THEORETICAL STUDIES OF EXCITED STATE
ELECTRON TRANSFER BETWEEN IRON-PORPHYRIN
AND TRINITROTOLUENE
By
CLINT BRADEN CONNER
Associates of Science
Rogers State University
Claremore, Oklahoma
1998
Bachelor of Science
Northeastern State University
Tahlequah, Oklahoma
2000
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
DOCTOR OF PHILOSOPHY
December, 2006
COPYRIGHT
By
Clint Braden Conner
December, 2006
THEORETICAL STUDIES OF
ELECTRON TRANSFER BETWEEN IRON-PORPHYRIN
AND TRINITROTOLUENE
Thesis Approved:
Dr. Timothy M. Wilson
Thesis Advisor
Dr. Harold J. Harmon
Dr. Gil Summy
Dr. Nick Materer
Dr. A. Gordon Emslie
Dean of the Graduate College
iii
ACKNOWLEDGMENTS
I would first like to thank my advisor Dr. Timothy Wilson for all his help during
my studies and my research. He has given me a better understanding of the principles
governing our research. I have also enjoyed the many conversations outside the realm
of physics. His encouragement has helped greatly through my time in graduate school.
In times I felt like giving up, unknowing to him, he would always encourage me at
just the right time to boost my confidence and keep me sane. I would like to extend
my thanks also to my committee members: Dr. Jim Harmon, Dr. Thomas Collins,
Dr. Gil Summy, and Dr. Nick Materer; all have pointed me in directions that I might
have not considered concerning my research.
I would like to further thank Dr. Jim Harmon for his financial support through
my research assistantship. He has given Dr. Wilson and me avenues to explore in
our research. I thank him for all the laughs and not putting up with the politi-cal
correctness. I thank him also for being blunt and not sugar coating the issues
discussed.
I want to especially thank Dr. Paul Westhaus for his endless support in my
education as one of my professors. His tireless work and great attention to detail in
my class work has greatly benefited my education. He has always been willing to
help in matters that he didn’t have to, thus taking much stress off of me. He always
made sure I had a teaching assistantship in the beginnings of my schooling, which I
was always grateful for. His help will certainly leave a positive lasting impression in
my life.
To my friends I would like to say thank you to Brian Timmons, Lisa Reilly,
Enkhbot Tesdinbaljir, Mammadur Rahaman, Kirk Haines, Marty Monigold, and
iv
Ryan Scott for our endless discussion about physics and matters outside of physics.
These friends of mine have made graduate school a much more enjoyable time when
it would have otherwise been extremely stressful. Thank you all for your support and
help that you have given me every step of the way. On the same note I would like
to thank another friend Tate Pope for his great conversation on religion and science.
He has on many occasions opened my mind and caused me to understand why I hold
the beliefs I do.
To the office staff in the physics department consisting of Cindi Raymond, Susan
Cantrell, and Stephanie Hall I say thank you for your help in all that you do. All of
you are some of the hardest working folks I have had the pleasure of getting to know
and enjoy. Thank you for making sure I got paid, had the right books to check out,
and had all the correct information for travelling to various locations.
Lastly, I thank my mom, dad, step dad, and the rest of my family for their
moral and financial support, as well as the endless encouragement they continuously
gave me through throughout my entire schooling. You all have been their in every
occasion and then some. Thank you for caring and making sure that I finished what
I started.
v
TABLE OF CONTENTS
Chapter Page
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. History . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . 4
2. THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2. Hartree-Fock . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1. Basis Sets . . . . . . . . . . . . . . . . . . . . . 9
2.3. Density Functional Theory . . . . . . . . . . . . . . . . . 12
2.4. Time-Dependent Density Functional Theory . . . . . . . 15
2.5. Configuration Interaction . . . . . . . . . . . . . . . . . . 18
2.6. Polarizable-Continuum Model . . . . . . . . . . . . . . . 19
3. CALCULATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2. Unsolvated FeP and FeTPP Complexes . . . . . . . . . . 22
3.2.1. Geometry Optimization FeP . . . . . . . . . . . 22
3.2.2. Geometry Optimization FeTPP . . . . . . . . . 26
3.3. Unsolvated Excited States of FeP and FeTPP . . . . . . 29
3.3.1. Unsolvated FeP Excited States . . . . . . . . . . 29
3.3.2. Unsolvated FeTPP Excited States . . . . . . . . 38
3.3.3. Charge and Spin Densities of Unsolvated
FeP and FeTPP . . . . . . . . . . . . . . . . . . . . 42
3.3.4. Comparison of Unsolvated FeP and FeTPP . . . 43
3.4. Solvated FeP Complexes . . . . . . . . . . . . . . . . . . 44
3.4.1. FeP Excited States . . . . . . . . . . . . . . . . 44
3.4.2. Solvated FeP Spin and Charge Densities . . . . 47
vi
Chapter Page
4. Excited State Electron Transfer . . . . . . . . . . . . . . . . . . . . . 49
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2. Thermodynamic Factors of Photoinduced Elec-tron
Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.1. Quenchers and Sensitizers . . . . . . . . . . . . 52
4.2.2. Enthalpy and Gibbs Free Energy . . . . . . . . 53
4.2.3. Redox Potentials . . . . . . . . . . . . . . . . . 55
5. Excited State Electron Transfer Results . . . . . . . . . . . . . . . . . 60
5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.2. Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.3. Thermodynamic Calculations . . . . . . . . . . . . . . . 64
6. Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 71
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
vii
LIST OF TABLES
Table Page
3.1. Table giving the optimized total energies of the FeP and FeTPP
multiplet system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2. Table showing the optimized bond lengths of the FeP multiplet system. 24
3.3. Table giving the optimized parameters of the FeP multiplet system. . 25
3.4. Table showing the optimized bond lengths of the FeTPP triplet
and quintet system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.5. Table giving the optimized parameters of the FeTPP triplet
and quintet system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.6. Table giving the TDDFT excitation data for the S=1 system of FeP. . 30
3.7. Table giving the CIS excitation data for the S=1 system of FeP. . . . 31
3.8. Table giving the TDDFT excitation data for the S=2 system of FeP. . 32
3.9. Table giving the CIS excitation data for the S=2 system of FeP. . . . 33
3.10. Table giving the TDDFT excitation data for the S=0 system of FeP. . 34
3.11. Table giving the TDDFT excitation data for the S=1 system
of FeP and the experimental results of FeTPP in benzene. . . . . . . 37
3.12. Table giving the TDDFT excitation data for the S=1 system of FeTPP. 39
3.13. Table giving the TDDFT excitation data for the S=2 system of FeTPP. 40
3.14. Table giving the TDDFT excitation data for the S=1 system
of FeTPP and the experimental results of FeTPP in benzene. . . . . 41
3.15. Table giving the Mulliken charge densities and spin densities
the triplet FeTPP and FeP. . . . . . . . . . . . . . . . . . . . . . . . 42
viii
Table Page
3.16. Table giving the TDDFT excitation data for the solvated S=1
system of FeP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.17. Table giving the TDDFT excitation data for the solvated S=2
system of FeP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.18. Table giving the Mulliken charge densities and spin densities
the triplet and quintet FeP. . . . . . . . . . . . . . . . . . . . . . . . 48
5.1. Table giving the occupied orbitals of the FeP and FeP+1 complexes. . 64
5.2. Table giving the Gibbs free energies for each molecule. . . . . . . . . . 69
5.3. Table giving the change in free energies for each set of reactions. . . . 69
ix
LIST OF FIGURES
Figure Page
1.1. Model of FeP molecule. This model is used as a reference dia-gram
for the optimized parameters. . . . . . . . . . . . . . . . . . . . 2
1.2. Model of FeTPP molecule. This model is used as a reference
diagram for the optimized parameters. . . . . . . . . . . . . . . . . . 3
2.1. Pictographs of the functions used to describe the orbitals. . . . . . . . 11
3.1. Absorbance spectrum of 6.7 μM FeTPPS in pH 7 buffer. Inset
are the Q-bands of FeTPPS. . . . . . . . . . . . . . . . . . . . . . . . 35
4.1. Excitation of the electron to the Frank-Condon state which
then relaxes to the equilibrated state. . . . . . . . . . . . . . . . . . . 51
4.2. Enthalpy changes of the donor complex. . . . . . . . . . . . . . . . . . 54
4.3. Energy diagram for photoinduced electron transfer. . . . . . . . . . . . 57
5.1. Model of solvated FeP triplet charge density map. This model
is used as a reference diagram for the atomic charges. . . . . . . . . . 62
5.2. Model of solvated FeP S=3/2 charge density map. This model
is used as a reference diagram for the atomic charges. . . . . . . . . . 63
5.3. Model of solvated TNT S=0 charge density map. This model
is used as a reference diagram for the atomic charges. . . . . . . . . . 65
5.4. Model of solvated ionic TNT S=1/2 charge density map. This
model is used as a reference diagram for the atomic charges. . . . . . 66
x
CHAPTER 1
INTRODUCTION
1.1 History
The study of porphyrins and porphyrin complexes is extensive due to their
biological importance, as well as their use in industrial applications and devices.
Some of these include their use as optical limiters, catalysts, sensors, actuators, and
molecular sieves.1 Two particular types of porphyrin of interest, shown in Figures 1.1
and 1.2, are those of the iron(II)-porphyrin (FeP) and iron(II)-tetraphenylporphyrin
(FeTPP), which have been studied extensively.
Much research has been performed on both FeP and FeTPP by many different
research groups. In these studies the actual ground state of the FeP system was
questioned in earlier work, centering on whether the ground state of the system was
an S=0, 1, or 2 for the d6 iron(II) complex.2–4 Numerous theoretical studies have
addressed this ambiguity and most have concluded computationally that the S=1
spin state is energetically favorable over the other spins for the ground state.5–7
Experimentally, there is still some question as to the exact ground state of the d6 FeP.
Like the computational studies for FeP, the experimental studies seek to determine
the ground state of the system by using various methods including: observing their
magnetic moments of iron(II)-tetraphenyl-porphyrin (FeTPP); measuring Fe-N bond
distances in FeTPP 8; and M¨ossbauer 8,9 and NMR spectroscopy of FeTPP 10 to name
a few. In each case the ground state was suggested by the data to be of intermediate
spin, S=1.
1
2
Figure 1.1. Model of FeP molecule. This model is used as a reference diagram for
the optimized parameters.
3
Figure 1.2. Model of FeTPP molecule. This model is used as a reference diagram for
the optimized parameters.
4
Our motivation for studying FeP and FeTPP is to look at iron-tetraphenyl-porphyrin-
sulfonate (FeTPPS) as a photocatalyst in degrading TNT and to deter-mine
a possible pathway by which this process occurs within the system of interest.
This requires detailed knowledge of the excited states of the system. In light of
the aforementioned arguments regarding the actual spin of the ground state and the
promising nature of using FeTPPS as a photocatalyst 11, we have studied the S=1
and S=2 iron(II) complexes. Since FeTPPS is a much larger system compared to
FeP and only slightly larger than FeTPP, in terms of the number of electrons to
account for, we studied the smaller system to interpret the experimental absorption
spectrum of FeTPPS. The question to be answered is if we can use the FeP results
to understand other properties of FeTPP and FeTPPS. In order to determine this,
however, we needed to know the excited state spectra for both spin states of both
iron(II) complexes, i.e. FeP and FeTPP.
The next step was to determine how the photocatalyst (FeP) mixed with 2,4,6-
trinitrotoluene (TNT) in an aqueous solutions interacts with the system as is discussed
by Harmon.11 It was believed that perhaps electron transfer was the preferred mech-anism
by which the demethylization of TNT occurred. The final set of products was
experimentally determined after this demethylization process by FeTPPS. While the
final products are known, the pathway is not yet understood. Therefore, the question
remains by what pathway does the process take? While this question may not be one
that can be answered in the scope of this work, we can determine whether or not the
electron transfer mechanism is a viable process for the system in question.
1.2 Outline of Thesis
The research herein is organized into three main groups. The first group dis-cusses
the theoretical aspects of the models employed in this research. The second
5
group is a discussion of the calculations and gives a comparison of these calcula-tions
to observations made experimentally. The third group gives a discussion on the
degradation pathway by examining the thermodynamics of the system.
As was mentioned, Chapter 2 introduces the theoretical framework in which
the research is built upon. Here a brief introduction to Hartree-Fock Theory (HFT),
density functional theory (DFT), and time-dependent density functional theory
(TDDFT), and the solvation method will be given. More importantly, a detailed
discussion of the density functional used through the calculations will be discussed.
In Chapter 3, the resulting calculations on the system will be given. The system
of FeTPP and FeP along with the case of the solvated system of FeP will be included.
These calculations of the different multiplets of the systems will be discussed and
compared to available experimental data given by other research groups as well as
our own.
Chapter 4 and 5 will include the thermodynamic considerations of electron
transfer within our system. Results from the calculations will be given and discussed
to show the possibility of electron transfer within the molecular systems used in this
research. Chapter 6 will then tie everything together as a summary will be given.
CHAPTER 2
THEORY
2.1 Introduction
Different levels of theory can be used for a variety of different applications.
For research presented here, HF and DFT were employed to find the lowest energy
state termed the ground state. For vertical transition states or electronic excitations,
configuration interaction of singles (CIS) and TDDFT were used. As the solvent was
included in the system, a method known as the polarizable-continuum model (PCM)
was used to model the solvent affects of the molecular system immersed in an aqueous
solution.
All the methods above were employed to obtain a comparison of the different
levels of theory with each other, i.e. comparing HF to DFT and CIS to TDDFT.
In addition to comparing with each other, each set of calculations will be compared
to the experimental results that are available. A brief discussion will be given to
HF, DFT, and PCM, with a more lengthy explanation on TDDFT and the density
functional used in this work involving DFT type calculations.
2.2 Hartree-Fock
The HF method is a model which treats each electron as though it lies in an
averaged potential field due to the other N −1 electrons. Therefore, the disadvantage
of HF is its inability to properly account for electron correlation within the model. In
addition, this method calculates the ground state energy as a function of the orbitals
6
7
in which the wavefunction is expanded i.e. the basis used. This will be shown to be
different from other methods included in this study, mainly the DFT methods.
The model begins with the Hamiltonian consisting of electrons and nuclei of the
molecular system as given by Szabo and Ostlund12 which is
H = −
N
Xi=1
1
2
∇2
i −
M
XA
=1
1
2MA
∇2
A −
N
Xi=1
M
XA
=1
ZA
riA
+
N
Xi=1
N
Xi<j
1
rij
+
M
XA
=1
M
XB=1
ZAZB
RAB
. (2.1)
Here N and M are the number of electrons and nuclei respectively. MA is the mass
ratio of nucleus A to the electron, and ZA represents the atomic number of nucleus
A.
Since we are interested in electronic excitations in the scope of this work, we
will utilize the electronic Hamiltonian. For this, we utilize the Born-Oppenheimer
approximation. This approximation treats the nuclei as stationary relative to the
electrons since the nuclei have a greater mass than the electrons. Therefore, the
electrons lie within the field of nuclei and the kinetic energy of the nuclei can be
neglected. This also leaves the last term in the above equation to be a constant.
Taking this approximation into consideration and also the fact that we are interested
in electronic energies, the full Hamiltonian can be reduced to give the electronic
Hamiltonian Helec as
Helec = −
N
Xi=1
1
2
∇2
i −
N
Xi=1
M
XA
=1
ZA
riA
+
N
Xi=1
N
Xi<j
1
rij
. (2.2)
For simplicity, Helec will now be written as H since the energies we are focusing on
will be the electronic energies and not the nuclear energies.
Once this electronic Hamiltonian H is written out, one can employ the use of
the HF equations, a set of nonlinear integro-differential equations that must be solved
numerically in an iterative fashion. The notation used is that of Szabo and Ostlund.12
These equations take the form
8
h(1)χa(1) + Xb
6=a
[Z dx2|χb(2)|2r−1
12 ]χa(1)
− Xb
6=a
[Z dx2χ∗
b(2)χa(2)r−1
12 ]χb(b) = εaχa(1) (2.3)
with h(1) written as
h(1) = −
1
2
∇2
1 −XA
ZA
r12
(2.4)
These equations essentially give the best spin orbitals χ that minimize the
electronic energy E0 given as
E0 = h 0|H| 0i =Xa
hχa|h|χai +
1
2Xa
b
hχaχbkχaχbi (2.5)
where
hχaχb||χaχbi = hχaχb|r−1
12 |χaχbi − hχaχb|r−1
12 |χbχai. (2.6)
The total wave function 0 is that formed by the optimized spin orbitals from
the HF equations that minimize the energy E0. The wave function must be antisym-metic
with respect to interchanging two electronic coordinates involving both space
and spin. The wave function necessary that includes this property can be obtained
from using the a Slater determinant. Therefore the wave function can be written as
(x1, x2, ..., xN) given by
(x1, x2, . . . , xN) = (N!)−1/2
χi(x1) χj(x1) . . . χk(x1)
χi(x2) χj(x2) . . . χk(x2)
...
...
...
χi(xN) χj(xN) . . . χk(xN)
(2.7)
9
which leads to stating that HF theory (HFT) is a single determinant theory. As can
be seen HFT is based on finding the optimum set of orbitals to construct a wave
function based on these orbitals i.e. the Slater determinant.
2.2.1 Basis Sets
When performing calculations on molecules one must use the expansion of the
MO to mathematically construct the wavefunction of the system. Since the wave-function
is an antisymmeterized product of MOs using a Slater determinant (Eq. 2.7),
then it becomes quite a task to computationally arrive at useable data. Therefore,
calculations must be performed on computers, and the method used computationally
is the basis set approximation. This is the case where the MOs are expanded in terms
of a chosen basis, as mentioned earlier in terms of the AOs. Many studies have been
done on basis sets such as those by Dunning and Huzinaga, Pople, and many more.13
However, the former two have gained popularity over other basis sets developed due
to extensive calculations done with both basis sets. There is an extensive amount of
work in which to compare the level of accuracy within a desired set of calculations.
The basis set used in these proceeding calculations consists of Gaussian Type
Orbitals (GTO) since gaussian functions can be integrated analytically, thus allowing
for less computational time to be used. There are drawbacks to using GTOs compared
to using Slater Type Orbitals (STO). In the limit as the electron-nuclear distance
approaches zero there should be a cusp because of the 1/r dependence of the coulombic
attraction between the nucleus and the electron. Another drawback is the rapid decay
of the GTOs at large distances. It is possible to remedy these behavioral problems by
using STOs as the basis functions. There is, however, a problem with using STOs as
the basis functions. This is due to the large number of multicentered integrals, which
take the form
hφAμ
φCν
|φBλ
φDσ
i = Z φ∗A
μ (r1)φ∗C
ν (r2)
1
r12
φBλ
(r1)φDσ
(r2). (2.8)
10
This is a multicentered integral, where φAμ
is a basis function on nucleus A, or centered
at ~R
A. For an STO 4-centered integral, one would need to evaluate the integral
numerically. However, when using a GTO 4-centered integral one can transform
these to a 2-centered integral which can then be solved analytically. Thus, one must
use an increased number of GTOs to describe a MO, compared to using STOs, to get
approximately the same behavior as using STOs. Even though there is an increased
number of GTOs used, the time spent evaluating these extra GTOs is considerably
less. So computationally speaking, GTOs are a more convenient basis then STOs in
which to express the MO.12
The types of basis functions used in my analysis are composed of s, p, and d
functions. These functions have the following form:
φ1s(α, r) = (8α3/π3)1/4e−αr2
(2.9)
φ2px(α, r) = (128α5/π3)1/4xe−αr2
(2.10)
φ3dxy (α, r) = (2048α7/π3)1/4xye−αr2
(2.11)
with each describing an s, p, and d AO respectively. The α term is the Gaussian
orbital exponent. For the s-function there is only one function available since the
angular momentum ℓ = 0. The p-functions have 3 possibilities: px, py, and pz (due to
ℓ = 1 giving mℓ = −1, 0, 1) to describe the 3 possible mℓ states. For the d-functions
there are 5 possible representations: dz2 , dx2−z2 , dxy, dyz, and dzx for the 5 mℓ states.
Visual representations of these functions have been illustrated in Fig. 2.114
In dealing with basis functions to describe molecular orbitals, it has been found
that that the behavior of these basis functions more accurately describes the core
electrons and less accurately describes the valence electrons. That is a problem for
quantum chemists because most chemical reactions are explained through the inter-action
of the valence electrons. In response to this problem, a method has been
11
Figure 2.1. Pictographs of the functions used to describe the orbitals.
12
developed in which the basis sets are contracted to better serve in the calculation
of the behavior of these valence electrons. The new basis utilizes what are called
contracted Gaussians. These take the form of
φCGF
μ (r − RA) =
L
Xp
=1
dpμφp(αpμ, r − RA) (2.12)
where φCGF
μ is the contracted Gaussian basis function, dpμ are the contraction coeffi-cients,
αpμ are the contraction exponents, and φp are the primitive Gaussians which
make up the contracted Gaussian function. L is the contraction length, or how many
contracted primitives there are within a contracted basis function, and are represented
by equations 2.9 - 2.11. The method used to define the contraction is more of an art
and will not be discussed here.
2.3 Density Functional Theory
Density functional theory is different compared to HFT in that DFT includes
within its framework the electron correlation energy. Another difference between
HFT and DFT is that DFT is based on the electron density in which all information
of the system is extracted through this electron density. As was mentioned in the
previous sections HFT is based on extracting the system’s information through the
wave function.
This method is centered on two theorems known as the Hohenberg-Kohn
theorems.15 The first theorem is stated as “The external potential v(r) is determined,
within a trivial additive constant, by the electron density ρ(r).”16 Therefore, in know-ing
the electron density ρ(r) one can know the electronic properties as well as the
wave function if so desired. The second theorem is stated as “For a trial density ˜ρ(r),
such that ˜ρ(r) ≥ 0 and R ˜ρ(r)dr = N,
E0 ≤ Ev[˜ρ] (2.13)
13
where Ev[˜ρ] is the energy functional.”16 The energy functional Ev[˜ρ] is written as
Ev[˜ρ] = T[ρ] + Vne[ρ] + Vee[ρ] = Z ρ(r)v(r)d(r) + FHK[ρ] (2.14)
FHK[ρ] = T[ρ] + Vee[ρ] = T[ρ] + J[ρ] + Exc[ρ] (2.15)
The term Vee[ρ] includes both classical J[ρ] and non-classical Exc[ρ] energies. It is
this non-classical energy that is important for it contains within it the exchange-correlation
energy Exc[ρ]. We can call this FHK[ρ], or more simply F[ρ], a universal
function of ρ(r).
The density which is sought after is the ground state density which minimizes
the energy E[ρ]. We can now recast this energy of the many electron system as
E[ρ] = Z ρ(r)v(r)dr + F[ρ] (2.16)
with F[ρ] given above.
The method of calculating the ground state energy as a function of the density
is known as the Kohn-Sham method.17 With this development, the system is initiated
by an energy that excludes interacting electrons i.e. a noninteracting system. This
loss of energy is picked up by the F[ρ] term in the energy expression above. The
kinetic energy is also somewhat simplified with the energy difference between the
exact energy expression T[ρ] and the approximate expression Ts[ρ] being picked up
also by the F[ρ]. Now F[ρ] is written as
F[ρ] = Ts[ρ] + J[ρ] + Exc[ρ] (2.17)
and the total energy expression can be rewritten as
E[ρ] = Ts[ρ] + J[ρ] + Exc[ρ] + Z v(r)ρ(r)dr (2.18)
14
Essentially, all the errors due to the approximations made are now contained within
the single exchange-energy correlation term Exc[ρ]. From the Euler equation comes
the effective potential veff (r) written as
veff (r) = v(r) +
δJ[ρ]
δρ(r)
+
δExc[ρ]
δρ(r)
= v(r) + Z ρ(r′)
|r − r′|
dr′ + vxc(r) (2.19)
where vxc(r) is known as the exchange-correlation potential. Now the idea is that
to improve the total energy E[ρ] all one needs to do is improve the quality of the
exchange-correlation potential vxc(r).
The Kohn-Sham development based on this exchange correlation potential and
electron density produces a set of equations that allows for one to calculate the ground
state in an iterative fashion. These equations adhere to the notation of Parr and
Yang16 which take the form
[−
1
2
∇2 + veff ]ψi = εiψi (2.20)
veff (r) = v(r) + Z ρ(r′)
|r − r′|
dr′ + vxc(r) (2.21)
ρ(r) =
N
Xi Xs
|ψi(r, s)|2 (2.22)
with the density constrained to the following condition
Z ρ(r)dr = N (2.23)
Once the ground state electron density has been found using the above equations, the
total energy of the ground state can be calculated using an expanded expression for
the energy E[ρ] given as
E[ρ] =
N
Xi
hψi| −
1
2
∇2 + veff (r)|ψii
15
−
1
2 Z ρ(r)ρ(r′)
|r − r′|
drdr′ + Exc[ρ] − Z vxc(r)ρ(r)dr (2.24)
One of the main difference between DFT and HF is that DFT is able to incor-porate
both the exchange and coulombic correlation effects. Hartree-Fock correctly
includes the exchange correlation, but incorrectly includes the coulomb correlation by
not including the interaction of electrons with unlike spins. DFT is definitely more
advantageous than HF due to DFT having a better scaling factor, with HF scaling as
N4 and DFT scaling as N3 where N is the number of electrons in the system. With
this in mind, it is therefore conceivable that DFT should give a better ground state
over HF.
2.4 Time-Dependent Density Functional Theory
Time dependent density functional theory is based on theorems by Runge and
Gross.18 It has been reviewed recently by Marques and Gross19 in which a detailed
discussion is given. I refer the reader to this review article for a detailed explanation
of TDDFT. In papers by Stratmann et al.20 and Lourderaj et al.21 TDDFT was
applied to molecular systems which is also the case of the research discussed herein.
The ideas behind the application of TDDFT to these molecular systems are discussed
there and their formulation of the TDDFT equations will be described in this section.
The derivation of the equations mainly follows what is given in both Stratmann et al.
and Lourdera et al. We begin by assuming a potential veff (r, t) for a non-interacting
system of particles
veff (r, t) = v(t) + vSCF (r, t) (2.25)
vSCF (r, t) = Z ρ(r, t)
|r − r′|
dr′ + vxc(r, t) (2.26)
16
which have the orbitals ψ(r, t) that produce the same charge density ρ(r, t) that the
system of interacting particles has. Given this assumption we can write down the
time-dependent Kohn-Sham equation
[−
1
2
∇2 + veff (r, t)]ψ(r, t) = i
∂
∂t
ψ(r, t). (2.27)
The exchange potential vxc(r, t) can be written as
vxc(r, t) =
δAxc[ρ]
δρ(r)
(2.28)
The Axc[ρ] term is the exchange correlation action functional which is over space and
time coordinates. This is essentially the time-dependent analogue of the Exc[ρ] in the
time-independent system. In making what is known as the adiabatic approximation,
which is were the potential varies slowly with time, the exchange-correlation potential
can be approximated by
vxc(r, t) =
δExc[ρt]
δρt(r)
= vxc[ρt](r) (2.29)
where the potential vxc[ρt] varies over space at some fixed time t.
As the system is in its ground state, a perturbation is introduced into the system
by an applied field δv(t) to first order giving
δveff (r, t) = δv(t) + δvSCF (r, t). (2.30)
The δvSCF (r, t) term is the linear response in the self-consistent field due to the change
in the charge density. This charge density is given in a frequency representation and
is written as
δρ(r, ω) =Xl
m
δPlm(ω)ψl(r)ψ∗
m(r) (2.31)
where the δPlm is the Kohn-Sham density matrix in the basis of unperturbed orbitals.
17
A new expression can be obtained for this density matrix if the density is divided
into two parts–the hole-particle (δPia) and particle-hole (δPai) parts. With this new
expression δρ(r, ω) becomes
δρ(r, ω) =Xl
m
δPai(ω)ψa(r)ψ∗
i (r) +Xl
m
δPia(ω)ψi(r)ψ∗
a(r) (2.32)
where i, j and a, b represent the occupied and unoccupied orbitals. The indices of l,
m, u, and v will represent dummy indices of general orbitals.
Using perturbation theory, we can write the response of the Kohn-Sham density
matrix to the applied field. This gives
δPlm(ω) =
nlm
(ǫl − ǫm − ω)
δveff
lm (ω) (2.33)
with nlm being the difference in occupation numbers. nlm = 1 for lm = ai
and nlm = −1 for lm = ia. We can describe the linear response of the self-consistent
field to the change in the charge density by defining a coupling matrix Klm,uv written
as
Klm,uv =
∂vSCF
lm
∂Puv
= Z Z ψ∗
l (r)ψm(r)
1
|r − r’|
ψv(r’)ψ∗
u(r)drdr’
+ Z Z ψ∗
l (r)ψm(r)
δ2Exc
|δρ(r) − δρ(r’)|
ψv(r’)ψ∗
u(r)drdr’. (2.34)
The derivative above is taken with respect to the ground state density. Noting that
the coupling matrix K has both the coulombic and exchange-correlation the potential
vSCF in δveff
lm can be expressed in terms of the coupling matrix and is given as
δvSCF
lm (ω) =Xb
j
Klm,bj(ω)δPbj(ω) +Xj
b
Klm,jb(ω)δPjb(ω). (2.35)
Given the above equations we are now in a position to calculate δPlm and δvSCF
lm self-consistently,
as they depend on the the linear response of the density matrix. Putting
18
together the expressions for veff (r, t), δPst(ω), and δvSCF
st (ω) with some algebra, we
have the coupled matrix equations given as
A B
B∗ A∗
X
Y
= ω
1 0
0 −1
X
Y
(2.36)
Here matrices A, B, X, and Y are give as
Aai,bj = δabδif (ǫa − ǫi) + Kai,bj (2.37)
Bai,bj = Kai,jb (2.38)
Xai = δPai(ω) (2.39)
Yai = δPia(ω) (2.40)
The electronic excitations are then given by the poles of δPlm(ω), and can also be
calculated by finding the eigenvalues ω from the matrix equation above.
2.5 Configuration Interaction
Time-dependent density functional theory is only one of a few ways in which to
calculated the vertical electronic excitation spectrum. In our work presented here we
also employed the method of CIS. In this method the wave function of the system is
composed of multiple excited determinants. So the exact wave function for any state
takes the form
| i = c0| 0i +Xr
a
cr
a| r
a + X r<b,r<s
crs
ab| rs
ab + . . . . (2.41)
19
The CIS method makes use of only the ground state and singly excited deter-minants,
i.e. | 0i and Pra cr
a| r
a. With the excitation energies calculated using only
singly excited determinants, there is no inclusion of correlation energy.
With an increased number of excited determinants used, a better ground state
energy and electronic excitation are calculated as it includes a better result due to
the inclusion of the correlation energy. That is to say including the doubly, triply,
etc. excited state determinants, the energies calculated become closer to the actual
ground state due to higher order correlation effects. What makes this method not as
advantageous as TDDFT is the fact that the higher order correlation effects become
very expensive computationally compared to TDDFT.
2.6 Polarizable-Continuum Model
Most, if not all, chemical reactions which are investigated involve some type of
solvent. Therefore, it stands to reason that the equations that are used to predict the
electronic transitions should be modified to include the solvent-solute interaction of
the system of interest. A model which takes this interaction into account is that of
the Polarizable-Continuum Model (PCM). Cossi and Barone22 develop the necessary
modifications to TTDFT to include solvent-solute interactions into the electronic
transitions. In this model, the molecules in the solvent system are treated quantum
mechanically, while the solvent is treated as a continuum.
”The physical picture underlying the PCM is based on a sharp partition be-tween
the solute (one or more molecules, described at the desired level of theory)
and the solvent, represented as a structureless infinite continuum, characterized by
its macroscopic dielectric constant and density.”22 This boundary is closed, and is
built by a spherical surface around the solute molecule. Inside the sphere containing
the solute the dielectric is 1 which is to represent that of a vacuum. Just outside
the cavity enclosing the solute the dielectric becomes that of the solvent used. For
20
instance if water is used as the solvent, as is the case in our study, the dielectric of
80.2 is used outside the sphere.
To account for the shape of the molecule, the cavity is generated by a number
of overlapping spheres about the atoms of the molecule. The formation of the cavity
and implementation is derived by Pascual-Ahuir et. al..23 The radii of the spheres
used have been optimized by Barone et. al.24 to give values of the solvation free ener-gies
of various molecules which agree with experimental values. Cossi and Barone22
implement their formulation of PCM in TDDFT in pyridazine, and pyrimidine in
different solvents. The authors show the calculated electronic excitations are in fairly
good agreement with experimental values.
CHAPTER 3
CALCULATIONS
3.1 Introduction
In this chapter, the results from the TDDFT and CIS calculations for the un-solvated
and solvatetd complexes will be given for the different multiplets of FeP.
For the FeTPP case, only the TDDFT results will be given, for the CIS method was
not used for this iron complex. For the solvated system, only the FeP complexes
implementing the TDDFT will be given for the different multiplets.
In previous works regarding the structure of four-coordinated FeP, two different
symmetries have been used. Most researchers have used a symmetry of D4h to model
the geometry of the molecule 2,5–8,10. Other researchers have used a less constrictive
geometry of D2h
25. In the work presented here, we used the D2h symmetry group in
an attempt to give the geometry more freedom to adjust during optimization while
still taking advantage of the high symmetry for computational convenience. Similar
optimizations were done by Matsuzawa et al. 26 employing the D2h symmetry group.
According to Kozlowski et al. 4 there are four possible electronic configurations
within the D4h symmetry group. Through data obtained by M¨ossbauer spectra 27
and magnetic data 8 on FeTPP, the accepted ground state is that of the 3A2g con-figuration.
This same configuration has also been proposed by Goff et al. 10 from
NMR spectra from the spatial symmetry. From our results of the D2h symmetry
group, we obtained a configuration of 3B3g(dxz, dyz)α(dxy, dz2)2 which corresponds to
a 3A2g(dxz, dyz)α(dxy, dz2)2 after transforming to the D4h symmetry group with the
z-axis perpendicular to the plane of the molecule. For the quintet, we obtained a
21
22
configuration of 5Ag(dxz, dxy, dyz, dy2−x2)α(dz2)2 for the D2h symmetry group, which
corresponds to the same configuration in the D4h symmetry group with the z-axis
perpendicular to the plane of the FeP molecule.
In earlier studies dealing with the four-coordinated FeTPP, some groups have
used an S4 symmetry group, or quasi-D2h, which is a ruffling of the porphyrin core
and displacement of the iron atom from the center of the four nitrogen bonds within
the center of the molecule 8,28,29. The use of the symmetry is due to the findings
of X-ray analysis of the structure. In our model of FeTPP, we employed the use of
D2h before optimization of the molecule. Further optimization of FeTPP gave a C2v
symmetry for the ground state due to the ”saddle” distortion of the main molecule
with the nitrogen atoms sticking out of the plane and the rotation of the phenyl rings
approximately 47◦ from the plane of the core macrocycle.
All calculations included in this analysis were initially started using unrestricted
HF employing the basis set 6-31G(d,p) for all the atoms. This was done to get a start-ing
point for the orbitals. Further optimizations were then carried out using Kohn-
Sham DFT 16 and the B3LPY 30 exchange-correlation density functional. The excited
states for the FeP were calculated using both CIS and TDDFT, while TDDFT was the
only method used for determining the excited states of FeTPP. All TDDFT calcula-tions
were approached using the adiabatic approximation as described by Stratmann
et al. 20. A comparison of CPU times for FeP and FeTPP could not be made since
calculations were done on different computer clusters. For FeP 456 basis functions
were used, and for FeTPP 896 basis functions were used.
3.2 Unsolvated FeP and FeTPP Complexes
3.2.1 Geometry Optimization FeP
We determined the optimized geometry for each of the S=0, 1, and 2 spin
states. For the optimizations we used the density functional B3LYP, and constrained
23
S=1 S=2 S=0
Energy (au) Energy (au) Energy (au)
FeP -2252.098 -2252.091 -2252.034
FeTPP -3176.326 -3176.317 -3176.259
TABLE 3.1. Tabulated are the optimized FeP and FeTPP total energies of the
different spin states.
the symmetry to D2h. For all three spin states, the iron atom was not displaced from
the center. Table 3.1 shows the total energy of these different spin states for FeP
and FeTPP. Rovira et al. 6 performed a similar study of geometrically optimizing
each spin state independently; however, they artificially displaced the iron atom 0.3°A
out of plane and allowed it to relax into the plane. This relaxation produced a
molecular geometry having the iron atom displaced from the plane by 0.08°A
. No
displacement of the iron atom from the molecule’s center was observed in the results
of our calculations. This is possibly due to finding a local minimum within the
geometry optimization process. In looking at this same issue, Kozlowski et al. 4 also
report no displacement of the iron atom from the center.
The optimized geometries obtained from our calculations for the unsolvated FeP
mulitplets are given in Table 3.2 and 3.3. In comparing the geometries of the different
spin states we can see a subtle difference in the bond lengths and bond angles between
the singlet and triplet. However, a more pronounced difference exists for some of the
parameters for the quintet compared to the other two multiplets. This is illustrated
in Tables 3.2 and 3.3 where the values refer to Figure 1.1. The bond length in the
quintet is longer than in the triplet, and the triplet is slightly longer than the singlet.
One might attribute this to the spin-spatial dependency arguing that the higher the
spin multiplicity, the greater the spatial occupation or spatial extension. However, a
recent theoretical study suggests this is not the case. Ugalde et al. 31 state that with
24
Bond Length Singlet (°A) Triplet (°A) Quintet (°A) Exp∗ (°A)
Fe - N2 1.996 1.998 2.056 1.972
Fe - N3 1.996 1.997 2.056 1.972
Fe - N4 1.996 1.998 2.056 1.972
Fe - N5 1.996 1.997 2.056 1.972
N2 - C6 1.394 1.393 1.374 1.379
C6 - C10 1.444 1.445 1.445 1.431
C10 - C12 1.366 1.366 1.364 1.353
C6 - C22 1.385 1.386 1.398 1.389
N3 - C14 1.394 1.393 1.374 1.379
C14 - C18 1.444 1.445 1.445 1.431
C18 - C19 1.366 1.366 1.364 1.353
C14 - C22 1.385 1.386 1.398 1.389
C8 - C12 1.444 1.446 1.445 1.440
C15 - C19 1.444 1.446 1.445 1.440
N2 - C8 1.394 1.393 1.374 1.384
N3 - C15 1.394 1.393 1.374 1.384
C8 - C24 1.385 1.386 1.398 1.395
C15 - C23 1.385 1.386 1.398 1.395
TABLE 3.2. Tabulated are the FeP bond lengths of the different spin states. ∗These
values are taken from FeTPP from reference 8.
25
Bond Angle Singlet (deg) Triplet (deg) Quintet (deg) Exp∗ (deg)
N3 - Fe - N4 90.0 90.0 90.0 90.01
Fe - N3 - C14 127.4 126.5 126.5 127.1
N3 - C14 - C22 125.3 125.1 125.1 125.2
C14 - C22 - C6 124.7 126.9 126.9 123.5
Fe - N3 - C15 127.4 126.5 126.5 127.4
N3 - C15 - C23 125.3 125.1 125.1 125.0
C19 - C15 - C23 124.4 125.4 125.4 124.5
N3 - C14 - C18 110.3 109.5 109.5 110.0
C14 - C18 - C19 107.1 107.0 107.0 106.9
C18 - C19 - C15 107.1 107.0 107.0 107.3
N3 - C15 - C19 110.3 109.5 109.5 110.3
C14 - N3 - C15 105.2 107.0 107.0 105.4
TABLE 3.3. Tabulated are the FeP bond angles of the different spin states. ∗These
values are taken from FeTPP from reference 8.
26
increasing multiplicity of the FeP there is a decrease in atomic radius, thus decreasing
the spatial occupation of the atom. No attempt was made in this study to resolve
the apparent disagreement.
Tables 3.2 and 3.3 also give the experimental results of FeTPP from X-ray data.
As can be seen, the three different spin states of FeP agree reasonably well with the
experimental findings for both the bond lengths and bond angles.
3.2.2 Geometry Optimization FeTPP
The same density functional used to optimize the structure of FeP was also used
with FeTPP, which was the B3LYP. Table 3.1 gives the total energies of FeTPP as
well. It is worth noting that for each spin state the energy difference between that
spin state and the ground state is very similar for both FeP and FeTPP. For instance,
the difference in energy between the triplet and quintet of FeP is 0.20eV and FeTPP
is 0.25eV. The difference between the triplet and the singlet is 1.74eV for FeP and
1.82eV for FeTPP. Therefore we see a similar ordering of the multiplets with similar
energy differences.
As with the FeP system, we compare the triplet and quintet states of the FeTPP
system. The parameters given are those of the marcocycle common in FeP and FeTPP
and the phenyl rings of FeTPP. Figure 1.2 shows the geometric setup in which the
bond lengths and bond angles are referenced in Table 3.4 and 3.5. The bond lengths
and bond angles for both the triplet and quintet are very close in magnitude with the
quintet bond lengths being slightly larger in almost every bond. There is consistent
deviation from the experimental values for the theoretical results within the phenyl
ring. Overall, however, our calculated results are in very good agreement with the
experimental findings.
27
Bond Length Triplet (°A) Quintet (°A) Exp∗ (°A)
Macrocycle
Fe - N2 1.993 2.057 1.972
Fe - N3 1.993 2.057 1.972
N2 - C6 1.383 1.377 1.379
C6 - C10 1.439 1.444 1.431
C10 - C12 1.358 1.362 1.353
C6 - C22 1.397 1.409 1.389
N3 - C14 1.383 1.377 1.379
C14 - C18 1.439 1.444 1.431
C18 - C19 1.358 1.362 1.353
C14 - C22 1.397 1.409 1.389
C8 - C12 1.439 1.444 1.440
C15 - C19 1.439 1.444 1.440
N2 - C8 1.383 1.377 1.384
N3 - C15 1.383 1.377 1.384
C8 - C24 1.397 1.409 1.395
C15 - C23 1.397 1.409 1.395
Phenyl Ring
C34 - C35 1.404 1.404 1.383
C35 - C36 1.395 1.395 1.394
C36 - C37 1.396 1.396 1.358
C37 - C39 1.396 1.396 1.367
C39 - C38 1.395 1.395 1.402
C34 - C38 1.404 1.404 1.374
C22 - C34 1.497 1.498 1.509
TABLE 3.4. Tabulated are the FeTPP bond lengths of the triplet and quintet spin
states. ∗These values are taken from FeTPP from reference 8.
28
Bond Angle Triplet (deg) Quintet (deg) Exp∗ (deg)
Macrocycle
N3 - Fe - N4 90.0 90.0 90.01
Fe - N3 - C14 127.4 126.4 127.1
N3 - C14 - C22 126.0 125.9 125.2
C14 - C22 - C6 122.9 125.1 123.5
Fe - N3 - C15 127.4 126.4 127.4
N3 - C15 - C23 126.0 125.9 125.0
C19 - C15 - C23 123.5 124.7 124.5
N3 - C14 - C18 110.6 109.4 110.0
C14 - C18 - C19 106.9 107.1 106.9
C18 - C19 - C15 106.9 107.1 107.3
N3 - C15 - C19 110.6 109.4 110.3
C14 - N3 - C15 105.1 106.9 105.4
Phenyl Ring
C22 - C34 - C35 120.7 120.8 121.1
C22 - C34 - C38 120.7 120.8 119.7
C34 - C35 - C36 120.7 118.4 119.5
C35 - C36 - C37 120.2 120.2 120.4
C36 - C37 - C39 119.6 119.6 120.7
C37 - C39 - C38 120.2 120.2 119.3
C39 - C38 - C34 120.7 120.8 120.5
C35 - C34 - C38 118.5 118.4 119.2
TABLE 3.5. Tabulated are the FeTPP bond angles of the triplet and quintet spin
states. ∗These values are taken from FeTPP from reference 8.
29
3.3 Unsolvated Excited States of FeP and FeTPP
3.3.1 Unsolvated FeP Excited States
We calculated the excited states and oscillator strengths of the FeP molecule
using two different levels of theory. Both CIS and TDDFT were used to obtain these
excited states. In Tables 3.6 and 3.7 for the S=1 state, and Tables 3.8 and 3.9 for the
S=2 state, we show the first few excited states obtained using both methods for the
FeP system respectively. In Table 3.10 we show only the S=0 excitations
Comparing the two TDDFT excitation spectra of the triplet and quintet, one
observes a very close correspondence between the electric dipole allowed transitions.
We believe this correspondence between the electric dipole allowed transitions is sig-nificant
in that for the experimental absorption spectrum in Figure 3.1, one cannot
be certain if the FeTPPS in aqueous solution is in the S=1 or the S=2 state. These
similarities can be seen by looking at the transition states. We noticed a 616nm
peak in the triplet excitation spectrum and a 618nm peak in the quintet excitation
spectrum – a very close correspondence. Again, in the triplet we see a 518nm peak
and a 523nm peak in the quintet giving a close correspondence with the 566nm peak.
Going down the list of data, we can see a similar trend in the excitation values of the
two systems.
In comparing the CIS excitation spectrum of the triplet and quintet, we see a
similar trend between the two systems as was seen within the TDDFT framework. For
instance, in the triplet there is a weak 869nm peak, which has a corresponding peak of
857nm in the quintet system. The trend continues through the electric dipole allowed
excitations. However, the difference in the corresponding peak values is greater in the
CIS comparison than in the TDDFT comparison, which is likely due to the neglect
of correlation and other approximations in the CIS approach.
Figure 3.1 shows an experimental absorption spectrum of iron(II)-tetraphenyl-porphyrin-
sulfonate (FeTPPS) in pH 7 buffer.32 The absorbance spectra of FeTPPS
30
Excitation Excitation Oscillator
Final Energy (eV) Wavelength (nm) Strength
Symm State Calculated Calculated Calculated
1B1g 0.4410 2811.37 0
1B2g 0.4412 2810.14 0
1B2u 1.7330 715.45 0.0003
1B1u 1.7330 715.44 0.0003
2B1u 2.0099 616.87 0.0028
2B2u 2.0100 616.85 0.0028
3B2u 2.3929 518.14 0.0016
3B1u 2.3929 518.13 0.0017
4B2u 3.1916 388.47 0.0028
4B1u 3.1922 388.40 0.0029
5B2u 3.3608 368.91 0.6464
5B1u 3.3608 368.91 0.6466
TABLE 3.6. TDDFT calculated excitation results for the FeP S=1 (3B3g) state.
31
Excitation Excitation Oscillator
Final Energy (eV) Wavelength (nm) Strength
Symm State Calculated Calculated Calculated
1B1g 0.2772 4473.30 0
1B2g 0.2772 4471.69 0
1B1u 1.4259 869.63 0.0039
1B2u 1.4263 869.38 0.0039
2B2u 2.2803 543.79 0.0098
2B1u 2.2804 543.76 0.0098
3B1u 2.5776 481.07 0.0403
3B2u 2.5777 481.05 0.0402
4B2u 3.8022 326.20 0.0041
4B1u 3.8022 326.13 0.0041
5B2u 4.7294 262.19 2.5540
5B1u 4.7296 262.18 2.5541
TABLE 3.7. CIS calculated excitation results for the FeP S=1 (3B3g) state.
32
Excitation Excitation Oscillator
Final Energy (eV) Wavelength (nm) Strength
Symm State Calculated Calculated Calculated
1B3g 0.4123 3007.00 0
1B1g 0.6083 2038.26 0
1B2g 0.6083 2038.17 0
1B2u 1.6814 737.40 < 10−4
1B1u 1.6815 737.36 < 10−4
2B1u 2.0044 618.57 0.0016
2B2u 2.0045 618.52 0.0016
3B2u 2.3630 524.68 0.0003
3B1u 2.3631 524.67 0.0003
4B2u 3.1974 387.76 0.0001
4B1u 3.1979 387.71 0.0001
5B2u 3.3564 369.39 0.6682
5B1u 3.3565 369.38 0.6684
6B2u 3.6943 335.61 0.0346
6B1u 3.6945 335.59 0.0347
TABLE 3.8. TDDFT calculated excitation results for the FeP S=2 (5Ag) state.
33
Excitation Excitation Oscillator
Final Energy (eV) Wavelength (nm) Strength
Symm State Calculated Calculated Calculated
1B2g 0.4736 2618.24 0
1B1g 0.4737 2617.69 0
1B3g 0.5142 2411.51 0
1B1u 1.4453 857.95 0.0010
1B2u 1.4455 857.83 0.0010
2B2u 2.2621 548.16 0.0030
2B1u 2.2622 548.14 0.0031
3B1u 2.5175 492.55 0.0357
3B2u 2.5175 492.55 0.0356
4B2u 3.7568 330.07 0.0002
4B1u 3.7572 330.03 0.0002
5B2u 4.7021 263.71 2.5909
5B1u 4.7023 263.70 2.5916
6B1u 5.2030 238.32 0.0022
6B2u 5.2031 238.32 0.0022
TABLE 3.9. CIS calculated excitation results for the FeP S=2 (5Ag) state.
34
Excitation Excitation Oscillator
Final Energy (eV) Wavelength (nm) Strength
Symm State Calculated Calculated Calculated
1B2u 1.8851 657.52 0.0121
1B3u 2.0262 611.91 0.0304
2B2u 2.6852 461.73 0.0474
2B3u 2.7942 443.72 0.0155
3B2u 3.1618 392.13 0.0004
4B2u 3.4545 358.90 0.1646
3B3u 3.5831 346.02 0.7163
5B2u 3.7389 331.60 0.0293
4B3u 3.9124 316.90 0.0384
6B2u 4.0475 306.32 0.6997
5B3u 4.3439 285.42 0.0886
7B2u 4.3685 283.82 0.3712
TABLE 3.10. TDDFT calculated excitation results for the FeP S=0 (1Ag) state.
35
Figure 3.1. Absorbance spectrum of 6.7 μM FeTPPS in pH 7 buffer. Inset are the
Q-bands of FeTPPS.
36
(Frontier Scientific Inc., Logan UT; used without further purification) was resolved
by a CARY 4000 UV-Vis spectrophotometer (Varian Instruments). We compare the
theoretical excitation spectrum of FeP calculated using TDDFT and CIS with this
spectrum. FeTPPS is slightly different from FeTPP and more so from FeP due to
the phenylsulfonate groups. However, the sulfonate groups only serve to make the
FeTPP soluble in water and have little or no effect on the absorption spectrum for the
FeTPP.33 In Figure 3.1 the largest peak can been seen at 407nm with small peaks at
566nm and 609nm. In the TDDFT theoretical absorption spectrum for the S=1 case
of FeP we have an excitation peak at 368nm and 388nm which we believe corresponds
to the 407nm Soret band of the absorption spectrum. The large oscillator strength
at 368nm is strong evidence of this correspondence between the theoretical spectrum
and experimental spectrum. The other absorption peaks appearing in Figure 3.1, i.e.
at 566nm and 609nm, correspond to peaks in the theoretical excitation spectrum at
518nm and 616nm respectively. The theoretical peak at 715nm has a relatively weak
oscillator strength and is not observed in the experimental spectrum.
Kobayashi and Yanagawa 34 have published an experimental spectrum of FeTPP
in benzene. Their spectrum shows a double peak structure (see figure 3 on page
452 in reference 32). Reverse saturable absorption (RSA) has been reported in the
tetraphenylporphyrins, including H2TPP, CoTPP, and ZnTPP 35 and in FeTPP 36. In
the cases of H2TPP, CoTPP, and ZnTPP, the RSA is associated with an intersystem
crossing from the singlet excited state to the ground state of the triplet spin system. In
the case of Fe(II)TPP, our results suggest it occurs between the triplet and quintuplet
spin systems. The decay from the 5Ag ground state of the S=2 spin system to the 3B3g
ground state of the FeTPP is both symmetry and spin forbidden. Thus, the excitation
of FeTPP will result in intersystem crossing and a build up of population in the S=2
system. This results in the simultaneous excitations of both spin systems appearing
in both of the experimental absorption spectra. From the excitation energies listed
in Tables 3.6 and 3.8 for the S=1 and S=2 spin systems of FeP, we calculated the
37
λ(Exp,nm)a E(Exp,eV) E(Exp,eV) λ(Calc,nm) E(Calc,eV) E(Calc,eV)
816.33 1.5190 0.0852 737.4, S=2 1.68 0.051
772.95 1.6042 715.45, S=1 1.7333
698.69 1.7747 0.0620 616.95, S=1 2.0099 0.0055
675.11 1.8367 618.64, S=2 2.0044
544.22 2.2785 0 524.76, S=2 2.3630 0.0299
- - 518.20, S=1 2.3929
444.44 2.7900 0.1627 369.39, S=2 3.3564 0.0044
419.95 2.9527 368.91, S=1 3.3608
TABLE 3.11. Comparison of the calculated TDDFT excited states of Fe.
a Values taken from Kobayashi and Yanagawa.34
projected energy difference between the two sets of peaks and compared these with
those of Kobayashi and Yanagawa 34. These are given in Table 3.11.
This is unlike the situation in H2TPP, CoTPP, and ZnTPP, where the intersys-tem
crossing occurs from the excited singlet state to the triplet ground state; we see
no evidence for a triplet-singlet intersystem crossing in FeTPP. The singlet excitation
energies given in Table 3.10 do not agree with the Harmon and Rahaman32 spectrum
(Figure 3.1) or that of Koybashi and Yanagawa 34.
For the first set of absorption peaks our calculated results are similar, however,
the calculated oscillator strengths are relatively weak. The energy difference between
the peaks theoretically is close to that experimentally. For the remaining peaks we
see a similar peak structure with similar energy differences. The experimental peak
at 544nm does not seem to show a double peak in that region. However, given the
width of the peak, it is possible the two peaks are too close together to be observed
individually. Our calculations for the S=1 and S=2 systems predict a double peak
structure.
38
3.3.2 Unsolvated FeTPP Excited States
The excitation spectrum of FeTPP as determined from the TDDFT results
gives a spectrum similar to that of FeP. The calculated results can also be compared
more directly to the experimental spectra of FeTPPS and FeTPP as was done for the
FeP system. Tables 3.12 and 3.13 contain the calculated theoretical dipole allowed
transitions for the triplet and quintet systems of FeTPP.
The results show a some close correspondence between the triplet and quintet
states of FeTPP. The small oscillator strengths associated with the 5(B1,B2) and
6(B1,B2) excitations of the triplet state and the 5(B2,B1) excitations of the quintet
state accounts for their negligible contribution to the absorption spectra. Note that
for the 9(B1,B2) and 10(B2,B1) excitations of the quintet state, we do not show
a corresponding excitation in the triplet state because we didn’t compute enough
excitation states for the triplet calculations.
Comparison of the experimental absorption spectrum of FeTPPS in Figure 3.1
to the theoretical excitation spectra in Tables 3.12 and 3.13 show a reasonably
close correspondence between the experimental and theoretical results. The observed
609nm peak corresponds to the calculated 629nm excitation of the triplet, and 647nm
and 623nm excitation of the quintet. The 647nm peak is smaller than the 623nm peak.
The observed peak of 566nm corresponds to the 541nm excitation of the triplet and
the 551nm excitation of the quintet. The largest peak seen at 409nm corresponds
to the 392nm and 386nm excitations of the triplet and the 395nm and 394nm exci-tations
of the quintet. We believe that the superposition of the triplet and quintet
broadens the peaks. It should be pointed out also that the absorbance intensity of the
observed peaks corresponds qualitatively to the oscillator strengths of the theoretical
excitations of those experimental peaks.
In comparing our calculated results for FeTPP with the experimental results
from Kobayashi and Yanagawa 34, we again see some agreement in the peak struc-ture.
Table 3.14 shows that the experimental energy difference between the double
39
Excitation Excitation Oscillator
Final Energy (eV) Wavelength (nm) Strength
Symm State Calculated Calculated Calculated
1B2 0.4607 2691.04 0
1B1 0.4607 2691.04 0
2B2 1.6492 751.79 < 10−4
2B1 1.6492 751.78 < 10−4
3B1 1.9704 629.24 0.0030
3B2 1.9704 629.24 0.0030
4B2 2.2909 541.21 0.0145
4B1 2.2909 541.21 0.0145
5B1 2.5272 490.61 0.0002
5B2 2.5272 490.61 0.0002
6B1 3.0033 412.83 0.0002
6B2 3.0033 412.83 0.0002
7B2 3.1584 392.54 0.8753
7B1 3.1584 392.56 0.8753
8B2 3.2112 386.10 0.1781
8B1 3.2112 386.10 0.1779
9B1 3.5810 346.22 0.0004
9B2 3.5810 346.22 0.0004
TABLE 3.12. TDDFT calculated excitation results for the FeTPP S=1 (3A2) state.
40
Excitation Excitation Oscillator
Final Energy (eV) Wavelength (nm) Strength
Symm State Calculated Calculated Calculated
1A2 0.4210 2945.12 0
1B2 1.5468 801.57 < 10−4
1B1 1.5472 801.36 < 10−4
2B2 1.9138 647.84 0.0014
2B1 1.9140 647.78 0.0014
3B1 1.9899 623.06 0.0065
3B2 1.9900 623.04 0.0065
4B2 2.2134 560.16 0.0001
4B1 2.2139 560.04 0.0001
5B1 2.2493 551.21 0.0191
5B2 2.2494 551.20 0.0192
6B1 3.1335 395.67 0.3249
6B2 3.1336 395.66 0.3259
7B1 3.1431 394.46 0.7575
7B2 3.1432 394.45 0.7565
8B2 3.5895 345.41 0.0260
8B1 3.5897 345.39 0.0260
9B1 3.6246 342.06 0.0011
9B2 3.6246 342.06 0.0011
10B2 3.6615 338.62 0.0221
10B1 3.6618 338.59 0.0222
TABLE 3.13. TDDFT calculated excitation results for the FeTPP S=2 (5A1) state.
41
λ(Exp,nm)a E(Exp,eV) E(Exp,eV) λ(Calc,nm) E(Calc,eV) E(Calc,eV)
816.33 1.5190 0.0852 801.5, S=2 1.547 0.102
772.95 1.6042 751.8, S=1 1.649
698.69 1.7747 0.0620 629.24, S=1 1.9704 0.0196
675.11 1.8367 623.06, S=2 1.9900
544.22 2.2785 0 551.21, S=2 2.2494 0.0415
- - 541.21, S=1 2.2909
444.44 2.7900 0.1627 394.46, S=2 3.1432 0.0152
419.95 2.9527 392.56, S=1 3.1584
TABLE 3.14. Comparison of the calculated TDDFT excited states of FeTPP and
the experimental absorption peaks of FeTPP in benzene.
a Values taken from Kobayashi and Yanagawa.34
peak structure agrees qualitatively with the theoretical energy difference. The first
set of experimental peaks (i.e. at 816nm and 772nm) are accounted for theoretically.
However, like the FeP system, the corresponding peaks are associated with small os-cillator
strengths, but still show a similar energy difference. As noted in the previous
section the observed peak of 544nm does not show a double peak structure. How-ever,
our results suggest a double peak with at 541nm and 551nm for S=1 and S=2
respectively. This is experimentally resolved in the spectrum observed by Kobayashi
and Yanagawa with FeTPP in pyridine solution which shows a double peak structure
with peaks at 565nm and 532nm giving a difference on the order of 0.14eV. For each
double peak seen experimentally (with the exception of the 816nm and 772nm peaks),
a large theoretical oscillator strength suggests a strong excitation.
As can be seen, the TDDFT results for the FeP and FeTPP structures and our
overall geometries are in good agreement with those previously published experimen-tal
and theoretical studies.
42
3.3.3 Charge and Spin Densities of Unsolvated FeP and FeTPP
For the triplet case of FeP and FeTPP we calculated the Mulliken spin density.
The calculations suggest nearly all the α-spin density is located on the central iron
atom. Around the core macrocycle there is a little α-spin density situated along the
carbon atoms bonded to the nitrogen atoms which show small β-spin densities. The
values for the densities are shown in Table 3.15, as well as the charge densities. From
the charge densities we can see the central FeN4 cluster of both iron complexes are
negative with the surrounding eight carbon atoms being positively charged.
S=1 FeTPP S=1 FeTPP S=1 FeP S=1 FeP
Atom Charge Density Spin Density Charge Density Spin Density
Fe 0.91 2.00 0.93 2.00
N2 -0.79 -0.05 -0.77 -0.05
N3 -0.79 -0.05 -0.77 -0.05
N4 -0.79 -0.05 -0.77 -0.05
N5 -0.79 -0.05 -0.77 -0.05
C6 0.33 0.02 0.35 0.02
C7 0.33 0.02 0.35 0.02
C8 0.33 0.02 0.35 0.02
C9 0.33 0.02 0.35 0.02
C14 0.33 0.02 0.35 0.02
C15 0.33 0.02 0.35 0.02
C16 0.33 0.02 0.35 0.02
C17 0.33 0.02 0.35 0.02
TABLE 3.15. Mulliken charge densities in units of electrons per atom and spin
densities for the triplet FeTPP and FeP.
43
3.3.4 Comparison of Unsolvated FeP and FeTPP
In the above subsections we showed the correspondence between the experimen-tal
absorption spectra of FeTPP and FeTPPS and the theoretical excitation spectra
for the two iron complexes (FeP and FeTPP). We are now in a position to compare
our results for the FeP and FeTPP complexes. Since the ground state is a triplet in
both systems we need only compare our results for the triplet of FeP and the triplet
of FeTPP.
We observe in Tables 3.6 and 3.12 the predicted absorption spectra for both
systems. For nearly each peak in the FeP triplet system there is a corresponding
peak in the FeTPP system. The exceptions in this case are the 715nm peak in the
FeP system. We cannot be certain about the 346nm peak in the FeTPP system
since we did not calculate excitations less than 350nm in the FeP calculations. In
general, we find a blue shift in the spectrum of the FeP molecule relative to the
FeTPP molecule. This is also seen for the quintet FeP and FeTPP as well. Both FeP
and FeTPP share the same electronic configuration suggesting even further similarity
between the two systems. We have already shown both the FeP and FeTPP results
compare well with experimental data for FeTPP and FeTPPS.
From the charge and spin densities in Table 3.15 we see a very similar structure
of the two characteristics for the FeP and FeTPP molecules. As can be seen there
is very little difference numerically between the two triplet systems. Therefore, from
our results, we conclude on the basis of similar electronic configurations, geometric
parameters, excitation spectra, spin and charge densities that FeP can act as a sub-stitute
for the FeTPPS system. However, since our actual system is in an aqueous
solution, it is necessary to study the solvent effects involving the interaction of the
iron complex and the solution.
44
3.4 Solvated FeP Complexes
3.4.1 FeP Excited States
The interaction of FeP with the solvent in the overall process we are study-ing
plays an important role in the electron transfer process. Smith et. al.37 shows
that there is an interaction between the iron complex with two water molecules–
each bonded to the iron atom along the axis perpendicular to the plane. Using the
B3LYP functional in their calculations, they conclude that the interaction changes
the electronic configuration of the the ground state to the quintet configuration with
the singlet configuration following close behind. They also conclude in terms of the
functional used, that the B3LYP most accurately describes and predicts the ground
state of there FeP-(H20)2 complex.
The effects of this solvent on the electronic structure of FeP was treated in these
calculations by way of the PCM method mentioned in Chapter 2. It has been shown
by Cossi and Barone22 that the PCM method gives better results with the explicit
inclusion of two water molecules (or molecules of the solvent type) bonded to the
solute molecule. In our calculations we did not explicitly include any water molecules.
Another member in our group has implemented this route and his calculations be used
as a comparison to what is done herein.
In the previous section, we showed that FeP can theoretically be substituted
for FeTPPS in future calculations due to similarities of various properties of it and
FeTPP. In Tables 3.16 and 3.17 we show the TDDFT results for the solvated triplet
and quintet FeP complexes. From this data we can see the two excited state spectra,
the triplet and quintet, are shifted slightly towards the blue end of the spectrum.
Therefore, we would expect very similar behavior in the large iron complex of FeTPP
and thus even the same in the larger FeTPPS. The 4(B2u,B1u) states the oscillator
strength has been diminished from 0.0002 in the unsolvated system to < 10−4 in
the solvated system. Therefore, we don’t necessarily see the double peak structure
45
Excitation Excitation Oscillator
Final Energy (eV) Wavelength (nm) Strength
Symm State Calculated Calculated Calculated
1B2g 0.37 3332 0
1B1g 0.37 3330 0
1B1u 1.75 707 0.0004
1B2u 1.75 707 0.0004
2B2u 2.03 610 0.0046
2B1u 2.03 610 0.0046
3B1u 2.41 515 0.0016
3B2u 2.41 515 0.0016
4B1u 3.22 386 0.0470
4B2u 3.22 386 0.0480
5B1u 3.27 379 0.8712
5B2u 3.27 379 0.8703
TABLE 3.16. Calculated TDDFT excitation results for the solvated FeP S=1 (3B3g)
state.
46
Excitation Excitation Oscillator
Final Energy (eV) Wavelength (nm) Strength
Symm State Calculated Calculated Calculated
1B3g 0.23 5389 0
1B1g 0.45 2732 0
1B2g 0.45 2732 0
1B2u 1.70 730 < 10−4
1B1u 1.70 730 < 10−4
2B1u 2.03 610 0.0024
2B2u 2.03 610 0.0024
3B2u 2.38 521 0.0001
3B1u 2.38 521 0.0001
4B2u 3.21 386 < 10−4
4B1u 3.21 386 < 10−4
5B2u 3.27 379 0.9865
5B1u 3.27 379 0.9867
6B2u 3.69 336 0.0268
6B1u 3.69 336 0.0269
7B2u 3.78 328 0.0181
7B1u 3.78 328 0.0179
TABLE 3.17. Calculated TDDFT excitation results for the solvated FeP S=2 (5Ag)
state.
47
at or around the 386nm peak as the 4(B2u,B1u) appears very weak. On the same
note, we do see for the 5(B2u,B1u) an increased oscillator strength compared to the
unsolvated system, which is seen experimentally to be a strong peak.
3.4.2 Solvated FeP Spin and Charge Densities
In looking at the charge and spin densities of the triplet and quintet in Ta-ble
3.18, we see very similar values for each atom making up the macrocycle, except
for the iron atom. The N4 cluster is negative, while the remaining macrocycle is posi-tive.
For the iron, we see most of the spin is located on the iron atom itself. Compared
to the values of the unsolvated system we see a very similar trend in values of the
charge and spin density.
48
S=1 FeP S=1 FeP S=2 FeP S=2 FeP
Atom Charge Density Spin Density Charge Density Spin Density
Fe 0.93 2.00 1.10 3.73
N2 -0.77 -0.05 -0.80 0.03
N3 -0.77 -0.05 -0.80 0.03
N4 -0.77 -0.05 -0.80 0.03
N5 -0.77 -0.05 -0.80 0.03
C6 0.32 0.02 0.33 0.02
C7 0.32 0.02 0.33 0.02
C8 0.32 0.02 0.33 0.02
C9 0.32 0.02 0.33 0.02
C14 0.32 0.02 0.33 0.02
C15 0.32 0.02 0.33 0.02
C16 0.32 0.02 0.33 0.02
C17 0.32 0.02 0.33 0.02
TABLE 3.18. Mulliken charge densities in units of electrons per atom and spin
densities for the triplet and quintet FeP.
CHAPTER 4
Excited State Electron Transfer
4.1 Introduction
In this chapter, we give the general idea behind excited state electron transfer
and discuss the topics associated with this process. The thermodynamics that govern
this process will be given. The PCM method will be applied to our system involving
the aqueous FeP-TNT interaction.
In Chapter 3 the theoretical excited state structure was given. Knowing the
excited states of a molecule or an atom, in part, can help determine if electron transfer
can occur. As will be shown below, the excitation energy is used to calculate the
change in Gibbs free energy of the system. Of course the excited state alone will not
tell you this – more knowledge is necessary. However, the excited state is a good
place to start in understanding an ET process. For now, the reader should know that
for an arbitrary chemical species, a letter will be used to designate the ground state,
for instance A. If the chemical complex is in its excited state, the designation A* will
be given.
In the ground state of a complex, the energy of each molecule is in its respective
minima. In other words, the geometry and electronic arrangement are in such a
configuration as to give the lowest energy of the system. The lowest possible vibration
level, denoted as the zero-point level, is normally more populated at room temperature
than the higher excited vibrational levels because the thermal energy of kbT is only
0.0257eV .38 This energy is enough to agitate the system into this lowest vibrational
state.
49
50
As with every complex there are vertical excited states (electronic excitations)
to which the system is promoted with the absorption of a photon. This promotes
the system from some vibrational state of the electronic ground state to another vi-brational
state in a higher energetic electronic state. During the the absorbance of
a photon by an electron, the nuclear geometry of the complex does not normally
change.38 The electron is much less massive then the nucleus and therefore the exci-tation
process can be approximated in this way (that the geometry doesn’t change
initially with an excitation occurring). This is the well known Frank-Condon princi-ple.
The time in which the vertical transition occurs is on the order of 10−16s. The
time for the nuclear vibrational transition is on the order 10−14 − 10−12s.38 In other
words, what is taking place is an absorption of a photon by an electron. This promotes
the electron to an excited state or what is known as a Frank-Condon state. After the
excitation occurs, the nuclear geometry relaxes due to the change in electron density
until an equilibrium is found. This relaxation lowers the energy slightly, compared
to the Frank-Condon state, and can be long-lived. This is illustrated in Fig. 4.1.39
It is the relaxed excited state that is responsible for participating in a photochemical
reaction.
Once a complex is in its excited state configuration, it can decay back into its
lower energy state in 3 different ways. The process can be nonradiative in which
case there is a release of heat due to lattice vibrations. Radiative processes involve
the emission of a photon through decay into lower states. Then there is fluorescence
which is the decay from the lowest vibrational state of the excited state into another
vibrational state of a lower electronic state, accompanied by light emission. Since
some of the energy of the excited state goes into the vibration of the lattice, there
will be a decrease in the energy emitted from the decay of the electronic excited
state to a lower state. Thus the wavelength for the emission process will be longer
corresponding to that for the absorption process. The difference in these wavelengths
is known as the Stokes shift.
51
Figure 4.1. Excitation of the electron to the Frank-Condon state which then relaxes
to the equilibrated state.
52
4.2 Thermodynamic Factors of Photoinduced Electron Transfer
4.2.1 Quenchers and Sensitizers
Thermodynamics describes the reaction between two molecules and can be used
to determine if the reaction will proceed forward. To better understand these reactions
a knowledge of the thermodynamics governing the reactions themselves must be taken
into account. The emphasis in this section will primarily be the energetics of reactions.
The notation used will follow that of Kavarnos.38
If a molecule is excited electronically and the excited state is long lived, it is
possible for another molecule in its ground state to interact with the excited molecule
and cause an electron transfer. The excited molecule will be known as the sensitizer
and the interacting ground state molecule will be known as the quencher. The sen-sitizer
can cause changes in the quencher through an electron transfer. The effect of
the quencher is to quench, or deactivate the excited molecule the sensitizer.
As mentioned above we have two complexes that are interacting with each other
which can be written as the following
D∗ + A −→
E
D + A∗ (4.1)
where D∗ represents the electron donor which is the excited state sensitizer. A is
the electron acceptor which is the quencher, and A∗ is the excited state acceptor.
Another way to see the reaction is to include the photoexcitation of the donor D.
Thus we have
Dm + An −→
kh
[Dm]∗ + An −→
ken
Dm+1 + An−1 (4.2)
The first reaction is the photoexcitation of the donor. Then the proceeding step is
the donor giving an electron from its excited state to the acceptor. This is seen as
the overall charge on the donor is increased by +1 and the acceptor is decreased by
−1, where m and n represent the initial charge of the molecule.
53
4.2.2 Enthalpy and Gibbs Free Energy
It is important to know the Gibbs free energy change which will tell us whether
or not the system will need energy to proceed or is a spontaneous process. The free
energy change accompanying a chemical process is given by
G = H − T S (4.3)
where G is the Gibbs free energy change and H is the heat of enthalpy. T and
S are the absolute temperature and change in entropy, respectively.
In the process of the donor system going from its ground state D to its ex-cited
state D∗ and then to its donor state D+, we can write the heat of enthalpy as
HD →D+ – an exothermic reaction. The process of the system going from its ground
state D to is donor state D+ gives the heat of enthalpy as HD→D+ – an endothermic
reaction.
This can be seen from Fig. 4.2 40 and the energy ED
00 can be written in terms of the
heats of enthalpy as
− HD −→D+ + HD−→D+ = ED
00. (4.4)
Using this equation in conjunction with the Gibbs free energy equation we can rewrite
ED
00 as
ED
00 = − GD −→D+ − T SD −→D+ + GD−→D+ + T SD−→D+. (4.5)
If we neglect the T S terms since their contribution will be small due to small
structural changes in going from D −→ D∗ −→ D+ thus making S negligible, the
free energy change is given by
GD −→D+ = GD−→D+ − ED
00. (4.6)
54
Figure 4.2. Enthalpy changes of the donor complex.
55
In keeping with the above prescription, we can find the free energy associated
with the reduced acceptor system A. We can write this as
GA −→A− = GA−→A− − EA
00. (4.7)
If we look at the half reactions of the donor and acceptor complexes, for which the
free-energy change was found, we can introduce the free-energy in terms of the redox
potential (also known as the electromotive force or emf) or Eredox. As is customary
the half-reactions will be written as reductions so that we have the following
D+ + e− −→ D (4.8)
A + e− −→ A−. (4.9)
Now in reference to the half-reactions we can write
G = −nFEredox. (4.10)
where n is the number of moles of e− transferred and F is number of electrons per
mole (known as the Faraday).
4.2.3 Redox Potentials
Now that we have the Gibbs free energy written in a compact form we can write
the redox potentials for the half-reactions involving the excited state. This gives
ED −→D+ = ED−→D+ + ED
00 (4.11)
EA −→A− = EA−→A− − EA
00. (4.12)
56
We can rewrite these redox potentials in a simpler form by using the notation
E0(D+/D) = −ED→D+ and E0(D+/D∗) = −ED →D+ and similar for the other po-tentials.
Now we can rewrite the potentials as
E0(D+/D∗) = E0(D+/D) − ED
00 (4.13)
E0(A∗/A−) = E0(A/A−) + EA
00. (4.14)
As was mentioned previously, the excited state is better suited for electron transfer
than its ground state. This is seen above where E0(D+/D∗) < E0(D+/D). In short,
the lower the E0(D+/D∗) of the complex the easier the donor can donate an electron
from its excited state.
Up to this point we have taken the individual energies of the donor and acceptor
complexes to develop the energetics. Now we are in a position to combine the two
complexes and find the energy necessary to determine the viability of electron transfer
within a solution. Figure 4.341 can be our guide to determine the free energy changes
associated with the electron transfer process from the excited state donor complex.
The free energy changes for the uphill reaction can be written as
Gup
el = GD→D+ + GA→A−. (4.15)
For the downhill reaction, which is where the ET takes place, we can write the
free energy change of the donor-acceptor complex as
Gdown
el = GD →D+ + GA→A−. (4.16)
If we rewrite our equation in terms of the reduced half reactions we obtain
Gdown
el = nF[E0(D+/D∗) − E0(A/A−)]. (4.17)
Now with the substitution of Eq.31 the above expression yields
57
Figure 4.3. Energy diagram for photoinduced electron transfer.
58
Gdown
el = nF[E0(D+/D) − E0(A/A−)] − ED
00. (4.18)
Changing units to kcal ·mol−1, using the fact that for most one-electron transfers we
have nF ∼ 1, and using direct substitution of the expression for E0(D+/D∗) leads to
Gdown
el = 23.06[E0(D+/D) − E0(A/A−)] − ED
00. (4.19)
where ED
00 is the excitation energy corresponding to the equilibrated energy ED
00
and is measured in eV s. The coefficient 23.06 kcal · mol−1 comes from (3.82929 ·
10−23kcal)NA where NA is Avagadro’s number.
Assuming now that instead of two initially charged complexes we have two
neutral complexes in which electron transfer will take place. After electron transfer
there will be two charge complexes D+ and A−. Due to the formation of this ion pair,
coulombic attraction between these final charged complexes will pull them together
thus releasing energy denoted by wp. Therefore, we should modify the Eq. 4.19 to
include this work term wp. We can write wp as
wp(kcal · mol−1) =
332(zD+zA−)
dccǫs
(4.20)
where we define zD+ and zA− as the molecular charges, ǫs as the static dielectric of the
solvent, and dcc as the center-to-center separation distance between donor-acceptor
complexes in °A
.38 The coefficient 332 comes from computing NAe2/4πε0 and changing
to kcal. With the inclusion of this modification we can arrive at what is known as
the Rehm-Weller equation42 given as
Gdown
el = 23.06[E0(D+/D) − E0(A/A−)] − wp − ED
00. (4.21)
In short, the Rehm-Weller equation states that Gel < 0 for spontaneous electron
transfer between uncharged reactants. This equation can be used as a tool to deter-mine
the energetic feasibility of ET taking place with the given system.43
59
In our ab initio study of the FeP-TNT system, we did not include the empirical
values for the oxidation and reducion potentials that are be used in Eq. 4.21. Instead,
however, we calculated the free energies of the molecules we believe are involved in
the electron transfer process. Therefore, Eq. 4.16 is modified to give us a working
equation that enables us to find the Gibbs free energy of the system by using the
individual molecular free energies. Substituting Eq. 4.6 into Eq. 4.16 and including
the the coulombic term wp gives us this more usable equation in terms of calculated
values of the free energies written as
Gdown
el = GD→D+ + GA→A− − ED
00 − wp. (4.22)
From Eq. 4.22 it can now be seen that we need only find the molecular free ener-gies
in the solvated system, the excited state involved in the electron transfer, and
the coulombic interaction of the solvated donor-acceptor complexes. This data is
presented in the next chapter.
CHAPTER 5
Excited State Electron Transfer Results
5.1 Introduction
In Chapter 4 the thermodynamic expressions needed to determine the Gibbs free
energy were developed. From this we can predict whether or not the thermodynamic
condition exists for electron transfer to proceed in the system we are studying, i.e.
Gdown
el < 0, by using Eq. 4.22. In this chapter, the probable reaction pathway will be
given, as well as the calculated free energies for each complex. From there, Eq. 4.22
will be used to show the possibility of electron transfer for this particular pathway of
our FeP-TNT system.
In this system, FeP acts as the sensitizer, and TNT acts as the quencher. Since
our previous calculations agree with previously published data on the matter of the
ground state of FeP being a triplet, we will use this triplet state as the basis for the
calculations done herein. Since both the reaction involving FeP and TNT occurs in
an aqueous solution we need to take into consideration the solvent effects.
This interaction is taken care of by using a method developed by Cossi and
Barone.22 The method used is called the PCM method which utilizes overlapping
spheres to generate a spherical cavity to account for the water solution in which the
system will be submerged, as was discussed in Chapters 2 and 3. It is, however,
possible that instead of electron transfer from FeP to TNT we can have just the
opposite take place–electron transfer from TNT to the FeP. Since visible light is
the driving force behind this process and FeP is excited with visible light as seen
in Fig. 3.1, FeP was initially thought of as being the excited molecule in which the
60
61
electron transfer process was initiated. Therefore, we did not investigate electron
transfer from TNT to FeP. In this thesis, only electron transfer from FeP to TNT will
be discussed and the proposed pathway is given by
FeP(S = 1) + TNT(S = 0)) −→
E
[FeP(S = 1)]∗ + TNT(S = 0)
−→ [FeP]+1(S = 3/2) + [TNT]−1(S = 1/2)
−→
O2
R − COOH(S = 0) + FeP(S = 1) + H2(S = 0)
−→ TNB(S = 0) + CO2(S = 0) (5.1)
where R represents the ligated trinitrobenzene, and TNB is just trinitrobenzene.
5.2 Calculations
The ionized iron complex has three different spin states possible for the ground
state. Our calculations show the S=3/2 state is the ground state with the S=1/2
being slightly above, and the S=5/2 being even higher in energy. These results are
consistent with previous work from Smith et. al..37 We believe this S=3/2 spin state
for FeP+1 is associated with the electron transfer pathway due to it being the ionized
ground state. In looking at the mapping of the electronic charge densities for both
the solvated FeP triplet in Fig. 5.1 and the FeP S=3/2 (ionic FeP) in Fig. 5.2 we can
see that the Fe atom is less negative in the ionized FeP. This is an indication that the
most of the charge associated with the electron transfer is coming from the iron with
a small contribution from the macrocycle. The electronic configuration can point us
in the direction as to which occupied orbital the transferred electron is coming from.
Table 5.1 shows the configurations FeP and FeP+1. From this, we can see that the
electron is predominately coming from the dz2 orbital.
Figures 5.3 and 5.4 show the effective charges at each atomic site using the
Mulikin population analysis. Comparing these charge density maps of the TNT and
TNT−1 shows how the charge distribution changes from the initial ground state to the
62
fep-s1-grd-solvent.log
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32(0.131)
28(0.135)
20(-0.145)
36(0.131)
31(0.131)
24(-0.213)
19(-0.145)
16(0.322)
12(-0.145)
8(0.323)
15(0.322)
4(-0.772)
10(-0.145)
34(0.131)
27(0.135)
3(-0.772)
23(-0.213)
6(0.323)
37(0.933)
7(0.323)
22(-0.213)
1(-0.772)
26(0.135)
35(0.131)
11(-0.145)
2(-0.772)
14(0.322)
5(0.323)
9(-0.145)
13(0.322)
18(-0.145)
21(-0.213)
30(0.131)
33(0.131)
17(-0.145)
25(0.135)
29(0.131)
Figure 5.1. Model of solvated FeP triplet charge density map. This model is used as
a reference diagram for the atomic charges.
63
fep+1-grd-1-solvent-opt.log
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30(0.152)
18(-0.132) 29(0.152)
17(-0.132)
26(0.157)
25(0.157)
14(0.340)
13(0.340)
22(-0.197)
21(-0.197)
2(-0.808)
34(0.152)
6(0.340)
33(0.152)
5(0.340)
10(-0.132)
9(-0.132)
3(-0.808)
37(1.510)
1(-0.808)
12(-0.132)
11(-0.132)
8(0.340)
36(0.152)
7(0.340)
35(0.152)
4(-0.808)
24(-0.197)
23(-0.197)
16(0.340)
15(0.340)
28(0.157)
27(0.157)
20(-0.132)
32(0.152) 19(-0.132)
31(0.152)
Figure 5.2. Model of solvated FeP S=3/2 charge density map. This model is used as
a reference diagram for the atomic charges.
64
dz2 dxy dxz dyz dx2−y2
FeP 2 2 1 1 0
FeP+1 1 2 1 1 0
TABLE 5.1. Table giving the occupied orbitals of the FeP and FeP+1 complexes.
ionic state. As can be seen, much of the charge is transferred to the oxygen atoms of
the NO2 groups. Atom 15N is the only nitrogen atom that appears to have noticeably
decreased in charge indicating a slight deposit of of the electronic charge, which is
in agreement with works done by Huang and Leszczynski.44 The map also shows the
atoms in the C-H bonds in the benzene ring have change more then the other carbon
atoms in the ring. The atoms in the C-CH3 bond show the ring-carbon atom to have
increased in negativity while the carbon atom in the CH3 group has become slightly
more positive then its initial counterpart. It is possible that this change in the charge
of the bonding atoms will weaken the C-CH3 bond.
We chose to focus our calculations on ET from FeP to TNT due to the exper-imental
evidence that showed FeP has the same excitation wavelengths as the light
being used to initiate the process. From the experimental results, it was found that
90% of the initial amount of TNT (in the TNT-FeTPPS) solution was deactivated
in a period of less than one hour.45 So the process has been verified to work exper-imentally.
However, the ET pathway by which the process takes place is not yet
understand and is the subject of this investigation.
5.3 Thermodynamic Calculations
For our system were were able to calculate the free energies for each molecule
we believe are involved in the process of electron transfer. Table 5.2 shows the free
energies associated with each component in the reaction given by Eq. 5.1.
65
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16O(-0.409)
14O(-0.389)
7H(0.207)
15N(0.406)
17O(-0.409)
3C(-0.046)
20H(0.173)
4C(0.244)
12N(0.375)
2C(0.179)
5C(-0.046)
8H(0.207)
13O(-0.391)
11O(-0.390)
1C(0.184)
6C(0.179)
9N(0.375)
18C(-0.352)
19H(0.145)
10O(-0.391)
21H(0.145)
Figure 5.3. Model of solvated TNT S=0 charge density map. This model is used as
a reference diagram for the atomic charges.
66
tnt--1-solvent-c1-opt.log
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14O(-0.424)
7H(0.155)
16O(-0.567)
12N(0.356)
13O(-0.430)
3C(-0.100)
20H(0.137)
15N(0.258)
2C(0.175)
4C(0.266)
17O(-0.568)
1C(0.110)
5C(-0.100)
11O(-0.424)
6C(0.175)
8H(0.155)
18C(-0.339)
9N(0.357)
19H(0.119)
21H(0.119)
10O(-0.430)
Figure 5.4. Model of solvated ionic TNT S=1/2 charge density map. This model is
used as a reference diagram for the atomic charges.
67
For simplicity, the reaction in Eq. 5.1 will be broken up into three parts: RXN
1, RXN 2, and RXN 3. Each reaction will be defined in the following way:
RXN 1 : FeP(S = 1) + TNT(S = 0) −→
E
[FeP(S = 1)]∗ + TNT(S = 0)
−→ [FeP]+1(S = 3/2) + [TNT]−1(S = 1/2) (5.2)
RXN 2 : [FeP]+1(S = 3/2) + [TNT]−1(S = 1/2) −→
O2
R − COOH(S = 0)
+FeP(S = 1) + H2(S = 0) (5.3)
RXN 3 : R − COOH(S = 0) −→ TNB(S = 0) + CO2(S = 0). (5.4)
In RXN 1, there are a few steps involved and the free energies are not calculated
for each step. The reason for this is that the donor free energy term in Gdown
el only
involves the ground state of FeP and its ionized state FeP+1. Therefore, even though
the excited state term exists in the reaction, it is accounted for in Eq. 4.22 by the
explicit excitation energy ED
00. By plugging in the values necessary for RXN 1, given in
Table 5.2, we can find out if electron transfer is likely to occur by taking into account
the thermodynamic condition of spontaneity for G < 0. Using Eq. 4.22 we see the
term wp which is the work term as mentioned previously. However, remembering
the work term is inversely proportional to the distance between the ionized donor
and acceptor, and is also subtracted from the total free energy, it can be seen that
this term only serves to make the free energy more negative with a smaller distance.
Calculations including the FeP and TNT molecules with the inclusion of a water cage
have been performed by Scofield46 whose calculations show, at this junction, that the
FeP is separated from the TNT by a couple of water molecules. The distance given
68
between the two complexes on average is on the order of 5.7°A
46 due to a slight tilt of
the TNT by 33.4◦ on the face to face orientation. Therefore, using a value of dcc=5.7
and εs=80.2 we find
wp =
14.40
dccεs
= 0.03 eV. (5.5)
Using this value for the free energy expression, we find
Gdown
el (eV ) = GD→D+ + GA→A− − ED
00 − wp = −2.23 eV. (5.6)
Therefore, we can see there is spontaneity in the the electron transfer from FeP*
to TNT. The excited energy used for ED
00 is that taken from the solvated excitation
spectrum of the triplet FeP in Table 3.6 and is the excited state with the largest
oscillator strength, i.e. the 5(B1u,B2u) state also known as the Q-band. Although
the equilibrated excited state of FeP participates in the electron transfer process,
the excitation energy used is not that of the relaxed excited state, but the Frank-
Condon state. In our calculations it is not possible to find the equilibrated energy
for this particular excited state using TDDFT because of the symmetry state held
by this excitation. Lower energy excitations also have the same symmetry and thus
optimizing this particular state will only optimize the first excited state with the same
symmetry and not the excited state we believe is involved in the electron transfer
process. However, we can estimate the value of the equilibrated excited state in a
limiting way by take the difference between the lowest allowed excitation and the
Q-band. This gives a value of 1.5 eV. In short, calculating the G for RXN 1 shows
in a somewhat simplified manner in which electron transfer is likely to occur.
As for whether or not the entire pathway proposed is an energetically favorable
pathway, we must also find the Gibbs free energy for RXN 2 and RXN 3. The
individual molecular free energies can be found from Table 5.2 as well. Table 5.3
shows a summary of these change in free energies. We can see from Table 5.3 that in
69
Gibbs Free
Molecule Energy (au)
FeP(S = 1) -2251.889359
FeP+1(S = 3/2) -2251.733281
TNT(S = 0) -884.976404
TNT−1(S = 1/2) -885.108219
O2(S = 1) -150.336406
H2(S = 0) -1.180059
R − COOH(S = 0) -1034.247947
TNB(S = 0) -845.693313
CO2(S = 0) -188.589561
TABLE 5.2. Gibbs free energies for each molecule in the electron transfer process.
Reaction G(eV)
RXN 1 -2.23
RXN 2 -3.80
RXN 3 -0.95
TABLE 5.3. Change in Gibbs free energies for each set of reactions in the electron
transfer process.
70
each set of the overall proposed reaction in Eq. 5.1, the processes is thermodynamically
favorable as given by the thermodynamic condition of spontaneity.
CHAPTER 6
Summary and Conclusion
In chapter 2 the theory for the first part of my calculations was presented mainly
dealing with DFT and TDDFT with a short discussion on the CIS, and PCM method
used to account for the aqueous solvent used. In chapter 3 I presented our results
involving different aspects of the calculations. We showed that our results involving
symmetry, geometry, energetic orderings, and electronic configurations are consis-tent
with those found by other groups. We also showed the excitations predicted
by TDDFT and CIS of the different multiplets of the unsolvated FeP and FeTPP
(TDDFT only) complexes. A comparison was given showing the theoretical excita-tions
of FeP and the experimental absorption spectrum FeTPPS. The excitations of
FeP showed there was a direct correlation to the experimental absorption spectrum of
FeTPPS, accounting for the observed peaks in our calculations through the electronic
dipole allowed transitions. From this we saw the theoretical excitation spectrum of
FeP was blue shifted by approximately 30 nm. We found from comparing the larger
FeTPPS complex to the smaller FeP complex that FeP was able to be to substituted
in for FeTPPS, in terms of computational convenience.
Calculations performed on the larger FeTPP complex (FeTPPS minus the sul-fonate
groups) show that FeP has a very similar charge density and spin density for
the the two molecules. In addition, the theoretical experimental spectra of the two
molecules further shows the similarities of the two molecules, suggesting more so that
FeTPPS can be substituted by FeP in terms doing calculations. We also presented
our results including the solvent for the FeP molecule, given in Chapterr 3. From
this we showed the theoretical excitation spectrum was slightly blue shift as a result
71
72
while maintaining the a qualitative agreement with the observed peaks. Therefore, it
is our belief that FeP can be used in place of FeTPPS in future calculations.
In chapter 4 I presented Gibb’s free energy condition of spontaneity. I showed
that instead of using redox potentials, we could in fact use the free energies resulting
from our calculations. Chapter 5 gives a suggested pathway in which to photodeac-tivate
TNT by way of FeTPPS. Using the free energy idea and a form of the Rehm-
Weller equation, chapter 5 gives the results of the free energies calculated within the
solvated system to more accurately account for solvent-solute interactions. It it shown
that the proposed pathway for the degradation of TNT is thermodynamically favor-able.
Taking these ideas into consideration and what has been presented in chapters 3
and 5, I believe the proposed pathway is the likely pathway for the process studied
by Harmon et. al..11,32
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VITA
Clint B. Conner
Candidate for the Degree of
Doctor of Philosophy
Thesis: THEORETICAL STUDIES OF EXCITED STATE ELECTRON
TRANSFER BETWEEN IRON-PORPHYRIN AND TRINITRO-TOLUENE
Major Field: Physics
Biographical:
Personal Data: Born in Tulsa, Oklahoma, on February 12, 1977, the son of
Mark Conner and Debi Foster.
Education: Graduated from Claremore High School, Claremore, Oklahoma in
May 1995; received Associates of Science degree in Mathematics from
Rogers State University, Claremore, Oklahoma in July 1998. Received
Bachelor of Science degree in Engineering Physics from Northeastern State
University, Tahlequah, Oklahoma in December 2000. Completed the re-quirements
for the Doctor of Philosophy degree with a major in Physics
at Oklahoma State University in December 2006.
Professional Memberships: American Physical Society.
Name: Clint B. Conner Date of Degree: December, 2006
Institution: Oklahoma State University Location: Stillwater, Oklahoma
Title of Study: THEORETICAL STUDIES OF EXCITED STATE ELEC-TRON
TRANSFER BETWEEN IRON-PORPHYRIN AND
TRINITROTOLUENE
Pages in Study: 75 Candidate for the Degree of Doctor of Philosophy
Major Field: Physics
Scope and Method of Study: The purpose of this study was to find and better un-derstand
the degradation of trinitrotoluene (TNT) by way of the photocatalyst
iron-tetraphenyl-porphyrin-sulfonate (FeTPPS) by looking at a possible path-way
in which the process might proceed. We studied a smaller iron complex
iron-porphyrin (FeP), due to the complexity of the larger molecule in terms
of size, to determine if future calculations involving FeTPPS could be handled
more effectively using the much smaller FeP. We used various levels of theory
in order to determine the viability of the process studied, and made use of our
computer cluster to carry out the necessary calculations.
Findings and Conclusions: Using FeTPPS as a photocatalyst has been shown to de-grade
or deactivate TNT experimentally. The analysis of our FeP system has
given us the theoretical excited state spectrum which is in good agreement with
that of the experimental spectrum. We calculated the different multiplets of the
iron complexes and found near degeneracies amongst the ground states. This
analysis shows a mixing of the multiplets in the system and that the excitation
of the iron complexes utilizes both the triplet and the quintet states. With the
good agreement between the theoretical results and the experimental results we
found that using the smaller iron complex FeP, we can account for the same
behavior as that of the larger iron complex FeTPPS. We then analyzed the ther-modynamics
of the process to determine the if the degradation process would
run. We did find that the pathway studied is certainly a possibility in which
the process can take. Therefore, we showed that we can use the smaller iron
complex to model a similar but larger iron complex, and that the process is
thermodynamically favored with the pathway we studied.
ADVISOR’S APPROVAL Dr. Timothy M. Wilson