Optoelectronic properties of one-dimensional molecular chains simulated by a tight-binding model

Studying optical properties of organic materials is important due to the rapid development of organic light-emitting diodes, solar cells, and photon detectors. Here for the first time we have performed tight-binding calculations for singlet excitons, in combination with first-principles calculations of the excited states in molecular dimers, to describe the optical properties of a zinc-phthalocyanine one-dimensional molecular chain. We have included the intra-molecule and charge-transfer excitations and the coupling between them. Our calculations have successfully interpreted a body of experimental UV-Vis optical spectra of transition-metal phthalocyanines. Compared with the previous ab initio calculations for a molecular dimer, the optical absorptions at the split peaks of the Q-bands can be comparable, which indicates the importance of the coupling between the intra-molecular and charge-transfer excitons.

at the split peaks of the Q-bands can be comparable, which indicates the importance of the coupling between the intra-molecular and charge-transfer excitons.
KEY WORDS: one-dimensional molecular chain; tight-binding model; UV-Vis spectra; Zinc phthalocyanine Successful commercial applications of organic light-emitting diodes (OLED) in mobile phones, tablet and TV, and rapid developments of organic solar cells (OSC) have stimulated an unprecedented volume of experimental and theoretical studies for the optoelectronic properties of organic materials [1,2]. From the fundamental perspective, the optical properties of organic materials can provide crucial information about the electronic structure and vibrational modes, indicating intrinsic electrical and structural properties and identifying potential materials for optoelectronic devices. The measurement of UV-Vis spectra is a general experimental spectroscopic methodology for analyzing the electronic structure and assessing the potential of organic materials for devices. The comparison between experiments and theoretical modelling is important for understanding the electronic-structure origin of the optical spectra, assessment of the potential of the organic materials for optical applications, and the development of new materials with advanced optical properties. We can elucidate the atomic-scale origin of the optical spectra from modeling electronic structure, providing solid bases for rational molecular tailoring for new materials with advanced optical properties. It is well known that molecular chains in organic crystals are self-assembled exclusively because the strong π − π electrostatic interactions for the stacking of planar molecules lead to one-dimensional (1D) chain [3,4]. The primary characteristics of the optical spectra of organic semiconductors such as transition-metal phthalocyanines (TMPc), can be approximated by onedimensional chains due to the weak electronic coupling between molecular chains. The main features of the optical properties of TMPc, including the Qband and B-band, originate from the one-dimensional chains in the compound. Serri, et. al have shown that the magnetic properties of cobaltphthalocyanines (CoPc) are strongly dependent on the 1D chain structure, which can eventually be attributed to the electronic structure of the CoPc single molecule [5]. Similarly, we could therefore understand most of the optical properties of the TMPc organic semiconductors based on the calculations of the optical properties of one-dimensional (1D) molecular chains.
The optoelectronic properties and crystal structures of TMPcs have been studied experimentally extensively. Two UV-Vis absorption bands of TMPcs, namely the B-band and Q-band respectively, have been observed. The main peak of the B-band is located at a wavelength of λ ~ 350 nm while the main peak of the Q-band at λ ~ 650 nm [7].
Usually the Q-band of most TMPcs is very broad, splitting into two peak s, for which the interpretation is controversial. Nevertheless, the splitting of the Q-band should originate from the inter-molecular interactions.
Although it is generally accepted that the splitting is due to the Davidoff splitting, however the effect of Davidoff splitting is too small if compared with the experimentally observed in the Q-band, which is in the order of 0.1 eV. On the other hand, the observed splitting in the Q-band can also be attributed to the intra-molecular (IM) and charge-transfer (CT) singlet excitations [8,9].
The optical properties of molecules or finite size nano-structure can be calculated by employing methods such as configuration interaction (CI) [10], time-dependent Hartree-Fock (TDHF) [11], time-dependent densityfunctional theory (TDDFT) [11], and GW-BSE [12].  [14]. Chan and his colleagues have been developing coupled-cluster methods to compute the excited states in solids [15]. To the authors' best knowledge, it is very difficult [14,15] to simulate the optical properties of periodic solids from first principles owing to (i) a large number of atoms in the unit cell and (ii) a large number of single-particle excited states need to be taken into account.
Recently TDDFT calculations have been performed for a TMPc dimer, which have shown good accuracy for the excitation energies, but the oscillator strengths for the low energy excitation in the Q-band are too weak to be comparable with experimental observations [16]. However, this low energy excitations at the Q-band are vital for organics-based optical devices, we have therefore established a new theoretical model to improve the description of the oscillator strengths at the Q-band for a molecular chains.
Here we propose an excitonic tight-binding model for a ZnPc 1D periodic chain, in combination with the TDDFT calculations of ZnPc dimer, accounting for the intra-molecule and charge-transfer excitations and the coupling between them. This model strategy is not only suitable for molecular chains, but also any excitonic systems in which a two-level system can be well separated from the higher excited states. The optical spectra of molecule-chain compound TMPc in general have been qualitatively interpreted; we can see the strengthening of the lower-energy band in the optical spectra when including the coupling between excitons. The rest of this paper falls into three parts. The first-principles calculations of a molecular dimer and tight binding model are addressed. The results will be analyzed and discussed, in which the optical spectra of TMPcs are qualitatively interpreted using the model of molecular chain. At the end some general conclusions are drawn.
We have performed TDDFT calculations using Gaussian 09 code [18] for a dimer of zinc-phthalocyanine (ZnPc) molecules (a typical non-magnetic molecule). For the ground-state calculations, we have used hybrid-exchange DFT to compute the Kohn-Sham wave functions and the total energies. We adopted the geometry of the dimer by following the chain structure of the molecular crystal. TDDFT has shown a reasonably accurate description of optical transitions for molecules [16]. Furthermore, the hybrid exchange density functional B3LYP has been chosen for the electron-electron correlations. As suggested by the previous calculations, B3LYP functional can give rather accurate optical spectra and exchange parameters [5,16].
However, the recent calculations have shown that the range-separated exchange-correlation functional such as CAM-B3LYP can provide much better results, especially for the charge-transfer excitations, which can be used in the future calculations [17]. The parameters of the optical excitations, including the excitation energies and oscillator strengths, will then be used in the tight-binding model calculations.
We employed the tight-bind model that takes into account the intramolecule and the nearest-neighboring charge-transfer excited states, as shown in Fig.1  We also show the couplings between them by using 1 t , 1 t , 2 t , 2 t , which are between the upper states (normally LUMO) and the lower states (normally HOMO). These can be approximately considered as the couplings between IM and CT excitons.
We can then solve the eigenvalue problems for each k-point (assuming the lattice constant here is 1) to obtain the band structure as well as the eigenvectors. We first solve the Hamiltonian to obtain eigenvectors for each  ~0.03 eV. The computed spectra are qualitatively consistent with most of the optical spectra for the transition-metal phthalocyanines [16,19]. In Figure   3(b), we have shown the experimental UV-Vis spectra for the ZnPc powder (red squares) and nanowires (blue triangles). When comparing Figure 3 (a) and (b), we can see the qualitative agreement between theory (red and blue curves) and experiment. In addition, the computed optical spectra strengthen the long-wavelength part of the Q-band, which improves the previous TDDFT dimer calculations [16]. On the other hand, we can also see the main discrepancy is at ~ 500 nm, where there is a shoulder in the experimental spectra. Moreover, between 700 nm and 800 nm, the experimental spectra are much broader than the theoretical predictions. This could be due to a few aspects, including the addition vibrational modes (not included in the current modelling), other low-lying excitations such as −d transitions, and further couplings between molecules along or between the chains. The methodology presented here can therefore be potentially used to fit the experimental optical spectra by varying the parameters including phenomenological couplings, Gaussian broadening, and other fitting parameters. Notice that the energies have been rigidly shifted by ~ 2 eV compared to the above Hamiltonian.
We also computed the excitonic band structure as shown in Figure 2 We can further to generalize our model Hamiltonian to investigate other parameters for inter-cell coupling t´ and intra-cell coupling t. From here, we adopt d = 0. Besides the uniform chains, we can study similar physics to the SSH model [6], but here for bosons. As shown in Fig.3  Moreover, we can generalize our model Hamiltonian to investigate other parameters for inter-cell coupling t´ and intra-cell coupling t which is analogue to the SSH model. We anticipate there would be more interesting physics related to topological properties.