An overview of the use of the Marcus Theory to calculate the energies of transition states. Contributed by: Elizabeth Greenhalgh, Amanda Bischoff, and Matthew Sigman, University of Utah, 2015

1. Organic Pedagogical Electronic
Network
Marcus Theory
Elizabeth Greenhalgh, Amanda Bischoff, and Matthew Sigman
University of Utah

2. Describing Electron Transfer Reactions
Marcus, R. A. The Nobel Prize in Chemistry 1992 1992, 69-92
Anslyn, E. V.; Dougherty, D.A (2006) Modern Physical Organic Chemistry. University Science Books
Marcus, R. A. The Journal of Chemical Physics 1956, 24(5), 966-978
Transition State Theory (TST) Overview:
• Developed by Henry Eyring in the 1930s
• Describes reaction rates focusing on geometry
of the transition state at the top of the energy
barrier
• Effective for describing bond breakage/bond
formation
The Problem: Reactions that do not involve
bond breakage/formation, i.e. electron transfer
reactions, involve little nuclear movement in the
transition state (rate of electron transfer is faster
than rate of molecular vibrations); thus, TST fails
and this necessitates a different model.
Background:
Morse potentials
Morse potentials describe potential
energy as a function of bond
distance. Marcus theory focuses on
a parabolic approximation of the
boxed portion.
Frank-Condon
principle
In an electronic
transition between two
reactants, the solvent
molecules do not
have time to
rearrange; thus, the
atomic configuration
and total energy of the
system remain the
same; only the
electronic state
changes.
Quantum Tunneling
Some small particles (electrons,
hydrogen atoms, etc.) can tunnel
through energy barriers rather than
going over them. This complicates
free energy calculations.

3. Marcus Theory and Implications
Marcus, R. A. The Nobel Prize in Chemistry 1992 1992, 69-92
Anslyn, E. V.; Dougherty, D.A (2006) Modern Physical Organic Chemistry. University Science Books
Why does Marcus theory work when TST
doesn’t?
TST focuses on orientation of reactants, while Marcus
theory accounts for the many configurations of the solvent
surrounding the inner coordination sphere; Marcus theory
finds the potential curves for different orientations of the
solvent molecules, and the point of intersection is the
point at which electron transfer occurs.
Marcus Theory
Overview of Marcus Theory:
• Developed by Rudolph A. Marcus in the 1950s
• Uses Morse potentials to describe reaction
coordinates
• This theory won the Nobel Prize in chemistry in
1992
Marcus Theory and Exergonicity
After aligning the Morse potentials along the intrinsic barrier, the
product potential is adjusted to compensate for the free energy
change of the reaction. The intersection of the curves defines the
energy of the transition state. A tunneling term is also included.
As the reaction becomes more
exergonic, the energy of the transition
state lowers; but, as it becomes still
more exergonic, the energy of the transition state rises! The region
beyond the zero point is called the “Marcus inverted region.”

4. Marcus Theory to Predict Hydrogen Atom Transfer
Mayer, J. Understanding Hydrogen Atom Transfer: From Bond Strengths to Marcus Theory, Acc. Chem. Res. 2011 44 (1), 36-46.
A H + A A + H A
kAH/A
(3)
The simplest form of the Marcus
equation for electron transfer (eq 1)
predicts the reaction barrier (∆G‡) from
the reaction driving force (∆G°) and
intrinsic barrier (λ) which is the energy
required to reorganize the reactants and
surrounding solvent without electron
transfer.
This equation, with a few assumptions,
can be rearranged to the Marcus cross
relation (eq 2). The kinetic information is
primarily in the rate constants for the
respective hydrogen-atom self
exchange reactions (such as eq 3).
Tests of the Marcus cross relation for HAT: log/log plot of observed versus
calculated HAT rate constant for a number of metal complexes reacting
with various substrates. The diagonal line illustrates kobs = kcalc. The
estimated errors on kcalc are typically ±1 log unit; they are larger for MeCN
reactions of Ru(O)bpy2py2+ because the BDFE is only available in H2O,
and smaller in three cases where KAH/B was measured directly.

5. Hydrogen Atom Transfer Continued
Schematic free energy surface for the reaction shown above through H bonding intermediates.
HAT from FeII(H2bip) to the
stable nitroxyl radical
TEMPO is very unusual in
that it becomes faster at
lower temperatures, with
∆H‡ = -2.7 ± 0.4 kcal mol-1.
The figure below shows the
cross relation quantitatively
predicts the cross rate
constants and the negative
temperature dependence.
The most temperature
dependent parameter is keq,
as the reaction is more
favorable at lower
temperatures causing the
Marcus cross relation to
have a temperature
dependence as well.
Mayer, J. Understanding Hydrogen Atom Transfer: From Bond Strengths to Marcus Theory, Acc. Chem. Res. 2011 44 (1), 36-46.

6. Problems
1. What does Marcus theory describe that transition state theory doesn’t?
a. Quantum tunneling
b. Electron transfer reactions
c. Transition state energies
d. All of the above
2. Which pair of potential curves would you expect to result in the reaction with the fastest rate?

7. Problems
3. Does the box on the adjacent diagram represent the intrinsic barrier,
the transition state, or both? What is the difference between the two?
4. Marcus cross relation relates what values?
a. Resonance
b. Induction
c. Reaction rates
d. All of the above
5. Marcus theory is based on the quantum mechanical principle
called_______.
6. True or False: Marcus cross relation can tell us information about
temperature.

8. This work is licensed under a
Creative Commons Attribution-
ShareAlike 4.0 International
License.
Contributed by:
Elizabeth Greenhalgh, Amanda Bischoff, and Matthew Sigman
University of Utah, 2015