Validity of density-functional-theory-based molecular modeling for UV/visible spectroscopy and rationale of panchromatic PbI64−(MeNH3+)4-structured molecular solar cells

Density-functional-theory-based molecular modeling verifies that perovskite solar cells (PSC) are composed of semiconducting and panchromatic layers constructed by van der Waals and Coulomb interactions (vdW&Clmb) between PbI64−(MeNH3+)4-derived species. Firstly we validate that DFT-based UV/vis spectral analysis is a useful approach for vdW&Clmb based on assignments of the fine-structured UV/vis spectrum of a benzene solution. The UV/vis spectral analysis of aggregated PbI64−(MeNH3+)4 proves that dimeric aggregates of [PbI64−(MeNH3+)4]2 have a panchromatic UV/vis spectrum of λ max ranging from 417 to 959 nm. Further analysis proves that the strong vdW&Clmb of PbI64−(MeNH3+)4 leads to unidirectional electron transport at the structured components such as the dimer of [PbI64−(MeNH3+)4]2, nc-TiO2/PbI64−(MeNH3+)4, and PbI64−(MeNH3+)4/spiro-OMeTAD. PbI64−(MeNH3+)4-structured solar cells should have a photoelectron diffusion length enhanced by the alignment of the frontier molecular orbitals in the structured PbI64−(MeNH3+)4 components, which supports remarkable short-circuit photocurrent, open-circuit voltage, and fill factor for the molecular-orbital-connected solar cell of HOTi9O18H/[PbI64−(MeNH3+)4]n/spiro-OMeTAD under solar-light irradiance.


Introduction
Density-functional-theory (DFT) 1,2) -based molecular modeling is a novel approach for verifying and predicting the molecular-orbital-based electronic properties of molecular aggregates in which the constituent molecules are aligned by van der Waals and Coulomb interactions (vdW&Clmb). [3][4][5] Recently, Persson and colleagues reported that DFT-based molecular modeling is a useful tool for predicting the electronic and energetic properties of crystalline materials in which crystalline structures play decisive roles in functionalities such as solar light conversion efficiency. 6) Molecules in crystalline lattice matrixes are aligned via vdW&Clmb. VdW&Clmb gives rise to London dispersion forces (<1-15 kcal=mol), dipole-dipole interactions (0.5-2 kcal=mol), and hydrogen bonding (1-12 kcal=mol). Molecules and ions have a tendency to form molecular aggregates with various sizes in the quasi-solid states. An accurate prediction of the geometrical configurations in molecular solids and aggregates is required. There have been theoretical investigations on the impact of the geometries on the electronic states by taking into account the intermolecular vdW interaction. The non-empirical or ab-initio vdWinclusive methods played important roles in predicting the atomic configurations of organic single crystals or organicmetal interfaces, which contributed to the understanding of the geometrical factors dominating electronic properties. [7][8][9][10][11] However, the computational complexity or cost of the nonempirical or ab-initio vdW-inclusive methods prevents one from theoretically investigating the electronic properties of realistic molecular aggregates that are of interest from the materials point of view. 12) In this work, to gain insights on the material properties, we employ the molecular mechanics optimization of the geometrical configurations taking vdW&Clmb into account, followed by the single-point electronic structure calculation within DFT.
The DFT calculation gives us useful information on molecular aggregates formed by vdW&Clmb, i.e., threedimensional geometrical configurations, their formation thermodynamics, molecular orbital configurations, surface electron density, and electronic energy structures. The DFTbased molecular modeling of the aggregates enables us to predict theoretical electromagnetic energy spectral data such as UV=vis, IR, FIR, and NMR spectral data.
For UV=vis spectral data, their absorption strength, molecular orbital components, and contribution to respective absorption peaks are extracted. UV=vis spectroscopy simulation with the time-dependent DFT 13,14) is likely to be more successful in anticipating changes in color resulting from changes in structure. 15) We previously reported that the slip-stacked zinc-phthalocyanine causes a redshift of the calculated and experimental absorption maximum of UV=vis spectra. 16) Before verifying the panchromatic color of so-called perovskite solar cells, we successfully attempt to validate the fact that the DFT-based modeling of molecular aggregates is useful for verifying and predicting the UV=vis absorption spectra of molecular vdW&Clmb aggregates. We first report that the DFT-based modeling of the vdW&Clmb aggregation of benzene verifies no large difference in calculated and experimental UV=vis spectra of a benzene solution.
Secondly, we attempt DFT-based modeling of PbI 6 4− (MeNH 3 + ) 4 and its vdW&Clmb aggregates to gain physicochemical insights on the high solar light harvesting efficiency of "so-called" perovskite solar cells (PSC), i.e., panchromatic color and electron diffusion length. The diffusion length of solar cells is defined by the electron lifetime and electron diffusion coefficient of the solar cell devices being affected by electron leakage at molecular heterojunctions. In the preceding paper, 3) we successfully attempted to verify the high efficiency of PbI 6 4− -based PSCs by the DFTbased modeling using PbI 6 4− in the X-ray crystalline structure, which is denoted as FOLLIB in Cambridge Structural Data (CSD). 17) However, FOLLIB-PbI 6 4− was found to be unsymmetrical in P-I bonds, giving unfavourable calculated UV spectra, i.e., λ max = 261 nm for FOLLIB-PbI 6

Theoretical methods
Single-point DFT calculations were performed using the B3LYP exchange-correlation functional 20) and 6-31G(d) basis set with Spartan'16. 21) Prior to the single-point DFT calculation for electronic properties, the geometries of the molecular aggregates were determined by the molecular mechanics simulation employing the Merck Molecular Force Field (MMFF). 22) The validity of the geometrical configurations of the molecular aggregates was confirmed by the single-point total energy calculation at the B3LYP-D3 23) =def 2-QZVP 24) level of theory. The basis set superposition error was eliminated by using the counterpoise correction. 25) The configurations of the molecular orbitals for vdW&Clmb molecular aggregates, molecular aggregation energy (ΔE ), dipole moment, and energy of the molecular orbitals such as LUMO (ELUMO), HOMO (EHOMO), and LUMO+1 [ELUMO (+1)] are obtained as key data. Here, the results are based on the single-point B3LYP=6-31G(d) calculation. It is found that the results obtained with B3LYP-D3=def 2-QZVP do not change the conclusion. The electron transfer gaps ΔE t (ELUMO-EHOMO) and ΔE t [ELUMO (+1)-ELUMO] are measures of the semiconducting electron transfer capability at the ground and excited states (electron accepted states, i.e., radical anion states) of molecules and molecular aggregates. [3][4][5] The chemical reactivities of the molecular species have been successfully investigated in terms of the change in electron density of the frontier orbitals upon electron addition=extraction, i.e., the Fukui function. [26][27][28][29] The semiconducting carrier mobility or electrical conductivity of the PSC materials has been reported. [30][31][32][33][34][35] For UV=vis analysis revealing the allowed transitions between molecular orbitals with wavelength and intensity, the timedependent DFT within the linear response regime was used. 13,14) The surface electron density and electrostatic potential map are used for understanding the vdW&Clmb aggregation as a new type of molecule without covalent bonding. For a review of the quantum mechanical methods used in Spartan, see Ref. 36.
The SPE-based oblique (C 6 H 6 ) 2 configuration, the most stable dimer configuration in this study, gives ΔE of −1.97 eV, which is lower than that of the orthogonal dimer by 0.3 eV. Given the previous theoretical studies reporting lower ΔE by using accurate wave function-based 38) or the non-local vdW-inclusive density functional 39) method, we might expect other configurations with lower ΔE. However, in this study, we focus on the electronic properties of the molecular aggregates in crystal matrixes or in solutions, rather than the detailed geometrical configurations of an isolated dimer.
The benzene trimer (C 6 H 6 ) 3 is investigated with DFT for SPE and EQG geometries obtained with the MMFF-based optimization of the oblique-oriented (C 6 H 6 ) 3 structure. The EQG-based structure is confirmed to give a higher exothermic formation energy than the SPE-based oblique dimer, and its molecular orbital energetics and UV=vis spectrum are shown in Figs. S5-1 and S5-2 in the online supplementary data at http://stacks.iop.org/JJAP/57/121602/mmedia.

Validity of density-functional-theory-based molecular modeling for UV/vis spectra of molecular aggregates
The examination of UV=vis spectra of benzene reveals that the thin film of a benzene solution (4°C) has a structured UV absorption in the range of λ = 200-210 nm and the benzene solution has a very weak but fine-structured UV absorption in the range from λ = 225 to 270 nm. 40) On the other hand, the crystalline structure of benzene was determined by X-ray crystallography (BENZEN18 in CSD 37) ). Interestingly, the packing unit consists of 7 orthogonal benzene dimers [totally, 14 benzene molecules of (C 6 H 6 ) 14 ]. We understand that benzene molecules aggregate with each other orthogonally via vdW&Clmb.
The benzene ring structure of EQG-based C 6 H 6 is almost comparable to that in BENZEN18 as shown in Fig. 1, but the distance between the benzene rings is closer in the orthogonal benzene dimer in BENZEN18 than in the EQG-based orthogonal benzene dimer. Crystal matrixes make the distance closer, and such aggregation never affects the UV=vis spectra in solution.
Further vdW&Clmb aggregations of benzene, i.e., SPEbased oblique dimer (C 6 H 6 ) 2 , EQG-based oblique trimer (C 6 H 6 ) 3 , trimer-derived tetramer (C 6 H 6 ) 4 , and tetramer (C 6 H 6 ) 4 as a dimer of orthogonal dimer, are molecularmodeled. The UV=vis spectra are shown in Fig. 2, along with their density structures. For comparison with the experi-(C 6 H 6 ) 2 optimized by DFT/MM  mental UV=vis spectra of vapor benzene, those of a thin amorphous state of benzene at 4.2 K were shown (Fig. 2). The strong absorption of vapor in the range between 170 and 190 nm is explained as due to a single benzene molecule, and the fine structured absorption of film-like solid benzene at 4.2 K as due to the amorphous state of mixtures consisting of at least dimers, a trimer, tetramers, a pentamer, and two kinds of hexamers of benzene aggregates. Agreement of the theoretical absorption spectra obtained at the TDDFT= B3LYP level of theory with experiments implies that the excited states of interest are dominated by the intramolecular valence excitations of a single benzene molecule, which are more or less perturbed by the molecular aggregation.
The DFT-based UV=vis spectra of pentamer (C 6 H 6 ) 5 and hexamer-1 and hexamer-2 of (C 6 H 6 ) 6 are shown in Fig. 3 with very weak structured UV=vis absorption spectra in the range of 220-270 nm. The DFT-based UV=vis absorption in the range of 224-230 nm appears even in the EQG-based single benzene, and the intensity increases with the number of aggregating benzene molecules. The metastable compact hexamer-2 gives the maximum UV=vis absorption at 230-231 nm. A benzene solution is a mixture of the vdW&Clmb aggregates, giving the UV=vis spectrum reflecting the amorphous states of benzene aggregation.
These results verify that the DFT-based calculation correctly simulates the aggregation of covalently bonded molecules such as benzene, giving molecular orbital energetics and UV=vis spectra as results of the degenerated HOMO and LUMO interactions of a single benzene, orthogonal benzene dimer, and oblique benzene trimer. We now validate that the DFT-based calculation of molecular aggregation is a novel approach for verifying the vdW&Clmb aggregation of covalent-bonded molecules as electron-density-based molecules, and then new molecular orbitals of the molecular aggregate affect the UV=vis spectra of the vdW&Clmb aggregation of a molecule such as benzene.  (Fig. 6). After introducing the HO − group to the aggregated [Ti 9 O 18 H] + , the interfacial structure is successfully molecular-modeled where all MeNH 3 + ions are not frozen. It is worth noting that the electron transfer gap between LUMO+1 and LUMO is 0.45 eV, suggesting the formation of semiconducting electronic structures at the interface.
3.4 Rationale of high efficiency of panchromatic PbI 6 4% (MeNH 3 + ) 4 -structured molecular solar cells We have found differences in chemical structures between the FOLLIB-based and the DFT-simulation-based PbI 6 4− . Surprisingly, their calculated UV=vis spectra are quite different in absorption maxima and intensity (Fig. 4). FOLLIB-PbI 6 4− consists of different Pb-I bond lengths, ranging from 3.195 to 3.765 Å. It is worth noting that the case is true for the orthogonal benzene dimer in BENZEN18. The average difference in the shortest separation distance observed between the EQG-based orthogonal benzene dimer and the one in BENZEN18 is calculated to be 0.518 Å (Fig. 1). The structural change may come from a kind of strain caused by crystalline matrixes. Considering that there is no such strain at amorphous states, the three-dimensional symmetrical structure of EQG(1)-PbI As mentioned in the preceding papers, [3][4][5] the energy transfer gap between LUMO and LUMO+ is a measure of the semiconducting molecular transfer under solar light irradiance. Photogenerated electrons convert LUMO and LUMO+1 into HOMO (SOMO) and LUMO of the resulting radical anion. The energy transfer gap of radical anions of three isomers is in the range of less than 0. In addition, we have predicted and verified the important roles of the semiconducting properties of heterojunctioninduced vdW&Clmb in dye-sensitized solar cells and photoelectrochemical water spliting. 5,18,19,42) (Fig. 6). Interestingly, HOMO locates between the HO group on the nc-TiO 2 model and the iodine atom on PbI 6 4− , verifying the interfacial vdW&Clmb interaction. The electron transfer gap of the radical anion of the structure is −0.45 eV, suggesting unidirectional electron transfer at the interface under photoenergized bias conditions. The case is also true for the interfacial structure between [PbI 6 4− (MeNH 3 + ) 4 ] and spiro-OMeTAD. In particular, the electron transfer gap of the radical anion of the structure is estimated to be −0.24 eV, verifying the unidirectional electron transfer at the interface as well (Fig. 7).
In general, a high open-circuit voltage (V oc ) of more than 1 eV is essential for a highly efficient PSC. To verify the facts, the theoretical maximum V oc is calculated from

Conclusions
We have shown that the DFT-based calculation can verify and predict the color of molecules in amorphous states, in which they aggregate by vdW&Clmb. On the basis of the DFT-based verification of vdW&Clmb molecular alignments and UV=vis spectroscopy, PSC is now visualized as a onemolecular wire by space-filling-model-based, electrostaticpotential-map-based, and LUMO-chain-based structures (Fig. 9). In other words, PSC can be regarded as a bundle of [PbI  quantum-dot-like or photosynthetic-like molecular structure rationalizes the large photoelectron diffusion length for the high photocurrent (J sc ), high V oc , and respectable fill factor (ff ) of the one-molecular-like wire of nc-TiO 2 =PbI