Approximations to the exchange-correlation functional

exchange and correlation parts (both local and non-local portions) with their own weighting factorsc₁–c₅, is the following

E_XC[ρ(r)] =c₁E_X^HF+c₂E_X^local+c₃E^non-^local

X +c₄E_C^local+c₅E^non-^local

C . (2.18)

Below, the formulas are given for some global hybrid functionals, which have been popular in the computational studies of PSC systems and have been also employed in this work. The popular three-parameter B3LYP[120, 121]implemented in the mod- ern quantum mechanical software combines the exchange and correlation terms de- rived from LSDA, fraction of the exact HF exchange (20%), and the Becke’s exchange functional B88[118]. The non-local and local correlations in B3LYP are provided by the LYP and VWN[122]correlation functionals, respectively:

E_XC^B3LYP=0.20E_X^HF+0.80E_X^LSDA+0.72E_X^B88+0.81E_C^LYP+0.19E_C^VWN (2.19)

with the parameters (c₁=0.20,c₂=1.00−c₁,c₃=0.72,c₄=0.81, andc₅=1.00−c₄) being derived by a fit to the set of experimental data. B3LYP has been very popu- lar functional and is commonly employed in the studies of PSC compounds even nowadays.

In the global hybrid functional PBE0 (also denoted by PBE1PBE1)[123, 124], which is mainly based on fundamental constants, the amount of the exact HF exchange (25%) has been defined on the basis of the perturbation theory rather than using empirically determined parameters:

E_XC^PBE0=1

4E_X^HF+3

4(E_X^LSDA+∆E_X^PBE) +E_C^PW+∆E_C^PBE. (2.20) The PBE0 functional uses the PBE GGA exchange[125, 126]and PW correlation functionals[127]. Due to its single parameter (c₂=c₃=1.00−c₁) derived from the theoretical considerations, PBE0 is also said to be the parameter-free functional.

It has resulted in relatively accurate electron densities for a set of the studied atomic

systems[128]and also for larger organic molecules including two to ten heavy atoms (e.g. carbon, oxygen, nitrogen, and sulfur)[129].

Despite the improved results provided by the global hybrid functionals due to the fragment of exact HF exchange when compared to the semilocal functionals, it is well known that they yield inaccurate description of some relevant molecular prop- erties of theπ-conjugated systems[26]. They tend to underestimate bond-length alternation (BLA) patterns[33], IEs[130], fundamental gaps[130, 131], and excited- state energies[9], while overestimating delocalization of the electron density[33], torsional barriers[132], and intermolecular electronic couplings[26]. These short- comings are mainly due to the MSIE[27, 28]in these functionals, which leads to the incorrect dependence of the asymptotic region ofν_XCwith distance. Basically, the decay of the global hybrid functionals is proportional to the amount of the HF exchange in them (-c₁/r), while the correct decay should be -1/r. The inclusion of 100% HF exchange would correct these issues, but semi-local exchange is also re- quired for the correct description of chemical bonding, etc. [26]. Thus, to utilize the advantages of HF and semi-local exchange, they should be both included in a functional.

Long-range corrected functionals

Restoring of the correct description of the asymptotic region of theν_XCcan be achieved by partitioning the Coulomb operator 1/r₁₂, which describes the interaction be- tween electrons 1 and 2. In the class of so-called range-separated hybrid (RSH) func- tionals[29, 133, 134], the Coulomb operator is divided into SR and LR components with a standard error function (erf, or its modified version):

1

r₁₂ = erf(ωr₁₂) r₁₂

⏞ ⏟⏟ ⏞

LR

+1−erf(ωr₁₂) r₁₂

⏞ ⏟⏟ ⏞

SR

(2.21)

whereωis the range-separation parameter (with a dimension of inverse length). The general form of the RSH functional is

E_XC^RSH=c_x,SRE^SR-^HF

x,SR +c_x,LRE^LR-^HF

x,LR + (1−c_x,SR)E_x,SR^DFT

+ (1−c_x,LR)E_x,LR^DFT+E_c^DFT. (2.22)

In the case of molecular systems, the RSH functionals referred to as LRC functionals are employed[29–32]. These functionals use the full non-local exchange and local correlation in the LR part, while treating the nonclassical interactions in the SR part by standard semi-local or global hybrid functionals. The idea is to take advantage of the semi-local or global hybrid functionals in the bonding region, while allow- ing for a correct treatment of the asymptotic region. In the calculations of solid- state systems, screened-exchange RSH functionals[135–137], where HF is used in SR and DFT in the LR, are employed instead. For the LRC functionals,c_x,LRin Equation 2.22 equals one, while for the screened-exchange RSH functionals it equals zero.

The use of default values ofω in the LRC functionals is not recommended, asω has proven to be a system-dependent parameter[130, 138]. Non-empirical tuning approaches for determining an optimalω value for the studied system have been developed by Kronik et al. [131, 138], which are based on the DFT’s analogous to the HF Koopman’s theorem (see Section 3.2.3). In addition to tuning ofω, the mixing of a small amount of the HF exchange into the SR part has been observed to result in improved prediction of optoelectronic properties of compounds employed in organic light emitting diodes[139]. The LRC functionalωB97X consists of full (100%) HF exchange at the LR part and a small fraction of the SR HF exchange[30]:

E_XC^ωB97X=E_X^LR−HF+c_xE^SR-HF

X +E^SR-B97

X +E_C^B97 (2.23)

withc_x=0.157706. Both theωB97X functional and its dispersion corrected version, ωB97X-D (see below), have been employed in this thesis.

In the LRC CAM-B3LYP functional[32], a generalized form of Equation 2.21, i.e.

a Coulomb-attenuation method (CAM), is used for the splitting of the Coulomb

operator into the SR and LR parts. On the contrary to the most of the LRC func- tionals, which include the full HF exchange in the LR part, the amount of the exact HF in CAM-B3LYP is 65% in the LR. This smaller than 100% amount of the exact HF can have consequences on the calculated results. However, as the CAM-B3LYP functional has been a popular choice in the previous studies of excited state character- istics, electronic coupling, and CT rate calculations of copolymer–fullerene systems, it has been also used in this thesis for comparison.

Dispersion corrected functionals

Weak dispersion interactions, which are generated by the fluctuating changes in the charge distribution around the molecular system caused by the movement of elec- trons, are known to impact descriptions of eD–eA interfaces[3, 45]. Thus, they should be taken into account in the calculations of the eD–eA interfacial complexes.

As both pure and global hybrid DFT functionals are not able to describe dispersion correctly, either new functionals including dispersion corrections have been devel- oped or the empirical dispersion term have been added to the existing functionals.

In the latter case, the general form of the total energy obtained with this kind of DFT-D scheme is[31]

E_DFT-^D=E_KS-^DFT+E_disp (2.24) where an empirical atomic-pairwise dispersion correction term (E_disp) is added to a Kohn-Sham (KS)-DFT part. For example, the form of theωB97X-D functional is otherwise the same as that ofωB97X (Equation 2.23), but with the additional un- scaled dispersion correction term added to the KS-DFT energy[31]. The dispersion corrected version of the global hybrid B3LYP, i.e. B3LYP-D, including the D3 ver- sion of Grimme’s dispersion with the original D3 damping function has been also employed in this work.

3 COMPUTATIONAL MODELS AND METHODS

The theoretical models of the studied PSC compounds and details of the calculation methods employed in this thesis are presented in this chapter. In Section 3.1, the models of the isolated eD and eA compounds and their interfacial complexes are in- troduced. Section 3.2 summarizes the main details of computational methods and software. More detailed information and all the equations employed in the calcula- tions can be found in the original publications.