Simple and accurate correlation of experimental redox potentials and DFTcalculated HOMO/LUMO energies of polycyclic aromatic hydrocarbons.
Dalvin D. Méndez-Hernández, Pilarisetty Tarakeshwar, Devens Gust, Thomas A. Moore, Ana L. Moore
and Vladimiro Mujica

Department of Chemistry and Biochemistry, Arizona State University, Tempe, Arizona 85287-1604

E-mail: ddmendez@asu.edu; tpilaris@asu.edu; gust@asu.edu, tmoore@asu.edu, amoore@asu.edu;
vmujica@asu.edu

Abstract: The ability to accurately predict the oxidation and reduction potentials of
molecules is very useful in various fields and applications. Quantum mechanical
calculations can be used to access this information, yet sometimes the usefulness of these
calculations can be limited because of the computational requirements for large systems.
Methodologies that yield strong linear correlations between calculations and
experimental data have been reported, however the balance between accuracy and
computational cost is always a major issue. In this work, linear correlations (with an R2
value of up to 0.9990) between DFT-calculated HOMO/LUMO energies and seventy
redox potentials from a series of fifty-one polycyclic aromatic hydrocarbons (obtained
from the literature) are presented. The results are compared to previously reported linear
correlations that were obtained with a more expensive computational methodology based
on a Born-Haber thermodynamic cycle. It is shown in this article that similar or better
correlations can be obtained with a simple and cheaper calculation.

Keywords: HOMO, LUMO, reduction, oxidation, linear correlation

I. Introduction
Determination of the reduction and oxidation potentials of a given molecule at an
interface is important in many research areas such as dye sensitized solar cells (DSSC),
organic photovoltaics (OPV), and artificial photosynthesis, where tuning the
thermodynamic properties of every component of a system is essential for it to work in

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the desired way [1-6]. The perturbation of the HOMO and LUMO energies (HLE) is also
of interest in the context of designing both novel and efficient molecules for harnessing
solar energy in artificial photosynthesis and photocatalysis [3-6]. Ideally one would like
to know with accuracy the redox properties of a compound of interest before embarking
on an often difficult, expensive and time-consuming synthetic procedure.

The simplest way to rationalize an approximate linear relationship between HLE and
oxidation/reduction potentials is to consider the Marcus Relation for interfacial electron
transfer

G

#

 G

r

0

 

2

(1)

4

where G # is the free energy of activation and  r G o and  are the reaction free energy
and the total reorganization energy, respectively. Solving for  r G 0 in equation (1) gives

 r G 0  4 G #  

Further we introduced the thermodynamic relation between

(2)

 rG o

and the

electrochemical cell potential

 rG 0  nF(E  E 0 )

(3)

The reorganization energy is generally decomposed into two contributions, i and 0 ,
coming from the molecule and from the solvent, respectively

  i  o

(4)

This decomposition leads to the following equation

2

 r G 0  4(i  o )G #  o  i

(5)

Up to this point equation (5) is formally exact. To proceed, one has to invoke that in
many reactions i = o [7] and that the first two terms in (5) are approximately constant
within a family of chemically related molecules, while i is proportional to the energy of
the HOMO or the LUMO depending on whether oxidation or reduction is considered [8].
Combining these two approximations and using equations (3) and (5) leads to the linear
relationship
EOX/RED  a b HOMO/LUMO

(6)

where EOX/RED is the experimental redox potentials,  HOMO/LUMO the calculated HLE, and a
and b are constants.

Maccoll showed for the first time in 1949 that a good correlation could be obtained
between Huckel LUMO energies and experimental reduction potentials [9]. Nevertheless,
in the computational electrochemistry literature [10-12], efforts to predict redox
potentials steer away from this methodology mainly because of the assumptions involved
(e.g. molecular and solvent reorganization energies are not included in the calculation).
Instead, a Born-Haber thermodynamic cycle is preferred as the free energies are directly
calculated and solvation energies are accounted for, which are significant as has been
observed when electrochemistry is done under different solvents and conditions [13].
Unfortunately, the main drawback for this methodology is that when applied to large
molecules (e.g. polycyclic aromatic hydrocarbons (PAHs), porphyrins, phthalocyanines)
the computations are very expensive.

To resolve this issue, Fry et al. have reported a methodology that accurately predicts the
redox potentials of PAHs with a simplified approach to the Born-Haber thermodynamic
cycle [14]. This methodology involves the optimization of molecules in both the ground
state and the reduced or oxidized state (radical anion or radical cation respectively) with a

3

continuum based solvent model in order to account for reorganization energies. This
approach removes the optimization step in the gas phase that is part of the Born-Haber
thermodynamic cycle, thus decreasing the number of calculations, from six to three (in
order to obtain both oxidation and reduction potentials). This methodology was tested
with fifty-one PAHs, whose experimental redox potentials (which were obtained from the
literature) were measured under similar conditions. They were able to obtain a very
strong correlation, up to R2 = 0.9981, between the calculated and the experimental values.

Before Fry, in 2008, Gillmore et al. introduced a further simplified methodology. It
consists of a molecular energy optimization in gas phase, followed by a single point
calculation in the presence of a continuum solvent model [15]. This increases again the
number of calculations to six (in order to obtain both oxidation and reduction potentials),
but the computational time is reduced because four of them (the ones that involve the
solvent) are single point calculations. They found a very good correlation between the
calculated energy difference and the reduction potentials of quinones, cyanoaromatics
and N-metyl heteroaromatic cations [15], and later expanded their methodology to PAHs,
flexible aromatic molecules and heterocyclic amines [16].

In searching for a methodology to calculated redox potentials accurately and that is
computationally less expensive than Gillmore’s approach but at least as accurate as Fry’s,
reported herein is a good correlation between HLE (calculated at the DFT level using the
B3LYP/6-31G(d) functional) and experimental redox potentials. This follows Maccoll’s
methodology and significantly reduces the complexity of the required calculations. The
advantage of this approach is that just one optimization calculation is required, thus it can
be applied to large molecules with a relatively low computational cost.

II. Computational method
To test this methodology and compare it to previously reported correlations, the library of
PAHs collected by Fry et al. [14], which include data from six different references, was
used. The literature redox potentials were measured in acetonitrile and are reported in V
vs. ferrocene/ferrocenium (Fc/Fc+), which was used as an internal reference.

4

All PAHs were optimized using Gaussian 09 [17] at the B3LYP/6-31G(d) level of theory
[18-23] in the gas phase and in the presence of acetonitrile simulated using the
Conductor-like Polarizable Continuum Model (CPCM) [24-25]. The calculated HLE
were plotted against the experimental oxidation and reduction potentials respectively.

III. Results
Plots for the calculated HLE of a series of fifty-one PAHs against their experimental
oxidation and reduction potentials are shown in Figure 1 and 2. When using acetonitrile
as the solvent a very strong correlation (R2=0.9989) is observed between the calculated
and experimental values. It is noteworthy that there is also a very strong correlation
between HLE and redox potentials when the molecules are optimized in the gas phase
(R2=0.9990). The reason why this is true in this specific case is because solvent
reorganization energies are very similar for the set of PAHs under study, resulting in a
cancelation of the error obtained by not including the solvent in the calculation. A
summary of the results obtained from the method used in this work as well as the results
obtained by Fry et al. can be found in Table 1.

The correlations obtained with the method used in this paper are comparable or better
than the ones obtained by Fry. When analyzing the oxidation and reduction potentials
together Fry obtained a mean absolute deviation (MAD) of 0.056 V while we get a MAD
of 0.046 V. When the data is split in two sets (only reduction or oxidation potentials),
Fry’s correlation is better than ours for both reductions and oxidations, but the error is of
the same order of magnitude for both approaches. In general, both methods are in very
good agreement with the experimental values, our method having the advantage that it
consists of only one optimization rather than three.

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Fig. 1 Plot of computed HOMO/LUMO energies (eV) using acetonitrile as the solvent vs.
experimental E1/2 avg. obtained from [14] and references therein.

6

Fig. 2 Plot of computed HOMO/LUMO energies (eV) in the gas phase vs. experimental
E1/2 avg. obtained from [14] and references therein.

slope (eV/V)

y-intercept (eV)

R2

MAD (V)

RMSDa (V)

x-interceptb (V)

All

1.1251

4.4264

0.9989

0.046

0.058

-3.93

Oxidation

1.1288

4.4311

0.9700

0.037

0.047

-3.93

Reduction

1.0444

4.2499

0.9882

0.039

0.053

-4.07

All

1.1222

4.2849

0.9990

0.046

0.056

-3.82

Oxidation

1.1318

4.2813

0.9740

0.035

0.043

-3.78

Reduction

1.0658

4.1616

0.9864

0.044

0.058

-3.90

All

0.8591

4.442

0.9981

0.056

---

-5.17

Oxidation

1.073

4.227

0.9847

0.032

---

-3.94

Reduction

0.947

4.64

0.9924

0.023

---

-4.90

This work
acetonitrile

gas phase

Reference [14]

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Table 1. Summary of the results from the linear correlations. a) Root-mean-square
deviation b) calculated with x-intercept = –b/a.

IV. Conclusions
A very strong linear correlation of DFT-calculated HLE and redox potentials of PAHs is
shown. It is important to emphasize that only one optimization calculation per molecule
is required to obtain both HLEs. In contrast, at least three (and up to six) optimizations
per molecule are required to obtain a theoretical approximation of the potentials by the
other methods described above. This can be a powerful tool to use when one needs to
quickly and accurately predict oxidation and reduction potentials of new compounds
given that one have a series of similar compounds whose redox potential are already
known.

The strong correlation obtained from HLE and redox potentials of PAHs was independent
of whether the solvent model was included in the calculations. This may not be the case
when including in the correlation other groups of molecules that have different solvent
reorganization energies. For future work this methodology will be extended to quinones,
cyanoaromatics, porphyrins, phthalocyanines and perylenes.

V. Acknowledgements
We would like to thank Dr. Jason G. Gillmore for very useful discussions and
suggestions. Thanks as well to John J. Tomlin for his suggestions, and all the members of
the Gust-Moore-Moore research group for inspirational conversations.
DDMH is supported by the National Science Foundation Graduate Research Fellowship
Program (NSF-GRFP) under Grant No. DGE-0802261 and by the More Graduate
Education at Mountain States Alliance (MGE@MSA) Alliance for Graduate Education
and the Professoriate (AGEP) National Science Foundation (NSF) Cooperative
Agreement No. HRD-0450137.
This work was supported as part of the Center for Bio-Inspired Solar Fuel Production, an
Energy Frontier Research Center funded by the U.S. Department of Energy, Office of
Science, Office of Basic Energy Sciences under Award Number DE-SC0001016.

8

Calculations were performed using the ASU Advance Computer Center Saguaro system.

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