest metal
oxidation state to act as a Lewis acid. In the systems studied here this requires a
transient RuV center, whereas many of the aforementioned catalysts top out at RuIv.
It is yet to be seen if any of the RuIV ions are suitably reactive for the AB mechanism.
Our work provides key support for the acid-base mechanism of water oxidation
catalysis and encourages new directions of research for the catalyst development community. Perhaps most importantly, our verification of the AB picture strongly encour-
ages further design of single-center catalysts, in contrast to the synthetically much
more difficult two-center paradigm. We offer our suggestion that the difficulty in
achieving 0-0 bonding in the AB picture can be mitigated by fine-tuning ligand
fields to lower the redox potential of the RuIV/Ruv couple, pre-orienting the reactant
water molecule in the first solvation shell, or alternatively by employing metals (such
as cobalt) that might be slightly more reactive in their IV oxidation state.
100
Chapter 4
A Polarizable QM/MM Explicit
Solvent Model for Computational
Electrochemistry in Water
4.1
Introduction
Electrochemistry, defined as electron transfer between a conducting electrode and a
molecule or ion in solution, allows us to explore the fascinating array of oxidationreduction (redox) chemistry underlying a great many processes in organic [15,57,68],
inorganic [26,126,160,204], and biological chemistry [83,113,185,259]. Additionally,
electrochemical processes play an indispensable role in the fields of energy storage
and conversion as the underlying mechanism for the operation of batteries [106,272]
and the pathway for conversion of electrical energy into chemical fuels [163]. In recent
years, with the assistance of theoretical models and powerful computers, simulations
have started to play an increasingly important role in the understanding of biochemical and energy-related redox processes [30,198,207,230,264,290,292,311].
The fundamental electrochemical property of a molecule and its associated solvent is the standard reduction potential E0 , which is proportional to the free energy
of reduction of the molecule in solvent and referenced to that of a standard electro101
chemical reaction, e.g. proton reduction at the normal hydrogen electrode (NHE).
The solute changes its electronic state during this process by virtue of gaining an
electron, and the solvent responds mostly via polarization, either by electron redistribution or molecular reorientation; a model for redox processes must then account
for both the electronic structure of the solute and the solvent response.
In general, the electronic structure of the solute is describable by the same quantum chemical methodologies commonly used in gas-phase calculations. In practice,
density functional theory (DFT) is often used to describe the electronic structure
of the solute, although fully ab-initio methods such as Hartree-Fock [21,268], multiconfigurational self-consistent field [46,71], second-order perturbation theory [17,47],
and coupled-cluster theory [45,53,54,61] have also been applied. For relatively large
systems (i.e. > 50 atoms), DFT provides a good compromise between accuracy and
computational cost. General-purpose density functionals such as B3LYP [25] and
PBE [216] have intrinsic errors on the order of 100-300 mV (about 2-7 kcal/mol) for
reaction energies of small molecules [78,79,200] and for gas-phase ionization potentials (IPs) and electron affinities (EAs) of a wide range of molecules [32, 144, 323]
including transition metal complexes [76,117,134,235]; this provides a rough upper
limit for the accuracy of Eo computations that utilize these functionals.
The other required component for computation of Eo is an accurate description
of the solvent. Solvent models can be broadly categorized into explicit and implicit
solvent models, which are distinguished by whether the model contains any discrete
solvent molecules. Implicit solvent models (ISMs) represent the solvent using a dielectric continuum with the solvent's experimentally determined dielectric constant,
which generates a reaction field in response to the solute electron density. Perhaps
the earliest such ISM is the Born model [34] which describes the dielectric energy
of a spherical ion placed in a spherical cavity surrounded by the dielectric continuum. This was refined by the Kirkwood-Onsager model [149,209] which includes the
dielectric energy of the higher-order multipole moments of the solute in a spherical
cavity. More recent models place the solute into a shaped cavity constructed from an
electronic density isosurface [99] or the union of small atom-centered spheres [152].
102
The solvation energy is then obtained by any of several numerical methods, including
solving the Poisson-Boltzmann equation on a grid in space [229], placing screening
charges on the cavity surface [72,152], or with an integral equation formalism [48].
Modern implementations of implicit solvent models include, but are not limited to,
IEF-PCM [48], C-PCM [21], COSMO [152], and the SMx series [75]; these models are
highly popular due to their simplicity and low cost [74]. Within an implicit solvent
framework, London dispersion [12] and excluded volume effects [196,312] can also be
described in addition to the dielectric response.
Simple protocols exist for the computation of EO in ISMs [16,22,162,165,197,241,
257,280] and highly accurate results have been obtained using DFT/ISM for a variety
of organic molecules [11, 16,104,257,297] and transition metal complexes [16,60,241]
in a variety of solvents. However, one intuitively expects the validity of ISMs to
vary from system to system as they do not describe many aspects of the solutesolvent interaction including solute-solvent correlations, charge transfer to solvent, or
hydrogen bonding effects. Thus, for example, ISMs provide an excellent description of
redox potentials in organic solvents [16]. However, in this paper we are interested in
the prediction of accurate reduction potentials for transition metals in water, and the
quality of ISM results here is quite mixed. For example, DFT/ISM computation of Eo
for transition metal aqua complexes suffer from large errors unless a second solvation
shell of water molecules is explicitly included in the electronic structure calculation
[133,165, 280]. Chiorescu and coworkers [60] reported a standard error of 200-600
mV for ISM computations of EO for a large number of ruthenium transition metal
complexes and found the largest errors when the solute is highly charged and contains
many hydrogen bond donors. Recent work in the Truhlar and Goddard groups based
on population analysis suggested that the second explicit solvation shell affects Eo by
charge transfer to the solute [44,175], but energy decomposition analyses in the HeadGordon and Gao groups indicated that population analysis often overestimates the
amount of charge transfer in hydrogen bonds [146,147,193]. To improve the accuracy
of ISMs, custom density functionals have been designed which contain empirical fitting
to solvent effects [107]; highly detailed solvent models tailored to a wide range of
103
physical properties of specific solvents have also been developed [75].
Explicit solvent models have a far greater capacity to capture the physical details
of the solvent. As an example, the SPC/E [27] classical water model not only correctly
reproduces the zero-frequency dielectric constant of water, but it also describes the
fine-grained structure of water and describes hydrogen bonding effects empirically
using a combination of electrostatic point charges and van der Waals interactions.
Classical explicit solvent molecules can be combined with a DFT treatment of the solute in a quantum mechanical
/
molecular mechanical (QM/MM) framework [3,321]
(4-1), or the entire system can be treated quantum mechanically. A downside to
these models is an immense increase in computational cost associated with thermally
sampling the many configurations of the explicit solvent. Despite these difficulties,
explicit solvent models have been applied to compute EO for systems of small size, including transition metal aqua complexes [31,194,322] and small organic molecules [59];
in these studies both solute and solvent were treated using DFT, with the exception
of Ref. 322, which uses QM/MM. QM/MM models have also been applied to compute EO for biological cofactors bound to enzymes [30,43,139,164,230,233,234] where
the highly heterogeneous environment of the enzyme active site demands an explicit
solvent treatment.
The main purpose of this article is to provide a general protocol for computing Eo
in aqueous solution using QM/MM explicit solvent and a full dynamical treatment
of both solute and solvent degrees of freedom. We begin by briefly summarizing the
methodologies for ISM and explicit solvent models associated with computing E0 . We
present a polarizable QM/MM explicit solvent model which significantly improves
the accuracy of Eo calculations for transition metal aqua complexes in water. We
provide a detailed analysis of the various physical effects in the QM/MM model
in order to pinpoint the source of the dramatic differences between the ISM and
QM/MM results, and by process of elimination we conclude that the explicit account
of hydrogen bonding effects is primarily responsible for the improved performance
of the QM/MM model. Finally, we summarize the strengths and weaknesses of our
model and suggest avenues for further improvement and applications.
104
Figure 4-1: Illustration of a QM/MM simulation of a [Fe(H 2 0) 6 ]3 + complex showing QM
solute (green, Fe; red, 0; white, H) and superimposed positions of MM solvent molecules
(red, 0; black, H). The geometry of the solute is held fixed. The positions of MM atoms
is an approximate representation of the probability distribution of nuclear positions, and
the short-range solvent structure (interpreted as a consequence of solute-solvent hydrogen
bonding) is visible.
4.2
Theory
The general procedure for computation of standard reduction potentials Eo is given
here. EO is related to the standard reduction free energy in solution by
-FEo = AGE)
where AG I
(4.1)
is the free energy change associated with reduction at standard con-
ditions and F is the Faraday constant. To obtain potentials referenced to NHE, the
absolute potential of the NHE is subtracted from the computed value of Eo; in this
work, we use a value of 4.43 V [231], although values ranging from 4.24 V [145] to
4.73 V [110] are also given in the literature. The implicit and explicit solvent models
contain major technical differences in the computation of AG(), so the following two
subsections will separately describe the theories corresponding to the two models.
105
4.2.1
Implicit Solvent Calculation of Standard Reduction Potentials
In ISMs, AGiEA is the difference between several components of the Gibbs free energy
computed separately for the reduced and oxidized species: [16,165]
AGSo) = AGg + AAGs1l,
AG)
= AESCF + AHT - TAS(g).
(4.2)
(4.3)
Here ESCF and AG,v are the electronic energy and free energy of solvation, respectively. In ISMs, the space surrounding the solute is filled with a dielectric continuum
which generates a reaction field in response to the solute electron density. This interaction yields the solvation free energy AG.Iv, and the difference in AG.1v values
for the two redox states is given by AAGo1v. Since the reaction field and the solute
electronic structure depend on one another, both entities are simultaneously iterated
to self-consistency and the sum of electronic and solvation free energies ESCF +AG,,Ov
is computed together. HT and -TS are the gas-phase enthalpy and entropy contributions to the Gibbs free energy, respectively. These terms are obtained from the
translational and rotational degrees of freedom, as well as the normal modes of the
molecule from a frequency calculation.
Since ISMs allow one to obtain all components of the free energy for a given
oxidation state from a single electronic structure calculation, it is computationally
efficient and thus generally preferred for calculations of E0 . The model makes several
important assumptions. One major assumption is that the solute is well described by
a continuum dielectric; the validity of this assumption depends on the strength of the
solute-solvent interaction, and is more likely to break down when solute-solvent hydrogen bonding is present. ISMs further assume that a mean-field model is appropriate;
the solute sees the average reaction field as opposed to having the solute wavefunction
and solvent polarization respond instantaneously to one another. Furthermore, the
flexibility of the solute is only treated in terms of its harmonic vibrations around the
106
minimum-energy geometry. These assumptions of ISMs can be investigated by using
explicit solvent models which provide a dynamical treatment and a highly detailed
description of the solvent.
4.2.2
QM/MM Calculation of Standard Reduction Potentials
In order to obtain the free energy difference AG between oxidized and reduced states
in the context of QM/MM molecular dynamics simulations, we use the formalism of
thermodynamic integration (TI) [102]. Within this framework, we first write down
a superposition of the system's potential energy function in its reactant and product
states using a tunable parameter A;
E(A) = AEox + (1 - A)Ere,
(4.4)
where we recover the reduced state when A = 0 and the oxidized state when A = 1.
The goal is to find the free energy difference between the two states: AF = F(A
=
1) - F(A = 0).
Differentiating the relation between the partition function Z and the free energy
with respect to A gives the following relationship between the free energy derivative
and the potential energy derivative:
dF
dA
d
dA
1 vdEi(A)
Z .
dA
E.(A)
(4dE(A))
dA
Thus, the free energy derivative at a given value of A is given by thermally averaging
the potential energy derivative using the canonical ensemble corresponding to E(A).
The potential energy derivative is given by the vertical energy gap (EG) between
reactant and product states,
dE(A)
= Eox - Ere.
dA
107
(4.6)
By this formalism, the free energy derivative dF/dA can be evaluated using sampling techniques such as the Metropolis algorithm [190]. TI consists of evaluating
dF/dA at several values of A between 0 and 1 and then numerically integrating to
evaluate AF;
(1
AF=
dF
dA-- o dA
]dA(-).
f1
dE
o
dA
(4.7)
Note that intermediate A values between 0 and 1 correspond to unphysical systems,
and carrying out these simulations can be a nontrivial task especially when the reactant and product have different numbers of atoms as is the case in protonation
/
deprotonation reactions [59,139,230, 233].
In the linear response (LR) approximation, the free energy derivative L is assumed to be linear in A, and AF can be evaluated by thermally averaging the vertical
EG at the reduced and oxidized states (i.e. at A = 1 and at A = 0).
AF -
fe
dA(-) = -((Eo - Ered) + (Eox - Ered)).
2
dA
(4.8)
Note that the LR approximation can be tested by evaluating L at intermediate
A values and performing TI [59]. In this Article, we have used the LR approximation throughout; some validation for the LR approximation is provided in Ref. 322,
in which full TI was performed on two transition metal complexes using a related
protocol.
4.3
Computational Methods
4.3.1
Implicit Solvent Model
All DFT/ISM calculations were performed with the B3LYP density functional [25]
and the conductor-like screening model (COSMO) [152] as implemented within TURBOMOLE version 5.10 [5]. The optimized atomic radii were used where available [248]
and Bondi's radii [33] were used for metal atoms. Geometry optimizations were performed in the gas phase using the SDD basis/pseudopotential combination [14] for
metals and the TZVP basis [247] for all other atoms; gas-phase frequency calcula108
tions were performed at the optimized geometries using the same level of theory.
Single-point energies in the solvent phase were computed at the gas-phase optimized
geometries using SDD for metals and cc-pVTZ [95] for all other atoms. The unrestricted spin formalism was used at all times, and the multiplicities of all complexes
were set to the experimentally determined values.
4.3.2
QM/MM Model
To prepare the simulation cell, the solute geometry was optimized in the gas phase
and placed in a 3.7 nm cubic box; ~ 1720 explicit solvent water molecules, modeled
using the SPC/E force field, were then added to fill the box. The solute-solvent
interaction was given by a QM/MM interaction potential as described in Ref. 3;
the Lennard-Jones parameters for the solute were taken from UFF [228].
solute hydrogens, the LJ radius was adjusted to 0.15
A;
For the
this value was determined
by running a QM/MM simulation with just one QM water molecule and optimizing
the LJ radius of the QM hydrogens to reproduce the typical hydrogen bond length
of 1.7
A between
MM waters. Dynamics were performed in the NVT ensemble with
a Nos6-Hoover thermostat; a 1.0 fs time step was used and the SHAKE constraint
algorithm [243] was applied to the MM water molecules. For each complex in each
oxidation state, the first 5 ps were used for thermal equilibration, and at least 20 ps
of QM/MM dynamics were generated in total.
We extracted configurations (snapshots) from the molecular dynamics trajectories
at 40 fs intervals to perform EG calculations. EG calculations were equivalent to
computing the vertical IP (EA) at simulation snapshots corresponding to the reduced
(oxidized) species; the result of the IP calculation was multiplied by -1 to obtain the
EG. The average EG from the oxidized and reduced trajectories were then combined
in 4.8 to obtain the free energy of reduction, AG E.
It is important to note here that the SPC/E water model implicitly accounts
for solvent polarizability in equilibrium molecular dynamics, but cannot capture electronic polarization effects that accompany vertical electron detachment
/
attachment.
We decided to explicitly add electronic polarization of the solvent in the form of polar109
izable Drude particles [156,157] in the EG calculations. For these calculations, a single
Drude particle with polarizability 1.0425'4 was attached to each water oxygen. To
ensure that the Drude particles account only for the electron detachment/attachment
events, we only included Drude polarization in the nonequilibrium term in the EG;
that is, when we calculated the EG for snapshots sampled from the Erd ensemble,
Drude particles were added for calculations of Ex and vice versa.
All QM/MM calculations were performed using the CHARMM (version c34b1)
[42] / Q-Chem (version 4.0) [252] interface [298]. The TZVP all-electron basis set
was used for all atoms. In the polarizable QM/MM calculations, the Drude particle
positions and Kohn-Sham wavefunction were self-consistently determined using a specialized dual- convergence procedure [89] that updates both quantities simultaneously
to minimize computational cost.
4.4
Results
We chose a set of small organic molecules, metallocene ("sandwich") complexes, and
octahedrally coordinated transition metal complexes from Ref. 16 to act as our control
set for the ISM; they comprise a fairly diverse set of redox-active small molecules in a
variety of solvents for which ISMs are known to provide a reasonably good description.
We also put together a test set of nine first-row transition metal complexes in
aqueous solvent, two of which also belong to the control set; EO was computed using
both ISM and QM/MM techniques for each complex in this set. These complexes are:
[Ti(H 2 0) 6 ]3+, [V(H 2 0) 6 ]3+, [Cr(H 2O) 6 ]3+, [Mn(H 2 0) 6 ]3+, [Fe(H 2 0)6 ]3 +, Fe[(CN) 6 ]3 ,
[Co(H 2 0) 6 ]3 +, [Co(NH 3 )6 ]3 +, and [Cu(H 20) 6 ]3 +. The experimental values of EO for
these nine complexes are given in the most recent edition of the CRC Handbook [2].
The nine aqueous transition metal complexes were chosen because they satisfied the
following conditions and were desirable for a study of this type; 1) they reversibly
underwent one-electron redox reactions without undergoing any concomitant chemical
reactivity, 2) they are sufficiently small that expensive QM/MM simulations were
relatively tractable, and 3) their experimental Eo values are accurately known.
110
Reduction Potentials
5
4 -
3 11
2 -+
Mn"
1
Fe""'
>1
Fe
C
CoMAI(NHA
-1 --
Organics RMSD=0.216V A
Sandwiches RMSD=0.372V +
Complexes RMSD=0.372V
Aqua Ions, PCM RMSD=1.484V +
Aqua Ions, QM/MM, B3LYP RMSD=0.357V a-.-'
4
5
1
2
3
-1
0
*uni
-2
3
-3
-2
E0 (expt, V vs. NHE)
Figure 4-2: Calculated values of Eo plotted against experiment. Gray: ISM calculations
of E0 for molecules chosen from Ref. 16 (organic molecules, triangles; sandwich complexes,
diamonds; octahedral complexes, circles). Blue crosses: ISM calculations performed on
aqueous transition metal complexes. Red crosses: QM/MM calculations performed on
aqueous transition metal complexes.
4.4.1
Standard Reduction Potentials in Implicit Solvent Model
Using the ISM, we reproduced the results from Ref. 16 for the control set (4-2, scatter
plot in gray). However, the result using the same methodology was generally poor for
the test set of aqueous transition metal complexes. The seven [M(H 20)6 ]3+ complexes
had a particularly large systematic error, where the ISM consistently overstimated
E0 by over 1.62 V (4-2, scatter plot in blue); the correct periodic trend was still
reproduced with a correlation of > 0.98.
4.4.2
Standard Reduction Potentials in QM/MM Model
We recomputed E0 using the QM/MM model (4-2, scatter plot in red). In these simulations, the QM region consisted of the transition metal atom and its six ligands, and
the solvent was treated classically. Zero-point vibrational effects, which contributed
< 0.05 V in the ISM calculations, were ignored. Using the QM/MM model, we found
111
a dramatic improvement in the agreement of our computed values with experiment;
the RMS deviation from experiment dropped by 1.6 V to 0.3 V, and the systematic
error for the [M(H 2 0) 6 ]3 + ions was nearly eliminated. In fact, all of the cations with
a +3 charge underwent a similar shift in their Eo values when QM/MM was used;
the difference between ISM and QM/MM Eo values for these eight complexes was
-1.49 ± 0.24 V. The Fe[(CN) 6 ]3 - ion, the only anion for which QM/MM EO values
were computed in this work, had a shift in the opposite direction of +1.33 V.
We also estimated the dependence of Eo on the density functional. In these calculations, the snapshots generated from B3LYP QM/MM were used and the functional
was only varied when calculating the EG. The deviations of the calculated Eo values
from experiment are given in 4-3; the functionals are ordered by their average deviation from experiment, and we can observe a strong correlation between the mean
error and the percent of exact exchange in the functional (BLYP, PBE, BP86, M06-L
0%; B3LYP 20%; PBEO 25%; M06 27%; M06-2X 54%.) This correlates with the
documented trend for calculated gas-phase IPs of transition metal atoms to increase
with the fraction of exact exchange [76,235,306,313], and confirms the usefulness of
using internal references for Eo computations using different functionals as suggested
in Ref. 241. We also observed that the standard deviation of the error is much larger
for functionals with much less or much more exact exchange compared to B3LYP
and PBE0; this reflects earlier reports that density functionals with 20%-30% exact
exchange gave the most accurate gas-phase IP values of transition metal complexes,
whereas functionals with significantly more or less exact exchange were much less
accurate. [306,313]. In the current study, we judged PBEO to give the most accurate
energy gaps out of all functionals tested, and we speculate that the results may be
further improved if PBEO were used for the QM/MM dynamics as well.
4.5
Discussion
The significant difference between standard reduction potentials computed using ISM
and QM/MM stems from the different physical descriptions of the solvent.
112
The
Functional Dependence of Reduction Potential
2 -
z
40
-1 --
-2
BLYP
PBE
BP86
M06-L B3LYP PBEO
Functional
M06
M06-2X
Figure 4-3: Mean and standard deviation of differences between calculated EO values and
experimental measurements. Functionals are ordered by mean error, and a correlation
between mean error and fraction of Hartree-Fock exchange is observed.
QM/MM description contains many physical effects that are absent in ISMs including solute flexibility, solute-solvent correlation, and hydrogen bonding. Furthermore,
there is a possibility that the errors in ISM are a consequence of incorrect parameterization of the atomic radii. In this section, we present a detailed analysis of the
different physical effects in the QM/MM model and their estimated contributions to
E0 . We found that solute flexibility and dispersion effects contribute a relatively small
amount to E0 . Some of the analyses required customized QM/MM simulations, which
were carried out for the {Fe(H 20)613 + complex as a case study. The relevant values
of EO for this complex are 2.35 V (calculated, ISM), 1.17 V (calculated, QM/MM),
and 0.77 V (experiment).
4.5.1
Atomic Radii in ISM
The most important parameters in a ISM are the dielectric constant of the solvent
and the solute atomic radii. While the dielectric constant is fixed, the choice of atomic
radii is not entirely unambiguous, and there are several sets of atomic radii including
113
UFF radii [228], Bondi's radii [33], and the optimized radii in TURBOMOLE [248].
We tested the effect of changing the hydrogen radii on the ISM Eo values for the
seven [M(H 2 0) 6 ]3 + complexes, and found that decreasing the hydrogen radii from 1.3
A to 0.7A changed the average deviation from experiment from +1.62 V to +0.61
V (lowering the radii further had no effect as the hydrogen radii were completely
contained within the oxygen radii.)
Decreasing the hydrogen radii for other molecules had a much smaller effect; the
change in Eo never exceeded 0.2 V when the hydrogen radii were eliminated from ferrocene and the organics. Thus, it seemed that the transition metal aqua complexes
were especially sensitive to the parameterization of hydrogen radii. Nevertheless, a
sizable systematic deviation from experiment still existed when the hydrogen radii
were tuned to zero, and from this we concluded that the residual errors associated
with ISM computations for these systems were not a consequence of improper parameterization, but rather the inability of the ISM to describe important physical
effects.
4.5.2
Solute Flexibility
A major difference between the ISM and QM/MM models is the treatment of solute
dynamics. ISM approximates the dynamics of the solute using harmonic vibrations
about the minimum energy geometry, while QM/MM provides a full dynamical treatment. The harmonic approximation may be valid for solutes with mostly stiff degrees
of freedom (e.g. anthracene, Ru(bpy) 3 , or Fe(CN) 6 ), but it is not an appropriate description for more flexible solutes with more complex conformational changes. In fact,
the QM/MM simulations indicated a significant degree of torsional freedom for the
aqua ligands on the [M(H 2 0) 6 ]3 + complexes, so it is conceivable that solute flexibility
effects could have a noticeable effect on E0 .
To investigate, we ran a QM/MM trajectory for the [Fe(H 2 0) 6 ]3 + complex with
the solute fixed at the gas-phase optimized geometry and calculated EO using the
same LR approximation. With the solute fixed, we computed a value of 1.07 V for
Eo; this constituted a 0.10 V difference from the result obtained from fully flexible
114
dynamics. This is very small when compared to the 1.18 V difference between the
ISM and QM/MM calculations; from this we concluded that the solute flexibility has
a relatively minor impact on EO for these complexes. In fact, an accurate estimation
of the size of the effect may not be possible from this study due to the statistical
errors of ~ 0.20 V in the QM/MM calculations. We do not expect our result to hold
for all redox systems, especially those with intramolecular hydrogen bonds or multiple
conformations (proteins being a prime example).
4.5.3
Solute-Solvent Correlation
The QM/MM model accounts for temporal correlations between the positions and orientations of solvent water molecules and the solute electron density, but ISM cannot
describe them as it is a mean-field model. To investigate, we performed an approximate mean-field QM/MM EO calculation on the frozen [Fe(H 2 0) 6 ]3 + complex. In this
simulation, correlations on timescales smaller than 10 ps were removed by averaging
the electrostatic field over fifty snapshots in a 10 ps window. This procedure only
removes correlation to first order as the snapshots themselves were generated using
full dynamics, in contrast to fully self-consistent methods [321,322]. The Drude particles were turned off for this test; this also introduces a large shift in the reduction
potential, so the results reported in this section are only intended to estimate the size
of correlation effects.
When the correlation effects were removed, the computed value of Eo increased
from 3.18 V to 3.42 V. This indicates that correlations contribute about 0.24 V, a
minor effect when compared to the difference of 1.18 V between the ISM and QM/MM
models and within the statistical error bars. In these simulations, the Eo values were
Z2 V higher compared to the actual QM/MM results due to the absence of Drude
particles; this illustrates the importance of including electronic (fast) polarization in
calculations of Eo and highlights the need for polarizable force fields in this type of
QM/MM model.
115
4.5.4
Hydrogen bonding effects
In the QM/MM model, electrostatic interactions between the solvent water molecules
and the ligands of the solute give rise to short-range structuring of the second solvation
shell and heterogeneous polarization of the solute electron density. These effects are
not captured by ISMs, and it is natural to associate these interactions with hydrogen
bonding, and we will typically make that association in what follows. By eliminating
dispersion and solute flexibility effects, we arrived at the conclusion that short-range
solvent structure - i.e. hydrogen bonding effects - are mainly responsible for the
difference between our ISM and QM/MM computed values of E0 .
We note that hydrogen bonding effects can also be treated using cluster-continuum
models that explicitly include the second solvation shell in the quantum calculation; [133, 165, 280] in these studies, adding the second shell dramatically improves
agreement with experiment.
These cluster-continuum models allow some charge
transfer to take place between the second solvation shell and the solute [44, 175],
a feature that is absent in the QM/MM model. The ability of our QM/MM model to
reproduce the experimental results seems to suggest that hydrogen bonding effects on
E0 can be appropriately described using classical electrostatics without the need to
invoke charge transfer; the relative contributions of electrostatics and charge transfer
effects to computed values of E0 is a topic worthy of further study. We also note that
a fully explicit solvent model has the advantage that larger, less symmetric solutes can
be treated on the same footing without having to carefully define a second solvation
shell.
4.6
Conclusions
In this Article we presented a QM/MM model for computing standard reduction
potentials in explicit solvent. Our model uses a combination of DFT and a polarizable force field for computing energy gaps associated with vertical electron attachment/detachment. The linear response approximation is applied to estimate the free
energy of reduction from the thermally averaged energy gaps in the oxidized and re116
duced states. Using this model, we computed Eo for a series of aqueous transition
metal complexes and obtained very good agreement with experimental results.
In an effort to locate the source of the dramatically different results between
the ISM and QM/MM models, we investigated the different physical effects in the
two models as possible contributing factors. We were able to make the following
conclusions:
* ISM calculations for transition metal aqua complexes are especially sensitive to
the choice of hydrogen radii parameters. However, the errors associated with
the ISM are not solely caused by poor parameter choice, but also stem from
physical effects missing from the model.
" Solute flexibility effects contribute a relatively small amount to EO calculated
using the QM/MM model (~ 0.1 V) for the transition metal aqua complexes,
which is smaller than the statistical noise.
" Dispersion effects contribute a relatively small amount to Eo calculated using
the QM/MM model (~ 0.2 V).
" By process of elimination, the largest physical effect responsible for the differences between the QM/MM and ISMs is due to short-range solvent structure,
most likely induced by solute-solvent hydrogen bonding.
We conclude with some caveats of the QM/MM model along with proposed improvements and extensions. The treatment of electronic polarizability in this study
was not ideal; electronic polarizability was implicit in the equilibrium MD simulations yet explicitly needed for computing the vertical energy gaps. A fully polarizable
QM/MM MD simulation may deliver a more satisfactory solution; this would treat
both equilibrium and ionized electronic states on the same footing, and also allow
us to study fluctuations of the energy gap for the purpose of determining Marcus
reorganization energies and corrections to the LR approximation.
As with all solvent models, proper calibration and parameterization is essential.
A classical explicit solvent model offers a lot of flexibility in its force field parameters,
117
and these parameters must be chosen such that the model properly reproduces important bulk properties of the solvent including but not limited to the dielectric constant.
The empirical van der Waals interaction in the QM/MM Hamiltonian must also be
parameterized to properly describe the London dispersion forces between solute and
solvent; this is a possible application for automatic force matching methods [289].
Finally, while the aqueous transition metal complexes presented here offer a reliable benchmark set for testing the QM/MM model, the vast majority of redox
processes in aqueous solution are accompanied by interesting chemistry such as proton transfer and dimerization. Any reaction chemistry that accompanies redox events
must be accounted for in the QM/MM model, and this also poses questions for the
validity of linear response. Aqueous systems that undergo reversible proton-coupled
electron transfer [59] including phenols, quinones, anilines and other small organics
will be studied in a future report.
118
Chapter 5
Direct-Coupling 02 Bond Forming
Pathway in Cobalt Oxide Water
Oxidation Catalysts
In the globally important problem of water oxidation, metal oxides have always played
an important role, from solid-state materials [150,151] to molecular catalysts containing metal-oxo clusters [38,317] and oxo-bridged metal dimers [108,166]. More recently,
a cobalt phosphate water oxidation catalyst [140,141,170,182,265] (CoPi) has generated particular interest due to its ease of synthesis, high activity at neutral pH, and
robustness in a wide range of operating environments [135,262,324], but the structure
and mechanism of catalysis are presently unclear.
In this Communication, we outline the results of a theoretical investigation in
which we study the 0--0 bonding pathways in a cobalt oxide cubane model based
on CoPi. We find a direct coupling pathway between two cofacial terminal CoIv(O)
oxo groups (5-1, outer ring) as a possible operative mechanism for 0-)
bonding.
A distinguishing charateristic of the proposed mechanism is the invocation of 0-0
coupling after only two out of four oxidation events in the overall cycle, avoiding the
energetically costly buildup of four CoIV redox equivalents. The third and fourth
oxidation events greatly reduce the energetic barriers for the catalytic turnover steps.
Our results are based upon a careful screening of many possible mechanisms and
119
verified by QM/MM calculations of standard reduction potentials (Eo) and free energy
barriers.
OH
I
OH
/
Cd"-- O
6(q0 SO)
/
(PT)
p 7,vs.NHE)
(q=
0s )
OH
0
O
C
H20
Co"
OH-Co" -O
---
O
E-1.39 V
5HOH
(q=O,S=I)
Cdv
O-Cd"
oH
Coll -O
co" -O
44,0
2H
3
_O
\
o
/
0----Cd,
(q=0,S=0 [AF])
AE-25oh
AG8- 2.7kcImol
/ -4(q=O)
2
Figure 5-1: Outer ring: Proposed mechanism of water oxidation catalysis; intermediates are labeled in bold face, with net charge and lowest-energy spin state in subscript
(AF denotes an open-shell electronic state with antiferromagnetic coupling). Standard reduction potentials (Eo) and free energy barriers for the first half of the cycle,
computed by QM/MM, are given. Terminal aqua ligands and the bottom half of
the compound are omitted for clarity. Center: MM dynamics trajectory of model
catalyst structure (held static) with superimposed positions of nearby explicit solvent
water molecules (oxygen: red spheres, hydrogen: black dots), highlighting the strong
hydrogen-bonding effects.
The model catalyst 1, shown in 5-1, contains four octahedrally coordinated Com
atoms (in teal) in a Co 4 0 4 cubane core. Each Co atom is coordinated to one hydroxo
ligand, two aqua ligands, and three bridging p-oxos. The compound is electrically
neutral. The geometry-optimized structure of 1 gives Co-O distances and Co-Co
distances in good agreement with Fourier-transformed XAS data on CoPi at short
distances [142,236] (Figure Sl). Our model does not contain phosphate ligands, as
the XAS interpretation does not suggest their structural role. This is supported by
the experimental finding that phosphate in this system is interchangeable with other
buffering electrolytes such as methyl phosphonate and borate [90,265], and plays the
120
role of a proton acceptor [266].
The single cubane model is smaller and simpler than the corner sharing cubane
[236] and molecular cobaltate cluster [142] models of CoPi proposed by experimental
groups, which were designed to agree with the finer details of the XAS spectra. Given
that the experimental system is amorphous, we aimed to use a minimal functional
model which contains the basic motifs of CoPi and can reproduce the chemical environment of the active site. Thus, the results of the current investigation may be
applied to the experimental system, while investigating the finer structural details of
the larger models would contribute to a more exhaustive description. Our model also
has similarities to proposed OEC structures [97] and other water oxidation catalysts
with metal-oxo cores [38,317], which makes this study helpful for understanding a
greater scope of systems.
We searched the configuration space of the singly and doubly oxidized compound
for stable O-O bonded structures. The energies of all relevant structures were computed using DFT; oxidation events, which we assume are proton-coupled, were simulated by deleting hydrogen atoms from 1. The pathway with the lowest barrier was
found to be direct coupling between cofacial Co'v(O) oxo groups after two oxidations
(3-+ 4).
During the course of our investigation, we examined many other 0-0
bonding
pathways which were all rejected based on unfavorable energetics. An O-O bond
between Colv(O) and CoIv(OH) groups was found (5-2, 3A), but the barrier was
much higher compared to coupling between two CoIv(O) groups. Nucleophilic attack
of a solvent water molecule on a terminal oxo (i.e. acid-base mechanism, 3B) was
observed only if at least one cobalt atom was oxidized to Cov(O); this was ruled out
for lack of experimental evidence for Cov. Earlier studies on ruthenium complexes
indicate that the acid-base mechanism requires a Ruv(O) intermediate [66,290], similar to our requirement for Cov here. No 0-0
bonding occurred between a solvent
water molecule and any CoIv(O) group, even when additional water molecules (up
to three) or phosphate groups were placed nearby as hydrogen bond/proton acceptors. We found that 3 was nearly isoenergetic to its nonreactive isomers containing
121
distal CoIv(0) groups (Table S1, Supporting Information), further indicating the
thermodynamic accessibility of 3. Some 0-0 bonded structures were found involving p-oxo groups (3C), but the activation barriers for liberating the resulting 02
molecule proved too great as it led to breakage of many Co-O bonds.
H
OH
--
C'" --
/~C~
AE48 kc~mol
H6kat .
kal
mO
0/
O0
0 \)
1, H
~
Co--O
H
wq__C
i/OH
O-
o--cow
CO7 o
3C
|
AE:=-6kcA~mAl:
E-
E~IAV
E*=-1.49
V
OH
3B
3A
2A
H/O
Coe - 0
Cov
11
Cov
0
O
Figure 5-2: Energetically disfavored structures and pathways. 2A: An alternative
protonation state of 2, containing a Colv bis-hydroxo; EO (labeled) is significantly
bonding pathhigher than that of the 1/2 couple. 3A, 3B, 3C: Alternative 0-0
ways with reaction energies and activation barriers labeled; the computations for 3B
included extra water molecules or phosphate anions as proton acceptors (not shown).
3B was ruled out because CoV was a prerequisite to reactivity.
We performed a series of constrained geometry optimizations, beginning from 3,
in which the two terminal CoIv(O) groups were brought together in 0.1
found that AE for 0-0
A steps.
We
bonding is -27.3 kcal/mol with an activation energy of 2.3
kcal/mol (5-3). We also found that the value of (S 2 ) went from 1.0 (broken symmetry)
to 0.0 (closed shell), indicating the formation of a single bond from two radicals; the
0-0
interaction is repulsive in the triplet state, further suggesting participation of
an open-shell singlet.
The broken-symmetry ground state of 3 contains antiferromagnetic coupling with
strong radical character on the oxo groups, similar to the Baik group's results for the
blue dimer [310] but involving different d orbitals [24]. A spin density plot is given in
Figure S4, suggesting that these CoIv(0) groups are formally closer to being Com"(0.)
radicals; however, we shall retain the Co'v(0) notation for simplicity. Molecular
orbital analysis of 3 and 4 (5-3 insets and Figure S2) shows formation of a o bond
along with doubly occupied 7r and -r* bonding orbitals for a total bond order of 1.
The direct coupling pathway reported here can be compared to the mechanistic
proposal for the two-center Ru-Hbpp catalyst proposed by the Llobet group [35]; a
key difference between our results and Ref. 35 is that direct coupling occurs after only
122
two out of four oxidations in the present study whereas direct coupling in Ru-Hbpp
requires all four oxidations. The early occurrence of 0-0 bonding in the cycle may
help to lower the overpotential by formally reducing the cobalt centers and avoiding
the energetically costly buildup of CoIv redox equivalents. Furthermore, we propose
that the third and fourth oxidation events will greatly facilitate subsequent steps
involving 02 release, which have presented significant activation barriers in previous
computational studies 312].
1
Co1" 2Co11 202 Energy, Spin Expectation Value
0Energy
-1
-
-5
HOMO: -6.1eV
a -12.1eV
-15
0.4
-20
0.2
-25 -u
0
3
2.8
2.6
2.4 2.2
2
Distance (A)
1.8
1.6
1.4
Figure 5-3: DFT energies and (S2 ) as a function of O-0 distance for the doubly
oxidized model catalyst, 3. Left inset: HOMO at roo =3.0 A. Right inset: o bonding
orbital at roo =1.4 A.
The subsequent 02 liberation step was also explored using DFT. The addition
of one water molecule directly to 4 can lead to displacement of the bridging 0-0
group coupled with intramolecular PT, resulting in terminal OOH and OH groups.
This process was found to be downhill but with a very high barrier (AE = -12.9
kcal/mol, Ea < 35 kcal/mol), as the transition state involves the breaking of an 0-H
bond and a Co-O bond. This imposing barrier was significantly lowered if the third
and/or fourth proton-coupled oxidation events in the overall cycle were allowed to
occur prior to water addition. When the water addition reaction was examined for
the once-oxidized (resp. twice-oxidized) equivalents of 4, we found that the energetics
were AE = -9.3 kcal/mol, Ea = 6.8 kcal/mol (resp. AE = -8.3 kcal/mol, E < 9
kcal/mol); the barrier height was significantly reduced. These results suggest that at
least one proton-coupled oxidation of 4 occurs before water addition.
123
Adding water to the once-oxidized or twice-oxidized equivalents of 4 displaces the
bridging 0-0 group without transferring a proton, resulting in terminal 00 and
OH 2 ligands; the combination of two oxidations and water addition thus leads to 5.
Adding a second water to (5) releases molecular dioxygen (AE = -13.4 kcal/mol,
Ea =_4.0 kcal/mol), leading to 6 which is simply an isomer of 1; the catalytic cycle
is then closed by intramolecular PT.
For 0-0
bonding to take place via the proposed cycle in 5-1, the steps must
be thermodynamically and kinetically viable, especially in comparison to alternative
pathways. Thus, it is vital to accurately compute EO for the various oxidation steps
and free energy barriers for the chemical steps. We found that an explicit solvent
model was essential for this task, as it accounts for hydrogen bonding effects which
are not present in gas-phase DFT or implicit solvent models. This is highlighted by
results from exploratory MM simulations, in which we found strong hydrogen bonding
effects (5-1, center). Furthermore, we found that the solute was likely to be highly
flexible (Figure S3), suggesting that configurational averaging might be necessary to
accurately evaluate E0 . The results of our explicit solvent calculations are presented
below.
We used QM/MM to compute the first two Eo values in explicit solvent. In this
framework, the catalyst (QM region, treated with DFT) interacts with the explicit
solvent molecules (MM region, treated with classical molecular mechanics) through a
QM/MM interaction potential [3]. Molecular dynamics trajectories were propagated
in the NVT ensemble at 300 K; we generated at least seven trajectories with length
>5 ps for each oxidation state. The free energy of oxidation was determined from the
thermally averaged vertical energy gaps using the formula AG =
j((IP)red + (EA).x)
within the linear response approximation [59]. Here, the subscript denotes the solute's
oxidation state within the dynamics trajectory. Eo was determined from AG by
subtracting 4.24 V for the absolute NHE potential and 59 mV per pH unit when PT
takes place. We computed EO for the Co"'/Co" couple in the hexaaquacobalt cation
as an internal standard and found a value of 1.74 V, a deviation of -0.18 V from
the experimental value of 1.92 V. We consider this to be the typical uncertainty of
124
computed E0 values in this report.
Our computed E0 values are reported in S1. The protonation state of the oxidized
species had a major effect on the E0 value for the first oxidation event. When the
oxidized species contained a terminal oxo group (1/2; first row), the E0 value was
0.67V lower compared to the alternative pathway leading to two terminal hydroxo
groups (1/2A; second row). This indicates that an aqua/oxo configuration is energetically more favorable than a bis-hydroxo configuration and increases the probability
for participation of terminal Corv(O) groups in the mechanism. Further oxidation of
2 leads to 3 and subsequent 0-0
coupling.
Table 5.1: Standard reduction potentials (E0 ) computed by QM/MM.
Reduced
1
1
2
Oxidized
2
2A
3
Eo(vs. NHE, pH 7)
0.82 V
1.49 V
1.39 V
We also investigated the free energy barrier to 0-0
bond formation using the
QM/MM dynamics trajectories of 3. 5-4 is a potential of mean force constructed
from the probability distribution of 0-0
distances. Two free energy minima are
visible, corresponding to the open and bonded configurations (3 and 4); the barrier
to bond formation is quite small at 2.7 kcal/mol, only slightly higher than the barrier
calculated using gas-phase DFT. Direct coupling occurred spontaneously (within 25ps) in three out of seven dynamics trajectories, demonstrating that 0-0
bonding
is a fairly rapid step. It should be noted that the true free energy minimum of (4)
is likely much deeper than that shown in Figure 3, but an accurate evaluation of the
well depth requires a much larger set of QM/MM data.
In summary, we examined the water oxidation mechanism of a cobalt oxide cubane
model of CoPi. The distinguishing features of the mechanistic cycle are as follows.
Two adjacent Co"1(OH) moieties are first oxidized to CoIv(O) via proton-coupled
oxidation events. Our computed values of Eo show that 1) the oxidized state heavily
favors the Colv(O) protonation state over Co'v(OH) 2 and 2) two adjacent Corv(O)
125
Co 2C'0 20
2
Free Energy Profile
65
-3
E
-4
WI
-2
3.5
3
2.5
2
1.5
0-0 Distance (A)
distance for
Figure 5-4: QM/MM free energy profile as a function of 0-0
Co 2 Com11 2 O2 (3-+ 4). Simulation snapshots are shown for the free energy minima
at roo =3.0 A and roo =1.4 A, indicated by triangles.
groups can be formed at experimentally applied potentials. After forming two adjacent CoIv(O) groups, direct coupling occurs to form an 0--0 bond with a low kinetic
barrier and a strong thermodynamic driving force. The third and fourth protoncoupled oxidation events greatly lower the barrier for adding two water molecules
and displacing one 02 molecule, affording a pathway for catalytic turnover.
The implications of our study in the context of ongoing experimental investigation are as follows. Recent electrochemical studies [266] have shown that the ratedetermining step is purely chemical in nature, preceded immediately by a one-electron,
one-proton minor equilibrium. In our proposed mechanism, the chemical steps are
the 0-0 direct coupling, the first water addition (leading to a terminal 00 group),
and the second water addition (displacing molecular 02). We found a very low barrier
for 0-0
direct coupling and slightly higher barriers for water addition and oxygen
displacement, suggesting that one of the latter two steps may be rate-determining.
To confirm this hypothesis, the redox potentials prior to water addition need to be
computed, because the oxidized species must be in minor equilibrium at experimental potentials. It is also important to investigate the role of intramolecular PT in
determining the reaction rate. These are topics of ongoing work.
Our work represents an early step towards understanding the catalytic mechanism
of this interesting system; while the direct coupling pathway found here is by no means
126
exclusive, the clarity of our theoretical findings suggests that this pathway deserves
further experimental investigation. The proposed mechanism can be tested on more
detailed models structures of CoPi as more structural data become available. We also
plan to apply the QM/MM methodology to compute the third and fourth Eo values
as well as free energy barriers for water addition and oxygen displacement, providing
quantitative verification of the entire cycle.
127
128
Chapter 6
Hybrid ensembles for improved
force matching
Molecular mechanics (MM) simulation is a powerful method for investigating the
dynamical behavior of complex atomistic systems, but its utility depends critically
on the accuracy of the empirical potential being used. A common approach is to
fit the parameters in the empirical potential to a high-accuracy reference potential
energy surface (PES), which can be obtained from quantum mechanical (QM) calculations. This approach, known as force matching [7,96,131,169,180,319] or potential
fitting [40, 224,263,275], aims to find the optimal parameters k that minimize an
objective function x 2 of the difference between a set of properties Q computed using
the reference PES, and the analogous properties M computed using the empirical
potential:
x2
f
JR3N
P(r)JX(r, k)|2dr.
(6.1)
Here, the norm squared of the difference vector X(r, k) = M(r, k) - Q(r, k) is integrated over the 3N-dimensional configuration space. X may contain energies, atomistic forces, and/or other computable quantities that one wishes to match.
The
integral is typically performed by quadrature, or sampling techniques such as the
Metropolis algorithm; in the latter case, P(r) is a probability density corresponding
to some ensemble.
129
The ensemble being sampled may be generated with the reference PES; we will call
this the QM ensemble and denote the corresponding objective function using XQM.
Accurate sampling of the QM ensemble is generally desirable but costly. Alternatively,
training configurations may be rapidly sampled from the empirical (MM) ensemble;
we will denote the corresponding objective function using X2gA.
Ischtwan and Collins,
[129] and more recently Akin-Ojo and Wang [7] have proposed a scheme in which
the MM ensemble is resampled after parameterization, with the process repeated
in generations until self-consistency.
Both types of force matching utilizing XQM
and X2MM have been applied widely to parameterize empirical potentials for gas-phase
molecules [129,172,226], condensed phase systems [7,132,319], and solids. [96,115,169,
211,271] The force matching method is not restricted to using MM for the empirical
potential or QM as the reference PES; in multiscale coarse graining methods [203], the
reference PES is an atomistic MM potential and is used to parameterize an empirical
potential for coarse-grained particles.
In this Communication, we first present a simple heuristic example where force
matching using either XAMM or XM does not optimally reproduce the QM PES.
Instead, configurations from both QM and MM ensembles are important for force
matching, and using either ensemble alone is insufficient. As a solution, we propose a
hybrid-ensemble approach that combines the probability densities of both ensembles
and generates a MM potential that optimally reproduces the QM PES in the heuristic
example. Finally, we demonstrate the effects of hybrid-ensemble force matching when
it is applied to parameterize MM potentials for a helium dimer and a water hexamer.
As an example, take both the QM PES and the MM potential to be hard repulsive
walls as illustrated in Fig. 6-1. The MM potential has one adjustable parameter
0
MM
which sets the location of the hard wall, and the optimal match is obtained when
0
MM
~aQM- We make the simplifying assumption that the wall height is
kT, such
that only the configurations where r > a are thermally accessible, and the repulsive
region is excluded from the integral in Equation 6.1.
Consider first the case where the QM PES has a hard wall at -Q = 50, and the
MM potential underestimates this value at aMM = 30; we will denote this potential
130
using MM< (Fig. 6-1a). The "error region", with finite X(r, (MM), is given by r c
QM M<-
Error rgion
-
(a) 12
1
0.8
QM Accessible
-
0.6
0.4
MM Accessible
0.2
0
0
20
'MM 4 0
'QM
80
60
100
Distance (arb.units)
(b) 1.2
C
Error region
-
0.8
Accessible
0.6 -QM
0.4
C
QM -
CO
U0.2--
OAMM
Accessible
0
0
20
40
aQM 60
GMM
80
i0
Distance (arb.units)
Figure 6-1: (a) QM and MM< potential energy curves. (b) QM and MM> potential energy
curves. The error region and thermally accessible regions are labeled using arrows.
{30, 50}; the QM-accessible region with finite PQM(r) is given by r > 50, and the MMaccessible region with finite PMM(r) is given by r > 30. The error region overlaps
with the MM-accessible region and contributes to XMM; thus, XMM is minimized by
increasing cMM until the optimal value, qMM = cQM, is obtained. However, the error
region does not overlap with the QM-accessible region and does not contribute to
XQM; in fact, any aMM < 50 minimizes XQM! Thus, minimizing XQM risks severely
underestimating oMM-
Now, consider the case where the initial guess is an overestimate (o-MM = 70),
denoted by MM> (Fig. 6-1b). The error region is now r E {50, 70}; the QM-accessible
region is the same (r > 50), and the MM-accessible region is r > 70. The error region
now overlaps with the QM-accessible region but not the MM-accessible region. The
fitting situation is reversed; there is a contribution to XQM but not to XMM
now expect minimizing X
M
We
to effectively optimize aMM, while any o-MM > 50 will
131
minimize X~M. Thus, in a realistic situation where the relationship between the QM
PES and the MM potential is unknown, we would expect force matching using X2M
and X2MM to respectively underestimate and overestimate o-; clearly, both choices are
inadequate. The problem is further worsened when approximate functional forms are
used for the MM potential.
The key observation is that the optimization fails when the chosen ensemble excludes configurations that are included in the other ensemble. Evidently, it is essential
for configurations from both ensembles to contribute to X2 in order to produce an
accurate MM potential. A simple treatment would be to construct x 2 using a hybrid
ensemble which combines the QM and MM probabilities:
2
j
(wPQm(r) + (1 - w)Pm(r))|X(r, k)| 2 dr.
(6.2)
JR3N
Here, w is an adjustable mixing parameter which mixes the QM and MM probabilities. In the hard-wall example, the entire error region always has a finite probability
density in x2, and minimizing x2 would effectively optimize oMM starting from any
initial value. Note that after obtaining a perfect fit (i.e. the QM PES and MM potential are exactly the same), the hybrid ensemble reduces to the standard QM or
MM ensemble.
Our main results in this Communication are based upon force-matching using
X2. We now demonstrate the application of hybrid-ensemble force matching to two
example systems; the helium dimer and water hexamer. The QM PES, in all cases, is
computed at the Hartree-Fock (HF)/3-21G level of theory using Q-Chem. [252] MM
simulations were performed using GROMACS. [86] Force matching was performed
using the ForTune force matching program developed within our group, which minimizes x2 using a Newton-Raphson algorithm in similar fashion to previously published
software packages. [41,288] We include both energy and force contributions in x2; each
component is inverse variance-weighted, and the force components are attenuated by
1 such that energy and force differences contribute approximately equally. In all
optimizations, a penalty proportional to |Ak| 2 is added to prevent large fluctuations
132
in the parameters when the change in x2 is small.
The examples presented in this Communcation are designed to highlight the effects
of using the hybrid ensemble for force-matching to the QM PES and not intended to
reproduce experimental properties. Thus, we will discuss the quality of the parameterization by direct comparison of the QM PES and MM potentials in the former
case, and by comparing radial distribution functions from QM and MM molecular
dynamics in the latter.
Our first example, the parameterization of an MM potential for a helium dimer,
is a direct computational realization of the hard-wall example above. In HF theory,
the interaction between helium atoms is well characterized by a van der Waals (vdW)
repulsive interaction. Our MM potential uses the Lennard-Jones (U) functional form:
ELJ(rb) =4c -
0
"ab
+ (")
rab
(6.3)
In all cases, the LJ well depth c is fixed at 0.01 kJ/mol, and only the X-intercept a
is optimized. Training configurations were obtained from a uniform grid of 800 He-He
internuclear separations ranging from 1.5
A to
4.0
A. X2M
and X2 M were computed
by performing a weighted sum over snapshots, with the appropriate Boltzmann probabilities: PQM,, = e-6EQM(r.) and PMM,s
-
e-EMM(r,). The hybrid x2 uses an average
probability with w set to 0.5: PH,s = !(PQM,s ± PMM,s).
Two initial MM potentials, MM< and MM>, were chosen such that they respectively underestimate and overestimate the position of the repulsive wall. For each
initial potential, we obtained three optimized MM potentials using the three objective
functions above; each optimized MM potential was obtained from three generations
of the iterative force matching scheme described earlier. Fig. 6-2 shows the QM PES,
initial MM potential, and the optimized MM potentials obtained using each of the
three objective functions. Here, MMMM stands for "optimized MM potential using
x2
The values of a- and x 2 for the three optimizations are given in Tables S1 and
S2.
From Fig. 6-2, the role reversal of QM vs. MM weights can be clearly seen. Start133
(a)
*
15 -
'
-.
'5
2
n')ai
MMMM*
MM M
MA~
MmH*
10 W5
0
1.5
2
2.5
3
4
3.5
Distance (A)
20
(b)
15
2
QM Potential
MM, (initial)
MMMM
MM
MMH
*
*
10 -
*
-
-
1
01.5
2
2.5
3
3.5
4
Distance (A)
Figure 6-2: Optimized MM potentials from force matching, starting from the initial guess
(black line) and fitting to the quantum PES (black dots). Initial guesses are MM< (a)
and MM> (b); optimized MM potentials using X~M (red), XQM (green), and XH (blue) are
shown.
ing from MM<, using X2M reproduces the quantum result while using XY2
underesti-
mates the solution; the opposite effect is observed when we start from MM>. In both
cases, using
xH
produces the correct solution, because the regions that contribute to
either XM and X2M are both counted.
It should be noted that in this example, all methods will eventually lead to the
correct result; the convergence will just be very slow, taking many generations. This
is because the repulsive interaction still contributes a very small amount to x2 , and
that all regions are perfectly sampled in this ideal case. When more parameters and
degrees of freedom are involved, changing the sampling method can lead to different
self-consistent MM potentials, as we describe in the following example.
In our second example, our system is a cluster of six water molecules; the intermolecular interactions in small water clusters is highly nontrivial [276] and MM
potential development for water remains a highly active field. We chose the flexible
134
SPC/E model [27] as our initial MM potential; the MM potential contains 7 empirical parameters. In each generation of force matching, the MM potential is used to
generate 3.0 ns of dynamics in the canonical ensemble (T = 300K), from which 3,000
snapshots are sampled at ips time intervals and added to the training data. We utilize WHAM [98] to include training data from past generations. A shallow harmonic
potential with force constant 0.01 kJ mol'
nm- 2 was applied to prevent divergent
trajectories.
The MM dynamics directly samples the MM ensemble, so in computing X2MM each
snapshot has an equal probability. In computing x , each snapshot's probability contains a constant from the MM ensemble, plus a non-Boltzmann factor corresponding
to that snapshot's probability in the QM ensemble:
s
X
1
,-(1QM(s)-EMM(r))
Z=(
SQM
+ (1-
)
IXrk)|
2
.
(6.4)
Here, S is the total number of snapshots, and SQM is the sum of all non-Boltzmann
factors.
Three objective functions were used with the following ensembles: X2 (W= 0.8),
xIo (w = 0.5), and
X2
(w = 0.0, pure MM). Due to large fluctuations in the non-
Boltzmann factors, X2oo (w
X80
=
1.0, pure QM) was not usable in force matching and
was used as a substitute. Force matching was performed for 19 generations, and
the final MM potentials (MM 80 , MM 5 o, MMO) were used to obtain radial distribution
functions (RDFs) for comparison with an 1.8 ns AIMD reference trajectory, generated
using the same temperature and harmonic potential. All three cases produced improved agreement with the AIMD RDFs compared to the initial SPC/E parameters;
however, none of the MM potentials reproduced the exact shape of the AIMD RDFs,
possibly due to the limitations of the functional form.
Fig. 6-3 shows the RDFs generated using the three optimized MM potentials
compared to the AIMD RDF. MM 80 underestimates the contact distances and peak
positions for all three atom pairs; notably, this occurred starting from an overestimated initial guess. This may have been caused by fitting to components of x 2 from
135
0.5
,A
I
(a) 0.4
simulation
.
MM ----
MMBO
MM50
0.3
0
0
CM
0.2
0.1
0
6
5
4
3
2
1
roo (A)
0.25
(b)
AIMD simulation
MMO
MM50
(b)MM
0.2
-
.
--
0.15
0
0.1
0.05
0
1
2
4
3
6
5
rOH (A)
0.25
AID simulation
(c) 02
MMO
MMso
O
0.15
a
-
MM50
0.1
0.05
1
2
4
3
5
6
rHH (A)
Figure 6-3: Radial distribution functions of water for (a) 0-0 distances, (b) 0-H
distances, and (c) H-H distances. Thirty AIMD RDFs of length 60 ps were combined to
produce the reference curve.
other intermolecular degrees of freedom (i.e. torques) which couple to this one. MMo
overestimates the distances, while MM 50 provides the best estimate with peak positions agreeing with the AIMD result to within 0.1
A. Thus, the hybrid-ensemble
force
matching leads to the most accurate MM potential in this more complex case and
corresponds well with the earlier examples.
We also performed force matching by sampling from the QM ensemble directly.
50,000 snapshots were sampled at ips intervals from the AIMD trajectories, and force
136
matching was used to obtain the optimized potential MM 100 . We find that MM 100
underestimates the approach distance, but not as severely as MMso; this result is
surprising as we expect MM 8 o to interpolate between MMo and MM 100 . This may be
due to differences in the sampling trajectories; the training data for MM 100 came from
direct AIMD, while the data for MM 80 came from an MM trajectory and required
non-Boltzmann weights. This indicates that more complete sampling may be achieved
by combining trajectories from both AIMD and MM dynamics.
In summary, using a single canonical ensemble to obtain training data for force
matching can lead to substantial errors in the optimized MM potential. In particular,
there is a risk of obtaining a poor fit if x 2 neglects regions which make large contributions to IX(r, k) 12 and have finite probability in either the QM or MM ensemble. As
a solution, we proposed constructing X2 from a hybrid ensemble which combines the
probabilities from both ensembles. We demonstrated the advantages of the method
by applying force matching to parameterize MM potentials for a helium dimer and a
water cluster. In both cases the anomalies associated with single-ensemble objective
functions X
and X2MM were shown, and the hybrid X provided the correct behavior.
Our finding makes intuitive sense: if training data is sampled from the MM ensemble,
the MM-accessible configurations will naturally be included in the fit. Meanwhile,
QM-accessible configurations that are MM-inaccessible should still be penalized for
their absence.
The hybrid ensemble may be applied to improve the parameterization of atomic
charges and vdW-type interactions (e.g. LJ, n-6 or Buckingham), which traditionally
are the most difficult to parameterize yet are essential for determining the dynamical
and bulk properties of any multimolecular system.
137
138
Chapter 7
Conclusions
After such a heavy dose of theory and computation, equations and simulations, it may
be time to come back to take a step back into reality and ask what all of our studies
have accomplished. What insight did we obtain from performing all of these studies?
Have we learned any knowledge that is actionable - i.e. could all of this information
actually lead an experimentalist to design an improved photovoltaic device or water
splitting catalyst?
Here are the main results of this thesis that pertain to experiment, condensed into
a few sentences:
" In a donor-acceptor interface between two organic semiconductor materials,
charge-transfer (CT) states may dissociate without a significant thermal barrier
due to a combination of band-bending effects and random fluctuations in CT
state energies.
" In a water oxidation cycle catalyzed by a single ruthenium center, the 0-0
bond forming step most likely occurs by nucleophilic attack of a solvent water
molecule on a highly oxidized Ruv(0) group via the acid-base mechanism.
* Two-center ruthenium catalysts are very likely to operate by the same mechanism as the one-center catalysts.
* In a water oxidation cycle catalyzed by the CoPi cobalt phosphate compound,
139
the 0-0 bond forming step may occur via a direct coupling mechanism between two terminal Corv(O) groups, and this may happen after only two oxidation events into the overall catalytic cycle.
Here are the main implications for theoretical method development:
" QM/MM simulations provide a highly suitable framework for simulating organic
semiconductor molecules in the environment of their disorderly neighbors.
" Explicit solvent models are essential for describing solvation in systems that
have strong solute-solvent hydrogen bonding effects.
" The force matching method, which is based upon fitting energies and atomistic
forces to high-accuracy ab initio data, depends critically on the canonical ensemble used to sample training configurations; the most suitable ensemble is a
linear combination of the fitting and reference ensembles.
The theoretical implications are clear; the results of our simulations provide much
guidance toward performing future simulations more accurately and more efficiently.
As always, we are reminded to be careful when making our approximations. Each
research problem requires a careful evaluation of the simulation technique and its
underlying assumptions; otherwise, one risks running into situations where the common approximations break down and the results are not meaningful (one example
is when we tried to compute the RuIv/Ruv standard reduction potential using implicit solvent models.) When the approximation breaks down, one must resort to
more detailed methods -
or build new models -
that are capable of describing the
essential aspects of the real system. On the other hand, in order to access larger systems and longer timescales, it is desirable to systematically improve upon the more
cost-effective approximations. The balance of cost vs. accuracy is an eternal problem
in theoretical chemistry, and the ability to make the right approximations and strike
the correct balance is a testament to the theoretical chemist's understanding of the
system.
140
Perhaps the experimental implications will also have their impact. The mechanism
for thermal dissociation of CT states is interestingly reminiscent of biological electron
transfer systems, in which charge separation is promoted by a downward-sloping
gradient that drives oppositely charged carriers away from each other. Perhaps such
an effect exists naturally at the donor-acceptor interface, and it may be possible to
engineer this built-in electric field in such a way that charge separation occurs rapidly
but without significant energy loss.
As for the mechanism of 0-0 bond formation, we found strong evidence for
two different coupling pathways. One pathway, the acid-base mechanism, involves
the nucleophilic attack of one oxygen on another; this requires a strong imbalance
between the electron densities of the two oxygen atoms, and generally necessitates
high-valent oxidation states. The direct coupling mechanism provides a relatively
easy route to 0-0
bond formation between two oxo groups containing some radical
character, but the main challenge is to synthesize systems where terminal oxo groups
can come together without strong electrical repulsions. Perhaps electrically neutral
systems, or ones that introduce oxo groups side-by-side rather than head-on, will fare
well in this design strategy.
The urgent need to refine our renewable energy technologies translates into a need
for deeper understanding of the underlying physical processes. When studying the
mechanisms of solar energy conversion and water oxidation in artificial systems, one
gains an appreciation for the fundamental challenges of the processes involved - for
example, the challenge of pulling apart a CT state where the electron and hole attract one another, or the challenge of focusing enough positive charge on a single
point to incite 0-0
bonding without wasting excessive amounts of energy in the
process. Suddenly, the biological processes that accomplish these energy conversion
processes daily are that much more amazing; the efficiency and rapidity with which
Photosystem II accomplishes energy transfer, charge separation, and water oxidation
can be described as fluid and graceful, compared to our crude artifical systems. We
may be able to learn some lessons from these intricate and precise living machines,
but to truly mimic them requires an unprecedented level of precision and dexter141
ity on the molecular scale. Perhaps in striving to achieve this goal, experimental
chemists and materials scientists can find many opportunities to pool their distinct
and complementary talents.
Theoretical chemistry has its relative strengths and weaknesses compared to experiment. Theory has the advantage of possessing unmatched spatial and temporal
resolution; while the most carefully designed X-ray diffraction experiment can only
resolve a protein structure to within 3 A a force field can provide structures resolved
down to the femtometer. Picosecond and femtosecond-timescale reactions are directly accessible by simulation yet require the most clever alignment and timing of
laser pulses in experiment. The weakness of theory lies in the constant tradeoff of
accuracy and computational cost. While an experiment is always a measurement of
reality, a theoretical simulation of increasingly large system sizes or long timescales
cannot help but deviate further from reality. Some observables are inherently more
difficult to compute than others, and perhaps some observables are best measured
while others are best computed. By recognizing the complementary abilities of theoretical and experimental chemistry, practitioners of the two disciplines eventually
learn how to more effectively help one another, and when theory and experiment
achieve a true mutual understanding the results are amazing indeed.
142
Appendix A
Supporting Information for
Chapter 2: Acid-Base Mechanism
for Ruthenium Water Oxidation
Catalysis
143
-2
-1
0
Computed reduction potential
1
2
Figure Si: Correlation diagram of calculated and experimental redox potentials for various
organic molecules and metallocenes as reported in Baik and Friesner [16]. No linear fit was
performed. R 2 = 0.98, MAE
=
0.150V
144
Table Sl: Redox potentials of relevant half-reactions vs. NHE. All potentials are standardized to pH 0 by applying a -0.059V/pH correction. For compounds 1 through 19, see Baik
and Friesner [16].
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Complex
Anthracene
Azobenzene
Benzophenone
Nitrobenzene
TCNQ
TCNQ
TTF
TTF
FeCp(H 3 COC) 2
FeCp 2
FeCp* 2
CoCp 2
Cr(Me 3 Ph)2
FeCpPh
CoCp* 2
FeCpPh*
FeCp*Ph*
RhCp 2
RhCp 2
CrCp* 2
CrCp*2
MnCp* 2
MnCp* 2
RuCp* 2
OsCp* 2
Couple
-1/0
-1/0
-1/0
-1/0
-2/-1
-1/0
0/+1
+1/+2
0/+1
0/+1
0/+1
0/+1
0/+1
0/+1
0/+1
0/+1
0/+1
-1/0
0/+1
-1/0
0/+1
-1/0
0/+1
0/+1
0/+1
145
E1
-1.692
-1.072
-1.528
-0.775
-0.247
0.641
0.366
1.248
1.133
0.544
-0.098
-0.634
-0.697
-1.013
-1.300
-1.340
-1.347
-2.067
-1.058
-1.512
-1.131
-2.073
-0.420
0.714
0.712
Ea*
-1.679
-1.119
-1.639
-0.909
-0.049
0.361
0.541
0.901
0.981
0.548
0.141
-0.669
-0.739
-1.119
-1.189
-1.329
-1.609
-1.939
-1.169
-2.059
-0.799
-2.259
-0.319
0.791
0.701
1.5
Computed reduction potential
Figure S2: Correlation diagram and linear fit of calculated and experimental redox potentials for various Ru-OH2 coordination complexes. R 2 = 0.55, MAE = 0.110V
146
Table S2: Redox potentials of relevant half-reactions vs. NHE. All potentials are standardized to pH 0 by applying a -0.059V/pH correction. For compounds 1 through 19, see
Dovletoglou and Meyer [92]. For compounds 20 through 22, see Masllorens and coworkers [179].
E
No.
Complex
1
[Ru(tpy)(acac)(H 2 0)] 2 +
2
3
[Ru(tpy)(C 2 0 4 )(H2 0)]2+
[Ru(tpy)(H 2 0) 3 ]2 +
4
6
7
8
9
10
trans-[Ru(tpy)(pic)(H 20)]2+
cis-[Ru(tpy)(pic)(H 2 0)]2 +
cis-[Ru(6,6'Me 2 (bpy) 2 (H2 0) 2]2 +
[Ru(tpy)(tmen)(H 2 0)] 2 +
[Ru(tpy)(phen)(H 2 0)] 2 +
cis-[Ru(bpy) 2 (py)(H2 0)]2+
[Ru(tpy)(bpy)(H 2 0)] 2 +
11
[Ru(tpy)(4,4'-CO 2 Et) 2 (bpy)(H 2 0)] 2 +
5
12
[Ru(tpy)(4,4'-Me2-bpy)(H 20)]
2+
cac
E
EOI/III
expt
0.617
0.456
0.747
0.679
0.813
0.840
0.810
0.823
0.860
1.030
0.917
0.849
0.923
E
calc
"1 E,111/1V
expt
1.030
0.939
1.117
1.002
1.134
1.210
1.100
1.113
1.100
1.210
1.043
1.177
1.203
1.010
1.150
1.207
1.252
1.240
1.250
0.853
0.937
1.070
1.140
1.321
1.250
1.180
1.270
1.036
1.310
1.259
1.450
0.938
1.120
1.231
1.260
13
cis-[Ru(bpy) 2 (AsPh 3 )(H2 0)] +
0.785
1.150
1.184
1.320
14
15
cis-[Ru(bpy) (biq) (PEt3 )(H2 0)]2+
[Ru(tpm) (4,4'-(N0 2 ) 2 (bpy)(H 2 0)] 2+
0.842
1.059
1.100
1.210
1.151
1.403
1.280
1.400
16
cis-[Ru(bpy) 2 (PEt3 )(H20)] 2 +
0.884
1.110
1.197
1.320
2
2+
17
cis-[Ru(bpy)(biq)(PPh 3 )(H2 0)]
0.836
1.130
1.010
1.350
18
19
cis-[Ru(bpy) 2 (P(i-Pr)3 )(H2 0)]2 +
cis-[Ru(bpy) 2 (SbPh)(H2 0)] 2 +
[Ru(CNC)(bpy)(H 2 0)]2+
0.782
0.699
0.805
1.100
1.170
1.100
1.013
1.224
1.245
1.330
1.450
1.150
cis-[Ru(CNC)(nBu-CN)(H 2 0)] 2+
tran9-[Ru(CNC) (nBu-CN) (H2 )] 2
0.966
1.199
1.080
10.180
1.102
1.306
0.940
1.180
20
21
22
147
148
Appendix B
Supporting Information for:
Direct-Coupling 02 Bond Forming
Pathway in Cobalt Oxide Water
Oxidation Catalysts
B.1
Methods
The B3LYP hybrid exchange-correlation functional [25] and the unrestricted spin formalism was used for all calculations involved. QM/MM calculations used the TZVP
basis set [247], while gas-phase DFT calculations used the LANL2DZ basis/ECP combination [123] for cobalt and 6-31G* [120] for all other atoms. All DFT calculations
were performed in Q-Chem [252], except for the small subset of DFT+PCM energy
comparisons reported in S1 which were performed in TURBOMOLE 5.10 [5].
All calculations were performed in the gas phase, except for the first two redox potentials and the 0-0 bonding step which used an explicit solvent model via
QM/MM, and the DFT+PCM computations which used the COSMO continuum
solvent model [248]. Reaction energies were typically obtained by taking energy differences of geometry-optimized structures; activation energies were determined using
149
a transition-state search. In cases where we were unable to determine the transition
state, an upper bound to the activation energy was given.
The QM/MM simulations were performed in CHARMM [42], and the exploratory
MM simulations were performed in GROMACS [86]. The explicit solvent was described by the SPC/E water model [27]. Van der Waals parameters for the solute
oxygen and hydrogen atoms were also taken from the SPC/E model; the cobalt VdW
parameters were taken from UFF [228].
To obtain starting configurations, the model catalyst was first placed into a standard box of 216 SPC/E water molecules and then propagated using a pure MM force
field (prepared using UFF and ESP-fitted charges) for 10.0 ns; frames were extracted
from the MM trajectory at 1.0 ns intervals to serve as starting points for the QM/MM
trajectory.
For each oxidation state, at least seven QM/MM trajectories were started; each
trajectory consisted of at least 2,500 x 2.0 fs time steps for a total of > 5.0 ps per
trajectory. The SHAKE algorithm [243] was applied to constrain the geometries of
the solvent molecules. A Nose-Hoover thermostat [208] was applied to keep the temperature constant at 300K. From the QM/MM trajectories, snapshots were extracted
at 20fs intervals for computation of vertical energy gaps; simulation time during the
first 1.0ps was excluded to allow the system to equilibrate.
Vertical energy gaps were calculated by performing QM/MM single-point energy
calculations on the simulation snapshots in the two different electronic states. Modeling the electronic polarization of the solvent was essential for calculating energy gaps.
For this purpose, we placed polarizable Drude particles at the locations of each water oxygen atom; the polarizability parameter was taken from the SWM4-DP water
model [157] and is equal to 1.042 A3 .
In computing the second proton-coupled redox potential leading up to 3, only
trajectory frames containing unbonded structures were considered. Repeating the
calculation using the 0-0
bonded frames reduces the second redox potential from
1.39V to 0.64V; while interesting, there is at present no experimental evidence for
such a highly concerted pathway.
150
Figure Sl: Gas-phase geometry optimized model catalyst (1 from Figure 1), with
four intramolecular distances labeled (Co-p-oxo, Co-OH, Co-Aq, Co-Co). The Co-O
and Co-Co distances are in agreement with previously published XAS data.
Table Sl: DFT+PCM computations of redox potentials for PCET from each of the
three Co"'(OH) groups of 2, indicating no energetic difference for creating a cofacial oxo group (on the top face) compared to a distal oxo group (on the bottom
face). The DFT+PCM calculations are intended for comparing cofacial and distal
deprotonations only, not for quantitative predictions of E0 .
Site (OH group)
Relative Energy (kcal/mol)
Cofacial
+0.00
Distal 1
Distal 2
-0.84
-0.81
151
HOMO
LUMO
-4.9 eV
-5.2 eV
-6.1 eV
-6.4 eV
-6.5 eV
-9.1 eV
HOMO
...
......
......
..
4..............................
-9.7 eV
-6.6 eV :
.- *..........
.
...
...........
-9.0 eV
........
.
.
-12.0 eV
Figure S2: Orbital diagram for twice-oxidized model catalyst in open position (3, on
left) and bonded position (4, on right). Only orbitals with O.3 occupancy on the oxo
groups are shown. The formation of a a bond is evident, as well as occupation of all
7r and lr* orbitals. Some orbitals (antibonding partners of -r on bottom left and o on
bottom right) are not shown, because they are high in the virtual-orbital manifold
and have very diffuse character.
152
Figure S3: MM dynamics trajectory of model catalyst 1 (freely moving) with superimposed positions of solute hydrogens (black dots) and nearby explicit solvent water
oxygens (red spheres). A high solute degree of freedom is evident. The hydrogenbonding effects are not as clearly visible as in Figure 1, due to free rotation of the
hydrogens.
153
Figure S4: Spin density plot of twice-oxidized model catalyst in open position (3).
Note the similarity between the spin density and the HOMO in S2, left column; this
indicates that the electronic ground state is a broken-symmetry singlet with metal d,
character. [24] There is no corresponding spin-density plot for the ground state of 4,
because it is a closed-shell singlet.
154
Table S2: XYZ coordinates for model catalyst 1, also shown in S1.
-1796.176824084 a.u.)
-0.957454
-0.989457
0.989450
0.957460
0.869280
0.892437
-0.892428
-0.869288
-2.969898
-3.153201
-3.194514
-0.942984
-0.944920
-1.861419
-1.055314
-0.178191
-3.064934
-3.318466
-2.724615
-0.997427
-0.371844
-0.537432
-1.324514
-1.096599
3.064912
3.318441
2.724603
0.997415
0.371820
0.537426
1.324497
1.096599
2.969906
3.153193
3.194491
0.942980
0.944908
1.861405
1.055317
0.178201
-0.989459
0.957455
-0.957465
0.989438
0.892414
-0.869301
0.869273
-0.892402
-0.997441
-0.537422
-0.371863
-3.064931
-2.724642
-3.318431
-1.324496
-1.096639
0.942889
1.861329
0.944870
2.969898
3.194511
3.153179
1.055252
0.178147
-0.942981
-1.861412
0.968850
-0.968875
-0.968880
0.968856
-0.923179
0.923164
0.923159
-0.923191
0.937312
1.776018
0.207158
1.118317
2.102971
0.920896
2.830373
3.181206
-1.118367
-0.921047
-2.103005
-0.937340
-0.207186
-1.776066
-2.830397
-3.181241
-1.118374
-1.055294
-0.920968
-2.103033
-0.937359
-0.207213
-1.776085
-2.830405
-0.178182
-3.181237
0.997421
0.537426
0.371814
0.937304
3.064894
2.724627
3.318413
1.324462
1.118327
2.102991
-0.944899
-2.969917
-3.194513
-3.153210
1.096595
155
1.776028
0.207165
0.920877
2.830376
3.181218
(E
=
Table S3: XYZ coordinates for 2. (E = -1795.499445861 a.u.)
-0.950839
-0.946004
0.995937
0.955038
0.894078
0.899422
-0.894665
-0.863500
-2.968590
-3.135453
-3.192035
-0.931960
-0.924874
-1.849885
-1.042559
-0.159947
-3.036130
-3.315357
-2.742840
-0.912930
-0.381683
-0.320931
-1.096923
3.044253
3.375406
2.654455
1.002461
0.373386
0.545290
1.242125
0.749305
2.968914
3.148930
3.188753
0.897405
0.885975
1.806575
1.011106
0.128330
-0.982050
0.945915
-0.976590
0.988254
0.877022
-0.868595
0.876436
-0.901700
-0.978499
-0.484960
-0.378564
-3.051628
-2.696659
-3.322336
-1.288704
-1.063696
0.922651
1.847584
0.846449
2.975145
3.160074
3.185282
0.944789
-0.876964
-1.786785
-0.823620
-2.980351
-3.211577
-3.130131
-1.081166
-0.319790
0.985514
0.473010
0.400693
3.054611
2.653200
3.364203
1.333086
1.088602
156
0.985806
-1.009899
-0.957786
0.980098
-0.905057
0.943178
0.909936
-0.899230
0.963921
1.787152
0.216668
1.167145
2.147113
0.990860
2.854418
3.192994
-1.199256
-1.080573
-2.161564
-0.951481
-0.135213
-1.695143
-2.800656
-1.201915
-1.101060
-2.169688
-0.927698
-0.203880
-1.776243
-2.816443
-3.174625
0.961659
1.769963
0.197421
1.173667
2.161956
1.017948
2.852896
3.178432
Table S4: XYZ coordinates for 3. (E
0.041234
0.013062
-1.375656
=
1.344526
-1.379106
-0.038875
-0.008905
0.001471
-1.273509
-0.019196
1.272848
1.273602
-1.228284
0.124605
0.073153
-0.035648
0.004660
1.335406
-1.651483
-0.884670
1.486003
0.687066
1.561665
1.118558
1.690316
-1.471006
-1.535195
-2.236962
1.531961
1.654941
2.266992
-1.637645
-1.459079
-2.297629
-2.786899
-2.661387
-2.241969
-2.584277
-2.976916
-1.767150
2.526311
2.873311
1.740161
2.843655
2.607854
2.548210
1.551361
-1.588826
-0.076720
-0.544923
0.178839
0.660390
2.774049
3.640699
2.510503
2.735309
2.483545
2.313468
-2.745423
-2.583461
-2.268097
-2.619562
-2.601373
-1.903982
-1.471864
-1.237663
-2.272716
1.663423
1.447377
2.158940
-1.637655
-1.377289
-2.186998
1.311980
1.452870
2.078634
157
-1794.7933790035
-1.046420
-1.044356
0.959436
1.023121
0.905287
0.879824
-0.882623
-0.948250
-2.817960
-2.816646
2.834341
3.203607
2.890332
3.158830
-1.109256
-1.134256
-2.054448
-1.268389
-2.218183
-0.830400
-1.119817
-2.081969
-0.685818
-1.273628
-2.245611
-1.166318
1.137698
2.114729
1.013993
1.141712
2.010736
1.379245
1.311319
2.187770
1.523246
1.105629
2.058466
0.577086
a.u.)
Table S5: XYZ coordinates for 4. (E = -1794.8369508192
-0.004495
0.015630
-1.337293
1.327093
0.008969
-0.017134
1.322930
-1.314573
-0.004284
0.033282
-1.506856
-0.729056
1.477667
0.692875
1.568818
1.506244
2.192914
-1.514795
-1.543365
-2.253882
1.526824
1.576985
2.263051
-1.554416
-1.482379
-2.177255
-2.721353
-2.546656
-2.174661
-2.567942
-2.817038
-1.766849
2.553558
2.799890
1.748370
2.707773
2.517643
2.162841
1.263545
-1.253273
-0.035797
0.025489
-1.322980
1.313925
0.011540
-0.003837
0.741420
-0.715149
-0.145706
-0.642946
0.123441
0.614373
2.475379
2.435710
1.743781
2.490892
2.017061
2.095851
-2.479303
-1.989246
-2.107999
-2.461765
-2.430934
-1.725971
-1.507946
-1.295780
-2.297342
1.576920
1.405578
2.144184
-1.591283
-1.426461
-2.154006
1.497655
1.277805
2.287630
158
-1.078059
-1.090605
1.092428
1.105269
0.913930
0.926173
-0.760981
-0.773891
-2.875514
-2.882163
2.985663
3.287066
3.000768
3.294201
-1.501934
-2.477034
-1.295843
-1.649992
-2.515584
-1.147832
-1.657400
-2.512659
-1.133542
-1.535128
-2.509615
-1.339746
1.311301
2.285679
1.137928
1.328669
2.259379
1.410225
1.340156
2.272671
1.415233
1.350152
2.320500
1.173980
a.u.)
Table S6: Initial XYZ coordinates for water addition to the once-oxidized equivalent
of 4. (E = -1870.6779521845 a.u.)
-0.040391
0.024008
-1.370048
1.369614
-1.213850
0.063658
0.216091
1.327123
0.031660
-0.072049
1.317351
-1.134390
1.457467
0.160757
-1.288992
0.074476
-0.000686
0.036510
-1.453585
-0.973074
1.413216
0.491204
1.533092
1.573995
2.158605
-1.580107
-1.988417
-2.212586
1.578388
1.242440
2.270345
-1.399307
-2.049262
-2.677424
-2.562038
-2.230421
-2.638239
-3.118103
-1.860009
2.677982
2.470707
2.196677
2.613984
2.400675
2.039504
0.169473
0.065208
-0.595035
0.716046
-0.603917
0.063550
-0.728237
0.301442
0.494637
2.664978
2.993806
1.904893
2.664054
2.311402
2.422740
-2.480579
-3.044332
-1.884292
-2.500320
-2.007056
-1.432334
-1.826324
-2.047441
1.690460
1.398590
2.180629
-1.235350
-1.215631
-2.006380
1.759395
1.566046
2.515310
-3.563891
-2.957851
-3.341132
159
-1.001245
-0.903982
1.108728
1.153527
0.986832
0.909477
-0.742605
-0.781560
-2.892056
-2.874928
2.978965
3.269707
3.029632
3.296791
-1.266202
-2.180369
-1.223224
-1.330036
-2.141088
-0.621056
-0.994960
-1.778657
-1.334628
-1.023028
-1.555236
1.054324
1.935972
0.380654
1.314355
2.113658
1.661605
1.444129
2.408667
1.080169
1.430417
2.396372
1.206084
-2.763055
-3.515006
-2.143055
Table S7: Transition state XYZ coordinates for water addition to the once-oxidized
equivalent of 4. (E = -1870.6671448537 a.u.)
0.007213
0.001758
-1.394406
1.347633
0.008914
-0.042088
1.304533
-1.241946
0.001939
-0.021046
-1.646566
-0.778298
1.492447
0.775113
1.614804
1.518781
2.238533
-1.443082
-2.032567
-1.922867
1.561859
1.329564
2.263881
-1.418269
-2.104117
-2.711898
-2.541544
-2.250098
-2.856414
-3.681209
-2.625846
2.685304
2.642006
2.136249
2.729697
2.605111
2.211996
0.172183
0.035793
-0.632124
1.278632
-1.415816
0.080977
0.183672
-1.116109
1.379714
-0.040166
-0.061329
1.228936
0.077634
0.184651
0.040853
0.476248
1.079508
2.543235
2.878112
1.787467
2.687565
2.334715
2.565885
-2.729206
-3.075533
-2.071618
-2.673305
-2.142092
-1.413638
-1.589582
-2.125064
1.508032
0.993836
1.505284
-1.296317
-1.138901
-2.091444
1.659450
1.591240
2.440240
-2.767196
-1.861858
-2.989600
160
-1.037219
-0.711434
1.078189
1.133037
1.096158
0.876802
-0.774929
-0.829310
-2.933238
-3.546542
2.927032
3.335852
3.002241
3.254823
-1.279456
-2.189715
-1.348510
-1.160990
-1.850413
-0.314263
-0.723686
-1.625180
-0.896902
-0.785040
-1.231621
1.187840
2.135623
0.638233
1.184831
1.114532
2.157995
1.555795
2.523013
1.406943
1.179869
2.170435
0.905189
-2.961231
-3.311843
-2.429830
Table S8: Final XYZ coordinates for water addition to the once-oxidized equivalent of 4; this is similar to 5, but with one additional electron and proton.
(E = -1870.6927839533 a.u.)
-0.054737
0.002245
-1.350764
1.379351
0.052600
-0.029280
1.246152
-1.267900
-0.138146
0.255528
-1.393712
-1.655373
1.495934
0.564650
1.541977
1.429746
2.178689
-1.516918
-2.108928
-1.998417
1.299636
0.484286
1.620333
-1.282050
-1.809960
-2.647897
-2.501409
-2.170375
-2.851212
1.211024
-1.515570
0.078693
0.183922
-1.112136
1.355282
-0.101188
-0.169235
1.315987
0.351736
0.548851
-0.200546
0.450407
0.608689
2.487719
2.802351
1.741629
2.629364
-1.095319
-0.681929
1.087941
1.057034
1.116179
0.764563
-0.878571
-0.813855
-3.028404
-3.830007
2.930782
3.487947
2.901801
3.170451
-1.350208
-2.265890
-1.405992
-1.164111
2.323866
2.473211
-3.078544
-3.620617
-3.314587
-2.933619
-1.872291
-0.321994
-0.355368
-0.199400
-1.243924
-3.114753
-1.106999
-1.440818
-1.839813
1.232780
-2.113034
1.488434
-3.665189
0.957774
-2.571693
2.781387
2.541694
2.368308
2.674708
2.488754
2.112004
0.070745
0.294166
-0.831591
1.467872
-1.199568
-1.084092
-2.033724
1.741906
1.673200
2.478429
-1.932544
-1.028943
-2.107071
161
-0.314000
2.106073
0.581065
1.158754
1.097823
2.136752
1.482126
2.434375
1.185271
1.136779
2.120427
0.830492
-2.640357
-3.070506
-2.955328
Table S9: Initial XYZ coordinates for water addition to 5. (E = -1946.477303537
a.u.)
0.176469
0.037273
-1.298467
1.433490
-0.013686
0.158968
1.356235
-1.128072
0.185651
0.921785
-1.475907
-1.859877
1.522584
0.588604
1.536912
0.718238
2.037787
-0.960554
-1.575643
1.482290
1.213435
2.275983
-1.434551
-1.858992
-2.824163
-2.568049
-3.658654
-2.038230
-2.547478
-1.944044
-3.205893
2.599252
2.430803
1.958320
2.897908
2.669252
2.512105
0.004815
0.368106
-3.554255
-4.113439
-2.665333
1.165284
-1.597741
0.070435
-0.129925
-1.276000
1.194299
-0.277815
-0.131646
1.101992
1.806020
0.077760
0.945934
0.064085
0.047038
2.713013
3.288921
2.813970
2.719048
2.693490
-3.019268
-3.065311
-2.449209
-2.953796
-2.886263
-1.269498
-0.989544
-0.812038
-2.576166
1.643011
2.251182
1.372145
-1.712764
-1.392509
-2.442472
1.250523
1.392388
1.999425
-2.109207
-1.393134
0.676740
0.255929
0.228247
162
-1.279800
-1.103771
0.718239
0.803636
0.764243
0.534673
-1.158627
-1.203175
-3.710930
-4.378014
2.593183
2.803142
2.652922
2.949853
-1.280604
-1.342565
-2.107349
-1.338534
-2.089403
-1.136743
-2.105970
-1.133143
-0.895082
-1.768716
1.035303
1.991382
0.833039
-0.219593
0.498360
-0.047020
-0.182343
1.278237
2.213522
1.163028
0.894087
1.844549
0.397240
-2.920345
-3.461593
-1.811798
-2.479903
-1.828749
Table S1O: Transition state XYZ coordinates for water addition to 5.
-1946.470836279 a.u.)
-0.056268
-0.114363
1.164991
-1.593282
-1.346957
1.389769
-0.039536
0.072328
0.010528
-0.133661
-1.308126
1.165329
1.165050
-0.247335
-1.324082
-0.164981
1.140660
0.473411
1.066338
-1.333498
-1.710955
1.607326
0.689723
1.253773
0.417819
1.657874
-1.257802
-1.816469
1.347095
1.004470
2.134039
-1.543330
-2.091656
-2.765818
-2.392914
-3.676543
-2.056758
2.200410
-0.019179
0.832230
0.022377
-0.024935
2.747632
3.299276
2.842638
2.649712
2.519838
-2.986008
-3.003001
-2.407654
-2.979944
-2.868823
-1.422254
-1.121867
-1.085931
-2.653194
-2.606066
1.599309
-2.113042
2.166830
-3.434336
2.609605
2.507587
1.355028
-1.701954
1.967546
2.822447
2.670249
2.380427
-0.229683
0.205605
-2.551892
-2.351875
-2.267661
-1.412368
-2.433894
1.280641
1.428505
2.014903
-2.028919
-1.317610
0.953674
0.435958
0.371021
163
-1.146839
-1.023662
0.924809
0.799730
0.855217
0.656229
-1.154353
-0.989811
-4.424805
-4.418275
2.817581
3.095001
2.638051
2.982613
-1.265123
-1.257374
-2.144873
-1.165126
-1.956829
-1.184874
-2.134235
-1.220022
-0.739865
-1.535841
1.411031
2.328543
1.369925
0.030739
0.800745
0.086888
0.354646
1.146759
2.101834
1.053206
0.780458
1.743479
0.307775
-2.857918
-3.353419
-3.093564
-3.887512
-2.346068
(E =
Table S1I: Final XYZ coordinates for water addition to 5; this is identical to 6+ 02(E = -1946.498637907 a.u.)
-0.424277
-0.408698
-1.421474
1.267844
-0.108903
-0.088727
0.865797
-1.635293
1.696768
2.013805
-1.185313
-1.539047
1.689725
0.822457
0.878153
0.087495
1.107078
-1.645564
-2.328049
1.152591
1.021715
1.802407
-1.807457
-2.489377
-2.729755
-2.229120
-3.649150
-2.145838
-2.714585
-2.295930
-3.563035
2.613643
2.557007
2.009311
2.651926
2.580406
2.144985
-0.846596
-0.084618
-0.768782
1.169340
-1.545405
-0.043552
-0.098383
-1.347581
-1.185210
-1.047114
1.069682
0.651275
0.806967
1.163849
0.634416
-0.222948
-0.178943
1.323335
2.192137
-0.122354
-1.237266
-0.809230
0.713951
0.046824
-0.046019
2.734694
3.320806
2.731179
2.648739
2.643186
-2.856852
-3.410885
-2.160097
-2.954067
-2.706363
-1.540337
-1.196733
-1.242778
-2.682521
1.508953
2.088415
1.229967
-1.609493
-1.278582
-2.376149
1.352502
1.471758
2.078190
-1.583869
-1.875251
-0.800770
0.924940
-0.150198
-1.691778
1.173171
164
-5.116477
-4.332111
2.959661
3.303994
2.473163
2.920403
-1.441166
-1.326451
-2.387984
-0.935191
-1.624739
-1.241505
-2.028187
-1.474466
-0.674173
-1.322289
1.723982
2.580826
1.821293
0.209973
1.133889
0.380936
0.752071
0.891757
1.845440
0.872675
0.515350
1.492920
0.099200
-2.926503
-3.452378
-3.142751
-3.138431
-3.311385
Appendix C
Supporting Information for:
Hybrid Ensembles for Improved
Force Matching
The supporting information contains parameters for the helium dimer and water
cluster examples, as well as RDFs for the water cluster obtained using the SPC/E and
AIMD-fitted MM potentials. A copy of the ForTune source code has been uploaded
to our group website at http://vanvoorhis.mit.edu/Main/Lee-PingWang.
Li
Figure Sl: Evolution of parameters during force matching for MMo (blue), MM 5 o (green),
and MM8 o (red).
165
0.25
AIMD simulation
MMO -
MM
0.2
e
-
SPC0.15
0.1
0.05
0I
1
2
4
3
rHH
5
(A)
Figure S2: Comparison of gHH using MMo, MM 50 , and the SPC/E initial guess.
0.25
0.2
0.15
0.1
0.05
0
1
4
3
2
5
6
rHH (A)
Figure S3: Comparison of gHH using MMo, MM 80 , and MM 1oo; the latter is fitted to an
AIMD trajectory.
Table Si: Force matching for the helium dimer, starting from MM<.
Gen.
MM
0
1
2
3
0.200
0.283
0.298
0.299
QM
0.200
0.206
0.213
0.222
x2
50
MM
QM
50
0.200
0.272
0.296
0.299
297.440
17.977
0.613
0.346
50.448
50.298
49.975
49.230
237.750
30.823
1.267
0.385
166
Table S2: Force matching for the helium dimer, starting from MM>.
Gen.
MM
0
1
2
3
0.360
0.344
0.334
0.326
oQM
0.360
0.304
0.299
0.299
50
MM
0.360
0.304
0.299
0.299
15.476
3.815
1.498
0.699
167
x2
QM
4106.800
2.841
0.410
0.409
50
3795.100
2.873
0.386
0.385
168
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