**4. TD-DFT methods in polymer solar cells**

TD-DFT has become the workhorse of quantum chemistry for computing excited-state characteristics and charge-transfer excitations in complex networks [53]. Linear and non-linear simulations may be popular due to their scalability and variety of methodologies. However, photovoltaic TD-DFT computations pose significant problems, notably with long-range CT interactions. Furthermore, although standard computer programs make TD-DFT easy to use, it is not a "black box" technique since approximations utilized in TD-DFT tools can occasionally produce major systemic mistakes in estimated results.

Making correct CT excitation energies with TD-DFT is problematic because approximate exchange-correlation (XC) potentials lack the unique features of exact Kohn–Sham (KS) potentials [54]. Thus, a non-local, exact-exchange contribution throughout the exchange-correlation kernel solves the charge-transfer problem in TD-DFT.

Many hybrid functionals (HF) have recently performed well in benchmark tests [55]. For example, Zhao and Truhlar created a functional (M06-2X) that contains the complete non-local exact-exchange contribution [56]. Prior studies investigated comprehensive adjustments for non-hybrid local adiabatic XC potentials targeting the CT problem utilizing constraint variational density functional. According to Ziegler et al. [57], the linear response approach produces qualitative variances between the time-dependent Hartree-Fock (TD-HF) and adiabatic local potentialbased TD-DFT excitation energies and also between the TD-DFT and SCF excitation energies. In the variational approach, the mix of occupied and empty orbitals is allowed above linear terms to determine transition densities. However, the recent surge of activity in the field gives reason for optimism.

Constrained DFT (CDFT) [58] is an alternative technique that uses a variational constraint approach to alleviate TD-DFT's drawbacks. To handle charge- and spinconstrained electronic states within the ground state KS DFT technique [59], CDFT was created. Implementation is computational. CDFT has been proven to be useful for long-range electron transmission [60–62]. While TD-DFT's thrilled states (valence states) are inaccessible to CDFT, many of TD-DFT's bothersome excited states are dealt with naturally in CDFT.

CDFT can be used in systems where the ground-state electron density must meet a threshold. The localization of electronic density in space indeed reduces the variation in electron number between donor and acceptor areas by two. To enforce the constraint, Lagrange multipliers are used in CDFT. To counteract selfinteraction mistakes caused by approximation XC functionals, CDFT uses semilocal functionals that are denser than the density. It defines diabatic states for electron transfer kinetics and chemical reaction rates. Using a ground state DFT functional, these limitations can also compute long-range charge transfer and lowlying spin states [61, 62].

However, this paradigm has several drawbacks. In CDFT, the electron density is partitioned by nuclear populations, affecting the constraint potential's shape. So the exact electron density partitioning is unknown. As a result, CDFT can only represent a subset of electronic excitations. Therefore, CDFT may not be the best method for simulating some diabatic circumstances. Also, a time-dependent optimal potential treatment of exact exchange can help describe charge-transfer excitations [63]. However, the exchange potential is too local to correctly predict the bandgap (i.e. HOMO–LUMO gap). Semi-quantitative hybrid functionals can forecast band gaps better than orbital dependent functionals. They use correlations to communicate precisely. Somewhat of focusing on density, we can improve approximation solution XC functionals in orbitals. They can also compensate for imperfections in selfinteraction, and exchange energy is naturally stated in orbitals.

Dispersion correction to KS-DFT is widely employed to handle long-range electron correlations that cause dispersion forces [64, 65]. Unfortunately, local DFT functionals overestimate dispersion forces, whereas non-local and hybrid DFT functionals underestimate them [64, 65]. Developing DFT functionals optimized for improved management of Vander Waals interactions, on the other hand, appears promising. The Vander Waals density functional (vdW-DF) [65] techniques successfully cope with London dispersion interactions. Modern dispersioncorrected DFT methods include empirical components because they function best for long-range interactions. However, standard density functionals perform well for close interactions. So any dispersion-correction method using DFT has to integrate the short and long-range asymptotic areas, which are both well understood individually.

The most widely used non-empirical method for determining dispersion energy for molecular systems is the vdW-DF approach [66]. Due to the charge transfer reliance of dispersion being incorporated via electron density, this technique naturally accommodates it. Regular adjustments also alter thickness. Despite its mathematical complexity, this approach can achieve a smooth transition between chemical binding at small distances and Vander Waals attraction at long distances.

The KS inherent DFT defect is the KS-dispersion DFT term. Adding damped inter-atomic potentials of the kind C6R6 to the KS-DFT energy appears to be another successful empirical way of accounting for dispersion [64]. This approach is substantially faster to calculate, has high numerical stability, and provides physical insight. Despite its semi-classical origin, it can supplement standard density functionals in treating long-range electron correlation. In addition, estimating supramolecular complex binding energies has been proved to be easier with this strategy.

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