Structure-aromaticity-spectroscopy relationship in conjugated polymers

In this study, an effort has been made to analyze the aromaticity of oligomers of phenylenes and thiophenes, with the presence and absence of linkers using Nucleus-Independent Chemical Shift (NICS) as a descriptor. The relation between HOMO–LUMO gaps, reorganization and excitation energies with respective NICS values has been employed to develop a structure-aromaticity-conjugation spectroscopy relationship (SACSR). Results show that HOMO–LUMO gaps/excitation energies of various model systems exhibit linear relationships with the inverse of the NICS values, indicating the possible existence of the SACSR.


Introduction
In 1931, Hückel introduced the concept of "aromaticity" in planar cyclic π-conjugated molecules with delocalized "4n + 2" π-electrons to account for their unique thermodynamic stability [1]. Later, in 1965 Breslow proposed that cyclic molecules with "4n" π-electrons are energetically unstable and antiaromatic [2]. The concept of aromaticity/antiaromaticity has been extended to clusters/polymers and organometallics in addition to organic and inorganic molecules [3][4][5][6]. In order to account for the nature of aromaticity in various types of molecular systems, a variety of rules following the Hückel rule have been proposed [7]. Notably, the concept of three-dimensional aromaticity was suggested to understand the structure and stability of fullerene, doped transition metal clusters, and carboranes [8][9][10]. It is now well-established that aromaticity is a highly important concept in the rationalization of the structure, stability, and reactivity of many cyclic molecules. A huge volume of research contributions has been made to understand this profound concept [11][12][13][14][15][16][17][18][19][20]. Several reviews have appeared in thematic review journals [8,18,[21][22][23][24]. It has been established that the majority of the molecules having aromatic character are highly symmetric in nature and exhibit more molecular orbital (MO) degeneracy with closed shell structures [25]. However, in numerous other molecular systems, various unconventional forms of aromaticity have been observed experimentally [26,27]. For example, Möbius aromaticity has been found in macrocycles and metallacycles [28][29][30][31][32][33][34][35][36]. Aromaticity in metal systems and σ-aromaticity of metal clusters in the solid state have been reported [37][38][39][40]. Particularly from the organic electronics perspective, the concept of aromaticity in electronically excited states of conjugated small molecules and polymers has been illustrated. Baird [7] showed that [4n]annulenes in the triplet excited (T 1 ) state Masiyappan Karuppusamy and Shyam Vinod Kumar Panneer have contributed equally to this work. adopt a planar conformation and aromaticity arising due to the electronic conjugation and associated energetic stabilization [41,42]. The applicability of the same has also been proposed for the S 1 state [43,44]. For example, the excited state aromaticity of benzene has been quantified by estimation of aromaticity and antiaromaticity of its ground (S 0 ) and excited (S 1 & T 1 ) states [45]. A similar investigation has been carried out for the excited states (S 1 and S 2 ) of various structures of cyclobutadiene. These findings imply that aromaticity plays an important role in modulating the properties of molecular systems. Specifically, understanding aromaticity in conjugated molecules is immensely useful in developing organic materials employed for thermally activated delayed fluorescence (TADF) and singlet fission (i.e., for tuning the singlet-triplet energy gap, E ST ). Bakouri et al. have reported design strategies for singlet fission in substituted fulvenes based on the aromaticity of their T 1 state [46]. Lin and Zhu have elicited the presence of adaptive aromaticity (i.e., a molecule exhibiting aromaticity in both lowest singlet and triplet states) in osmapentalene, which enables singlet fission [47].
Several studies have been carried out to develop structure-activity and structure-property relationships with the help of aromaticity [11][12][13][14][15][16][17][18][19][20]48]. However, aromaticity cannot be experimentally determined as no single direct parameter measures the term "aromaticity". It can be defined through various indirect quantities based on energetic, geometric, or magnetic properties. The main factors affecting the structure of conjugated polymers and, thereby highest occupied molecular orbital (HOMO)-lowest unoccupied molecular orbital (LUMO) energy gap ( E g ) can be qualitatively described as [18] where E topol g is the energy contribution arising from the linear, branched and dendrimer structure of the polymer, E Δr g is the component of energy from bond length alternation (BLA), E g is arising from torsion, E arom g represents the energy due to the presence of an aromatic ring which limits π-delocalization outside the ring, E sub g is the change in energy of the system due to the substitution of various groups and E error g denotes energetics related to geometry defects, disorder and inter-chain interactions. Among these factors, the BLA plays a key role in both conjugated and extended systems. The BLA is defined as: where R long and R short are the average bond lengths of the longest and shortest bonds in the cyclic system, respectively, and it is connected to an empirical relationship as given below.
where c altern is the proportionality constant. Brédas has explored similar relationships by considering model systems [49,50]. Essentially, the changes in the bond length describe both aromatic and quinoid structures and hence the energy gap. This descriptor intrinsically bridges the profound concepts of the present study related to aromaticity and conjugation.
Aromaticity is defined using the harmonic oscillator model of aromaticity (HOMA) index, HOMO-LUMO gap, resonance stabilization energy, aromatic stabilization energy obtained from homodesmotic equations, magnetic anisotropy, and magnetic exaltation (i.e., diamagnetic susceptibility anisotropies). From the conceptual density functional theory (CDFT) perspective, the usefulness of various global reactivity descriptors in understanding aromaticity has been reviewed [51,52]. Chattaraj and coworkers have systematically delineated the electronic structure principles and aromaticity [53]. In this context, both wave function theory (WFT) and density functional theory (DFT)-based methods have been widely employed to understand the term aromaticity [11,14,54]. DFT-based information approach has also been employed to unravel the aromaticity of various systems [55,56].
Experimentally, 1 H chemical shifts are used as a criterion for characterizing aromatic and antiaromatic compounds [57,58]. For example, the higher deshielding of the benzene protons with reference to the vinyl protons of cyclohexene is a manifestation of the molecular ring current induced by an external magnetic field [57]. Schleyer and co-workers have used absolute magnetic shielding calculated at ring centers of heavy atoms as an aromaticity/antiaromaticity criterion which is described as a Nucleus-Independent Chemical Shift (NICS) [17]. The negative value of the calculated NICS defines the aromaticity of the compounds, while the positive value describes the antiaromaticity of the compounds. In the context of excitation spectra of conjugated molecules, both ground and excited state aromaticity have been used to understand the spectral properties. Specifically, excitedstate aromaticity reversal has received much attention from researchers [59][60][61]. Stanger and co-workers have used NICS as a descriptor for aromaticity to derive the relationship between the HOMO-LUMO gap and adiabatic ionization potential of polycyclic aromatic hydrocarbons (i.e., oligomers of benzene, pyrrole, furan, and thiophene) [62]. In this seminal contribution, the relationship was based on the consideration that there is an intricate balance between the aromaticity of the entire system and its conjugation. Stuyver et al. reported that a reduction of aromaticity increases conductivity [63]. Brédas reported a connection between quinoid character and the HOMO-LUMO gap using bond length alternation as an aromaticity index [64].
In the context of organic opto-electronic and photovoltaic materials, the formation of excimers due to the interaction between the ground state of the molecule with an excited state molecule is also implicated through the diffusion of charges. For example, the presence of aromatic excimers has been reported in pyrenes [65,66]. The strong interaction between the two fragments of the excimer has contributions from localized excitations (LE), ion pair, and charge transfer [67].
Prompted by the success of the above-stated salient findings, the relationship between the calculated vertical excited energy of selected model compounds (as shown in Fig. 1) with the NICS index has been attempted to develop the SACSR. The optimum balance between the conjugation and aromaticity of the individual unit as well as the overall structure of the oligomer play a decisive part in the stabilization of the system. Thus, selecting appropriate descriptors is critical. It can be seen that the selected models contain several ring systems with conjugation. However, there are some concerns on the summation of NICS values in fused ring systems [62]. Bultinck et al. have applied the Polansky index and generalized population analysis for polycyclic aromatic hydrocarbon [68]. They have discussed several issues relating to the applicability of the local aromaticity of benzenoid rings in substituted benzene [69,70]. Stanger and coworkers have developed a NICS-based method for analyzing local and global ring currents in conjugated multi-ring systems [71,72]. They have highlighted that the current density of anthracene may be considered as a local benzoic current and two naphthalenic currents. The synergetic effect of these two currents leads to one "global" anthracenic current. Intuitively, the strong induced field at the center of anthracene can be considered as local and global currents within the compound. Motivated by the seminal contributions of Stanger and co-workers, we have undertaken this study to explore SACSR. Therefore, in the development of a relationship, the sum of the NICS values of rings has been chosen as a descriptor and the correlation between the calculated spectral properties and the sum of the NICS values of all rings in the oligomers has been derived.

Computational details
The chemical structures of various model systems considered in this investigation are depicted in Fig. 1. The models are classified into two groups, based on phenylene and thiophene-containing units, as given below: Thiophene-based oligomers • oligo(thiophene) (OT n : n = 1-5) • oligo(thiophene vinylene) (OTV n : n = 1-5) • oligo(thiophene ethynylene) (OTE n : n = 1-5) • oligo(thiophene phenylene) (OTP n : n = 1-5) In this work, the trans conformers of the model systems were considered for investigation. The geometry of all the molecules was optimized using DFT(B3LYP)/6-31G(d) method [73][74][75][76][77]. The optimized geometries were characterized for minimum energy nature based on zero imaginary frequency criterion. Initially, we calculated the NICS(0) and NICS(1) values using Schleyer's approach [17], which involves including a ghost atom at the centroid and at a distance of 1 Å from the centroid of various rings in the model systems at DFT(B3LYP)/6-311G(d,p) level. As it is known, these values contain contributions from π and σ electrons. In order to remove the effect of σ-electrons, hydrogen atoms were added to the ring systems of selected molecules. As suggested by Stanger et al. [62,78] the total NICS(0) som and NICS (1)  where NICS(r) Del ZZ , and NICS(r) som ZZ represent the ZZ-component of the NICS value of the delocalized system under investigation, the ZZ-component of the NICS value for the system after the addition of hydrogen atoms to remove the effect of σ-electrons and the NICS value of the system after correcting for the effect of σ-electrons, respectively. The NMR keyword was used as implemented in the Gaussian package [79] to determine the shielding values. The calculated NICS values were used for the development of SACSR of the conjugated systems. The vertical excitation energies of all these model systems were calculated in the gas phase at ωB97XD/6-31G(d,p) level employing time dependent-DFT (TD-DFT) approach [80][81][82][83].
It is now possible to predict the charge-transporting nature of π-conjugated systems with comparable accuracy with experiments by employing the incoherent hopping model. In this approach, during charge transport, an organic molecule is considered to undergo multiple energetic reorganizations between its neutral and ionic states. A schematic representation of the potential energy surfaces of a molecule changing from a neutral to a cationic state is illustrated in Fig. 2. Hence, the relative difference in energy between a molecule's neutral and ionic states would describe the molecule's chargetransporting tendency. According to the Marcus-Hush model of charge transport [84][85][86][87][88][89][90][91][92], the rate of charge transfer is expressed using the relation: where, k h∕e is the rate at which charge is transferred between adjacent molecules, h∕e is the hole/electron reorganization energy of the molecule, V is the transfer integral between an adjacent molecule, h is the Planck's constant, k B is the Boltzmann constant, T is the temperature at which the charge transport occurs. From the above equation, the charge transport rate mainly depends on V and h∕e . Further, the h∕e is considered to have two components: external reorganization energy due to solvation and internal reorganization energy due to thermodynamic restructuring of the molecular energy.
Although both V and h∕e are equally significant to understand the charge transport rate; in the present study, we have only estimated the internal reorganization energy for further discussion. The hole and electron reorganization energies can be calculated by employing the following equations: where E 0 S 0 is the total energy of the neutral molecule calculated using the optimized geometry of the neutral molecule, E + S + is the total energy of the cationic state of the molecule calculated using the optimized geometry of the cation of the molecule, E − (S − ) is the total energy of the anionic state of the molecule calculated using the optimized geometry of the anion of the molecule, E + S 0 is the total energy of the cationic state of the molecule calculated using the optimized geometry of the neutral molecule, E 0 S + is the total energy of the neutral state of the molecule calculated using the optimized geometry of the cation of the molecule, E − S 0 is the total energy of the anionic state of the molecule calculated using the optimized geometry of the neutral molecule, and E 0 (S − ) is the total energy of the neutral state of the molecule calculated using the optimized geometry of the anion of the molecule. The reorganization energy calculations were carried out at the DFT(B3LYP)/6-31G(d) level. All the calculations were performed using the Gaussian 16 (Revision A.03) software package [79]. The GaussView program was used to build and visualize the model systems [93]. All the plots were obtained using the OriginPro 2016 software package [94].

Results and discussion
It is well known that aromaticity is a concept, conjugation is a structural consequence, and their combined effect can be observed in spectroscopy. Hence, the structural parameters    Figs. 3 and 4, respectively. It can be observed that the backbone of the vinylene (OPV n & OTV n ) and ethynylene linked (OPE n & OTE n ) oligomers are planar compared to other model systems considered in the study due to restriction of dihedral angles and reduction in steric hindrance between the adjacent rings of the molecules by vinylene and ethynylene linkers along the backbone. However, the introduction of these π-linkers makes subtle changes in the bond lengths of the corresponding molecule. These geometrical changes may have a profound impact on the overall aromaticity of the system. Also, the introduction of these π-linkers improves the overall conjugation of the molecule. The reduction in the aromaticity of organic molecules would lead to a decrease in the HOMO-LUMO gap, which in turn enhances the conductivity.
Similarly, the HOMO-LUMO gap varies as a function of an inverse of the number of oligomers in a chain which, in turn, is directly related to the end-to-end distance of oligomers from the geometrical perspective. The end-to-end distance reflects conjugation in oligomers. Hence, it is intuitively possible to expect a linear relationship between the aromaticity and conjugation of a molecular system. The OP n series of oligomers (Fig. 3) exhibit non-planar geometries due to the steric hindrance between the ortho hydrogens, which results in the rotation of alternate phenyl units, which affects their electronic properties. Also, it can be observed from Fig. 4 that although the α-linked OT n oligomers display a planar geometry, a bent structure of the molecule is visible where the curvature becomes significant with the increase in the number of repeating units. In the case of OPT n and OTP n systems having alternating similar rings exhibit non-planar geometry. However, distortion becomes prominent with an increase in the number of thiophene units, and the oligomer adopts a curvy geometry. For example, in the case of the OTP 3 molecule, the presence of phenylene linkers inhibits the curling. The curvature in OPT n , is noticeable in the OPT 3 oligomer. However, the higher number of phenylene rings, compared to those in OTP n , curtails the curvature on the addition of more repeating units.   The isosurface plots of HOMO and LUMO of the considered oligomers are presented in Figs. 5 and 6. They reveal that the electron density of the molecules is delocalized on the entire backbone of the system. Also, with an increase in the oligomer length, the concentration of electron density is high at the central regions in contrast to the peripheral units. Specifically, the incorporation of phenylene units instead of thiophene moieties in the molecular backbone significantly affects the delocalization of electrons.     (Figures S1 and S2). It can be seen that the oligomerization significantly affects the aromaticity of the individual units and hence the overall local aromaticity of the models. However, scrutiny of the results could not explain the higher aromaticity of the benzene ring when compared to thiophene. For example, the calculated NICS(0) values for benzene and thiophene are −8.89 and −13.88 ppm, respectively. This may be due to the multidimensional nature of aromaticity [68]. Hence, the NICS(r) som ZZ model has been used to quantify the aromaticity of these systems. The NICS(r) som ZZ and ∑

NICS(r)
som ZZ values quantify the local and global aromaticity of constituent rings and overall oligomer.
The NICS values calculated employing Stanger's approach [62] Figures S3 and S4, respectively. Analysis of the results shows that the ZZ-component of peripheral rings is more aromatic when compared to central units, which is in close correspondence with the previous investigation [62].  (1) som ZZ values of the thiophene-based oligomers are due to: (1) the marginal overlap of 3p orbitals of sulfur with carbon's π-system and (2) the weaker diatropic currents in the thiophene units [78].
It can be seen from the NICS values (Tables S3 and S4) of each ring of the model systems that not all rings contribute equally. Also, it can be observed that, with an increase in the chain length, the NICS(0) som ZZ and NICS(1) som ZZ of the individual rings decreases with reference to monomeric units (OP and OT). Further, it is possible to observe odd and even effects in their NICS(0) som ZZ and NICS(1) som ZZ values. In addition, the results elicit that the average NICS(0) som ZZ and

NICS(1) som
ZZ value per ring decreases on elongation of the chain of the oligomer in accordance with the earlier reports [62,96].
Among the selected models, the OPT n and OTP n exhibit more negative NICS(0) som ZZ and NICS(1) som ZZ values, which can be observed in Tables S3 and S4, implying the presence of strong diatropic ring currents. This trend is followed by phenylene-based oligomers which have more negative NICS(r) som ZZ values, compared to the thiophene-based analogues. Apart from the common understanding of better aromaticity in benzene than in thiophene, it can also be seen that their oligomers carry the same trend on elongation. Analysis of systems with linkers elicits that the ethynylene linkers-based oligomers (OPE n & OTE n ) display enhanced diatropic ring currents compared to their vinyl counterparts.
This indicates that the triple bond of the ethynylene group facilitates the conjugation resulting in stronger diatropic ring currents across the oligomer than the double bond of the vinylene bridges.

Spectroscopic properties and aromaticity
Conventionally, the electronic properties of oligomers are related to the inverse of the number of repeating units in the oligomers [97]. The variation of the HOMO-LUMO gaps as a function of the inverse of the number of repeating units (1/n) has been plotted in Fig. 7. A linear relationship between the HOMO-LUMO gap and n is evident. It can be observed from the figure that the OPT n and OTP n oligomers exhibit comparable HOMO-LUMO gaps and

Reorganization energy and aromaticity
It is well known from Marcus theory [90] that the reorganization energy is a parameter that describes the rate of charge transfer. Therefore, it is interesting to unravel the inter-relationship between reorganization energy and aromaticity. From the conceptual density theory perspective, higher HOMO-LUMO gaps imply higher hardness and less chemical reactivity. Therefore, there is a need for external energy to activate the molecule. From the spectroscopic context, higher HOMO-LUMO gaps are related to higher reorganization energy and vice-versa. The relationship between the hole ( h ) and electron ( e ) reorganization  h and e values than their vinylene (OPV n ), ethynylene (OPE n ) and thiophene (OPT n ) bridged counterparts. A similar trend can be observed for the thiophene-based model systems. In the case of phenylene and thiophene oligomers, the correlation between reorganization energy and the inverse of ∑ NICS(r) som ZZ is poor, which calls for further detailed investigation. These results imply that a molecule with higher aromaticity exhibits higher reorganization energy. Therefore, a highly aromatic system may display a low charge transfer character. In general, the high (low) HOMO-LUMO gap indicates high (low) aromaticity and hence associated conductivity.

Conclusions
The relationship between the structure, aromaticity, and spectroscopic properties has been investigated for a series of oligomers based on phenylene and thiophene. A quantitative conclusion has been arrived between the aromaticity, as measured by NICS, and HOMO-LUMO gaps and excitation energies. The linear relationship between aromaticity and opto-electronic properties facilitates the rational design of oligomers for various applications with desired opto-electronic properties. Specifically, the OPT n and OTP n systems are more aromatic than other systems. Therefore, incorporation of either OP or OT influences the aromatic characteristics of any oligomer and hence associated properties which are highly useful in the modulation of optoelectronic properties of oligomers. A comparison of results shows that ethynylene linkers-based systems are more aromatic than molecules inter-connected with vinylene units. However, from the conjugation point of view, vinylene linkers are more preferable. In summary, the intertwined nature