**5. Application to DFT and TD-DFT methods in polymer solar cells**

Many properties of interest in OPVs can be calculated or developed using DFT based computational methods, including bandgap, optical absorption, intramolecular and intermolecular charge transfer, exciton binding energy, charge transfer integral (to quantify electronic coupling), reorganization energy, and the rate of charge transfer and recombination in D–A compositors. In addition, in natural systems, the effects of the surrounding OPV media on these properties should be addressed. Although the impact of the medium varies depending on its nature and the nature of the transitions involved, low-cost polarization continuum solvation models can be utilized to account for solvation. In such situations, the solvent is treated as a continuous with a static dielectric constant, polarizing and polarized by the solute.

The following is a discussion of the theoretical methodologies and computational techniques mentioned above to foster a greater understanding of the connection between chemical structures and the optical and electrical properties of D–A systems about the rational design of OPV devices.

#### **5.1 Bandgap engineering**

A range of experimentally observed methods utilized may or may not indicate appreciable quantities in diverse contexts. Band gaps (see infra) are essential features to consider when evaluating conducting polymers. Controlling band gaps can improve the electroluminescence of OLEDs or the light absorption efficiency of photovoltaic cells [67]. To make organic polymers with good nonlinear optical response [67] or semiconductors having high electrical conductivity [68], materials with tiny band gaps are sought.

The phrase "bandgap" has multiple meanings. An infinite periodic system's electrical structure is called a "band." Also examined are monomers of conjugated polymers and oligomers of various sizes. The term bandgap also refers to a finite method feature that converges to the infinite periodic (band-structure) limit with oligomer size. The "gap" is the difference in electronic energy levels. Such as the energy difference between the highest occupied molecular orbital (HOMO) and the lowest unoccupied one, or computed energy gaps MO or CO (LUMO) [69]. Lowest optically permissible electronic excitation energy A visible energy gap (also known as an adiabatic or vertical optical gap) is an EO. Also, adiabatic or vertical electron attachment/detachment has associated energies like electron affinity (EA), ionization potential (IP) and electronegativity (EF) = IP-EA. Orbital energies and eigenvalues in KS DFT and HF may or may not be visible. However, approximations in actual computations might lead to big mistakes (see below). Furthermore, an orbital energy gap cannot reflect both EO and EF. Although the interpretation of the orbital energy gap as the fundamental gap is accurate in principle, the multiple approximations in the functionals make this interpretation useless in practice.

Band gaps often exhibit a roughly linear dependence on 1/m, allowing extrapolation to m ! ∞, 1/m ! 0. Generally, the quality of calculated results for different properties strongly relies on the physical models employed [70]. Extrapolating the infinite-periodic limit from a sequence of oligomer simulations requires prudence. Band gaps can diverge from linearity in 1/m [71]. These can also compute band gaps for polymers with unlimited chain lengths. PBCs have the advantage of not requiring further computations or extrapolation. However, PBC quantum chemistry algorithms are less functional than their molecular counterparts, which is a disadvantage.

The cheap computational cost of DFT and TD-DFT allows for the study of large systems. In contrast, the most commonly used functionals for molecules, such as generalized gradient approximations (GGA) and global hybrid GGA functionals,

result in the incorrect asymptotic behavior of a potential, significant delocalization errors, and lack of derivative discontinuity, all of which negatively impact o They showed that the band gaps computed with different hybrid functionals differ significantly, indicating that present hybrid functionals do not yield proper band gaps [72–74]. The link between projected orbital energy gaps and measured band gaps absorption energies is also poor for nonhybrid density functionals. Therefore, typical hybrid and meta-hybrid functionals are not acceptable for assessing the performance of organic photovoltaics, according to Savoie et al. Other hybrid functional analyses found striking agreement with measured optical gaps [75]. Many studies suggest that DFT and TD-DFT have limited predictive potential in this critical field [74]. Using hybrid functionals to integrate eX solves some of the DFT difficulties in extended systems [76]. However, estimated transfer integrals in organic semiconductors are sensitive to eX fraction [77].

#### **5.2 Intramolecular and intermolecular charge transfer**

To explain the transport properties, the charge transfer rate between donor and acceptor moieties could be calculated. Using DFT calculations of the electronic coupling, reorganization energy, and free energy difference associated with one electron transfer from donor to acceptor at the high-temperature limit, we can determine a rate using Marcus's formula [78, 79]. These predictions are then discussed in light of DFT's intrinsic flaws, such as overestimation of actual ground state energy, failure of the basis set to represent the system, and the inability of the functional to approximate critical interactions, to name a few. When it comes to charge transfer, however, there is another problem that is often overlooked.

A system where an extra account is localized on a single molecular unit is impossible to simulate using conventional DFT. As a result, calculations are frequently performed first on the charge donor, then on the charge acceptor, with or without the charge transferred. The overall energy of the system is calculated by adding the energies of the individual components. This is true when a considerable distance separates the donor and acceptor, and the electron density distribution of one entity has no effect on the electron density distribution of the other [79]. The size of the organizations participating in the charge transfer and the quantity and location of the surplus charge should naturally establish this limit. More sophisticated (and consequently more expensive) techniques, such as charge-constrained DFT [80], have been proposed for situations where this limit can never be achieved, such as intramolecular charge transfer [79]. However, standard ground state DFT remains the approach of choice for intermolecular charge transfer in solar cells, which usually involves larger molecular assemblies.

#### **5.3 Extion binding energy**

Exciton binding energy (Eb) is an essential element in polymer electronics and fundamental polymer physics, and it has been a source of debate for a long time. For example, a big Eb is required for a light-emitting polymer so that charge recombination takes precedence. Both the semiempirical model study [81] and the DFT/ LDA Bethe–Salpeter equation (BSE) or GW approaches [82] have relied heavily on theoretical research of Eb of conjugated polymers. The semiempirical model can be addressed nearly precisely for the electron correlation, but the results depend highly on the parameters, even though a qualitative comprehension has been reached [81].

Although the BSE or GW approach is first-principles, it is unclear if the Hohenberg–Kohn–Sham framework might accommodate these many-body adjustments at the Green's function level.

*Applications of Current Density Functional Theory (DFT) Methods in Polymer Solar Cells DOI: http://dx.doi.org/10.5772/intechopen.100136*

Furthermore, the final results differ from one another [82]. We highlight two recent advances in quantum chemistry: (i) Time-Dependent Density Functional Theory (TD-DFT) and its successful application to the lowest-lying excited states [83], and (ii) the hybrid GGA Becke three-parameter Lee–Yang–Parr (B3LYP) functional for the quantitative prediction of chemical and electronic structures [82]. In examining the excitation processes in conjugated systems, the combination of the two has proven quite effective [84]. Tretiak et al. [85] demonstrated that hybrid density functionals might simulate excitonic phenomena and provide satisfactory Eb findings. Pure local spin density approximation (LSDA), generalized gradient approximation (GGA, such as BLYP, BP86, BPBE, PBEPBE, BPW91), meta-GGA (such as PBEKCIS), and hybrid density functionals (H-GGA, such as O3LYP, B3LYP, B972, PBE1PBE) are used to optimize the ground-state geometries of the molecules at the DFT level.

#### **5.4 Electron transfer parameters**

The ET parameters are now frequently calculated using ab initio quantum chemistry methods. In classical Marcus theory [86], two critical parameters determine the temperature-dependent kinetics of electron transfer: the driving force and the reorganization energy. The activation energy, G#, is calculated as follows:

$$
\Delta G^\text{\\\(\lambda\)} = \frac{\left(\lambda + \Delta G^0\right)}{4\lambda} \tag{2}
$$

which then can be used in the Arrhenius relationship for the rate constant

$$k\_{ET} = A \exp\left(\frac{\Delta G^{\circ}}{k\_B T}\right) \tag{3}$$

The Boltzmann constant is denoted by kB. Note that Eq. (3) is essentially classical, and when quantum effects are relevant, the temperature dependence of Eq. (3) breaks down [87].

Although free energies will be used in natural systems, ab initio calculations usually overlook entropy changes and instead use potential energy. When the reactant and product structures are known, G° may readily determine the difference between respective equilibrium energies. However, calculating is more complicated since it involves nonequilibrium energy.

Abs Initio algorithms have difficulties determining since it is not a ground-state attribute. In the adiabatic representation, the ground state potential energy curve is lower than the excited state potential energy curve. To compute, one must first know the energy of the product state at the reactant state's equilibrium structure. Excited-state energies are more difficult to calculate than ground-state energies. TD-DFT methods provide good excited-state energies (e.g., up to 100 atoms). The energy of long-range CT states in TD-DFT is underestimated [88, 89], limiting its use in ET reactions.

In ET research, constrained DFT has various advantages. For starters, restricted DFT makes accessing diabatic states and calculating Marcus parameters a breeze. Second, from ground-state computations, constrained DFT generates diabatic conditions. Excited-state calculations are avoided. Third, the equilibrium structures of the reactant and product states are obtained by optimizing constrained geometry. Third, the quality of diabatic potential energy curves from limited DFT is superior to adiabatic curves from DFT to optimize in adiabatic conditions because fractional charge systems are more susceptible to self-interaction errors [90]. Fourth, the

localization of an unpaired electron is forced via constrained DFT. As a result, restricted DFT energy values are more precise. Limited DFT cannot be used for ET reactions involving a locally excited state as a ground-state approach. To investigate such responses, TD-DFT and limited DFT could be utilized. Our method, in particular, is before the electron source and acceptor. In systems in which the donor and acceptor are not separated, this can be a problem. As a result, our method is now the most effective for long-range ET responses, which is why it was created.

The coupling constant should be computed to obtain correct adiabatic energies from diabatic ones, making limited DFT more useful. In the adiabatic representation, the two curves create an upper and lower curve, with the energy gap at qc being twice the electronic coupling constant Hab, which is a significant coupling constant in nonadiabatic dynamics. DFT overestimates Hab, which could lead to erroneous RobinDay class III compound assignment. The limited DFT methodology for calculating high-quality diabatic energies could also be used to forecast exact Hab values. One of the difficulties is that constrained DFT techniques do not provide the proper wave function. Hab is the union of two independent wave functions with no equivalent in static density-dependent observables. As a result, some estimates are required to extract Hab from constrained DFT. We are now doing an active study on this topic, and the results will be released soon. Hab can also be employed with restricted DFT to study the issue of degenerate charge transfer states. It can also tackle problems similar to those that the restricted open-shell Kohn-Sham approach can solve [45].

#### **5.5 Scharber's model-electronic properties**

They are using density functional theory and Scharber's model to forecast the power conversion efficiency of organic solar cells. The scientific community has long sought improved polymers with excellent power conversion efficiency. Because polymer synthesis and device production take time, a guide would help find the best polymers. They published a simple model in 2006 that outlines how to estimate the power conversion efficiency of bulk heterojunction solar cells, and they claimed that these devices could attain 10% power conversion efficiencies. To evaluate a polymer's photovoltaic potential, Scharber's model requires knowledge about energy levels. Commonly, cyclic voltammetry is used to obtain these energy levels after polymer production. Modern theoretical tools like density functional theory come into use. These technologies can theoretically predict polymer properties before they are made.

The DFT has been extensively used to develop, explain, and predict the features of present and future organic solar cells [91]. Even though the model predicts certain qualities like open-circuit voltage (VOC) and short-circuit current density (JSC), one may wonder if the model estimates some attributes more precisely than others when combined with density functional theory. The dependability of theoretical computations is critical for understanding and predicting device attributes. There have been extensive research on oligomers [92] and crystals [93], but few comparisons of computations on polymers with experimental evidence. These highest values for power conversion efficiency can be derived by integrating density functional theory determined attributes with Sharber's model.

#### *5.5.1 Scharber's model*

This is equivalent to the maximum power density output of the device divided by the total power density receiving from the Air Mass 1.5 solar spectrum [94], which is 1000 W/m2 . The device's power density comprises the open-circuit

*Applications of Current Density Functional Theory (DFT) Methods in Polymer Solar Cells DOI: http://dx.doi.org/10.5772/intechopen.100136*

voltage, short-circuit current density, and fill factor (FF). According to Scharber's model, the VOC is connected to the difference between the acceptor's LUMO and the donor's HOMO. The VOC is obtained by subtracting 0.3 eV from the energy level difference. This shift was discovered empirically and is linked to residual carrier binding energy and interface effects [95]. An integral of external quantum efficiency (EQE) multiplied by the number of photons from the Air Mass 1.5 sun spectrum at all frequencies. For energies below and above the donor's optical band gap (Eopt), the EQE is just a step function with a value of 0%. The fill factor is FF = 0.65 for all devices. Other EQE and FF assumptions can be made if desired. For example, the EQE could be determined by investigating the Kohn-Sham joint density of states, revealing the frequency-dependent absorption cross-section behavior. In this situation, the polymer layer is thick enough to absorb any photon over the optical band gap, and the film shape essentially limits the EQE. The following equations describe Scharber's mode.

$$PCE = \frac{V\_{OC} J\_{SC} FF}{1000 W/m^2} \tag{4}$$

$$\text{LUMO}\_{donor} > \text{LUMO}\_{acceptor} + \text{0.3eV} \tag{5}$$

$$eV\_{OC} = LUMO\_{acceptor} - LUMO\_{donor} \tag{6}$$

$$E\_{\rm opt} = \text{LUMO}\_{domer} - \text{HOMO}\_{domer} \tag{7}$$

$$EQE(o) = \mathbf{0.65} \times \Theta(\hbar o - E\_{opt})\tag{8}$$

$$J\_{\rm SC} = \int EQE(o) \times \#photons\_{AirMax1.5}(o) do \tag{9}$$

As seen in Eq. (5), this model implies a 0.3 eV energy difference between the donor and acceptor Lumos to enable effective charge transfer. This LUMO offset should not be confused with the 0.3 eV empirical shifts for Eq. (6). So Eopt = 0.6 eV is the maximum value for eVOC.

#### **5.6 Photoabsorption spectrum**

Time-dependent DFT has surpassed all other methods for calculating organic compounds' excitation energies and optical characteristics in the last decade. Visual features like absorption spectra and optical band gap can be used to validate structures further. To better understand the electronic transitions of polymer monomers, used TD-DFT/CAMB3LYP/with varied basis set levels to perform quantum calculations on electronic absorption spectra in the gaseous phase and solvent. Aside from the bandgap, the computational prediction of whole spectrum excitation energies and cross-sections above the bandgap is equally crucial to solar cell efficiency. The excited electron tends to decay toward the conduction level (or the LUMO level) before being injected into the anode due to vibronic (molecular dyes) or photonic (solid dyes) contact with the environment, resulting in thermalization loss of the cell efficiency. The choice of time-dependent TD-DFT can be critical in accurately reproducing absorption, especially when using donor-acceptor dyes with charge-transfer excitations, as in the current study, where range corrected functionals become a viable option. The band maximum (λmax) is an apparent essential feature of the absorption spectra.

These are significant for optical properties of polymers in polymer solar cell applications, ranging from TD-DFT methodologies to predicted absorption wavelengths (λmax), oscillator strengths (f), and vertical excitation energies (E).
