Prof. Dr Neri Alves

Prof. Dr Christine Lartigau-Dagron

Dr Kathleen Moineau-Chane Ching

Prof. Dr Marystela Ferreira

Prof. Dr Augusto

Keywords: DFT, HOMO, LUMO, B3LYP, electron donor, electron acceptor, polyfullerenes. CITOLINO, L. V DFT, HOMO, LUMO, B3LYP, donneur d'électrons, accepteur d'électrons, polyfullerènes. CITOLINO, L DFT, HOMO, LUMO, B3LYP, doador de elétrons, aceitador de elétrons, polifulerenos

My family, my parents José and Evanilda, my siblings João and Ana

LIST OF FIGURES

3,3,13,13tetrakis(4-hexylphenyl)-7,10,17,20-tetrathiahexacyclo[9.9.0.0 2,9 .0 4,8 .0 12,19 .0 14,18 ] icosa-1(11),2(9),4(8), 5,12(19),14(18),15-heptaen-6-yl]methylidene]-3-oxoinden-1ylidene]propanedinitrile : O6T-4F The population growth and global economic activities result in an increasing expansion in the production, distribution and consumption of energy, which leads to increased efforts in scientific studies to develop alternatives regarding the use of fossil fuels and their impacts on the environment [1,2]. With more than 80% of the energy coming from fossil fuels, along with its depletion and increased CO2 emissions caused by its combustion, sustainable energy sources are already in use, such as solar energy and wind energy, to help supply the energy demand [3,4]. Thus, the development of devices that can convert solar energy constitutes a very important Currently, photovoltaic devices based on inorganic materials, mainly in crystalline Si, lead the world market with higher stability and efficiencies [15]. The multijunction devices present high efficiencies, however they use several expensive inorganic materials, mainly III-V semiconductors, and present a complex architecture, leading to high production costs [16,17]. In emerging devices, perovskite solar cells show promising efficiencies, but are based on heavy metals, such as lead, making them toxic to the environment [18,19]. Still in emerging devices, there is a great interest in organic solar cells due to the study and use of low-cost organic materials and simple solution processing method, in addition to the flexibility of the devices and large production volume [20,21].

Using photoactive heterojunctions, OSCs have been widely studied and highlighting with devices reaching efficiency around 16-18% [22,23]. The photovoltaic effect in OSCs occurs by means of the electronic structure of the arrangement of carbon and heteroatoms on the molecular level, which leads to different electron donor (p-type) or acceptor (n-type)

characteristics for each molecule and the species are locally and globally neutral, most of the time [24]. The desired semiconductor character of these organic materials is assured by the Peierls distortion mechanism, which creates the energy gaps necessary for ultraviolet-visible (UV-Vis) light absorption [24]. Then, under illumination, following a π → π * transition, neutral excitons can be formed either in one or both materials (Figure 2) [25,26]. If one has a large enough energy offset between the lowest unoccupied molecular orbital (LUMO) of the p-type material and the LUMO of the n-type one, the exciton will dissociate via an ultrafast electron transfer into long-lived charged species called polarons (namely, an electron on the n-type molecule and a hole on the p-type one) [25,26]. The same process can happen when the highest occupied molecular orbital (HOMO) offsets are large enough as well [25,26]. In order to avoid exciton recombination, the bulk heterojunctions (BHJs), i.e., a blend of an electron donor and an electron acceptor material is the key component in OSCs for improvement of the efficient [27,28,29].

Figure 3 shows a typic energy levels configuration of donor and acceptor materials and the charge cascade processes in the device [28]. In this type of heterojunction (Figure 4), small electron donor and acceptor domains are mixed, guaranteeing short distances for diffusion, and leading to efficient dissociation of excitons at the interface of these domains before recombination [27,28,29]. The difference in HOMO and LUMO between donor and acceptor layers creates electrostatic forces at the interface [27,28,29]. Give an appropriate of materials, such differences generate an electric field that leads to the efficient break-up of excitons into electrons and holes [27,28,29]. After exciton splitting, free electrons and holes are transported in the acceptor and donor networks (Figure 2 and 4) [27,28,29]. The free electrons are then accepted by the material with lower LUMO level and holes by the material with higher HOMO [27,28,29]. The transportation of charge carriers in organic semiconductors mostly takes place by hopping from one localized state to the next, until reaching to their respective electrodes [27,28,29]. In the past decades, a conjugated polymer and a fullerene derivative have been the most commonly used electron donor and electron acceptor, respectively [32,33,34]. Recently, nonfullerene acceptor materials, particularly small molecules and oligomers, have emerged as a promising alternative to replace fullerene derivatives [35,36,37]. Compared to fullerenes, these new acceptors are generally synthesized from diversified, low-cost routes based on building block materials with extraordinary chemical, thermal, and photostability [35,36,37].

So, the use of electron donating polymers combined with fullerene and non-fullerene electron acceptors in active layers of OSCs has shown significant efficiency in devices [38]. Fu et al. showed that active ternary layers using donor polymers with fullerenes and non-fullerenes acceptors showed higher efficiency than in binary active layers (donor polymers only with nonfullerenes) [39]. The optical absorption of fullerene is complementary to that of polymer and nonfullerene, in addition to bringing higher stability to the device when exposed to the environment In this context, the computational quantum chemistry plays an important role for understanding and designing new organic materials to optimize OSCs performance [41].

Distinct molecular modelling techniques can be chosen considering the type of system under study and the properties that are required to be described [41]. Thereby, several approaches exist to treat either small (a few dozen atoms) or big molecules (several hundred atoms), ranging from electronic structure-based ab initio methods to molecular mechanics ones [42]. Such approaches will be discussed in the next section.

Molecular modelling techniques

As our studies were restricted to methods involving the electronic structure of molecules, the advantages and disadvantages of some widely used methods at this level of structure are presented here.

The study of the electronic structure of molecules requires the knowledge of the solutions of the Schrödinger equation [43]. The impossibility to reach the analytical solutions of this integrodifferential equation has led theoretical chemists to develop a whole strategy and a set of methods allowing to reach and calculate the main observables with a precision that is constantly increasing with the development of the computational capacity [43].

In the Born-Oppenheimer (BO) approximation the basic assumption is that of the independent particle model which proposes to write at order zero an eigenfunction of the adiabatic electronic Hamiltonian (stationary state) as a product of functions depending on only one electronic variable (spin-orbits) [43]. This hypothesis completed the necessity to antisymmetrize the simple product of spin-orbitals, and on the other hand, the adoption of the model of the double occupation of space functions, is at the basis of the self-consistent field (SCF) model whose objective is the determination of molecular orbitals [43].

In particular, the Restricted Hartree-Fock (RHF) method allows to account for multielectron interactions by a model that stipulates that each electron is in an average field of the others (independent-particle model) always in the hypothesis of the double occupation of the molecular orbitals [44]. From these approximations it results an error on the evaluation of the electronic energy which, while being weak (a few percent of the total electronic energy), has disastrous consequences on the calculation of quantities such as orbital energies, binding energy, etc [44]. Unlike the RHF solution which corresponds, as we have just recalled, to the description of a Slater determinant made up of N/2 doubly occupied orbitals (the other orbitals being empty), the exact solution that we are trying to describe by taking into account the correlation is in fact made up of an infinite number of configurations that can be thought of as excitations with respect to the N-electron RHF determinant in an infinitely large base. This comes out of the fact that only the occupied orbitals are used in the energy minimization process of the N electron system. For a more appropriate description one should also consider the other (virtual) states also occupied. And, of course, it is impossible to work in a complete or infinite basis.

Various methods are then proposed by theoretical chemists to evaluate as accurately as possible this correlation. Three types of approaches are used: 1) the methods aiming at solving or in interaction (Coupled Pair Theory) [47,48].

On the contrary, in the Multiconfigurational Self-Consistent Field (MCSCF) methods, the objective is to optimize with the use of a variational method both the monoelectronic functions used and the multiconfigurational development of the eigenfunction of the electronic Hamiltonian [49,50,51]. Among the numerous developments proposed, the Complete Active Space Self-Consistent Field (CASSCF) approaches lead to the development of the multiconfigurational wave function on the totality of the configurations generated by the set of all possible excitations in a restricted space of occupied and virtual molecular orbitals (active space) [52].

However, the dimensional aspect of the systems selected by the experimenters in the OSCs processes (without mentioning the effects of the environment, the solvent, etc, which are generally neglected and considerably increase the dimension of the system to be modelled) constrain the researchers in the use of fast approaches, although more approximated, but able of treating electronic correlation like the Density Functional Theory (DFT) [53].

Within the DFT approach which is actually the most known one, defining LUMO (or whichever other orbital) is not a problem often overlooked by modelers [54]. This is due to the fact that DFT is not a wave function method and, hence, it does not lead to explicit orbitals, but to Kohn-Sham (KS) ones [54,55]. These KS orbitals have been, for a long time, viewed as an auxiliary concept, not necessarily meaningful and just a way to build the total density [54,55].

This is so based on the fact that, in the process of construction of these orbitals, an approximative exchange-correlation potential has to be used, and it may keep reality away of their definition [54,55] approach which is certainly one of the approaches that allows the study of higher dimensional systems and which is among the most used to "estimate" the energy of multi-configuration states in 62,63,64]. Indeed, its implementation does not require any additional methodological input to what is commonly found in a standard quantum chemistry program.

The principle is to approximate the "multi-deterministic" state by a single Slater determinant, which allows the use of 62,63,64]. This determinant, which is not an eigenvector of the S 2 operator, is said to be broken symmetry [61,62,63,64]. The only way to obtain such a determinant is to use the unrestricted formalism [61,62,63,64]. This formalism allows the space parts of the α and β orbitals to be different, unlike the restricted formalism [61,62,63,64]. Nevertheless, this formalism suffers from the spin contamination problem which limits the accuracy of such an approach compared to the multi-determinant approaches cited above [61,62,63,64]. The Yamaguchi method is the most widely used the BS approach at present [65,66,67,68]. The simplest of the open layer states consists of two electrons located in two orbitals leading to the description of a singlet state, called open shell system, and a triplet state. The Yamaguchi method is based on the approximation that the unique determinant BS sought is a pure linear combination between the singlet state and the triplet state and that the spin polarization introduced by the unrestricted formalism is negligible in the triplet state [65,66,67,68]. Thus, the BS determinant can be expressed very easily and cheaply [65,66,67,68].

Another advantage of these approaches is that it is commonly accepted that for the most classical functionals in the molecular domain (hybrid-Generalized Gradient Approximation (GGA) such as B3LYP, PBE0, etc), the BS approach with the Yamaguchi formula represents more or less correctly the open shell system state and at lower computational cost [65,66,67,68].

However, this should not make us forget the main limitation of this approach. One may wonder what is the meaning of using the mean value of the S 2 operator in DFT. Indeed, this value is most often computed as in wave function theory on the determinant KS by a 2-body operator. This determinant is that of a fictitious system without interaction and this average value should normally be calculated on the wave function of the real system. Experience shows that the BS method is efficient when the difference between the average value obtained for the KS determinant and the real system is small, which is unfortunately not the case in the systems we have studied.

A last alternative to the BS methods exists, and it is the Spin- the reader to the considerations mentioned above).

Objectives

The main objective of this work is set out the general considerations for the use of DFT approaches in particular when one has to describe one or more electronic states of more or less strongly correlated electrons. Heisenberg [75,76].

The HJE is an extremely powerful and sophisticated method for solving the equations of motion in analytical mechanics [77]. The method is based on the search for a canonical transformation that is capable of making the transformed Hamiltonian identically zero [77].

HJE is particularly important as it is the only mathematical formulation of mechanics in which the motion of a particle can be represented as a wave, and for this reason, HJE is considered the closest approximation of classical mechanics to quantum mechanics [77,78]. The complete and time-dependent HJE is given by:

𝜕𝑆 𝜕𝑡 = -𝐻(𝑞 𝑖 , 𝑝 𝑖 , 𝑡) = -𝐻 (𝑞 𝑖 , 𝜕𝑆 𝜕𝑞 𝑖 , 𝑡) (2.1)
where L is the Lagrangian of the system. The Lagrangian is the kinetic energy minus the potential energy of the system. The 𝑆 is a generating function that depends on spatial coordinates (𝑞 𝑖 ), momenta (𝑝 𝑖 ) and time (𝑡) [77,79]. The generating function 𝑆(𝑞 𝑖 , 𝑡) is central to analytical mechanics, and while everything that can be known about the system can be derived from it, it is defined by [77,79]:

𝑆 = ∫ 𝐿(𝑞 𝑖 , 𝑝 𝑖 , 𝑡)𝑑𝑡 𝑡 2 𝑡 1 (2.2)
From 𝑆(𝑞 𝑖 , 𝑡) it is possible to obtain the canonical moments of system as [77,79]:

𝑝 𝑖 = 𝜕𝑆 𝜕𝑞 𝑖 (2.3)
and the Hamiltonian as [77,79]:

𝐻 = - 𝜕𝑆 𝜕𝑡 (2.4)
In case of conservative systems, the Hamiltonian is the total energy E (sum of kinetic and potential energies), which is constant [77,79]. Thus, applying the Equation 2.4 to the hydrogen atom, and considering 𝜕𝑆 𝜕𝑡 ⁄ = -𝐸, the Equation 2.1 becomes [77,78,79]

𝐻 (𝑞, 𝜕𝑆 𝜕𝑞 ) = 𝐸 (2.5)
which is also called the time-independent HJE or the energy conservation equation [77,78,79].

Applying a logarithmic function 𝛹 spatial which is the logical equivalent of 𝑆 in Equation 2.5

[78, 79]

𝑆 = 𝑖ħ𝑙𝑛(𝛹) (2.6)
being 𝛹 defined by [78,79]:

𝛹 = exp (- 𝑖𝑆 ħ ) (2.7)
Since Equation 2.7 is considered to be the amplitude of matter waves, the finitude and definition of can be assumed by a normalization condition [77,78,79]:

∫|𝛹| 2 𝑑𝑉 = 1 (2.8)
This 𝛹 is called a wavefunction [78,79]. Substituting Equation 2.6 in Equation 2.1, using this wavefunction, into the time-dependent and time-independent HJEs in Equations 2.1 and 2.5 gives [78,79]:

𝐻 ̂𝛹 = 𝑖ħ 𝜕𝛹 𝜕𝑡 (2.9)
and 𝐻 ̂𝛹 = 𝐸𝛹 (2.10) respectively. Equations 2.9 and 2.10 are called the time-dependent and time-independent Schrödinger equations, respectively, and the Hamiltonian is replaced by the Hamiltonian operator acting on spatial functions [78,79]. Because of its definition in terms of the principal function 𝑆, 𝛹 also has no physical meaning and cannot be measured, but contain all system information [78,79]. Considering a one-dimensional and time-independent system of particles, the Hamiltonian operator is [78,79]:

𝐻 ̂= ∑ 𝑝 𝑖 2 2𝑚 𝑖 𝑖 + 𝑉 = ∑ 1 2𝑚 𝑖 𝑖 ( 𝜕𝑆 𝜕𝑞 𝑖 ) 2 + 𝑉 = -∑ ħ 2 2𝑚 𝑖 𝑖 1 |𝛹| 2 ( 𝜕𝛹 𝜕𝑞 𝑖 ) 2 + 𝑉 = 𝐸 (2.11)
For energy conservation, it is obtained [78,79]:

-∑ ħ 2 2𝑚 𝑖 𝑖 ( 𝜕𝛹 𝜕𝑞 𝑖 ) 2 + (𝑉 -𝐸)|𝛹| 2 = 0 (2.12)
Using the variational method, the stationary condition is obtained under the normalization condition [78,79]:

-∑ ħ 2 2𝑚 𝑖 ( 𝜕 2 𝛹 𝜕𝑞 𝑖 2 ) 𝑖 + (𝑉 -𝐸)𝛹 = 0 (2.13)
For a multidimensional system:

-∑ ħ 2 2𝑚 𝑖 𝑖 ∇ 𝑖 2 𝛹 + (𝑉 -𝐸)𝛹 = 0 (

The many-body Schrodinger equation

The physical properties of matter at the atomic level are described using the formalism of quantum mechanics [80]. In the case of optoelectronic properties, for example, the wave functions of electrons are fundamental to describe the frontier energy levels in molecular orbitals [85]. Thus, a multielectron atom systems are considered as a many-body problem [86].

Thus, the interactions of a system with 𝑁 electrons and 𝑀 nuclei, with coordinates {𝑟 ⃗ 𝑖 } = (𝑟 ⃗ 1 , 𝑟 ⃗ 2 , … , 𝑟 ⃗ 𝑁 ) and {𝑅 ⃗⃗ 𝛼 } = (𝑅 1 , 𝑅 2 , … , 𝑅 ⃗⃗ 𝑀 ), respectively, can be described by the nonrelativistic Hamiltonian operator [87]:

𝐻 ̂= 𝑇 ̂𝑛 + 𝑇 ̂𝑒 + 𝑉 ̂𝑛𝑛 + 𝑉 ̂𝑛𝑒 + 𝑉 ̂𝑒𝑒 (2.16)
The terms 𝑇 ̂𝑛 and 𝑇 ̂𝑒 represent the kinetic energy of nuclei and electrons, respectively,

given by:

𝑇 ̂𝑛 = ∑ - 1 2𝑀 𝑀 𝑀 𝛼<1 ∇ 𝑅 ⃗⃗ 𝛼 2 , 𝑇 ̂𝑒 = ∑ - 1 2 𝑁 𝑖=1 ∇ 𝑟 ⃗ 𝑖 2 (2.17)
where, 𝑀 𝑀 is the mass of the 𝑀-th nucleus. Atomic units are a convenient system of units commonly used in electronic property calculations, so, 𝑚 𝑒 = ħ = 𝑒 = 1, where 𝑚 𝑒 and 𝑒 are the electron's mass and charge, respectively; furthermore, it is also 4𝜋𝜀 0 = 1. The Coulomb interactions between nucleus-nucleus (𝑉 ̂𝑛𝑛 ), nucleus-electron (𝑉 ̂𝑛𝑒 ) and electron-electron (𝑉 ̂𝑒𝑒 ) are given by:

𝑉 ̂𝑛𝑛 = 1 2 ∑ ∑ 𝑍 𝛼 𝑍 𝛽 |𝑅 ⃗⃗ 𝛼 -𝑅 ⃗⃗ 𝛽 | 𝑀 𝛽=1 𝑀 𝛼=1 𝛼≠𝛽 , 𝑉 ̂𝑛𝑒 = ∑ ∑ - 𝑍 𝛼 |𝑅 ⃗⃗ 𝛼 -𝑟 ⃗ 𝑖 | 𝑁 𝑖=1 , 𝑀 𝛼=1 𝑉 ̂𝑒𝑒 = 1 2 ∑ ∑ 1 |𝑟 ⃗ 𝑖 -𝑟 ⃗ 𝑗 | 𝑁 𝑗=1 𝑁 𝑖=1 𝑗≠1
(2.18) with 𝑍 𝛼 and 𝑍 𝛽 the atomic number of nuclei, 𝑅 ⃗⃗ 𝛼 and 𝑟 ⃗ 𝑖 the positions of nuclei and electrons, respectively. In the interaction of the system, neglecting the nuclear motions is one of the most important approaches, as it makes possible the decoupling of the nuclear and electronic parts in the time-independent Schrödinger equation with the application of the Hamiltonian (Equation 2.16) [88]. As the atomic nucleus is heavier than the electron (𝑚 𝑝 𝑚 𝑒 ⁄ ≈ 1836), it is considered that electrons move much faster than nuclei, so in the absence of nuclei movement there is no exchange of thermal energy with the external environment [88]. This approximation is adiabatic and known as the BO approximation [88]. With electronic movement being considered instantaneous in relation to nuclear movement, decoupling is given by:

𝜓({𝑟 ⃗ 𝑖 }, {𝑅 ⃗⃗ 𝛼 }) = 𝜓 𝑛 ({𝑅 ⃗⃗ 𝛼 })𝜓 𝑒 ({𝑟 ⃗ 𝑖 }, {𝑅 ⃗⃗ 𝛼 ′ }) (2.19)
where 𝑅 ⃗⃗ 𝛼 ′ indicates the parametric dependence of the wavefunction on the position of the nuclei.

After decoupling and applying only the wavefunction of the electronic part to the Schrödinger equation, a new Hamiltonian is written:

𝐻 ̂𝑒 = 𝑇 ̂𝑒 + 𝑉 ̂𝑛𝑒 ({𝑟 ⃗ 𝑖 }, {𝑅 ⃗⃗ 𝛼 ′ }) + 𝑉 ̂𝑒𝑒 ({𝑟 ⃗ 𝑖 }) (2.20)
With the BO approximation, the interaction term of the nuclei 𝑉 ̂𝑛𝑛 , which can be considered constant, is omitted from the Equation 2.20 [88]. In addition, once electrons have a much higher velocity than the nucleus, the nucleus is considered fixed and generates an effective potential under which the electrons move. Thus, there is a new time-independent Schrodinger equation for the electronic part to be solved:

𝐻 ̂𝑒𝜓 𝑒 ({𝑟 ⃗ 𝑖 }, {𝑅 ⃗⃗ 𝛼 ′ }) = 𝐸 𝑒 𝜓 𝑒 ({𝑟 ⃗ 𝑖 }, {𝑅 ⃗⃗ 𝛼 ′ }) (2.

The Hartree-Fock method

In 1928, in order to simplify the solution to the Schrodinger equation for N-electron atoms, Hartree proposed a solution method using BO approximation and assuming that each electron interacts only with the average potential of the other electrons [90]. As this approximation assumed independent electrons, the Hamiltonian operator is divided into terms for the different electrons and the wave function 𝜓(𝑟 ⃗ 1 , 𝑟 ⃗ 2 , … , 𝑟 ⃗ 𝑖 ) is represented as the product of different electronic wave functions 𝜙 𝑖 (𝑟 ⃗ 𝑖 ) [90]:

𝜓(𝑟 ⃗ 1 , 𝑟 ⃗ 2 , … , 𝑟 ⃗ 𝑖 ) = 𝜙 1 (𝑟 ⃗ 1 )𝜙 2 (𝑟 ⃗ 2 ) … 𝜙 𝑖 (𝑟 ⃗ 𝑖 ) (2.22)
With the variational method, which minimizes the energy value of each electron, the Schrödinger equation for obtaining the set of wave functions of an electron 𝜙 𝑖 is represented as

𝐻 ̂(𝑟 ⃗ 𝑖 )𝜙 𝑖 (𝑟 ⃗ 𝑖 ) = 𝜀 𝑖 𝜙 𝑖 (𝑟 ⃗ 𝑖 ) (2.23)
Since the eigenvalue of the Equation 2.23, 𝜀 𝑖 , is interpreted as an eigenenergy for the movement to the i-th electron, the total wave function can be obtained by solving the eigenequation for each electron. Thus, in Hartree method, the total self-energy is the sum of the self-energy corresponding to different electronic movements,

𝐻 ̂𝜓 = (𝜀 1 + 𝜀 2 + ⋯ + 𝜀 𝑖 )𝜙 1 𝜙 2 𝜙 𝑖 = 𝜀𝜓 (2.24)
However, the Hartree method does not consider that the wave function of electronic motion must be antisymmetric [89]. Since electrons are indistinguishable, and to satisfy the Pauli exclusion principle, the fermion particles must be represented by antisymmetric functions in terms of changing coordinates [91]. This is because the wave functions must be zero for the case where the same electron, with the same spin, occupies the same orbital. As a result, electrons always have antisymmetric wave functions [92,93]. The new electron-electron interaction resulting from antisymmetrization is called the exchange interaction [89]. Thus, the antisymmetric wave function can be written as a determinant, as proposed, independently, by Heisenberg and Dirac [92,93].

In 1929, Slater introduced the normalized determinant that guaranteed the antisymmetry of a many-electrons wave function [94]. In the Slater determinant, each row represents an electron and each column an spin-orbital. So, for the case of two or more electrons, the antisymmetric wavefunction is written by

𝜓(𝑟 1 , 𝑟 2 , … , 𝑟 𝑁 ) = 1 √𝑁! | 𝜙 1 (𝑟 1 ) 𝜙 1 (𝑟 2 ) … 𝜙 1 (𝑟 𝑁 ) 𝜙 2 (𝑟 1 ) ⋮ 𝜙 2 (𝑟 2 ) ⋮ … ⋱ 𝜙 2 (𝑟 𝑁 ) ⋮ 𝜙 𝑁 (𝑟 1 ) 𝜙 𝑁 (𝑟 2 ) … 𝜙 𝑁 (𝑟 𝑁 ) | (2.25)
and its determinant is

𝜓(𝑟 1 , 𝑟 2 , … , 𝑟 𝑁 ) = 1 √𝑁! 𝑑𝑒𝑡|𝜙 1 (𝑟 1 )𝜙 2 (𝑟 2 ) … 𝜙 𝑁 (𝑟 𝑁 )| (2.26)
Then, in 1930, Fock applied the Slater determinant to the Hartree method and proposed the Hartree-Fock (HF) method [95]. Thus, the HF method is based on the HF equation (Equation 2.27) in which the total antisymmetric wave function is the combination of wave functions of each electron individually, and uses an operator that acts on each individual wave function, and with that, we obtain individual energies for each electron or orbital. This operator is called the closed shell Fock operator and is given by Equation 2.28.

𝐹 ̂𝜙𝑖 = 𝜖 𝑖 𝜙 𝑖 (2.27)

𝐹 ̂= ℎ ̂+ ∑ (2𝐽 ̂𝑗 𝑛=𝑁 2 ⁄ 𝑗 -𝐾 ̂𝑗) (2.28)
From the variational principle, in the Equation 2.28, 𝐽 ̂𝑗 and 𝐾 ̂𝑗 are called the Coulomb operator and the exchange operator, respectively, which are defined as Then, using the HF equation in closed shell systems, the orbital energy 𝜖 𝑖 is represented as

𝐽 ̂𝑗(𝑟 1 )𝜙 𝑖 (𝑟 1 ) = ∫ 𝑑 3 𝑟 2 𝜙 𝑗 * (𝑟 2 )𝜙 𝑗 (𝑟 2 ) 1 𝑟 12 𝜙 𝑖 (𝑟 1 ) (2.29) 𝐾 ̂𝑗(𝑟 1 )𝜙 𝑖 (𝑟 1 ) = ∫ 𝑑 3 𝑟 2 𝜙 𝑗 * (𝑟 2 )𝜙 𝑖 (𝑟 2 ) 1 𝑟 12 𝜙 𝑗 (𝑟 1 ) (2.
𝜖 𝑖 = ∫ 𝑑 3 𝑟 1 𝜙 𝑖 * (𝑟 1 )𝐹 ̂𝜙𝑖 (𝑟 1 ) = ℎ 𝑖𝑖 + ∑ (2𝐽 𝑖𝑗 𝑛=𝑁 2 ⁄ 𝑗 -𝐾 𝑖𝑗 ) (2.31)
where 𝐽 𝑖𝑗 (𝑟 1 ) = ∫ 𝑑 𝜙 𝑗 (𝑟 1 )𝜙 𝑖 (𝑟 2 ). For this orbital energy, the total electronic energy is written as

𝐸 = 2 ∑ ℎ 𝑖𝑖 𝑛 𝑖 + ∑(2𝐽 𝑖𝑗 𝑛 𝑖,𝑗 -𝐾 𝑖𝑗 ) = ∑(𝜖 𝑖 𝑛 𝑖 + ℎ 𝑖 ) (2.32)

Basis sets

The HF equations can be solved numerically for atoms, but this method has been shown to be computationally inadequate for polyatomic molecules. To incorporate the anisotropic nature of molecular orbitals originated from chemical bonds, the polarization functions are added to the basic sets, in addition to the divided valence basis. In Pople-type basis functions, the inclusion of polarization functions for heavy atoms is represented by an asterisk "*", such as "6-31G*", or letter d in parentheses (d). While two asterisks "**" or letter p in parentheses (p), indicate that the polarization functions p are also added to light atoms (hydrogen and helium).

Another common addition to basic sets is the addition of diffuse functions to account for loosely bound electrons. Adding diffuse functions is shown by a plus sign "+" in Pople-type basis functions. "6-311+G(d)" augments the sp diffuse functions by mixing s and p orbitals for all atoms except hydrogen atoms, and "6-311++G(2df, 2pd)" adds two diffuse functions d and one f to all atoms, except hydrogen, and two orbital functions p and one d for hydrogen.

In general, the equations of the HF method are solved using the SCF [89]. Basically, the SCF method follows these steps: 1) starts setting the molecular geometry (coordinates of atomic nuclei); 2) choose basis functions; 3) calculate one and two-electron integrals; 4) initial guess coefficients {𝐶 𝑖 }, 5) compute the Fock matrix 𝑭; 6) calculate Fock matrix; 7) solve Roothan equations for 𝐶 𝑖 ; 8) if SCF converges, calculate molecular properties or, if no, return to step 5 and continue with the calculation. This method is known as the SCF-HF method. Despite the simplicity of the procedure, it soon became clear that solving this equation is not trivial for usual molecular electronic systems.

Density Functional Theory

Thomas-Fermi-Dirac method

The solution of the HF equation, using the formalism of wave function manipulation, can be used for the electronic description of a system. Since the wavefunction for a single electron depends on three spatial variables plus one of spin, the solution for systems with a large number of electrons requires a high computational cost, due to the scaling of computational time for this type of solution.

A second totally different approach for calculating the electronic structures of atoms was initially developed by Thomas (1926) and [START_REF] Fermi | Eine statistische Methode zur Bestimmung einiger Eigenschaften des Atoms und ihre Anwendung auf die Theorie des periodischen Systems der Elemente[END_REF] treating electron density as a fundamental variable [103,104]. This approach sought to show that the kinetic energy and potential energy of the electrons could be directly related to the electron density, 𝑛(𝑟 ⃗), instead of the electron wave function, and thus the number of variables in the system depends only on three spatial coordinates, i.e., it is independent of the number of electrons in the system.

The starting point used by Thomas was that an atom can be approximated to a uniform electron gas model, where in electronic movements in solid crystals, electrons are uniformly distributed in the proportion of two per unit cell of 𝐿 side and volume 𝐿 3 in addition to containing 𝑁 non-interacting electrons evenly distributed for a grand total of 𝑁 𝑒 electrons [103]. Thus, the total kinetic energy of electrons is a functional of electron density 𝑛(𝑟 ⃗) given by

𝑇 𝑇𝐹 [𝑛(𝑟 ⃗)] = 𝐶 𝐹 ∫ 𝑑 3 𝑟𝑛 5 3 ⁄ (𝑟 ⃗) (2.36)
where

𝐶 𝐹 = 3 10 (3𝜋 2 ) 2 3 ⁄ .
Deriving the same kinetic energy functional obtained by Thomas, Fermi completes the Thomas method, now known as the Thomas-Fermi (TF) method, using a Fermi statistic at the absolute zero point [104]. Thus, the total energy functional in the TF method for one atom is

𝐸 𝑇𝐹 [𝑛(𝑟 ⃗)] = 𝑇[𝑛(𝑟 ⃗)] + 𝑉 𝑛𝑒 [𝑛(𝑟 ⃗)] + 𝑉 𝑒𝑒 [𝑛(𝑟 ⃗)]
(2.37)

The TF method proved to be promising, but it did not formulate the terms of exchange (obtained in the HF model) and correlation, thus leading to the impossibility of calculations in real electronic states. Then, in 1930, Dirac proposed the first exchange functional of electron density 𝑛(𝑟 ⃗), taking the distribution of electrons to be that of a uniform gas in each region where the atom is divided as in the . Thus, Dirac expresses the exchange contribution as a local density approximation (LDA) where the term depends on the electron density at one point

𝐸 𝑋 𝐿𝐷𝐴 = - 3 2 ( 3 𝜋 ) 1 3 ⁄ ∫ 𝑑 3 𝑟𝑛 4 3 ⁄ (𝑟 ⃗) (2.38)
The Thomas-Fermi-Dirac (TFD) method cannot establish the uniqueness of the solutions and the existence of density functionals, and cannot reproduce chemical bonds quantitatively or qualitatively. Thus, this method was despised until the mid-1960s, but it is considered a springboard for the modern DFT [89].

Hohenberg-Kohn formalism

In 1964, the concept of the TF method was revived and the total energy was fully proposed (relating classical and quantum interactions) and expressed in terms of electron density by two Hohenberg-Kohn (HK) theorems [106]. From these theorems the ground state properties of a system can be expressed in terms of electron density, thus creating the bases of the DFT, and are presented below.

Theorem 1: For each electron density there is only a single external potential, 𝑉 𝑛𝑒 , associated, i.e., the external potential is a unique functional of the electron density.

Theorem 2: A universal functional for energy 𝐸[𝑛(𝑟 ⃗)] can be defined in terms of electron density. The ground state density is the global minimum of this functional.

From the first theorem, in an electronic state, the number of electrons 𝑁 per unit volume in a given state is the electron density

𝑁 = ∫ 𝑛(𝑟 ⃗)𝑑 3 𝑟 (2.39)
The first theorem states that the ground state properties of the many-electron system depend only on electron density. Where, knowing the external potential, the terms of the Hamiltonian operator will be fixed as well as the electron density, consequently all observable properties of a quantum system can be determined, just like in the wave function. Among the observable properties is the total electronic energy, which is density dependent, which can be expressed by

𝐸[𝑛(𝑟 ⃗)] = 𝑇 𝑒 [𝑛(𝑟 ⃗)] + 𝑉 𝑛𝑒 [𝑛(𝑟 ⃗)] + 𝑉 𝑒𝑒 [𝑛(𝑟 ⃗)] (2.40)
where 𝑉 𝑒𝑒 [𝑛(𝑟 ⃗)] lists all interactions between electrons.

The kinetic energy and interaction energy of a non-relativistic Coulomb system are described by universal operators, so Hohenberg and Kohn grouped all the functionals from Equation 2.40 that depended directly on the external potential into a single term called the HK functional, being a universal functional because electron density determines 𝐸[𝑛(𝑟 ⃗)] and 𝑉 𝑛𝑒 [𝑛(𝑟 ⃗)]. So, the total electronic energy can be rewritten as:

𝐸[𝑛(𝑟 ⃗)] = 𝑉 𝑛𝑒 [𝑛(𝑟 ⃗)] + 𝐹 𝐻𝐾 [𝑛(𝑟 ⃗)]
(2.41)

The second theorem states that the correct ground-state density for a system is one that minimizes the total energy through the functional 𝐸[𝑛(𝑟 ⃗)]. Thus, by the variational principle, when the Hamiltonian is applied to an electronic density, other than that of the ground state, the system energy will always be higher than the ground state energy 𝐸[𝑛(𝑟 ⃗)] > 𝐸 0 [𝑛 0 (𝑟 ⃗)]. To find the ground state electron density, the Lagrange multiplier method is used, setting 𝛿𝐸[𝑛(𝑟 ⃗)] = 0 with the restriction that the total number of electrons is constant as 𝑛(𝑟 ⃗) is varied.

The variation theorem implies that the variation in electron density subject to restriction corresponds to an extreme, in this case, the minimum point:

Kohn-Sham Formalism

Although the HK theorems are stated as the fundamental theorems of quantum chemistry, based on electron density, they say nothing about how to obtain the ground-state density, of which all observable properties of the system are functional. Then, in 1965, Kohn and Sham were the first to present a method for calculating the electronic structure of systems involving many particles using 𝐸[𝑛(𝑟 ⃗)] [107]. In this new method, it was considered that for each real system of interacting particles there would be an auxiliary system composed of noninteracting particles under the action of an effective potential capable of maintaining the spatial arrangement of these particles so that its electronic density is identical to that of the real system.

So, the KS method is a variational approach using the electron-electron interaction potential of the density functional to give the ground state energy, molecular orbitals and orbital energies by means of the Lagrange multiplier method.

By the TF model it is possible to calculate the kinetic energy as a function of the electronic density, considering a gas of non-interacting electrons, and knowing the wave function of the system, the determination of the kinetic energy of the system becomes a much simpler process. Given these characteristics, Kohn and Sham proposed a model of union of electronic density with the wave function, where the total energy of the system could be rewritten as:

𝐸[𝑛(𝑟 ⃗)] = 𝑇 0 [𝑛(𝑟 ⃗)] + 𝑉 𝑛𝑒 [𝑛(𝑟 ⃗)] + 𝑉 𝐻 [𝑛(𝑟 ⃗)] + 𝐸 𝑋𝐶 [𝑛(𝑟 ⃗)] (2.44)
where 𝑇 0 [𝑛(𝑟 ⃗)] represents the kinetic energy of a non-interacting electron gas, having a density equal to that of the real system, and 𝐸 𝑋𝐶 [𝑛(𝑟 ⃗)] indicates the exchange and correlation functional of an interacting system with density 𝑛(𝑟 ⃗). Both the TF and the KS methods use the premise of a non-interacting electron gas, but the determination of kinetic energy differs, because the KS method considers the antisymmetric characteristic of the function. Thus, the wavefunction of the system is described through the representation of a Slater determinant. By the variational principle, both the energy and the electronic density of the ground state of the non-interacting auxiliary system can be obtained through eigenvectors 𝜙 𝑖 (𝑟 ⃗) and eigenvalues 𝜖 𝑖 of the monoelectronic HF equations. Thus, Kohn and Sham proposed to introduce monoelectronic orbitals, called KS orbitals (𝜙 𝑖 𝐾𝑆 (𝑟 ⃗)), into DFT to effectively describe the kinetic energy term of a non-interacting system. Through these orbitals, the kinetic energy of the system can be calculated by The diagonal 𝜺 matrix is an energy-levels matrix, whose diagonal elements are MO energy levels 𝜙, corresponding to the MOs 𝜀 𝑖 . 𝑭 is an energy-elements matrix, the Fock matrix, whose elements are

𝑇 0 [𝜌(𝑟 ⃗)] = ∑ ∫ 𝜙 𝑖 * 𝐾𝑆 (𝑟 ⃗) ( -∇ 2 2 ) 𝑁 𝑖 𝜙 𝑖 𝐾𝑆 (𝑟 ⃗)𝑑 3 𝑟 (2.
𝐹 𝑝𝑞 = ℎ 𝑝𝑞 + ∑ 𝑃 𝑟𝑠 ⟨𝑝𝑟|𝑞𝑠⟩ 𝑛 𝑏𝑎𝑠𝑖𝑠 𝑟,𝑠=1 + (𝑉 𝑋𝐶 ) 𝑝𝑞 (2.53)
where

(𝑉 𝑋𝐶 ) 𝑝𝑞 = ∫ 𝑑 3 𝑟𝜒 𝑝 * (𝑟)𝑉 𝑋𝐶 𝜒 𝑞 (𝑟 ⃗) (2.54)
The KS method is solved by the SCF method, since the effective KS potential depends on the electron density, which in turn depends on the KS orbitals. The SCF method is solved for nonlinear equations through the following steps: 1) provide an initial (trial) value for the electron density, 𝑛 (1) (𝑟 ⃗), which allows to obtain the initial effective potential, 𝑉 𝑒𝑓𝑓 (1) (𝑟 ⃗); 2) solve the KS equation with 𝑛 (1) (𝑟 ⃗) and obtain single electron wave functions 𝜙 𝑖 𝐾𝑆(1) (𝑟 ⃗); 3) calculate the electron density based on the wavefunctions of a single electron and 4) From this moment on, a new electron density, 𝑛 (2) (𝑟 ⃗), can be calculated using the new KS orbitals 𝜙 𝑖 𝐾𝑆(2) (𝑟 ⃗), and consequently, a new effective potential, 𝑉 𝑒𝑓𝑓 (2) (𝑟 ⃗), can be determined. This procedure is applied repeatedly until a minimum condition of electronic density difference is found, or another convergence condition adopted is satisfied, such as the variation of the total energy.

Once the ground state electronic density is found, the total electronic energy of the system can be calculated using the eigenvalues of the KS equation, through:

𝐸[𝑛(𝑟 ⃗)] = ∑ 𝜖 𝑖 𝐾𝑆 𝑁 𝑖=1 - 1 2 ∬ 𝑛(𝑟 ⃗ ′ )𝑛(𝑟 ⃗) |𝑟 ⃗ ′ -𝑟 ⃗| 𝑑 3 𝑟𝑑 3 𝑟 ′ -∫ 𝑉 ̂𝑋𝐶 (𝑟 ⃗)𝑛(𝑟 ⃗)𝑑 3 𝑟 + 𝐸 𝑋𝐶 [𝑛(𝑟 ⃗)] (2.55)
The sum being made over all occupied orbitals and all 𝑁-KS orbitals obtained by the monoelectronic Schrodinger equation.

Exchange-correlation functional

The exchange-correlation functional is very important in the fundamentals of the KS method, because it is from these specific forms of functions that all observable properties of a quantum system are obtained exactly [89]. Thus, it becomes necessary the existence of an ideal functional, which describes all exchange and correlation interactions, however, this functional is far from being determined [89]. So, this is the only part of the KS method that is based on approximations, in which the known functionals are determined [89]. The physical properties obtained by known functionals depend directly on the class of approximations, which try to minimize errors in the description of exchange-correlation energy, as well as correct the kinetic energy of a non-interacting system and self-interaction [89]. Thus, the more efficient these approximations are, the better results will be obtained through DFT calculations.

Density Functional Approximations

The LDA is the simplest functional for exchange-correlation energy, which is based on an uniform electron gas (UEG) model as the basis of its foundation. UEG is a system of electrons with constant density everywhere. Thus, the of exchange-correlation energy can be written as

𝐸 𝑋𝐶 [𝑛(𝑟 ⃗)] = ∫ 𝑛(𝑟 ⃗)𝜀 𝑋𝐶 𝑢𝑛𝑖𝑓 [𝑛(𝑟 ⃗)]𝑑 3 𝑟 (2.56)
where 𝜀 𝑋𝐶 𝑢𝑛𝑖𝑓 [𝑛(𝑟 ⃗)] is the exchange-correlation energy per electron for an UEG. So, Kohn and Sham proposed, in the original paper on the density functional theory, the first approximations for 𝜀 𝑋𝐶 𝑢𝑛𝑖𝑓 [𝑛(𝑟 ⃗)] and that the energy density (non-homogeneous) is the energy (exchange plus correlation) by electron of a homogeneous electron gas

𝐸 𝑋𝐶 [𝑛(𝑟 ⃗)] = ∫ 𝑛(𝑟 ⃗)𝜀 𝑋 𝑢𝑛𝑖𝑓 [𝑛(𝑟 ⃗)]𝑑 3 𝑟 + ∫ 𝑛(𝑟 ⃗)𝜀 𝐶 𝑢𝑛𝑖𝑓 [𝑛(𝑟 ⃗)]𝑑 3 𝑟 (2.57)
The 𝜀 𝑋 𝑢𝑛𝑖𝑓 [𝑛(𝑟 ⃗)] can be explicitly written with the exchange term being obtained by substituting the wave function that describes a non-interacting electron gas system in 2.15, resulting in

𝜀 𝑋 𝑢𝑛𝑖𝑓 [𝑛(𝑟 ⃗)] = -3𝑒 2 𝑘 𝑓 4𝜋 (2.58)
The problem with this functional is in determining the correct value for the correlation part 𝜀 𝐶 𝑢𝑛𝑖𝑓 [𝑛(𝑟 ⃗)], as there is no exact LDA correlation functional. As alternatives, several approximate LDA correlation functionals have been suggested, for example, in quantum chemistry, the Vosko-Wilk-Nusair (VWN) functional is widely used and is calculated based on Monte Carlo quantum methods of a gas of uniform density and in Padè's interpolation [108].

LDA tends to underestimate the exchange energy and overestimate the correlation energy by assuming that the density is the same everywhere [109]. Errors due to the exchange and correlation parts tend to compensate each other to some extent, caused by the fact that for any density the LDA satisfies a number of so-called sum rules [110,111,112]. To solve the problem of inhomogeneity in the real electron density of the electron, it is necessary to perform density expansions in terms of the gradient and higher order derivatives [112]. These expansions are called generalized gradient approximations (GGA) which consider not only the density at a position, but also the density gradient around that point [112]. Thus, the GGA functional has a semi-local character and the exchange-correlation energy is written as

𝐸 𝑋𝐶 [𝑛(𝑟 ⃗)] = ∫ 𝑓[𝑛(𝑟 ⃗), ∇ ⃗ ⃗⃗ 𝑛(𝑟 ⃗)]𝑑 3 𝑟 (2.59)
where the function 𝑓 is an "exchange enhancement factor" that modifies the LDA expression according to the density variation in the neighborhoods of a considered point. Thus, several descriptions for the GGA 𝑓 function have been proposed, among them we can highlight the introduction of the B88 functional proposed by Beck, and then the PW91 functional developed by Perdew and Wang, the first GGA functional used in large scale [113,114,115].

𝐸 𝑋 𝐵88 [𝑛] = 𝐸 𝑋 𝐿𝐷𝐴 [𝑛] -𝛽 ∑ ∫ 𝑛 𝜎 4 3 ⁄ 𝑥 𝜎 2 (1 + 6𝛽𝑥 𝜎 𝑠𝑖𝑛ℎ -1 𝑥 𝜎 ) 𝜎 𝑑 3 𝑟 ⃗ (2.60)
where 𝛽 is the only parameter with value equal to 0.0042. where 𝑘 𝑓 = (3𝜋 2 𝑛) 1 3 ⁄ and 𝑘 𝑠 = (4𝑘 𝑓 /𝜋) 1 2 ⁄

. 𝐸 ̅ 𝑐 𝑃𝑊-𝐿𝐷𝐴 it is the integral core of the PW-LDA correlation functional. In the GGA term of this functional, there are 11 fundamental constants in total, which are determined to satisfy various exact exchange and correlation functional conditions, incorporate the inhomogeneity of the system and at the same time maintain the good characteristics of the LDA functional. However, several problems such as overparameterization and the complicated form of 𝐸 𝑋𝐶 , caused new GGA functionals to be proposed.

The (2.67)

𝐴 = 𝛽 𝛾 [𝑒𝑥𝑝 (- 𝐸 ̅ 𝑐 𝑃𝑊-𝐿𝐷𝐴 [𝑛] 𝛾𝜙 2 𝑛 ) -1] -1
(2.68)

𝜙 = 1 2 [(1 + 𝜁) 2 3 ⁄ + (1 -𝜁) 2 3 ⁄ ] (2.69)
where 𝛾 = (1 -ln 2)/𝜋 2 = 0.031091, 𝛽 = 0.066725 e 𝜁 = 𝑛 𝛼 -𝑛 𝛽 𝑛 𝛼 + 𝑛 𝛽 ⁄ .

Hybrid functionals

The hybrid functionals, so called because they contain an additional part of exact exchange energy, were created to improve the description of the physical properties of a system such as bond length, ionization energy, vibrational frequencies, among others [89]. Hybrid functionals connect the HF exchange integral with the GGA exchange functionals at a constant ratio, based on the concepts of adiabatic connection, which causes the KS energies of the independent electron model to bond with those of the fully interactive electron

𝐸 𝑋 = ∫ 𝑑𝜆𝐸 𝑋 𝜆 ≈ 𝐸 𝑋 𝐺𝐺𝐴 + 𝜆(𝐸 𝑋 𝐻𝐹 -𝐸 𝑋 𝐺𝐺𝐴 ) 1 0 (2.70)
Hybrid functional term should be understood from the ansatz that the exact energy exchange is situated between the GGA energy exchange functional and the HF exchange integral, and not as the combination of the HF exchange integral with exchange functionals

[89].

The hybrid functional B3LYP, the first proposed hybrid functional, is the most frequently used functional among in all functionals in quantum chemistry calculations and has reasonable accuracy at affordable computational costs [117,118]. This functional use three parameters such as the mixing ratios to form the adiabatic connections between a fixed global fraction 20% HF exchange integral, and the LDA exchange functional and between the LYP-GGA correlation functional and the LDA correlation functional, and to match the GGA attenuated term of the B88 exchange functional

𝐸 𝑋𝐶 𝐵3𝐿𝑌𝑃 = 𝐸 𝑋𝐶 𝐿𝐷𝐴 + 𝑎 1 (𝐸 𝑋 𝐻𝐹 -𝐸 𝑋 𝐿𝐷𝐴 ) + 𝑎 2 ∆𝐸 𝑋 𝐵88 + 𝑎 3 (𝐸 𝐶 𝐿𝑌𝑃 -𝐸 𝐶 𝑉𝑊𝑁-𝐿𝐷𝐴 ) (2.71)
The LYP correlation functional is derived from the Colle-Salvetti energy correlation functional and is a key component of the B3LYP functional, providing very accurate correlation energies in molecular property calculations [119]. In this functional, five semi-empirical parameters, 𝑎 = 0.04918, 𝑏 = 0.7628, 𝑐 = 0.58, 𝑑 = 0.8 and 𝑞 = 2.29 are contained, without fundamental constant, and is written as

𝐸 𝑐 𝐿𝑌𝑃 [𝑛, ∇𝑛, ∇ 2 𝑛] = -∫ 𝑑 3 𝑟 𝑎 1 + 𝑑𝑛 -1 3 ⁄ {𝑛 + 𝑏𝑛 -2 3 ⁄ [𝐶 𝐹 𝑛 5 3 ⁄ -2𝑡 𝑊 + 1 9 (𝑡 𝑊 + 1 2 ∇ 2 𝑛)] 𝑒𝑥𝑝(-𝑐𝑛 -1 3 ⁄ )} (2.72)
where

𝛽 = 𝑞𝑝 1 3 ⁄ (2.73) 𝑡 𝑊 = 1 8 ( |∇𝜌| 2 𝜌 -∇ 2 𝑛) (2.74) 𝐶 𝑓 = 3 10 (3𝜋 2 ) 2 3 ⁄ (2.75)
As shown, the hybrid functionals contain the HF exchange integral, which is not formulated from the electron density, so it is not strictly correct to use the hybrid functionals within the framework of the KS method, which depends on formulated functionals from the electronic density. However, Levy et al. solved this problem by extending the restricted search formulation to the one using functionals containing the HF exchange integral and suggested the generalized KS method as an extension of the KS method [120,121].

As exchange energy contributions are generally much higher than the correlation ones, the non-Coulomb part of exchange functionals, typically decreases very quickly in relation to distance, becoming imprecise at large distances, and inappropriate for calculate some properties such as binding energies van der Waals, electronic excitation spectra, optical response properties and orbital energies [122]. Thus, it is necessary to consider several types of corrections in the exchange functionals to improve the description of the physical properties of a quantum system. Then, range-separated hybrid functionals developed a correction in which the exchange interactions are divided into short-range (SR) and long-range (LR) parts, and then a general exchange functional and the HF exchange integral are adopted in the short-range calculations and long range, respectively. The separation of the SR and LR term is given through the general function (2.79)

𝑏 𝜎 = 𝑒𝑥𝑝 (- 1 4𝑎 𝜎 2 ) -1
(2.80)

𝑐 𝜎 = 2𝑎 𝜎 2 𝑏 𝜎 + 1 2
(2.81)

The LR part of the exchange interaction is expressed with the HF exchange integral

𝐸 𝑋 𝐿𝑅-𝐻𝐹 = - 1 2 ∑ ∑ ∑ ∫ ∫ 𝜓 𝑖𝜎 * (𝑟 ⃗ 1 )𝜓 𝑗𝜎 * (𝑟 ⃗ 1 ) 𝑜𝑐𝑐 𝑗 𝑜𝑐𝑐 𝑖 𝜎 × erf (𝜔𝑟 12 ) 𝑟 12 𝜓 𝑖𝜎 (𝑟 ⃗ 2 )𝜓 𝑗𝜎 (𝑟 ⃗ 2 ) 𝑑 3 𝑟 ⃗ 1 𝑑 3 𝑟 ⃗ 2 (2.82)
where 𝜓 𝑖𝜎 is the i-th molecular σ-spin orbital. The correlation functional uses 20% of the VWN LDA parameterization along with 80% of the LYP GGA functional, similarly to the B3LYP functional.

In empirical functionals, LR correction has also been applied and often produces more accurate results for properties, including atomization energies, than the originals in reference set calculations [124,125]. The first significant advance was made by Becke with the B97 (2.89)

where, 𝑅 𝑟 is the sum of the van der Waals radii of a pair of atoms 𝑖𝑗, and the only nonlinear parameter 𝑎, which controls the strength of the dispersion corrections.

Time-Dependent Kohn-Sham Method

When electrons are under the influence of a generic time-dependent potential, 𝑉 𝑒𝑥𝑡 (𝑟 ⃗, 𝑡), the electronic Hamiltonian of system, 𝐻 ̂𝑒(𝑟 ⃗, 𝑡), also becomes time-dependent and describes a variety of physical and chemical situations, including atoms, molecules and solids, in arbitrary time-dependent electric or magnetic fields, scattering experiments, etc [127]. Thus, the timedependent density functional theory (TD-DFT) extends the basic ideas of the stationary ground state of DFT to the treatment of more general time-dependent excitations or phenomena. TD-DFT can be seen as an alternative formulation of time-dependent quantum mechanics, its basic variable is the electron density of system, 𝑛(𝑟 ⃗, 𝑡) [127].

Runge-Gross theorem

With the HK theorems being the basis of DFT, for the description of periodically time- where the operator 𝑗(𝑟 ⃗) is given by

𝑗(𝑟 ⃗) = - 1 2𝑖 {[∇𝛹 ̂ * (𝑟 ⃗)]𝛹 ̂(𝑟 ⃗) -𝛹 ̂ * (𝑟 ⃗)[∇𝛹 ̂(𝑟 ⃗)]} (
𝑑 𝑑𝑡 [𝑗 ⃗(𝑟 ⃗, 𝑡) -𝑗 ⃗′(𝑟 ⃗, 𝑡)] 𝑡=𝑡 0 = 𝑛 0 (𝑟 ⃗)∇[𝑉(𝑟 ⃗, 𝑡 0 ) -𝑉′(𝑟 ⃗, 𝑡 0 )] (2.100)
As the external potentials are constrained and expandable in Taylor series with respect to the time coordinate around the initial moment 𝑡 0 , we obtain

𝑉(𝑟 ⃗, 𝑡) = ∑ 𝑐 𝑘 (𝑟 ⃗)(𝑡 -𝑡 0 ) 𝑘 ∞ 𝑘=0 (2.101)
where the coefficients 𝑐 𝑘 (𝑟 ⃗) are given by

𝑐 𝑘 (𝑟 ⃗) = 1 𝑘! 𝜕 𝑘 𝜕𝑡 𝑘 𝑉(𝑟 ⃗, 𝑡)| 𝑡=𝑡 0 (2.102)
In addition, the function 𝑢 𝑘 (𝑟 ⃗) is define

𝑢 𝑘 (𝑟 ⃗) = 1 𝑘! 𝜕 𝑘 𝜕𝑡 𝑘 [𝑉(𝑟 ⃗, 𝑡 0 ) -𝑉′(𝑟 ⃗, 𝑡 0 )]| 𝑡=𝑡 0 (2.103)
If the difference between the two potentials is more than a purely time-dependent function, at least one of the expansion coefficients in Taylor expansion around 𝑡 0 will differ by more than one constant

∃ 𝑘≥0 : 𝑢 𝑘 (𝑟 ⃗) ≠ 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (2.104)
If the above condition is satisfied from 𝑘 = 0, that is, the two potentials, 𝑉(𝑟 ⃗, 𝑡) and 𝑉 ′ (𝑟 ⃗, 𝑡), differ since 𝑡 0 . This implies that the left-hand derivative in the equation x is nonzero, since ∇[𝑉(𝑟 ⃗, 𝑡 0 ) -𝑉′(𝑟 ⃗, 𝑡 0 )] ≠ 0. The two current densities 𝑗 ⃗ and 𝑗 ⃗′, will therefore be different for 𝑡 > 𝑡 0 . If 𝑘 is much higher than zero, the equation of motion is applied 𝑘 + 1 times producing the result

𝑑 𝑘+1 𝑑𝑡 𝑘+1 {𝑗 ⃗(𝑟 ⃗, 𝑡) -𝑗 ⃗′(𝑟 ⃗, 𝑡)}| 𝑡=𝑡 0 = 𝑛 0 (𝑟 ⃗)∇𝑢 𝑘 (𝑟 ⃗) (2.105)
The right member of the above equation is nonzero, which again implies that 𝑗 ⃗(𝑟 ⃗, 𝑡) ≠ 𝑗 ⃗′(𝑟 ⃗, 𝑡) for 𝑡 > 𝑡 0 . From the continuity equation,

𝜕 𝜕𝑡 𝑛(𝑟 ⃗, 𝑡) = -∇𝑗 ⃗(𝑟 ⃗, 𝑡) (2.106)
If the above equation is written for the system and prime system and take the difference

Time-dependent Kohn-Sham Equations

The formalism developed by KS has been successful in determining the fundamental state properties in DFT. Therefore, trying to generalize them is the first step towards obtaining the desired properties. As mentioned in the previous subsection, the RG theorem broadly restates that any observable can be calculated if the electron density is known. However, is not told how to calculate this quantity, so, similarly to the Section 2.3.2, a generalization of the KS equations is searched to give all the information about the system. In the time-dependent case, the natural extension of the KS equation has the following form

[- 1 2 ∇ 𝑟 2 + 𝑉 ̂𝑒𝑓𝑓 (𝑟 ⃗, 𝑡)] 𝜙 𝑖 (𝑟 ⃗, 𝑡) = 𝑖 𝜕 𝜕𝑡 𝜙 𝑖 (𝑟 ⃗, 𝑡) (2.111)
where 𝑉 ̂𝑒𝑓𝑓 (𝑟 ⃗, 𝑡) is given by

𝑉 ̂𝑒𝑓𝑓 [𝑟, 𝑡; 𝜌(𝑟, 𝑡)] = 𝑉 𝑒𝑥𝑡 [𝑟, 𝑡] + ∫ 𝑑 3 𝑟 𝜌(𝑟 ′ , 𝑡) |𝑟 -𝑟 ′ | - 𝛿𝑆 𝑋𝐶 [𝜌] 𝛿𝜌(𝑟, 𝑡) (2.112)
The Equation 2.111 using the effective potential in Equation 2.112 is called the timedependent Kohn-Sham (TD-KS) equation [128].

In DFT, 𝑉 𝑋𝐶 [𝜌] is normally written as a derivative of the exchange-correlation energy, as can be seen in Equation 2.51. This result was obtained via total energy minimization, but in the case of TD-DFT this is no longer true [127]. In time-dependent systems, total energy is not a conserved quantity, so there is no variational principle. However, there is the action, a quantity analogous to energy, defined as

𝑆 = ∫ 𝑑𝑡𝜙(𝑡) (𝑖 𝜕 𝜕𝑡 -𝐻 ̂) 𝑡 1 𝑡 0 𝜙(𝑡) (2.113)
and is representable as a density functional, 𝑆[𝜌], and can be decomposed into

𝑆[𝜌] = ∫ 𝑑𝑡𝜙 * (𝑡) [𝑖 𝜕 𝜕𝑡 -(𝑇 ̂+ 𝑉 ̂𝑒𝑒 )] 𝑡 1 𝑡 0 𝜙(𝑡) -∫ 𝑑𝑡 𝑡 1 𝑡 0 ∫ 𝑑 3 𝑟𝜌(𝑟, 𝑡)𝑉 𝑒𝑥𝑡 (𝑟, 𝑡) 𝑡 1 𝑡 0 (2.114)
Two properties of 𝑆[𝜌] are obtained: (i) when performing the functional derivative of the action with respect to 𝜙 * (𝑡) and equaling it to zero, the time-dependent Schrödinger equation is obtained. Thus, the function 𝜙(𝑡) that makes the action stationary will be the timedependent solution of the Schrödinger equation. It should be noted that here there is no "minimum principle", as in the case of DFT, but rather a "stationary principle". (ii) At the solution point, always have 𝑆[𝜌] = 0.

Van Leeuwen using the Keldish formalism, defined a new action that would circumvent the causal problems contained in the theory. So, with the new action, the potential for exchange and correlation could be found [127]. The form of this potential is as follows

𝛿𝑆 𝑋𝐶 [𝜌] 𝛿𝜌(𝑟, 𝑡) ≈ 𝑉 𝑋𝐶 [𝜌](𝑟, 𝑡) = 𝛿𝐸 𝑋𝐶 [𝜌] 𝛿𝜌(𝑟, 𝑡) (2.115)
where 𝑆 𝑋𝐶 is the exchange-correlation part of the action integral and is approximated based on the BO approximation. is approximately the negative of the energy of its HOMO. The proof assumes that the orbitals of the neutral system and those of the resulting ionized system with an electron removed are the same, and that there is no relaxation of the orbitals of the latter. For orbital 𝜙 𝑖 , the orbital energy of the HF equation is represented by

𝜖 𝑖 = ∫ 𝑑 3 𝑟𝜙 𝑖 * (𝑟 ⃗) 𝐹 ̂𝜙𝑖 (𝑟 ⃗) = ℎ 𝑖 + ∑(2𝐽 𝑖𝑗 -𝐾 𝑖𝑗 ) 𝑛 𝑗 (2.128)
where ℎ 𝑖 , 𝐽 𝑖𝑗 and 𝐾 𝑖𝑗 are the one-electron and two-electron integrals, and in this case, the total energy is provided as

𝐸 0 = ∑ ℎ 𝑖 𝑛 𝑖 + ∑(2𝐽 𝑖𝑗 -𝐾 𝑖𝑗 ) 𝑛 𝑖<𝑗 = ∑ 𝜖 𝑖 𝑛 𝑖 -∑(2𝐽 𝑖𝑗 -𝐾 𝑖𝑗 ) 𝑛 𝑖<𝑗 (2.129)
Moreover, the energy after removing one electron from orbital 𝜙 𝑖 derived as

𝐼𝑃 = 𝐸 ′ -𝐸 0 = -ℎ 𝑖 -∑(2𝐽 𝑖𝑗 -𝐾 𝑖𝑗 ) = -𝜖 𝑖 𝑛 𝑗 (2.130)
This indicates that the occupied orbital energies are the corresponding negative of the ionization potentials (IP) [132]. The Koopmans' theorem is also established for unoccupied orbitals, being the unoccupied orbital energies are the corresponding negative of the electron affinities [132].

Considering the energy after adding one electron to an unoccupied orbital as

𝐸 ′′ = 𝐸 0 + ℎ 𝑎 + ∑(2𝐽 𝑎𝑗 -𝐾 𝑎𝑗 ) 𝑛 𝑗 (2.131)
Therefore, the electron affinity, which is the energy difference from 𝐸 ′′ to 𝐸 0 , is proven to be

𝐸𝐴 = 𝐸 0 -𝐸 ′ = -ℎ 𝑎 -∑(2𝐽 𝑎𝑗 -𝐾 𝑎𝑗 ) = 𝑛 𝑗 -𝜖 𝑎 (2.132)

Janak's Theorem

In analogy to HF theory, and according to Koopmans' theorem, the variation in the total energy of a molecule in relation to occupancy coincides with the HF eigenvalue of its spinorbital 𝜙 𝑖 . Regardless of the detailed form of the exchange-correlation functional, it happens in parallel for the DFT theory due to the Janak's theorem [133]. With Janak's theorem, the possibility of describing the energies of excited states also takes shape according to the DFT formalism. Strictly speaking, DFT theory is applicable to the ground state of a system and a situation in which the occupancy number of one (or some) of the states is modified by an infinitesimal amount, which is not necessarily a ground state. The generalization due to Janak's theorem consists in the introduction of occupation numbers of the {𝑛 𝑖 } orbitals for each state.

The 𝑛 𝑖 are allowed occupancy numbers with 0 ≤ 𝑛 𝑖 ≤ 1. Therefore, the relationship between the total electronic energy and orbital energy is proven to be

𝜕𝐸 𝜕𝑛 𝑖 = 𝜖 𝑖 (2.147)
This equation indicates that the derivative of the total electronic energy in relation to the occupancy number of an orbital is identical to the orbital energy, this is called Janak's theorem, and from it is observed that only the highest occupied (HO) orbital can be fractionally, and that all orbitals with 𝜖 𝑖 < 𝜖 𝐻𝑂 must have full occupancy [133]. The energy linearity theorem for fractional occupations proves that the total electronic energy varies linearly as a function of its fractional occupation number, that is,

𝐸 (𝑛 + 𝑝 𝑞 ) = 𝑝 𝑞 𝐸(𝑛 + 1) + 𝑞 -𝑝 𝑞 𝐸(𝑛) (2.148)
and combined with Janak's theorem, the physical meaning of the orbital energies is obtained [134]. Figure 5 clearly indicates that if the total electron energy meets the energy linearity theorem, it is proved by Janak's theorem that the HOMO and LUMO energies are identical to the corresponding negative of the ionization potential and the electron affinity, respectively Source: Elaborated by the author.

CHAPTER 3

Materials

Electron donor-acceptor organic polymers

In the last few decades, considerable progress has been made in the development of BHJ in OSCs based on a blend of a p-type organic semiconductor as donor and an n-type organic semiconductor as acceptor. In general, to produce efficient OSCs, one needs donor and acceptor materials with high charge carrier mobility, complementary absorption bands in the Vis-NIR range, and a small energy offset to minimize voltage losses. In this context, seven molecules described in the next sections, with unique or alternating moieties of electron-rich donor and electron-deficient acceptor have been studied in this work, which ones are seen to be promising materials for photovoltaic application.

Electron donor polymers

Low-bandgap polymers with high HOMO energy level are used as electron donor material of OCSs for harvest more sunlight then reaching high efficiency and present the same structure: a donor core (electron-rich unit) and an acceptor core (electron-deficient unit). This structure is also known as donor-acceptor (D-A) structure.

The poly [(2,6-(4,8-bis(5-(2-ethylhexyl) Source: Elaborated by the author. Source: Elaborated by the author.

Electron acceptor polymers

Polyfullerenes

The association between the electronic properties of fullerene and the mechanical properties, the stability, as well as the processibility of polymers gives a promising mixture that is interesting in photovoltaic applications [143]. Over the years, polyfullerenes have been proposed in the role of n-type material in all-polymer blends and as electron-selective-layer (ESL) in devices [143]. Here was studied the properties of new classes of soluble main-chain polyfullerenes (Figure 13), HSS-8 (short sidechain) and HSS-16 (long sidechain), synthesized by Dr. Roger Hiorns' group (CNRS-IPREM). All previous information about these polyfullerenes are contained in the work of Santos Silva et al. [143]. Source: Elaborated by the author.

material [144,145,146]. The spread material, such as a fatty acid, a polymer or others, is initially dissolved in a volatile solvent, which evaporates after the solution is spread over water [144,145,146]. When the monomolecular layer is compressed through mobile barriers, the molecules orient themselves with the hydrophobic parts rising from the surface, generating a highly organized film in the condensed state [144,145,146]. Figure 15 represents the organization of a Langmuir film. With film compression, the phases of the Langmuir film can be characterized through isotherms of surface pressure versus average molecular area (Figure 16), also known as π-A isotherms [144,145,146]. During film compression, it is possible to characterize three distinct phases of the film: gas phase, in which there is no interaction between molecules; liquid phase, in which the molecules begin to interact with each other; and the solid (condensed) phase, with the molecules arranged in a regular arrangement forming a monomolecular thin film [144,145,146]. When the film is compressed beyond the solid phase, the molecules can clump together disorderly on top of each other causing the film to collapse, thus losing its monomolecular shape To produce ultrathin films, the LB technique is widely used [144,145,146]. However, in the case of some conjugated polymers, the deposition of uniform LB films becomes a problem of great difficulty due to the stiffness of the thin film in the aqueous subphase, which leads to a poor-quality deposition of the film monolayers on solid substrates [144,145,146].

An alternative to this problem would be the use of the LS technique, because, in addition to allowing a better transfer, it also allows a faster transfer of the film monolayers to the solid substrate [144,145,146].

In this work, the Langmuir and LS films were fabricated using a Langmuir KSV trough model 5000. The first procedure for fabricating the films is to clean the trough and compression barriers, using chloroform to avoid contamination in subphase. Therefore, the liquid subphase is poured over the trough, which in this work was approximately 1350 mL of ultrapure water.

The water is obtained from Millipore's water purification system, with a resistivity of 18.2 MΩ.

With the end of the previous procedures, the fabrication of the films starts from the process of spreading the solution.

The solutions were made using the materials HSS8 and HSS16, and the solvents were chloroform and xylene. For fabricate the films, the concentration of the solutions for both, HSS8 and HSS16, were 0.2 mg/mL. For a higher quality in the fabricating process, the control parameters are provided in the KSV trough software. The parameters are: volume of the subphase, concentration of the solution, volume to be spread from the solution, molecular mass of the material to be deposited. In concentration used is considered the mass of the respective polyfullerene and the total volume of the solution. Thus, the solutions were spread in the subphase with the aid of a micro syringe, with the total volume spread equal to 750 µL. After spreading, a waiting time of 15 minutes is necessary, so that the chloroform or xylene evaporates, and thus begins the compression of the barriers. Through the pressure isotherm of the film present in the subphase, a pressure analysis is performed to choose the deposition pressure. In this case, the deposition pressure is equivalent to the point where the film is in the condensed phase with monomolecular thickness. The depositions of the thin film layers on the substrates were carried out at a constant surface pressure equal to 30 mN/m. Symmetrically, the monolayers were compressed by the barriers at a speed of 10 mm/min. For UV absorption were deposited 5 layers onto quartz and for cyclic voltammetry measurements were deposited 30 layers onto ITO.

Optical Characterization in Ultraviolet-Visible (UV-Vis)

One way of optical characterization of thin films is through the measurement of radiation absorption of the sample in the ultraviolet (UV) and visible (Vis) regions of the electromagnetic spectrum, being the regions between 190 and 800 nm where the molecules undergo electronic transitions [148,149].

The transitions are related to the wavelength (λ) of absorption, while the intensity is dependent on the probability of the transition occurring when the molecular system and the radiation interact [148,149]. The electrons in a molecule involved in the transitions are nonbonding paired electrons (n electron), such as in N, O, S and halogens, and the electrons in π orbitals, in double or triple bonds [148,149]. These are the most easily excited electrons and are responsible for most of the electronic spectra in the UV and visible regions [148,149].

The molecules that contain π electrons or non-bonding electrons can absorb energy in the range of ultraviolet or visible light to excite these electrons to unoccupied anti-bonding orbitals, which correspond to the energy level of the excited state (σ* or π orbitals) [148,149].

Therefore, the absorption of radiation results in an electronic transition to an anti-bonding orbital. The most common transitions are from π or n orbitals to π* orbitals, and these are represented by π → π* and n → π* transitions [148,149].

For the UV-Vis characterization of thin films, deposited on ITO substrates, through UV-Vis optical absorption spectroscopy, a VarianCary 100 spectrophotometer was used. The spectrophotometer basically consists of two lamp that produces different wavelengths (deuterium lamp -200 to 400 nm and tungsten lamp -400 to 700 nm + NIR), a selector (diffraction grating) that selects the wavelengths, the sample holder and a detector to determine the intensity with which the sample is traversed for the different wavelengths of incident light, providing, per wavelength, the absorption of light. These measurements were taken from the incidence of light in the range of 200 to 800 nm. The results are presented in the form of unit absorbance.

Cyclic Voltammetry Measurements

Electrochemistry is a powerful tool to probe reactions involving electron transfers [149,150]. Among electrochemical techniques, CV is a powerful and popular technique commonly employed to investigate the reduction and oxidation processes of molecular species. By the measurement of the electric current as a function of the applied voltage, under proper conditions that promote the polarization of a reference electrode, or a work electrode [149,150]. CV is a potential-controlled "reversal" electrochemical experiment, whereas a stationary electrode, immersed in a solution without stirring, provides a current as a function of a triangle-shaped excitation potential signal [149,150]. A linear voltage sweep between two defined values is initially applied. It first goes to the maximum and then returns to the initial value at the same rate [149,150]. When the applied potential polarizes the electrode surface positively, oxidation occurs, while the opposite direction (negatively polarizes) occurs the reduction of the species present at the electrode interface [149,150]. The oxidation and reduction currents are limited by mass transport of analyte from the bulk to the diffuse double layer interface [149,150].

The 'duck-shaped' plot generated by cyclic voltammetry is called a cyclic voltammogram [149,150]. The Figure 18 shows important variables in a cyclic voltammogram as the cathodic peak potential (Epc), the anodic peak potential (Epa), the cathodic peak current (ipc), and the anodic peak current (ipa) [149,150]. In the ipa current reaches peak maximum (point c) for oxidation at the Epa, and the process for reduction mirrors that for the oxidation, only with an opposite scan direction and an ipc at the Epc (point f) [149,150]. Source: Elaborated by the author.

In the literature, it is found that the TD-DFT/CAM-B3LYP method can better describe the excited states than B3LYP method [89,127]. So, a series of calculations was performed by DFT/CAM-B3LYP and TD-DFT/CAM-B3LYP methods using 6-31G(d) basis set, and the results are present in Tables 3 and4. Using TD-DFT/CAM-B3LYP method, in donors, the MSE for LUMO energies is equal to 2.33 eV, and for acceptors, the MSE for LUMO is 1.57 eV.

Thus, for this set of molecules, the correlation for TD-DFT/CAM-B3LYP method is much poor than B3LYP. Source: Elaborated by the author.

As the CAM-B3LYP functional failed to predict the values of HOMO and LUMO, it was necessary to use another functional. The problem in predict LUMO values may be related to what can be described as the many-electron self-interaction error (MSIE) or the localization/delocalization error, wherein this error directly influences the DFT description of the ground and excited states [163,164]. Where, one of the major issues of global hybrid functionals with a fixed fraction of HF exchange is to find a good trade-off between semilocal and HF exchange [163,164]. On the other hand, full HF exchange is needed for a complete correction of self-interaction and, thus, a correct description of the asymptotic region of the xc potential [163,164]. On the other hand, however, semilocal exchange is known to mimic shortrange static correlation effects that are important for chemical bonding [163,164]. So, global hybrids must choose between semilocal and HF exchange. One way to try to correct this error is using range-separated hybrid functionals that have the correct asymptotic behavior and system-specific range-separation parameters [163,164]. Therefore, a series of calculations were carried out using the ωB97-XD functional with 6-31G(d) basis set, in addition to TD-DFT calculations.

The results from HOMO and LUMO values obtained by ωB97-XD and TD-ωB97-XD are presented in Tables 5 and6. For both donor and acceptor molecules, the MSE values for HOMO and LUMO was the same for independent and time-dependent approaches. In the donor molecules, the MSE for HOMO energies is equal to -1.68 eV and for LUMO is 2.84 eV. For the acceptors, the MSE for HOMO energies is -1.48 eV and for LUMO is 2.05 eV. Even with all corrections presented by ωB97-XD functional, the results of the correlations are poorer than B3LYP and CAM-B3LYP functionals. For all these series of calculations, the MSE values for HOMO and LUMO of donor and acceptor molecules were the same for time-independent and time-dependent approaches. As the materials are based on PCBM, it is compelling to start with the interpretation of PCBM isotherms because it is possible to find many literature reports regarding fullerene-type monolayers [165,166]. According to the literature, in PCBM isotherms, it is possible notice the absence of the usual sharp phase transitions of the typical molecule isotherms, and this is a strong indication that this is a liquid-expanded isotherm (according to Harkins' classification), so, the isotherms show a strong indication of the formation disordered aggregates onto the aqueous subphase [165,166]. The cross-surface area of PCBM is 100 Å 2 and the estimated area from the π-A isotherms range from 9.0 to 30 Å 2 ; this means that there is a strong aggregation of PCBM regardless of the solvent used [165,166].

These results may be due the functional group of the molecule generally is oriented toward the water, a vertical build-up of the molecules in the air-water interface, and this can be a major contribution for the area, and therefore the apparent values are reduced [165,166]. As reported by Roncaselli et al., the mean area for the PCBM is 23 and 9 Å 2 for chloroform and xylene solutions, respectively, suggesting that water surfaces are less appealing to the PCBM when coming from these solutions [166]. Surprisingly, even when in good non-polar solvents such as xylene, one cannot ignore the possibility that PCBM readily forms very small nanoaggregates of several units that arises from the strong fullerene-fullerene interactions, hindering the formation of a true monolayer [166].

In Figure 40, it is possible to see the pristine polyfullerenes with isotherms attained at the Langmuir trough. The area per monomer found for HSS8 was 48 and 12 Å 2 for chloroform and xylene solution, respectively, while, for HSS16 was 126 and 33 Å 2 for chloroform and xylene solution, respectively. In regard to the isotherms, it can be observed that the polymers, like the PCBM, can be classified as a liquid-expanded type, not achieving the phase related to a condensed phase either [165,166]. However, the overall isotherm behaviors are quite divergent from one another. The HSS8, both chloroform and xylene solution, presents no distinct phase transitions and has a rather steady upward trend until it reaches the collapse pressure. The collapse pressure for chloroform is 62 mN/m, while, for xylene, the isotherm does not reach the collapse pressure. While the HSS16, the collapse pressure for chloroform is 52 mN/m, and, again, for xylene, the isotherm does not reach the collapse pressure. A possible explanation is that the arrangement and interaction of these materials on the aqueous surface change according to the functional group of each material, which is generally water-oriented, explaining the differences between the areas obtained in the analysis of isotherms, with HSS16

showing the largest area and HSS8 showing the intermediate area in relation to the area of the 108 PCBM [165,166]. Through Tables 20 and21, a relationship can be established between the solvent used and the molecular area obtained in the study of isotherms. The use of different solvents changes the arrangement of molecules because these materials have a low solubility in chloroform, so we found a relatively larger area, while for the xylene solvent, these areas decreased considerably, indicating higher solubility and lower level of aggregation [166]. According to the DFT/6-31+G(d) results presented in Figure 41, HSS8 and HSS16 have structural differences. HSS16 has an elongation (i) in the 'X' direction of about 34% compared to HSS8; (ii) in the 'Y' direction, has an elongation direction of about 19% compared to HSS8; and (iii) in the 'Z' direction, HSS8 has a strong shortening of about 6%. Table 22 presents the areas of the monomers calculated based on the DFT results. To avoid 'crossing' each other, the arms of the HSS16 molecules (although surrounding mostly the fullerene) start to extend. We found that the calculated molecular areas are all higher than the ones estimated in the π-A isotherms of Figure 40 and between the PCM calculations and the experiments is, therefore, to be found elsewhere and in particular in the absence of explicit consideration of the first solvation layer in the PCM model.

Because of its smaller size and the presence of chlorine, chloroform is, of the two solvents used in this work, the one that will most likely favor the setting up of these interactions by inserting itself more easily into the structures of the HSS8 and HSS16 molecules, automatically increasing the volume and the surfaces of these systems [166].

Cyclic Voltammetry

CV is a sophisticated tool to analyze the redox processes of a molecular species, and Source: Elaborated by the author.

In the cyclic voltammograms shown in Figure 42, it is possible to observe that the cast film of HSS8 solubilized in chloroform has three oxidation peaks at -0.41, 0.22 and 1.18 V, and a reduction peak at -0.61 V. In the cast film solubilized in xylene, it has two oxidation peaks at 0.15 and 1.18 V, and a reduction peak at -0.50 V. The LS film of HSS8 solubilized in chloroform (Figure 43) has two oxidation peaks at 0.19 V and 1.25 V, and a reduction peak at -0.54 V, while the LS film solubilized in xylene (Figure 43) shows the same pattern, with two reduction peaks at 0.21 and 1.25 V, and a reduction peak at -0.57 V.

For the HSS16 films, from the cyclic voltammograms in Figure 44, it is possible to observe that the cast film solubilized in chloroform has two oxidation peaks at 0.17 and 1.36

V, and a reduction peak at -0.59 V. The cast film solubilized in xylene shows the same pattern, however, with two oxidation peaks at 0.11 and 1.21 V, and a reduction peak at -0.57 V. The LS film solubilized in chloroform (Figure 45) has three oxidation peaks at -0.50, 0.21 and 1.18 V, and a reduction peak at -0.62 V, while the LS film solubilized in xylene (Figure 45) has two reduction peaks at 0.18 and 1.18 V, and a reduction peak at -0.50 V.

Since the measurements were carried out in solution and not under vacuum, the values must be corrected for each type of reference electrode and solution that was employed. The From the voltammograms, the 𝐸 𝐻𝑂𝑀𝑂 calculated for the cast films of the HSS8, solubilized in chloroform and xylene, for the first oxidation peak is the same for both and equal to -4.23 eV, and for the second oxidation peak also the same for both and equal to -5.16 eV.

This same pattern is observed for the first oxidation peak in the LS films of HSS8, where in both chloroform and xylene, the 𝐸 𝐻𝑂𝑀𝑂 calculated for is-4.12 eV, for the second oxidation peak is -5.28 and -5.37 eV for HSS8 in chloroform and xylene, respectively. The HSS16 films also shows the same pattern as HSS8 films for the first oxidation peak, where for the cast films solubilized in chloroform and xylene, the 𝐸 𝐻𝑂𝑀𝑂 calculated is -4.16 eV, and for the second oxidation peak is -5.36 and -5.17 eV in chloroform and xylene, respectively. For LS films solubilized in chloroform, the 𝐸 𝐻𝑂𝑀𝑂 calculated is -4.17 and -5.20 eV for the first and second oxidation peak, respectively, and in xylene is -4.15 and -5.28 eV for the first and second oxidation peak, respectively.

It was not possible to estimate the electronic affinity in the region of the cathodic potential of the voltammograms (related to the LUMO of the considered material) due to the difficulty to accurately determine the potentials' onset for the second reduction peak. In spite of the electrochemical bandgap being usually higher than the optical bandgap in conjugated polymers, an outcome generally attributed to the formation of free ions in the electrochemical experiment rather than a neutral excited state [172]. So, the optical bandgap can be used in many cases as an approximation for electronic bandgaps, this was already reported for some low-bandgap polymers.

Thence, the optical bandgap energy (𝐸 𝑔𝑎𝑝 𝑜𝑝𝑡 ) was obtained from the UV-Vis absorption Source: Elaborated by the author.

Comparing with PCBM, in its spectrum has three feature peaks of fullerene derivatives, the [6-6]-addition at 260 nm, 330 nm (C60) and 430 nm ([6,6]-addition in C60) [174,175]. In LS films, for both solvents, it is possible to observe the peak referring to C60, and the broad absorption is an overlapping between the [6,6]-addition in C60 and 1,4-bis adducts (445 nm), that would result from the ATRAP due to steric effects [143]. The broad absorption is also indicative of the presence of 1,2-adducts (400 nm) [143]. The spectra show more peaks related to PCBM, it is related to the large proportion of PCBM contained in the materials can overlapping many of the electronic properties of oligomers [143,176].

The use of chloroform and xylene, as solvents, showed a difference in the position of the absorption peaks. This show that the solvent influences not only the aggregation as shown in the study of isotherms, but also the absorption, in which he found that the propensity to form aggregates depends mainly on solvent polarization and has a direct relationship with the maximum absorption [177,178]. Due to the organization of the films, these peaks and bands are more evident in the LS films, and not in the cast films [179].

The 𝐸 𝑔𝑎𝑝 𝑜𝑝𝑡 determined for the LS film of HSS8 solubilized in chloroform and xylene (Figure 47) were 2.38 eV (λonset = 520 nm) for both. For HSS16, the LS film solubilized in chloroform (Figure 49) were 2.32 eV (λonset = 535 nm), and for film solubilized in xylene (Figure 49) were 2.34 eV (λonset = 530 nm). For cast films of HSS8 solubilized in chloroform (Figure 47) were 2.38 eV (λonset = 520 nm), and for the film solubilized in xylene (Figure 47)

were 2.32 eV (λonset = 534 nm). For HSS16, the cast film solubilized in chloroform (Figure 49)

were 2.30 eV (λonset = 542 nm), and for film solubilized in xylene (Figure 49) were 2.32 eV (λonset = 535 nm). From the 𝐸 𝐻𝑂𝑀𝑂 and 𝐸 𝑔𝑎𝑝 𝑜𝑝𝑡 computed heretofore, it is conceivable to estimate the LUMO energy level, 𝐸 𝐿𝑈𝑀𝑂 , by means of the relation 𝐸 𝐿𝑈𝑀𝑂 = 𝐸 𝐻𝑂𝑀𝑂 + 𝐸 𝑔𝑎𝑝 𝑜𝑝𝑡 . Table 23 depicts the values aforementioned attained for the films, that will be used to estimate the energy diagrams. Since there is no previous literature from these exact polyfullerenes, to estimate the energy of HOMO and LUMO levels of the monomers, were also carried out the theoretical calculations using DFT/B3LYP/6-31+G(d) method for optimizations of geometries and energies. Calculations were performed using the Gaussian 09 program package. DFT calculations were combined with the Grimme's D3 method to account for longer range dispersion interactions.

According to the DFT/6-31+G(d) results, for HSS8 monomer in vacuum, the 𝐸 𝐻𝑂𝑀𝑂 is -5.36 eV, and the 𝐸 𝐿𝑈𝑀𝑂 is -3.13 eV. Considering the chloroform effects, the 𝐸 𝐻𝑂𝑀𝑂 and 𝐸 𝐿𝑈𝑀𝑂 is -5.37 and -3.09 eV, respectively. In vacuum, for HSS16 monomer, the 𝐸 𝐻𝑂𝑀𝑂 and 𝐸 𝐿𝑈𝑀𝑂 is -5.36 and -3.18 eV, respectively, and in chloroform, the 𝐸 𝐻𝑂𝑀𝑂 and 𝐸 𝐿𝑈𝑀𝑂 is -5.35 and -3.13 eV, respectively. Comparing with experimental results, it is possible observe that the theoretical results are close to the experimental values calculated from the second oxidation peak, and the discrepancy between experimental values could be related to the CV measurements contains experimental errors and it is noticeable that solid-state packing effects are not included in the DFT calculations, which tend to affect the HOMO and LUMO energy levels in a thin film compared to an isolated monomer as considered in the calculations [180,181,182,183].

CHAPTER 7 7. Conclusion

In this work, in a series of DFT calculations in singlet state, with different functionals and basis set, the B3LYP functional together with 6-311G(d,p) basis set give the best values for HOMO and LUMO, and 6-31G(d) can be used to provide good values for energy levels from frontier molecular orbitals and saving computational resources. For donors, the correlation between the LUMO calculated with experimental results is too poor, while for acceptors, is acceptable. Faced to the problem in DFT calculations predict accurately LUMO values, DFT calculations in triplet states give good correlation for donors, where the LUMO energy is most accurately approximated from the αHOMO energy.

Through the π-A isotherms of polyfullerenes Langmuir films, a strong indication of the formation of disordered aggregates in the aqueous subphase is shown, even when in good nonpolar solvents, such as xylene, one cannot ignore the possibility that the PCBM readily form nanoaggregates. Thus, a relationship is established between the solvent used and the molecular area obtained in the study of isotherms. The use of different solvents changes the arrangement of molecules, indicating higher solubility and lower level of aggregation.

The influence of solvent is also evidenced both in UV-Vis and CV measurements. In UV-Vis measurements was found that the propensity to form aggregates depends mainly on the polarization of the solvent and is directly related to the maximum absorption and difference in the position of the absorption peaks. And in CV measurements, the type of solvent influences the oxidation and reduction peaks of the materials, consequently, in the HOMO and LUMO energy levels.

In the junction between DFT calculations and CV and UV-Vis measurements allowed the study of optoelectronic properties. The DFT/B3LYP/6-31+G(d) method provide LUMO values close to experimental values, thus being an important tool for comparing results, since there is no previous literature from these exact polyfullerenes.

Figure 1 .

 1 Figure 1. The highest efficiencies for solar cells from 1976 to the present.............................

Figure 2 .

 2 Figure 2. Explanation of basic operational physics that is unique to OSCs. ...........................

Figure 3 .

 3 Figure 3. Schematic diagram showing the energy level of the materials and the ...................

Figure 4 .

 4 Figure 4. Schematic diagram of the BHJ in 2D and 3D in an OSC. .......................................

Figure 5 .

 5 Figure 5. Schematic diagram of total electronic energy as a function of fractional occupation number variation, Δn. ...............................................................................................................

Figure 6 .

 6 Figure 6. Molecular structure of PBDB-T. ..............................................................................

Figure 7 .

 7 Figure 7. Molecular structure of PM6. ....................................................................................

Figure 8 .

 8 Figure 8. Molecular structure of PTB7-Th. .............................................................................

Figure 9 .

 9 Figure 9. Molecular structure of PC71BM. ..............................................................................

Figure 10 .

 10 Figure 10. Molecular structure of Y6. .....................................................................................

Figure 11 .

 11 Figure 11. Molecular structure of ITIC. ..................................................................................

Figure 12 .

 12 Figure 12. Molecular structure of O6T-4F. .............................................................................

Figure 13 .

 13 Figure 13. Molecular structure of di-bromo monomer. ...........................................................

Figure 14 .

 14 Figure 14. Langmuir trough .....................................................................................................

Figure 15 .

 15 Figure 15. Schematic representation of Langmuir principle. ..................................................

Figure 16 .

 16 Figure 16. π-A isotherms of a phospholipid in different phases..............................................

Figure 17 .

 17 Figure 17. LB deposition (to the left) and LS deposition (to the right). .................................. Figura 18. The 'duck-shaped' plot generated by cyclic voltammetry. ....................................

Figure 19 .

 19 Figure 19. An electrochemical cell setup for CV experiments. ...............................................

Figure 20 .

 20 Figure 20. Optimized structure of PBDB-T monomer. ...........................................................

Figure 21 .

 21 Figure 21. HOMO (to the left) and LUMO (to the right) orbitals of PBDB-T monomer. ......

Figure 22 .

 22 Figure 22. Optimized structure of PM6 monomer...................................................................

Figure 23 .

 23 Figure 23. HOMO (to the left) and LUMO (to the right) orbitals of PM6 monomer. ............

Figure 24 .

 24 Figure 24. Optimized structure of PTB7-Th monomer. ..........................................................

Figure 25 .

 25 Figure 25. HOMO (to the left) and LUMO (to the right) orbitals of PTB7-Th monomer. .....

Figure 26 .

 26 Figure 26. Optimized structure of Y6 monomer. ....................................................................

Figure 27 .

 27 Figure 27. HOMO (to the left) and LUMO (to the right) orbitals of Y6 monomer. ...............

Figure 28 .

 28 Figure 28. Optimized structure of ITIC monomer. ..................................................................

Figure 29 .

 29 Figure 29. HOMO (to the right) and LUMO (to the left) orbitals of ITIC monomer. ............

Figure 30 .

 30 Figure 30. Optimized structure of O6T-4F monomer. ............................................................

Figure 31 .

 31 Figure 31. HOMO (to the right) and LUMO (to the left) orbitals of O6T-4F monomer. .......

Figure 32 .Figure 33 .Figure 34 .Figure 35 .Figure 36 .Figure 37 .Figure 38 .Figure 39 .Figure 42 .Figure 43 .Figure 44 .Figure 45 .Figure 46 .Figure 47 .Figure 48 .Figure 49 .

 32333435363738394243444546474849 Figure 32. Optimized structure of PC 71 BM monomer............................................................. 93 Figure 33. HOMO (to the right) and LUMO (to the left) orbitals of PC71BM monomer. ...... 93 Figure 34. Diagram energy obtained by DFT methods for donor molecules. ......................... 97 Figure 35. Diagram energy obtained by TD-DFT methods for donor molecules. .................. 97 Figure 36. Diagram energy obtained by DFT methods for acceptor molecules. ..................... 98 Figure 37. Diagram energy obtained by TD-DFT methods for acceptor molecules. .............. 98 Figure 38. Energy diagram comparing HOMO and LUMO values in singlet and triplet states for donor molecules. ............................................................................................................... 104 Figure 39. Energy diagram comparing HOMO and LUMO values in singlet and triplet states for acceptor molecules. ........................................................................................................... 105 Figure 40. π-A isotherms of HSS8 (to the up) and HSS16 (to the down). ........................... 106 Figure 41. Theoretical DFT model of the disposition of HSS8 and HSS16 monomers in different orientations............................................................................................................... 109 Figure 42. Cyclic voltammogram for cast films of HSS8 solubilized in chloroform (to the left) and xylene (to the right).......................................................................................................... 111 Figure 43. Cyclic voltammogram for LS films of HSS8 (30 layers) solubilized in chloroform (to the left) and xylene (to the right)....................................................................................... 111 Figure 44. Cyclic voltammogram for cast films of HSS16 solubilized in chloroform (to the left) and xylene (to the right).......................................................................................................... 111 Figure 45. Cyclic voltammogram for LS films of HSS16 (30 layers) solubilized in chloroform (to the left) and xylene (to the right)....................................................................................... 112 Figure 46. Absorption spectra for cast films of HSS8 solubilized in chloroform (to the left) and xylene (to the right). ............................................................................................................... 114 Figure 47. Absorption spectra for LS films of HSS8 solubilized in chloroform (to the left) and xylene (to the right). ............................................................................................................... 114 Figure 48. Absorption spectra for cast films of HSS16 solubilized in chloroform (to the left) and xylene (to the right).......................................................................................................... 115 Figure 49. Absorption spectra for LS films of HSS16 solubilized in chloroform (to the left) and xylene (to the right). ............................................................................................................... 115

  Photovoltaic: PV

  study and a vast potential for application, since solar energy is the only inexhaustible source of energy on Earth[5]. One of the tools for converting solar energy into electrical energy are solar cells or photovoltaic devices[6]. Consequently, studies aimed at improving the efficiency of devices in converting solar energy into electrical energy become essential[6].The basic principle of operation of a solar cell is the photovoltaic effect that was first observed by the French physicist Alexandre EdmondBecquerel (1820Becquerel ( -1891) ) in 1839[7, 8].Becquerel observed the generation of electric current illuminating platinum electrodes, covered with silver bromides or silver chloride, immersed in electrolyte solution[7, 8]. The first solar cell, consisting of a layer of selenium covered by a thin film of gold, was fabricated and studied by Charles Fritts in 1884, but had a very low efficiency [9]. The first solar cell with reasonable efficiency was developed at Bell Laboratories, in 1953, through studies of energy sources and diffusion in the solid state at p-n junction by Pearson, Fuller and Chapin [10]. The solar cell was fabricated using silicon (Si) and achieved an efficiency of around 6% [10]. In the 1960s, with the advancement of technology, the efficiency of a silicon solar cell, with AM1 illumination, was 12% to 13%. The violet cell reached an efficiency of 16% in 1973 [11]. A Sibased solar cell using a non-reflecting light system (black cell), in 1974, showed an efficiency between 18% and 19% [11]. Nowadays, Geisz and coworkers fabricated in the laboratory an inorganic multijunction solar cell (six-junction III-V) reaching an efficiency of 47.1% [12]. The device has six junctions of active layers based on gallium arsenide, a very expensive material in relation to common Si, in addition to a complex and relatively expensive fabrication process due to the much slower vapor deposition processes used [12]. Therefore, efforts continue in the search and development of low-cost solar cells. The technological evolution of solar cells is usually divided into three categories called as first, second and third generation devices [6]. The first-generation one is based on solar cells fabricated using monocrystalline and polycrystalline Si wafers [6]. The devices in this generation present an efficiency of up to 24% and are well established in the market, however they need a large amount of Si in the wafers, which raises their costs [6]. The second generation one innovates with the fabrication of flexible devices using thin film technology of semiconductor materials such as amorphous Si, cadmium telluride (CdTe), and copper indium gallium selenide (CIGS) [6]. This generation introduces new features such as new fabrication techniques reduced amount of material leading to reduce costs compared to the first generation [6]. The third generation one brings the exploration of new materials and architectures in solar cells, seeking to lower production costs and higher efficiencies [6]. Some examples of such devices are multijunction solar cells (MJSCs), dye sensitized solar cells (DSSCs), quantum dotsensitized solar cells, organic solar cells (OSCs), and perovskite solar cells (PSCs) [6]. The chart in Figure 1 shows the highest efficiencies for solar cells from 1976 to the present, being divided into 5 categories (multifunctional cells, single junction gallium arsenide cells, crystalline silicon cells, thin film technologies, photovoltaic (PV) emerging), and subdivided into 28 subcategories [13]. The chart is maintained and updated every year by the National Renewable Energy Laboratory which is a national laboratory of the U.S. Department of Energy [14].

Figure 1 .

 1 Figure 1. The highest efficiencies for solar cells from 1976 to the present.

Figure 2 .

 2 Figure 2. Explanation of basic operational physics that is unique to OSCs.

Figure 3 .

 3 Figure 3. Schematic diagram showing the energy level of the materials and the charge cascade process in a BHJ.

Figure 4 .

 4 Figure 4. Schematic diagram of the BHJ in 2D and 3D in an OSC.

[

  39]. Meng et al. fabricating a device with two layers of BHJs, each combining different electron donor and acceptor polymers, achieved an efficiency of 17.3% [40]. In the first layer, a mixture of donor and non-fullerene acceptors was used, and in the second layer the donor polymer was mixed with fullerene and non-fullerene acceptor [40]. The final overlap of the absorbances showed that the materials absorbed a large part of the solar spectrum [40]. However, the device required complex and careful fabrication with several layers, including a zinc oxide (ZnO) one to ensure charge transfer between the two layers and the electrodes [40].

  variably the Schrodinger equation in the eigenstates by using a basis of Slater determinants or configurations. These are Configuration Interaction (C.I.) type approaches [45]; 2) perturbation methods requiring the definition of a minimal basis and a partition of the Hamiltonian operator adapted to ensure the convergence of the perturbation series in a Rayleigh-Schrodinger type approach [46]; 3) Pairwise methods which consist in evaluating the correlation energy of pairs of electrons assumed without interactions (Independent Electron Pair Approximation -IEPA)

  2.14) here the moment 𝑝 𝑖 is transformed into the moment operator 𝑝 𝑖 = -𝑖ħ∇ 𝑖 , which produces discrete energy levels under the boundary condition of the potential 𝑉 to make the wave functions finite [78, 79]. The discrete energy values and the corresponding wave functions are called eigenvalues and eigenfunctions, respectively [78, 79]. Since the Schrödinger equation was developed, various realistic and idealistic interpretations of the wavefunction has been vigorously discussed [78]. Schrödinger originally regarded the wavefunction as a description of real physical wave, where the physical existence of a matter consists only of waves [78, 80]. As this interpretation has many serious problems, other interpretations soon emerged. Born proposed the probabilistic interpretation of the wave function, becoming the mainstream interpretation of the wavefunction today [78, 80]. According to Born, what has direct physical meaning been not spatial-temporal wave functions, which may even be imaginary functions [81]. The physical meaning is in the magnitude |𝛹(𝑟 ⃗, 𝑡)| 2 [81]. The squared modulus of the wave function gives the probability density probability of finding the particle at point 𝑥, at time 𝑡: 𝜌(𝑥, 𝑦, 𝑧, 𝑡) = 𝑑𝑃(𝑟 ⃗, 𝑡) 𝑑𝑉 = |𝛹(𝑟 ⃗, 𝑡)| 2 = 𝛹 * (𝑟 ⃗, 𝑡)𝛹(𝑟 ⃗, 𝑡) (2.15) where 𝑑𝑃(𝑟 ⃗, 𝑡) is the probability that a single particle is at position 𝑟 ⃗, inside the volume 𝑑𝑉 = 𝑑𝑥𝑑𝑦𝑑𝑧 (in Cartesian coordinates), in the time 𝑡, per unit of 𝑑𝑉. This interpretation is considered idealist (orthodox position) and only focuses attention on the existence probability in discussions on the wave-particle relation, where the observations not only disturb what is to be measured, they produce it, and the particle assume a definite position [78, 80]. Eventually, the Copenhagen school, led by Bohr, Heisenberg and Pauli, advocated Born's probabilistic interpretation [82]. Some alternative realistic interpretations of the wave function have been proposed and widely studied, as for example, the Schrödinger's cat and Everett's many-worlds interpretation [83, 84].

  21)Although the Schrödinger equation for the electronic part (Equation 2.21) is simpler, its analytical solution is not yet possible (quantum many-body problem), except for extremely simple systems such as the hydrogen atom and hydrogen-like atoms[80]. The numerical solution is hardly viable due to the large number of variables, with three spatial coordinates per electron, in addition to the spin[87]. Thus, over the years, new quantum methods have been proposed to reduce the cost of this type of calculation and obtain information from 𝑁-electron atom systems[89].

  30) When the Coulomb operator acts on the wave function it generates the Coulomb integral [89]. The Coulomb integral is the potential energy of electrostatic repulsion between two electrons in the same orbital [89]. Coulomb interactions are described as classical mechanical interactions between electrically charged particles [89]. When the exchange operator acts on the wave function, the exchange integral is generated, where the electron is in an orbital on one side of the operator and a different orbital on the other side [89]. Unlike Coulomb interaction, exchange interactions are purely quantum interactions, have no classical analogue and are related to the antisymmetry of the wave function [89]. Due to these interactions, the total electronic energies are reduced, stabilizing the electronic states. A significant proportion of the exchange interactions are absorbed by the self-interactions of the electrons themselves [89]. Consequently, these self-exchange interactions remove the Coulomb potential of an electron [89]. The exchange interactions increase orbital overlap to produce attractions between orbitals [89]. As a result, these attractions lead to the delocalization of electron distribution, which causes the far-reaching nature of exchange interactions [89].

  𝛿 {𝐸[𝑛(𝑟 ⃗)] -𝜇 [∫ 𝑛(𝑟 ⃗)𝑑 3 𝑟 -𝑁]} = 0 (2.42) Equation 2.43 is the fundamental equation of the DFT, being the third part of the equality written as a function of the external potential.

  45) Since all 𝜙 𝑖 𝐾𝑆 (𝑟 ⃗) are electron density functionals, Equation 2.45 is an explicit functional of the orbitals and an implicit one of the electron density, since the electron density unequivocally determines the KS orbitals. Of the other terms in Equation 2.44, the nuclearelectron interaction potential, 𝑉 𝑛𝑒 [𝑛(𝑟 ⃗)], is given by 𝑉 𝑛𝑒 [𝑛(𝑟 ⃗)] = ∫ 𝑉 ̂𝑛𝑒 𝑛(𝑟 ⃗)𝑑 3 𝑟 = -∑ ∫ 𝑍 𝛼 𝑛(𝑟 ⃗) |𝑅 ⃗⃗ 𝛼 -𝑟 ⃗| 𝑀 𝛼=1 𝑑 3 𝑟 (2.46) and 𝑉 𝐻 [𝑛(𝑟 ⃗)] is the Hartree potential calculated by 𝑉 𝐻 [𝑛(𝑟 ⃗)is responsible for the classical electrostatic interaction between electrons, and should not be confused with the electron-electron interaction presented in the HK formalism of Equation 2.40, since 𝑉 𝑒𝑒 [𝑛(𝑟 ⃗)] considers the contributions of 𝑉 𝐻 [𝑛(𝑟 ⃗)] and nonclassical terms. The term 𝐸 𝑋𝐶 [𝑛(𝑟 ⃗)] gets the name of exchange-correlation energy, and considers the exchange term, the correlation term, the kinetic energy correction term and the electrons self-interaction correction term. 𝐸 𝑋𝐶 [𝑛(𝑟 ⃗)] = (𝑇[𝑛(𝑟 ⃗)] -𝑇 0 [𝑛(𝑟 ⃗)]) + 𝑉 𝐶 [𝑛(𝑟 ⃗)] + 𝑉 𝑋 [𝑛(𝑟 ⃗)] (2.48) It is possible to minimize the energy functional again, to obtain the ground-state electronic density, by means of the Lagrange multipliers, with the link that the electronic density integral over all space is equal to the number of electrons in the system. Mathematically, this idea is represented by 𝛿 {𝐸[𝑛(𝑟 ⃗)] -𝜇 [∫ 𝑛(𝑟 ⃗)𝑑 3 𝑟 -𝑁]} = 0 (2.49) where, 𝜇 is the Lagrange multiplier, and have the physical sense of the chemical potential of the system. So, the solution of the Equation. 2.49, leads to the equation called the KS equation ̂𝑒𝑓𝑓 (𝑟 ⃗)] 𝜙 𝑖 𝐾𝑆 (𝑟 ⃗) = 𝜖 𝑖 𝐾𝑆 𝜙 𝑖 𝐾𝑆 (𝑟 ⃗) (2.50) with 𝑉 ̂𝑒𝑓𝑓 (𝑟 ⃗) being the effective KS potential composed by Hartree potential, the electronnucleus interaction and the exchange-correlation potential, with the exchange-correlation potential given by the first derivative of the exchange-correlation energy functional with respect KS equation is also transformed into a matrix equation based on the Roothaan method. Similar to the HF equation, a Kohn-Sham-Roothaan (KSR) equation is written as 𝑭𝑪 𝒊 = 𝜺𝑺𝑪 𝒊 (2.52) The 𝑪 𝒊 matrix is the coefficients matrix, whose elements are the weighting factors 𝐶 𝑝𝑖 that determine to what extent each basis function ϕ (roughly, each atomic orbital on an atom) contributes to each MO 𝜙. 𝑺 matrix is the overlap matrix, whose elements are overlap integrals 𝑆 𝑖𝑗 which are a measure of how well pairs of basis functions (roughly, atomic orbitals) overlap.

  ⃗, 𝑡) -𝑛′(𝑟 ⃗, 𝑡)} = -∇{𝑗 ⃗(𝑟 ⃗, 𝑡) -𝑗 ⃗′(𝑟 ⃗, 𝑡)} (2.107) With the expression involving the 𝑘-th derivative of the external potential. Taking (𝑘 + 1) times in the time derivative in the equation above, 𝑑 𝑘+2 𝑑𝑡 𝑘+2 {𝑛(𝑟 ⃗, 𝑡) -𝑛′(𝑟 ⃗, 𝑡)}| 𝑡=𝑡 0 = -∇ 𝑑 𝑘+1 𝑑𝑡 𝑘+1 {𝑗 ⃗(𝑟 ⃗, 𝑡) -𝑗 ⃗′(𝑟 ⃗, 𝑡)}| 𝑡=𝑡 0 (2.108) substituting the expression y, 𝑑 𝑘+2 𝑑𝑡 𝑘+2 {𝑛(𝑟 ⃗, 𝑡) -𝑛′(𝑟 ⃗, 𝑡)}| 𝑡=𝑡 0 = -∇{𝑛 0 (𝑟 ⃗)∇𝑢 𝑘 (𝑟 ⃗)} (2.109) by the restriction of 𝑢 𝑘 (𝑟 ⃗) ≠ 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, ∇{𝑛 0 (𝑟 ⃗)∇𝑢 𝑘 (𝑟 ⃗)} ≠ 0 (2.110) therefore, 𝑛(𝑟 ⃗, 𝑡) ≠ 𝑛′(𝑟 ⃗, 𝑡). Thus, from the proof, the Runge-Gross theorem is then the analogue of the Hohenberg-Kohn theorem for the time-dependent system [128].

Figure 5 .

 5 Figure 5. Schematic diagram of total electronic energy as a function of fractional occupation number variation, Δn.

  Figure 6.

Figure 6 .

 6 Figure 6. Molecular structure of PBDB-T.

Figure 7 .

 7 Figure 7. Molecular structure of PM6.

Figure 8 .

 8 Figure 8. Molecular structure of PTB7-Th.

Fullerenes

  are a family of large all-carbon cage molecules [138]. The most abundant and stable form of fullerene is buckminster fullerene (C60) first discovered in 1985 by Kroto, Heath, O'Brien, Curl and Smalley during experiments on the mechanism of formation of longchain carbon molecules in interstellar space and circumstellar shells [138]. The low solubility makes fullerenes difficult to handle [34]. It is well-known that this problem can be in part solved by the chemical functionalization of this carbon allotrope [34]. So, fullerenes (Figure 9) and their derivates have been predominately used as electron acceptor of OSCs, because of their isotropic charge transport, high electron mobility and high LUMO energy level [34].

Figure 9 .

 9 Figure 9. Molecular structure of PC 71 BM.

Figure 11 .

 11 Figure 11. Molecular structure of ITIC.

Figure 12 .

 12 Figure 12. Molecular structure of O6T-4F.

Figure 13 .

 13 Figure 13. Molecular structure of di-bromo monomer.

Figure 15 .

 15 Figure 15. Schematic representation of Langmuir principle.

Figure 17 .

 17 Figure 17. LB deposition (to the left) and LS deposition (to the right).

Figura 18 .

 18 Figura 18. The 'duck-shaped' plot generated by cyclic voltammetry.

Figure 19 .

 19 Figure 19. An electrochemical cell setup for CV experiments.

Figure 20 .

 20 Figure 20. Optimized structure of PBDB-T monomer.

Figure 21 .

 21 Figure 21. HOMO (to the left) and LUMO (to the right) orbitals of PBDB-T monomer.

Figure 32 .

 32 Figure 32. Optimized structure of PC 71 BM monomer.

Figure 33 .

 33 Figure 33. HOMO (to the right) and LUMO (to the left) orbitals of PC71BM monomer.

  After a series of calculations using different functionals and maintaining the 6-31G(d) basis it was possible to set up a diagram energy for independent (Figure34and 36) and timedependent (Figure35and 37) values.

Figure 34 .

 34 Figure 34. Diagram energy obtained by DFT methods for donor molecules.

Figure 35 .

 35 Figure 35. Diagram energy obtained by TD-DFT methods for donor molecules.

Figure 39 .

 39 Figure 39. Energy diagram comparing HOMO and LUMO values in singlet and triplet states for acceptor molecules.

Figure 41 .

 41 Figure 41. Theoretical DFT model of the disposition of HSS8 and HSS16 monomers in different orientations.

  these processes provide a way to investigate the electronic bandgap diagram of the chosen material[168, 169]. Whereas the HOMO and LUMO of a material are related to the ionization potential (by oxidation potential) and the electron affinity (by reduction potential), respectively[168, 169]. Figures 42 to 45 show the cyclic voltammograms for cast and LS films of the HSS8 and HSS16 materials, solubilized in chloroform and xylene, on an ITO substrate, in a 0.1 mol/L solution of TBAP in acetonitrile, the sweeping was carried out at a scan rate of 50 mV/s.

Figure 42 .

 42 Figure 42. Cyclic voltammogram for cast films of HSS8 solubilized in chloroform (to the left) and xylene (to the right).

Figure 43 .

 43 Figure 43. Cyclic voltammogram for LS films of HSS8 (30 layers) solubilized in chloroform (to the left) and xylene (to the right).

Figure 44 .

 44 Figure 44. Cyclic voltammogram for cast films of HSS16 solubilized in chloroform (to the left) and xylene (to the right).

Figure 45 .

 45 Figure 45. Cyclic voltammogram for LS films of HSS16 (30 layers) solubilized in chloroform (to the left) and xylene (to the right).

  voltammograms show the ionization potential of the thin films at the point where the start of the current increase (onset) occurs before the first oxidation peak. For the HSS8/Cast/Chloroform and HSS16/LS/Chloroform films, the onset before the second oxidation peak was used, due to the difficulty in accurately determine the onset before the first oxidation peak. The energy level for the HOMO can be calculated through the relation 𝐸 𝐻𝑂𝑀𝑂 = 113 -(𝐸 𝑂𝑋 ′ + 4.40), and for the Ag/AgCl/KCl electrode, the correction of 4.40 eV was made [170,171].

  measurements, the wavelength related to the optical bandgap is attained by the edge of the absorption spectrum of the film, and allows the calculation of the 𝐸 𝑔𝑎𝑝𝑜𝑝𝑡 [173]. The 𝐸 𝑔𝑎𝑝 𝑜𝑝𝑡 can h represents the Planck's constant in eV•s and c the speed of light in vacuum, in m/s.In cast films, the absorption spectrum of HSS8 in chloroform (Figure46) has two peaks at 265 and 355 nm, while in xylene (Figure46) has three peaks at 227, 277 and 348 nm. Both films have a broad absorption maximum between 420 and 540 nm, which is possibly due to the presence of two or more absorption bands in that region of the spectrum. For HSS16 in chloroform (Figure48), the absorption spectrum has three peaks at 235, 270 and 350 nm, while in xylene (Figure48) also has three peaks, but at 233, 275 and 348 nm. For both solvents, the 114 broad absorption maximum is in the same range as for the HSS8. The absorption spectrum of LS films of HSS8 in chloroform (Figure47) has three peaks at 200, 270 and 345 nm, while in xylene (Figure47) also has three absorption peaks, but with values at 200, 270 and 348 nm.Both films have a broad absorption maximum between 400 and 500 nm. For HSS16 in chloroform (Figure49), has thee peaks at 200, 267 and 340 nm, and in xylene (Figure49) has thee peaks at 209, 267 and 348 nm. Both films have a broad absorption maximum between 400 and 500 nm.

Figure 46 .

 46 Figure 46. Absorption spectra for cast films of HSS8 solubilized in chloroform (to the left) and xylene (to the right).

Figure 47 .

 47 Figure 47. Absorption spectra for LS films of HSS8 solubilized in chloroform (to the left) and xylene (to the right).

Figure 48 .

 48 Figure 48. Absorption spectra for cast films of HSS16 solubilized in chloroform (to the left) and xylene (to the right).

Figure 49 .

 49 Figure 49. Absorption spectra for LS films of HSS16 solubilized in chloroform (to the left) and xylene (to the right).

Table 1 .

 1 HOMO and LUMO values obtained by DFT/B3LYP/6-31G(d) method. ................. 93

Table 2 .

 2 HOMO and LUMO values obtained by TD-DFT/B3LYP/6-31G(d) method. .......... 94

Table 3 .

 3 HOMO and LUMO values obtained by DFT/CAM-B3LYP/6-31G(d) method. ...... 95

Table 4 .

 4 HOMO and LUMO values obtained by TD-DFT/CAM-B3LYP/6-31G(d) method. 95

Table 5 .

 5 HOMO and LUMO values obtained by DFT/ωB97-XD/6-31G(d) method.............. 96

Table 6 .

 6 HOMO and LUMO values obtained by TD-DFT/ωB97-XD/6-31G(d) method. ...... 96

Table 7 .

 7 HOMO and LUMO values from DFT/B3LYP/6-31G(d,p) method. ......................... 99

Table 8 .

 8 HOMO and LUMO values from TD-DFT/B3LYP/6-31G(d,p) method. .................. 99

Table 9 .

 9 HOMO and LUMO values from DFT/CAM-B3LYP/6-31G(d,p) method. .............. 99

Table 10 .

 10 HOMO and LUMO values from TD-DFT/CAM-B3LYP/6-31G(d,p) method. .... 100

Table 11 .

 11 HOMO and LUMO values from DFT/ωB97-XD /6-31G(d,p) method................. 100

Table 12 .

 12 HOMO and LUMO values from DFT/TD-ωB97-XD/6-31G(d,p) method. .......... 100

Table 13 .

 13 HOMO and LUMO values from DFT/B3LYP /6-311G(d,p) method. .................. 101

Table 14 .

 14 HOMO and LUMO values from DFT/TD-B3LYP/6-311G(d,p) method. ............ 101

Table 15 .

 15 HOMO and LUMO values from DFT/CAM-B3LYP/6-311G(d,p) method. ........ 101

Table 16 .

 16

  SUMMARY CHAPTER 1 ............................................................................................................................. 1. Introduction .......................................................................................................................... 1.1 Organic solar cells........................................................................................................... 1.2 Molecular modelling techniques ..................................................................................... 1.3 Objectives ....................................................................................................................... CHAPTER 2 ............................................................................................................................. 2. Methodology ......................................................................................................................... Langmuir and Langmuir-Schaefer Films ................................................................. 4.1.2 Optical Characterization in Ultraviolet-Visible (UV-Vis) ....................................... 4.1.3 Cyclic Voltammetry Measurements ......................................................................... CHAPTER 5 ............................................................................................................................. 5. Orbital energy modeling for electron donor and acceptor molecules .................................. CHAPTER 6 ........................................................................................................................... 6. Polyfullerenes ..................................................................................................................... 6.1 Langmuir-Schaefer thin films .......................................................................................

	CHAPTER 1
	1. Introduction
	1.1 Organic solar cells

2.1. The Schrodinger equation .............................................................................................. 2.2 The many-body Schrodinger equation ............................................................................ 2.3 The Hartree-Fock method ............................................................................................... 2.3.1 Basis sets .................................................................................................................. 2.4 Density Functional Theory ............................................................................................. 2.4.1 Thomas-Fermi-Dirac method ................................................................................... 2.4.2 Hohenberg-Kohn formalism .................................................................................... 2.4.3 Kohn-Sham Formalism ............................................................................................ 2.4.3 Exchange-correlation functional .............................................................................. 2.4.4 Time-Dependent Kohn-Sham Method .................................................................... 2.4.5 Orbital Energy .......................................................................................................... CHAPTER 3 ............................................................................................................................. 3. Materials ............................................................................................................................... 3.1 Electron donor-acceptor organic molecules .................................................................... 3.1.1 Electron donor molecules ......................................................................................... 3.1.2 Electron acceptor molecules..................................................................................... 3.2 Polyfullerenes ................................................................................................................. CHAPTER 4 ............................................................................................................................. 4.1 Fabrication of thin films and characterization .................................................................... 4.1.1 6.2 Cyclic Voltammetry ...................................................................................................... CHAPTER 7 ........................................................................................................................... 7. Conclusion .......................................................................................................................... REFERENCES .......................................................................................................................

  . A complete and extensive discussion on the evolution of this concept is done by Stowasser and Hoffmann, but we can jump to the general conclusion achieved by

	Baerends et al., who argue that these orbitals are very suitable for qualitative, chemical
	applications [55, 56]. That is why care must be taken if one wants absolute LUMO energy
	values when designing new molecules [57]. Furthermore, identifying the open-circuit voltage
	(V OC ) via the LUMO energies relies on a one-electron picture (orbital relaxation is not
	considered) and on the validity of the Koopmans' theorem [58].
	In the same time, the calculation of the LUMO energies, can be done experimentally by
	electrochemical methods, as cyclic voltammetry (CV), and the values so-obtained do not
	considering approximations to the energies of this orbital as well. The usual procedure consists
	of measuring the onset reduction potential of the molecule to calculate LUMO energy and
	correlate it with the electron affinity of a standard molecule, assuming the validity of
	Koopmans' theorem [59].
	McCormick et al. reported in their very useful work on conjugated polymers that the
	use of DFT methods could be justified in the OSCs context [60]. We believe that the approach
	proposed by McCormick is even the better compromise disponible available today between
	theoretical DFT LUMO estimation and experimental CV LUMO energy measurement. In order
	to situate the limits avoided by McCormick and collaborators by the use of DFT in the OSCs
	context it is necessary look at to the strengths and weaknesses of each of the computational
	techniques disponible for the modeler.
	Post-DFT treatments appropriate to describe open shell states exist and allow to solve
	all or part of the problems above mentioned. Let us first mention the Broken-Symmetry (BS)

  functions used to represent molecular orbitals being written in terms of functions that represent atomic orbitals, thus forming the. The linear combination of atomic orbitals-molecular orbitals (LCAO-MO) approximation was proposed byLennard-Jones, in 1929 [97]. Subsequently, Roothan's method was initially implemented for computational calculations in HF methods. It is noteworthy thatHall, also in 1951, independently suggested this method, which is also called the.In LCAO-MO approximation the number of molecular orbitals (molecular eigenfunctions) is equal to the number of atomic orbitals (molecular eigenfunctions) included in the linear expansion. In a sense, 𝑛 𝐴𝑂 atomic orbitals combine to form 𝑛 𝑀𝑂 molecular orbitals, which can be numbered 𝑝 = 𝑖 to 𝑛 𝐴𝑂 and which may not all be the same. The expression (linear expansion) for the {𝜙 𝑖 } molecular orbital can be expanded as: Gaussian-type function centered at a different point, and this dramatically speeds up twoelectron integral calculations[99]. Based on this theorem, four-center integrals can be reduced to finite sums of two-center integrals and, in a next step, to finite sums of one-center integrals, and this dramatically speeds up two-electron integral calculations compared to the Slater orbitals, outweighing the extra cost of having more basic functions usually needed in a Gaussian calculus.However, in the hydrogen atom wave function, the real atomic orbitals are close to the Slater-type function, 𝑒𝑥𝑝(-𝜁𝑟), thus, as proposed by Boys, the use of the contracted Gaussiantype base function, which is linear combinations of Gaussian-type functions (Equation2.35), are needed to simulate STOs, as a single Gaussian-type function gives a poor representation of a Slater-type function.𝜒 𝑝 (𝑟 -𝑅 𝐴 ) = ∑ 𝑐 𝜇𝑝 𝑒𝑥𝑝(-𝛼 𝜇𝑝 |𝑟 -𝑅 𝐴 | 2 ) (𝑥 3 , 𝑦 3 , 𝑧 3 , 𝑥 2 𝑦, 𝑥 2 𝑧, 𝑥𝑦 2 , 𝑦 2 𝑧, 𝑦𝑧 2 , 𝑥𝑧 2 , 𝑥𝑦𝑧)𝑒𝑥𝑝(-𝛼𝑟 2 ) contracted functions for valence orbitals. The notation for split valence basis sets, that arise from Pople-type basis functions, is typically X-YZWG[102]. In a minimal or single zeta basis set (SZ), there is one set of basis functions for each subshell occupied in the ground state of the atom. The double, triple, quadruple, quintuple, and hextuple zeta base sets

	are defined similarly and are in abbreviated forms denoted by DZ, TZ, QZ, 5Z, and 6Z. For
	example, in base 6-31G, "6-31" indicates the extent of contraction and division, where "6"
	means the use of contracted base functions from 6 primitive functions for central orbitals and
	"31" means the use of doubly divided valence basis functions combining contracted basis
	functions from three primitive functions with one uncontracted basis function for valence
	orbitals.
	𝜙 𝑖 (𝑟) = 𝐶 1𝑖 𝜒 1 + 𝐶 2𝑖 𝜒 2 + 𝐶 3𝑖 𝜒 3 + ⋯ + 𝐶 𝑝𝑖 𝜒 𝑝	(2.33)
	or
	𝑛 𝐴𝑂
	𝜙 𝑖 (𝑟) = ∑ 𝜒 𝑝 (𝑟)𝐶 𝑝𝑖
	For contracted Gaussian base functions, several types of functions have been suggested
	integrals involving this type of function generate computational difficulties. In fact, the "four-[100, 101]. The minimal basic functions, e.g., STO-LG contain only primitive functions needed
	center, two-electron integral problem" was once considered one of the greatest problems in for each atom. The letter L represents the number of primitive Gaussian functions used to
	quantum chemistry [85]. represent each Slater-type orbital in the atom.

Then, in 1951, Roothaan proposed

a method, known as the Roothaan method, to solve the computational difficulty of implementing the HF method for molecular calculations

[96]

. In this method, Roothaan solved this problem by suggesting linear combinations of atomic orbitals through the HF equations, with the 𝑝=𝑖 (2.34) being 𝑛 𝐴𝑂 is the number of atomic orbitals. The expansion coefficient 𝐶 𝑝𝑖 is called the molecular orbital coefficient and weight the contributions of the atomic orbitals to the molecular orbitals to be adjusted to get the best 𝜙 𝑖 (𝑟). Although {𝜒 𝑝 } is essentially the set of atomic orbitals, it is more efficient and general to use base functions modeling atomic orbitals. Thus, starting from the Hartree-Fock-Roothaan method, an incessant search for finite sets of bases that accurately describe the physical and chemical properties of atomic and molecular systems became intense.

The basic functions chosen are usually real and Slater or Gaussian type. A choice of base functions can be the Slater-type orbitals (STOs) which adequately describe the electronic behavior in regions near and far from the nucleus, and have been extensively used in atomic calculations. However, for molecular calculations, STOs are not desired, since multicentric On the other hand, multicentric integrals involving Gaussian-type functions are easier to evaluate. Gaussian type basis functions were suggested by

[START_REF] Boys | Electronic wave functions-I. A general method of calculation for the stationary states of any molecular system[END_REF]

, and are widely used in quantum chemistry calculations, as they have a specific product rule for them ("Gaussian Product Theorem"), where the product of Gaussian-type functions is given by a 𝜇 (2.35) In this equation, the original Gaussian-type functions are called primitive functions to distinguish them from contracted ones. Primitive functions are specified only by the orbital exponent, 𝛼 𝜇𝑝 , contracted coefficient, 𝑐 𝜇𝑝 and coordinate vector of the center of the function, 𝑅 𝐴 . Primitive functions generally have a standardized form and are represented analogously to spherical harmonic functions to those corresponding to s, p, d, and f atomic orbitals as [89] s function: 𝑒𝑥𝑝(-𝛼𝑟 2 ) p function: (𝑥, 𝑦, 𝑧)𝑒𝑥𝑝(-𝛼𝑟 2 ) d function: (𝑥 2 , 𝑦 2 , 𝑧 2 , 𝑥𝑦, 𝑦𝑧, 𝑧𝑥)𝑒𝑥𝑝(-𝛼𝑟 2 ) f function: Since the valence orbitals are responsible for chemical bonds, while the central orbitals hardly participate in bonds, the study of atomic orbitals can be divided into core and valence region. Thus, split valence basis functions use a contracted Gaussian-type function for central orbitals and multiple

  𝜔𝑟 12 ) is the error function, and the first term refers to the LR and the second to the SR, and the value of 𝜔 delimits the interface between the SR and LR terms.

					1 𝑟 12	=	1 -erf (𝜔𝑟 12 ) 𝑟 12	+	erf (𝜔𝑟 12 ) 𝑟 12	(2.76)
	where erf (For instance, the CAM-B3LYP functional is a LR corrected functional hybrid that uses
	the Coulomb-attenuating method (CAM) [123], using
			1 𝑟 12	=	1 -[𝛼 + 𝛽 • erf(𝜔𝑟 12 )] 𝑟 12	+	𝛼 + 𝛽 • erf(𝜔𝑟 12 ) 𝑟 12	(2.77)
	as a replacement for Equation 2.76 to perform the LR correction for the B3LYP hybrid
	𝐸 𝑋 𝑆𝑅 = -	1 2	∑ ∫ 𝑛 𝜎 4 3 ⁄ 𝐾 𝜎 𝜎	× {1 -	8 3	𝑎 𝜎 [√𝜋𝑒𝑟𝑓 (	1 2𝑎 𝜎	) + 2𝑎 𝜎 (𝑏 𝜎 -𝑐 𝜎 )]} 𝑑 3 𝑟 ⃗	(2.78)
	where 𝑎 𝜎 , 𝑏 𝜎 e 𝑐 𝜎 are
							𝑎 𝜎 =	1 2 ⁄ 𝜇𝐾 𝜎 6√𝜋𝑛 𝜎 1 3 ⁄

functional. The exchange functional is a mix of exact exchange, i.e., inclusion of the LR HF exchange integral at a constant ratio, and SR DFT exchange, but unlike B3LYP, the exact exchange ratio to DFT varies in different regions of the molecule. The degree of mixing of exact exchange and DFT is controlled by the 𝛼 and 𝛽 parameters, whose values used are 0.19 and 0.46 for 𝛼 and 𝛽, respectively. Specifically, the SR part of the exchange interaction is incorporated by modifying the usual exchange functional form

  The derivative of the exchange-correlation potential in terms of electron density, 𝑓 𝑋𝐶 , is called the exchange correlation-integral kernel. Define the response function of the electron density, 𝜒 𝐾𝑆 , for the infinitesimal change in KS potential, 𝛿𝑉 𝐾𝑆 , as𝛿𝜌(𝑟 1 , 𝑡 1 ) = ∬ 𝑑𝑡 2 𝑑 3 𝑟 2 𝜒 𝐾𝑆 [𝜌 𝑠𝑡𝑎𝑡 ] (𝑟 1 , 𝑟 2 , 𝑡 2 -𝑡 1 )𝛿𝑉 𝐾𝑆 (𝑟 2 , 𝑡 2 )Note that this response function is Fourier-transformed (𝑡 → 𝜔). What is important is that this response function has poles in the excitation energies. Casida proposed that the pole energies of the response function in Equation2.119, i.e., excitation energies, can be calculated Ionization energies (IE) and electron affinities (EA) can be calculated simply as the energy difference between the neutral and the ion[132]. In HF theory, approximate IEs can be obtained by applying Koopmans' theorem which says that the energy required to remove an electron from an orbital is the negative of the orbital energy[132]. Thus, the IE of a molecule

	𝐾 𝑖𝑎,𝑗𝑏 𝜎𝜏 = ⟨𝑖𝑏|𝑎𝑗⟩ 𝜎𝜏 + ∬ 𝑑 3 𝑟 1 𝑑 3 𝑟 2 𝜙 𝑖𝜎 * (𝑟 1 )𝜙 𝑏𝜏 * (𝑟 2 )𝑓 𝑋𝐶 (𝑟 1 , 𝑟 2 )𝜙 𝑎𝜎 (𝑟 1 )𝜙 𝑗𝜏 (𝑟 2 )	(2.127)
	by solving the following simultaneous matrix equations [131]: 2.4.5 Orbital Energy
	∑[𝛿 𝜎𝜏 𝛿 𝑖𝑗 𝛿 𝑎𝑏 (𝜖 𝑎𝜎 + 𝜖 𝑖𝜎 + 𝜔) + 𝐾 𝑖𝑎,𝑗𝑏 𝜎𝜏 ]𝑋 𝑗𝑏 𝜏 + 𝐾 𝑖𝑎,𝑗𝑏 𝜎𝜏 𝑋 𝑗𝑏 𝜏 2.4.5.1 Koopmans' Theorem	= 0	(2.120)
	𝑗𝑏𝜏
	and
	∑[𝛿 𝜎𝜏 𝛿 𝑖𝑗 𝛿 𝑎𝑏 (𝜖 𝑎𝜎 -𝜖 𝑖𝜎 + 𝜔) + 𝐾 𝑎𝑖,𝑏𝑗 𝜎𝜏 ]𝑋 𝑏𝑗 𝜏 + 𝐾 𝑎𝑖,𝑗𝑏 𝜎𝜏 𝑋 𝑗𝑏 𝜏	= 0	(2.121)
	𝑗𝑏𝜏
	In these equations, the spins of the orbitals (𝜎, 𝜏, 𝜎 ′ ≠ 𝜎) are explicitly displayed for
	purposes of accuracy. The simultaneous matrix equations are also represented for singlet and
	triplet excitations as follows
	therefore, represented as	𝛺𝑭 𝑖𝑎𝜎 = 𝜔 𝑖𝑎 2 𝑭 𝑖𝑎𝜎		(2.122)
	𝑉 𝑋𝐶 [𝜌](𝑟 1 , 𝑡 1 ) = 𝑉 𝑋𝐶 𝛺 𝑖𝑎𝜎,𝑗𝑏𝜏 𝑠𝑖𝑛𝑔𝑙𝑒𝑡 = 𝛿 𝜎𝜏 𝛿 𝑖𝑗 𝛿 𝑎𝑏 (𝜖 𝑎𝜎 -𝜖 𝑖𝜎 ) 2 𝑠𝑡𝑎𝑡 [𝜌](𝑟 1 ) + ∬ 𝑑𝑡 2 𝑑 3 𝑟 2 𝑓 𝑋𝐶 [𝜌 𝑠𝑡𝑎𝑡 ] (𝑟 1 , 𝑟 2 , 𝑡 2 -𝑡 1 )𝛿𝜌(𝑟 2 , 𝑡 2 ) (2.116) + 2(𝜖 𝑎𝜎 -𝜖 𝑖𝜎 ) 1 2 ⁄ (𝐾 𝑖𝑎,𝑗𝑏 𝜎𝜎 + 𝐾 𝑖𝑎,𝑗𝑏 𝜎𝜎 ′ )(𝜖 𝑏𝜏 -𝜖 𝑗𝜏 ) 1 2 ⁄ (2.123)
	and	𝑓 𝑋𝐶 [𝜌 𝑠𝑡𝑎𝑡 ](𝑟 1 , 𝑟 2 , 𝑡 2 -𝑡 1 ) =	𝛿𝑉 𝑋𝐶 (𝑟 1 , 𝑡 1 ) 𝛿𝜌(𝑟 2 , 𝑡 2 )	𝜌=𝜌 𝑠𝑡𝑎𝑡 |	(2.117)
	𝛺 𝑖𝑎𝜎,𝑗𝑏𝜏 𝑡𝑟𝑖𝑝𝑙𝑒𝑡 = 𝛿 𝜎𝜏 𝛿 𝑖𝑗 𝛿 𝑎𝑏 (𝜖 𝑎𝜏 -𝜖 𝑖𝜏 ) 2
		+ 2(𝜖 𝑎𝜎 -𝜖 𝑖𝜎 ) 1 2 ⁄ (𝐾 𝑖𝑎,𝑗𝑏 𝜎𝜎 + 𝐾 𝑖𝑎,𝑗𝑏 𝜎𝜎 ′ )(𝜖 𝑏𝜏 -𝜖 𝑗𝜏 )	1 2 ⁄	(2.124)
	where 𝐹 𝑖𝑎𝜎 is the response coefficient matrix given by
		𝐹 𝑖𝑎𝜎 = (𝜖 𝑎𝜎 -𝜖 𝑖𝜎 ) -1 2 ⁄ (𝜒 𝑖𝑎𝜎 -𝜒 𝑎𝑖𝜎 )	(2.118) (2.125)
	In this definition, the response function is given by Green function theory as 𝜒 𝐾𝑆 (𝑟 1 , 𝑟 2 , 𝜔) = 2 lim 𝜂→0+ 𝜙 𝑖 * (𝑟 1 )𝜙 𝑎 (𝑟 1 )𝜙 𝑖 (𝑟 2 )𝜙 𝑎 * (𝑟 2 ) 𝑛 𝑣𝑖𝑟 𝑛 𝑜𝑐𝑐 𝜒 𝑖𝑎𝜎 (𝜔) = 𝑛 -1 × ∫ 𝑑 3 𝑟𝜙 𝑖𝜎 * (𝑟) 𝛿 (2 ∑ 𝐽 ̂𝑖 + 𝑉 𝑋𝐶 ) (𝑟, 𝜔)𝜙 𝑎𝜎 (𝑟) 𝜔 + (𝜖 𝑎𝜎 -𝜖 𝑖𝜎 ) 𝑗 ∑ ∑ [ 𝜔 -(𝜖 𝑎 -𝜖 𝑖 ) + 𝑖𝜂 𝑎 𝑖 -𝜙 𝑖 (𝑟 1 )𝜙 𝑎 * (𝑟 1 )𝜙 𝑖 * (𝑟 2 )𝜙 𝑎 (𝑟 2 ) 𝜔 -(𝜖 𝑎 -𝜖 𝑖 ) + 𝑖𝜂 ] 𝜖 𝑖 is the i -th orbital energy, and 𝐾 𝑖𝑎,𝑗𝑏 𝜎𝜏 is provided as	(2.126) (2.119)

Applying the TD-KS equation to linear response theory, excitation energies can be calculated and assigned to corresponding transitions

[130]

. Following the RG theorem, it is assumed that only a weak perturbation, 𝛿𝑉 𝑒𝑥𝑡 , is added to the external potential. Under this assumption, it is interpreted that the electron density also undergoes an infinitesimal change, 𝛿𝜌(𝑟, 𝑡), in the stationary part, 𝜌 𝑠𝑡𝑎𝑡

[130]

. The exchange-correlation potential, 𝑉 𝑋𝐶 , is,

Table 1

 1 sets up the results from DFT/B3LYP/6-31G(d) for donors and acceptors.

Table 1 .

 1 HOMO and LUMO values obtained by DFT/B3LYP/6-31G(d) method. Elaborated by the author.Using the mean signed error (MSE) is possible determine how well our theoretical energy values estimated match the experimental energy values from literature. In the donor molecules, the MSE for HOMO energies is equal to -0.213 eV and for LUMO is 1.13 eV. For the acceptors, the MSE for HOMO energies is -0.165 eV and for LUMO is 0.46 eV. Analyzing, for both donor and acceptor molecules, B3LYP gives good correlation to experimental HOMO energies, and for the LUMO, B3LYP gives better estimations than in donors, however the correlation is still poor. This situation is not surprising given all problems that the quantum methods have to calculate virtual orbitals.An alternative to try better estimate the LUMO values is to perform TD-DFT calculations. Table2presents the results from TD-DFT/B3LYP/6-31G(d). In donors, the MSE for LUMO energies is 1.09 eV and for acceptors is 0.46 eV, so, despite the TD-DFT calculations, B3LYP continues giving a poor correlation for LUMO values.

	Method			Molecules
	DFT/B3LYP/6-
	31G(d)	PBDB-T PM6	PTB7-Th	Y6	ITIC OT6-4F PC71BM
	HOMO (eV)	-5.02	-5.12	-5.12	-5.55	-5.46	-5.37	-5.55
	LUMO (eV)	-2.35	-2.46	-2.27	-3.51	-3.35	-3.52	-3.51
			Source:

Table 2 .

 2 HOMO and LUMO values obtained by TD-DFT/B3LYP/6-31G(d) method.

	Method			Molecules
	TD-
	DFT/B3LYP/6-
	31G(d)	PBDB-T PM6	PTB7-Th	Y6	ITIC OT6-4F PC71BM
	HOMO (eV)	-5.02	-5.12	-5.13	-5.56	-5.46	-5.37	-5.56
	LUMO (eV)	-2.35	-2.46	-2.39	-3.50	-3.35	-3.52	-3.50

Table 3 .

 3 HOMO and LUMO values obtained by DFT/CAM-B3LYP/6-31G(d) method.

	Method TD-				Molecules
	DFT/CAM-
	B3LYP/6-	PBDB-T	PM6	PTB7-Th	Y6	ITIC OT6-4F PC71BM
	31G(d)
	HOMO (eV)	-6.30	-6.43	-6.39	-6.60	-6.54	-6.44	-6.62
	LUMO (eV)	-1.19	-1.23	-1.06	-2.51	-2.30	-2.50	-2.11
			Source: Elaborated by the author.

Table 4 .

 4 HOMO and LUMO values obtained by TD-DFT/CAM-B3LYP/6-31G(d) method.

	Method TD-				Molecules
	DFT/CAM-
	B3LYP/6-	PBDB-T	PM6	PTB7-Th	Y6	ITIC OT6-4F PC71BM
	31G(d)
	HOMO (eV)	-6.20	-6.66	-6.40	-6.60	-6.54	-6.44	-6.62
	LUMO (eV)	-1.30	-1.39	-1.30	-2.51	-2.30	-2.50	-2.11

Table 5 .

 5 HOMO and LUMO values obtained by DFT/ωB97-XD/6-31G(d) method.

	Method
					Molecules
	DFT/ωB97-
	XD /6-31G(d) PBDB-T	PM6	PTB7-Th	Y6	ITIC OT6-4F PC71BM
	HOMO (eV)	-6.91	-7.025	-7.00	-7.15	-7.12	-7.00	-7.24
	LUMO (eV)	-0.64	-0.745	-0.56	-2.012 -1.80	-2.00	-1.68
			Source: Elaborated by the author.

Table 6 .

 6 HOMO and LUMO values obtained by TD-DFT/ωB97-XD/6-31G(d) method.

	Method				Molecules
	DFT/TD-
	ωB97-XD /6-
	31G(d)	PBDB-T	PM6	PTB7-Th	Y6	ITIC OT6-4F PC71BM
	HOMO (eV)	-6.91	-7.025	-7.00	-7.15	-7.12	-7.00	-7.24
	LUMO (eV)	-0.64	-0.745	-0.56	-2.012 -1.80	-2.00	-1.68
			Source: Elaborated by the author.

Table 7 .

 7 HOMO and LUMO values from DFT/B3LYP/6-31G(d,p) method.

	Method				Molecules
	DFT/
	B3LYP/6-
	31G(d,p)	PBDB-T	PM6	PTB7-Th	Y6	ITIC OT6-4F PC71BM
	HOMO (eV)	-5.03	-5.13	-4.41	-5.56	-5.47	-5.37	-5.6
	LUMO (eV)	-2.36	-2.47	-2.29	-3.52	-3.36	-3.53	-3.06
			Source: Elaborated by the author.

Table 8 .

 8 HOMO and LUMO values from TD-DFT/B3LYP/6-31G(d,p) method.

	Method
					Molecules
	DFT/ TD-
	B3LYP/6-
	31G(d,p)	PBDB-T	PM6	PTB7-Th	Y6	ITIC OT6-4F PC71BM
	HOMO (eV)	-5.03	-5.13	-4.41	-5.56	-5.47	-5.37	-5.6
	LUMO (eV)	-2.36	-2.47	-2.29	-3.52	-3.36	-3.53	-3.06
			Source: Elaborated by the author.

Table 9 .

 9 HOMO and LUMO values from DFT/CAM-B3LYP/6-31G(d,p) method.

	Method				Molecules
	DFT/ CAM-
	B3LYP/6-
	31G(d,p)	PBDB-T	PM6	PTB7-Th	Y6	ITIC OT6-4F PC71BM
	HOMO (eV)	-6.30	-6.30	-5.92	-6.61	-6.54	-6.45	-6.62
	LUMO (eV)	-1.20	-1.47	-1.29	-2.52	-2.31	-2.70	-2.12
			Source: Elaborated by the author.

Table 10 .

 10 HOMO and LUMO values from TD-DFT/CAM-B3LYP/6-31G(d,p) method.

	Method DFT/ TD-				Molecules
	CAM-
	B3LYP/6-	PBDB-T	PM6	PTB7-Th	Y6	ITIC OT6-4F PC71BM
	31G(d,p)
	HOMO (eV)	-6.30	-6.30	-5.92	-6.61	-6.54	-6.45	-6.62
	LUMO (eV)	-1.20	-1.47	-1.29	-2.52	-2.31	-2.70	-2.12
			Source: Elaborated by the author.

Table 11 .

 11 HOMO and LUMO values from DFT/ωB97-XD /6-31G(d,p) method.

	Method
					Molecules
	DFT/ ωB97-
	XD/6-31G(d,p)	PBDB-T	PM6	PTB7-Th	Y6	ITIC OT6-4F PC71BM
	HOMO (eV)	-6.93	-7.03	-7.00	-7.16	-7.12	-7.00	-7.24
	LUMO (eV)	-0.65	-0.757	-0.57	-2.016 -1.81	-2.00	-1.68
			Source: Elaborated by the author.

Table 12 .

 12 HOMO and LUMO values from DFT/TD-ωB97-XD/6-31G(d,p) method.

	Method
					Molecules
	DFT/ TD-
	ωB97-XD /6-
	31G(d,p)	PBDB-T	PM6	PTB7-Th	Y6	ITIC OT6-4F PC71BM
	HOMO (eV)	-6.93	-7.03	-7.00	-7.16	-7.12	-7.00	-7.24
	LUMO (eV)	-0.65	-0.757	-0.57	-2.016 -1.81	-2.00	-1.68
			Source: Elaborated by the author.

Table 13 .

 13 HOMO and LUMO values from DFT/B3LYP /6-311G(d,p) method.

	Method
					Molecules
	DFT/
	B3LYP/6-311G(d,p)	PBDB-T	PM6	PTB7-Th	Y6	ITIC OT6-4F PC71BM
	HOMO (eV)	-5.30	-5.38	-5.36	-5.98	-5.69	-5.61	-5.98
	LUMO (eV)	-2.60	-2.67	-2.47	-3.76	-3.58	-3.76	-3.46
			Source: Elaborated by the author.

Table 14 .

 14 HOMO and LUMO values from DFT/TD-B3LYP/6-311G(d,p) method.

	Method
					Molecules
	DFT/ TD-
	B3LYP/6-311G(d,p)	PBDB-T	PM6	PTB7-Th	Y6	ITIC OT6-4F PC71BM
	HOMO (eV)	-5.30	-5.38	-5.36	-5.98	-5.69	-5.61	-5.98
	LUMO (eV)	-2.60	-2.67	-2.47	-3.76	-3.58	-3.76	-3.46
			Source: Elaborated by the author.

Table 15 .

 15 HOMO and LUMO values from DFT/CAM-B3LYP/6-311G(d,p) method.

	Method
					Molecules
	DFT/ CAM-
	B3LYP/6-
	311G(d,p)	PBDB-T	PM6	PTB7-Th	Y6	ITIC OT6-4F PC71BM
	HOMO (eV)	-6.47	-6.30	-6.68	-6.83	-6.76	-6.65	-6.62
	LUMO (eV)	-1.52	-1.47	-1.36	-2.77	-2.54	-2.90	-2.12

Source: Elaborated by the author.

Table 16 .

 16 HOMO and LUMO values obtained by DFT/TD-CAM-B3LYP/6-311G(d,p) method.

	Method
	DFT/ TD-				Molecules
	CAM-
	B3LYP/6-	PBDB-T	PM6	PTB7-Th	Y6	ITIC OT6-4F PC71BM
	311G(d,p)
	HOMO (eV)	-6.47	-6.30	-6.68	-6.83	-6.76	-6.65	-6.62
	LUMO (eV)	-1.52	-1.47	-1.50	-2.77	-2.54	-2.90	-2.12
			Source: Elaborated by the author.

Table 17 .

 17 Table 17. HOMO and LUMO values obtained by DFT/ωB97-XD/6-311G(d,p) method.

	Method
					Molecules
	DFT/ωB97-
	XD/6-
	311G(d,p)	PBDB-T	PM6	PTB7-Th	Y6	ITIC OT6-4F PC71BM
	HOMO (eV)	-7.071	-7.27	-7.23	-7.35	-7.29	-7.21	-7.24
	LUMO (eV)	-0.88	-0.888	-0.74	-2.22	-1.99	-2.18	-1.68
			Source: Elaborated by the author.

Table 18 .

 18 HOMO and LUMO values obtained by DFT/TD-ωB97-XD/6-311G(d,p) method.

	Method
					Molecules
	DFT/TD-
	ωB97-XD/6-
	311G(d,p)	PBDB-T	PM6	PTB7-Th	Y6	ITIC OT6-4F PC71BM
	HOMO (eV)	-7.071	-7.27	-7.23	-7.35	-7.29	-7.21	-7.24
	LUMO (eV)	-0.88	-0.888	-0.74	-2.22	-1.99	-2.18	-1.68

Source: Elaborated by the author.

Table 19

 19 set up all MSE results from HOMO and LUMO from all DFT calculations. It is possible

	observe that, for all calculations, the calculated LUMO energy, consistently gives values that
	are less negative (higher lying) than those experimentally determined. From MSE values, the
	B3LYP functional together with 6-311G(d,p) basis set give the best values for HOMO and
	LUMO, but the problem of obtaining more accurate values for LUMO remains, even for
	approaches based on 6-311G(d,p) basis set. So, such results highlight that 6-31G(d) small basis
	set can provide good values for energy levels from frontier molecular orbitals, in addition to
	saving computational resources.

Table 19 .

 19 The MSE results from HOMO and LUMO from all DFT calculations.

			MSE
		Donors		Acceptors
	Functional/Basis set	HOMO (eV) LUMO (eV) HOMO (eV) LUMO (eV)
	B3LYP/6-31G(d)	-0.213	1.13	-0.165	0.46
	B3LYP/6-31G(d,p)	-0.443	1.11	-0.147	0.65
	B3LYP/6-311G(d,p)	-0.047	0.91	-0.167	0.29
	CAM-B3LYP/6-31G(d)	-1.073	2.33	-0.90	1.57
	CAM-B3LYP/6-31G(d,p)	-0.873	2.17	-0.91	1.51
	CAM-B3LYP/6-311G(d,p)	-1.183	2.04	-1.07	1.345
	ωB97-XD/6-31G(d)	-1.68	2.84	-1.48	2.055
	ωB97-XD/6-31G(d,p)	-1.40	2.83	-1.48	2.052
	ωB97-XD/6-311G(d,p)	-1.89	2.654	-1.63	2.41
		Source: Elaborated by the author.
	The same problem in estimating LUMO values for conjugated polymers was reported
	by McCormick. Where they presented a series of DFT/B3LYP computations for 22 different
	conjugated polymer model compounds, and the MSE for LUMO energies was equal to 0.59

eV. Therefore, an alternative presented was calculate the molecular orbital energy in the triplet state, with the molecules being optimized in the neutral singlet ground state and then the single point energy being calculated in the triplet state, both were calculated with B3LYP/6-311G(d).

Table 20 .

 20 Solubility values for C60 in different solvents.

	Solvent	Solubility of C60
		(mg/ml)
	Chloroform	0.16
	Xylene	5.2
	Source: Elaborated by the author.

Table 21 .

 21 Area per molecule for different types of polyfullerene solutions.

	Solution	PCBM	Area per molecule (Å 2 ) HSS8 HSS16
	Chloroform	23	48	126
	Xylene	9	12	33
		Source: Elaborated by the author.
	Since there is no previous literature from these exact polymers, to estimate monomer
	areas, were carried out the theoretical calculations using DFT/B3LYP/6-31+G(d) method for
	optimizations of geometries and energies. Calculations were performed using the Gaussian 09
	program package. DFT calculations were combined with the Grimme's D3 method to account
	for longer range dispersion interactions. Using the same parameters, Roncaselli et al. carried
	out the calculations for C60 and PCBM [166].

  Table 21, indicating that both materials aggregate in the Langmuir trough. In spite of this, the areas obtained by DFT are smaller in 'XY' and 'XZ' planes than that in the 'YZ' plane, which suggests an orientation closer to a 'XY' or 'XZ' configuration against the air-water interface. Another important point is the volume occupied by these materials; HSS8 is 42% smaller than HSS16, which is a significant difference. These results go in the same direction as those obtained for.

Table 22 .

 22 Structural parameters, areas and volumes of C60, PCBM, HSS8 and HSS16 monomers extracted by DFT calculations. Elaborated by the author.Complementary calculations were carried out under an implicit model of the solvent (Polarizable Continuum Model, PCM), making it possible to implicitly average the degrees of freedom of the solvent and to treat the electrostatic interactions (generally dominant in the solvation) for C60, PCBM, HS88 and HSS16 systems. These new calculations show that neither the volumes nor the calculated surfaces are modified whatever the considered the chloroform.The results from chloroform effects, reveal an important information, i.e., that it is not the electrostatic interactions that govern the behavior of the HSS8 and HSS16 molecules in the solutions in our experiments as xylene and chloroform. The dielectric constant of chloroform and xylene is 4.809 and 2.562, respectively [167]. The origin of the observed differences

		Structural parameters (Å)	Area in different orientations (Å 2 )	Volume (Å 3 )
	Materials	X	Y	Z	XY	YZ	XZ	XYZ
	C60	7.02	6.90	6.88	48.4	47.5	48.3	333.2
	PCBM	14.79	9.82	6.96	145.2	68.4	102.9	1010.9
	HSS8	12.24	15.58	16.62	190.7	258.9	203.4	3169.4
	HSS16	18.52	19.18	20.82	355.2	399.3	385.6	7395.5
			Source:

Table 23 .

 23 Values of oxidation potential, 𝐸 𝑔𝑎𝑝 𝑜𝑝𝑡 , 𝐸 𝐿𝑈𝑀𝑂 and 𝐸 𝐻𝑂𝑀𝑂 for polyfullerene thin films.

	Thin film	Solvent	𝑬 𝑯𝑶𝑴𝑶 𝑶𝑿𝑰 (eV)	𝑬 𝑳𝑼𝑴𝑶 𝑶𝑿𝑰 (eV)	𝑬 𝑯𝑶𝑴𝑶 𝑶𝑿𝑰𝑰 (eV)	𝑬 𝑳𝑼𝑴𝑶 𝑶𝑿𝑰𝑰 (eV)	𝒐𝒑𝒕 𝑬 𝒈𝒂𝒑 (eV)
	HSS8/Cast Chloroform	-4.23	-1.85	-5.16	-2.78	2.38
	HSS8/Cast	Xylene	-4.23	-1.91	-5.16	-2.84	2.32
	HSS8/LS	Chloroform	-4.12	-1.74	-5.28	-2.90	2.38
	HSS8/LS	Xylene	-4.12	-1.74	-5.37	-2.99	2.38
	HSS16/Cast Chloroform	-4.16	-1.86	-5.36	-3.06	2.30
	HSS16/Cast	Xylene	-4.16	-1.84	-5.17	-2.85	2.32
	HSS16/LS Chloroform	-4.17	-1.85	-5.20	-2.88	2.32
	HSS16/LS	Xylene	-4.15	-1.81	-5.28	-2.94	2.34
			Source: Elaborated by the author.

The funding agencies CAPES, CNPq, FAPESP and INEO for all the financial support, which was of fundamental importance for the accomplishment of this work; All those who, directly or indirectly contributed to the development of this PhD research.

The performance of organic electronic devices based on conjugated polymers, such as in the mobility of charge carriers, is influenced by the orientation of the molecules and the molecular aggregation morphology of the polymer. Within this context, the Langmuir, Langmuir-Blodgett (LB) and Langmuir-Schaefer (LS) techniques stand out, as they provide higher organization at the molecular level and control over the thickness and uniformity of the thin film [144,145,146]. Within this context, the polyfullerenes were studied using the Langmuir and LS techniques.

A Langmuir trough (Figure 14) is used to fabricate the thin films, which is basically composed of a teflon trough top (3), which inside is filled with a liquid subphase, two mobile barriers (2), also made of teflon, for symmetrical compression of the film [144,145,146]. The trough is also equipped with a surface pressure sensor (4), barrier position detection (1 and 6) and a dipping mechanism (5) used to transfer the film to a solid substrate where it is possible to control the speed of vertical deposition [144,145,146]. Langmuir films are monomolecular films obtained by spreading a small amount of a material consisting of amphiphilic molecules on a liquid surface [144,145,146]. Amphiphilic molecules have both hydrophilic and hydrophobic parts. The hydrophilic part (head) is responsible for spreading while the hydrophobic part (tail) is responsible for the flotation of the [144,145,146]. In Figure 16, the points G, L 1 , L 2 and S represent the gaseous phase, liquidexpanded state, the liquid-condensed state and solid state, respectively. The successive deposition of monolayers of a Langmuir film on the same solid substrate can be performed using the LB and LS techniques (Figure 17). In the LB technique, the film is transferred onto the substrate by immersion and vertical elevation, while in the LS technique, the deposition is carried out by horizontal contact of the substrate with the monolayer [144,145,146]. To ensure better deposition, the substrate is slowly brought to the interface with the stabilized Langmuir film [144,145,146]. Then, the substrate is also slowly raised, and with this technique, the hydrophobic part of the molecule is in contact with the substrate [144,145,146]. 31G(d,p) basis set, were -5.49 and -3.37 eV, and by CV measurements, carried out by Lin et al.,respectively [141,160]. As 6-31G(d) is a small basis set, it has some limitations to describe the energy of the frontier molecular orbitals. So a new series of calculations were carried out using larger basis set, such as: 6-31G(d,p) and 6-311G(d,p), in B3LYP, CAM-B3LYP and ωB97-XD functionals.

Tables 7 -18 summarize the results coming from all these series of calculations.

The MSE was equal to -0.13 eV for energy of the αHOMO of the triplet state, thereby, giving the best correlation with experiment, and, so, αHOMO energy of the triplet state energy is most accurately approximated from the LUMO. So, another alternative was carried out calculations in the triplet state for our systems. The study of PCBM and its derivatives using Langmuir techniques aims to characterize the nanostructure of the material which is spread as very thin layers at the water-air interface.

Figure 40 shows the surface pressure versus the mean area isotherms (π-A) of HSS8 and HSS16, both of them previously solubilized in either chloroform or xylene (0.2 mg/mL). Source: Elaborated by the author.