Poly(3-alkylthiophenes) (P3[Alkyl]T) exhibit high mobility and efficiency of formation of polaronic charge carriers generated by light absorption, thus finding applications in field effect devices. Excited states of *π*-stacked dimers of tetra-thiophene oligomers (T4), infinite isolated polythiophene (PT) chains, and P3[Alkyl]T crystals are modeled using configuration interaction singles (CIS) calculations. Excited states in cofacial T4 dimers are mostly localized Frenkel states except for two low energy charge transfer (CT) exciton states, which become the ionization potential and electron affinity levels of T4 molecules at large dimer separation. The lowest excited states in infinite, isolated PT chains and P3[Alkyl]T crystals are intra-chain excitons where the electron and hole are localized on the same chain. The next lowest excited states are interchain, CT excitons in which the electron and hole reside on neighboring chains. The former capture almost all optical oscillator strength and the latter may be a route to efficient formation of polaronic charge carriers in P3[Alkyl]T systems. Changes in optical absorption energies of T4 dimers as a function of molecular separation are explained using CIS calculations with four frontier orbitals in the active space. Shifts in optical absorption energy observed on going from isolated chains to P3[Alkyl]T lamellar structures are already present in single-particle transition energies induced by direct *π*–*π* interactions at short range. The electroabsorption spectrum of T4 dimers is calculated as a function of dimer separation and states that are responsible for parallel and perpendicular components of the spectrum are identified.

## I. INTRODUCTION

Charge separation in conjugated polymers is a key step in the formation of free carriers in organic photovoltaic materials.^{1} A sequence of light absorption by a chromophore, followed by energy transfer to a charge transfer (CT) state, and subsequent dissociation of the electron and hole as charged polarons seems a likely scenario for carrier formation in many systems. Frequently, this requires a donor–acceptor heterojunction to provide the driving force for formation of the charge transfer state. The case of regioregular poly(3-alkylthiophenes) (P3[Alkyl]T) is one where charged polarons form with high efficiency (up to 30%) following photoexcitation without the need for a heterojunction, although it has been suggested that poorly stacked regions in otherwise regular P3[Alkyl]T are responsible for charge separation. Indeed, the fate of optical excitations in P3[Alkyl]T is strongly affected by its microcrystalline structure.^{2} Charged polaron formation following light absorption in P3[Alkyl]T^{3–8} occurs when the polymer exists in a lamellar structure, rather than single chains in solution.^{6} Efficiencies of 11%,^{8} 20%,^{4} and 30%^{6} have been reported. The spectrum of polaron states in polythiophene (PT) has been investigated by experiment^{5,9} and theory.^{10,11}

The primary motivation of this work is to understand CT states in photosensitizers in organic photovoltaic materials such as P3[Alkyl]T. First of all, we consider a minimal model for CT states in a symmetric dimer using a 4 × 4 Hamiltonian. This is extended to configuration interaction singles (CIS) calculations of optical absorption spectra of a cofacial tetra-thiophene (T4) dimer as a function of dimer separation and P3[Alkyl]T systems in the crystalline state, using the same basis set and Hamiltonian. CIS calculations on the P3[Alkyl]T systems show that a low energy CT state exists in which the electron and hole are located on neighboring chains. Finally, the electroabsorption (EA) spectrum of the T4 dimer is calculated as a function of dimer separation. Most EA calculations in the literature^{12,13} use a model Hamiltonian approach. Here, we apply a first-principles CIS approach to the T4 dimer. The sum over states (SOS) formalism used (*vide infra*) is complex and it is prudent to apply the method to the molecular case and extrapolate results to the crystalline case, in the first application. A feature previously observed in EA spectra of P3[Alkyl]T by Österbacka *et al.*^{3} and attributed to a CT state is reproduced in our calculations.

Crystalline morphologies of P3[Alkyl]T polymers have been reviewed in Refs. 14 and 15. Here, we refer to a number of P3[Alkyl]T polymers according to the alkyl side chain group, X, as X = H (hydrogen), Et (ethyl), Bu (butyl), Hex (hexyl), Oct (octyl), and Dec (decyl). X-ray structures of P3[Hex]T have been reported by a number of groups. A key feature of thiophenes with long side chains is that they form lamellar structures with short interchain distances (less than 4 Å), which gives them their properties of strong tendency to form polaron charge carriers, high hole mobilities, etc. The structure from a combined nuclear magnetic resonance (NMR) and x-ray diffraction study by Dudenko and co-workers^{16} was used for crystalline P3[X]T calculations in this work. The short distance between *π*-stacked chains in P3[Alkyl]T led Brown and co-workers^{5} to assert that charge carriers in these systems “exhibit quasi-two-dimensional characteristics.” The crystal structure of polythiophene (PT), which lacks side chains, has been found to be a herringbone structure^{17} with two PT chains per unit cell.

Many-body *GW* and Bethe–Salpeter equation (BSE) calculations on periodic polythiophenes have been reported for isolated infinite chains (ICs)^{18–21} and the herringbone structure of PT.^{22} Here, we report configuration interaction singles [CIS, exactly equivalent to time-dependent Hartree–Fock theory in the Tamm–Dancoff approximation (TDA)] calculations on isolated chain PT and on the *π*-stacked P3[X]T structure with X = H, Et, and Bu. The authors are not aware of previous *GW*/BSE calculations, which include the electron–hole attraction, on the *π*-stacked structure in the crystal phase. It is interesting, therefore, that we find charge transfer (CT) excitons in which the electron and hole are localized on neighboring chains. Quantum classical/INDO studies of CT states in PT aggregates have been used to argue that CT states can be stabilized by a dielectric background.^{23} Wang and co-workers^{11} found exciton states with various degrees of charge transfer in poly-phenylene vinylene (PPV) cofacial oligomers.

Multi-configuration calculations (including the CIS method used here) naturally incorporate states with CT character in the spectrum of excited states, even when the system under study is a symmetric, cofacial dimer. The simplest possible Hamiltonian that can describe these states contains even and odd combinations of the HOMO and LUMO of each molecule in the dimer to yield a 4 × 4 Hamiltonian. Two of the states that result from this Hamiltonian are Frenkel (FR) excitons, which are mainly localized on one molecule or other in the dimer. The other two are CT excitons in which the electron and hole mainly reside on distinct molecules. In each case, however, symmetry dictates that all excited states belong to an irreducible representation of the system point group. Hence, both FR and CT excited states consist of linear combinations of the local configurations just described. In the limit of large separation, these states evolve into degenerate, FR or CT excitons. The latter are also known as IP-EA states since they consist of electron transfer from the HOMO of one molecule to the LUMO of the other and their energies are differences in the electron affinity and ionization potential of the system, Δ = IP − EA. CT states differ from other states in that they are stabilized by intermolecular Coulombic interactions and their energies have an approximate Δ − 14.4/*r* energy dependence, where the latter term is the interaction energy of two point charges of magnitude, *e*, in eV.

Optical properties and device physics of polythiophenes have been reviewed in Refs. 24 and 25. Fundamentals of linear and nonlinear optical absorption and excited state energies have been reported.^{7,12,26} Optical absorption and emission spectra of P3[Alkyl]T aggregates have been modeled using a generalized Holstein Hamiltonian, which includes electron–phonon interactions.^{27} H and J aggregates display distinct behavior, depending on the sign and magnitude of the dipolar coupling between polymer chains.^{28} Yamagata and co-workers^{29} performed calculations on octathiophene (T8) dimers and showed that the sign of the dipolar coupling and, therefore, the type of aggregate predicted depended on lateral shifts of the T8 dimers—cofacial dimers favored an H aggregate while a shift along the PT chain direction of 2.5 Å resulted in the strongest J coupling. The variation in coupling was attributed to wave function overlap rather than Coulombic coupling through space.

The origin of the difference in optical absorption threshold in P3[Alkyl]T in solution vs thin films has been investigated in detail by experiment^{6,9} and simulation using a Holstein–Hubbard model applied to molecular aggregates with random disorder.^{27,30} Cook and co-workers^{6} studied the evolution of optically excited states of P3[Hex]T in solution and in thin films by transient absorption spectroscopy. The onset of optical absorption in chlorobenzene solution (around 2.3 eV) is higher than in thin films (around 2.0 eV). This may be attributed, for example, to a noncoplanar chain structure, or it may be due to *π*–*π* interactions in the condensed phase. In solution, P3[Hex]T has a long-lived *S*_{1} state that has a fluorescence quantum yield of 0.33 and otherwise decays via a triplet state. In thin films, however, a fast (∼600 ps) fluorescent decay with low quantum yield (0.02) and a slow, bimolecular process (∼300 ns) attributed to polaron recombination were found.^{6}

Electroabsorption (EA) spectroscopy is commonly used to probe CT states in organic matter.^{13} CT states involve transfer of an excited electron over larger distances than is normal in an optical transition and the charge transition density is small since the occupied and virtual states in the transition are well separated in space. On the other hand, CT states may have a large permanent dipole moment, which can interact with an external electric field, resulting in a Stark shift of an otherwise weak transition that can be observed in EA spectroscopy. The EA spectra for the 1^{1}*B*_{u} absorption in P3[Alkyl]T from several groups^{5,12,26} are found to be in good agreement.

The remainder of this article is organized as follows: A 4 × 4 Hamiltonian model for CT and FR states in a symmetric dimer is outlined, molecular and periodic CIS methods used in this work and computational details are described, and the sum over states (SOS) model for electroabsorption spectroscopy are given in Sec. II. Results for the variation in singlet and triplet state energies in the T4 dimer as a function of intermolecular separation for the 4 × 4 Hamiltonian and full CIS calculations of excited state energies and linear optical absorption spectra are given. CIS calculations of the wave functions and linear absorption spectra of infinite polymers are compared to T4 oligomer calculations. This is followed by results of SOS electroabsorption calculations for the T4 dimer as a function of separation. A summary and conclusions are presented in the Sec. IV.

## II. THEORY

### A. Model for CT and FR excited states

The simplest model that can be constructed for charge delocalization and separation in excited states of a molecular dimer requires in- and out-of-phase combinations of the HOMO(*H*) and LUMO(*L*) of the isolated molecules. In the dimer, these generally order energetically with the in-phase combination slightly lower in energy, with a small energetic splitting at large separation. These orbitals are even or odd about a mirror plane between the molecules in the dimer. Occupied orbitals localized on either molecule in the dimer are denoted *A* and *B* and virtual orbitals are denoted *C* and *D* (Fig. 1). Dimer *H* − 1 and *H* − 0 orbitals are $(A+B)/2$ and $(A\u2212B)/2$ and dimer *L* + 0 and *L* + 1 orbitals are $(C+D)/2$ and $(C\u2212D)/2$. For simplicity, it is assumed that intermolecular distances are large enough that overlaps ⟨*A*|*B*⟩ and ⟨*C*|*D*⟩ are negligible and we consider a minimal Hamiltonian for triplet states of the dimer in which a CIS Hamiltonian matrix element is given by

Here, *ɛ*_{ai} is the HOMO–LUMO gap (assumed the same for *H* − 0 → *L* + 0 and *H* − 1 → *L* + 0, etc.). Chemist’s notation is used for two-electron integrals, (*ij*|*kl*), in which orbitals *i* and *j* share the same electron coordinate and *k* and *l* share the other electron coordinate. Diagonal and off-diagonal two-electron matrix elements have the following approximate forms:

Each diagonal element has the same (*ij*|*ab*) two-electron part and each nonzero off-diagonal part is the same when intermolecular overlaps ⟨*A*|*B*⟩ and ⟨*C*|*D*⟩ are neglected. Transitions in the Hamiltonian are ordered as *H* − 1 → *L* + 0, *H* − 1 → *L* + 1, *H* − 0 → *L* + 0, *H* − 0 → *L* + 1. The odd/even symmetry of the molecular wave functions with respect to the *σ*_{h} mirror plane means that only four off-diagonal terms in the Hamiltonian are nonzero. Diagonalization of the 4 × 4 Hamiltonian,

yields two pairs of degenerate states with eigenvalues $\epsilon ai\u2212(AA|CC)+(BB|DD)/2$ and $\epsilon ai\u2212(AA|DD)+(BB|CC)/2$. The corresponding wave functions for the former eigenvalue are

These states are linear combinations in which the excited electron and hole reside on the same molecules in each configuration and combine with different phases. These are low energy FR1 and FR2 Frenkel states. The corresponding wave functions for the eigenvalue $\epsilon ai\u2212(AA|DD)+(BB|CC)/2$ are

These states are in- and out-of-phase combinations in which the excited electron and hole reside on either molecule, i.e., high energy “IP-EA” or CT1 and CT2 charge transfer states. The (*AA*|*DD*) + (*BB*|*CC*) terms in this eigenvalue are Coulombic electron–hole attraction terms for electron and hole on either molecular site. Intramolecular (*AA*|*CC*) and (*BB*|*DD*) integrals in the lower eigenvalue are independent of molecular separation, to a first approximation, whereas intermolecular (*AA*|*DD*) and (*BB*|*CC*) integrals in the upper eigenvalue vary with separation approximately as 14.4/*r* eV Å^{−1}, where *r* is the intermolecular separation in Å. For separations less than 5 Å, overlap of *π* systems leads to a splitting of the FR and IP-EA states with different phases owing to splitting of the single-particle gap energies at the self-consistent field (SCF) level. This is shown in Sec. III where a 4 × 4 CIS Hamiltonian is diagonalized as a function of T4 dimer separation. The H − 1 and H − 0 orbitals of a T4 dimer belong to the *b*_{u} and *b*_{g} irreducible representations of the *C*_{2h} point group (Fig. 1). The L + 0 and L + 1 orbitals belong to the *a*_{g} and *a*_{u} representations. The wave functions in Eqs. (2)–(5) are, therefore, excited states of *B*_{g} symmetry (FR1 = $[|AC\u3009\u2212|BD\u3009]/2$ and CT2 = $[|BC\u3009\u2212|AD\u3009]/2$) or *B*_{u} symmetry (FR2 = $[|AC\u3009+|BD\u3009]/2$ and CT1 = $[|BC\u3009+|AD\u3009]/2$). The lowest state of $Ag1$ symmetry contains configurations that are products of *b*_{u} and *b*_{u} (|(*H* − 1)(*L* + 2)⟩) and *a*_{u} and *a*_{u} (|(*H* − 2)(*L* + 1)⟩) orbitals and the lowest state of $Au1$ symmetry contains configurations that are products of *b*_{g} and *b*_{u} (|(*H* − 1)(*L* + 2)⟩) and *b*_{u} and *b*_{g} (|(*H* − 1)(*L* + 3)⟩) orbitals. These are only present when the active space is expanded beyond the limited space in this simple model.

### B. Electroabsorption spectroscopy

Electroabsorption (EA) spectroscopy is a technique in which the difference in the optical absorption spectrum of a thin film induced by a low frequency applied electric field is measured. It can be modeled using the Liptay equation,^{31} which is a classical approach based on molecular polarizability derivatives with respect to external field frequency or using a quantum mechanical SOS third-order optical susceptibility.^{32,33} The connection between these approaches has been discussed by Saito and co-workers.^{34}

The third-order susceptibility^{32,34,35} contains one-photon terms,

and two-photon terms,

whose third-order resonant terms are given in Eqs. (6) and (7). There are additional one- and two-photon terms with lower-order resonant terms.^{34} In the above expressions, the frequencies satisfy *ω*_{σ} = *ω*_{1} + *ω*_{2} + *ω*_{3}, *N* is the number density of molecules, *S*_{T} is a permutation operator which permutes pairs (−*ω*_{σ}, **p**), (*ω*_{1}, **e**_{1}), (*ω*_{2}, **e**_{2}), (*ω*_{3}, **e**_{3}) (see Ref. 33, Eq. 4.120), Ω_{mg} is the transition frequency for the ground state |*g*⟩ to excited state |*m*⟩, and *μ*_{gm} = ⟨*g*|** μ**|

*m*⟩ is the transition dipole moment connecting those states. The two-photon terms contain differences in permanent dipole moments of excited states, |

*l*⟩, and the ground state, $\mu \u0304lm=\u27e8l|\mu |m\u27e9\u2212\u27e8g|\mu |g\u27e9\delta lm$. The EA spectrum is obtained from the imaginary part of

*χ*

^{(3)}(−

*ω*;

*ω*, 0, 0), where

*ω*

_{σ}=

*ω*

_{1}=

*ω*is the incident light frequency and two zero frequencies,

*ω*

_{2}=

*ω*

_{3}= 0, represent an applied electrostatic field, which is typically around 10

^{7}V m

^{−1}in experiment. Unit vectors

**p**,

**e**

_{1},

**e**

_{2}, and

**e**

_{3}give directions of the applied and resultant electric fields.

### C. CIS methods and computational details

CIS calculations were performed using the Exciton code,^{36–38} which is a Gaussian orbital code for HF, CIS, time-dependent Hartree Fock (TDHF), and BSE calculations on finite and periodic systems.^{38} Two-electron integrals in CIS are calculated using a density-fitting method.

Fractional coordinates for P3[Hex]T in the *P*2_{1}/*c* space group (No. 14) were adopted from Ref. 16. Side chains were edited to yield P3[X]T structures (X = H, Et, Bu), which were optimized using the Crystal code^{39} with a B3LYP density functional.^{40,41} Lattice parameters were kept fixed at P3[Hex]T experimental values^{16} (16.4 × 7.6 × 7.7 Å^{3}) with a monoclinic angle of 87°. A 2 × 12 × 4 k-point mesh was used for these calculations. The P3[H]T calculation was performed using all 18 valence states and 40 virtual states. P3[X]T calculations with X = Et and Bu were performed with 20 valence states and 19 virtual states per k-point.

Fractional coordinates for an isolated chain in the *P*2/*c* space group (No. 10) were also optimized with lattice parameters fixed at 7.6 × 10.0 × 20.0 Å^{3} and a monoclinic angle of 90.0°. This calculation was performed with several sizes of active space in a 12 × 4 × 2 k-point mesh. Excitation energies given in Table I show the change in predicted S_{1} excitation energy with the number of valence and virtual states in the active space. Fractional coordinates for each system are given in the supplementary material, Tables S1–S4.

Valence . | Virtual . | S_{1} Energy (eV)
. |
---|---|---|

10 | 11 | 2.90 |

18 | 18 | 2.84 |

18 | 29 | 2.80 |

18 | 40 | 2.78 |

Valence . | Virtual . | S_{1} Energy (eV)
. |
---|---|---|

10 | 11 | 2.90 |

18 | 18 | 2.84 |

18 | 29 | 2.80 |

18 | 40 | 2.78 |

Calculations on T4 dimers and crystalline polythiophene systems were performed using a modified Pople 6-311G* basis for S^{42–44} and modified def2-TZVP^{45} and def2-SVP^{45} basis sets for C and H, respectively. Linear independence of diffuse exponent Gaussian basis functions in periodic systems limits the smallest exponent that can be used. Modified basis sets used are given in Tables S5–S7 of the supplementary material. All valence states and 230 virtual states were included in the CIS active space for T4 dimer calculations. Basis sets were obtained from the EMSL database.^{46}

## III. RESULTS

### A. CT and FR states in 4 × 4 Hamiltonian

In Sec. II A, a simple model for FR and CT triplet states originating from four frontier orbitals in a T4 dimer was outlined. In order to evaluate the model numerically, the CIS Hamiltonian for a *π*-stacked T4 dimer was diagonalized in the active space described in Sec. II A for singlets and triplets and several intermolecular separations, ranging from 3.2 to 10.0 Å. Energies of the 1^{1}*B*_{g}, 1^{1}*B*_{u}, 2^{1}*B*_{g}, and 2^{1}*B*_{u} singlet states and 1^{3}*B*_{g}, 1^{3}*B*_{u}, 2^{3}*B*_{g}, and 2^{3}*B*_{u} triplet states are shown in Fig. 2, along with variations in single-particle energy differences. Triplet state energies and wave function coefficients are given in Tables S8–S11 in the supplementary material in the MO product basis and in the FR $[|AC\u3009\xb1|BD\u3009]2$ and CT $[|BC\u3009\xb1|AD\u3009]2$ bases [Eqs. (2)–(5)].

The 4 × 4 CIS Hamiltonian has four roots at each intermolecular separation corresponding to two FR and two CT states. At 10.0 Å, the states correspond to pure FR states on each molecule, combined with ± phases and a very small splitting owing to their weak interaction at that range. Below 5 Å, these wave functions begin to deviate from pure FR character (Fig. 2, lower panel). The lowest FR state begins to mix in a significant amount of CT character. The in-phase (1^{3}*B*_{g}) combination becomes dominated by the *H* − 0 → *L* + 0 transition because the single-particle gap for this transition, *ɛ*_{ai}, is reduced in the SCF calculation at short range (Fig. 2, upper panel). The wave function for this state at 3.2 Å is 0.248|(*H* − 1)(*L* + 1)⟩ + 0.968|(*H* − 0)(*L* + 0)⟩, which deviates considerably from coefficients of $1/2$, the wave function for the FR1 state at large separation. On the other hand, single-particle gaps for the *H* − 1 → *L* + 0 and *H* − 0 → *L* + 1 transitions remain relatively constant at short intermolecular separations and the wave function for the FR2 (1^{3}*B*_{u}) state, 0.748|(*H* − 1)(*L* + 0)⟩ + 0.664|(*H* − 0)(*L* + 1)⟩ remains much closer to that for the FR2 state at large separation. Splitting of the two FR states at short intermolecular distances can be seen in Fig. 2, lower panel.

There is a concomitant splitting of the CT states and upward deviation of the *H* − 1 → *L* + 1 transition energy at short intermolecular separations. CT state 2 acquires a significant amount of FR character (2^{3}*B*_{g}, Table S11) and deviates upward for intermolecular separations below 4 Å. At large separation, the CT states evolve into the two IP-EA states whose energies are almost degenerate at 10.0 Å. Singlet FR states in Fig. 2 have a similar behavior to the corresponding triplet states but are shifted upward by half of the (*AA*|*CC*) intramolecular two-electron integral, which is ∼4.44 eV and independent of molecular separation. This integral does not enter the energy of the CT states; hence, they are degenerate, except at short range where splitting of single-particle gaps causes them to deviate to higher or lower energy.

### B. Excited states in T4 dimers

CIS calculations on singlet states of *π*-stacked T4 dimers with C_{2h} symmetry (Fig. 1) were performed using the same basis sets as those reported in Sec. III A but with the much larger active space described in Sec. II C. Excitation energies of the first four states of each irreducible representation vs separation are shown in Fig. 3. The lowest excited states are four FR states, one of each symmetry type, whose transitions with the largest weights are *H* − 0 → *L* + 0 and *H* − 1 → *L* + 1 ($Bg1$), *H* − 0 → *L* + 1 and *H* − 1 → *L* + 0 ($Bu1$), *H* − 0 → *L* + 2 and *H* − 1 → *L* + 3 ($Au1$), and *H* − 1 → *L* + 2 and *H* − 0 → *L* + 3 ($Ag1$) transitions. Lowering of the $Bg1$ and $Au1$ states for separations below 4 Å arises from changes in single-particle excitation energies (Fig. 3) similar to those in the 4 × 4 model Hamiltonian. The 1^{1}*B*_{g}, 1^{1}*B*_{u}, 2^{1}*B*_{u}, and 2^{1}*B*_{g} states are the FR and CT states found in the 4 × 4 Hamiltonian, now calculated in a much larger Hamiltonian.

The first CT states are the 2^{1}*B*_{u} and 2^{1}*B*_{g} states indicated by black and red diamonds in Fig. 3. They show a similar splitting to the two CT states in the 4 × 4 model Hamiltonian, and for separations between 3.6 and 5.2 Å, they are the states immediately above the four lowest FR states. The next CT states are a pair of $Ag1$ and $Au1$ symmetry. The state of $Ag1$ symmetry deviating downward at short range and around 6 eV is the first CT state with $Ag1$ symmetry.

At large separation, the 2^{1}*B*_{g} and 2^{1}*B*_{u} CT states separate into |*AD*⟩ ± |*BC*⟩ ionic configurations. They are nearly degenerate beyond a separation of 4.5 Å. Consequently, when an electrostatic field is applied perpendicular to the molecular planes and the interaction energy of the CT dipole moment exceeds the small CT state splitting, the wave functions break symmetry and the excited states separate into two states where the dipole moment of the excited state lies parallel or antiparallel to the applied field. In this case, the wave functions are |*AD*⟩ and |*BC*⟩. Figure 4 shows the magnitudes of the permanent dipole moments of the CT states when the CIS calculation is performed in the presence of a field of ∼10^{7} V m^{−1}, applied perpendicular to the molecular planes. T4 dimer CT excited states begin to acquire a dipole moment for a separation of 4.4 Å and this reaches the full *e* times dimer separation by 5.2 Å, after which the dipole moment grows linearly with separation. The dipolar interaction energy, −**p** · **E**, has a magnitude of 2.6 meV for full transfer of charge *e* between molecules when they are separated by 5 Å. This is the order of magnitude of the splitting energy when the molecules acquire a dipole moment in the applied electric field.

### C. Linear optical absorption in T4 and P3[X]T polymers

In this section, we present CIS calculations of the linear optical absorption spectra for a single T4 molecule and for infinite P3[X]T polymer crystals with X = H, Et, and Bu. Most many-body calculations on infinite PT systems have been performed on isolated, infinite polymers with hydrogen^{18,19} or small alkyl chain^{20,21} substituents. van der Horst and co-workers^{22} performed a study of crystalline PT,^{17} which has a herringbone structure. On the other hand, experimental work is commonly done using P3[X]T polymers with unsaturated alkyl chains including X = Oct^{26} and Hex.^{7,16} P3[Hex]T has a 3D crystalline *π*-stacked, *P*2_{1}/*c* structure. In one structural study (that used in this work), the lattice constant in P3[Hex]T^{16} perpendicular to PT chains was 7.7 Å, so that the distance between chains is 3.85 Å and there is a small tilt angle and axial shift between chains. In another epitaxial study of the structure of P3[Hex]T, the perpendicular interchain separation was found to be 3.4 Å and the tilt angle was 26°.^{47}

The T4 monomer and dimer and PT crystal structures belonging to *P*2/*m* and *P*2_{1}/*c* space groups each have *C*_{2h} point symmetry. The $Bu1$ and $Au1$ irreducible representations are the optically active representations for C_{2h} point symmetry. (*x*, *y*) transform as $Bu1$ and *z* transforms as $Au1$. Transition dipole moments from the 1^{1}*A*_{g} ground state lie parallel (perpendicular) to thiophene molecular planes in $Bu1$ ($Au1$) transitions. Transitions between excited states are considered later in Sec. III D. In this case, dipole allowed transitions between $Bg1$ and $Bu1$ or $Ag1$ and $Au1$ states are polarized perpendicular to the molecular plane while transitions between $Bg1$ and $Au1$ or $Ag1$ and $Bu1$ are polarized in the molecular plane.

The CIS linear optical absorption spectrum for a T4 molecule is shown in Fig. 5 for the applied optical field in the molecular (*x*, *y*) plane. Single T4 $Bu1$ absorptions at 3.62, 5.70, and 6.34 eV compare to T4 dimer 1^{1}*B*_{u}, 3^{1}*B*_{u}, and 6^{1}*B*_{u} excitations at 3.72, 5.73, and 6.55 eV, respectively. The 2^{1}*B*_{u} CT state at 5.38 eV in the T4 dimer is absent from the single molecule and has a very small transition strength. The HF IP-EA gap at large separation is 7.79 eV and represents the asymptotic limit of the $Bu1$ and $Bg1$ IP-EA states. There are no other $Bu1$ states with large transition moments up to that limit.

S_{1} CIS excitation energies of a series of PT oligomers as a function of the reciprocal of the chain length are shown in Fig. S1 in the supplementary material and extrapolated to infinite chain length. The CIS excitation energy of an isolated, infinite PT chain is 2.78 eV, which agrees well with the extrapolated limit for the series of PT oligomers used.

Excitonic hole wave functions for the first excited states of PT infinite polymers with IC and P3[X]T crystal structures, with the electron localized at the center of the figure are shown in the upper panel of Fig. 6. Envelopes of both states are very similar, indicating that interchain interaction has limited effect on S_{1} exciton localization in the intra-chain exciton. The hole wave function extends over fewer than 10 thiophene units. Imaginary parts of the dielectric functions of PT IC, P3[H]T, P3[Et]T, and P3[Bu]T are shown in the lower panel of Fig. 6. It is immediately evident that the PT IC has a higher excitation energy than the P3[X]T systems. The latter have strong interchain interactions as they are only 3.85 Å apart.

Excitation energies from CIS calculations in this work and *GW*/BSE calculations in the literature are compared in Table II. In the PT IC, the CIS S_{1} state occurs at 2.78 eV, when all valence states and 40 virtual states per k-point are included in the calculation (convergence with respect to number of virtual states is given in Table I). S_{1} excitation energies in P3[H]T, P3[Et]T, and P3[Bu]T are significantly lower, at 2.11, 2.16, and 2.16 eV, respectively. The HF bandgap in the isolated chain is 6.70 eV and this is reduced to 5.51 and 5.64 eV in P3[H]T and P3[Et]T, respectively, which accounts for the significant reduction in excitation energy. The top panel of Fig. 3 shows that the HF gap in T4 dimers decreases by over 0.5 eV when the molecules are less than 4 Å apart. van der Horst *et al.* found that local-density approximation (LDA) HOMO and LUMO values shifted by a combined 0.80 eV when two T1 monomers were *π*-stacked with a separation of 4 Å.^{22}

Method . | Structure . | S_{1}, S_{2}
. | S_{3}, S_{4}
. | S_{5}, S_{6}
. |
---|---|---|---|---|

GW/BSE-TDA^{a} | PT IC | 1.73 | 2.26 | |

GW/BSE-TDA^{a} | HPT | 1.49, 1.50 | 1.53, 1.53 | |

GW/BSE-TDA^{b} | PT IC | 1.48 | ||

GW/BSE-TDA^{c} | PT IC | 1.0 | ||

CIS^{d} | PT IC | 2.78, 5.29 | 5.57, 5.57 | 5.86, 5.86 |

CIS^{d} | P3[H]T | 2.11, 2.19 | 3.63, 3.66 | 3.70, 3.70 |

CIS^{d} | P3[Et]T | 2.16, 2.21 | 3.65, 3.67 | 3.69, 3.70 |

Expt.^{e} | P3[Oct]T | 2.0 | ||

Expt.^{f} | P3[Hex]T | 1.8 |

Method . | Structure . | S_{1}, S_{2}
. | S_{3}, S_{4}
. | S_{5}, S_{6}
. |
---|---|---|---|---|

GW/BSE-TDA^{a} | PT IC | 1.73 | 2.26 | |

GW/BSE-TDA^{a} | HPT | 1.49, 1.50 | 1.53, 1.53 | |

GW/BSE-TDA^{b} | PT IC | 1.48 | ||

GW/BSE-TDA^{c} | PT IC | 1.0 | ||

CIS^{d} | PT IC | 2.78, 5.29 | 5.57, 5.57 | 5.86, 5.86 |

CIS^{d} | P3[H]T | 2.11, 2.19 | 3.63, 3.66 | 3.70, 3.70 |

CIS^{d} | P3[Et]T | 2.16, 2.21 | 3.65, 3.67 | 3.69, 3.70 |

Expt.^{e} | P3[Oct]T | 2.0 | ||

Expt.^{f} | P3[Hex]T | 1.8 |

These S_{1} energies compare well with experimental values of 2.0 eV in highly ordered P3[Oct]T films^{26} and 1.8 eV for P3[H]T in a thin film.^{7} CIS excitation energies are expected to overestimate experimental excitation energies. On the other hand, *GW*/BSE calculations in the TDA, which used a density functional starting point, tend to underestimate experimental values. Calculations by van der Horst and co-workers found S_{1} excitation energies of 1.73 eV for an isolated chain and 1.49 eV^{22} in a calculation on the crystalline PT structure^{17} with a herringbone structure. Interaction between PT chains in that structure is weaker: The shift of 0.80 eV for LDA eigenvalues in *π*-stacked T monomers reduced to 0.21 eV when the monomers were rearranged in the herringbone structure. Hence, a smaller reduction in excitation energy on going from the isolated chain to the crystal is expected in that case (0.24 eV vs 0.67 eV in the CIS calculations reported here). More recent *GW*/BSE calculations^{20,21} also find S_{1} excitation energies below experimental values.

S_{1} and S_{2} and S_{3} and S_{4} excitations in PT crystals occur in pairs with a small energy splitting arising from odd vs even combinations of electron–hole pairs on neighboring polymer chains. 1^{1}*B*_{u} excitations carry most of the oscillator strength. The CIS calculation on the isolated chain found the only low-lying excitation to be S_{1} at 2.78 eV. In P3[H]T, low-lying states are found at 2.11, 2.19, 3.63, 3.66, 3.69, and 3.69 eV. Very similar excitation energies are found in P3[Et]T and P3[Bu]T (Table II and Fig. 6). In the lowest excitation state, the electron and hole are essentially confined to the same chain (left panel, Fig. 7). In the higher states, the electron and hole are more delocalized over pairs of neighboring chains (right panel, Fig. 7). The S_{2} and S_{3} excitons are dark and the S_{1} and S_{4} excitons are bright. S_{1} and S_{2} and S_{3} and S_{4} states were found to be split by 0.01 and less than 0.01 eV, respectively, in a *GW*/BSE calculation,^{22} compared to splittings of 0.08 in the herringbone structure and 0.03 in this work. A dark state of $Ag1$ symmetry in P3[Oct]T has been postulated at 2.5 eV on the basis of two-photon absorption^{26} and at 2.67 eV on the basis of EA.^{12}

### D. T4 electroabsorption spectra

EA spectra are proportional to the imaginary part of the third-order susceptibility, *χ*^{(3)}(−*ω*; *ω*, 0, 0). Susceptibilities calculated using the SOS method for all fields in the *x* direction and for finite frequency fields in the *x* direction and static fields in the *z* direction are shown in Fig. 8. Both one- and two-photon terms [Eqs. (6) and (7)] contribute to Im$\chi xxxx(3)$. The one-photon term yields a blue-shifted EA signal that is smaller than the two-photon red-shifted signal, giving the overall red-shifted, first derivative line shape shown in the left column of Fig. 8. There is little dependence of the EA response on dimer separation, if fields are applied along the long molecular axis, as might be expected.

The EA spectrum for a thin film of regioregular P3[Hex]T redrawn from Ref. 3 is shown in Fig. 9. The line shape for the 1^{1}*B*_{u} transition around 2 eV is quite different from the first derivative line shape of the 1^{1}*B*_{u} peak at 3.62 eV in the T4 EA spectrum (Fig. 8). Leiss and co-workers^{12} showed that inclusion of vibronic coupling in models of EA spectra of polymers, including polythiophene, changes a simple first derivative line shape into the 1^{1}*B*_{u} line shape in Fig. 9. The authors of Ref. 3 also noted a shoulder, which they attributed to the *mA*_{g} excited state (where m is a small, unknown integer), and a second derivative line shape around 2.75 eV, which they attributed to a CT state (Fig. 9).

The EA spectrum of P3[Dec]T polymer films was measured^{12} and shown to be dominated by a Stark shift of the 1^{1}*B*_{u} state at 2.01 eV, as well as features fitted to excited states of $Ag1$ symmetry at 2.67 and 3.27 eV. The sign of the EA signal depends on the competition of the one- and two-photon terms in which the intermediate transition returns the electron to the ground state [Eq. (6)] or to a different intermediate state [Eq. (7)]. Since the ground state is connected only to $Bu1$ states in dipole allowed transitions, the intermediate state is either a state of $Ag1$ symmetry (in *xxxx* polarization) or a state of $Bg1$ symmetry (in *xxzz* polarization). Recall that $Ag1$ to $Bu1$ transitions have the transition moment in the molecular (*x*, *y*) plane and $Ag1$ to $Bg1$ transitions have the transition moment parallel to the *z* direction. Based on the sign of the EA signal arising from transitions to the 1^{1}*B*_{u} state, Liess and co-workers surmised^{12} that there must be one or more transitions to states of $Ag1$ symmetry at relatively low energy. In order to have a positive EA signal, the |*μ*_{01}|^{2}|*μ*_{1x}|^{2} two-photon term must exceed the |*μ*_{01}|^{4} one-photon term, where in Liess’ notation, 0 is the ground state, 1 the 1^{1}*B*_{u} state, and *x* an intermediate excited state of $Ag1$ symmetry.

Dipole transition matrix elements between states of $Ag1$ symmetry (including the 1^{1}*A*_{g} ground state) and the 1^{1}*B*_{u} state are given in the top section of Table III. The dipole transition matrix element between the 1^{1}*A*_{g} ground state and 1^{1}*B*_{u} excited state is ∼10 D in the isolated molecule (infinite dimer separation) and reduces to around 9.5 D at small intermolecular separation. For dimer separations of 3.6 Å and above, the 3^{1}*A*_{g} and 4^{1}*A*_{g} states, which lie around 6.2 and 6.3 eV in CIS calculations, are the main intermediate state contributors to the $\chi xxxx(3)$ EA signal (Table III). These correspond to matrix elements $\mu \u0304lm$ in Eq. (7), where *l* is the 1^{1}*B*_{u} state and *m* is an excited state of $Ag1$ symmetry (or *μ*_{1x} in Liess’ notation). The 1^{1}*B*_{u} to 3^{1}*A*_{g} transition dipole moment has a value of 11.21 D at a separation of 4.0 Å, allowing the two-photon term to exceed the one-photon term and yield a positive EA signal.

Separation (Å) . | |||||||||
---|---|---|---|---|---|---|---|---|---|

. | 3.2 . | 3.6 . | 4.0 . | 4.5 . | 4.8 . | 5.2 . | 6.0 . | 6.8 . | ∞ . |

1^{1}A_{g} | 9.53 | 9.61 | 9.63 | 9.71 | 9.79 | 9.79 | 9.84 | 9.84 | 10.07 |

2^{1}A_{g} | 2.01 | 1.42 | 1.37 | 1.32 | 1.07 | 1.32 | 1.32 | 1.22 | 1.14 |

3^{1}A_{g} | 1.09 | 10.09 | 11.21 | 11.03 | 11.11 | 10.96 | 10.75 | 10.55 | 10.52 |

4^{1}A_{g} | 11.13 | 4.93 | 4.22 | 7.12 | 7.19 | 7.40 | 7.70 | 8.01 | 8.13 |

1^{1}B_{g} | 0.74 | 0.48 | 0.28 | 0.13 | 0.08 | 0.05 | 0.00 | 0.00 | 0.02 |

2^{1}B_{g} | 0.02 | 0.20 | 1.09 | 0.92 | 0.79 | 0.61 | 0.00 | 0.00 | 0.00 |

3^{1}B_{g} | 0.13 | 1.22 | 0.18 | 0.05 | 0.05 | 0.10 | 0.28 | 0.08 | 0.00 |

4^{1}B_{g} | 0.08 | 0.10 | 0.02 | 0.00 | 0.04 | 0.00 | 0.00 | 0.00 | 0.00 |

5^{1}B_{g} | 1.35 | 0.00 | 0.00 | 0.04 | 0.02 | 0.02 | 0.00 | 0.00 | 0.00 |

Separation (Å) . | |||||||||
---|---|---|---|---|---|---|---|---|---|

. | 3.2 . | 3.6 . | 4.0 . | 4.5 . | 4.8 . | 5.2 . | 6.0 . | 6.8 . | ∞ . |

1^{1}A_{g} | 9.53 | 9.61 | 9.63 | 9.71 | 9.79 | 9.79 | 9.84 | 9.84 | 10.07 |

2^{1}A_{g} | 2.01 | 1.42 | 1.37 | 1.32 | 1.07 | 1.32 | 1.32 | 1.22 | 1.14 |

3^{1}A_{g} | 1.09 | 10.09 | 11.21 | 11.03 | 11.11 | 10.96 | 10.75 | 10.55 | 10.52 |

4^{1}A_{g} | 11.13 | 4.93 | 4.22 | 7.12 | 7.19 | 7.40 | 7.70 | 8.01 | 8.13 |

1^{1}B_{g} | 0.74 | 0.48 | 0.28 | 0.13 | 0.08 | 0.05 | 0.00 | 0.00 | 0.02 |

2^{1}B_{g} | 0.02 | 0.20 | 1.09 | 0.92 | 0.79 | 0.61 | 0.00 | 0.00 | 0.00 |

3^{1}B_{g} | 0.13 | 1.22 | 0.18 | 0.05 | 0.05 | 0.10 | 0.28 | 0.08 | 0.00 |

4^{1}B_{g} | 0.08 | 0.10 | 0.02 | 0.00 | 0.04 | 0.00 | 0.00 | 0.00 | 0.00 |

5^{1}B_{g} | 1.35 | 0.00 | 0.00 | 0.04 | 0.02 | 0.02 | 0.00 | 0.00 | 0.00 |

*χ*^{(3)}(−*ω*; *ω*, 0, 0) susceptibilities for T4 dimers with the static electric field applied in the *z* direction and the optical responses measured in the *x* direction, Im$\chi xxzz(3)$, are shown in the right column of Fig. 8 as a function of T4 dimer separation, along with the responses of a single T4 molecule. For the *xxzz* susceptibility, only the two-photon term in Eq. (7) contributes. In the one-photon term [Eq. (6)], the 1^{1}*A*_{g} ground state must couple directly to all virtual states that make a contribution. This limits one-photon term virtual states to $Bu1$ symmetry and coupling in the *x* direction only. In the two-photon term, an electric field in the *z* direction couples $Bu1$ and $Bg1$ states, hence one of the virtual states in this case is of $Bg1$ symmetry.

Dipole transition matrix elements between states of $Bg1$ symmetry and the 1^{1}*B*_{u} state are given in the lower section of Table III. There is moderately strong coupling of the 1^{1}*B*_{u} state to the 1^{1}*B*_{g} FR state, but the strongest coupling is to the CT state of $Bg1$ symmetry, which tracks from 2^{1}*B*_{g} to 5^{1}*B*_{g} as the T4 dimer separation is reduced. The impact of the 2^{1}*B*_{u} and 2^{1}*B*_{g} CT states is to strengthen the first derivative, *xxzz* response at 3.7 eV when the dimer separation is small and to introduce a new second derivative response at the CT state excitation energy (Fig. 8, right panel). The latter ranges between 5 and 6 eV, depending on dimer separation and becomes weaker with increasing dimer separation. The response from an isolated molecule in the molecular plane with the static electric field applied perpendicular to the molecular plane (*xxzz* response) is about 2000 times weaker than the *xxxx* response, with all fields in the molecular plane, since there is no $Bg1$ CT state in that case.

In the infinite *π*-stacked polymer, the CT states occur at lower energy. CT energies presented in Table II are predicted to occur around 3.6 eV. A feature at 2.75 eV in the EA spectrum of P3[Hex]T has been attributed to a CT exciton [Fig. 4(A), Ref. 3, redrawn here as Fig. 9]. The shape of the signal is similar to the second derivative feature of the CT derived peak in Fig. 8. CIS is expected to overestimate energies of states; therefore, it seems likely that the experimental feature does originate in a CT state where the electron and hole reside on neighboring chains.

## IV. SUMMARY AND CONCLUSIONS

We have reported CIS calculations on bulk P3[Alkyl]T and T4 monomer and cofacial dimer systems. A simplest possible model for CT states in a symmetric dimer was presented in Sec. II A. CIS calculations of excited state energies in cofacial T4 dimers as a function of dimer separation found the first six excited states at separations around 4 Å to consist of four low energy Frenkel states and two CT states. (The separation of P3[Alkyl]T chains in the bulk crystal is just under 4 Å.) CIS calculations on bulk P3[Et]T and P3[H]T crystals show intra-chain Frenkel excitons as the lowest energy bright excitons having the majority of the optical oscillator strength. The next excited states are interchain CT excitons in which the excited electron and hole are mainly on neighboring chains. These states are analogous to CT states found in T4 dimers. The two lowest singlet excited states in CIS calculations on P3[H]T are bright and dark intra-chain excitons at 2.11 and 2.19 eV. The third and fourth singlet excited states occur at 3.63 and 3.66 eV. The fifth state at 3.70 eV is the next bright exciton but with a very weak absorption cross section compared to the first bright state. In both bulk crystals and T4 dimers, the fifth and sixth singlet excited states are the lowest CT states. In the crystal, they both occur at 3.70 eV, and in the T4 dimer at 4.0 Å separation, they occur at 5.18 and 5.30 eV. Excited states observed by two-photon absorption at 2.5 eV^{26} or at 2.67 and 3.27 eV by EA spectroscopy have been proposed as low-lying states of $Ag1$ symmetry. These values compare to the 2^{1}*A*_{g} state in the T4 dimer CIS calculations at 4.85 eV.

The CIS calculations reported here are equivalent to *GW*/BSE calculations in the TDA with no screening of the HF exchange interaction (included in a *GW* calculation) and none in the electron–hole attraction (included in a screened BSE calculation). Omission of screening in both cases results in a degree of self-cancellation, but the CIS method typically yields excitation energies in conjugated organic molecules greater than experimental values.

The EA spectrum of the T4 monomer and cofacial dimers was reported as a function of dimer separation. In the *xxxx* field configuration where input and output optical fields and static fields in an EA experiment are aligned with the long T4 molecular axis, the EA signal is dominated by the ground state to 1^{1}*B*_{u} transition and intermediate transitions to higher states of $Ag1$ symmetry. This EA feature does not depend strongly on T4 dimer separation as each excited state involved is an intramolecular Frenkel excited state. In contrast, in the *xxzz* field configuration, where input and output optical fields are aligned with the long T4 molecular axis and static fields are aligned with an axis that passes through both molecules, the EA signal is dominated by the ground state to the 2^{1}*B*_{u} CT excited state and an intermediate transition to the 2^{1}*B*_{g} CT excited state. At a dimer separation of 4 Å, the 2^{1}*B*_{u} CT state energy is 5.38 eV. The small splitting of the 2^{1}*B*_{u} and 2^{1}*B*_{g} states results in a second derivative line shape for this feature in the EA spectrum, which is about five times weaker than the main 1^{1}*B*_{u} feature in the *xxxx* field configuration.

Calculations reported here and experimental observations of efficient polaron generation following light absorption make it seem likely that there are low-lying CT excitons in P3[Alkyl]T that are important in generation of charge carriers in these lamellar systems. Creation of such CT excitons is widely believed to be an important intermediate step in free carrier generation in organic photovoltaics.^{48} It may be possible to increase the sensitivity of EA to CT states in highly oriented P3[Alkyl]T films by applying the static field along an axis perpendicular to the polymer chains that passes through multiple polymer chains and the optical field parallel to the chains. The CT line shape is found to be a second derivative line shape in our calculations and in a feature observed in experimental EA spectra of P3[Dec]T.^{3}

## SUPPLEMENTARY MATERIAL

The supplementary material includes the following: (1) lattice parameters and fractional coordinates of periodic polythiophene systems reported in Sec. III C; (2) basis sets used for calculations on both the T4 dimer and periodic polythiophene systems; (3) tables containing wave function coefficients and excitation energies for the T4 dimer as a function of separation in a 4 × 4 model Hamiltonian; and (4) extrapolation of thiophene oligomer S_{1} excitation energies to the infinite polymer.

## ACKNOWLEDGMENTS

S.R.S. acknowledges financial support provided by the Irish Research Council through a Government of Ireland Postdoctoral Fellowship, Grant No. GOIPD/2020/792. Calculations were performed on the Kelvin cluster maintained by the Trinity Centre for High Performance Computing, funded through grants from Science Foundation Ireland and the Irish Higher Education Authority and the Kay cluster maintained by the Irish Centre for High End Computing in Project No. TCPHY174b.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Smruti Ranjan Sahoo**: Conceptualization (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). **Charles H. Patterson**: Conceptualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.