From Monomer Sequence to Charge Mobility in Semiconductor Polymers via Model Reduction

A model reduction scheme for polymer semiconductors is presented that can be utilized to compute intra‐chain charge‐carrier mobility from the monomer sequence. The reduced model can be used in conjunction with any quantum dynamics approach, but it is explored here assuming that transport takes place through incoherent hopping events between states of different degrees of delocalization. The procedure is illustrated by considering 28 realistic polymers for which a quantitative correlation is established between charge localization characteristics and charge mobility. The data set helps in establishing plausible ranges for all the microscopic parameters of the model and it can therefore be used to determine the maximum plausible improvement in mobility. The reduced model is also used to provide some insight on the observation that the highest mobility polymers do not have very broad valence bands: there is indeed a range of the inter‐monomer coupling for which this parameter has little effect on the mobility.


Introduction
Over the years, semiconducting organic polymers have evolved as the active-layer materials in a range of technological applications like field-effect transistors, [1,2] light-emitting diodes, [3] and organic photovoltaic devices. [4] Despite of these large number of current as well as proposed technological applications like in photocatalytic, [5,6] thermoelectric, [7,8] bioelectronic, [9] and neuromorphic devices, quantitative models that link the chemical structures of the conjugated polymers with their chargecarrier mobilities are not yet available. The charge-carrier's motion in polymeric semiconductors is a complex process [10][11][12] due to the anisotropy of charge transport along the chain (intrachain transport) from that between neighboring chains (interchain transport). The traditional consensus is that the chargecarrier transport is primarily mediated through transport within DOI: 10.1002/adfm.202303234 spatially close semi-crystalline domains which are interconnected by long "tie chains". [11,13] However, recent demonstrations of high charge-carrier mobility in disordered, donor-acceptor copolymers, [14][15][16][17][18] comparable to that within highly ordered polymers like PBTTT, have highlighted that short-range intermolecular order is sufficient for efficient long-range charge transport. [11] The relatively rigid backbones of these donor-acceptor copolymers result in lower energetic disorders [19] and in consequence, extended persistence lengths; consequently, a small number of interchain contacts can successfully result in 3D percolation network. [20] Within these relatively rigid structures, the intra-chain transport becomes the major contributor to charge mobility. [21] Intra-chain charge-carrier transport in organic polymers is commonly attributed to a sequence of charge hopping events between localized states which stems from the static disorder in electronic coupling, [22][23][24], i.e., these localized states are quite similar in nature to the eigenstates observed in Anderson localization due to off-diagonal disorder. [25,26] In the organic polymer community, it is generally agreed that the static disorder in the electronic coupling, not the polaronic effects, localizes the charge-carriers over different chain segments due to higher magnitude of the inter-monomer electronic coupling as compared to the monomer's reorganization energy. [22,27,28] The soft nature of the polymer backbone results in twisting and/or bending of the polymer chains in the bulk phase, which introduces disorder in the electronic coupling between neighboring, relatively rigid, monomer units (Figure 1a). Molecular dynamics (MD) simulations of amorphous polymers reveal that the conformation of the polymer chain does not change with time in the timescale of carrier transport (≈ 10 ns) and the localized orbitals retains their shape and localization for hundreds of nanoseconds; [28] consequently, within the timescale of charge-carrier dynamics, the bulk polymer can be assumed to be in a glassy phase. Such large, yet slow conformational deformations, which cause overall change in the polymer shape usually lead to the static disorder in electronic couplings. [22] Furthermore, in this glassy phase, the dihedral angle between neighboring monomers cannot evolve freely but rapidly fluctuates around an equilibrium position imposed by the interactions with the neighboring chains. These thermal oscillations of the dihedral angles introduce additional dynamic disorder in the electronic couplings which in turn couples the abovementioned localized states. [22][23][24] Therefore, at the zeroth-level of Figure 1. a) Schematics of the model reduction procedure developed within this work. IDTBT is shown as a prototypical polymer. The conformation of the polymer chain (top) is represented by a sequence of dihedral angles k between the rigid conjugated fragments IDT and BT, shown in the middle panel. One electronic state |k〉 per rigid fragment (HOMO) is considered and the electronic coupling with the nearest-neighbor k is a function of k (bottom panel). Typical fitting of the torsional potential profile b) and electronic coupling c) of the corresponding monomer pair with analytical functions (detailed in text below). Blue filled circles represent the first-principles calculated datapoints while the yellow solid lines represent the fitting curve. Subfigures (b) and (c) provide visual aid to the model reduction procedure as well as information on the dihedral angles sampled for the fitting procedure. In addition, they attribute an immediate impression of the quality of the typical fitting, contrary to the simple mention of the standard error or quality of the fitting. transport description, each polymer chain can be defined as a sequence of monomer units along with the sequence of quasifrozen dihedral angles between adjacent monomer units which dictates the band structure of the localized states. [29] Although a large number of theoretical studies have been conducted on the charge-carrier dynamics in polymeric semiconductors, they are somewhat divided either between accurate descriptions of benchmark systems [19,27,28,[30][31][32][33][34][35][36][37] (e.g., P3HT, PBTTT, IDTBT) or model studies with empirical parameter values entering into the kinetic master equations. [12,21,[38][39][40][41][42] While the first group of studies are accurate enough for a selected set of polymers, they cannot be generalized for a larger set for rapid in silico screening. On the other hand, the second category of studies fails to explicitly correlate key transport characteristics like charge mobility with chemical structures of the polymers. In recent years, a number of strategies have been employed to improve the de-scription of the polymer density of states (DOS) from empirical Gaussian or exponential functions by quantum-chemically constructing them from a model Hamiltonian, [22,23,43] yet the parameters employed, although similar to the typical values observed in polymeric semiconductors, are empirical and do not relate to specific polymer systems. In addition, high throughput screenings of large database of known molecular semiconductors have revealed that the best known materials are not always ideal with respect to key properties, and a modular approach to construct these materials can enhance the charge mobility by tenfold. [44] However, modular construction of polymer chains followed by first-principles quantum chemical studies are limited to the calculation of energy levels [45] or properties that can serve as proxies of the charge mobility like orbital localization of DOS slope and the valence band edge; [29,46] similar fragment-based method has also been applied within GAMESS-US package [47] to www.advancedsciencenews.com www.afm-journal.de probe the electronic structures of organic polymers and chargetransport materials. [48][49][50][51] Therefore, what is missing in the literature is a generalized strategy to establish a rapid connection between chemical structures and transport properties of polymers which can be extracted from simple electronic structure calculations.
It is the primary objective of the present article to bridge this gap by developing a robust scheme for model reduction of polymer chains. Within this model reduction scheme, we make the necessary assumption that the conformation of the chain is determined by the torsional potentials of the neighboring monomers, rather than the inter-chain interactions, which is clearly valid for amorphous polymers, and we generate a model Hamiltonian which can be studied in principle with any choice of quantum dynamics methods. The other objective of this article is the study of transport characteristics for a sample of polymers in the limit of hopping transport, a computationally efficient approach that can rationalize quantitatively the difference between similar materials, [52] and is therefore suitable for virtual screening purposes. Though the model Hamiltonian developed is structurally alike to that proposed in several earlier works by one of the co-authors, [23,43] the main novelty of the current model reduction scheme is that the prediction of intrachain mobilities for diverse polymer systems is quantitative: we can predict how far can the mobility be pushed by optimizing the parameters, within a chemically plausible range, which is also determined by this study considering a relatively large number of polymers. We also make a quantitative connection between charge mobility and localization length for these polymers; the localization length has been known in the community as a good measure of charge transport efficiency, [27,29] but a quantitative relation with charge mobility has not been worked out till now.

Model Reduction Scheme for Polymer Chains
We begin the model reduction of semiconducting polymers by observing that conjugated polymers are made of relatively rigid monomers which retain their local structures regardless of the overall chain conformation. [22] A recent exploration of the electronic structures of 36 donor-acceptor copolymers has confirmed that the valence band orbital of semiconducting polymer can be described, with very few exceptions, as a linear combination of the highest occupied molecular orbitals (HOMOs) localized on individual monomer. [46] Consequently, the first step of our model reduction scheme is the construction of an electronic structure model of the polymer chain considering only one orbital per monomer, the HOMO of each monomer as we are focusing exclusively on hole transport (see bottom panel of Figure 1a).
The polymer chain can be represented by a list of neighboring monomers along with the list of dihedral angles between them as mentioned in section I (see top and middle panel of Figure 1a). We assume that the distributions of these dihedral angles in the polymer chain are uncorrelated and have the same distributions of dimers in gas phase; such an assumption can be considered as a reasonable starting point, but it can be systematically improved by performing MD simulations of the polymers (classical or ab initio depending on the polymer size) to extract the effect of other intra-as well as inter-molecular in-teractions, important in the bulk phase, which have been neglected herein, and to determine the actual dihedral angle distributions in presence of other chains. We also note that within homopolymers like P3HT, the torsional potential profiles of oligomers usually differ from that in dimer due to extended conjugation and steric hindrance. [53,54] However, as we are primarily interested in donor-acceptor type hole-conducting copolymers, in which the effective electronic coupling between non-adjacent monomers with highest HOMO energy is mediated by superexchange interaction, [55] the effective coupling strength in these polymers is smaller compared to that in homopolymers. Consequently, the extent of modulation of charge-transfer integral values due to extended conjugation is minimal within these polymers, and we anticipate that it will have negligible impact on the calculated mobility values. In addition, a generalized analytical function that can well-represent the electronic couplings ( k ) between localized orbitals on neighboring monomers (k and k + 1) as a function of the corresponding dihedral angle ( k ) is required since distortions along the polymer backbone and subsequent disorder in the electronic couplings leads to localization of the charge-carriers. [22,30] Analytical expressions of torsional potential and electronic coupling as a function of the dihedral angle are beneficial to represent the model with few parameters and could be used for implementing more advanced quantum dynamics methodologies like fewest switches surface hopping. [30,31,56] We begin by considering the set of 36 polymers screened in Ref. [46] as our primary set of polymers (the polymers are shown in Figure S1, Supporting Information) to determine the analytical descriptors of the dimer torsional potentials and electronic couplings. We have avoided abbreviated names of the polymers, except for the most well-documented ones, e.g., "P3HT", "IDTBT", and "CDTBT" since there is no congruity in naming the polymers within the existing literature. While the IUPAC nomenclature of these polymers is too long and complicated, some of the experimental reports designate them with coded abbreviations like polymer/molecule/compound. Therefore, to avoid confusion, we have designated the polymers with a numbered index while the structural repeating units are explicitly shown in the Supporting Information. These polymers correspond to hole-transport and ambipolar types and consist of 33 unique monomer units, with 7 having chemically inequivalent connection sites. [46] We build a database of 153 monomer pairs, 83 corresponding to the monomer pairs within the above mentioned 36 polymers along with 70 related monomer pairs. The torsional potential profile of each of the monomer pairs is calculated within the density functional theory (DFT) at the B3LYP/6-311G+** level while the torsion-dependent HOMO-HOMO coupling profile for the same is worked out within the localized molecular orbital method (LMOM), as detailed in Refs. [29] and [46].
The DFT-calculated torsional potentials of the set of monomer pairs are fitted within the cosine Fourier series with 5 coefficients, [57] considering the double minima features around 0°/180°in most of the torsional profiles while adhering to the statistical rule of thumb (one parameter for ten data points) for regression; [58] since dimer torsional potentials are symmetric about planar configurations ( = 0°/180°), qualitatively similar results can be obtained with shorter Fourier expansions except www.advancedsciencenews.com www.afm-journal.de for cosine Fourier series with only 2 coefficients.
A m is the m-th expansion coefficient in the cosine Fourier expansion while is represented in radian units. We find that the median root mean square error (RMSE) of the fitting for the set of monomer pairs is 2 meV (Figure 1b shows a typical fitting of this quality and Figure S2 (Supporting Information) shows the distribution of RMSE). There are 12 monomer pairs out of 153 with a relatively large RMSE in the range of 6-10 meV, which is probably still satisfactory and could be improved systematically by extending the cosine series (the fittings are shown in Figure S3, Supporting Information).
The most commonly used [29,30] analytical expression to describe the functional dependence between k and k is a sinusoidal function that we utilize in the present model reduction scheme.
where (0) k is the maximum electronic coupling parameter between monomers in planar configuration. Employing Equation 2, we fit the torsional angle dependent electronic couplings for the 153 monomer pairs and find that the median RMSE of fitting of electronic couplings of the set of monomer pairs is 0.03 eV; although, the magnitude of median RMSE is higher than that for the torsional potential fitting, median (0) k is ≈ 2 orders of magnitude larger compared to median RMSE. In Figure 1c, we show the fitting for a sample monomer pair while the distribution of RMSE of fitting is given in the Supporting Information (see Figure S4, Supporting Information). The proposed fitting model can predict the electronic coupling of the dimers both at planar as well as non-planar conformation with similar accuracy, though highly non-planar conformations of the dimers are thermodynamically improbable. In addition, we do not find any correlation between accuracy of fitting and magnitude of electronic coupling for the corresponding dimers.
The simple relation between k and k , shown in Equation 2, does not hold when the monomer fragment has negligible coupling with the neighbor moiety due to symmetry reason (e.g., pyridine moiety, with connecting sites at (2,5) positions, [46] has ntype HOMO (natural bond orbital analysis [59] within GAUSSIAN-16 package [60] reveals that the lone pair of electrons on the nitrogen atom has the maximum contribution, ≈77%); see Figure  S5, Supporting Information), or because of the presence of nodal plane at the connecting carbon atoms (e.g., HOMO of 1,3-difluorobenzene with connecting sites at (2,5) positions; [46] shown in Figure S6, Supporting Information). In these cases, the assumption that the valence band is primarily dominated by the HOMOs of the monomer fragments does not hold. While generalization of the model with multiple frontier molecular orbitals is possible, in this work we prefer to keep the reduced model as simple as possible to allow comparison. In addition, we find 8 monomer pairs where the HOMO-HOMO couplings are less sensitive to torsional deformations (shown in Figure S7, Supporting Information) and such effect has been identified earlier for monomer pairs with five-membered rings and pronounced single bond-double bond character within the ring. [61] For these monomer pairs, fitting of the HOMO-HOMO coupling profile within a sinusoidal function proves unsatisfactory, yet the fitting can be improved by considering a larger cosine Fourier expansion (shown in Figure S7, Supporting Information).
In summary, we exclude for simplicity from further consideration the two polymers (#7 and #32 in Figure S1, Supporting Information) where the monomers' HOMOs are insufficient to describe the valence orbitals. We also excluded polymers #27 and #31, because of insufficient quality of the torsional potential fitting, and polymers #22, #24, #25, and #26 because of insufficient quality of the HOMO-HOMO coupling fitting. In Tables T1 and  T2 (Supporting Information), we have given the fitting parameters corresponding to the monomers and monomer pairs of this full set of 28 polymers respectively, while in Table T3 (Supporting Information), we have tabulated the corresponding properties of the polymers. Since this set of 28 polymers roughly represents the full chemical sphere of amorphous polymers, we deduce that the present reduction scheme should be applicable at the early stages of high throughput screening studies which can be followed by more advanced quantum chemical calculations to exclude false positive results, if any. In relation to polymers erroneously predicted to be of high-or low-mobility (false positive and negative) the error does not derive from the model reduction scheme since all approximations are checkable at the time of performing the model reduction. The limitations are in the model itself and are mainly due to the exclusion of inter-chain transport and approximated generation of the chain conformations.
The reduced models we have employed in the present work are controlled approximations and, indeed, there are no fundamental limitations preventing the use of more flexible fitting schemes (spline functions, higher-order Fourier expansion, etc.). However, ad hoc solutions that will address the problem for the few polymers that cannot be fitted within the present model reduction scheme may not be general. The most convenient approach will be to explore more general fitting schemes when a large dataset is considered, e.g., at the early stages of high-throughput screening study but this can be considered just a minor adjustment of the methodology.
Construction of the zeroth-order model Hamiltonian: The zerothorder Hamiltonian of specific polymer chain is constructed in a modular way by determining the monomer sequence and using the list of earlier calculated fitting parameters (torsional potential as well as electronic coupling) for neighboring monomer pairs. For a particular chain conformation, the Hamiltonian is given by, The on-site (HOMO) energy values of the monomer fragments ( k ) are calculated within the LMOM method. [29,46] Since we assume that the polymer chains remain in a glassy state, as previously mentioned in the Introduction, the overall chain conformation, i.e., the sequence of quasi-frozen dihedral angles between neighboring monomers ({ k,frozen }) are determined employing the torsional potential parameters of the corresponding monomer pairs; for each monomer pair (k, k + 1), k,frozen is chosen from the Boltzmann distribution at 300 K. In effect, our method of generating dimer conformations automatically accounts for the preference between syn-and anti-conformations and no further geometry relaxation is introduced after the chain conformation is generated.

Charge-Carrier Dynamics within Hopping Model
The charge-carrier dynamics model presented below is an adaptation of that introduced in Ref. [22] and adopted also in Refs. [23] and [43]. Within this model, charge-carrier's motion is usually considered as a series of hopping events between localized eigenstates ({|i〉}) which are eigenstates of the zeroth-order Hamiltonian given in Equation 3. It is also assumed that the process follows a first-order reaction kinetics, and the coupling of the electronic degrees of freedom with the nuclear vibrations equilibrates the charge-carrier between successive hopping events. From that point of view, we consider a Holstein-type local electron-phonon coupling term (Equation 4) as the first-order perturbation to the zeroth-order Hamiltonian given in Equation 3.
The k-th monomer's HOMO energy gets linearly modulated with the intra-monomer, dimensionless coordinate Q k,m along vibrational mode m by the local electron-phonon coupling strength g k,m . On simplification, the extra stabilization of the charge (reorganization energy) localized in state |i⟩ = ∑ k C k,i |k⟩, due to this perturbative term, is given by (similar to that in Ref. [22]), where k is the total reorganization energy if the charge has been fully localized on monomer k only (detailed derivation of Equation 5 is given in the Supporting Information). In essence, reorganization energies of the localized states create an additional energy barrier for charge-carrier's motion on top of the dominant energetic disorder. In bulk polymers, at experimentally relevant temperature, dynamic disorder of the electronic couplings remains small, and consequently they can be considered as a smaller perturbation to the zeroth-order Hamiltonian. Given the small values of the torsional angle fluctuations Δ k (Δ k = k − k,frozen ) in the constrained environment of the bulk polymers, only the leading term of the non-local electron-phonon coupling is included in the present study. [62] Thus, the effective non-local electron-phonon coupling term in the total Hamiltonian can be represented as, where M k is the non-local electron-phonon coupling strength due to thermal fluctuation of k . M k is derived by taking the derivative of k with respect to k , Dynamic disorder in electronic couplings (non-adiabatic coupling) induces hopping between localized states; [23,24] the hopping rate between localized states |i⟩ = ∑ k C k,i |k⟩ and |j⟩ = ∑ k C k,j |k⟩, within the Fermi golden rule formalism, can be represented by (full derivation is given in the Supporting Information), [63] In Equation 8, 〈|M ij | 2 〉 is the average of squared electronic coupling between the electronic states |i〉 and |j〉, [64] FCWT is the Franck-Condon factor and temperature-weighted density of states, [65] and ℏ is the reduced Planck's constant. 〈|M ij | 2 〉 is given by, where ⟨Δ 2 k ⟩ is the mean squared torsional fluctuation of the single bond between monomer k and k + 1; on the other hand, FCWT (ΔE ij ) is the component in Equation 8, which describes the functional dependence of k ij on the reorganization energy of hopping ( ij = i + j ) and the energy gap between the localized states In Equation 10, we assume that ij is the sum of a classical component ( due to low-frequency nuclear modes and a quantum component ( (Q) ij = f Q ij ) due to one effective high-frequency mode ℏ ≈ 1450 cm −1 . f Q = 0.4 is the reorganization energy partition factor which is close to the value emerges from electronic structure calculations and has been utilized in a previous study. [23] In addition, w and w′ are the initial and final vibrational states along the high-frequency mode, P(w) is the Boltzmann population of the initial vibrational state, and FC ij,ww′ is the Franck-Condon overlap integral between the vibronic states |i, w〉 and |j, w′〉. [63] The hopping rate expression (Equation 8) looks structurally similar to the Marcus-Levich-Jortner (MLJ)-type rate expression. [66] However, it is worth to note that -1) unlike MLJ theory, the coupling element (Equation 9) is non-adiabatic coupling between localized states, not electronic coupling which is difficult to define for intra-chain charge transport. The nonadiabatic coupling between the localized states stems through "inducing" vibrational modes [63] which are anharmonic in nature (low frequency modes). Since we consider only the linear term of the non-adiabatic coupling with respect to displacement, this approximation leads to a similar-looking coupling term, that within MLJ theory, in the hopping rate expression. 2) The reorganization energy ij in FCWT (ΔE ij ) (Equation 10) originates from "accepting" vibrational modes, [63] which are primarily C = C stretching modes and are harmonic in nature. Thus, this factor is analogous to that within MLJ theory, where all vibrational modes (classical and quantum) are considered harmonic.
Employing these calculated hopping rates, the intra-chain charge-carrier mobility ( ) is calculated within low charge-carrier and weak electric field approximation within a standard master equation approach. [22,23,43] We assume that the electric field modifies the electrostatic potential linearly along the chain (rod-like chain model [21] ).
In real polymer systems both intra-chain and inter-chain components contribute to charge transport, though extent of the contribution depends on the conformation of the chains. [21] With the assumptions that the characteristic time-scale of inter-chain hopping (t inter ) is much larger than that of intra-chain hopping (t intra ) (t inter ≫ t intra ) and the displacement due to significantly infrequent inter-chain jumps contributes very little to the final charge-carrier displacement from the initial position, the charge diffusion coefficient in 3D can be roughly defined as D 3D = 〈R 2 (t inter )〉/6t inter , where 〈R 2 (t inter )〉 is the mean squared displacement of the charge in the characteristic inter-chain charge-carrier hopping time. For sufficiently rigid polymers where the charge primarily moves along the chain backbone, 〈R 2 (t)〉 is related to the diffusion coefficient in 1D (D 1D ) via 〈R 2 (t inter )〉 = 2D 1D t inter , since the charge motion is similar to the random walk problem in 1D. [67] This leads to the final conclusion that D 3D = D 1D /3 and in turn, through Einstein relation, the average charge-carrier mobility in bulk is 1/3 of the average intra-chain mobility.
Equation 11 is not strictly valid for all realistic polymers which are semiflexible rather than rigid or in case the inter-chain hopping frequency is comparable to the intra-chain counterpart (e.g., for short polymer chains) but can be used as an initial guide to corre-late intra-chain mobility with the experimentally measured mobility. More complicated analytical relationships between charge diffusivity and polymer geometric structure could be introduced and they reveal that the motion becomes more sub-diffusive as the chain flexibility increases and the inter-chain hopping become more important for very flexible (or very short) chains. [21]

Discussion
Simulation of the bulk polymers requires considering a large number of polymer chains; for the screening of the selected polymers, we calculate the mobility values of each polymer considering 25 realistic, random conformations of the corresponding polymer chain unless otherwise stated. We start the discussion considering the change in the intra-chain mobility as a function of the polymer chain length. To probe the effect we consider polymer #3 (4,4-dimethyl-4H-cyclopenta[2,1-b:3,4-b']dithiophene-cobenzo[c] [1,2,5]thiadiazole; shown in the inset of Figure 2a and Figure S1, Supporting Information) as the model polymer; the choice is made based on the calculated mobility values ( ≈ 0.1 cm 2 V −1 s −1 ) which remain close to the earlier predicted maximum mobility within hopping description (Ref. [52]) and it is expected that it will carry the signatures of both hopping-assisted mobility and high charge mobility. We find that the parameters of the electronic structure (energy gap and mean localization length of the contributory localized states) converge rapidly as the chain length is increased (as shown for the localization length in Figure 2a and for the energy gap in Figure S8, Supporting Information). However, the distribution of mobilities obtained considering 100 random configurations of the corresponding polymer chain evolves more slowly toward lower mobilities as the chain length is increased (Figure 2b). This is expected for dispersive transport in 1D systems where the probability of having transport-limiting deeper traps increases with longer chains; [12,68] herein, mobility values decrease and converge to a stable distribution around 4000 monomer units (Figure 2b). Yet, for a comparison with experimental mobility values, it is probably convenient to consider realistic polymer lengths -material characterization of recently synthesized semiconducting polymers reveals that the average length of the polymer chains remains within 100 − 500 monomer units. [69][70][71][72] Consequently, for further analysis of transport properties, we consider polymer chains with 200 monomers and compute the average mobility values (〈 〉).
The distributions of the intra-chain mobilities show that the numerical value varies over several orders of magnitude (Figure 2b), and in turn, the arithmetic mean can get dominated by mobility values at the higher end of the distribution. Consequently, the arithmetic mean of the calculated mobilities can portray the charge-transport efficiency in bulk polymer incorrectly. On the other hand, the geometric mean of a distribution suffers less from the skewness of the distribution, and thus may better represent the charge-transport efficiency within a particular polymer in bulk. Hence, in the following, we report the geometric mean values [23,43] of the intra-chain hole mobility for the screened polymers. We also find that the mobility distributions for most of the polymers (25 out of 28) pass the Shapiro-Wilk test [73] of lognormality and therefore, the geometric means are good measures of these mobility distributions.
We find that the calculated mobilities are comparable to the experimentally measured values (μ exp ≈ 1 − 10 cm 2 V −1 s −1 ) [2] for polymers with high 〈 〉; these high-mobility polymers primarily consist of fused aromatic rings and the average length of the structural repeating unit per monomer (l SRU ) is mostly ≥ 8 Å (see Figure 3; indacenodithiophene (IDT) and cyclopentadithiophene (CDT) based copolymers also have disordered microstructures in the bulk). However, the experimentally observed mobilities are much larger than the calculated ones for polymers with low 〈 〉. It is likely that for polymers that reside at the lower end of the computed mobility values, inter-chain hopping has to be considered within the transport model which has been ignored in the present model. In connection, we also ob-serve that majority of the polymers in this latter category usually form large aggregates in the bulk with significant intermolecular − stacking between polymer chains, [74][75][76][77] and therefore, inter-chain hopping may be as frequent as intra-chain hopping in these systems. It is, however, possible to analyze the results to derive possible insights on how to increase the intra-chain hole mobility. A first observation is that the localization length at the band edge (E Edge ) correlates very strongly (rank correlation coefficient ≈ 0.9) with the computed mobility as shown in Figure 3. The energydependent localization length at the band edge for each polymer is computed employing method similar to that given in Ref. [29]. Therefore, this observation implies that the localization length (or the extent of delocalization) at band edge can be chosen as a suitable charge-transport proxy parameter at the zeroth-level screening. Yet, the not so simple linear relationship between the charge mobility and localization length further flags the relevance of an efficient model reduction scheme for polymer materials.
A recent high-throughput screening of organic copolymers (Ref. [46]) has investigated the polymer structural features which enhances charge delocalization along the polymer backbone and determined the optimal choices for achieving large localization length. Since the localization length at the band edge has a strong positive correlation with charge-carrier mobility (Figure 3), we also expect the same structural features, e.g., 1) minimal number of monomers in the polymer structural repeating unit, [46] 2) monomer HOMOs with large energy mismatch, [43,78] 3) stiffer polymer backbone (smaller torsional entropy [46] ), and eventually 4) low static disorder in the electronic coupling, to be essential to achieve high intra-chain charge-carrier mobility. This has been confirmed within our study where the high mobility polymers (〈 〉 ≳ 1.0 cm 2 V −1 s −1 ) satisfy all of the above criteria (see Table  T3, Supporting Information).
Thanks to the connection proposed here between model Hamiltonian and mobility we can look specifically at the effect on charge mobility of the transfer integral magnitude since it has been proposed that small transfer integral could be (counterintuitively) beneficial for transport. [43,46] Indeed, over the last thirty years, the state-of-the-art organic polymers have seen a steady decrease in electronic bandwidth (MEH-PPV > P3HT > PBTTT > IDTBT) and it was suggested this was due to the reduced activation energy for transport in narrower bands. [43] There must be a limit to this trend as the mobility should vanish for vanishing transfer integrals. To clarify this matter, we consider the parameters corresponding to polymer #36 (highest mobility polymer screened; Ref. [79]) and compute the mobility as function of (0) (Figure 4a). In agreement with Ref. [43] we find that increasing (0) is detrimental to charge transport due to the trapping of the charge-carrier at deep trap states away from the band edge, and in effect from the reduced number of longer range hopping; yet, we also find that the at very small (0) the effective coupling strength between localized states (|M ij | 2 ) diminishes significantly, and this mutual interplay dictates the optimal electronic coupling strength for high-mobility polymers. The range of coupling which is close to ideal is 0.15-1.25 eV which is very similar to that observed for the high mobility polymers in our screening. Therefore, the most suitable bandwidth for high charge-carrier mobility has more-or-less been achieved and this parameter does not play too critical a role during further optimization of modular polymers.
The natural next step is to check whether we can optimize the parameters further to build modular polymer with larger mobility values. For this purpose, we begin with the parameter set corresponding to the highest mobility polymer found during the screening (polymer #36). Improvement of the reorganization energies of the constituting monomers from k = 0.11 eV to 0.08 eV (smallest of the parameter values obtained for the screened polymers) improves the mobility only by 0.5%. However, significant improvement in mobility is achieved by reducing the static disorder in electronic coupling which can be quantitatively defined by /〈 〉 where 〈 〉 and are the mean and standard deviation in HOMO-HOMO coupling respectively. Reducing /〈 〉 up to the realistic limit (0.03 -smallest value obtained for the screened polymers) we find that the charge-mobility can be enhanced almost fivefold compared to that obtained for polymer #36 during our screening ( Figure 4b); the largest computed 〈 〉, within the realistic range of key parameters, is ≈ 30 cm 2 V −1 s −1 for (0) = 1.66 eV, /〈 〉 = 0.03, Δ = 2.3 eV, and k = 0.08 eV with only 2 monomers in the structural repeating unit of the polymer (〈 〉 scales trivially with the length of the structural repeating unit [23] ). The shift of the mobility maxima with reduced /〈 〉 toward slightly larger (0) (Figure 4b) indicates that with reduced disorder in the electronic coupling we approach more toward the band transport of the charge-carrier.
Finally, we also probe T-dependence of 〈 〉 for the same set of 28 polymers within the approximation that the dynamic disorder in electronic coupling is T-dependent while the static disorder in electronic coupling remains unaffected with the variation in temperature. This is not a drastic approximation since the glass transition temperatures of known polymers usually reside between − 50°C and 200°C. [80] Each of these 28 polymers screened here exhibit T-activated, Arrhenius-type behavior and we find that the room temperature mobility values of the polymers exhibit a strong negative correlation with the activation energies (E a ) (Pearson's correlation coefficient ≈ −0.97, see Figure 5). Our findings are in line with the previous experimental [81,82] and theoretical [23]  reports of T-dependence of charge-carrier mobility in polymers which concluded that the effect of various physical parameters within a polymer chain can be effectively combined in a single activation energy parameter. The results also indicate that the reduced model is working flawlessly within the hopping description of charge transport and validate the overall model reduction scheme for polymers. The only outlier (see Figure 5) corresponds to polymer #1, which has a significantly small electronic coupling strength ( (0) ≈ 0.1 eV) that gives rise to a small activation barrier but much reduced hopping probability, and thus a much lower mobility value than expected (also see Figure 4a).

Conclusion
In this work, we have developed an efficient model reduction scheme for polymer semiconductors which can be utilized to compute intra-chain charge-carrier mobility in a computationally inexpensive way. The model captures the main chemical details, and it can be used to compare rapidly different polymers or to perform virtual screenings. We have illustrated the procedure by examining the chemical structure-charge mobility relationship in 28 realistic polymers. The data set helps in establishing plausible ranges for all the microscopic parameters of the model and it can therefore be used to determine whether mobility can be further optimized. We observe for example that the current optimal polymers can be further improved to achieve a fivefold increase in mobility by reducing the torsional static disorder to the smaller value observed in the data set. Possible strategies to achieve this are by incorporating more planar and fused aromatic moieties and promoting non-covalent locks between successive monomer units.
One of the main advantages of the reduced model is the possibility of addressing the role of individual parameters that cannot be changed in isolation in the experiment. We use the reduced model to provide some insight on the observation that the highest mobility polymer does not have very broad valence bands, i.e., the electronic coupling along the chain is not critical for high mobility polymers. We find that there is indeed a range of intermonomer transfer integrals where the transfer integral has little effect on the mobility with everything else being the same. This result is indeed surprising and suggests that this quantity does not require major optimization through chemical or structural engineering as many authors in the community usually assume. An additional advantage of constructing reduced models is the possibility of performing more accurate quantum dynamics calculations if desired. We expect for example that for very high mobilities one may want to consider models that retain some element of coherent transport [83,84] and for very low mobility one should include the possibility for trapped charges to hop to a different chain.
Finally, a common theme of recent literature is that the localization length at the valence band edge (alternatively the slope of the tail of the DOS) correlates well with the charge-carrier mobility. Here, we have provided a quantitative relation between localization length and hole mobility (and in turn, between DOS tail slope, which shows strong positive correlation with localization length, [46] and hole mobility), thus providing a firmer ground for using localization length (computed by theoreticians) or DOS tail slope (measured by experimentalists) as proxy charge-transport parameter to explain specific observations.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.