Impact of dynamic electrostatic disorder on hole mobility in rubrene: a nonadiabatic molecular dynamics investigation

Abstract

High-mobility organic molecular crystals such as rubrene are important materials for organic electronics, yet a quantitatively predictive description of their charge transport properties remains challenging. Direct mixed quantum-classical nonadiabatic molecular dynamics simulations provide a promising route by explicitly propagating the charge carrier wavefunction, without assuming a specific transport mechanism. However, previous large-scale simulations of apolar molecular crystals have commonly neglected dynamic electrostatic disorder, since evaluation of electrostatic interactions is computationally demanding and the approximation appears plausible for apolar systems. Here, we use the damped shifted-force (DSF) real-space electrostatic summation method, combined with an efficient addition-subtraction scheme, to include dynamic electrostatic disorder in fragment orbital-based surface hopping (FOB-SH) simulations of room-temperature hole transport in rubrene. We find that electrostatic interactions increase the reorganization energy for (hypothetical) nearest neighbour hopping by 29 and 39 meV along the a and b-directions, respectively, relative to a baseline value of 152 meV obtained without electrostatics. In FOB-SH simulations, electrostatic interactions lead to increased site energy disorder, reducing the spatial extent of the hole wavefunction, as measured by a decrease in the inverse participation ratio from 13 to 9, and lowering the predicted mobility along the high-mobility direction from 35 to 21 cm² V⁻¹ s⁻¹, in close agreement with experiment.

We show that dynamic electrostatic disorder is significant even in apolar crystals such as rubrene, reducing hole delocalization and bringing mobilities obtained from nonadiabatic molecular dynamics into close agreement with experiment.

1. Introduction

Organic semiconductors have attracted sustained interest as active materials for flexible electronics, thin-film transistors, light-emitting diodes, and photovoltaic devices.^1–3 Their performance in these applications is often limited by the mobility of charge carriers, making the development of predictive theoretical approaches for charge transport a central goal in the field. Rubrene is a particularly important benchmark system in this context: high-quality rubrene single crystals exhibit among the highest hole mobilities reported for molecular organic semiconductors.^4–6 Rubrene has also been used as a platform to explore spin-transport^7,8 and triplet-upconversion.^9,10

Charge transport in high-mobility molecular crystals such as rubrene occurs in an intermediate regime between the limiting cases of small-polaron hopping and band transport.^11–13 In this regime, the charge carrier is neither fully localized on a single molecule nor completely delocalized over the crystal. Instead, the picture that has emerged over recent years is one of transient delocalization,^14,15 in which the charge carrier wavefunction forms a dynamic polaron delocalized over several molecules, with its spatial extent continuously modulated by molecular and lattice vibrations and transitions between the densely spaced valence or conduction band states. Here, it is important to note that the (partial) localization is not caused by a change in bond length in response to the presence of a charge carrier, as is the case for small polaron formation in oxides,¹⁶ but due to the strong thermal fluctuations of electronic coupling and on-site energy in organic materials (i.e. off-diagonal and diagonal electron–phonon coupling). This underlies the characteristic band-like decrease of mobility with temperature in ultrapure high mobility organic crystals,¹⁵ in contrast to the activated transport commonly observed for small polarons in oxides.

Accurately describing this regime requires a method that does not impose either a fully localized hopping picture or a fully delocalized band picture. Approaches based on kinetic rate theories, such as kinetic Monte Carlo (KMC) simulations using Marcus or Marcus–Levich–Jortner rates,^17,18 are computationally efficient, but assume that charge carriers are localized and move through a sequence of incoherent hopping events. They are therefore ill-suited to describe high-mobility molecular crystals such as rubrene, where carrier delocalization plays an important role. A second class of methods explicitly treats the coupled electronic and nuclear dynamics, including Ehrenfest dynamics,^19,20 time-dependent density matrix renormalization group (TD-DMRG),²¹ quantum Monte Carlo techniques,²² and, more recently, advanced surface-hopping schemes such as MASH.²³ Although these approaches provide a more faithful description of charge transport, their computational cost has generally restricted the most accurate quantum-dynamical treatments to one-dimensional or otherwise simplified model systems. Furthermore, many existing studies lack a fully atomistic description of the vibrational degrees of freedom, with environmental and electrostatic effects often incorporated only implicitly through model parametrization. These considerations motivate the use of methods that combine nonadiabatic dynamics with an efficient atomistic description of molecular motion in extended crystal models.

The fragment orbital-based surface hopping (FOB-SH) method is a promising approach to this end.²⁴ FOB-SH is a fully atomistic method in which an excess charge carrier is propagated quantum mechanically in a basis of molecular fragment orbitals, while the nuclei evolve classically according to fewest-switches surface hopping.²⁵ The resulting time-dependent electronic Hamiltonian contains diagonal site energies and off-diagonal electronic couplings, both of which fluctuate due to thermal nuclear motion. This allows FOB-SH to capture the competition between electronic coupling, which promotes spatial delocalization of the charge carrier, and energetic disorder arising from local and nonlocal electron–phonon coupling, which tends to localize the charge carrier.

In previous work,²⁶ FOB-SH simulations were used to compute hole mobility in rubrene using a molecular model in which atomic partial charges were set to zero, thereby neglecting Coulombic electrostatic interactions. This approximation is a plausible starting point for apolar molecular crystals such as rubrene, while also being computationally attractive because conventional Ewald-type evaluation of Coulombic interactions becomes a significant bottleneck for the large system sizes required in FOB-SH simulations. However, neglecting electrostatic interactions removes electrostatic contributions to the reorganization energy and suppresses electrostatic fluctuations of molecular site energies. Since such site energy fluctuations promote dynamic localization of the charge carrier, it remains unclear to what extent their omission affects carrier delocalization and charge mobility.

Here, we address this question by incorporating electrostatic interactions into FOB-SH simulations of hole transport in rubrene. We employ the damped shifted-force (DSF) real-space electrostatic summation method,²⁷ using a recently developed addition–subtraction scheme that enables efficient evaluation of the electrostatic contributions to the site energies and forces required in FOB-SH simulations.²⁸ This approach avoids the prohibitive cost of conventional Ewald summation when site energies must be evaluated for many possible charge-localized states, making large-scale nonadiabatic simulations with electrostatic interactions feasible.

We first quantify how Coulombic electrostatics affects the reorganization energy for hole transfer in rubrene, using both the DSF method and smooth particle-mesh Ewald²⁹ (SPME) calculations as a reference. We show that electrostatic interactions moderately increase the reorganization energy of rubrene (by about 20%) by enhancing fluctuations of the vertical energy gap between charge-localized diabatic states. Spectral analysis of the vertical energy gap fluctuations reveals contributions from both low-frequency lattice motions and higher-frequency intramolecular modes. We then perform FOB-SH simulations including DSF electrostatics and compare the resulting transport properties with earlier simulations without electrostatics.²⁶ Inclusion of electrostatic interactions increases site energy disorder, reduces the spatial delocalization of the hole wavefunction, and lowers the predicted room-temperature hole mobility, bringing it into close agreement with experimental results of Podzorov et al.⁴ These results demonstrate that electrostatic site energy disorder is essential for quantitatively predictive simulations of charge transport in molecular organic semiconductors, even if they are composed of apolar molecules such as rubrene.

2. Results

2.1. Reorganization energy

We first investigate the effect of electrostatic interactions on the reorganization energy for hole transfer in rubrene, a central quantity controlling charge carrier dynamics. We emphasize that hole transfer from one rubrene molecule to another is a fictitious process, since the hole is partially delocalized over several molecules in this material, but one that is important to investigate theoretically because it defines the reorganization energy (eqn (5) and (6)).

The reorganization energy measures the energetic cost associated with nuclear relaxation upon charge transfer and reflects the strength of local electron–phonon coupling. It contains an internal contribution from structural relaxation of the molecule carrying the charge, as well as an external contribution from the response of the surrounding crystalline environment. Coulombic interactions are therefore expected to affect the reorganization energy, in particular its external contribution, by modifying the electrostatic response of the crystal environment to changes in charge localization.

To quantify this effect, the reorganization energy for hole transfer from molecule M to neighbour N in rubrene was calculated using three treatments of Coulombic interactions: a zero-charge model, in which all atomic partial charges were set to zero; the DSF method; and the SPME method. For the latter two methods, polarization effects were included implicitly by using scaled partial charges, as described in Section 4.2. In each case, reorganization energy was computed according to eqn (5) and (6) from the distribution of vertical energy gaps ΔE_{M → N} (eqn (4)) over a molecular dynamics trajectory in the diabatic state corresponding to the hole being localized on molecule M (see Section 4.3 for details). Furthermore, reorganization energies were computed for hole transfer from a reference molecule M to neighbouring molecules along two distinct transfer directions, denoted P and T. For each direction, two symmetry-related neighbours of M were considered: P1 and P2 for the P direction, and T1 and T2 for the T direction. These molecules are illustrated in Fig. 1, where molecule M is coloured in red, P1/P2 in blue, and T1/T2 in brown.

Fig. 1. Top-down view of the high-mobility (a) and (b) plane of crystalline rubrene, showing the molecular pairs used for the reorganization energy calculations. The central molecule shown in red is the molecule on which the hole is localized in the reference diabatic state. The neighbouring molecules used to define vertical energy gaps are highlighted in blue for the P-type dimers (P1, P2) and brown for the T-type dimers (T1, T2). The P and T dimers are stacked along the high- and low-mobility transfer directions, respectively.

The computed reorganization energies λ^st and λ^var for the different treatments of Coulombic interactions are listed in Table 1. Reported P- and T-direction values for the reorganization energy are averages over the two corresponding dimers, while values for the individual dimers are given in Section S1. In the zero-charge model, the reorganization energy is isotropic with respect to hole transfer direction, with λ^st = 152 meV and λ^var = 155 meV for both P and T. This is expected because, without electrostatics, the reorganization energy is dominated by intramolecular reorganization, which is independent of the hole transfer direction. Including electrostatics moderately increases the reorganization energy, with both DSF and SPME giving values larger than the zero-charge model by approximately 20–40 meV, depending on the transfer direction and treatment of electrostatic interactions.

Table 1. Computed reorganization energies for hole transfer in rubrene using different treatments of electrostatic interactions; a zero-charge model (i.e., no Coulombic interactions), the DSF method and the SPME method.

Method	Dimer typea	λ st/meVb	λ var/meVc
Zero-charge	P	152	155 ± 2
Zero-charge	T	152	155 ± 5
DSF	P	175	174 ± 3
DSF	T	182	187 ± 4
SPME	P	181	190 ± 3
SPME	T	191	197 ± 3

^a P and T denote the neighbour directions considered for hole transfer from molecule M. Reported values are averages over two symmetry-related neighbouring molecules, see Fig. 1. Values for the individual dimers are reported in Section S1.

^bStokes shift reorganization energy, λ^st, eqn (5). All statistical uncertainties for λ^st are smaller than 1 meV.

^cFluctuation-derived reorganization energy, λ^var, eqn (6), and statistical uncertainty (see Section 4.3 and Section S1).

The two estimates, λ^st and λ^var, remain within approximately 5% of one another in all cases, indicating that the vertical energy gap fluctuations are reasonably well described by linear response³⁰ (in the linear response limit, λ^st = λ^var). The DSF values are systematically lower than the corresponding SPME values, but recover most of the electrostatic increase relative to the zero-charge model. Finally, including Coulombic interactions introduces a small directional anisotropy that is absent in the zero-charge model, with the T direction showing a slightly larger reorganization energy than the P direction for both electrostatic methods.

To resolve how different vibrational frequencies contribute to the fluctuation-derived reorganization energy, we computed the spectral density of the vertical energy gap fluctuations, S(ω)/ω (eqn (9); see Section 4.3). Fig. 2(a) and (b) show S(ω)/ω for the P and T transfer directions, respectively, for each treatment of Coulombic interactions. The peak positions are very similar for the P and T directions, indicating that the same vibrational modes dominate the energy-gap fluctuations in both cases.

Fig. 2. Spectral density functions of vertical energy gap fluctuations and cumulative reorganization energies for hole transfer, comparing different treatments of Coulombic interactions; zero-charge model in green, the DSF method in blue and the SPME method in red. (a) and (b) Spectral densities, computed using eqn (9), for transfer along the P and T directions, respectively. Insets highlight the low-frequency region from 0 to 250 cm⁻¹. (c) and (d) Cumulative fluctuation-derived reorganization energies up to frequency ω, λ^cum(ω), computed using eqn (10), for transfer along the P and T directions, respectively. The dotted horizontal lines indicate the corresponding λ^var values obtained from the full energy-gap distribution. (e) and (f) Electrostatic contributions to the cumulative reorganization energy Δλ^cum(ω) = λ^cum_j(ω) − λ^cum_zero-charge(ω) (j = DSF, SPME in blue and red, respectively), for transfer along the P and T directions, respectively. Dotted lines indicate the values of Δλ^var = λ^var_j − λ^var_zero-charge (j = DSF, SPME in blue and red, respectively).

The insets highlight the low-frequency region below 200 cm⁻¹. In the zero-charge model, the spectral density in this region is negligible, whereas both DSF (blue) and SPME (red) show a pronounced low-frequency contribution and are in close agreement. This low-frequency enhancement indicates that Coulombic interactions increase the coupling between the localized charge and low-frequency nuclear motions of the rubrene crystal.

To quantify how spectral contributions accumulate into the total fluctuation-derived reorganization energy (eqn (6)), Fig. 2(c) and (d) show the cumulative reorganization energy, λ^cum(ω), for the P and T transfer directions, respectively. As defined in eqn (10), λ^cum(ω) is obtained by integrating S(ω)/ω up to frequency ω, and therefore gives the portion of λ^var arising from vertical energy gap fluctuations including all modes up to ω.

In Fig. 2(c) and (d), λ^cum(ω) increases through a series of step-like changes that coincide with peaks in the spectral density (panels (a) and (b)). The DSF and SPME curves follow the same overall profile as the zero-charge model, indicating that most of the dominant vibrational features responsible for site energy fluctuations are present without Coulombic interactions. However, several peaks are either absent or significantly weaker in the zero-charge model, leading to a slower accumulation of λ^cum(ω) compared with the electrostatic simulations, for instance in the low-frequency range 0–200 cm⁻¹ as discussed above (see insets of panels (a) and (b)).

The comparison with the zero-charge model can be seen more directly in Fig. 2(e) and (f), which plot the excess cumulative reorganization energy, Δλ^cum(ω) = λ^cum_j(ω) − λ^cum_zero-charge(ω) (j = DSF, SPME. These difference curves show that the electrostatic enhancement of λ^var is distributed over several distinct frequency ranges, with prominent contributions below 200 cm⁻¹, around 700–750 cm⁻¹, and near 1550 cm⁻¹. For the P direction, the SPME calculation shows an additional increase at higher frequencies.

The difference plots also clarify where the DSF and SPME treatments begin to diverge. For both transfer directions, the first noticeable separation appears around 700–750 cm⁻¹. In the T direction, this discrepancy remains approximately constant over higher frequencies, whereas in the P direction it increases further at higher frequencies. Thus, DSF captures most of the electrostatic contribution to the cumulative reorganization energy, while slightly underestimating the SPME result in selected frequency ranges.

2.2. Hole transport from FOB-SH simulations

To investigate how Coulombic interactions affect hole transport in rubrene, we performed FOB-SH simulations at 300 K using the DSF electrostatics scheme. These results are compared directly with previously reported FOB-SH simulations for the same system using the zero-charge model.²⁶ Both sets of simulations used the same supercell size, 50 × 13 × 1 unit cells, corresponding to 1300 electronically active molecules in the high-mobility (a, b) plane.

At the microscopic level, charge transport is governed by a competition between terms in the electronic Hamiltonian that promote delocalization and those that induce dynamic localization. The off-diagonal matrix elements, (H_kl, k≠l), describe electronic couplings between neighbouring molecules and drive spatial delocalization of the charge carrier. By contrast, thermal fluctuations of the diagonal site energies, (H_kk), and off-diagonal couplings constitute dynamic energetic disorder, which tends to localize the carrier.¹² The site energy fluctuations are directly related to the vertical energy-gap fluctuations and hence to the fluctuation-derived reorganization energy, eqn (6), which was shown in Section 2.1 to moderately increase by Coulombic interactions. We therefore first examine how the inclusion of electrostatics modifies the distributions of electronic couplings and site energies before analyzing the resulting charge carrier delocalization and mobility.

Table 2 summarizes the distributions of the electronic Hamiltonian matrix elements obtained from the FOB-SH simulations, including the mean electronic couplings along the a and b crystallographic directions, 〈J_d〉 (d = a, b), their root-mean-square fluctuations, σ_d, and the root-mean-square fluctuations of the site energies, σ_E. The table also reports the average inverse participation ratio (IPR), 〈IPR〉, which measures the spatial extent of the hole wavefunction, together with σ_IPR, the root-mean-square fluctuation of the IPR over the ensemble of trajectories. Finally, the table reports hole mobilities along the a and b directions, μ_a and μ_b, obtained from the linear regime of the time-dependent mean-square displacement (MSD) using the Einstein–Smoluchowski relation (eqn (16) and (18)).

Table 2. Thermal averages and root-mean-square fluctuations of electronic couplings and site energies, average and root-mean-square fluctuations of the IPR, timescales associated with transient delocalization events, and hole mobility obtained from FOB-SH simulations at 300 K with and without electrostatics.

	〈Ja〉a	σ a a	〈Jb〉a	σ b a	σ E a	〈IPR〉b	σ IPR b	t r c	τ r c	μ a d	μ b d
Zero-charge modele	106.8	28.2	−15.6	6.7	63.5	13.0	10.0	9	71	34.8 ± 1.5	6.5 ± 0.4
DSF	103.4	27.8	−16.4	6.7	69.2	8.9	8.0	10	89	21.1 ± 2.0	5.0 ± 0.2
Experimentf										20.2	7.7

^aMean value of electronic couplings, 〈J_a(b)〉, and root-mean-square fluctuations of electronic couplings, σ_a(b), and site energies, σ_E, in meV.

^bAverage inverse participation ratio of the hole wavefunction, 〈IPR〉 (eqn (19)) and corresponding root-mean-square fluctuation over all trajectories, σ_IPR, neglecting the first 500 fs of dynamics.

^cAverage duration of a transient delocalization event, t_r (fs), defined as time intervals during which the IPR of a trajectory is greater than 〈IPR〉 +σ_IPR, and inverse rate of transient delocalization events, τ_r (fs).

^dCharge carrier mobility from FOB-SH, eqn (18), along the a and b crystallographic directions, in cm² V⁻¹ s⁻¹.

^eFOB-SH simulations without electrostatics, from ref. 26.

^fExperimental results from ref. 4.

The mean electronic couplings are only weakly affected by the inclusion of DSF electrostatics, remaining within 5% of the zero-charge values along both crystallographic directions. Similarly, the root-mean-square fluctuations of the electronic couplings, σ_a and σ_b, change by less than 2%. The dominant effect of Coulombic interactions is instead an increase in the site energy disorder: σ_E increases by 9%, from 63.5 meV in the zero-charge model to 69.2 meV with DSF electrostatics.

This increase in site energy disorder can be rationalized directly from the reorganization energies reported in Section 2.1. If the thermal fluctuations of adjacent molecular site energies, E₁ and E₂, are assumed to be statistically equivalent and approximately uncorrelated, Cov(E₁, E₂) ≈ 0, then the variance of the vertical energy gap is

σ_ΔE² = Var(E₂ − E₁) ≈ 2σ_E² 1

Combining this relation with eqn (6) gives

2

Using the values in Table 1, λ^var = 155 meV for the zero-charge model and an orientation-averaged value of λ^var = 180.5 meV for DSF, this expression predicts σ_E = 63.3 meV and 68.3 meV, respectively. These estimates agree closely with the site energy fluctuations sampled directly in the FOB-SH simulations, 63.5 meV and 69.2 meV. We note that this comparison is not fully consistent because the relation eqn (2) assumes that both σ_E and λ^var are obtained from ground state MD simulation in a charge localized state, whereas the σ_E value reported here are obtained from FOB-SH non-adiabatic dynamics. Evidently this difference does not play a significant role here.

This agreement shows that the increase in site energy disorder observed in the transport simulations is quantitatively consistent with the electrostatic increase in the reorganization energy. Thus, the inclusion of Coulombic interactions primarily modifies the diagonal energetic disorder experienced by the hole, while leaving the distributions of electronic couplings largely unchanged.

The enhanced site energy disorder has a direct impact on the spatial extent of the charge carrier wavefunction. Fig. 3(a) shows the time evolution of the IPR (eqn (19)), which measures the number of molecular sites over which the hole wavefunction is delocalized. In both the zero-charge and DSF simulations, the IPR, averaged over all FOB-SH trajectories, increases rapidly during the first ≈200 fs, reflecting quantum relaxation from the initially localized state (see Section 4.5). At later times, the wavefunction remains substantially more delocalized in the zero-charge model, with an average IPR of 13, compared to 9 when DSF electrostatics are included.

Fig. 3. Overview of results from FOB-SH simulations of hole transport in rubrene at 300 K employing DSF electrostatics (blue) alongside results from ref. 26 using the zero-charge model (green). (a) Time evolution of the average IPR, eqn (19). Horizontal black dashed lines indicate the average value of the IPR over the interval 500–900 fs. (b) and (c) Mean-square displacement of the charge carrier wavefunction, eqn (16), as a function of time along the a and b crystallographic directions, respectively. Linear fits used to calculate the charge carrier mobility are shown in black dashed lines. (d) Polar representation of the charge carrier mobility with experimental values from ref. 4 in the red dash-dotted line. The a and b crystallographic directions correspond to 0° and 90°, respectively.

This reduction in charge carrier delocalization is reflected in the MSD of the hole wavefunction. Fig. 3(b) and (c) show the MSD, averaged over all FOB-SH trajectories, along the a and b crystallographic directions, respectively. In both directions, the MSD grows more slowly when electrostatic interactions are included, indicating reduced charge diffusion. Consequently, the mobilities obtained from eqn (18) are lower for the DSF simulations than for the zero-charge model.

Fig. 3(d) shows the calculated mobility tensor in polar representation, µ(θ) = µ_acos² θ + µ_bsin² θ, for the zero-charge and DSF simulations, together with the experimental values reported by Podzorov et al.⁴ Without Coulombic interactions, the calculated mobility along the high-mobility a direction is 34.8 cm² V⁻¹ s⁻¹,²⁶ overestimating the experimental value of 20.2 cm² V⁻¹ s^−1 4 by a factor of ≈1.7. Including DSF electrostatics reduces this value to 21.1 cm² V⁻¹ s⁻¹, bringing the simulation into near-quantitative agreement with experiment.

Along the low mobility b direction, both models slightly underestimate the experimental mobility. However, including electrostatics improves the overall anisotropy of the mobility tensor. The anisotropy ratio decreases from μ_a/μ_b = 5.4 in the zero-charge model to 4.2 with DSF electrostatics, which is closer to the experimental ratio of 2.6.⁴

The inclusion of electrostatic interactions does not qualitatively change the underlying transient delocalization mechanism. To quantify this, we identify transient delocalization events as time intervals during which the IPR of a given trajectory exceeds 〈IPR〉 +σ_IPR. The values of 〈IPR〉 and σ_IPR are listed in Table 2, along with the average duration of transient delocalization events, t_r, and their inverse event rate, τ_r, i.e., a measure of the average time between two transient delocalization events. Both t_r and τ_r are very similar in the zero-charge and DSF simulations: t_r increases only slightly from 9 to 10 fs, while τ_r increases from 71 to 89 fs. Thus, while electrostatic interactions reduce the average spatial extent of the hole wavefunction by increasing site energy disorder, the transient delocalization mechanism is preserved.

3. Conclusions

In this work, we have investigated the role of Coulombic electrostatic interactions in determining the reorganization energy and hole mobility of rubrene at 300 K. Including electrostatics leads to a moderate increase in the reorganization energy relative to the zero-charge model, reflecting additional electrostatic contributions to vertical energy-gap fluctuations. Spectral analysis of these fluctuations shows that this increase is not confined to a single frequency range, but arises from enhanced coupling across both low-frequency nuclear motions and higher-frequency intramolecular vibrations. Benchmarking against SPME calculations further shows that the computationally efficient real-space DSF scheme captures the main electrostatic contribution to the reorganization energy, while slightly underestimating the SPME reference.

The increase in reorganization energy translates directly into enhanced site energy fluctuations in explicit FOB-SH simulations of hole transport. In contrast, the mean electronic couplings and their thermal fluctuations are only weakly affected by the inclusion of DSF electrostatics. The dominant effect of Coulombic interactions is therefore to increase diagonal energetic disorder, which reduces the spatial extent of the hole wavefunction. This is reflected in a decrease of the average IPR from 13 in the zero-charge model to 9 with DSF electrostatics.

This enhanced localization has a substantial effect on the predicted charge carrier mobility. Previous FOB-SH simulations using the zero-charge model yielded a hole mobility of 34.8 cm² V⁻¹ s⁻¹ along the high-mobility a direction, overestimating the experimental reference by a factor of approximately 1.7. Including electrostatic site energy disorder through the DSF scheme reduces this value to 21.1 cm² V⁻¹ s⁻¹, in close agreement with the experimental value of ≈20 cm² V⁻¹ s⁻¹. The inclusion of electrostatics also improves the predicted mobility anisotropy. Overall, these results show that electrostatic site energy disorder is essential for quantitatively predictive simulations of hole transport in rubrene.

4. Methods

4.1. Electrostatic interactions

Electrostatic interactions were treated using the DSF method proposed by Fennell and Gezelter,²⁷ which provides a computationally efficient alternative to conventional Ewald summation for periodic systems. In the DSF approach, the long-range Coulomb interaction is represented by a damped real-space interaction that is smoothly shifted to zero at a finite cutoff radius (R_cut). The electrostatic energy is written as

3

where q_i and q_j are the atomic charges, R_ij is the distance vector between atoms i and j, and n denotes lattice translation vectors under periodic boundary conditions, with the ^' symbol indicating that i = j is excluded when n = 0. R^DSF is the damped shifted interaction kernel including force shifting and real-space truncation, whose full expression is given in ref. 28.

Compared to Ewald summation, the DSF method avoids reciprocal-space calculations entirely and enables efficient evaluation of electrostatic site energies for multiple charge-localized states, leading to near-linear scaling with system size while retaining good accuracy for condensed-phase systems.^27,28 In particular, the DSF formalism allows us to use an efficient addition–subtraction scheme (introduced in ref. 28) through which the electrostatic energy of a given charge-localized state is decomposed into the electrostatic energy of the fully neutral system plus correction terms accounting for the molecule carrying the excess charge. This decomposition reduces the computational overhead associated with calculating all N_mol charge states from O(N_mol) to approximately O(N⁰_mol), i.e., to a near system-size independent prefactor relative to the cost of a single electrostatic calculation.

Benchmark calculations performed in ref. 28 demonstrated that DSF reproduces Ewald electrostatic energies and forces with good accuracy when suitable damping parameters and cutoff radii are employed. Using R_cut = 16 Å and a damping parameter α = 0.09 Å⁻¹, mean unsigned errors in electrostatic energies and forces remained below approximately 5% relative to Ewald summation.

4.2. Implicit polarizable force field via charge scaling

In conventional fixed-charge force fields, the absence of electronic polarization is known to lead to an overestimation of electrostatic interactions, resulting in exaggerated site energy fluctuations and reorganization energies.^31,32 In particular, the outer-sphere contribution to the reorganization energy is typically overestimated. These effects are relevant in nonadiabatic molecular dynamics simulations of charge transport, where electrostatic disorder directly influences charge delocalization and mobility.

To account for electronic polarization effects in FOB-SH simulations while retaining the computational efficiency of a nonpolarizable force field, an implicit polarization scheme based on charge scaling was employed, following the strategy introduced in ref. 28. The atomic charges of the neutral molecular state, obtained from a DFT calculation of a rubrene molecule in vacuum, were uniformly scaled by a factor γ according to q^n,FF_i → q̃^n,FF_i = γq^n,FF_i, while the charges of the charged molecular state were defined as q^c,FF_i → q̃^c,FF_i = γq^n,FF_i + Δq^DFT_i, where Δq^DFT_i = q^c,DFT_i − q^n,DFT_i is the charge difference obtained from DFT calculations between charged and neutral states of a rubrene molecule in vacuum. Atomic charges for rubrene were obtained by fitting the electrostatic potential (ESP) using the Merz–Kollman scheme.³³ The ESP was computed for the optimized geometries of neutral and cationic rubrene in vacuum. All calculations were performed at the B3LYP/6-311G(d) level of theory.

Physically, charge scaling mimics dielectric screening arising from the fast electronic polarization of the surrounding medium. Reducing the magnitude of the electrostatic interactions decreases both the amplitude of site energy fluctuations and the external contribution to the reorganization energy.

The optimal value of γ can be determined by matching the total reorganization energy obtained from scaled-charge simulations to the corresponding value computed using an explicitly polarizable force field with induced dipoles, as done in ref. 28. For apolar systems such as anthracene, a value of γ = 0.80 was found to reproduce accurately the reorganization energies and electrostatic fluctuations obtained from polarizable molecular dynamics simulations, while maintaining a significantly lower computational cost. The same value of γ was adopted here for rubrene.

4.3. Reorganization energy

The reorganization energy for hole transfer from molecule M to a neighbouring molecule N was computed from the distribution of vertical energy gaps evaluated along a classical molecular dynamics trajectory in which the hole was localized on molecule M. The vertical energy gap is defined as

ΔE_{M → N}(t) = E_N(R(t)) − E_M(R(t)), 4

where R(t) denotes a nuclear configuration sampled from the trajectory in which the hole was localized on molecule M. Here, E_M(R(t)) is the potential energy of the system with the hole localized on molecule M, while E_N(R(t)) is the potential energy of the same nuclear configuration with the hole localized on molecule N.

The Stokes shift reorganization energy is given by³⁰

λ^st = 〈ΔE_{M → N}〉_M, 5

where 〈…〉_M denotes an average over nuclear configurations sampled on the diabatic state with the hole localized on molecule M. The reorganization energy can also be computed based on the fluctuations of the vertical energy gap,³⁰

6

with

δΔE_{M → N}(t) =ΔE_{M → N}(t) − 〈ΔE_{M → N}〉_M. 7

In the linear response regime, λ^st and λ^var are identical.

For each treatment of electrostatics, the system was first equilibrated to 300 K on the diabatic state where molecule M is in the +1 charge state for 500 ps in the NVT ensemble using a Nosé–Hoover thermostat. Production trajectories were then generated on this diabatic state for 1 ns in the NVE ensemble, with configurations saved every 4 fs. E_N(R(t)) was computed by single-point energy recalculations over the saved trajectory by transferring the cationic parameters from molecule M to molecule N for the molecules N = P1, P2, T1, T2 depicted in Fig. 1. In practice, localization of the hole on a given molecule was represented by assigning that molecule force-field parameters corresponding to the +1 charge state, with all other molecules described by parameters corresponding to the neutral state.

For each transfer direction, the reported reorganization energy was obtained by computing λ^j, j = st, var, separately for the two symmetry-related dimers and averaging the two values. The quoted uncertainty combines the block-bootstrap sampling uncertainty within each trajectory with the residual scatter between the two symmetry-related dimers. Further details of the uncertainty estimate, together with the individual dimer values, are provided in the Section S1.

To analyze the frequency dependence of the energy gap fluctuations, we computed the autocorrelation function

C(t) = 〈δΔE_{M → N}(0)δΔE_{M → N}(t)〉_M. 8

The spectral density of vertical energy gap fluctuations is given by

9

where β = (k_BT)⁻¹. The cumulative fluctuation-derived reorganization energy was then evaluated as

10

By construction, integration over all frequencies recovers the fluctuation estimate,

11

For the spectra shown in Fig. 2(a) and (b), autocorrelation functions were computed from 15 ps sliding windows advanced in 4 fs increments and averaged over all windows. The resulting averaged autocorrelation function was tapered with a Hann window to reduce finite-window artefacts in S(ω)/ω. For the cumulative reorganization energies shown in Fig. 2(c) and (d), the autocorrelation function was computed over the full trajectory and cosine transformed without applying a Hann taper, so that the integrated spectral weight is consistent with λ^var.

4.4. Fragment orbital-based surface hopping

Charge transport simulations were performed using FOB-SH, a mixed quantum–classical dynamics method based on fewest-switches surface hopping.^24,34 In this approach, a single excess charge carrier is propagated quantum mechanically in a basis of orthogonalized, time-dependent frontier orbitals associated with the molecules of the crystal. For hole transport, the wavefunction is expanded in the orthogonalized HOMOs of the molecular fragments,

12

where R(t) denotes the nuclear configuration.

The charge carrier evolves under the following time-dependent Hamiltonian,

13

where ε_k are site energies and H_kl are electronic couplings. Site energies are evaluated using a classical force field by assigning the charged-state parameters to molecule k, with all other molecules in the neutral state. Electronic couplings were computed using the analytic overlap method (AOM),³⁵ in which the electronic coupling is assumed to scale linearly with the overlap between fragment orbitals projected onto a minimal Slater-type orbital basis. The proportionality constant was obtained by fitting to reference couplings computed using projector-operator-based diabatization.³⁶

The wavefunction coefficients u_l(t) are propagated according to the time-dependent Schrödinger equation in the (quasi-diabatic) site basis,

14

where are the nonadiabatic coupling elements in the site basis. At each nuclear time step, nuclei are propagated classically on a single active adiabatic surface. Stochastic hops between adiabatic states occur according to the fewest-switches surface-hopping criterion.²⁵ After a surface hop, energy conservation is enforced by rescaling the nuclear velocities along the direction of the nonadiabatic coupling vector (NACV). If the available nuclear kinetic energy is insufficient to conserve the total energy, the hop is rejected and the velocity component along the NACV is reversed.^34,37

Several extensions to the standard surface-hopping algorithm were employed, including decoherence correction, trivial-crossing detection, and suppression of decoherence-induced spurious long-range charge transfer. These corrections are required to obtain converged mobilities, detailed balance, and internally consistent charge dynamics; detailed descriptions are given in ref. 34 and 37.

4.5. FOB-SH simulation details

Interatomic interactions in rubrene were described using a force field based on the general amber force field (GAFF),³⁸ with selected parameters re-optimized to better reproduce the thermal distribution of electronic couplings obtained from ab initio MD simulations,³⁹ see ref. 26 for details. Electrostatic interactions were treated as explained in Sections 4.1 and 4.2. A supercell of 50 × 13 × 1 rubrene unit cells was equilibrated to 300 K for 200 ps using a time step of 1 fs in the NVT ensemble using a Nosé–Hoover thermostat. Initial positions and velocities for the swarm of FOB-SH trajectories were drawn from snapshots separated by 0.5 ps from a subsequent 250 ps MD trajectory in the NVE ensemble. A total of 460 FOB-SH trajectories were propagated. For each FOB-SH trajectory, the wavefunction was initialised on a single molecular site (i.e. diabatic state) located at the corner of the simulation box and propagated for 900 fs in the NVE ensemble using an MD time step of 0.05 fs and an electronic time step of 0.01 fs for integration of eqn (14) employing the Runge–Kutta algorithm to 4th order. The initial 200 fs of dynamics, corresponding to quantum relaxation from the initial diabatic state to the partially delocalized polaronic state, was neglected in all analysis. Average IPR values and transient delocalization statistics were computed over the interval 500–900 fs. Results for the model without Coulombic electrostatics, denoted the “zero-charge model”, are reproduced from ref. 26. In that work, the same supercell size, electronically active region, initialization procedure, and analysis protocol were used, with averages taken over 686 FOB-SH trajectories.

4.6. Calculation of charge mobility and IPR

Hole mobilities are obtained from FOB-SH trajectories from the mean-square displacement (MSD) of the hole wavefunction, given by

15

16

where Ψ_n(t) is the hole wavefunction in trajectory n, N_traj is the number of FOB-SH trajectories, and α, β denote Cartesian coordinates along the crystallographic a and b directions. In the discretized expression, eqn (16), the hole position is represented by the molecular centres of mass, with α_k,n(t) the α-coordinate of molecule k in trajectory n, and |u_k,n(t)|² the corresponding hole population. The quantity α_0,n is the initial centre-of-charge position of the hole wavefunction along the α direction in trajectory n, with β_0,n defined analogously.

The diffusion tensor is obtained from the long-time slope of the MSD, and the mobility tensor is then calculated using the Einstein–Smoluchowski relation,

17

18

where e is the elementary charge, k_B is the Boltzmann constant, and T is the temperature.

Delocalization of the time-dependent hole wavefunction is quantified using the inverse participation ratio (IPR),

19

where M is the number of molecules in the electronically active region.

Author contributions

J. E.: conceptualization, data curation, formal analysis, investigation (lead), methodology, visualization, writing – original draft, writing – review & editing. S. G.: conceptualization, investigation (supporting), methodology, software, writing – review & editing. J. B.: conceptualization, funding acquisition, resources, methodology, supervision, writing – review & editing.

Conflicts of interest

There are no conflicts to declare.

Data availability

All data needed to evaluate the conclusions in this paper are present in the paper and/or the supplementary information (SI). Supplementary information is available. See DOI: https://doi.org/10.1039/d6tc01991a.