## 1. Introduction

## 2. Effect of Aggregation When Modeling Optoelectronic Properties

#### 2.1. Absorption Spectra and Band Alignment

_{XC}is the exchange-correlation energy [52]. For example, the effect of errors in orbitals is much stronger on TD-DFT excitation energies than it is on orbital energies [53]. These issues carry into TD-DFTB.

**k**is the wavevector; and

**q**is the photon polarization vector. The real part ${\u03f5}_{r}\left(\omega \right)$ of the dielectric function is then computed from the Kramers–Kronig relation and then the absorption spectrum (absorption coefficient):

_{B}T, where k

_{B}is the Boltzmann constant and T the temperature) and negligible effect on band alignment; see Figure 2.

#### 2.2. Charge Transport

_{ij}is the overlap integral between the wavefunctions of states i and j: $\u27e8{\psi}_{i}\left|{V}_{ij}\right|{\psi}_{j}\u27e9$, where V

_{ij}is the coupling (Coulomb interaction) term. The driving force could be approximated by the differences between corresponding single electron state energies. The Marcus equation depends strongly and non-linearly on the driving force and on intermolecular separation. The overlap integral J

_{ij}strongly depends on mutual position of molecules and its proper estimate requires explicit aggregate state modeling. J

_{ij}can in certain cases be estimated from the orbital splitting (e.g., for nearly-isotropic interactions and when molecular sites are similar) [63], but in general, more involved methods need to be used, such as dimer projection [64,65] which was also used by us in our studies of charge transport in fullerene derivatives. The dimer projection method has an advantage over the orbital splitting approach in that it does not make the isotropic approximation [63]. In the dimer projection method, the charge transport integral is computed as $\u27e8{\psi}_{i}\left|{F}_{ij}\right|{\psi}_{j}\u27e9$ where the states ${\psi}_{i}$, ${\psi}_{j}$ are represented by frontier orbital energies of two isolated molecules in the dimer, and ${F}_{ij}$ are elements of the Fock matrix $\mathit{F}=\mathit{S}\mathit{C}{\u03f5}_{KS}{\mathit{C}}^{-1}$ computed from the overlap matrix

**S**, the matrix of orbital coefficients

**C**, and the vector of Kohn–Sham energies ${\u03f5}_{KS}$ [63,64,65].

^{12}s

^{−1}) was in good agreement with available literature [66,67]

^{12}s

^{−1}(highest 3.98 × 10

^{12}s

^{−1}). The important conclusion was that the electron transfer rate does not noticeably drop compared to pure C60 (on the order of 5 × 10

^{12}s

^{−1}) crystals (for comparison, the rate was 1.8 × 10

^{13}s

^{−1}for pure C70).

## 3. Effect of Aggregate State When Modeling Organic Battery Materials

#### 3.1. Insertion-Type Materials

_{y,y+m}Host) are energies of the host in the corresponding charge states and E(M) the energy per atom of bulk M, n is the number of electrons transferred per unit M (n = 1 for Li or Na, n = 2 for Mg etc.), and F is the Faraday constant.

^{+}potential, gives an estimate of the voltage in a specific type of battery.

#### 3.2. p-Type Materials

^{+}, Na

^{+}) does not insert into the cathode material during battery discharge. Instead, the metal cations reversibly coordinate to the anions of the salt in the electrolyte, such as ClO

_{4}

^{−}or PF

_{6}

^{−}. The cathode material is reversibly oxidized by coordination and de-coordination to the anions [6]. The voltage is largely determined by the oxidation potential or the HOMO, and HOMO design, through the choice of key building blocks and functional groups, is an important component of the material’s design. This type of material is naturally more suited to realize high voltage (and therefore high energy density) organic cathodes. Most proposed p-type materials are polymers. Even common and non-expensive polymers such as polythiophene or polyaniline (PANI) can be used as cathode materials. This practically important and promising class of materials poses difficulties in ab initio modeling. Modeling of solid polymers requires relatively large simulation cells; moreover, these materials are often amorphous or semi-amorphous, which requires large simulation cells (on the order of 10

^{3}atoms or more) and consideration of multiple conformations.

_{4}

^{−}. In [46], we estimated the voltage curve of PANI based on single-molecule DFT calculations of the oxidation potential of PANI using small oligomers. In [45], we computed the voltage-capacity curve of solid amorphous PANI in a study that combined force field MD (to pre-optimize and sort multiple structures by energy), DFTB (to optimize a pre-selected number of structures), and DFT (to benchmark DFTB). As each simulation cells contained on the order of 10

^{3}atoms and dozens of structures had to be considered at different degrees of oxidation of PANI (different concentrations of counter-anions), the use of DFTB was critical for feasibility. The report in [45] presented the first relatively large-scale ab initio model of the voltage-capacity curve of a solid amorphous polymeric cathode.

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CPU | central processing unit |

DFT | density functional theory |

DFTB | density functional tight binding |

DOS | density of states |

GGA | generalized gradient approximation |

HOMO | highest occupied molecular orbital |

LUMO | lowest unoccupied molecular orbital |

MD | molecular dynamics |

MOF | metal-organic framework |

OLED | organic light-emitting diode |

OSC | organic solar cell |

PANI | polyaniline |

PLED | perovskite light-emitting diode |

PSC | perovskite solar cell |

TCNE | tetracyanoethylene |

TD | time-dependent |

UV-VIS | ultraviolet and visible |

vdW | van der Waals |

XRD | X-ray diffraction |

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**Figure 1.**Absorption spectra of C60 clusters (TD-DFT(B) and polarizability-based method) and solid (dipole approximation) computed with different methods. TD-DFT(B) curves end due to a finite number of excited states. The bottom right panel shows changes to the bandstructure induced by aggregation. Reproduced from [43] with the permission of AIP Publishing.

**Figure 2.**The computed formation energies (

**a**) of C60, C70, and the mixture of C70 and C60; and a zoom in into the density of states (DOS) of the mixture (

**b**) showing separate C60 and C70 contributions. Adopted with permission from the Supporting Information of [42]. Copyright (2018) American Chemical Society.

**Figure 3.**Top: absorption spectra of thiophene oligomers computed with TD-DFT (top left) and the polarizability-based method (top right), with B3LYP functional. Bottom: absorption spectra of different pentacene clusters (see [44] for definitions of the clusters) cut out of the crystal structure computed with TD-DFT (bottom left) and the polarizability-based method (bottom middle), with B3LYP functional. Bottom right: absorption spectra of solid pentacene and different pentacene clusters cut out of the crystal structure computed with the dipole approximation, with PBE functional. See [44] for details. Reproduced from Ref. 44, with the permission of AIP Publishing.

**Figure 4.**Correlations between HOMO and LUMO energies, and intermolecular distances (between centers of mass) during molecular dynamics in C60 and C70. Pearson R

^{2}coefficients are also shown. First appeared in [48].

**Figure 5.**Left: computed interaction energy of Li with a TCNE molecule, with a hybrid functional. Different symbols correspond to different configurations. The distance from the black curve to the cohesive energy of Li gives an estimate of voltage. Adapted from [38] with permission from the PCCP Owner Societies. Right: computed voltage-capacity curves for lithiation of solid TCNE (red curve and axis) and LiTCNE MOF (black curve and axis), using a GGA functional. Adapted from [37] published by the PCCP Owner Societies.

**Figure 6.**Voltage-capacity curve of sodium benzene tricarboxylate computed with a cluster model shown in the insert. See [47] for details.

**Figure 7.**Top left: the structure of the oligomeric model of PANI showing coordination of ClO

_{4}

^{−}anions. Top right, black curve: the computed voltage profile (for a Li ion battery) from the oligomeric model. Reproduced from [46] with permission from Elsevier. Bottom left: a simulation cell of solid PANI with intercalated ClO

_{4}

^{−}. Bottom right: the computed voltage profile for Li and Na ion batteries from the solid state model for PANI (curve with round symbols) and CN-functionalized PANI (curve with rhombic symbols). Reproduced from [45] with permission from the PCCP Owner Societies.

**Table 1.**HOMO and LUMO energies at the equilibrium geometry (“Equil.”), their expectation values over MD trajectories (“<…>”), standard deviations of HOMO and LUMO distributions over MD trajectories (“σ”), and reorganization energies λ and driving forces ΔG

_{eq}for electron and hole transport in C60 and C70 computed in [48]. All values are in eV.

Equil. | <…> | σ | λ | ΔG_{eq} | ||
---|---|---|---|---|---|---|

LUMO | C60 | −3.98 | −4.09 | 0.023 | 0.132 | 0.15 |

C70 | −4.04 | −4.16 | 0.027 | 0.123 | 0.15 | |

HOMO | C60 | −5.60 | −5.47 | 0.019 | 0.167 | 0.23 |

C70 | −5.54 | −5.48 | 0.020 | 0.138 | 0.34 |

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