dottorato di ricerca in nanoscience and …modiﬁcation of ruthenium bipyridyl complex with a...

DOTTORATO DI RICERCA IN

NANOSCIENCE AND NANOTECHNOLOGY

XXII CICLO

Sede Amministrativa

Universita degli Studi di MODENA e REGGIO EMILIA

TESI PER IL CONSEGUIMENTO DEL TITOLO DI

DOTTORE DI RICERCA

Time Dependent DFT Investigation of

Optical Properties and Charge Dynamics

in Light-Harvesting Assemblies

Candidato: Nicola Spallanzani

Relatori: Prof.ssa Franca Manghi

Dr. Carlo Andrea Rozzi

a Chiara e Greta

i

ii

As far as the laws of mathematics refer to reality, they are not certain;

and as far as they are certain, they do not refer to reality.

A. Einstein, “Geometry and Experience”, January 27, 1921.

Contents

1 Introduction 1

1.1 Third Generation Photovoltaics . . . . . . . . . . . . . . . . . 3

2 Time-Dependent Density Functional Theory 11

2.1 An introduction to DFT . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 The Hohenberg-Kohn Theorem . . . . . . . . . . . . . 12

2.1.2 Degenerate Ground States and v-Representability . . . 14

2.1.3 The Kohn-Sham Equations . . . . . . . . . . . . . . . 15

2.2 Excited States . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.1 The Runge-Gross Theorem . . . . . . . . . . . . . . . . 18

2.2.2 Time Dependent Kohn-Sham Equations . . . . . . . . 21

2.3 Photo-absorption Spectra . . . . . . . . . . . . . . . . . . . . 22

3 Carotenoid-Porphyrin-C60 Molecular Triad 25

3.1 Photochemistry of Carotenoporphyrin-Fullerene Triads . . . . 30

3.2 Why TDDFT? . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Ground State 35

4.1 Geometry Optimization . . . . . . . . . . . . . . . . . . . . . 35

4.2 Convergence Tests . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3 Ground State Electronic Structure . . . . . . . . . . . . . . . 39

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 Excited States 45

5.1 Propagation Scheme . . . . . . . . . . . . . . . . . . . . . . . 45

5.2 Photoabsorption cross-section . . . . . . . . . . . . . . . . . . 46

iii

iv CONTENTS

5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6 Charge Transfer Dynamics 53

6.1 Photo-induced Charge SeparationMechanism . . . . . . . . . . 54

6.2 Simulation of the Charge Transfer Process . . . . . . . . . . . 56

6.3 Charge-density evolution . . . . . . . . . . . . . . . . . . . . . 57

6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Bibliography 68

Chapter 1

Introduction

The quest for efficient ways to exploit renewable energy sources has become

one of the key problems of our times. The idea of a solid-state device capable

of transforming solar light energy into electric potential energy is quite old,

and enormous progress has been made in optimizing traditional junction-

based devices with respect to their thermodynamical efficiency and produc-

tion cost [1].

Figure 1.1 shows the energy band diagram of a standard p–n junction

Figure 1.1: Energy band diagram of a p–n junction with energy conversion

loss processes.

1

2 Chapter 1. Introduction

Figure 1.2: Solar electricity costs as function of module efficiency and cost.

solar cell. The right-hand side of the figure represents the n-type region of

the device where a semiconductor is doped by adding some amount of an

element with more valence electrons than the semiconductor. The left-hand

side represents a p-type region obtained by doping the same semiconductor

with an element with less valence electrons. When these two types of doped

semiconductors are in contact, once the equilibrium is reached the junction

is to behave as an elementary photovoltaic unit. As shown in Figure 1.1, a

photoexcited electron in the p side of the junction is under the influence of

an electrochemical potential gradient that moves the electron to the n side.

The n-type region is well predisposed to accept electrons due to the presence

of the dopant that yields a conduction band where electrons can flow easily

to and from cells contacts. The mobility of the charge carriers produces the

current flow.

Figure 1.2 shows the fully amortized cost of electricity as a function of

the efficiency and cost of an installed photovoltaic module for several classes

of devices. The single-band gap solar cell modules, like the one described

above, lie in zone I, whereas zones II and III are related to second and third

1.1. Third Generation Photovoltaics 3

generation devices respectively. The theoretical efficiency limit for even an

optimal single-band gap solar conversion device is 31%, because photons with

an energy below the band gap of the absorber material cannot generate a

hole-electron pair, and so their energy is not converted to useful output and

the absorbed energy is dissipated into hat. Photons with an energy larger

than the band gap energy rapidly release heat to the ions lattice and only a

fraction of the energy can be converted into useful electric potential. In addi-

tion a carrier able to reach the p-n junction could lead to the recombination

process, since one electron and one hole recombine and thereby annihilate

the associated free charge. The carriers that recombine do not contribute to

the generation of electrical current.

Recently with the up-coming of nanosciences and nanotechnologies, the

study and the control of matter at an atomic and molecular scale (generally

structures of the size 100 nanometers or smaller), entirely new classes of

materials have been proposed as building blocks for next-generation solar

cells. Among these nanostructured semiconductors, organic-inorganic hybrid

and supra-molecular assemblies appear to be the most promising [2].

The problems concerning the manipulation of matter at nano size were

first illustrated by the physicist Richard Feynman in the famous lecture

“There’s Plenty of Room at the Bottom” at an American Physical Society

meeting at Caltech on December 29, 1959. He pointed that as the sizes get

smaller, we would have to reconsider the physics of the system because the

relative strength of various forces would change. Gravity would become less

important, surface tension would become more important as well as Van der

Waals attraction, etc. This opened the doors to a new science dedicated at

the study of the “nano-world”, the nanoscience.

1.1 Third Generation Photovoltaics

In green plants photosynthesis light-harvesting complexes, systems that ab-

sorb photons with very high efficiency, are able to capture solar energy and

to store it in a charge-separated state. The solar energy is first used to create

an excited state of the molecule, and it is later transferred in a reaction cen-

ter. This reaction center is composed by several electron acceptor molecules


(a) (b)

Figure 1.3: (a) Gold nanoparticle functionalized with pyrene (from Ref. [2]).

(b) Binding of fluorophore-linked gold nanoparticles to nanostructured film

using a bifunctional surface modifier (from Ref. [3]).

disposed in a chain. The first molecule transfers its excited electron to the

next creating an electron transport chain. The solar energy is thus converted

into potential energy in terms of charge-separation.

Organic-Inorganic Hybrid Assemblies

Organic-inorganic nanohybrids can be developed by mimicking the photoin-

duced electron-transfer process of natural photosynthesis [3], tailoring the

optoelectronic properties of metal nanoparticles. This nanohybrids are made

by assembling monolayers of organic molecules containing functional groups,

such as amines, thiols, isothiocynate and silanes, on three-dimensional surface

of metal nanoparticles, such as gold nanoparticles. Such monolayer protects

the metal cluster and can be functionalized with a chromophore monolayer.

This type of spherical shape hybrid clusters exhibit efficient light-harvesting

capability and suppresses undesirable energy transfer quenching of singlet-

excited-state of the chromophores due to the fact that they are densely packed

on metal nanoparticle surfaces (Figure 1.3(a)).

The nonmetallic property of metal nanoparticles can also be utilized to

accept and then to transfer the electrons from an excited sensitizer to another

acceptor molecule bound to the surface [3]. For example, a gold-nanoparticle


(a) (b)

Figure 1.4: Design of semiconductor-metal nanocomposites using (a) coupled

(from Ref. [4]) and (b) core–shell (from Ref. [5]) geometry.

could behave like a bridge for a chromophore and a TiO2 surface, in which

a bifunctional surface-linking molecule such as mercaptopropionic acid was

used to link the gold nanoparticle to the TiO2 surface (thiol group to gold

and carboxylic group to TiO2) (Figure 1.3(b)).

Semiconductor/Metal Nanocomposites

Of particular interest is the use of semiconductor nanostructures for solar

hydrogen production by the photocatalytic splitting of water. An example

are the TiO2–Gold composite nanoparticles [4], where the gold nanoparticles

capped with organic molecules exhibit unusual redox activity by readily ac-

cepting electrons from the charged semiconductor TiO2 nanostructure. The

Au-mediated electron transfer to an electron acceptor, such as C60, can be

followed by introducing Au nanoparticles to UV-irradiated TiO2 suspension

(Figure 1.4(a)).

This type of coupled metal–semiconductor structure is obtained by expos-

ing both metal and oxide surfaces to reactants, products, and the medium.

Corrosion or dissolution of the noble metal particles during the operation of

photocatalytic reaction is problematic. A better synthetic design would be

to employ the metal as a core and a semiconductor photocatalyst as a shell

[5]. For example, the Ag@TiO2 core–shell clusters are able to store electrons

under UV irradiation and discharge them on demand in the dark (Figure


(a) (b)

Figure 1.5: (a) Schematic of operation of a dye-sensitized electrochemical

photovoltaic cell (from Ref. [6]). (b) Ru-complex adsorbed on TiO2 nanopar-

ticle (from Ref. [7]).

1.4(b)).

Photoinduced charge separation in a semiconductor nanoparticle can be

greatly improved by coupling it with another semiconductor particle having

favorable energetics. By selecting a small band gap semiconductor, one can

also harvest photon in the visible, and by controlling the particle size of quan-

tum dots, one can vary the energetics of the particles. Opportunities exist to

develop ordered assemblies of small and large band gap semiconductors and

harvest photons over a wide spectral range of the visible and infrared light

with better charge separation.

Dye-Sensitized Solar Cells

The first example of solar cells in which the units responsible of the photo-

excitation and of the charge collection are separated, are the so-called dye-

sensitized (or Gratzel) solar cells [8, 6] (Figure 1.5(a)). The dye-modified

semiconductor films provide an efficient method to mimic the photosynthetic

process. The charge separation in this case is facilitated by a semiconduc-

tor particle. In this devices the nanostructured TiO2 films modified with

a ruthenium complex (Figure 1.5(b)) exhibit photoconversion efficiencies in


the range of 11%, which is comparable to that of amorphous silicon-based

photovoltaic cells. When the electrode is illuminated with visible light, the

sensitizer molecules absorb light and inject electrons into the semiconductor

particles. These electrons are then collected at the conducting glass surface

to generate anodic photocurrent. The charge injection from excited sensitizer

into semiconductor nanoparticles is an ultrafast process occurring on the time

scale of femtoseconds to nanoseconds. The redox couple (e.g., I−3 /I−) present

in the electrolyte quickly regenerates the sensitizer. The iodide is regenerated

in turn by the reduction of triiodide at the counter-electrode, the circuit be-

ing completed via electron migration through the external load. The voltage

generated under illumination corresponds to the difference between the Fermi

level of the electron in the solid and the redox potential of the electrolyte.

Overall the device generates electric power from light without suffering any

permanent chemical transformation.

Recently this type of cells have been subject of several studies toward fine-

tuning the performance of electrode structure, pretreatment of TiO2 surface,

modification of ruthenium bipyridyl complex with a variety of functional

groups, regenerative redox couples, and electrolyte medium. Despite this

large body of work, the maximum attainable efficiency has remained in the

range of 10-11%.

Donor-Acceptor Assemblies

In the class of supra-molecular assemblies the donor-acceptor dyads and tri-

ads are object of careful experimental study. In fact they are considered an

ideal mimic for the primary photo-synthetic process, which basically consists

of a photo-excitation followed by a charge-transfer between the component

units [9]. The research for ideal components of light-harvesting dyads has

led to porphyrin-C60 assemblies [10, 11] (Figure 1.6(a)).

Porphyrin molecules show extensive absorption of visible light, and an

optical gap of approximately 2 eV and are therefore ideal photo-reaction

centers and electron donors. C60, on its side, is the ideal electron acceptor.

It has a particularly deep triply degenerate LUMO at approximately -4.3 eV,

capable of accommodating six electrons from neighboring molecules. More-


(a) (b)

Figure 1.6: (a) Porphyrin-C60 dyad (from Ref. [9]) and (b) oligo(p-

phenylene vinylene)-C60 dyad (from Ref. [14]).

over the highly conjugated nature of its bonds, together with a cage radius

of 3.45 A, stabilizes the extra charges in a very efficient way, so that little

repulsive forces are felt by the excess electrons.

It is interesting the use of a porphyrin-C60 dyad as a light receptor in

the spectral sensitization of nanostructured tin oxide semiconductor (SnO2)

electrode [12, 13]. In the sensitization of wide band gap semiconductors,

the energy difference between the conduction band edge of an n-type semi-

conductor film and the oxidation potential of an excited dye present as an

adsorbate provides a driving force for interfacial state charge injection.

From a study of the photo-induced electron transfer process in poly-

mer/fullerene blends [15] derives the idea of oligo(p-phenylene vinylene)-

fullerene dyads [14] (Figure 1.6(b)). The most relevant difference is that,

for covalently bound dyads, intramolecular energy and electron transfer pro-

cesses compete on short timescales after initial excitation. In fact, intramolec-

ular photoinduced charge transfer in these dyads in solution occurs via a two-

step mechanism that involves an initial photoinduced intramolecular singlet-

energy transfer from the oligomer to the fullerene moiety, followed by an

intramolecular charge transfer reaction. But, a second important difference

with polymer/fullerene blends is that the lifetime of the chargeseparated

state in a dyad in solution is much shorter.

A further step toward solar cells based on these components consisted in

transforming the donor-acceptor unit into a donor-bridge-acceptor unit. This

switching from dyads to triads has the main purpose of obtaining a better


charge separation and a longer life time of the final (excited) state, that will

be the subject of the present study.

Outline

The thesis is organized as follows: in the first part of chapter 2 is introduced

the basic formalism of the Density Functional Theory (DFT) illustrating

the scheme proposed by Kohn and Sham that is based on the Hohenberg-

Kohn theorem. Then, in the second part of the chapter, we show how to

extend the theory on time dependent fenomena. In fact Time-Dependent

Density Functional Theory (TDDFT), based on the Runge-Gross theorem,

allow us to describe time dependent fenomena like photon-absorption and

quantum dynamics. In chapter 3 is introduced the system studied with a

large overview of the photochemistry of Carotenoporphyrin-Fullerene Triads

from an experimental point of view. From chapter 4 it starts the description

of our calculations with the discussion of the results. In particular the chapter

4 is focused on the ground state properties of the system while chapter 5

introduce the study of the excited states with the analysis of the photo-

absorption spectra. The thesis concludes in the chapter 6 with the most

relevant novelty, the study of the charge transfer dynamics that is the crucial

step in the realization of third-generation photovoltaics.

Chapter 2

Time-Dependent Density

Functional Theory

The aim of this chapter is to introduce the basic formalism of the Density

Functional Theory (DFT) and its time-dependent extension (TDDFT). We

start section 2.1 with the statement of two important theorems that are the

foundation of the DFT. In section 2.2 we illustrate how to extend the basic

idea of DFT to time-dependent phenomena.

2.1 An introduction to DFT

The problem of the determination of ground state properties of a quan-

tum many-body system is formulated in terms of the solution of the static

Schrodinger equation. This procedure is applicable only for small systems.

As pointed by W. Kohn in his Nobel Lecture, in January 28 1999, for systems

with a number of interacting electrons N > O(10), we have a so called “ex-

ponential wall” that makes the traditional wavefunction method unfeasible.

The aim of DFT is to replace the many-body wave-function, a very com-

plex mathematical object that depends on 3N variable, with a simple func-

tion, the charge density, that depends solely on the three spatial coordinates.

The advantage is clear in particular from a computational point of view. In

this section we illustrate the scheme proposed by Kohn and Sham that is

based on the Hohenberg-Kohn theorem.

11

12 Chapter 2. Time-Dependent Density Functional Theory

2.1.1 The Hohenberg-Kohn Theorem

The Hohenberg-Kohn (HK) theorem [16] states that the ground state expec-

tation value of any observable O is a unique functional of the exact ground

state density n(r)

〈Ψ[n]|O|Ψ[n]〉 = O[n] .

To prove the HK theorem we start considering a many-fermion system

characterized by a non-relativistic time-independent Hamiltonian H, com-

posed by a kinetic contribution T , a Coulomb (two-particle interaction) con-

tribution W , and an external potential V that belongs to a set V of local

one-particle potentials

H|Ψ〉 = (T + W + V )|Ψ〉 = Egs|Ψ〉 , V ∈ V . (2.1)

The states |Ψ〉 are collected in a set Ψ of antisymmetric ground state

wavefunctions, that solves the fermions eigenvalue problem, and defines the

surjective map

C : V → Ψ .

Calculating the ground state density

n(r) = N∑

σ2...σN

∫

dr2 . . .∫

drN |Ψ(rσ, r2σ2, . . . , rNσN )|2 ,

where N is the number of fermions

N =∑

σ

∫

dr3nσ(r) ,

another surjective map is generated collecting the ground state densities in

the set N

D : Ψ → N .

Now we demonstrate that the maps C and D are also injective. For the

map C we consider the eigenvalue problem of a ground state |Ψ′〉

H ′|Ψ′〉 = (T + W + V ′)|Ψ′〉 = E ′gs|Ψ

′〉 , V ′ ∈ V , (2.2)

2.1. An introduction to DFT 13

where the Hamiltonian differs from the one in (2.1) only by the external po-

tential. With the assumption |Ψ〉 = |Ψ′〉 and subtracting the two Schrodinger

equations (2.1) and (2.2) we obtain

(V − V ′)|Ψ〉 = (Egs − E ′gs)|Ψ

′〉 ,

and

V = V ′ + (Egs − E ′gs) . (2.3)

The last equation shows that the potentials differ only by an additive

constant, thus they are equivalent. For the map D we start to calculate the

ground state energy as the expectation value of the Hamiltonian, and for

|Ψ〉 6= |Ψ′〉 we get

Egs = 〈Ψ|H|Ψ〉 < 〈Ψ′|H|Ψ′〉 = 〈Ψ′|H ′ + V + V ′|Ψ′〉

then

Egs < E′gs +

∫

n′(r)[v(r) − v′(r)]d3r .

A similar procedure for E ′gs leads to

E ′gs < Egs +

∫

n(r)[v′(r) − v(r)]d3r ,

and additioning the last two inequalities with the assumption n(r) = n′(r)

we obtain

Egs + E ′gs < Egs + E ′

gs . (2.4)

The contradiction (2.4) proves that the maps C and D are also injective

and thus bijective. In particular the unique inversion of the map D

D−1 : n(r) → |Ψ[n]〉

proofs the Hohenberg-Kohn (HK) theorem.

In particular, if our observables are the kinetic energy and the two-particle

interaction, we can define the functional

FHK [n] = 〈Ψ[n]|T + W |Ψ[n]〉


that is universal, i.e., valid for any number of fermions and does not depend

on the one-particle external potential. The energy functional now can be

written as

Ev0[n] = 〈Ψ[n]|T + W + V0|Ψ[n]〉 = FHK [n] +

∫

d3rv0(r)n(r) (2.5)

and another important statement, of the HK theorem, shows its variational

character: the exact ground state density can be determined by minimization

of the energy functional

E0 = minn∈N

Ev0[n] . (2.6)

2.1.2 Degenerate Ground States and v-Representability

In the case of a system with degenerate ground states we can define a subset

ΨV as a linear combination of the states that solve the eigenvalue problem

(2.1), related to a potential V ∈ V , where the union of all subset gives the

set Ψ. It is also possible to define a subset NV that groups all the ground

state densities related to a potential V ∈ V , where the union of all subset

gives the set N .

In this case the relation C cannot be considered a map, because one

potential now is associated with more then one ground state. Moreover the

map D is now not invertible.

We can overcome the problem proving that (CD) is a proper map and it

is invertible. Kohn [17] showed that the ground state energy corresponding

to the unique external potential V [n] of a given density n(r) is the same for

all the degenerate ground states |Ψi〉 ∈ ΨV

〈Ψi|T + W + V [n]|Ψi〉 = 〈Ψj |T + W + V [n]|Ψj〉 ≡ E .

Thus

FHK [n] = E −∫

n(r)v([n]; r)d3r

is an unique functional of the density and the relations (2.5) and (2.6) are

conserved.


A function n(r) is termed pure-state v-representable if it is the density

of a (possibly degenerate) ground state of the Hamiltonian (2.1) with some

suitably chosen local external potential v(r) (the number of particles, N , and

their mutual interaction w(r, r′) being specified). But not all “reasonably

well behaved” non-negative functions are pure-state v-representable. A first

extension of the domain of the functional FHK [n] was proposed by Levy and

further investigated by Lieb [18] and Levy [19]. The functional is defined as

FLL[n] = infΨ→n

〈Ψ|T + W |Ψ〉 ,

where the notation Ψ → n indicates that the minimum must be searched

over all antisymmetric, normalized N -particle functions Ψ(r1σ1, . . . , rNσN )

whose density equals the prescribed function n(r). As a consequence of the

Rayleigh-Ritz principle FLL[n] = FHK [n] for all pure-state v-representable

functions n(r), thus

Ev0[n] = FLL[n] +

∫

n(r)v0(r)d3r .

2.1.3 The Kohn-Sham Equations

The HK theorem establishes a one-to-one correspondence between a given

local external potential v(r) and the ground state density n(r) of a N -fermion

system. Now we need a scheme to make the theorem suitable for solving

actual quantum-mechanical problems.

To achieve this Kohn and Sham introduce an auxiliary system of N non-

interacting particles denoted with the letter “s” and described by the same

density as the interacting one n(r) = ns(r).

Following the HK theorem there exists a potential vs(r) generated by a

given density n(r), thus the single-particle orbitals obtained by solving the

Schrodinger equation(

−h2

2m∇2 + vs(r)

)

ϕi(r) = εiϕi(r) (2.7)

with

n(r) =N∑

i=1

|ϕi(r)|2 , (2.8)


are unique functional of the density, ϕi(r) = ϕi([n]; r) as well as the non-

interacting kinetic energy

Ts[n] =N∑

i=1

∫

ϕ∗i (r)

(

−h2

2m∇2

)

ϕi(r)d3r .

Now considering an interacting system, with external potential v0(r) and

ground state density n0(r), we can rewrite its total energy functional in terms

of the non-interacting kinetic energy

Ev0[n] = Ts[n] +

∫

d3rv0(r)n(r)

+1

2

∫ ∫

d3rd3r′n(r)w(r, r′)n(r′) + Exc[n] ,

where the exchange-correlation (xc) functional is defined as

Exc[n] = FHK [n] −1

2

∫ ∫

d3rd3r′n(r)w(r, r′)n(r′) − Ts[n] .

Defining the xc potential as

vxc([n0]; r) =δExc[n]

δn(r)

∣

∣

∣

∣

∣

n0

we can obtain the auxiliary potential, the so called Kohn-Sham potential,

vs,0(r) = v0(r) +∫

d3r′w(r, r′)n0(r′) + vxc([n0]; r) . (2.9)

Equations (2.7) (2.8) and (2.9) allow us to apply the Kohn-Sham theorem:

the exact ground state density n(r) of an arbitrary interacting system can

be obtained by the self-consistent solution of the following set of equations

(

−h2

2m∇2 + v(r) +

∫

d3r′w(r, r′)n(r′) + vxc([n]; r)

)

ϕi(r) = εiϕi(r)(2.10)

with

n(r) =N∑

i=1

|ϕi(r)|2 .


Finally the ground state total energy of the interacting system is given

by

E0 =N∑

i=1

εi −1

2

∫ ∫

d3rd3r′n(r)w(r, r′)n(r′)

+Exc[n] −∫

vxc([n]; r)n(r)d3r .

This expression contains no approximations and it is able, in principle,

to predict all the ground state properties of a system provided we can write

the exact exchange and correlation functional expression.

Unfortunately we don’t know the exact formulation of the functional

Exc[n], in all real-world applications approximations are therefore always

required. In the present thesis we will use the oldest and most widely used

approximation, the local density approximation (LDA). LDA is based on the

concept that the electron density can be considered locally constant, thus

the xc energy is written as

Exc[n]LDA =∫

d3r εHEG(n)|n=n(r) ,

where εHEG(n) is the xc energy per unit volume of the homogeneous electron

gas. From this expression the xc potential is readily found as

vLDAxc ([n]; r) =

δ

δnεHEG(n)|n=n(r) .

For systems with a truly slowly varying density one can expect reasonable

results from the LDA. On the other hand one finds that rather favorable

results are also obtained for strongly inhomogeneous systems when the KS

scheme is employed with an xc potential derived from ELDAxc [n]. The method

works surprisingly well in many cases in spite of some deficiencies such as

the inadequate cancellation of self-interaction contributions in the long-range.

As a consequence of this drawback, for instance, the local xc potential does

not exhibit the correct asymptotic behavior proportional to 1/r for localized

systems.


One reason for the success of the LDA seems to be the fact that the xc

energy, if expressed in terms of an xc hole xc(r, r′) as

Exc[n] =1

2

∫

d3r∫

d3r′n(r)w(r, r′)xc(r, r′) ,

is determined by the spherical average of xc(r, r′) alone. The xc hole in LDA

is spherical and appears to be a good representation of the spherical average

of the true xc hole, in part due to a rather subtle cancellation of errors from

the exchange and the correlation parts.

2.2 Excited States

This section is dedicated to the study of excited states. The simplest approx-

imation in the calculation of the excited energies is to take the differences

between the ground state KS eigenvalues. However is important to empha-

size that the KS eigenvalues, obtained from (2.10), do not have a physical

meaning, thus this procedure is not entirely justifiable. In particular, as

mentioned in section 2.1, with DFT we have a scheme to obtain the expec-

tation value of a ground state observable as a functional of the ground state

density. Thus we are not able to predict correctly the expectation value of

a time dependent observable only with the stationary density. Several tech-

niques have been proposed to overcome this problem, and one of the most

successful for the calculation of excitations in finite system is the TDDFT.

In this section we state the fundamental theorem of TDDFT, and outline a

procedure to extract absorption spectra from the solution of TDDFT equa-

tions. Application to the real-world system of interest here is presented in

Chapter 5.

2.2.1 The Runge-Gross Theorem

The Runge-Gross (RG) theorem [20] is the time dependent extension of the

HK theorem. It states that, given the initial state of a many-fermion system,

there is a one-to-one correspondence between the time dependent external

potential and the time dependent density

v(r, t) ↔ n(r, t) .

2.2. Excited States 19

The proof of the RG theorem, requires the demonstration that if two

potentials, v(r, t) and v′(r, t), differ by more than a purely time dependent

function c(t), they cannot produce the same time dependent density n(r, t).

For simplicity we consider external potentials that are Taylor expandable

with respect to the time coordinate around the initial time t0

v(r, t) =∞∑

k=0

ck(r)(t − t0)k ,

where

ck(r) =1

k!

∂k

∂tkv(r, t)

∣

∣

∣

∣

∣

t=t0

.

Now we define the function

uk(r) =∂k

∂tk[v(r, t) − v′(r, t)]

∣

∣

∣

∣

∣

t=to

, (2.11)

where

∃k≥0 : uk(r) 6= constant

is true for the initial assumption v(r, t) 6= v(r, t) + c(t). We start to demon-

strate this first showing that the current densities, j and j′, generated by v

and v′, are also different. In second step we prove that j 6= j′ implies n 6= n′.

The equations of motion for the current density in the primed and un-

primed systems are

id

dtj(r, t) = 〈Ψ(t)|

[

j(r), H(t)]

|Ψ(t)〉

id

dtj′(r, t) = 〈Ψ′(t)|

[

j(r), H ′(t)]

|Ψ′(t)〉 .

Using the same initial state, at t0 the wave-functions, the densities, and the

current densities are equal in the primed and unprimed systems. Then taking

the difference between the lasts equations for t = t0 we obtain


id

dt[j(r, t) − j′(r, t)]t=t0 = 〈Ψ0)|

[

j(r), H(t0) − H ′(t0)]

|Ψ(t)〉

= 〈Ψ0)|[

j(r), v(r, t0) − v′(r, t0)]

|Ψ(t)〉

= in0(r)∇[v(r, t0) − v′(r, t0)] .

The right-hand side of the equation differs from zero by assumption, and the

two current densities will consequently deviate for t > t0, as well as, using

the function (2.11) for k > 0,

dk+1

dtk+1[j(r, t) − j′(r, t)]t=t0 = n0(r)∇uk(r) .

This again implies that j(r, t) 6= j′(r, t) for t > t0.

Now we will make use of the continuity equation

∂

∂tn(r, t) = −∇ · j(r, t)

for the primed and unprimed systems and taking the difference, we arrive

at

∂

∂t[n(r, t) − n′(r, t)] = −∇ · [j(r, t) − j′(r, t)] .

As before, using the function (2.11) for k > 0, we obtain at t = t0

∂k+2

∂tk+2[n(r, t) − n′(r, t)]t=t0 = −∇ ·

∂k+1

∂tk+1[j(r, t) − j′(r, t)]t=t0

= −∇ · [n0(r)∇uk(r)] .

We conclude the demonstration noting that the last step of the above

equation could not be zero by the assumption uk(r) 6= const, then n 6= n′

that proof the RG theorem.

As well as in the static case we need now a scheme to make the theorem

suitable for solving actual time dependent quantum-mechanical problems.

2.2. Excited States 21

2.2.2 Time Dependent Kohn-Sham Equations

The time dependent KS equations are an exact reformulation of the time

dependent Schrodinger equation

i∂

∂tϕi(r, t) =

[

−∇2

2+ vKS(r, t)

]

ϕi(r, t) , (2.12)

where vKS(r, t) is the potential operating on an auxiliary system of non-

interacting fermions, and the density of the interacting system is obtained

from the KS orbitals

n(r, t) =occ∑

i

|ϕi(r, t)|2 . (2.13)

We can rewrite the KS potential as the sum of three terms

vKS = vext(r, t) + vH(r, t) + vxc(r, t) ,

where, vext(r, t) is the external potential, vH(r, t) is the classical electrostatic

interaction, the so called Hartree potential, and vxc(r, t) is the Exchange and

Correlation potential.

This last term in DFT is formally obtained as functional derivative of

the Exchange and Correlation energy. But since we don’t know its exact

expression, we are obliged to introduce approximations.

Time-dependent exchange-correlation functionals

The exact xc potential has a functional dependence on n(r, t) that is nonlocal

in space and time, i.e., contains information about the previous history of

the system, including its initial state. However, almost all present applica-

tions of TDDFT employ an adiabatic approximation, ignoring all functional

dependence of the xc potential on prior time-dependent densities.

In fact we can obtain the time dependent xc potential using the so called

adiabatic LDA (ALDA) that considers the xc potential as a functional local

in time

vALDAxc (r, t) = vLDA

xc [n](r)|n=n(r,t) .


For most applications, especially regarding finite systems with no memory

effects the use of adiabatic functionals is well justified. Major failures are

on the other hand evident when ALDA is applied to infinite systems. For

systems where the electron density is not slowly varying it is necessary to go

beyond ALDA; for a discussion of the role of non adiabatic approximations

see for example Refs. [21, 22, 23].

2.3 Photo-absorption Spectra

We conclude this chapter showing how the TDDFT allows us to obtain photo-

absorption spectra by simplifying the perturbation theory or by propagating

in real-time the time dependent KS equations.

Linear Response Theory

From perturbation theory the quantity that allows us to calculate the optical

absorption spectrum is the linear change of the density, i.e. the response

of the system when perturbed by a weak time dependent external electric

potential δvext(r, t). At the first order of a perturbative expansion the time

dependent perturbation of the density is related to the perturbing potential

by the following expression

δn(r, ω) =∫

dr′χ(r, r′, ω)δvext(r′, ω) ,

where χ(r, r′, ω) is the linear density-density response function of the system.

Unfortunately, the evaluation of χ through perturbation theory is a very

demanding task. TDDFT is able to simplify this process calculating the

induced density using the KS auxiliary system

δn(r, ω) =∫

dr′χKS(r, r′, ω)δvKS(r′, ω) ,

where χKS(r, r′, ω) is the density-density response function of a system of

non-interacting electrons. Thus applying first order perturbation theory to

KS equations (2.10), χKS can be expressed in term of the ground state KS

eigenvalues εi and eigenfunctions ϕi

χKS(r, r′, ω) =∑

ij

(fi − fj)ϕi(r)ϕ

∗j(r)ϕj(r

′)ϕ∗i (r

′)

ω − (εi − εj) + iη,

2.3. Photo-absorption Spectra 23

where fi is the occupation number of the KS orbitals.

We can now define the dynamical polarizability in terms of the linear

response function as

αij(ω) = −∫ ∫

drdr′xiχKS(r, r′, ω)xj .

Finally the photo-absorption cross-section is proportional to the imaginary

part of the dynamical polarizability averaged over the three spatial directions

σ(ω) =4πω

c

1

3ℑ[Trαij(ω)] , (2.14)

where c is the velocity of light.

Full solution of the KS equations

If we don’t want to restrict the perturbation to a weak field, or if we want to

study both linear and non-linear response functions, a very efficient method

consists in the direct solution of the time dependent KS equations (2.12).

In Quantum Mechanics a time dependent problem is mathematically defined

as an initial value problem. Using as starting point the KS ground state,

obtained from a DFT calculation, we can perturb it with an instantaneous

electric field δυext = −k0xνδ(t), where xν = x, y, z. All frequencies of the

system are excited with equal weight, and at time t = 0+ we have

ϕi(r, t0+) = eik0xν ϕi(r) ,

where ϕi(r) are the KS ground state wave-functions. Now the orbitals are

propagated in time using the KS time-evolution operator U

ϕi(r, tf) = U(tf , t0+)ϕi(r, t0+) = T exp

[

−i∫ tf

t0+

dτHKS(τ)

]

ϕi(r, t0+) .(2.15)

Once the desired propagation time is reached we can obtain the change

of the density through equation (2.13) as

δn(r, t) = n(r, t) − n(r) , (2.16)


where n(r) is the ground state density of the system. With a Fourier transfor-

mation in frequency space of the induced density the dynamical polarizability

can be obtained directly from its definition:

ανν(ω) = −1

k0

∫

d3rxνδn(r, ω) ,

Finally the photo-absorption cross-section can be calculated by equation

(2.14).

Approximation of the time-evolution operator

In order to reduce the error in propagation from t0+ to tf , the time interval

is splitted into smaller sub-intervals of duration ∆t. The orbitals are then

propagated from tn to tn + ∆t, step by step, until to reach tf . The simplest

approximation to the time-evolution operator U is a direct expansion of the

exponential in a power series of ∆t

U(t + ∆t, t) ≈k∑

l=0

[−iH(t + ∆t/2)∆t]l

l!+ O(∆tk+1) . (2.17)

Unfortunately, this expression does not retain the unitarity of the operator,

and it makes propagation unstable. To remedy the defect it is possible to

use the enforced time-reversal symmetry method [24]. This procedure relies

on the fact that we can obtain the state at time t + ∆t/2 either by forward

propagating the state a t by ∆t/2, or by backward propagating the state at

t + ∆t. In fact we can write

ϕ(t + ∆t/2) = U(t + ∆t/2, t)ϕ(t)

= U(t − ∆t/2, t + ∆t)ϕ(t + ∆t) ,

that imposing the inverse relation U−1(t + ∆t, t) = U(t − ∆t, t) becomes

ϕ(t + ∆t) = U(t + ∆t/2, t + ∆t)U(t + ∆t/2, t)ϕ(t) . (2.18)

The whole procedure amounts to the following steps: i) obtain an estimate of

KS wave-functions at time t+∆t using a “low quality” formula (for example

the expression (2.17) expanded to third or forth order); ii) construct an

approximation to H(t + ∆t) and U(t + ∆t/2, t + ∆t); iii) apply the equation

(2.18). This procedure leads to a very stable propagation.

Chapter 3

Carotenoid-Porphyrin-C60

Molecular Triad

In this thesis we study the electronic structure, the absorption spectrum, and

the charge dynamics of the Carotenoid-Porphyrin-C60 supra-molecular triad

(abbreviated hereafter as triad or C-P-C60), depicted in Figure 3.1.

Figure 3.1: Structure formula for C-P-C60.

As mentioned in Chapter 1, this type of system is considered an ideal

mimic for the primary photosynthetic process, that occurs in green plant and

bacterial photosynthetic membrane. In these biological systems, the mem-

brane is composed by a lipid bilayer containing several protein components,

each of which has a specific role in the photo-induced process. An antenna

component captures the solar photon energy. The photo-excitation in the an-

tenna migrates to a reaction center where it is converted to chemical energy

in the form charge separation across the bilayer in a series of steps involv-

25

26 Chapter 3. Carotenoid-Porphyrin-C60 Molecular Triad

ing several donor-acceptor cofactors. The photosynthesis culminates in the

production of ATP, the natural energy storage. Thus the question is: what

are the design requirement for an artificial antenna plus reactor center? A

minimal design includes an electron donor chromophore that absorbs visible

light, and an electron acceptor. In this molecular assembly triad a covalent

bond joins a diaryl-porphyrin, the donor chromophore, with a pyrrole-C60,

the acceptor, and provides electronic coupling that allows photoinduced elec-

tron transfer. The β-carotenoid is an additional electron donor moiety and

it is joined to the diaryl-porphyrin by an amide connector. The role of the

β-carotenoid is to prolong the lifetime of the charge separated final state by

removing the hole from the porphyrin.

Carotenoids

The carotenoids are compounds made by eight isoprenic units and contain-

ing rings and some conjugated double bonds. They are insoluble in water,

but soluble in fats and in hydrocarbon solvents; all the carotenoids, which

generally have colors from yellow to red, are widespread in nature, both in

plants and animals.

Lycopene is a precursor of carotenoids, from which all those known by

various changes such as reduction of part of the double bonds, the cycliza-

tion, the isomerization of double bonds position and the introduction of oxy-

genated functional groups. Note that in the lycopene all the double bonds are

in trans configuration and that, except at the two extremities of the molecule,

they are all conjugated. This extended system of conjugation is responsible

for the color of these compounds, which are the pigments of many plants.

Figure 3.2: Structure formula for lycopene.

27

The carotene, the pigment in carrots, is particularly interesting because

it is closely related to vitamin A, a diterpenoid essential for growth, health

of the membranes and the view. In carrots are present in abundance two

isomers of carotene, which differ from each other for the position of one of

the double bonds, and both, in biological assays, show strong activities as

vitamin A.

Figure 3.3: Structure formula for β-carotene and α-carotene.

Figure 3.4: Structure formula for vitamin A.

The way in which carotenes derived from lycopene by cyclization of both

ends it is evident. β-carotene, in which all the double bonds are conjugated, it

is more abundant that α-carotene in which one double bond is not conjugated

with the others. The molecule of vitamin A is half of the carotene molecule

and it has an alcoholic function, and, unlike the crystalline carotene that has

an intense red-orange color, it is pale yellow.


Nitrogen Heterocycles

The pyrrole, which has a pentagonal ring structure with nitrogen in one of

the corners, plays an important role in the chemistry of living organisms.

The conversion of light energy supplied by the sun in the energy of chem-

ical bonds of carbohydrates, synthesized by green plants, occurs by means

of compounds known as chlorophylls. The fundamental structural charac-

teristics of chlorophylls is a system of four pyrroles rings held together by

bridges, each of which contains one carbon atom. This ring system, which is

called porphin, is also present in heme, associated with proteins hemoglobin

and myoglobin, which are responsible for the transport and accumulation of

oxygen in body tissues of warm-blooded animals.

Figure 3.5: Structure formula for porphin.

The porphin, with four nitrogen atoms of the rings pyrrole, directed to-

ward the center of its extensive ring system, complexes effectively with metal

ions. In heme the ion is iron (II), in the chlorophylls it is the magnesium

(II), and these metal ions can also complex with additional bonds above and

below the heterocyclic plane. Cyclic porphin systems can have different sub-

stituents on peripheral atoms; at substituted porphins is given, however, the

general name of porphyrins.

The ring system of porphyrins is very stable and has aromatic character;

the extended conjugation present in these compounds is responsible for their

intense blue color. Porphin and heme have 22 π electrons, but only 18 of these

are part of a ring system for which is possible to write resonance structures.

29

Figure 3.6: Resonance structure formula for porphyrins.

Buckminsterfullerene

In 1985, Harold Kroto of the University of Sussex and Richard Smalley of

the Rice University have announced the discovery of stable structures con-

sisting of only carbon atoms [25], different from graphite and diamond. The

more abundant and more stable of the new structures was a C60 molecule

(720 amu), and since then has been the subject of intense research. The C60,

when it is isolated from soot, is a mustard-colored solid, soluble in common

organic solvents, especially in solvents such as benzene and toluene, which

form magenta solutions. C60 has a very high melting points and, although

they are soluble in organic solvents, dissolve very slowly, which shows that

its molecules are packed very well in its respective crystal lattices.

The spectrum of Nuclear Magnetic Resonance (NMR) of carbon-13 of C60

shows a single peak at 142.68 ppm in the aromatic region of the spectrum,

indicating that all the C60 carbon atoms are equivalent to each other. This is

only possible with a highly symmetrical, confirmed by the high melting point

and by compact crystal structure packaging of C60. The proposed structure

is the family of football, whose technical name is truncated icosahedron; in

this structure each carbon atom is at the junction of two esatomic rings and

one pentatomic ring, and each esatomic ring contain three double bonds.

To C60 species has been given the name Buckminsterfullerene in honor of

Buckminster Fuller, inventor of the architectural structures called geometri-

cal domes, but it is more familiarly called “buckyball”. The term fullerenes

has been chosen to indicate the class of structures consisting of penta- and

esatomic rings, for example, the C70 is a fullerene.


Figure 3.7: Buckminsterfullerene (C60).

Chemists know how to easily get the C60, by evaporating graphite with

lasers, and how to purify it, and this has allowed to study the C60 from a

chemical point of view. The studies made so far indicate that the C60 is

not exactly aromatic such as benzene, for example, the NMR peak is more

typical for a tensioned aromatic structure.

Electrophilic compounds shall be added easily to double bonds. Further-

more, from chlorinated C60 it is possible, for nucleophilic substitution, add

aliphatic chains.

3.1 Photochemistry of Carotenoporphyrin-Fullerene

Triads

The observed absorption spectrum of a similar triad in a toluene solution is

shown in Figure 3.8. Superimposed on the bands at 629, 576, and 411 nm

that are caracteristic of the porphyrin is the broad carotene absorption with

maxima at 509, 477, and 448 nm. The weak fullerene absorption throughout

the visible with a long-wavelength maximum at 705 nm is also present, but

the amplitude is too low to be seen in the figure. The absorption spectrum is

essentially a linear combination of the spectra of porphyrin, carotenoid, and

3.1. Photochemistry of Carotenoporphyrin-Fullerene Triads 31

Figure 3.8: Absorption spectra in toluene solution at 292 K of a triad [26]

(—), and a model porphyrin (- - -), carotenoid (− · − · −), and fullerene

(− · ·− · · −). Also shown is a linear combination of the spectra of the model

compounds that approximates the spectrum of the triad (· · ·).

fullerene compounds. Thus, electronic interaction among the chromophores

is too weak to cause significant spectral perturbations. D. Kuciauskas et

al. [26] have observed in benzonitrile and 2-methyltetrahydrofuran small

(≤3 nm) spectral shifts to longer wavelengths.

Steady-state fluorescence emission studies of a triad in polar solvents,

with excitation at wavelengths where the porphyrin and fullerene absorb,

demonstrated very strong quenching of the emission from both the porphyrin

and fullerene chromophores. This is ascribed to rapid photoinduced electron

transfer involving either excited state. Excitation of the carotenoid moiety

did not lead to the observation of any carotenoid fluorescence, as the fluo-

rescence quantum yields of carotenoids of this type are ≤ 10−4. Studies of

carotenoporphyrin molecules have shown that singlet-singlet energy transfer

from the carotenoid to the porphyrin is negligible, and that charge-separated

species that might be generated by carotenoid excitation are not observed.


Figure 3.9: Relevant transient species created after excitation of triad 2 and

interconversion pathways. The energies of the charge-separated states are

estimated from the cyclic voltammetric data obtained in polar solvents, as

reported in Ref. [26]

Thus, the carotenoid is photochemically inert in the singlet manifold of the

triad.

Transient absorption experiments were performed to learn more about

electron-transfer processes in a P-C60 dyad and a C-P-C60 triad by Kuci-

auskas et al. [26]. The samples were excited at 590 nm with 200-fs laser

pulses. Their conclusion were that the first excited singlet state of the por-

phyrin moiety in the dyad, 1P-C60, decays at least in part by photoinduced

electron transfer to yield the P+-C−60 charge-separated state. The same result

was obtained for the triad and, after the formation of C-P+-C−60, the tran-

sient absorption experiments revealed a strong new absorption assigned to

the carotenoid radical cation of the C+-P-C−60 charge-separated state.

The electron-transfer events leading to the final state in the triad are

shown in Figure 3.9. Excitation at 590 nm yields mostly C-1P-C60, which

decays by step 3 to yield C-P+-C−60. The fullerene excited singlet state also

undergoes photoinduced electron transfer to give the same species. Electron

3.2. Why TDDFT? 33

transfer from the carotenoid (step 4) gives the final C+-P-C−60 state. Kuci-

auskas et al. have observed the formation of the carotenoid radical cation as

a function of time measured at 970 nm in 2-methyltetrahydrofuran at 292,

200, 120, and 77 K. They relate formation time constants of 59 ps (292 K),

90 ps (200 K), 247 ps (120 K), and 615 ps (77 K).

The time-scale of the transient absorption experiments discussed above

is too short to allow observation of the decay of the C+-P-C−60 state. There-

fore, transient absorption spectroscopy on the nanosecond time-scale was

employed by Kodis et al. [27] to investigate the long-lived state. The decay

of the absorption maximum at ∼980 nm is fitted by an exponential function

with a time constant of 57 ns. The decay is accompanied by a complemen-

tary exponential rise (fitted by an exponential function with a 57 nm time

constant) of absorbance at 550 nm. Absorbance at this wavelength is due to

the carotenoid triplet state, and its rise indicates the formation of 3C-P-C60

by charge-recombination of C+-P-C−60. The decay of this absorbance can be

observed on an even longer time-scale, and has a time costant of 4.9 µs in

absence of oxygen.

3.2 Why TDDFT?

While many experimental properties of porphyrin-C60 based dyads and tri-

ads are known, less is known from an ab-initio theoretical point of view. The

main reason is that the large size of the systems yields traditional accurate

quantum-chemistry methods such Configuration Interaction (CI) impractical

or infeasible. Less accurate but computationally efficient methods such as

DFT have successfully been employed to describe the ground state of such

compounds [28], but unfortunately, the prediction of excited state properties,

like the optical absorption spectrum, from the ground state Kohn-Sham or-

bitals and eigenvalues usually leads to results that are in disagreement with

the observed spectra. In particular the role of the interfaces between different

parts of the triad and between the molecule and the solvent are not under-

stood in spite of the fact that they are of paramount relevance to the problem

of improving the energy conversion efficiency of next generation solar cells.

TDDFT is a rigorous and in principle exact method that allows the calcu-


lation of excited-states through an extension of DFT to the domain of time-

dependent external potentials [20], and therefore allows us to get a much

better insight on all those properties that are most relevant for the photoab-

sorption process, and the subsequent transfer of charge, that are crucial steps

in any photovoltaic device.

Chapter 4

Ground State

Our first goal is to study the ground state properties of the molecule, and to

compare its electronic structure with the electronic structure of the separated

components. For all our calculations we used an optimized geometry obtained

by Baruah et al. [28]. In this structure the molecule can be entirely embedded

in a box approximately 55 × 16 × 20 A wide, the longest side being along

the x axis. As it is also clear following a steric argument, the porphyrin

ring is perpendicular to the aryl rings, which, in turn, are coplanar with

the carotenoid. The optimized geometry shows a slight bent in the carotene

chain (see Figure 4.1). In the calculations of the separated moieties the same

geometries used in the triad were employed.

4.1 Geometry Optimization

The C-P-C60 system is composed by 207 atoms with 632 valence electrons.

Its structure corresponds to the formula C132H68N6O. The geometry opti-

mization was carried out using the LBFGS scheme [29, 30], by Baruah et

al. [28]. Initially the geometries of the separate units were optimized be-

fore constructing the triad. After forming the composite molecular triad,

its geometry was optimized again. Since the molecule has a quite floppy

structure, two different geometries were optimized. In the first geometry, the

three subunits were attached in such a way that the resultant triad has an

elbow shaped structure. Its energy after optimization turned out to be higher

35

36 Chapter 4. Ground State

Figure 4.1: Orthogonal projections of the optimized geometry of C-P-C60.

From top to bottom: xy, xz and yz projections.

by about 0.6 eV and also the dipole moment for the final charge separated

structure is much smaller. The linear geometry as shown in Figure 4.1 was

found to be lower in energy and all subsequent calculations were carried out

on this geometry. In the optimization stage, the process was carried out until

the smallest force was 0.003 eV/A and RMS (root mean square) force was

0.076 eV/A. It can be pointed out here that NMR (nuclear magnetic reso-

nance) studies on carotenoid-porphyrin dyad shows a structure similar to the

linear one [31]. Similar structure was found by Smirnov et al. as the lowest

energy structure based on a molecular mechanics level calculation [32]. We

4.2. Convergence Tests 37

would like to point out that the molecule has a long chainlike structure and

can exhibit substantial temperature induced structural deformation. Since

our ground-state study is aimed at understanding the zero-temperature state,

such calculations are beyond the scope of present work. On the other hand

the temperature will be significant in the charge transfer calculations (see

Chapter 6).

4.2 Convergence Tests

In a real-space based approach the choice of the grid mesh is a critical param-

eter, and the consistency of the calculations must be checked with a careful

set of convergence tests. A cartesian grid was chosen with uniform spacing of

the mesh. We analyze the convergence of the eigenvalues, of the last 6 occu-

pied states and of the first 4 unoccupied states as a function of the distance

between grid points.

The first test was performed to determine the volume necessary to enclose

the states of the molecules. In the open source code octopus [33, 34] it is

possible to choose a particular box shape that is composed putting a sphere

of radius R over all the atoms. The union of all these spheres makes the

box shape that enclose the molecule. As boundary conditions we imposed

the wave-functions to be zero at the surface of the box. This approximation

is adequate only if the surface of the box is far enough from the system,

because if the box is too small the wave-functions will be forced to go to

zero unnaturally. Moreover if the box is too large, will increase the memory

requirement and the computational time because of the large number of grid

points. The test analyzes the convergence of the states changing the values

of radius sphere, and for all the moieties a good convergence was obtained

for value of R ≥ 4.0 A.

The second test was performed analyzing the energy of the states changing

the value of the mesh spacing. This test allow us to find out the largest value

of spacing for which ground state quantities are expect to converge. Figure

4.2 shows that the convergence of the systems was maintained until a spacing

of 0.26 A.


−7−6−5−4−3−2−1

0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32

ener

gy [e

V]

spacing [Å]

β−carotenoid−7−6−5−4−3−2−1

ener

gy [e

V]

Diaryl−porphyrin−8

−7

−6

−5

−4

−3

ener

gy [e

V]

Fullerene (C60)

Figure 4.2: Plots of the last six occupied orbitals (red lines) and first four

unoccupied orbitals (green-dashed lines) for the three moieties of the triad

at different values of spacing.

4.3. Ground State Electronic Structure 39

4.3 Ground State Electronic Structure

We have thus performed a plain DFT calculation at the LDA level, using

the Perdew-Zunger correlation functional [35]. The core electrons were rep-

resented via Troullier-Martins pseudo-potentials [36], and the Interpolating

Scaling Functions method [37] was used to efficiently solve the Poisson’s equa-

tion in the calculation of the Hartree potential. All the simulations in the

present work, a part from the geometry optimization, were performed using

the open source code octopus on a Linux cluster (IBM BCX/5120) com-

posed on Opteron Dual Core 2.6 GHz processors, maintained by CINECA

Bologna. All the quantities were discretized in real-space using a uniform

mesh with 0.2 A spacing that guarantees the convergence of the total energy

of the system [34]. This value of spacing, coupled with a radius R = 4.0 A

of the atomic spheres that compose the simulation box, gives 862,017 points

on the inner mesh, and 1,287,768 points adding the boundary mesh.

In Table 4.1 we report the ground state energies of the highest occupied

molecular orbitals (HOMOs) and the lowest unoccupied molecular orbitals

(LUMOs) for the isolated moieties and for the full triad. The detailed energy

levels for other orbitals are depicted in Figure 4.3.

Comparing the values of the HOMO and LUMO energies of the triad

with those of the isolated moieties we notice that the energy of the HOMO

in the triad and in the isolated β-carotenoid differ by only 40 meV, as well

as the energy of the LUMO in the triad and in the isolated C60. Moreover,

looking at Figure 4.4 and 4.5 we observe that it is possible to find a close

correspondence between many of the orbitals localized on each molecule,

molecules ǫHOMO (eV) ǫLUMO (eV)

β-carotenoid -4.47 -3.49

Diaryl-porphyrin -5.15 -3.31

pyrrole-C60 -5.89 -4.42

Triad -4.51 -4.38

Table 4.1: Energy of HOMO and LUMO respectively for β-carotenoid,

Diaryl-porphyrin, pyrrole-C60 and the whole triad.


-6.5

-6

-5.5

-5

-4.5

-4

-3.5

-3E

nerg

y [e

V]

βcar Dpor pyr-C60 Triad

Figure 4.3: Ground state energy levels of the triad and of the separated moi-

eties: β-carotenoid (βcar), Diaryl-porphyrin (Dpor), and Pyrrole-C60 (pyr-

C60).

and orbitals of the full triad. In particular the HOMO wave-function of the

triad is localized on the carotenoid, and it corresponds to the HOMO of the

isolated β-carotenoid, while the LUMO of the triad is fully localized on the

C60 and it corresponds to the LUMO of the C60 alone. The same is for the

HOMO-2 and LUMO+4 of triad that correspond respectively to the HOMO

and LUMO of the isolated Diaryl-porphyrin.

In some other cases the correspondence between the localization of or-

bitals remains, but some degeneracies are lifted, for example the triply degen-

erate C60 LUMO is split into three separate levels in the triad. Finally some

orbitals present a clear mixed character (Figure 4.5), such as the LUMO+5,

LUMO+6, LUMO+7 with mixed P-C60 character, or HOMO-6, with mixed

C-P character. These orbitals however do not appear to have a major role

4.4. Conclusions 41

Figure 4.4: Ground state energy levels of Diaryl-porphyrin and Fullerene.

Some orbitals of the two moieties are depicted.

in the photo-excitation process (see below in Chapter 6).

4.4 Conclusions

From the discussion above we conclude that the electronic structure of the

ground state of the triad is well approximated by the overlap of the electronic

structures of the isolate molecules (Figure 4.3). It is noteworthy that this

property is not an artefact of the local density approximation adopted here,

but it is also an experimentally observed property of the system [27]. This

point will become more clear with the analysis of the excited-state properties

in Chapter 5.

In Chapter 1 we illustrated the basic working mechanism of a single-


Figure 4.5: Ground state energy levels of the triad. Some orbitals of the

triad are depicted.

junction cell, and we mentioned that the necessary condition for the electrons

migration is that the valence band of a side of the junction falls into the band

gap of the other side. All the examples of third-generation solar devices

presented in the Chapter 1 are based on the same idea, where the sides

of the junction are replaced by electron donor/acceptor moieties creating

assemblies. In the triad presented here the porphyrin is the donor moiety

and the fullerene is the acceptor moiety.

There are two reason because this type of assembly is considered a good

candidate for third-generation photovoltaics. The first could be detected only

looking the electronic structures of the electron donor, the dyaril-porphyrin,

and of the electron acceptor, the Fullerene. In Figure 4.4 we can notice

4.4. Conclusions 43

the same type of structure of the p–n junction, where the HOMO of the

porphyrin, at -5.15 eV, falls into the HOMO-LUMO gap of the fullerene, gap

of about 1.81 eV, from the HOMO at -6.14 eV to the LUMO at -4.33 eV. Then

the above mentioned necessary condition is fulfilled because the photoexcited

electron can migrate to the almost degenerate LUMO of the C60.

The second important consideration is the presence of mixed P-C60 char-

acter orbitals in the triad at an energy close to the LUMO+4 at -3.39 eV

(see Figure 4.5). This is the state supposed to be the end point of the pho-

toexcitation, and it corresponds to the LUMO of the porphyrin moiety. For

example, the shape of the LUMO+5 and LUMO+6, respectively at only

28 meV and 64 meV above the LUMO+4, suggests the possibility that the

electron migration could be made easier by the presence of these mixed or-

bitals.

As mentioned above, the β-carotenoid is an additional electron donor

moiety. His role will be better explained in Chapter 6.

Chapter 5

Excited States

In this Chapter we show the application of TDDFT to the calculation of the

optical absorption spectra of the triad and of its components. The method

used, illustrated in Section 2.3, is recalled and deepened before the results

are analyzed.

5.1 Propagation Scheme

Starting from the ground state Kohn-Sham orbitals ϕj , the system is in-

stantaneously perturbed by a weak external electric potential of magnitude

k0 along the principal cartesian direction (i.e. by applying the external po-

tential δυext(r, t) = −k0xνδ(t)). The magnitude of the perturbation is kept

small in order to keep the dipolar response linear [34]. In this way all the

frequencies of the system are excited with the same weight. The initially

perturbed states therefore become

ϕi(r, t0+) = eik0xν ϕi(r) .

Now the orbitals are propagated in time using the time-evolution operator

U ,

ϕi(r, tf) = U(tf , t0+)ϕi(r, t0+) = T exp

[

−i∫ tf

t0+

dτHKS(τ)

]

ϕi(r, t0+) ,

45

46 Chapter 5. Excited States

and the new wave-functions are used to solve the time dependent KS equa-

tions

i∂

∂tϕi(r, t) =

[

−∇2

2+ vKS(r, t)

]

ϕi(r, t) .

In order to reduce the error in propagation from t0+ to tf , the time inter-

val is splitted into smaller sub-intervals of duration ∆t. The orbitals are then

propagated from tn to tn + ∆t, step by step, until to reach tf . For our cal-

culations we use, to approximate the operator U , the enforced time-reversal

symmetry method (see Section 2.3). This procedure leads to a very stable

propagation.

The numerical discretization of the propagation equations requires that

a finite difference equation is defined on a discretized grid in both space

and time. This lead to a dependence of the time step on the size of space

mesh. In addition the time-step depends on the method used to numerically

calculate the exponential of the Hamiltonian in the propagation operator.

For the triad we used a fourth order Taylor expansion that forces us to use

a small time-step (1.69 as). For the isolated molecules we use the Lanczos

method [38] that is compatible with a larger time-step (7.89 as), resulting

in a window of approximately 8 eV for the described spectral range. The

KS orbitals were propagated for 15 fs in total, that corresponds to a spectral

resolution of about 0.15 eV.

Finally the dynamical polarizability is obtained from

ανν(ω) = −1

k0

∫

drxνδn(r, ω),

where δn(r, ω) is the Fourier transform of n(r, t) − n(r, t = 0). The photo-

absorption cross section is related to the imaginary part of the polarizability

through the expression

σ(ω) =4πω

c

1

3Im[Tr (α(ω))].

5.2 Photoabsorption cross-section

We start our description of the optical properties of the system analyzing the

absorption spectra of the separated parts of the triad. Figure 5.1 reports the

5.2. Photoabsorption cross-section 47

2 2.5 3 3.5 4 4.5 5 5.5 6

Abs

orpt

ion

cros

s se

ctio

n

Photon energy [eV]

EXPTDDFT

2 2.1 2.2 2.3 2.4

a

b

Figure 5.1: Comparison between theoretical TDDFT optical absorption in

diaryl-porphyrin and observed free-base porphyrin absorption as in [39]. In

the inset the region between 2.0 to 2.4 eV is magnified, and the vertical lines

are the transition energies obtained by linear-response calculations.

comparison between theoretical optical absorption in diaryl-porphyrin and

the observed optical absorption in a free-base porphyrin [39]. The TDDFT

spectra is obtained with a real time propagation. In order to get a plausible

attribution of the single-excitation states involved in the process of photo-

excitation, we have also performed a linear response calculation [40, 41] of

the absorption spectrum on the diaryl-porphyrin moiety.

The spectra obtained by the two methods are quite similar, even if they

are not expected to be identical. In particular the analysis of the absorption

of the diaryl-porphyrin allows us to clarify that, even if an intense shoulder

only appears in the UV region around 3.3 eV, the so called B-band of the por-

phyrin, the actual very weak onset of absorption, the Soret Q-band, is in the

orange-yellow visible region at approximately 2.2 eV. In the inset of Figure


3 3.5 4 4.5 5 5.5 6 6.5

Abs

orpt

ion

cros

s se

ctio

n

Photon energy [eV]

EXPTDDFT

0.5 1 1.5 2 2.5

Abs

orpt

ion

cros

s se

ctio

n

Photon energy [eV]

Figure 5.2: Left: comparison between theoretical TDDFT optical and ob-

served absorption for fullerene as in [42]. Right: comparison between the

β-carotenoid main absorbance peak as observed in [43] and the calculated

one.

5.1, that magnifies the region between 2.0 to 2.4 eV, are reported the transi-

tion energies obtained by linear response calculation. The excitation named

“a” at 2.14 eV is composed at 60% by the transition HOMO→LUMO+1 and

at 38% by HOMO-1→LUMO. The excitation “b” at 2.32 eV is composed at

56% by the transition HOMO→LUMO and at 42% by HOMO-1→LUMO+1.

The remaining 2% for both the excitation is due to other transition with a

not relevant weight. These two transitions were observed also in the spectra

obtained by time propagation (thick line). The non-symmetric shape of the

large peak at about 2.2 eV suggest that a longer propagation could resolve

two peaks.

In Figure 5.2 left, it is shown the comparison between theoretical TDDFT

optical and observed absorption for fullerene as in [42]. In these spectra we

5.2. Photoabsorption cross-section 49

1 2 3 4 5 6 7 8

Abs

orpt

ion

cros

s se

ctio

n

Photon energy [eV]

Triadβcar

Dporpyr-C60

Figure 5.3: Calculated absorption spectrum of the triad, and the isolated

pyrrole-C60, Diaryl-porphyrin and β-carotenoid.

can notice a good correspondence of the peaks, provided a mean red-shift

of about 0.28 eV is taken into account. The right plot shows the compar-

ison between the β-carotenoid main absorbance peak as observed in [43],

at approximately 1.24 eV and the calculated one. Here a blue-shift is visi-

ble of about 0.47 eV. For an in depth discussion about the effect of exciton

confinement in molecular chains see also Ref. [44].

The TDDFT photo-absorption cross-section of the triad and of the sep-

arated molecules is shown in Figure 5.3. We can easily distinguish one dom-

inant peak at 1.7 eV, several small features between 2.0 and 3.0 eV, another

peak at 3.3 with a shoulder at 3.6 eV, and a group of peaks between 4.0 and

8.0 eV.

The decomposition of the spectrum into optical densities of the isolated

moieties clearly shows again that the total spectrum is very well approxi-

mated by the sum of the spectra of the parts. In particular the main feature


0 1 2 3 4 5

Abs

orpt

ion

cros

s se

ctio

n

Photon energy [eV]

EXPTDDFT

DFT

Figure 5.4: Comparison between theoretical and observed optical absorption

for the full C-P-C60. DFT is the calculated spectrum as in [28]. TDDFT

present work. EXP depicts the observed spectrum as reported in [27].

at 1.7 eV appears to be entirely originated by the β-carotenoid, which has

a weak contribution from the rest of the spectral range, apart from a small

feature at 2.3 eV that partially obscures the weak Soret Q-band of the diaryl-

porphyrin. The B-band of the diaryl-porphyrin is the main contribution at

3.2 eV, and the diaryl-porphyrin has another prominent peak in the ultra-

violet region at 6.6 eV. The contribution of pyrrole-C60 counts many peaks,

notably two isolated at 5.3 and 5.9 eV, and several more from 4.0 to 5.0 eV.

The comparison of our calculations with the experimentally observed ab-

sorption in similar triads clearly shows that TDDFT improves on DFT in

describing the main features of the spectra. In Figure 5.4 we compare our

data with the absorption reported in Ref. [27], that refers to a triad in 2-

methyltetrahydrofuran solution. We observe a good one-to-one correspon-

dence between the calculated and the observed peaks, whose positions ap-

5.3. Conclusions 51

pear overestimated in the calculation by an overall rigid shift of about 0.3

eV. We do not investigate here the possible role of the environment, that

might well be responsible for the rigid shift, but is unlikely to qualitatively

change the shape of the spectrum. In fact triads made of slightly different

components, or immersed in different solvents, often have very similar spec-

tra (see Ref. [45]). In particular the prominent porphyrin peak around 3 eV

and the wide band between 2.0 and 3.0 eV seem to be common features in all

the experimental data. The transitions approximately at 2.0 eV corresponds

to the calculated transitions in the diaryl-porphyrin alone, and they are also

responsible for the optical excitation of the triad as a whole.

Also notice that, despite the apparent superposition of the main exper-

imental peak to a peak in the DFT calculation, the nature of the latter is

predicted to be of C60 character, while the corresponding TDDFT feature at

3.5 eV correctly is attributed to the porphyrin (see decomposition in Figure

5.3). The experimental data range in Refs. [27, 45] are limited above approx-

imately 2.0 eV, but as mentioned above, the β-carotenoid absorption does

not appear to be relevant for the photo-conversion in the triad.

5.3 Conclusions

In summary, we have addressed the fundamental problem of describing the

creation of electron-hole pairs and the efficient charge separation in a supra-

molecular assembly of interest for third generation solar devices. Since not

all the effects leading to the final-state charge separation are fully known the

accurate description of the excited states of such an object is a crucial step in

order to understand the details of the photo-absorption and charge-transfer

processes occurring in the system.

We have calculated the optical absorption cross section of such a large

system using TDDFT as a rigorous ground for the excited states dynamics.

In addition to the computational challenge, our calculations demonstrate that

the simple ground-state DFT description of the system is not able to capture

the correct shape of the photo-absorption spectrum. In contrast we have

shown that the main features of the TDDFT spectrum are in good agreement

with the experimental data, and the analysis of the total absorption in terms


of the absorption of the isolated moieties indicates that even at the TDDFT

level (at least in the weak field limit) the component molecules in the triad

do not appear to strongly interact.

Finally, in Figure 4.5 we have shown the energy levels for the ground

state of the full triad, and we have emphasized the two orbitals, HOMO-2

and LUMO+4, localized on the porphyrin moiety. From the analysis of the

electronic structures we supposed that these two orbitals are responsible for

the optical excitation. From the analysis of the spectra of the porphyrin we

find that the orbitals that could be involved in the photo-excitation process

are the HOMO-1, HOMO, LUMO and LUMO+1 of the moiety. This allow us

to think that also in the triad there are other orbitals involved in the photo-

excitation, in particular considering the presence of the orbitals with mixed

P-C60 character. As concluding remarks we must recall that the 15 fs time

span of our simulation is unlikely to account for the complete charge transfer

process, that occurs on a different time scale, but it is able to accurately

describe the photo-excitation of the molecule. Nevertheless we have been

able to get a valuable insight about the charge transfer from the careful

examination of the energy levels and of the calculated absorption spectrum

of the system.

Chapter 6

Charge Transfer Dynamics

In Chapters 4 and 5 we have analyzed respectively the ground state and

optical properties of the triad and of its parts. In this chapter we address

the charge transfer dynamics in the whole triad.

The charge transfer dynamics is a crucial step in the realization of third-

generation photovoltaics devices such as Gratzel or donor-acceptor based

systems, and it happens in a subsequent step respect to the photo-excitation

process. Also the time scales are different.

We stress again the needed features of a good candidate for third-generation

photovoltaics. It should be composed at minimum by two units each with

a different role in the photo-induced charge separation mechanism. The

first unit is a chromophore that absorbs visible light and the second unit is

an electron acceptor capable to provide electronic coupling that allows the

photo-induced electron transfer. The central idea is to separate the light

absorption process from the charge collection process, mimicking natural

light harvesting procedures in photosynthesis. In order to make solar cells

efficient, intense visible absorption in addition to rapid charge transfer are

required. The efficiency is determined by two main factors: i) the maximum

photocurrent density, related to the charge transfered rate from the chro-

mophore (donor) HOMO to the acceptor unoccupied orbitals (probably the

LUMO), and ii) the open circuit potential, related to the energy difference

between the HOMO of the electron donor to the LUMO of the acceptor. In

addition to those, other two features make a device a good candidate, namely

53

54 Chapter 6. Charge Transfer Dynamics

a long mean lifetime of the charge separated state and good photochemistry

stability.

All the properties described above are present in the Carotenoid-Porphyrin-

C60 triad, where an additional unit takes the role of a second electron donor.

The function of the carotene is described in the next sections.

6.1 Photo-induced Charge Separation

Mechanism

Resting on the careful examination of the structure, levels, and orbital local-

ization we concluded in Chapter 4 that the triad has the perfect electronic

configuration for a photo-induced charge separation. This occurs thanks to

the fact that the HOMOs and LUMOs of the moieties are localized in the

respective sub-molecule, and right positioned. In particular the HOMO of

the porphyrin falls into the energy range of the HOMO-LUMO gap of the

fullerene, and the presence of mixed P-C60 character orbitals in the triad at

an energy close to the LUMO of the porphyrin makes the electron migration

easier.

We are now able to achieve a theoretically motivated view of the photo-

excitation process, and of the subsequent charge-transfer that can be sum-

marized in the following chain of steps:

C-P-C60 + hν → C-P∗-C∗60

C-P∗-C∗60 → C-P+-C∗−

60 → C+-P-C−60

The first step corresponds to the optical transition from HOMO-2 to

an orbital between LUMO+4 and LUMO+7. We call this excited state

C-P∗-C∗60. The electron in a previously unoccupied orbital on the porphyrin

is now under the influence of the high affinity of the C60, and it is easily

accommodated on the almost-degenerate orbitals LUMO+1, LUMO+2 at

0.14 eV above LUMO localized on the C60. In the assembled triad the three-

fold degeneracy of the C60 T1u LUMO is broken due to the presence of the

Pyrrole and the LUMO of the triad is a not degenered state. We have

6.1. Photo-induced Charge Separation

Mechanism 55

Figure 6.1: Photo-induced charge separation mechanism scheme, ground

state levels (red occupied, green unoccupied) for the triad and its moieties.

The scheme shows the photo-excitation step and the two steps representing

the electron and hole migration.


then reached a configuration C-P+-C∗−60 . The last step consists in a hole

delocalization from the Porphyrin HOMO to the Carotenoid HOMO to give

the final state C+-P-C−60. Obviously only the study of the full dynamics of

the whole process can support the latter two steps interpretation, which is

inferred here on the basis of the analysis of the energy levels in Figure 6.1,

and on the electrochemical properties of the moieties. Despite the fact that

the lifetime of the final state is estimated to be in the order of tens of ns, it is

possible to get an accurate description of the photo-excitation, and possibly

charge-transfer processes that occur on a much faster time-scale.

As we have already stressed, it is the almost non-interacting character

of the moieties, demonstrated by the shape of the absorption spectrum, and

the nature of the molecular orbitals, that makes it possible to describe this

process in terms of orbitals localized on each component. In fact the charge-

transfer states associated with this molecule are not dipole allowed transitions

due to the fact that the hole on the (donor) carotene and the particle-state

on the C60 (acceptor) molecule are essentially non overlapping.

6.2 Simulation of the Charge Transfer Pro-

cess

In order to focus on the charge transfer process we have chosen as the initial

state for the propagation a single-particle state, presumably close to the

real excited state after a photon absorption. Namely the state obtained by

occupating the LUMO+4 of the triad with one electron, taken from the state

HOMO-2. As mentioned above these two orbitals are almost fully localized

on the porphyrin. This excitation corresponds to an energy increase of about

1.85 eV.

To reduce the computational time for this demanding task, we decided

to use a mesh spacing of 0.24 A, an acceptable value that guarantees the

convergence of the total energy of the system (see Section 4.2 about the tests

performed to obtain a good simulation grid). In addition we increase the

radii, of the spheres that compose the volume of the simulation box, from

4.0 A to 5.0 A. This to be sure to contain the orbitals also for long time

6.3. Charge-density evolution 57

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70

Ele

ctro

nic

Cha

rge

[ele

ctro

ns]

time [fs]

a b

c d

pyrrole-C60Carotenoid-Porphyrin

Figure 6.2: Fraction of electrons transfered to the pyrrole-C60 part (black

line) from the Carotenoid-Porphyrin part (cyan line) of the triad as a function

of time after excitation.

propagations.

We have performed calculations both with clamped and mobile ions, in

order to investigate the role of ion-electron dynamics coupling in the charge-

transfer process. In the mobile case we have assigned random velocities to

the atoms following a Boltzmann distribution with a temperature of 300 K.

6.3 Charge-density evolution

In order to obtain evidence of the charge-transfer process we have partitioned

the simulation box into three regions matching the moieties boundaries along

the longitudinal axis of the molecule. Figure 6.2 shows the evolution of

the excess electronic charge contained in the volume including the pyrrole-

C60 and in the remaining part of the molecule. The black line represents


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 2 4 6 8 10 12 14 16

Ele

ctro

nic

Cha

rge

[ele

ctro

ns]

time [fs]

dynamical ionsclamped ions

Figure 6.3: Comparison between charge transfer dynamics obtained through

calculations with dynamical ions (solid line) or clamped ions (dashed line)

for the pyrrole-C60 part.

the fraction of electronic charge transferred into a volume that contain the

pyrrole-C60 moiety as a function of time. From this plot it is easy to see

that there is an increase of the electronic charge into this volume, which

corresponds to a decrease of electronic charge into the volume containing the

carotenoid and the porphyrin. Less than 70 fs are needed to transfer 0.9 e to

the fullerene.

The fact that the charge transfer is tied to the dynamics of the ions is

visible in the Figure 6.3. The solid line shows the charge transferred to

fullerene when the ions are allowed to move freely, while the dashed line

shows the charge transfer if the triad’s ions are hold fixed. In the last case

the fraction of electron transferred to the fullerene volume after 17 fs is too

small to be considered a trend rather than a fluctuation. This plot clearly

shows that the energy recombination due to a rearrangement of the ions plays

6.3. Charge-density evolution 59

(a) time = 9.31 fs

(b) time = 17.69 fs

(c) time = 41.74 fs

(d) time = 68.15 fs

Figure 6.4: Plots of the time dependent perturbation of the density δn(r, t)

(see equation (2.16)) at different values of time. The letters “a”, “b”, “c”,

and “d” are referred at the points of the Figure 6.2.


a major role in the process.

In Figure 6.4 we can see the time dependent perturbation of the density

δn(r, t) at different values of the time propagation. The pictures are snap-

shots of an animation visible following the link:

http://www.quantumsolids.unimore.it/animations/triad_density.avi.

It is obtained by computing the difference between the charge density at time

t and the density of the KS ground state (see equation (2.16)). The magenta

isosurface corresponds to a positive difference and the green to a negative

difference. In the figure it is evident an increase of the magenta color during

the time evolution on the outer fullerene region. It means an increase of

charge density in this region. Form Figure 6.4(a) to Figure 6.4(b) it is also

possible to notice a decrease of the charge density on the porphyrin region

in terms of increase of the green isosurface. Form Figure 6.4(c) to Figure

6.4(d) is visible a charge polarization, that starts at about 40 fs, where in

the region between the fullerene and the porphyrin an emptying of charge is

observable. Since we know from Figure 6.2 that a net charge is transfered

from the porphyrin to the fullerene, we interpreted these changes of signs as

a local polarization of the moieties.

6.4 Conclusions

In order to clarify the spectral weight of different excitations in the charge

fluctuations, we have performed a Fourier analysis of the charge variation on

the fixed volumes. We have decomposed the charge density ρ(t) into a linear

term and an oscillating term

ρ(t) −→ At + B + f(t) .

Then we have performed a Discrete Fourier Transform of the oscillating term

alone

f(t) −→ dft[f(t)] = F (ωn) .

The functions in the frequency space are plotted in Figure 6.5. In all

the three regions it is possible to see frequencies of relevant amplitude in the

http://www.quantumsolids.unimore.it/animations/triad_density.avi

6.4. Conclusions 61

0

50

100

150

200

0 0.5 1 1.5 2 2.5 3 3.5 4

Am

pliti

de

energy [eV]

pyrrole-C60×50

0

50

100

150

200

Am

pliti

de

Porphyrin×50

0

50

100

150

200

Am

pliti

de

Carotene×50

0 0.05 0.1

0.15 0.2

3 3.2 3.4 3.6

0 0.05 0.1

0.15 0.2

3 3.2 3.4 3.6

0 0.05 0.1

0.15 0.2

3 3.2 3.4 3.6

Figure 6.5: Discrete Fourier transform of the excess of electronic charge con-

tained in the volumes matching the moieties. In the vertical axis is reported

the amplitude of the Fourier coefficients.

energy region up to 1 eV. These small energetic frequencies could be associ-

ated to vibrational modes of the molecule. In addition, in the fullerene and

carotene regions relevant frequencies from 3.0 to 3.5 eV are also present that

could be assigned to electronic modes. As an assessment of our procedure in

Figure 6.6 we have reconstructed the signal back to the time domain. Using


-0.2

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70

Ele

ctr.

Cha

rge

[e]

time [fs]

pyrrole-C6050 harmonic components10 harmonic components

-2

-1.6

-1.2

-0.8

-0.4

0

0.4

Ele

ctr.

Cha

rge

[e]

Porphyrin50 harmonic components10 harmonic components

-0.4-0.2

0 0.2 0.4 0.6 0.8

1E

lect

r. C

harg

e [e

] Carotenoid50 harmonic components10 harmonic components

0 2.5 5

Figure 6.6: Harmonic reconstruction of the excess of electronic charge con-

tained in the volumes matching the moieties.

only the first 10 frequencies we are able to obtain a good matching with the

main features of the original function. Moreover we needed the first 50 fre-

quencies to reproduce properly also the small fluctuations visible in the inset

of the fullerene plot in the Figure 6.6.

6.5. Summary 63

The next step in the study of the charge transfer dynamics is the attribu-

tion of the frequencies used for the reconstruction to the mode (rotational,

vibrational, electronic, etc. . . ) of the molecule. This could help us to bet-

ter understand the role of the ions dynamic in the charge transfer process.

Some of these modes are visibly indispensable for the charge transfer (see

Figure 6.3) and non influential in the photo-excitation process, probably due

to the different time scale in which these process are brought to completion.

In addition, with a single-particle excitation in a single state (the LUMO+4

of the triad), probably we have provided the amount of energy necessary for

the excitation of other transitions. In the Figure 6.6, the plot concerning the

carotene shows an increase of charge comparable with that on the fullerene.

This could be due to the excitation of some electronic transitions that involve

the orbitals localized on the carotene. For this reason further calculations

could be done performing an excitation with partial occupation of the states

with mixed P-C60 character.

6.5 Summary

In Chapter 4 we showed the theoretical calculations concerning the electronic

structure of the triad and its parts. In particular, it is seen that the energy

levels of the moieties have the necessary requirements for the construction of

a photovoltaic device.

In Chapter 5 we compared the experimental absorption spectra with the-

oretical spectra, showing a good description for the parts. In addition, for the

whole triad we compare our results with the spectra of a previous theoretical

work based solely on the DFT. In this way is possible to see not only the

good agreement of our calculations with the experimental spectrum, as well

as the TDDFT, as compared to DFT, best describes the optical property of

a system like this one.

After the study of the ground state properties and optical properties, of

the triad and its parts, it is possible to state that the system as a whole acts

as the sum of its parts, each of which has a specific role in the photo-induced

charge separation mechanism that makes this molecule a good candidate

for third generation solar energy conversion devices. This fact has allowed


us to sketch a diagram of all the states involved in the photo-absorption

and charge transfer processes, and indicates that the design of solar devices

based on similar assemblies can be performed by individually addressing the

active components, and then chemically joining them into a unit capable of

transforming the energy of an incoming photon into electric potential energy

stored into a 50 A long dipole.

Finally, the most relevant novelty of this thesis work is described in Chap-

ter 6, where has been made the study of the dynamics of charge transfer

through the moieties of the triad. So we have shown that TDDFT is able

to produce a good dynamic even using local potentials, and that the time-

domain analysis of this process reveals the tight strong interaction between

the dynamics of the ions and the evolution of the electronic cloud in the

molecule.

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