Theoretical basis of energy and electron transfer

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Theoretical basis of energy and electron transfer
EPA summerschool, Egmond, June 2003
Gert van der Zwan (zwan_at_few.vu.nl) Department
of Analytical Chemistry and Applied
Spectroscopy Vrije Universiteit, Amsterdam
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Contents.
Day 1 Polarization properties of solvents

• Statics and dynamics of dielectrics
• Onsager, Lorentz, Debye, Lippert-Mataga
• Dynamic Stokes shift

Day 2 Reaction rates and Marcus theory

• From transition state theory to solvent
  coordinates
• Non-equilibrium free energy surfaces
• Electron transfer and Marcus theory

Day 3 Molecules in solution and energy transfer

• How molecules feel the world and each other
• Excitonic interaction, coherent transfer
• Dipole interaction and the Förster rate

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Lippert-Mataga with electronic contributions
Electronic part of the reaction field in ground
(ig) or excited (ie) state
Dipolar part of the reaction field in ground
(ig) or excited (ie) state
4
Reaction rates and Marcus theory (1)
Transition state theory
5
Reaction rates and Marcus theory (2)
Kramers theory
H.A. Kramers (1894-1952)
The Kramers rate is always lower than the
rate calculated with TST. The reason is that now
it is not sufficient for the particle to reach
the top of the barrier, it has to have sufficient
velocity to go over it, or it will be sent back
to the reactant well.
Reaction coordinate
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Intermezzo Brownian motion (1)
Random walks and diffusion
R. Brown (1773-1858)
½
½
n
n1
n-1
n-2
n2
t1
t2
t4
t0
An example of a Random Walk
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Intermezzo Brownian motion (2)
Langevin and related equations
P. Langevin (1872-1946)
v
A particle drawn through a fluid experiences drag
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Intermezzo Brownian motion (3)
The Fokker-Planck equation
M. Planck (1858-1947)
Classical mechanics
Dissipative
Quantum Liouville
Classical Liouville
add
Fokker-Planck
dissipation
(?)
add
dissipation
Adding dissipation in a formal way is usually
done by coupling the system to a bath, and then
projecting out the bath modes. The equilibrium
state (density function or operator) can of
course be calculated using (classical or
quantum) statistical mechanics.
(Redfield, Brownian oscillator)
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Reaction rates and Marcus theory (3)
Langevin approach to Kramers
x
The reactive rate k can always be found from
10
Reaction rates and Marcus theory (4)
Beyond Kramers
?
x(t),v(t)
For a high frequency barrier the reaction can be
over before the environment realizes it.
?
Solve for reactive frequency
Frictional model ?(?)
x(t),v(t)
Only for slow barrier transitions the steady
state friction applies.
The solvent is probed at the frequency of the
barrier transition
11
Reaction rates and Marcus theory (5)
An example charge transfer reactions (1)
Solve dielectric boundary value problem i.e. find
A(?)
Calculate dielectric friction
Solution of the BV problem
Solve for the reactive frequency
Get rate
(K0, K1, I0 Besssel functions)
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Reaction rates and Marcus theory (6)
An example charge transfer reactions (2)
Important parameters
Time dependent dielectric friction on the moving
charge. ?e is the longitudinal relaxation time.
The coupling of charge motion to solvent motion
makes it impossible to concentrate just on the
reaction coordinate. The solvent coordinate is
actively involved in the reaction.
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Reaction rates and Marcus theory (6)
An example charge transfer reactions (3)
Transmission coefficient ? in the region of
strong solvent forces. If the solvent were
frozen, the particle would end up in a well,
rather than on top of a barrier. Repeated
oscillations in the well eventually relax it,
and the reaction can go.
Transmission coefficient ? for weak solvent
forces (?lt1). Kramers underestimates the rate
considerably, the reason is the the friction
never reaches it full potential.
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Reaction rates and Marcus theory (7)
An example Cl-CH3Cl CH3ClCl-
? e
The rate of change of charge along the reaction
coordinate determines the solvent force on the
reacting system
Free energy profile along the reaction coordinate
15
Reaction rates and Marcus theory (8)
An example Cl-CH3Cl CH3ClCl-
The main thing to note from this table is that
Kramers is way off, and that the nonadiabatic
(frozen solvent) regime applies very well to this
reaction.
These results, and those on the previous sheets,
are from MD experiments done in Kent Wilsons
group. J. Chem. Phys. 86, (1987), 1356 1377.
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Reaction rates and Marcus theory (9)
Equilibrium and non-equilibrium solvation
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Reaction rates and Marcus theory (10)
Equations of motion
I moment of inertia of reactive dipoles Is
moment of inertia of the solvent dipoles ?, ?
coupling constants ?s friction on the solvent
dipoles
The solvent dipoles can be viewed as a
representation of a polarizable solvent, in the
overdamped limit it becomes equivalent to the
Debye model.
The coupling between reactive and non-reactive
modes means that the reactive coordinate also
involves solvent motion.
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Reaction rates and Marcus theory (10)
Parameters and regimes
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Reaction rates and Marcus theory (11)
Parameters and regimes
20
Reaction rates and Marcus theory (12)
Back to the electrons
R. Marcus (1923-)
?(?)
If the charge is changed instantaneously, the
order in the dielectric does not change, hence
the entropy remains constant, only the internal
energy changes
e
The dipoles create a reaction potential
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Reaction rates and Marcus theory (13)
The solvent perspective
?R(t)
?R
Reaction potential
?R,eq
Debye relaxation
0
time
Time dependence
sudden charge change
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Reaction rates and Marcus theory (14)
Non equilibrium free energy curves
e
Acceptor
Donor
The reaction coordinate can be equated with the
amount of charge transfered. The nuclear
coordinates correspond to polarization in the
solvent.
XD nuclear coordinate equilibrated with donor
charge distribution. XA nuclear coordinate
equilibrated with acceptor charge distribution.
At the crossing point XC it does not matter for
the solvent where the electron is.
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Reaction rates and Marcus theory (14)
Calculation of the crossing point
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Reaction rates and Marcus theory (15)
Electron transfer rates
? G0
In reality other factors also play a role in the
electron transfer problem distance between
donor and acceptor, diffusion can be rate
limiting step. The inverted regime has only been
observed in rigid systems, such as proteins.
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Reaction rates and Marcus theory (16)
Duttons rule
In the photosynthetic reaction center (in this
case of bacteria) a number of electron transfer
reactions take place. By modifying amino acids in
the right places, ? G0 can be changed.
The distance dependence of the rate depends on
the environment. Proteins behave like other
solvents.
Dutton log10kET13-0.6(R-3.6)-3.1(? G0?)2/?
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Reaction rates and Marcus theory (17)
Remaining questions
Obviously there is a distance effect, as shown by
Duttons rule, and the observation that the
inverted regime is hard to observe since the rate
between unfixed donor-acceptor pairs is often
detemined by the diffusion rate.
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Reaction rates and Marcus theory (18)
What have we done?
General (?)
TST solvent only modifies energies
Kramers steady state dynamics
Solvent dynamics
(non)-adiabatic
Electrons (only?)
Polarization cage
Marcus theory
Diffusive
Coupled quantum/classical dynamics
Many others, see lit. list
Effective mass